Whitson PVT SPE Phase Behaviour Monograph

PHASE BEHAVIOR

CURTIS H. WHITSON AND MICHAEL R. BRULÉ

MONOGRAPH VOLUME 20 HENRY L. DOHERTY SERIESSPE

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PHASE BEHAVIOR

Curtis H. WhitsonProfessor of Petroleum Engineering

U. Trondheim, NTHand

FounderPERA a/s

and

Michael R. BruléPresident and Chief Executive Officer

Technomation Systems Inc.

First PrintingHenry L. Doherty Memorial Fund of AIME

Society of Petroleum Engineers Inc.

Richardson, Texas2000

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SPE Monograph SeriesThe Monograph Series of the Society of Petroleum Engineers was established in 1965 byaction of the SPE Board of Directors. The Series is intended to provide authoritative,up-to-date treatment of the fundamental principles and state of the art in selected fields oftechnology. The Series is directed by the Society’s Monograph Committee. A committeemember designated as Monograph Editor provides technical evaluation with the aid of theReview Committee. Below is a listing of those who have been most closely involved with thepreparation of this monograph.

Monograph Review CommitteePeter G. Christman, Shell Intl. E&P B.V., Monograph EditorDavid F. Bergman, Amoco Production Co.W. David Constant, Louisiana State U.A.S. Cullick, Landmark Graphics Corp.Gustave A. Mistrot III, Mistrot & Assocs.Teresa G. Monger-McClure, Marathon Oil Co.Franklin M. Orr Jr., Stanford U.Robert R. Wood, Shell Intl. E&P B.V.Aaron A. Zick, Zick Technologies

Monograph Committee (2000)Mary Jane Wilson, WZI, ChairpersonJesse Frederick, WZIRussell T. Johns, U. of Texas, AustinMedhat Kamal, Arco E&P TechnologyMark Miller, U. of Texas, AustinKen Newman, CTES L.S.Dan O’Meara Jr., U. of OklahomaDavid Underdown, Chevron Production Technology Co.

AcknowledgmentsMany people contributed to the production of this monograph. It is first and foremost theproduct of the authors. I am sure that the effort was more significant than either author hadanticipated, but they persevered and should be proud of the book they wrote. I want to thankR.R. Wood, who initiated the project, chose the authors, and formed a distinguished reviewcommittee. I succeeded Rob in 1990 and coordinated the efforts of A.A. Zick, G.A. Mistrot,T.G. Monger-McClure, D.F. Bergman, A.S. Cullick, and W.D. Constant, who reviewed everychapter from their own unique perspectives. F.M. Orr contributed significant reviews onselected chapters. It was a pleasure to work with such a talented group of engineers. I amconfident that we kept the focus of the monograph on use by the working engineer. The bookis meant to serve as a reference. As such, I hope it will be a valuable addition to the library ofevery petroleum engineer working in phase behavior.

Peter G. Christman

Copyright 2000 by the Society of Petroleum Engineers Inc.Printed in the United States of America. All rights reserved.This book, or any part thereof, cannot be reproduced in anyform without written consent of the publisher.

ISBN 1-55563-087-1

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Dedication

To Morris Muskat, a pioneer in the field of reservoir engineering,who made important contributions in the area of phase behavior.

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Acknowledgments

We thank the SPE editorial staff, the Monograph Review Committee members, ourprofessional colleagues, our students, and the petroleum industry at large for valuableassistance and input toward the completion of this monograph. In particular, we thankthe two technical editors, Rob R. Wood and Peter G. Christman, and our staff editor,Flora Cohen.

We have been strongly influenced by the pioneering phase-behavior research ofDonald Katz, Muz Standing, and Ken Starling and the many others who have madeinvaluable contributions to the field. The scientific contributions of these engineers andtheir coworkers, together with contributions from the community of petroleum andchemical engineers, have laid the foundation for the material selected, synthesized, andpresented in this monograph. We hope that all contributors have been correctly citedand given due credit for their contributions.

We are confident that the material contained herein is valuable for dealing withengineering problems affected by phase behavior, both today and in the future. We usethe technology presented in this monograph daily to solve problems for the industryand as the basis of our long-term research.

Curtis H. Whitson

Michael R. Brulé

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Table of Contents

Chapter 1—Introduction 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.1 Purpose 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Historical Review 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Objectives 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Scope and Organization 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Nomenclature and Units 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 2—Volumetric and Phase Behavior of Oil and Gas Systems 5. . . . . . . . . . . . . . . . . . . . . . . .

2.1 Introduction 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Reservoir-Fluid Composition 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Phase Diagrams for Simple Systems 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Retrograde Condensation 11. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Classification of Oilfield Systems 12. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 3—Gas and Oil Properties and Correlations 18. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1 Introduction 18. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Review of Properties, Nomenclature, and Units 18. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Gas Mixtures 22. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Oil Mixtures 29. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 IFT and Diffusion Coefficients 38. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 K-Value Correlations 40. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 4—Equation-of-State Calculations 47. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.1 Introduction 47. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Cubic EOS’s 47. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Two-Phase Flash Calculation 52. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Phase Stability 55. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Saturation-Pressure Calculation 62. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Equilibrium in a Gravity Field: Compositional Gradients 63. . . . . . . . . . . . . . . . . . . . 4.7 Matching an EOS to Measured Data 65. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 5—Heptanes-Plus Characterization 68. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.1 Introduction 68. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Experimental Analyses 68. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Molar Distribution 70. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Inspection-Properties Estimation 77. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Critical-Properties Estimation 80. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Recommended C7� Characterizations 83. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Grouping and Averaging Properties 83. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 6—Conventional PVT Measurements 88. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.1 Introduction 88. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Wellstream Compositions 88. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Multistage-Separator Test 91. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Constant Composition Expansion 93. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Differential Liberation Expansion 95. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Constant Volume Depletion 97. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 7—Black-Oil PVT Formulations 109. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.1 Introduction 109. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Traditional Black-Oil Formulation 109. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Modified Black-Oil (MBO) Formulation 110. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Applications of MBO Formulation 116. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Partial-Density Formulation 118. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Modifications for Gas Injection 119. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 8—Gas-Injection Processes 121. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.1 Introduction 121. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Miscibility and Related Phase Behavior 122. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Lean-Gas Injection 128. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Enriched-Gas Miscible Drive 131. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 CO2 Injection 135. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 9—Water/Hydrocarbon Systems 142. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.1 Introduction 142. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Properties and Correlations 142. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 EOS Predictions 150. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Hydrates 151. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Appendix A—Property Tables and Units 162. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Appendix B—Example Problems 172. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Appendix C—Equation-of-State Applications 193. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Appendix D—Understanding Laboratory Oil PVT Reports 209. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Nomenclature 220. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Author Index 225. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Subject Index 230. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

INTRODUCTION 1

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This monograph covers a wide range of topics related to phase behav-ior. Phase behavior is the behavior of vapor, liquid, and solids as afunction of pressure, temperature, and composition. In this mono-graph, “vapor” is used interchangeably with “gas,” “liquid” refers tooil and water, and “solids” include hydrates, asphaltenes, and wax.

We are concerned primarily with the volumetric behavior andcomposition of phases, including density and isothermal compress-ibility, and component distribution in each phase. For a mixture witha known composition, we need to determine the vapor/liquid equi-librium (VLE), including saturation conditions over a wide range oftemperatures and pressures. Transport properties are also needed forflow calculations (e.g., viscosity in Darcy’s law and molecular dif-fusion coefficients in Fick’s law).

Phase behavior has many applications in petroleum engineering.The reservoir engineer relies on pressure/volume/temperature(PVT) relations to calculate oil and gas reserves, production fore-casts, and the efficiency of enhanced oil recovery (EOR) methods.Most reservoir calculations require PVT properties at reservoir tem-perature. Production engineers use phase behavior data for surfaceseparator design and to calculate flow in pipe, where such calcula-tions are made over a range of temperatures from surface to reser-voir conditions. Petroleum engineering calculations generally aremade at temperatures from 60 to 350°F and at pressures from about15 to 15,000 psia.

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Gibbs1,2 and van der Waals3 stated the basic theory of phase behav-ior in the the late 1800’s and early 1900’s. They formulated the con-cepts and mathematical relations necessary to describe phase be-havior. Katz and Rzasa4 published a comprehensive review of phasebehavior literature from before 1860 to 1945. Muckleroy5 pub-lished a bibliography covering 1946 to 1960, and other bibliogra-phies exist for work in phase behavior over the past 30 years.*

Experimental data on reservoir fluids were scarce before the late1930’s, when Katz et al.4,6-39 at the U. of Michigan, Sage and col-leagues40-73 at the California Inst. of Technology, and Eilerts etal.74-78 at the U.S. Bureau of Mines (USBM) began significant re-search programs. For 10 years, during the 1950’s, a large amount ofhigh-quality experimental data was compiled on reservoir fluids.During the past 40 years, most phase behavior data have been mea-sured by commercial service laboratories and major oil companies.

*SPE Reprint Series No. 15 Phase Behavior gives a recent update of earlier bibliogra-phies.

These data have been used for engineering studies of primary deple-tion, waterflood evaluation, and gas-injection studies.

Correlation of phase behavior data began in the 1940’s, with nota-ble work by Standing and Katz,17,18 Bicher and Katz,25 Stand-ing,79,80 Eilerts,78 Kennedy and colleagues,81-85 and others. Al-though equations of state (EOS’s) had been available for more than50 years (since van der Waals3 published the first cubic EOS in1873) it was necessary to rely mostly on tables, figures, and chartcorrelations, such as nomograms. These correlations provided reli-able property estimates for engineering calculations through the1970’s. Subsequently, empirical equations representing thesegraphical correlations were developed and programmed for calcula-tors and computer applications.

With the introduction of electronic computers in the late 1940’s,application of complicated thermodynamic models became pos-sible. In 1949, Muskat and McDowell86 published one of the earliestpapers in the SPE/AIME Transactions on applications of this newgeneration of computers. These authors solved the two-phase flashcalculation with fixed K values for multistage separator design.

Not until Redlich and Kwong87 introduced their classic cubic EOSin 1949 was it generally accepted that volumetric properties could beaccurately predicted by use of theoretical models. Considerable ad-vances were made in the 1950’s toward correlating volumetric prop-erties of pure components with multiconstant EOS’s.88 By the early1960’s, there was considerable activity in the application of sophisti-cated thermodynamic models to multicomponent VLE calculations,although most of this activity was in process engineering.

In the 1960’s and 1970’s, Starling,89 Soave,90 and Peng and Ro-binson91 proposed several important modifications of existingEOS’s. Petroleum engineering EOS applications started seriously inthe late 1970’s and early 1980’s, when EOS-based compositionalreservoir simulators were introduced.92,93 At the same time, severalmethods were proposed for EOS fluid characterization of reservoirfluids, in particular for heptanes and heavier components.94-96 Fi-nally, in the 1980’s, supercomputers appeared and special solutiontechniques were developed for compositional simulators,93 therebymaking possible full-field, EOS compositional simulation.

Today’s standard treatment of phase behavior in reservoir simula-tion is still based on formation volume factors (FVF’s) and surfacegas/oil ratios (GOR’s). This will probably remain true for manyyears, in part because many problems can be solved adequately witha simple PVT formulation and in part because many petroleum engi-neers are not familiar with more complicated EOS models. Thismonograph treats both simple and complicated methods for estimat-ing phase behavior. We suspect that the more complicated PVT

2 PHASE BEHAVIOR MONOGRAPH

models will gradually become the standard, eventually replacingmany of the simpler correlations.

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This monograph provides the petroleum engineer with a tool tosolve problems that require a description of phase behavior and spe-cific PVT properties. These problems include calculating the FVFto determine original oil and gas in place and GOR’s, design of “op-timal” surface separator conditions, and description of near-criticalphase behavior resulting from the injection of a gas that developsmiscibility with a reservoir oil.

Because of the dramatic evolution in computer technology, petro-leum engineers can now study such phenomena as developed misci-bility,97 compositional gradients,98 and near-critical phase behav-ior99 with more sophisticated models. The quality of these modelsis sensitive to the EOS fluid characterizations. This monographpresents phase behavior concepts used in petroleum engineeringand state-of-the-art technology for more complex phase behaviormodels, such as cubic EOS’s. We hope the monograph will serve itspurpose for many years to come.

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The scope of this monograph is limited mostly to two-phase, gas/oilphase behavior. Multiphase and vapor/solid phase behavior are dis-cussed only briefly. Phase behavior related to chemical (surfactantand polymer) flooding is not covered because a detailed descriptionwould necessarily reduce coverage of problems more commonlyencountered in petroleum engineering. We also think that this sub-ject should be covered in a separate publication specifically withinthe context of chemical flooding technology.

Chaps. 2 and 3 review the “nuts and bolts” of phase behavior prin-ciples, relevant PVT properties, and methods to solve most petro-leum engineering problems. Useful correlations are presented forthe most common PVT properties.

Chap. 4 discusses cubic EOS’s, including the two-phase flash,saturation-pressure, and phase-stability calculations and numericalmethods used to solve these VLE calculations. The problem of “tun-ing” an EOS to match measured PVT data is also addressed.

Chap. 5 describes the characterization of heavy components(“heptanes plus”) in reservoir fluids for EOS applications. Exper-imental and mathematical methods describing the heptanes-plusmaterial are presented, including splitting C7+ into petroleum frac-tions, estimating critical properties, and grouping an extended fluidcharacterization into a reduced number of pseudocomponents.

Chap. 6 covers laboratory measurements of PVT properties andtheir application in engineering calculations. The standard PVTstudies include constant composition (mass) expansion, differentialliberation, constant-volume depletion, and the multistage separatortest. Separator and bottomhole sampling methods for establishingwellstream composition are also discussed.

Chap. 7 describes the black-oil PVT formulation and its extensionto gas condensates, volatile oils, and gas-injection processes. Theblack-oil PVT formulation uses FVF’s and solution gas/oil ratios torelate phase and volumetric properties at reservoir conditions to sur-face volumes.

Chap. 8 reviews the importance of phase behavior to gas-injec-tion EOR processes. These processes include vaporizing, condens-ing, and the combined condensing/vaporizing miscible-drive mech-anisms. CO2 immiscible and miscible drives and nitrogen injectionare also reviewed.

Chap. 9 covers the behavior of water/hydrocarbon phase and vol-umetric behavior, including mutual solubilities, water FVF andcompressibility, and the treatment of hydrates.

Appendix A gives tables of component properties, various otheruseful tables, and unit conversion factors. Appendix B includesmore than 20 worked examples that range from simple calculationsof ideal gas properties to detailed step-by-step EOS calculations fora ternary system. Appendix C gives two detailed EOS fluid charac-terizations, one for a gas condensate and another for a slightly vola-tile oil. Appendix D is a set of notes by M.B. Standing on under-standing laboratory-oil PVT reports. These notes clearly belong

with Chap. 6, Conventional PVT Experiments, and are included asa supplement to the discussion in that chapter.

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SPE-approved symbols are used throughout the monograph. Someof these symbols will be unfamiliar even to the seasoned SPE reader(as they are confusing even to the authors!).

One of the most significant changes in nomenclature that we haveintroduced is the use of different subscripts for surface and reservoirphases. Traditionally, o, g, and w are used for oil, gas, and water atreservoir and at surface conditions, a practice that was difficult tofollow in Chaps. 6 and 7. We have therefore introduced the sub-scripts o, g, and w for surface phases, retaining o, g, and w for res-ervoir phases. A better solution to this problem was not apparent,particularly because some quantities required subscripts for bothreservoir and surface phases—e.g., the gravity of surface gas pro-duced from reservoir oil (written �go in this monograph). To avoidconfusion in the property correlations in Chap. 3, gas and oil specif-ic gravities are still written �g and �o (instead of �g and �o) becausespecific gravity is always reported at standard conditions.

We use customary oilfield units (psi, ft3 and bbl, °F and °R, andlbm). The oilfield unit for mass is pound, written “lbm” to avoidconfusion with pounds force, written “lbf.” Pounds force is neverused explicitly in this monograph. Conversion factors to SI units areincluded at the end of each chapter, and Appendix A provides a com-prehensive discussion of units and unit conversion tables.

Standard conditions are defined in this monograph as 60°F and14.7 psia. We recognize that standard pressure varies geographical-ly and the calculation of surface gas volumes in some areas must usethe locally defined value for standard pressure. To accomplish this,some constants given in the monograph must be recalculated.

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1. Gibbs, J.W.: The Collected Works of J. Willard Gibbs, Yale U. Press,New Haven, Connecticut (1948) 1.

2. Gibbs, J.W.: On the Equilibrium of Heterogeneous Substances, C.Works (ed.), Yale U. Press, New Haven, Connecticut (1928) Chap. 1.

3. van der Waals, J.D.: Continuity of the Gaseous and Liquid State of Mat-ter (1873).

4. Katz, D.L. and Rzasa, M.J.: Biblography of Hydrocarbons Under Pres-sure 1860–1946, University Microfilms Inc. (1946).

5. Muckleroy, J.A.: Biblography on Hydrocarbons, 1946–1960, Gas Pro-cessors Assn. (1962).

6. Katz, D.L. and Hachmuth, K.K.: “Vaporization Equilibrium Constants ina Crude-Oil Natural Gas System,” Ind. & Eng. Chem. (1937) 29, 1072.

7. Katz, D.L.: “Application of Vaporization Equilibrium Constants to Pro-duction Engineering Problems,” Trans., AIME (1938) 127, 159.

8. Katz, D.L., Vink, D.J., and David, R.A.: “Phase Diagram of a Mixtureof Natural Gas and Natural Gasoline Near the Critical Conditions,”Trans., AIME (1939) 136, 106.

9. Katz, D.L. and Singleterry, C.C.: “Significance of the Critical Phenom-ena in Oil and Gas Production,” Trans., AIME (1939) 132, 103.

10. Katz, D.L. and Saltman, W.: “Surface Tension of Hydrocarbons,” Ind.& Eng. Chem. (January 1939) 31, 91.

11. Katz, D.L. and Kurata, F.: “Retrograde Condensation,” Ind. & Eng.Chem. (June 1940) 32, No. 6, 817.

12. Wilcox, W.I., Carson, D.B., and Katz, D.L.: “Natural Gas Hydrates,”Ind. & Eng. Chem. (1941) 33, No. 5, 662.

13. Katz, D.L.: “High Pressure Gas Measurement,” Refiner and NaturalGasoline Manufacturer (June 1942).

14. Carson, D.B. and Katz, D.L.: “Natural Gas Hydrates,” Trans., AIME(1942) 146, 150.

15. Kurata, F. and Katz, D.L.: “Critical Properties of Volatile HydrocarbonMixtures,” Trans., AIChE (1942) 38, 995.

16. Katz, D.L.: “Possibilities of Secondary Recovery for the OklahomaCity Wilcox Sand,” Trans., AIME (1942) 146, 28.

17. Standing, M.B. and Katz, D.L.: “Density of Natural Gases,” Trans.,AIME (1942) 146, 140.

18. Standing, M.B. and Katz, D.L.: “Density of Crude Oils Saturated withNatural Gas,” Trans., AIME (1942) 146, 159.

19. Katz, D.L.: “Prediction of the Shrinkage of Crude Oils,” Drill. & Prod.Prac. (1942) 137.

20. Matthews, T.A., Roland, C.H., and Katz, D.L.: “High Pressure Gas Mea-surement,” Proc., Natural Gas Assn. of America (NGAA) (1942) 41.

INTRODUCTION 3

21. Weinaug, C.F. and Katz, D.L.: “Surface Tension of Methane-PropaneMixtures,” Ind. & Eng. Chem. (1943) 35, No. 2, 239.

22. Bicher, L.B. Jr. and Katz, D.L.: “Viscosities of the Methane-PropaneSystem,” Ind. & Eng. Chem. (1943) 35, 754.

23. Katz, D.L., Monroe, R.R., and Trainer, R.P.: “Surface Tension of CrudeOils Containing Dissolved Gases,” Trans., AIME (1943) 155, 624.

24. Standing, M.B. and Katz, D.L.: “Vapor/Liquid Equilibria of NaturalGas/Crude Oil Systems,” Trans., AIME (1944) 155, 232.

25. Bicher, L.B. Jr. and Katz, D.L.: “Viscosity of Natural Gases,” Trans.,AIME (1944) 155, 246.

26. Katz, D.L., Brown, G.G., and Parks, A.S.: “NGAA Report on SamplingTwo-Phase Gas Streams from High Pressure Condensate Wells,” Proc.,NGAA (September 1945).

27. Katz, D.L. and Beu, K.L.: “Nature of Asphaltic Substances,” Ind. &Eng. Chem. (February 1945) 37, 195.

28. Katz, D.L.: “Prediction of Conditions for Hydrate Formation in NaturalGases,” Trans., AIME (1945) 160, 140.

29. Poettman, F.H. and Katz, D.L.: “CO2 in a Natural Gas Condensate Sys-tem,” Ind. & Eng. Chem. (1946) 38, 530.

30. Brown, G.G. et al..: Natural Gasoline and the Volatile Hydrocarbons,NGAA, Tulsa, Oklahoma (1948) 24–32.

31. Kobayashi, R. and Katz, D.L.: “Methane-n-Butane-Water System in Two-and Three-Phase Regions,” Ind. & Eng. Chem. (1948) 40, No. 5, 853.

32. Unruh, C.H. and Katz, D.L.: “Gas Hydrates of Carbon Dioxide/Meth-ane Mixtures,” Trans., AIME (1949)186, 83.

33. Rzasa, M.J. and Katz, D.L.: “The Coexistence of Liquid and VaporPhases at Pressures Above 10,000 psi,” Trans., AIME (1950) 189, 119.

34. Kobayashi, R. et al.: “Gas Hydrates Formation with Brine and EthanolSolutions,” Proc., 30th Annual Convention of NGAA (1951).

35. Katz, D.L. and Williams, B.: “Reservoir Fluids and Their Behavior,”Amer. Soc. Petr. Geology Bulletin (February 1952) 36, No. 2, 342.

36. Katz, D.L.: “Possibility of Cycling Deep Depleted Oil Reservoirs AfterCompression to a Single Phase,” Trans., AIME (1952) 195, 175.

37. Kobayashi, R. and Katz, D.L.: “Vapor-Liquid Equilibria for Binary Hy-drocarbon-Water Systems,” Ind. & Eng. Chem. (1953) 45, No. 2, 440.

38. Donnelly, H.C. and Katz, D.L.: “Phase Equilibria in the Carbon Diox-ide-Methane System,” Ind. & Eng. Chem. (1954) 46, 511.

39. Katz, D.L. et al.: Handbook of Natural Gas Engineering, McGraw-HillBook Co. Inc., New York City (1959).

40. Reamer, H.H., Sage, B.H., and Lacey, W.N.: “Volumetric Behavior ofHydrogen Sulfide,” Ind. & Eng. Chem. (1950) 42, 140.

41. Sage, B.H. and Olds, R.H.: “Volumetric Behavior of Oil and Gas fromSeveral San Joaquin Valley Fields,” Trans., AIME (1947) 170, 156.

42. Olds, R.H., Sage, B.H., and Lacey, W.N.: “Partial Volumetric Behaviorof the Methane-Carbon Dioxide System,” Fundamental Research onOccurrence and Recovery of Petroleum, API, Dallas (1943) 44.

43. Reamer, H.H. et al.: “Phase Equilibria in Hydrocarbon Systems—Volu-metric Behavior of Ethane-Carbon Dioxide System,” Ind. & Eng.Chem. (1945) 37, 688.

44. Sage, B.H. and Lacey, W.N.: “Partial Volumetric Behavior of theLighter Paraffin Hydrocarbons in the Gas Phase,” Drill. & Prod.Prac. (1939) 641.

45. Sage, B.H. and Lacey, W.N.: “Thermodynamic Properties of the LightParaffin Hydrocarbons and Nitrogen,” API Research Project 37, mono-graph, API, New York City (1950).

46. Sage, B.H., Hicks, B.L., and Lacey, W.N.: “Partial Volumetric Behav-ior of the Lighter Hydrocarbons in the Liquid Phase,” Drill. & Prod.Prac. (1938) 402.

47. Sage, B.H. and Lacey, W.N.: “Apparatus for Determination of Volumet-ric Behavior of Fluids,” Trans., AIME (1948) 174, 102.

48. Reamer, H.H., Sage, B.H., and Lacey, W.N.: “Volumetric and PhaseBehavior of the Methane-Propane Systems,” Ind. & Eng. Chem.(1950) 42, 534.

49. Sage, B.H., Lacey, W.N., and Schaafsma, J.G.: “Phase Equilibria in Hy-drocarbon Systems: Methane-Propane Systems,” Ind. & Eng. Chem.(1934) 26, 214.

50. Reamer, H.H., Sage, B.H., and Lacey, W.N.: “Volumetric and Phase Be-havior of the Methane-n Butane-Decane System,” Ind. & Eng. Chem.(1951) 43, 1436.

51. Sage, B.H. and Lacey, W.W.: Volumetric and Phase Behavior of Hydro-carbons, Gulf Publishing Co., Houston (1949).

52. Reamer, H.H., Sage, B.H., and Lacey, W.N.: “Phase Equilibria in Hy-drocarbon Systems,” Ind. & Eng. Chem. (June 1951) 43, 1436.

53. Reamer, H.H., Sage, B.H., and Lacey, W.N.: “Phase Behavior in Hydro-carbon System,” Ind. & Eng. Chem. (1951) 43, 2515.

54. Olds, R.H. et al.: “Phase Equilibria in Hydrocarbon Systems. The Bu-tane-Carbon Dioxide System,” Ind. & Eng. Chem. (1949) 41, 475.

55. Reamer, H.H. and Sage, B.H.: “Phase Equilibria in Hydrocarbon Sys-tems—Volumetric and Phase Behavior of the n-Decane-CO2 System,”J. Chem. Eng. Data (1963) 8, 508.

56. Reamer, H.H., Fiskin, J.M., and Sage, B.H.: “Phase Equilibria in Hy-drocarbon Systems: Phase Behavior in the Methane-n-Butane-DecaneSystem at 160°F,” Ind. & Eng. Chem. (December 1949) 41, 2871.

57. Reamer, H.H., Sage, B.H., and Lacey, W.N.: “Phase Equilibria in Hy-drocarbon Systems—Volumetric and Phase Behavior of the Methane-n-Heptane System,” Ind. & Eng. Chem. (1956) 1, 29.

58. Sage, B.H., Webster, D.C., and Lacey, W.N.: “Phase Equilibria in Hy-drocarbon Systems,” Ind. & Eng. Chem. (1936) 28, 1045.

59. Reamer, H.H., Sage, B.H., and Lacey, W.N.: “Phase Equilibria in Hy-drocarbon Systems—Volumetric and Phase Behavior of the Methane-Cyclohexane System,” Ind. & Eng. Chem. (1958) 3, 240.

60. Sage, B.H. and Lacey, W.N.: “Effect of Pressure Upon Viscosity ofMethane and Two Natural Gases,” Trans., AIME (1938) 127, 118.

61. Sage, B.H., Yale, W.D., and Lacey, W.N.: “Effect of Pressure on Viscos-ity of n-Butane and i-Butane,” Ind. & Eng. Chem. (1939) 31, 223.

62. Sage, B.H. and Lacey, W.N.: “Gravitational Concentration Gradients inStatic Columns of Hydrocarbon Fluids,” Trans., AIME (1939) 132, 120.

63. Reamer, H.H., Sage, B.H., and Lacey, W.N.: “Volumetric and Phase Be-havior of the Methane-n-Butane-Decane System,” Ind. & Eng. Chem.(1947) 39, 77.

64. Reamer, H.H., Sage, B.H., and Lacey, W.N.: “Volumetric and Phase Be-havior of the Methane-n-Butane-Decane System,” Ind. & Eng. Chem.(1952) 44, 1671.

65. Sage, B.H., Lacey, W.N., and Schaafsma, J.G.: “Behavior of Hydrocar-bon Mixtures Illustrated by a Simple Case,” API Bulletin (1932) 212, 119.

66. Sage, B.H.: Thermodynamics of Multicomponent Systems, ReinholdPublishing Co. (1965)

67. Sage, B.H. and Lacey, W.N.: Volumetric and Pha.se Behavior of Hydro-carbons, Stanford Press, Stanford, Connecticut (1939).

68. Sage, B.H. and Reamer, R.H.: “Volumetric Behavior of Oil and GasFrom the Rio Bravo Field,” Trans., AIME (1941) 142, 179.

69. Olds, R.H., Sage, B.H., and Lacey, W.N.: “Phase Equilibria in Hydro-carbon Systems. Composition of the Dew-Point Gas of the Methane-Water System,” Ind. & Eng. Chem. (1942) 34, No. 10, 1223.

70. Reamer, H.H. et al.: “Phase Equilibria in Hydrocarbon Systems. Com-position of the Dew-Point Gas in the Ethane-Water System,” Ind. &Eng. Chem. (1943) 35, No. 7, 790.

71. Reamer, H.H. et al.: “Phase Equilibria in Hydrocarbon Systems. Com-positions of the Coexisting Phases of n-Butane-Water System in theThree-Phase Region,” Ind. & Eng. Chem. (1944) 36, No. 4, 381.

72. Reamer, H.H., Sage, B.H., and Lacey, W.N.: “Phase Equilibria in Hy-drocarbon Systems. n-Butane-Water System in the Two-Phase Re-gion,” Ind. & Eng. Chem. (1952) 44, No. 3, 609.

73. Sage, B.H. and Lacey, W.N.: “Some Properties of the Lighter Hydrocar-bons, Hydrogen Sulfide, and Carbon Dioxide,” API Research Project37, monograph, API, New York City (1955).

74. Eilerts, C.K.: “The Reserve Fluid, Its Composition and Phase Behav-ior,” Oil & Gas J. (1 January 1947) 63.

75. Eilerts, C.K.: “Gas Condensate Reservoir Engineering, 1. The ReserveFluid, Its Composition and Phase Behavior,” Oil & Gas J. (1 February1947) 63.

76. Eilerts, C.K., Carlson, H.A., and Mullen, N.B.: “Effect of Added Nitro-gen on Compressibility of Natural Gas,” World Oil (June 1948) 129.

77. Eilerts, C.K. et al.: “Phase Relations of a Gas-Condensate Fluid atLow Temperatures, Including the Critical State,” Pet. Eng. (February1948) 19, 154.

78. Eilerts, C.K.: Phase Relations of Gas Condensate Fluids, Monograph10, USBM, American Gas Assn., New York City (1957) I and II.

79. Standing, M.B.: “Vapor-Liquid Equilibria of Natural Gas-Crude OilSystems,” PhD dissertation, U. of Michigan, Ann Arbor, MI (1941).

80. Standing, M.B.: “A Pressure-Volume-Temperature Correlation for Mix-tures of California Oils and Gases,” Drill. & Prod. Prac. (1947) 275.

81. Alani, G.H. and Kennedy, H.T.: “Volumes of Liquid Hydrocarbons atHigh Temperatures and Pressures,” Trans., AIME (1960) 219, 288.

82. Kennedy, G.C.: “Pressure-Volume-Temperature Relations in CO2 at Ele-vated Temperatures and Pressures,” Amer. J. Sci. (April 1954) 252, 225.

83. Kennedy, H.T. and Bhagia, N.S.: “An EOS for Condensate Fluids,” JPT(September 1969) 379.

84. Little, J.E. and Kennedy, H.T.: “A Correlation of the Viscosity of Hy-drocarbon Systems with Pressure, Temperature and Composition,”SPEJ (June 1968) 157; Trans., AIME, 243.

85. Nemeth, L.K. and Kennedy, H.T.: “A Correlation of Dewpoint PressureWith Fluid Composition and Temperature,” SPEJ (June 1967) 99;Trans., AIME (1967) 240.

4 PHASE BEHAVIOR MONOGRAPH

86. Muskat, M. and McDowell, J.M.: “An Electrical Computer for SolvingPhase Equilibrium Problems,” Trans., AIME (1949) 186, 291.

87. Redlich, O. and Kwong, J.N.S.: “On the Thermodynamics of Solutions,V: An Equation of State. Fugacities of Gaseous Solutions,” Chem. Rev.(1949) 44, 233.

88. Benedict, M., Webb, G.B., and Rubin, L.C.: “An Empirical Equationfor Thermodynamic Properties of Light Hydrocarbons and Their Mix-tures, I. Methane, Ethane, Propane, and n-Butane,” J. Chem. Phy.(1940) 8, 334.

89. Starling, K.E.: “A New Approach for Determining Equation-of-StateParameters Using Phase Equilibria Data,” SPEJ (December 1966) 363;Trans., AIME, 237.

90. Soave, G.: “Equilibrium Constants from a Modified Redlich-KwongEOS,” Chem. Eng. Sci. (1972) 27, No. 6, 1197.

91. Peng, D.Y. and Robinson, D.B.: “A New-Constant EOS,” Ind. & Eng.Chem. Fund. (1976) 15, No. 1, 59.

92. Coats, K.H.: “An EOS Compositional Model,” SPEJ (October 1980)363; Trans., AIME, 269.

93. Young, L.C. and Stephenson, R.E.: “A Generalized Compositional Ap-proach for Reservoir Simulation,” SPEJ (October 1983) 727; Trans.,AIME, 275.

94. Yarborough, L.: “Application of a Generalized Equation of State to Pe-troleum Reservoir Fluids,” Equations of State in Engineering and Re-

search, K.C. Chao and R.L. Robinson Jr. (eds.), Advances in ChemistrySeries, American Chemical Soc. (1978) 182, 386–439.

95. Whitson, C.H.: “Characterizing Hydrocarbon Plus Fractions,” SPEJ(August 1983) 683; Trans., AIME, 275.

96. Pedersen, K.S., Thomassen, P., and Fredenslund, A.: “Characterizationof Gas Condensate Mixtures,” C7 Fraction Characterization, L.G.Chorn and G.A. Mansoori (eds.), Advances in Thermodynamics, Tay-lor & Francis, New York City (1989) 1.

97. Zick, A.A.: “A Combined Condensing/Vaporizing Mechanism in theDisplacement of Oil by Enriched Gases,” paper SPE 15493 presentedat the 1986 SPE Annual Technical Conference and Exhibition, New Or-leans, 5–8 October.

98. Schulte, A.M.: “Compositional Variations Within a Hydrocarbon Col-umn Due to Gravity,” paper SPE 9235 presented at the 1980 SPE Annu-al Technical Conference and Exhibition, Dallas, 21–24 September.

99. Coats, K.H.: “Simulation of Gas Condensate Reservoir Performance,”JPT (October 1985) 1870.

� &�� '��

°F (°F�32)/1.8 �°Cpsi �6.894 757 E�00�kPa

VOLUMETRIC AND PHASE BEHAVIOR OF OIL AND GAS SYSTEMS 1

��

��

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Petroleum reservoir fluids are naturally occurring mixtures of natu-ral gas and crude oil that exist in the reservoir at elevated tempera-tures and pressures. Reservoir-fluid compositions typically includehundreds or thousands of hydrocarbons and a few nonhydrocar-bons, such as nitrogen, CO2, and hydrogen sulfide. The physicalproperties of these mixtures depend primarily on composition andtemperature and pressure conditions. Reservoir temperature canusually be assumed to be constant in a given reservoir or to be a weakfunction of depth. As oil and gas are produced, reservoir pressuredecreases and the remaining hydrocarbon mixtures change in com-position, volumetric properties, and phase behavior. Gas injectionalso may change reservoir-fluid composition and properties. Katzand Williams1 give an excellent review of reservoir fluids and theirgeneral behavior under different operating conditions.

The hydrocarbon phases and connate water sharing the pore vol-ume (PV) in a reservoir are in thermodynamic equilibrium. Strictlyspeaking, hydrocarbons and water should be treated simultaneouslyin phase-behavior calculations. At typical reservoir conditions, the ef-fect of connate water on hydrocarbon phase behavior can usually beneglected. Water can, however, affect the total-system phase behavior(for example, when hydrates form from natural-gas/water mixtures).

This chapter covers only two-phase, vapor/liquid phase behavior.Chap. 8 briefly covers three- and four-phase systems (vapor/liquid/liquid and vapor/liquid/liquid/solids) for low-temperature CO2/oiland rich-gas/oil mixtures, and Chap. 9 gives the behavior of vaporand solids related to hydrates.

Sec. 2.1 introduces the composition of petroleum reservoir fluidsand emphasizes their chemical complexity. Because reservoir fluidsare made up of many components, a detailed quantitative analysisis difficult to perform. Organic compounds found in reservoir fluidsare expressed by a general formula that classifies even high-molec-ular-weight compounds containing sulfur, nitrogen, and oxygen.This chapter also gives a historical review of the American Petro-leum Inst. (API) -supported projects that defined many of the com-pounds known today.

Simple one- and two-component phase behavior can be helpful indescribing the effects of pressure, temperature, and composition onthe reservoir-fluid phase behavior. Sec. 2.2 presents pressure/temper-ature ( p-T), pressure/volume ( p-V), and pressure/composition ( p-x)phase diagrams of simple systems. The behavior of these idealizedsystems is qualitatively similar to the behavior of complex reservoirfluids, as Sec. 2.3 shows.

Retrograde condensation is perhaps the most unusual phase be-havior that petroleum reservoir fluids exhibit.* Sec. 2.4 discussesthe definition of retrograde condensation and the effect of retro-grade condensation on the behavior of gas-condensate reservoirs.

Petroleum reservoir fluids can be divided into five general cate-gories, in increasing order of chemical complexity: dry gas, wet gas,gas condensate, volatile oil, and black oil. However, the phase be-haviors of gas condensates and volatile oils are considerably morecomplex than those of black oils. The component distribution in areservoir fluid, not simply the number of components, determineshow close a fluid is to a critical state. Complex phase behavior is typ-ically associated with systems that are “near critical”: systems thatusually contain 10 to 15 mol% of components that are heptanes andheavier (C7+).

Since the early 1930’s, experimental data have been measured on-fluids of each type listed above. Sec. 2.5 defines each fluid type byits p-T diagram. Also, general characteristics of reservoir fluids aresummarized in terms of composition and surface properties, such asGOR and stock-tank-oil gravity.

��

The nature and composition of a reservoir fluid depends somewhaton the depositional environment of the formation from which thefluid is produced. Geologic maturation also influences reservoir-fluidcomposition. Several theories offer explanations for the origin andformation of petroleum over geologic time; no single theory sufficesto explain how oil and gas were formed in all reservoirs. One theoryportrays reservoirs as giant high-temperature/high-pressure reactorswith catalytic rock surfaces that slowly convert deposited organicmatter into oil and gas. Other theories hypothesize that oil and gaswere formed from bacterial action on deposited organic matter. Otherinvestigators maintain that oil and gas may be formed in the samegeologic formation but that each fluid migrates to “traps” at differentelevations because of fluid-density differences and gravity forces.

Crude oil and natural gas are composed of many chemical com-pounds with a wide range of molecular weights. Some estimates2-4

suggest that perhaps 3,000 organic compounds can exist in a single

*Historically, retrograde condensation has been considered the most complex phase-behav-ior phenomenon observed by reservoir fluids. Perhaps equally intriguing are the phenomenaof strong compositional gradients, the condensing/vaporizing miscible mechanism (Chap. 8),asphaltene precipitation, and low-temperature, multiphase CO2 behavior.

2 PHASE BEHAVIOR

TABLE 2.1—COMPOSITION AND PROPERTIES OF SEVERAL RESERVOIR FLUIDS

Composition (mol%)

Component Dry Gas Wet Gas

Gas

Condensate

Near-Critical

Oil Volatile Oil Black Oil

CO2 0.10 1.41 2.37 1.30 0.93 0.02

N2 2.07 0.25 0.31 0.56 0.21 0.34

C1 86.12 92.46 73.19 69.44 58.77 34.62

C2 5.91 3.18 7.80 7.88 7.57 4.11

C3 3.58 1.01 3.55 4.26 4.09 1.01

i-C4 1.72 0.28 0.71 0.89 0.91 0.76

n-C4 0.24 1.45 2.14 2.09 0.49

i-C5 0.50 0.13 0.64 0.90 0.77 0.43

n-C5 0.08 0.68 1.13 1.15 0.21

C6(s) 0.14 1.09 1.46 1.75 1.61

C7 + 0.82 8.21 10.04 21.76 56.40

Properties

MC7�130 184 219 228 274

�C7�0.763 0.816 0.839 0.858 0.920

KwC712.00 11.95 11.98 11.83 11.47

GOR, scf/STB ∞ 105,000 5,450 3,650 1,490 300

OGR, STB/MMscf 0 10 180 275

�API 57 49 45 38 24

�g 0.61 0.70 0.71 0.70 0.63

psat, psia 3,430 6.560 7,015 5,420 2,810

Bsat, ft3/scf or bbl/STB 0.0051 0.0039 2.78 1.73 1.16

�sat, �� 9.61 26.7 30.7 38.2 51.4

reservoir fluid. The lighter and simpler compounds are produced asnatural gas after surface separation, whereas the heavier and morecomplex compounds form crude oil at stock-tank conditions. Table2.1 gives typical oilfield molar compositions for reservoir mixtures.The heavier components are usually lumped into a “plus” fractioninstead of being identified individually. Chap. 5 discusses methodsof quantifying and characterizing the components that make up theplus fraction—usually heptanes-plus.

Natural gas is composed mainly of low-molecular-weight alka-nes (methane through butanes), CO2, hydrogen sulfide, nitrogen,and, in some cases, lesser quantities of helium, hydrogen, CO, andcarbonyl sulfide.5 Most crude oils are composed mainly of hydro-carbons (hydrogen and carbon compounds). The broad spectrum oforganic compounds found in petroleum during the formation ofcrude oil also includes sulfur, nitrogen, oxygen, and trace metals.Tars and asphalts are solid or semisolid mixtures that include bitu-men, pitch, waxes, and resins. These high-molecular-weight com-plex colloidal suspensions exhibit non-Newtonian rheology.

The temperature and pressure gradients in a formation may causereservoir-fluid properties to vary as a function of depth. “Composi-tional grading” is the continual change of composition as a functionof depth.6-8 In compositional grading, reservoir temperature may benear the critical temperature of reservoir fluid(s) at certain depths inthe reservoir. Physically, the thermodynamic forces of individualcomponents in a near-critical mixture are of the same order of mag-nitude as gravity forces that tend to separate the lighter from theheavier components. The result can be a transition from an undersat-urated gas condensate at the highest elevation to an undersaturatedoil at the lowest elevation, with or without a visible phase transitionfrom gas to oil (gas/oil contact).

In petroleum refining, crude oil is often categorized according toits base and the hydrocarbon series (paraffin, naphthene, or aromat-ic) it contains in the highest concentration. Figs. 2.1 and 2.29 illus-trate the types and relative amounts of hydrocarbon series that canbe found in typical petroleum-refinery products. Nelson3 gives afull account of basic hydrocarbon chemistry and test methods that

have been used for many years to determine petroleum compositionand inspection properties for refining purposes. The more commontest methods include paraffin, naphthene, and aromatic; saturates,aromatics, resins, and asphaltenes; and Strieter (asphaltenes, resins,and oils) analyses; oil gravity in °API; Reid vapor pressure; true-boiling-point distillation; flash, fire, cloud, and pour points; color;and Saybolt and Furol viscosities. Chap. 5 discusses some of thesemethods that are used in petroleum engineering.

The empirical formula CnH2n�hSaNbOc can be used to classifynearly all compounds found in crude oil. The largest portion ofcrude oil is composed of hydrocarbons with carbon number, n, rang-ing from 1 to about 60, and h numbers ranging from h��2 for low-molecular-weight paraffin hydrocarbons to h��20 for high-mo-lecular-weight organic compounds (e.g., polycyclic aromatichydrocarbons). Occasionally, sulfur, nitrogen, and oxygen substitu-tions occur in high-molecular-weight organic compounds, with a,b, and c usually ranging from 1 to 3.2,10

Over the past 60 years, petroleum chemists have identifiedhundreds of the complex organic compounds found in petroleum.Beginning in 1927, Rossini and others11,12 conducted a lengthy in-vestigation of the composition of petroleum [API Research Project6 (API 6)] to develop and improve petroleum-refining processes. Ittook API 6 investigators almost 40 years to elucidate the composi-tion of a single midcontinent crude oil from Well No. 6 in South Pon-ca City, Oklahoma.

Because compounds with carbon numbers �12 could not be iso-lated from crude oils, during 1940–66, API Research Project 42 fo-cused on synthesizing and characterizing model hydrocarbons withhigh molecular weights. These model compounds were used foridentifying compounds that could not be isolated from crude oil. Acrude oil compound with analytical responses that matched those ofa synthesized model compound was inferred to have a similar chem-ical structure.

Other API projects13 followed API 6, and increasingly more com-plex petroleum compounds were identified. API 48 focused on sul-fur compounds, API 52 on nitrogen compounds, and API 56 on or-


Fig. 2.1—Petroleum products identified according to carbon number.

ganometallic compounds. API 60 extended the work of API 6 toinclude petroleum heavy ends.

In 1975, API stopped sponsoring basic research into the composi-tion of petroleum. From 1975 to 1982, the petroleum engineeringindustry made additional advances in analytical techniques mainlybecause of the synfuels effort. The most sophisticated analyticaltechniques now in use include highly selective solvent extrac-tion14-16; simulated distillation; gel permeation, high-performanceliquid,17 and supercritical chromatography18; and mass infrared,13C nuclear magnetic resonance,19 and Fourier-transform infraredspectroscopy. The American Chemical Soc. Div. of PetroleumChemistry provides a comprehensive review of this area of researchevery 2 to 3 years.

Table 2.220 shows an example of a crude-oil distillate classifiedby h number (in the general formula CnH2n�hSaNbOc) and prob-able structural type, which determines the range of possible n num-bers. Within and across each hydrocarbon class, many isomers shareh and n numbers. The alkane (paraffin) series (h�2) has completelysaturated hydrocarbon chains that are chemically very stable. Thealkene (olefin) and alkyne (acetylene) series (h�0 and h��2) arecomposed of unsaturated, straight-chain hydrocarbons. Because al-kenes and alkynes are reactive, they are not usually found in natural-ly occurring petroleum deposits.

The naphthene series (h�0), saturated-ring or cyclic compounds,are found in nearly all crudes. The aromatic or “benzene” series(h��6) are unsaturated cyclic compounds. Low-boiling-point aro-matics, which are also reactive, are found in relatively low concentra-tions in crude oil. Heavier crude oils are characterized by unsaturatedpolycyclic aromatic hydrocarbons with increasingly negative h num-

bers. As molecular weight increases, these compounds assume vary-ing degrees of fused-ring saturation, with occasional hydrocarbonside chains. Sulfur, nitrogen, and oxygen can be substituted in thefused hydrocarbon rings to form heterocyclics or can occupy variouspositions on side chains.21 Metals, such as nickel and vanadium, canform organometallic compounds (porphyrins) in crude oil.2,10

Asphalts, bitumens, and tars are complex colloidal mixtures of car-boids, carbenes, asphaltenes, and maltenes (resins and oils). Micellarstructures of carboids, carbenes, and asphaltenes are formed by aro-matic polycondensation reactions and are maintained in colloidal sus-pension by the maltenes. These fractions are separated according totheir solubility or lack of solubility in certain low-molecular-weightsolvents, such as propane, pentane, n-hexane, and carbon disulfide.

Fig. 2.316 shows a hypothetical chemical structure of an asphal-tene. The bracket around the structure implies that the structure isrepeated three times. Although asphalt mixtures are complex incomposition and rheology, they follow certain molecular-weightdistributions that can be characterized as discussed in Chap. 5.

Understanding the nature of asphaltenes is important in petroleumengineering because, even in low concentrations, asphaltenes canmarkedly affect reservoir-fluid phase behavior.22 Because asphal-tenes are polar and hydrogen bonding, they alter reservoir wettabilityby adsorbing onto the rock surface.23 This alteration of reservoirwettability may affect capillary pressure, relative-permeability rela-tions, residual oil saturations, waterflood behavior, dispersion, andelectrical properties. Figs. 2.2 and 2.3 vividly show that the composi-tion of crude oil is considerably more complex than the CnH2n�2straight-chain models commonly thought of as “oil.” This complexity

4 PHASE BEHAVIOR

Fig. 2.2—Summary of hydrocarbons to be expected in crude-oil fractions (from Neumann et al.9).

should be borne in mind when modeling the phase behavior of com-plex reservoir fluids, particularly in gas-injection projects.23,24

��! �� "��#��

The dependence of volumetric and phase behavior on temperature,pressure, and composition is similar for simple (two- and three-component) and complex (multicomponent) systems. Traditionally,the introduction to phase behavior of complex reservoir fluids startswith a description of the vapor-pressure and volumetric behavior ofsingle components. The introduction then proceeds to the behaviorof two- and three-component systems, and finally to the behavior ofcomplex multicomponent systems. Part of the rationale for this pro-cession lies in the Gibbs phase rule.25,26

The Gibbs phase rule states that the number of intensive variables(i.e., degrees of freedom), F, that must be specified to determine thethermodynamic state of equilibrium for a mixture containing n com-ponents distributed in P phases (gas, liquid, and/or solid), is

F � n � P � 2 . (2.1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Intensive (thermodynamic) variables, such as temperature, pres-sure, and density, do not depend on the amount of material in the sys-tem. Extensive variables, such as flow rate, total mass, or liquid vol-ume, depend on the extent of the system.

To attain equilibrium requires that no net interphase mass transfercan occur. Thus, the temperatures and pressures of the coexisting

phases must be the same and the chemical potentials of each compo-nent in each phase must be equal. A more stringent definition ofphase equilibrium includes other forces in addition to chemical po-tential (e.g., gravity and capillarity).

On the basis of Eq. 2.1, for a two-phase, single-component system,F�1 and only temperature or pressure needs to be specified to deter-mine the thermodynamic state of the system. For a two-phase, two-component system, F�2 and both temperature and pressure need tobe specified to define the thermodynamic state of the mixture. Two-phase binary systems allow one to focus on the effect of temperatureand pressure on the composition and the relative amounts of each ofthe two phases, regardless of the composition of the overall mixture.

The Gibbs phase rule implies that as the number of componentsincreases to n in a two-phase mixture, n�2 composition variablesmust be specified in addition to temperature and pressure. If morethan two phases are present, then n�P variables must be specifiedin addition to temperature and pressure. Because reservoir fluidscomprise many components, the number of variables that must bedefined to determine the state of a reservoir fluid is conceptually un-manageable. Therefore, simple systems are often used to model thebasic volumetric and phase behavior of crude oil mixtures.

Note that the phase rule must be modified if other potential fieldsare considered. For example, if the force of gravity is considered, as


TABLE 2.2—DISTRIBUTION OF h SERIES FROM 698 TO995°F DISTILLATE OF SWAN HILLS CRUDE OIL (Ref. 20)

Mass h Series Probable Type

�12 Naphthalenes

�14 Naphthenonaphthalenes and/or biphenyls

�16 Dinaphthenaphthalenes and/or

naphthenobiphenyls

�18 Trinaphthenaphthalenes and/or

dinaphthenobiphenyls

�20 Tetranaphthenaphthalenes and/or

trinaphthenobiphenyls

�22 Pentanaphthenaphthalenes and/or

tetranaphthenobiphenyls

�24 Hexanaphthenaphthalenes and/or

pentanaphthenobiphenyls

�26 Heptanaphthenaphthalenes and/or

hexanaphthenobiphenyls

�28 Octanaphthenaphthalenes and/or

heptanaphthenobiphenyls

�4S Tricyclic sulfides

�6S Tetracyclic sulfides

�8S Pentacyclic sulfides

�10S Hexacyclic sulfides

�8S Thiaindanes/thiatetralins

�10S Naphthenothiaindanes/thiatetralins

�12S Dinaphthenothiaindanes/thiatetralins

�14S Trinaphthenothiaindanes/thiatetralins

�10S Benzothiophenes

�12S Naphthenobenzothiophenes

is done when calculating compositional variation with depth, thephase rule is F�n�P�3.7

2.3.1 Single-Component Systems. The p-T curve shown in Fig. 2.4is a portion of the vapor-pressure curve for a typical hydrocarboncompound. Above and to the left of the curve, the hydrocarbon be-haves as a liquid; below and to the right, the hydrocarbon behavesas a vapor. Saturated liquid and vapor coexist at every point alongthe vapor-pressure curve. The curve ends at the critical temperatureand critical pressure of the hydrocarbon (the “critical point”). Fig.2.5 shows a 3D PVT diagram of a pure compound.

The critical temperature of a single component defines the tem-perature above which any gas/liquid mixture cannot coexist, regard-less of pressure. Similarly, the critical pressure defines the pressureabove which liquid and vapor cannot coexist, regardless of tempera-ture. Along the vapor-pressure curve, two phases coexist in equilib-rium. At the critical point, the vapor and liquid phases can no longerbe distinguished, and their intensive properties are identical.

For a multicomponent system, the definition of the critical pointis also based on a temperature and pressure at which the vapor andliquid phases are indistinguishable. However, for a single-compo-nent system, the two-phase region terminates at the critical point. Ina multicomponent system, the two-phase region can extend beyondthe system’s critical point (i.e., at temperatures greater than the criti-cal temperature and pressures greater than the critical pressure).

Fig. 2.627 illustrates the continuity of gas and liquid phases for purecomponents. In this figure, the darker shading corresponds to higherdensity. A sharp contrast in phase densities is readily apparent alongthe vapor-pressure curve. As temperature increases along the vapor-pressure curve, the discontinuity becomes harder to discern, until fi-nally, at the critical point, the contrast in shading is hardly noticeable.Qualitatively, the behavior described by the shading in Fig. 2.6 is thesame for multicomponent mixtures in the undersaturated region.

Fig. 2.3—Hypothetical structure of a petroleum asphaltene (afterSpeight and Moschopedis14).

Fig. 2.4—p-T diagram for a single component in the region of va-por/liquid behavior near the critical point ( pc�critical pressureand Tc�critical temperature).

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Phase changes do not have to take place abruptly if certain tem-perature and pressure paths are followed. A process can start as asaturated liquid and end as a saturated vapor, with no abrupt changein phase. The path D–A–E–F–G–B–D in Fig. 2.4 is an example ofa process that changes phases without crossing the vapor-pressurecurve. Pure components actually exist as a saturated “liquid” and“vapor” only along the vapor-pressure curve. At other pressures andtemperatures, the component only behaves “liquid-like” or “vapor-like,” depending on the location of the system temperature and pres-sure relative to the system’s critical point. Katz28 suggested callinga pure substance “single-phase fluid” at pressures greater than thecritical pressure. Strictly speaking, the terms liquid-like and vapor-like should be used to describe undersaturated fluids.

6 PHASE BEHAVIOR

Fig. 2.5—Three-dimensional schematic of the PVT surface of apure compound (source unknown).

Fig. 2.726 shows a p-V diagram for ethane. The area enclosed bythe saturation envelope represents the two-phase region. The areato the left of the envelope is the liquid-like region, and the area tothe right is the vapor-like region. Point C represents the criticalpoint. The saturation curve to the left of the critical point (from PointA to Point C) defines the bubblepoint curve, along which the com-

ponent is a saturated liquid. Similarly, the saturation curve to theright of the critical point (Point B to Point C) defines the dewpointcurve, along which the component is a saturated vapor.

For any temperature less than the critical temperature, successivedecreases in volume will elevate the pressure of the vapor until the“dewpoint” (vapor pressure) is reached (Point B on Fig. 2.7). Atthese conditions, the component is a saturated vapor in equilibriumwith an infinitesimal amount of saturated liquid. Further decreasesin the volume at constant temperature will result in proportionate in-creases in the amount of saturated liquid condensed, but the pressuredoes not change (i.e., the system pressure remains equal to the vaporpressure). While more liquid is being formed, the total volume (atPoint D) is being reduced. However, the densities and other inten-sive properties of the saturated vapor and saturated liquid remainconstant as a consequence of the Gibbs phase rule.

A simple mass balance further shows that the ratio of liquid to va-por equals the ratio of Curve B–D to Curve D–A. Further decreasesin volume will condense more liquid until the bubblepoint isreached. At the bubblepoint, the system is 100% saturated liquid inequilibrium with an infinitesimal amount of saturated vapor. Furtherdecreases in volume beyond the bubblepoint are accompanied by alarge increase in pressure because the liquid is only slightly com-pressible. This is indicated by the nearly vertical isotherms on theleft side of Fig. 2.7. In the undersaturated vapor region on the rightside of the diagram, a large change in volume reduces pressure onlyslightly because the vapor is highly compressible.

2.3.2 Two-Component Systems. Two-component systems areslightly more complex than single-component systems becauseboth temperature and pressure affect phase behavior in the saturatedregion. Two important differences between single- and two-compo-nent systems exist. The saturated p-T projection is represented by aphase envelope rather than by a vapor-pressure curve, and the criti-

Fig. 2.6—Continuity of vapor and liquid states for a single component along the vapor-pressure curveand at supercritical conditions (after Katz and Kurata27).

00 100 200 300 400 500 600

Temperature, °F

3,000

2,000

1,000


Fig. 2.7—p-V diagram for ethane at three temperatures (fromStanding26).

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cal temperature and critical pressure no longer define the extent ofthe two-phase, vapor/liquid region. Fig. 2.829 compares the p-T andp-V behavior of pure compounds and mixtures.

Fig. 2.926 is a p-T projection of the ethane/n-heptane system fora fixed composition. For a single-component system, the dew- andbubblepoint curves are one in the same; i.e., they coincide with thevapor-pressure curve. In a binary (or other multicomponent) sys-tem, the dew- and bubblepoint curves no longer coincide, and aphase envelope results instead of a vapor-pressure curve. To the leftof the phase envelope, the mixture behaves liquid-like, and to theright it behaves vapor-like.

For binary or other multicomponent systems, the critical tempera-ture and pressure are defined as the point where the dew- and bub-blepoint curves intersect. At this point, the equilibrium phases arephysically indistinguishable. Also, in contrast to the single-compo-nent system, two phases can exist at temperatures and pressuresgreater than the critical temperature and pressure. The highest tem-perature at which two phases can coexist in equilibrium is definedas the cricondentherm (Tangent b–b in Fig. 2.9). Similarly, the high-est pressure at which two phases can coexist is defined as the cricon-denbar (Tangent a–a).

In the single-phase region, vapor and liquid are distinguishedonly by their densities and other physical properties. The region justbeyond the critical point of a mixture has often been called the “su-percritical” or “dense-fluid” region. Here, the fluid is considered tobe neither gas nor liquid because the fluid properties are not strictlyliquid-like or vapor-like.

Kay30 measured the phase behavior of the binary ethane/n-hep-tane system for several compositions, as Fig. 2.10 shows. On the leftside of this figure, the curve terminating at Point C is the vapor pres-sure of pure ethane; the curve on the right, terminating at Point C7,is the vapor pressure of pure n-heptane. Points C1 through C3 arethe critical points of ethane/n-heptane mixtures at different com-positions. The dashed line represents the locus of critical points forthe infinite number of possible ethane/n-heptane mixtures. Eachmixture composition has its own p-T phase envelope.

The three compositions shown, which are 90.22, 50.25, and 9.78wt% ethane, represent a system that is mainly ethane, a system thatis one-half ethane and one-half n-heptane (by weight), and a systemthat is mainly n-heptane, respectively. Several interesting featuresof binary and multicomponent systems can be studied from thesethree mixtures. As composition changes, the location of the criticalpoint and the shape of the p-T phase diagram also change.

Note that the critical pressures of many (but not all) mixtures arehigher than the critical pressures of the components composing the

mixture. With a mixture composed mainly of ethane, the criticalpoint lies to the left of the cricondentherm. Such a system is analo-gous to a reservoir gas-condensate system. As the percentage ofethane in the mixture increases further, the critical point of the sys-tem approaches that of pure ethane.

The critical point for the mixture composed mostly of n-heptanelies below the cricondenbar. This system is analogous to a reservoirblack-oil system. As the percentage of n-heptane increases, the criti-cal point of the mixture approaches that of pure n-heptane. Withequal percentages of ethane and n-heptane, the critical pressure isclose to the cricondenbar of ethane and n-heptane. As the concentra-tion of each component becomes similar, the two-phase region be-comes larger.

Other binaries provide additional insight into the effect of widelydiffering boiling points of the components making up the system.Fig. 2.1131 shows the vapor pressure of several hydrocarbons andthe critical loci of their binary mixtures with methane. As the boilingpoints of the methane/hydrocarbon binary become more dissimilar,the two-phase region becomes larger and the critical pressure in-creases. For binaries with components that have similar molecularstructures, the loci of critical points are relatively flat.

2.3.3 Multicomponent Systems. Phase diagrams for naturally oc-curring reservoir fluids are similar to those for binary mixtures. Fig.2.125 is the first p-T phase diagram measured for a complex gas-condensate system. This p-T diagram is particularly useful becauseit exhibits oil-like to gas-like behavior over a range of typical reser-voir temperatures, from 80 to 240°F. Katz and coworkers32 used aglass-windowed cell to measure the distribution of gas and liquidphases throughout the two-phase region and near the mixture’s criti-cal point. Fig. 2.135 shows isotherms of volume percent vs. pressurethat were measured to determine the two-phase boundary and thevolume-percent quality lines in the p-T diagram in Fig. 2.12.

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Kurata and Katz33 give the most concise and relevant discussion ofretrograde phenomena related to petroleum engineering. In 1892,Kuenen34 used the term “retrograde condensation” to describe theanomalous behavior of a mixture that forms a liquid by an isother-mal decrease in pressure or by an isobaric increase in temperature.Conversely, “retrograde vaporization” can be used to describe theformation of vapor by an isothermal increase in pressure or by anisobaric decrease in temperature. Neither form of retrograde behav-ior occurs in single-component systems.

Fig. 2.14 is a constant-composition p-T projection of a multicom-ponent system. The diagram shows lines of constant liquid volumepercent (quality). Although total composition is fixed, the respec-tive compositions of saturated vapor and liquid phases change alongthe quality lines. The bubblepoint curve represents the locus of100% liquid, and the dewpoint curve represents the locus of 0% liq-uid. The bubble- and dewpoint curves meet at the mixture criticalpoint. Lines of constant quality also converge at the mixture criticalpoint. The regions of retrograde behavior are defined by the lines ofconstant quality that exhibit a maximum with respect to temperatureor pressure. Fig. 2.14 shows that for retrograde phenomena to occur,the temperature must be between the critical temperature and thecricondentherm. Fig. 2.1535 illustrates the liquid volumetric behav-ior of a lean gas-condensate system measured by Eilerts et al.35-37

Fig. 2.12 shows the p-T diagram of a reservoir mixture that wouldbe considered a gas condensate if it had been discovered at a reservoirtemperature of, for example, 200°F and an initial pressure of 2,700psia. For these initial conditions, if reservoir pressure drops below2,560 psia from depletion, the dewpoint will be passed and a liquidphase will develop in the reservoir. Liquid dropout will continue toincrease until the pressure reaches 2,300 psia, when a maximum of25 vol% liquid will have accumulated. According to Fig. 2.12, furtherpressure reduction will revaporize most of the condensed liquid.

These comments assume that the overall composition of the res-ervoir mixture remains constant during depletion, a reasonable as-sumption in the context of this general discussion. In reality, howev-

8 PHASE BEHAVIOR

Fig. 2.8—Qualitative p-T and p-V plots for pure fluids and mixtures; Vc�critical volume (after Edmister and Lee29).

er, the behavior of liquid dropout and revaporization differs fromthat suggested by constant-composition analysis. The retrogradeliquid saturation is usually less than the saturation needed to mobi-lize the liquid phase. Because the heavier components in the originalmixture constitute most of the (immobile) condensate saturation,the overall molecular weight of the remaining reservoir fluid in-creases during depletion. The phase envelope for this heavier reser-voir mixture is pushed down and to the right of the original phasediagram (Fig. 2.16); the critical point is shifted to the right towarda higher temperature. It is not unusual that a retrograde-condensatemixture under depletion will reach a condition where the overallcomposition would exhibit a bubblepoint pressure if the reservoirwere repressured (i.e., the overall mixture critical temperature be-comes greater than the reservoir temperature). This change in over-all reservoir composition results in less revaporization at lower pres-sures. Fig. 2.17 shows the difference between constant-compositionand “depletion” liquid-dropout curves.

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One might assume that the name used to identify a reservoir fluidshould not influence how the fluid is treated as long as its physicalproperties are correctly defined. In theory this is true, but in practicewe are usually required to define petroleum reservoir fluids as either“oil” or “gas.” For example, regulatory bodies require the definitionof reservoir fluid for well spacing and determining allowable pro-duction rates and field-development strategy (e.g., unitization).

The classification of a reservoir fluid as dry gas, wet gas, gas con-densate, volatile oil, or black oil is determined (1) by the location ofthe reservoir temperature with respect to the critical temperature andthe cricondentherm and (2) by the location of the first-stage separa-tor temperature and pressure with respect to the phase diagram ofthe reservoir fluid. Fig. 2.18 illustrates how four types of depletionreservoirs for the same hydrocarbon system are defined by the loca-tion of the initial reservoir temperature and pressure.


Fig. 2.9—p-T diagram for a C2/n-C7 mixture with 96.83 mol%ethane (from Standing26).

Fig. 2.11—p-T diagram for various hydrocarbon binaries illus-trating the effects of molecular-weight differences on critical-point loci (after Brown et al.31).

Fig. 2.10—p-T diagram for the C2/n-C7 system at various con-centrations of C2 (after Kay30).

A reservoir fluid is classified as dry gas when the reservoir temper-ature is greater than the cricondentherm and surface/transport condi-tions are outside the two-phase envelope; as wet gas when the reser-voir temperature is greater than the cricondentherm but the surfaceconditions are in the two-phase region; as gas condensate when thereservoir temperature is less than the cricondentherm and greater thanthe critical temperature; and as an oil (volatile or black oil) when thereservoir temperature is less than the mixture critical temperature.

For a given reservoir temperature and pressure, Fig. 2.1938 showsthe spectrum of reservoir fluids from wet gas to black oil expressedin terms of surface GOR’s and oil/gas ratios (OGR’s). A more quan-titative classification is also given in Fig. 2.19 in terms of molarcomposition, by use of a ternary diagram. In the near-critical region,gas condensates have a C7+ concentration less than �12.5 mol%and volatile oils fall between 12.5 to 17.5 mol% C7+.

Retrograde gas-condensate reservoirs26,39 typically exhibitGOR’s between 3,000 and 150,000 scf/STB (OGR’s from about 350to 5 STB/MMscf) and liquid gravities between 40 and 60°API. Thecolor of stock-tank liquid is expected to lighten from volatile-oil togas-condensate systems, although light volatile oils may be yellow-ish or water-white and some condensate liquids can be dark brown

Fig. 2.12—p-T diagram for a gas-condensate system (after Katzet al.5).

10 PHASE BEHAVIOR

Fig. 2.13—Volume isotherms for the gas-condensate p-T dia-gram in Fig. 2.12 (after Katz et al.5)

Fig. 2.14—Hypothetical p-T diagram for a gas condensate show-ing the isothermal retrograde region.

or even black. Color has not been a reliable means of differentiatingclearly between gas condensates and volatile oils, but in general,dark colors indicate the presence of heavy hydrocarbons.

In some cases, for condensate recovery from a surface process fa-cility, the reservoir fluid is mistakenly interpreted to be a gas con-densate. Strictly speaking, the definition of a gas condensate de-pends only on reservoir temperature. The definition of a reservoirfluid as wet or dry gas depends on conditions at the surface. Thismakes differentiation between dry and wet gas difficult because anygas can conceivably be cooled enough to condense a liquid phase.

The classification of a fluid as an oil is unambiguous because theonly requirement is that the reservoir temperature be less than the

Fig. 2.15—Liquid volume (expressed as a liquid/gas ratio) behavior for a lean-gas-condensatesystem (from Eilerts et al.35).

BUBBLEPOINT


Fig. 2.16—Change in phase envelope during the depletion of a gas condensate.

critical temperature. However, the distinction between a black oiland a volatile oil is more arbitrary. Generally speaking, a volatile oilis a mixture with a relatively high solution gas/oil ratio. Volatile oilsexhibit large changes in properties when pressure is reduced onlysomewhat below the bubblepoint. In an extreme case, the oil volumemay shrink from 100 to 50% with a reduction in pressure of only 100psi below the bubblepoint. Black-oil properties, on the other hand,exhibit gradual changes, with nearly linear pressure dependence be-low the bubblepoint.

Volatile oils typically yield stock-tank-oil gravities greater than35°API, surface GOR’s between 1,000 and 3,000 scf/STB, andFVF’s (see Formation Volume Factors in Chap. 6) greater than �1.5RB/STB. Solution gas released from a volatile oil contains signifi-cant quantities of stock-tank liquid (condensate) when this gas isproduced to the surface. Solution gas from black oils is usually con-sidered “dry,” yielding insignificant stock-tank liquids when pro-duced to surface conditions.

Fig. 2.17—Retrograde volumes for constant-composition andconstant-volume depletion experiments.

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CCE has stronger revaporization at lowpressures because of greater (initial)mass of gas remaining in cell

For engineering calculations, the liquid content of released solu-tion gas is perhaps the most important distinction between volatileoils and black oils. This difference is also the basis for the modifica-tion of standard black-oil PVT properties discussed in Chap. 7. Areasonable engineering distinction between black oils and volatileoils can be made on the basis of simple reservoir material-balancecalculations. If the total surface oil and gas recoveries calculated bya reservoir material balance with the standard black-oil PVT for-mulation are sufficiently close to the recoveries calculated by acompositional material balance, the oil can probably be considereda black oil (see Chap. 7). If calculated oil recoveries are significantlydifferent, the reservoir mixture should be treated as a volatile oil byuse of a compositional approach or the modified black-oil PVTproperties outlined in Chap. 7. Several researchers40,41 have shownthat a compositional material balance for depletion of volatile-oilreservoirs may predict from two to four times the surface liquid re-

Fig. 2.18—p-T diagram of a reservoir fluid illustrating differenttypes of depletion reservoirs.

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12 PHASE BEHAVIOR

Fig. 2.19—Spectrum of reservoir fluids in order of increasingchemical complexity from wet gas to black oil (from Cronquist38).

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covery predicted by conventional material balances that are basedon the standard black-oil PVT formulation.

Fluid samples obtained from a new field discovery may be instru-mental in defining the existence of an overlying gas cap or an under-lying oil rim. Referring to Fig. 2.20, if the initial reservoir pressureequals the measured bubblepoint pressure of a bottomhole or re-combined sample, the oil is probably saturated at initial reservoirconditions. This implies that an equilibrium gas cap could exist atsome higher elevation. Likewise, if the initial reservoir pressure isthe same as the measured dewpoint pressure of a reservoir gas sam-

Fig. 2.20—p-T phase diagram of a gas-cap fluid in equilibriumwith an underlying saturated oil.

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ple, the gas is probably saturated at initial reservoir conditions, andan equilibrium oil could exist at some lower elevation.

Discovery of a saturated reservoir fluid will usually require fur-ther field delineation to substantiate the presence of a second equi-librium phase above or below the tested interval. This may entailrunning a repeat-formation-tester tool to determine the fluid-pres-sure gradient as a function of depth, or a new well may be requiredupdip or downdip to the discovery well. Representative samples ofsaturated fluids may be difficult to obtain during a production test.42

Standing26 discusses the situation of an undersaturated gas conden-sate sampled during a test where bottomhole flowing pressure dropsbelow the dewpoint pressure. The produced fluid, which is not rep-resentative of the original reservoir fluid, may have a dewpointequal to initial reservoir pressure. This situation would incorrectlyimply that the reservoir is saturated at initial conditions and that anunderlying oil rim may exist.

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1. Katz, D.L. and Williams, B.: “Reservoir Fluids and Their Behavior,”Amer. Soc. Pet. Geologists Bulletin (February 1952) 36, No. 2, 342.

2. Smith, H.M. et al.: “Keys to the Mystery of Crude Oil,” Trans., API,Dallas (1959) 433.

3. Nelson, W.L.: Petroleum Refinery Engineering, fourth edition,McGraw-Hill Book Co. Inc., New York City (1958).

4. Nelson, W.L.: “Does Crude Boil at 1400°F?,” Oil & Gas J. (1968) 125.5. Katz, D.L. et al.: Handbook of Natural Gas Engineering, McGraw-Hill

Book Co. Inc., New York City (1959).6. Muskat, M.: “Distribution of Non-Reacting Fluids in the Gravitational

Field,” Physical Review (1930) 35, 1384.7. Sage, B.H. and Lacey, W.N.: “Gravitational Concentration Gradients in

Static Columns of Hydrocarbon Fluids,” Trans., AIME (1939) 132, 120.8. Schulte, A.M.: “Compositional Variations Within a Hydrocarbon Col-

umn Due to Gravity,” paper SPE 9235 presented at the 1980 SPE Annu-al Technical Conference and Exhibition, Dallas, 21–24 September.

9. Neumann, H.J., Paczynska-Lahme, B., and Severin, D.: Compositionand Properties of Petroleum, Halsted Press, New York City (1981).

10. Thompson, C.J., Ward, C.C., and Ball, J.S.: “Characteristics of World’sCrude Oils and Results of API Research Project 60,” Report B-7, Ener-gy R&D Admin. (ERDA) (1976).

11. Rossini, F.D.: “The Chemical Constitution of the Gasoline Fraction ofPetroleum—API Research Project 6,” API, Dallas (1935).

12. Rossini, F.D. and Mair, B.J.: “The Work of the API Research Project onthe Composition of Petroleum,” Proc., Fifth World Pet. Cong. (1954) 223.

13. Miller, A.E.: “Review of American Petroleum Institute Research Proj-ects on Composition and Properties of Petroleum,” Proc., Fourth WorldPet. Cong. (1955) 27.

14. Speight, J.C. and Moschopedis, S.E.: “On the Molecular Nature of Pe-troleum Asphaltenes,” Trans., Advances in Chemistry, AmericanChemical Soc. (1981) 195, 1.

15. Speight, J.G., Long, R.B., and Trowbridge, T.D.: “Factors Influencingthe Separation of Asphaltenes from Heavy Petroleum Feedstocks,”Fuel (1984) 63, 616.

16. Speight, J.G. and Pancirov, R.J.: “Structural Types in Petroleum As-phaltenes as Deduced from Pyrolysis/Gas Chromatography/MassSpectrometry,” Liquid Fuels Technology (1984) 2, No. 3, 287.

17. Such, C., Brulé, B., and Baluja-Santos, C.: “Characterization of a RoadAsphalt by Chromatographic Techniques (GPC and HPLC),” J. LiquidChrom. (1979) 2, No. 3, 437.

18. Fetzer, J.C. et al.: “Characterization of Carbonaceous Materials UsingExtraction with Supercritical Pentane,” report, Contract No. DOE/ER/00854-29, U.S. DOE (1980).

19. Helm, R.V. and Petersen, J.C.: “Compositional Studies of an Asphalt andIts Molecular Distillation Fractions by Nuclear Magnetic Resonance andInfrared Spectrometry,” Analytical Chemistry (1968) 40, No. 7, 1100.

20. Dooley, J.E. et al.: “Analyzing Heavy Ends of Crude, Swan Hills,” Hy-dro. Proc. (April 1974) 53, 93.

21. Dooley, J.E. et al.: “Analyzing Heavy Ends of Crude, Comparisons,”Hydro. Proc. (Nov. 1974) 53, 187.

22. Katz, D.L. and Beu, K.L.: “Nature of Asphaltic Substances,” Ind. &Eng. Chem. (February 1945) 37, 195.

23. Monger, T.G. and Trujillo, D.E.: “Organic Deposition During CO2 andRich-Gas Flooding,” SPERE (February 1991) 17.

24. Bossler, R.B. and Crawford, P.B.: “Miscible-Phase Floods May Precip-itate Asphalt,” Oil & Gas J. (23 February 1959) 57, 137.


25. Gibbs, J.W.: The Collected Works of J. Willard Gibbs, Yale U. Press,New Haven, Connecticut (1948).

26. Standing, M.B.: Volumetric and Phase Behavior of Oil Field Hydrocar-bon Systems, SPE, Richardson, Texas (1977).

27. Katz, D.L. and Kurata, F.: “Retrograde Condensation,” Ind. & Eng.Chem. (June 1940) 32, No. 6, 817.

28. Katz, D.L. and Singleterry, C.C.: “Significance of the Critical Phenom-ena in Oil and Gas Production,” Trans., AIME (1939) 132, 103.

29. Edmister, W.C. and Lee, B.I.: Applied Hydrocarbon Thermodynamics,second edition, Gulf Publishing Co., Houston (1984) I.

30. Kay, W.B.: “The Ethane-Heptane System,” Ind. & Eng. Chem. (1938)30, 459.

31. Brown, G.G. et al.: Natural Gasoline and the Volatile Hydrocarbons,NGAA, Tulsa, Oklahoma (1948) 24–32.

32. Katz, D.L., Vink, D.J., and David, R.A.: “Phase Diagram of a Mixtureof Natural Gas and Natural Gasoline Near the Critical Conditions,”Trans., AIME (1939) 136, 165.


34. Kuenen, J.P.: “On Retrograde Condensation and the Critical Phenome-na of Two Substances,” Commun. Phys. Lab. U. Leiden (1892) 4, 7.

35. Eilerts, C.K.: Phase Relations of Gas Condensate Fluids, Monograph10, USBM, American Gas Assn., New York City (1957) I and II.

36. Eilerts, C.K.: “Gas Condensate Reservoir Engineering, 1. The Re-serve Fluid, Its Composition and Phase Behavior,” Oil & Gas J. (1February 1947) 63.

37. Eilerts, C.K. et al.: “Phase Relations of a Gas-Condensate Fluid atLow Tempertures, Including the Critical State,” Pet. Eng. (February1948) 19, 154.

38. Cronquist, C.: “Dimensionless PVT Behavior of Gulf Coast ReservoirOils,” JPT (May 1973) 538.

39. Moses, P.L.: “Engineering Applications of Phase Behavior of Crude Oiland Condensate Systems,” JPT (July 1986) 715.

40. Lohrenz, J., Clark, G.C., and Francis, R.J.: “A Compositional MaterialBalance for Combination Drive Reservoirs with Gas and Water Injec-tion,” JPT (November 1963) 1233; Trans., AIME, 228.

41. Reudelhuber, F.O. and Hinds, R.F.: “Compositional Material-BalanceMethod for Prediction of Recovery From Volatile Oil Depletion DriveReservoirs,” JPT (1957) 19; Trans., AIME, 210.

42. Fevang, Ø. and Whitson, C.H.: “Accurate In-Situ Compositions in Pe-troleum Reservoirs,” paper SPE 28829 presented at the 1994 EuropeanPetroleum Conference, London, 25–27 October.

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�API 141.5/(131.5��API� �g/cm3

bbl �1.589 873 E�01�m3

ft3 �2.831 685 E�02�m3

�F (�F�32)/1.8 ��Cgal �3.785 412 E�03�m3

lbm �4.535 924 E�01�kgpsi �6.894 757 E�00�kPa

18 PHASE BEHAVIOR

��

� ��

��

Chap. 3 covers the properties of oil and gas systems, their nomencla-ture and units, and correlations used for their prediction. Sec. 3.2covers the fundamental engineering quantities used to describephase behavior, including molecular quantities, critical and reducedproperties, component fractions, mixing rules, volumetric proper-ties, transport properties, and interfacial tension (IFT).

Sec. 3.3 discusses the properties of gas mixtures, including cor-relations for Z factor, pseudocritical properties and wellstream grav-ity, gas viscosity, dewpoint pressure, and total formation volumefactor (FVF). Sec. 3.4 covers oil properties, including correlationsfor bubblepoint pressure, compressibility, FVF, density, and viscos-ity. Sec. 3.5 gives correlations for IFT and diffusion coefficients.Sec. 3.6 reviews the estimation of K values for low-pressure ap-plications, such as surface separator design, and convergence-pres-sure methods used for reservoir calculations.

��

3.2.1 Molecular Quantities. All matter is composed of elements thatcannot be decomposed by ordinary chemical reactions. Carbon (C),hydrogen (H), sulfur (S), nitrogen (N), and oxygen (O) are examplesof the elements found in naturally occurring petroleum systems.

The physical unit of the element is the atom. Two or more ele-ments may combine to form a chemical compound. Carbon dioxide(CO2), methane (CH4), and hydrogen sulfide (H2S) are examples ofcompounds found in naturally occurring petroleum systems. Whentwo atoms of the same element combine, they form diatomic com-pounds, such as nitrogen (N2) and oxygen (O2). The physical unitof the compound is the molecule.

Mass is the basic quantity for measuring the amount of a substance.Because chemical compounds always combine in a definite propor-tion (i.e., as a simple ratio of whole numbers), the mass of the atomsof different elements can be conveniently compared by relating themwith a standard. The current standard is carbon-12, where the elementcarbon has been assigned a relative atomic mass of 12.011.

The relative atomic mass of all other elements have been deter-mined relative to the carbon-12 standard. The smallest element ishydrogen, which has a relative atomic mass of 1.0079. The relativeatomic mass of one element contains the same number of atoms asthe relative atomic mass of any other element. This is true regardlessof the units used to measure mass.

According to the SI standard, the definition of the mole reads “themole is the amount of substance of a system which contains as manyelementary entities as there are atoms in 0.012 kilograms of car-

bon-12.” The SI symbol for mole is mol, which is numerically iden-tical to the traditional g mol.

The SPE SI standard1 uses kmol as the unit for a mole where kmoldesignates “an amount of substance which contains as many kilo-grams (groups of molecules) as there are atoms in 12.0 kg (incor-rectly written as 0.012 kg in the original SPE publication) of car-bon-12 multiplied by the relative molecular mass of the substanceinvolved.”

A practical way to interpret kmol is “kg mol” where kmol is nu-merically equivalent to 1,000 g mol (i.e., 1,000 mol). Otherwise, thefollowing conversions apply.

1 kmol � 1,000 mol� 1,000 g mol� 2.2046 lbm mol

1 lbm mol � 0.45359 kmol� 453.59 mol� 453.59 g mol

1 mol � 1 g mol� 0.001 kmol� 0.0022046 lbm mol

The term molecular weight has been replaced in the SI system bymolar mass. Molar mass, M, is defined as the mass per mole(M�m/n) of a given substance where the unit mole must be consis-tent with the unit of mass. The numerical value of molecular weightis independent of the units used for mass and moles, as long as theunits are consistent. For example, the molar mass of methane is16.04, which for various units can be written

M� 16.04 kg/kmol� 16.04 lbm/lbm mol� 16.04 g/g mol� 16.04 g/mol

3.2.2 Critical and Reduced Properties. Most equations of state(EOS’s) do not use pressure and temperature explicitly to define thestate of a system, but instead they generalize according to corre-sponding-states theory by use of two or more reduced properties,which are dimensionless.2

Tr� T�Tc , (3.1a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

GAS AND OIL PROPERTIES AND CORRELATIONS 19

Fig. 3.1—Reservoir densities as functions of pressure and tem-perature.

pr� p�pc , (3.1b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Vr� V�Vc , (3.1c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

and �r� ��c, (3.1d). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

where �r� 1�Vr. Absolute units must be used when calculatingreduced pressure and temperature. pc, Tc, Vc, and �c are the truecritical properties of a pure component, or some average for a mix-ture. In most petroleum engineering applications, the range of re-duced pressure is from 0.02 to 30 for gases and 0.03 to 40 for oils;reduced temperature ranges from �1 to 2.5 for gases and from 0.4to 1.1 for oils. Reduced density can vary from 0 at low pressures toabout 3.5 at high pressures.

Average mixture, or pseudocritical, properties are calculatedfrom simple mixing rules or mixture specific gravity.3,4 Denotinga mixture pseudocritical property by �pc, the pseudoreduced proper-ty is defined �pr� ��pc. Pseudocritical properties are not approxi-mations of the true critical properties, but are chosen instead so thatmixture properties will be estimated correctly with corresponding-states correlations.

3.2.3 Component Fractions and Mixing Rules. Petroleum reser-voir mixtures contain hundreds of well-defined and “undefined”components. These components are quantified on the basis of mole,weight, and volume fractions. For a mixture having N components,i� 1, . . . , N, the overall mole fractions are given by

zi�ni

�N

j�1

nj

�mi�Mi

�N

j�1

mj�Mj

, (3.2). . . . . . . . . . . . . . . . . . . . . . .

where n�moles, m�mass, M�molecular weight, and the sum ofzi is 1.0. In general, oil composition is denoted by xi and gas com-position by yi.

Weight or mass fractions, wi, are given by

wi�mi

�N

j�1

mj

�ni Mi

�N

j�1

nj Mj

, (3.3). . . . . . . . . . . . . . . . . . . . . . . .

where the sum of wi� 1.0. Although the composition of a mixtureis usually expressed in terms of mole fraction, the measurement ofcomposition is usually based on mass, which is converted to molefraction with component molecular weights.

For oil mixtures at standard conditions (14.7 psia and 60°F), thetotal volume can be approximated by the sum of the volumes of indi-vidual components, assuming ideal-solution mixing. This results in

the following relation for volume fractions xvi, based on componentdensities at standard conditions �i or specific gravities �i.

xvi�mi��i

�N

j�1

mj��j

�ni Mi��i

�N

j�1

nj Mj�� j

�xi Mi��i

�N

j�1

xj Mj��j

�xi Mi�� i

�N

j�1

xj Mj�� j

, (3.4). . . . . . . . . . . . . . . . . . . . . . . . . . . . .

where the sum of xvi is unity.Having defined component fractions, we can introduce some

common mixing rules for averaging the properties of mixtures.Kay’s5 mixing rule, the simplest and most widely used, is given bya mole-fraction average,

��N

i�1

zi�i . (3.5). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

This mixing rule is usually adequate for molecular weight, pseudo-critical temperature, and acentric factor.6 We can write a generalizedlinear mixing rule as

��

��

N

i�1i�i

�N

i�1

�i

, (3.6). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

where �i is usually one of the following weighting factors: �i� zi,mole fraction (Kay’s rule); �i� wi, weight fraction; or �i� xvi,volume fraction. Depending on the quantity being averaged, othermixing rules and definitions of �i may be appropriate.7,8 For exam-ple, the mixing rules used for constants in an EOS (Chap. 4) can bechosen on the basis of statistical thermodynamics.

3.2.4 Volumetric Properties. Density, �, is defined as the ratio ofmass to volume,

�� m�V, (3.7). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

expressed in such units as lbm/ft3, kg/m3, and g/cm3. Fig. 3.1 showsthe magnitudes of density for reservoir mixtures. Molar density,�M, gives the volume per mole:

�M� n�V. (3.8). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Specific volume, v^, is defined as the ratio of volume to mass and isequal to the reciprocal of density.

v^ � V�m� 1��. (3.9). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Molar volume, v, defines the ratio of volume per mole,

v� V�n� M�� 1��M , (3.10). . . . . . . . . . . . . . . . . . . . .

and is typically used in cubic EOS’s. Molar density, �M, is given by

�M� 1�v� ��M, (3.11). . . . . . . . . . . . . . . . . . . . . . . . . . . . .

and is used in the formulas of some EOS’s.According to the SI standard, relative density replaces specific

gravity as the term used to define the ratio of the density of a mixtureto the density of a reference material. The conditions of pressure andtemperature must be specified for both materials, and the densitiesof both materials are generally measured at standard conditions(standard conditions are usually 14.7 psia and 60°F).

�� psc, Tsc�

�ref�psc, Tsc�

, (3.12a). . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20 PHASE BEHAVIOR

Fig. 3.2—Reservoir compressibilities as functions of pressure.

�o��o�sc

��w�sc

, (3.12b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

and �g��g�sc

��air�sc

. (3.12c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Air is used as the reference material for gases, and water is used asthe reference material for liquids. Specific gravity is dimensionless,although it is customary and useful to specify the material used as areference (air�1 or water�1). In older references, liquid specificgravities are sometimes followed by the temperatures of both the liq-uid and water, respectively; for example, �o� 0.82360�60, where thetemperature units here are understood to be in degrees Fahrenheit.

The oil gravity, �API, in degrees API is used to classify crude oilson the basis of the following relation,

�API�141.5�o� 131.5 (3.13a). . . . . . . . . . . . . . . . . . . . . . . . . .

and �o�141.5

�API� 131.5, (3.13b). . . . . . . . . . . . . . . . . . . . . . . .

where �o�oil specific gravity (water�1). Officially, the SPE doesnot recognize �API in its SI standard, but because oil gravity (in de-grees API) is so widely used (and understood) and because it isfound in many property correlations, its continued use is justifiedfor qualitative description of stock-tank oils.

Isothermal compressibility, c, of a fixed mass of material is de-fined as

c�� 1V�Vp�

T

�� 1v^�v^p�

T

�� 1v �vp�

T

, (3.14). . . . .

where the units are psi�1 or kPa�1. In terms of density, �, and FVF,B, isothermal compressibility is given by

c� 1��p�

T

� 1B�Bp�

T

, (3.15). . . . . . . . . . . . . . . . . . . . .

where B is defined in the next section. Fig. 3.2 shows the variationin compressibility with pressure for typical reservoir mixtures. Adiscontinuity in oil compressibility occurs at the bubblepoint be-cause gas comes out of solution. When two or more phases are pres-ent, a total compressibility is useful.8,9

3.2.5 Black-Oil Pressure/Volume/Temperature (PVT) Proper-ties. The FVF, B; solution gas/oil ratio, Rs; and solution oil/gas ratio,rs, are volumetric ratios used to simplify engineering calculations.Specifically, they allow for the introduction of surface volumes ofgas, oil, and water into material-balance equations. These are notstandard engineering quantities, and they must be defined precisely.These properties constitute the black-oil or “beta” PVT formulaused in petroleum engineering. Chap. 7 gives a detailed discussionof black-oil properties.

Fig. 3.3—Reservoir FVF’s as functions of pressure.

FVF, or simply volume factor, is used to convert a volume at ele-vated pressure and temperature to surface volume, and vice versa.More specifically, FVF is defined as the volume of a mixture at spe-cified pressure and temperature divided by the volume of a productphase measured at standard conditions,

B�Vmixture

�p, T �

Vproduct�psc, Tsc�

. (3.16). . . . . . . . . . . . . . . . . . . . . . . . . .

The units of B are bbl/STB for oil and water, and ft3/scf or bbl/Mscffor gas. The surface product phase may consist of all or only part ofthe original mixture.

Primarily, four volume factors are used in petroleum engineering.They are oil FVF, Bo; water FVF, Bw; gas FVF, Bg; and total FVFof a gas/oil system, Bt, where

Bo�Vo

(Vo)sc�

Vo

Vo, (3.17a). . . . . . . . . . . . . . . . . . . . . . . . . . . .

Bw�Vw

(Vw)sc�

Vw

Vw, (3.17b). . . . . . . . . . . . . . . . . . . . . . . . . .

Bg�Vg

�Vg�sc

�Vg

Vg, (3.17c). . . . . . . . . . . . . . . . . . . . . . . . . . . .

and Bt�Vt

(Vo)sc�

Vo� Vg

(Vo)sc�

Vo� Vg

Vo; (3.17d). . . . . . . . . . .

and the total FVF of a gas/water system is

Btw�Vt

(Vw)sc�

Vg� Vw

Vw. (3.17e). . . . . . . . . . . . . . . . . . . . . .

In Eq. 3.17, Vo�oil volume at p and T ; Vg�gas volume at p andT ; Vw�water/brine volume at p and T ; Vo�(Vo)sc�stock-tank-oil volume at standard conditions; Vw� (Vw)sc�stock-tank-wa-ter volume at standard conditions; and Vg��Vg�sc

�surface-gasvolume at standard conditions.

Because gas FVF is inversely proportional to pressure, a recipro-cal gas volume factor, bg (equal to 1/Bg), is sometimes used, wherethe units of bg may be scf/ft3 or Mscf/bbl. Fig. 3.3 shows FVF’s oftypical reservoir systems. Inverse oil FVF, bo (equal to 1/Bo) is alsoused in reservoir simulation.

Wet gas and gas-condensate reservoir fluids produce liquids at thesurface, and for these gases the surface product (separator gas) con-sists of only part of the original reservoir gas mixture. Two gasFVF’s are used for these systems: the “dry” FVF, Bgd, and the “wet”FVF, Bgw (or just Bg). Bgd gives the ratio of reservoir gas volume tothe actual surface separator gas. Bgw gives the ratio of reservoir gasvolume to a hypothetical “wet” surface-gas volume (the actual sepa-rator-gas volume plus the stock-tank condensate converted to anequivalent surface-gas volume). Chap. 7 describes when Bgd andBgw are used. The standard definition of Bg� (psc�Tsc)(ZT�p) (seeEq. 3.38) represents the wet-gas FVF.


Fig. 3.4—Solution gas/oil ratios for brine, Rsw, and reservoir oils,Rs , and inverse solution oil/gas ratio for reservoir gases, 1/rs , asfunctions of pressure.

When a reservoir mixture produces both surface gas and oil, theGOR, Rgo, defines the ratio of standard gas volume to a referenceoil volume (stock-tank- or separator-oil volume),

Rgo��Vg�sc

(Vo)sc�

Vg

Vo(3.18a). . . . . . . . . . . . . . . . . . . . . . . . . . . .

and Rsp��Vg�sc

(Vo)sp�

Vg

(Vo)sp(3.18b). . . . . . . . . . . . . . . . . . . . . .

in units of scf/STB and scf/bbl, respectively. The separator condi-tions should be reported when separator GOR is used.

Solution gas/oil ratio, Rs, is the volume of gas (at standard condi-tions) liberated from a single-phase oil at elevated pressure and tem-perature divided by the resulting stock-tank-oil volume, with unitsscf/STB. Rs is constant at pressures greater than the bubblepoint anddecreases as gas is liberated at pressures below the bubblepoint.

The producing GOR, Rp, defines the instantaneous ratio of the to-tal surface-gas volume produced divided by the total stock-tank-oilvolume. At pressures greater than bubblepoint, Rp is constant andequal to Rs at bubblepoint. At pressures less than the bubblepoint,Rp may be equal to, less than, or greater than the Rs of the flowingreservoir oil. Typically, Rp will increase 10 to 20 times the initial Rs

because of increasing gas mobility and decreasing oil mobility dur-ing pressure depletion.

The surface volume ratio for gas condensates is usually expressedas an oil/gas ratio (OGR), rog.

rog�(Vo)sc

�Vg�sc

�Vo

Vg� 1

Rgo. (3.19). . . . . . . . . . . . . . . . . . . . . .

The unit for rog is STB/scf or, more commonly, “barrels per million”(STB/MMscf). To avoid misinterpretation, it should be clearly spe-cified whether the OGR includes natural gas liquids (NGL’s) inaddition to stock-tank condensate. In most petroleum engineeringcalculations, NGL’s are not included in the OGR.

The ratio of surface oil to surface gas produced from a single-phase reservoir gas is referred to as the solution oil/gas ratio, rs. Atpressures above the dewpoint, the producing OGR, rp is constantand equal to rs at the dewpoint. At pressures below the dewpoint, rp

is typically equal to or just slightly greater than rs; the contributionof flowing reservoir oil to surface-oil production is negligible inmost gas-condensate reservoirs.

In the definitions of Rp and rp, the total producing surface-gasvolume equals the surface gas from the reservoir gas plus the solu-tion gas from the reservoir oil; likewise, the total producing surfaceoil equals the stock-tank oil from the reservoir oil plus the conden-sate from the reservoir gas. Fig. 3.4 shows the behavior of Rp, Rs,and 1�rs as a function of pressure.

Fig. 3.5—Reservoir viscosities as functions of pressure.

3.2.6 Viscosity. Two types of viscosity are used in engineering cal-culations: dynamic viscosity, �� and kinematic viscosity, �. The def-inition of � for Newtonian flow (which most petroleum mixturesfollow) is

��gc

du�dy, (3.20). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

where ��shear stress per unit area in the shear plane parallel to thedirection of flow, du/dy�velocity gradient perpendicular to theplane of shear, and gc�a units conversion from mass to force. Thetwo viscosities are related by density, where �� .

Most petroleum engineering applications use dynamic viscosity,which is the property reported in commercial laboratory studies.The unit of dynamic viscosity is centipoise (cp), or in SI units,mPas, where 1 cp�1 mPas. Kinematic viscosity is usually re-ported in centistoke (cSt), which is obtained by dividing � in cp by� in g/cm3; the SI unit for � is mm2/s, which is numerically equiva-lent to centistoke. Fig. 3.5 shows oil, gas, and water viscosities fortypical reservoir systems.

3.2.7 Diffusion Coefficients. In the absence of bulk flow, compo-nents in a single-phase mixture are transported according to gradi-ents in concentration (i.e., chemical potential). Fick’s10 law for 1Dmolecular diffusion in a binary system is given by

ui�� Di j�dCi�dx�, (3.21). . . . . . . . . . . . . . . . . . . . . . . . . .

where ui�molar velocity of Component i; Dij�binary diffusioncoefficient; and Ci�molar concentration of Component i� yi�M,where yi�mole fraction; and x�distance.

Eq. 3.21 clearly shows that mass transfer by molecular diffusioncan be significant for three reasons: (1) large diffusion coefficients,(2) large concentration differences, and (3) short distances. A com-bination of moderate diffusion coefficients, concentration gradi-ents, and distance may also result in significant diffusive flow. Mo-lecular diffusion is particularly important in naturally fracturedreservoirs11,12 because of relatively short distances (e.g., small ma-trix block sizes).

Low-pressure binary diffusion coefficients for gases, Doij, are in-

dependent of composition and can be calculated accurately fromfundamental gas theory (Chapman and Enskog6), which are basical-ly the same relations used to estimate low-pressure gas viscosity. Nowell-accepted method is available to correct Do

ij for mixtures at highpressure, but two types of corresponding-states correlations havebeen proposed: Dij� Do

ij f(Tr, pr) and Dij� Doij f(�r).

At low pressures, diffusion coefficients are several orders of mag-nitude smaller in liquids than in gases. At reservoir conditions, thedifference between gas and liquid diffusion coefficients may be lessthan one order of magnitude.

3.2.8 IFT. Interfacial forces act between equilibrium gas, oil, andwater phases coexisting in the pores of a reservoir rock. These forces

22 PHASE BEHAVIOR

are generally quantified in terms of IFT, �� units of � are dynes/cm(or equivalently, mN/m). The magnitude of IFT varies from �50dynes/cm for crude-oil/gas systems at standard conditions to �0.1dyne/cm for high-pressure gas/oil mixtures. Gas/oil capillary pres-sure, Pc, is usually considered proportional to IFT according to theYoung-Laplace equation Pc� 2��r, where r is an average pore ra-dius.13-15 Recovery mechanisms that are influenced by capillarypressure (e.g., gas injection in naturally fractured reservoirs) willnecessarily be sensitive to IFT.

�� !��

This section gives correlations for PVT properties of natural gases,including the following.

1. Review of gas volumetric properties.2. Z-factor correlations.3. Gas pseudocritical properties.4. Wellstream gravity of wet gases and gas condensates.5. Gas viscosity.6. Dewpoint pressure.7. Total volume factor.

3.3.1 Review of Gas Volumetric Properties. The properties of gasmixtures are well understood and have been accurately correlatedfor many years with graphical charts and EOS’s based on extensiveexperimental data.16-19 The behavior of gases at low pressures wasoriginally quantified on the basis of experimental work by Charlesand Boyle, which resulted in the ideal-gas law,3

pV� nRT, (3.22). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

where R is the universal gas constant given in Appendix A for vari-ous units (Table A-2). In customary units,

R� 10.73146psia � ft3

°R� lbm mol, (3.23). . . . . . . . . . . . . . . . . .

while for other units, R can be calculated from the relation

R� 10.73146�punit

psia�� °R

Tunit��Vunit

ft3�� lbm

munit� . (3.24). . . . . . . .

For example, the gas constant for SPE-preferred SI units is given by

R� 10.73146 ��6.894757kPapsia� � �1.8 °R

K�

� �0.02831685 m3

ft3� � �2.204623 lbm

kg�

� 8.3143kPa m3

K kmol. (3.25). . . . . . . . . . . . . . . . . . . . . . . .

The gas constant can also be expressed in terms of energy units (e.g.,R�8.3143 J/molK); note that J�Nm�(N/m2)m3�Pam3. Inthis case, the conversion from one unit system to another is given by

R� 8.3143�Eunit

J�� K

Tunit�� g

munit� . (3.26). . . . . . . . . . . . . . . .

An ideal gas is a hypothetical mixture with molecules that arenegligible in size and have no intermolecular forces. Real gasesmimic the behavior of an ideal gas at low pressures and high temper-atures because the mixture volume is much larger than the volumeof the molecules making up the mixture. That is, the mean free pathbetween molecules that are moving randomly within the total vol-ume is very large and intermolecular forces are thus very small.

Most gases at low pressure follow the ideal-gas law. Applicationof the ideal-gas law results in two useful engineering approxima-tions. First, the standard molar volume representing the volume oc-cupied by one mole of gas at standard conditions is independent ofthe gas composition.

�vg�sc� vg�

�Vg�scn �

RTscpsc

�10.73146(60� 459.67)

14.7

� 379.4 scf�lbm mol

� 23.69 std m3�kmol . (3.27). . . . . . . . . . . . . . . . . . .

Second, the specific gravity of a gas directly reflects the gas molecu-lar weight at standard conditions,

�g��g�sc

��air�sc

�Mg

Mair�

Mg

28.97

and Mg� 28.97 �g . (3.28). . . . . . . . . . . . . . . . . . . . . . . . . . . .

For gas mixtures at moderate to high pressure or at low tempera-ture the ideal-gas law does not hold because the volume of the con-stituent molecules and their intermolecular forces strongly affect thevolumetric behavior of the gas. Comparison of experimental datafor real gases with the behavior predicted by the ideal-gas law showssignificant deviations. The deviation from ideal behavior can be ex-pressed as a factor, Z, defined as the ratio of the actual volume of onemole of a real-gas mixture to the volume of one mole of an ideal gas,

Z�volume of 1 mole of real gas at p and T

volume of 1 mole of ideal gas at p and T,

(3.29). . . . . . . . . . . . . . . . . . . .

where Z is a dimensionless quantity. Terms used for Z include devi-ation factor, compressibility factor, and Z factor. Z factor is used inthis monograph, as will the SPE reserve symbol Z (instead of the rec-ommended SPE symbol z) to avoid confusion with the symbol zused for feed composition.

From Eqs. 3.22 and 3.29, we can write the real-gas law includingthe Z factor as

pV� nZRT, (3.30). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

which is the standard equation for describing the volumetric behav-ior of reservoir gases. Another form of the real-gas law written interms of specific volume (v^ � 1��) is

pv^ � ZRT�M (3.31). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

or, in terms of molar volume (v� M��),

pv� ZRT. (3.32). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Z factor, defined by Eq. 3.30,

Z� pV�nRT, (3.33). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

is used for both phases in EOS applications (see Chap. 4). In thismonograph we use both Z and Zg for gases and Zo for oils; Z withouta subscript always implies the Z factor of a “gas-like” phase.

All volumetric properties of gases can be derived from the real-gas law. Gas density is given by

�g� pMg�ZRT (3.34). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

or, in terms of gas specific gravity, by

�g� 28.97p�g

ZRT. (3.35). . . . . . . . . . . . . . . . . . . . . . . . . . . .

For wet-gas and gas-condensate mixtures, wellstream gravity, �w,must be used instead of �g in Eq. 3.35.3 Gas density may rangefrom 0.05 lbm/ft3 at standard conditions to 30 lbm/ft3 for high-pressure gases.

Gas molar volume, vg, is given by

vg� ZRT�p, (3.36). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

where typical values of vg at reservoir conditions range from 1 to 1.5ft3/lbm mol compared with 379 ft3/lbm mol for gases at standardconditions. In Eqs. 3.30 through 3.36, R�universal gas constant.


Fig. 3.6—Standing-Katz4 Z-factor chart.

PseudoreducedTemperature

1 January 1941

Gas compressibility, cg, is given by

cg��1Vg�Vg

p�

� 1p�

1Z�Zp�

T

. (3.37). . . . . . . . . . . . . . . . . . . . . . . . . .

For sweet natural gas (i.e., not containing H2S) at pressures less than�1,000 psia, the second term in Eq. 3.37 is negligible and cg� 1�pis a reasonable approximation.

Gas volume factor, Bg, is defined as the ratio of gas volume at spe-cified p and T to the ideal-gas volume at standard conditions,

Bg� �psc

Tsc� ZT

p . (3.38). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

For customary units ( psc�14.7 psia and Tsc�520°R), this is

Bg� 0.02827 ZTp , (3.39). . . . . . . . . . . . . . . . . . . . . . . . . . .

with temperature in °R and pressure in psia. This definition of Bg

assumes that the gas volume at p and T remains as a gas at standardconditions. For wet gases and gas condensates, the surface gas willnot contain all the original gas mixture because liquid is produced

after separation. For these mixtures, the traditional definition of Bg

may still be useful; however, we refer to this quantity as a hypotheti-cal wet-gas volume factor, Bgw, which is calculated from Eq. 3.38.

Because Bg is inversely proportional to pressure, the inverse vol-ume factor, bg� 1�Bg, is commonly used. For field units,

bg in scf�ft3� 35.37p

ZT(3.40a). . . . . . . . . . . . . . . . . . . . . .

and bg in Mscf�bbl� 0.1985p

ZT. (3.40b). . . . . . . . . . . . . .

If the reservoir gas yields condensate at the surface, the dry-gasvolume factor, Bgd, is sometimes used.20

Bgd� �psc

Tsc��ZT

p �� 1Fgg

�, (3.41). . . . . . . . . . . . . . . . . . . . . . .

where Fgg�ratio of moles of surface gas, ng , to moles of wellstreammixture (i.e., reservoir gas, ng); see Eqs. 7.10 and 7.11 of Chap. 7.

3.3.2 Z-Factor Correlations. Standing and Katz4 present a general-ized Z-factor chart (Fig. 3.6), which has become an industry stan-dard for predicting the volumetric behavior of natural gases. Manyempirical equations and EOS’s have been fit to the original Stand-ing-Katz chart. For example, Hall and Yarborough21,22 present an

24 PHASE BEHAVIOR

accurate representation of the Standing-Katz chart using a Carna-han-Starling hard-sphere EOS,

Z� ppr�y, (3.42). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

where � 0.06125 t exp[� 1.2(1� t)2], where t� 1�Tpr.The reduced-density parameter, y (the product of a van der Waals

covolume and density), is obtained by solving

f(y)� 0�� ppr�y� y2� y3� y4

(1� y)3

� (14.76t� 9.76t2� 4.58t3)y2

� (90.7t–242.2t2� 42.4t3)y2.18�2.82 t, (3.43). . . . . . . . .

withdf(y)dy�

1� 4y� 4y2� 4y3� y4

(1� y)4

� (29.52t� 19.52t2� 9.16t3)y

� (2.18� 2.82t)(90.7t� 242.2t2� 42.4t3)

� y1.18�2.82 t . (3.44). . . . . . . . . . . . . . . . . . . . . . . . .

The derivative Z/p used in the definition of cg is given by

�Zp�

T

� ppc 1y� ppr�y2

df(y)�dy� . (3.45). . . . . . . . . . . . . . . . . . .

An initial value of y�0.001 can be used with a Newton-Raphsonprocedure, where convergence should be obtained in 3 to 10 itera-tions for �f(y)� � 1� 10�8.

On the basis of Takacs’23 comparison of eight correlations repre-senting the Standing-Katz4 chart, the Hall and Yarborough21 and theDranchuk and Abou-Kassem24 equations give the most accuraterepresentation for a broad range of temperatures and pressures. Bothequations are valid for 1� Tr� 3 and 0.2� pr� 25 to 30.

For many petroleum engineering applications, the Brill andBeggs25 equation gives a satisfactory representation (�1 to 2%) ofthe original Standing-Katz Z-factor chart for 1.2� Tr� 2. Also,this equation can be solved explicitly for Z. The main limitations arethat reduced temperature must be �1.2 (�80°F) and �2.0(�340°F) and reduced pressure should be �15 (�10,000 psia).

The Standing and Katz Z-factor correlation may require specialtreatment for wet gas and gas-condensate fluids containing signifi-cant amounts of heptanes-plus material and for gas mixtures withsignificant amounts of nonhydrocarbons. An apparent discrepancyin the Standing-Katz Z-factor chart for 1.05� Tr� 1.15 has been“smoothed” in the Hall-Yarborough21 correlations. The Hall andYarborough (or Dranchuk and Abou-Kassem24) equation is recom-mended for most natural gases. With today’s computing capabili-ties, choosing simple, less-reliable equations, such as the Brill andBeggs25 equation, is normally unnecessary.

The Lee-Kesler,26,27 AGA-8,28 and DDMIX29 correlations for Zfactor were developed with multiconstant EOS’s to give accuratevolumetric predictions for both pure components and mixtures.They require more computation but are very accurate. These equa-tions are particularly useful in custody-transfer calculations. Theyalso are required for gases containing water and concentrations ofnonhydrocarbons that exceed the limits of the Wichert and Azizmethod.30,31

3.3.3 Gas Pseudocritical Properties. Z factor, viscosity, and othergas properties have been correlated accurately with corresponding-states principles, where the property is correlated as a function of re-duced pressure and temperature.

Z� f�pr , Tr�

and �g ��gsc� f�pr , Tr�, (3.46). . . . . . . . . . . . . . . . . . . . . . . . .

Fig. 3.7—Gas pseudocritical properties as functions of specificgravity.

where pr� p�pc and Tr� T�Tc. Such corresponding-states rela-tions should be valid for most pure compounds when componentcritical properties pc and Tc are used. The same relations can beused for gas mixtures if the mixture pseudocritical properties ppc

and Tpc are used. Pseudocritical properties of gases can be estimatedwith gas composition and mixing rules or from correlations basedon gas specific gravity.

Sutton7 suggests the following correlations for hydrocarbon gasmixtures.

TpcHC� 169.2 � 349.5�gHC� 74.0�2gHC (3.47a). . . . . . . . . . .

and ppcHC� 756.8� 131�gHC � 3.6�2gHC. (3.47b). . . . . . . . . .

He claims that Eqs. 3.47a and 3.47b are the most reliable correla-tions for calculating pseudocritical properties with the Stand-ing-Katz Z-factor chart. He even claims that this method is superiorto the use of composition and mixing rules.

Standing3 gives two sets of correlations: one for dry hydrocarbongases ( �gHC� 0.75),

TpcHC� 168� 325�gHC� 12.5�2gHC (3.48a). . . . . . . . . . . . . .

and ppcHC� 667� 15.0�gHC� 37.5�2gHC, (3.48b). . . . . . . . . .

and one for wet-gas mixtures ( �gHC� 0.75),

TpcHC� 187� 330�gHC� 71.5�2gHC (3.49a). . . . . . . . . . . . . .

and ppcHC� 706� 51.7�gHC� 11.1�2gHC. (3.49b). . . . . . . . . .

The Standing correlations are used extensively in the industry; Fig.3.7 compares them with the Sutton correlations. The Sutton and theStanding wet-gas correlations for Tpc give basically the same results,whereas the three ppc correlations are quite different at �g� 0.85.

Kay’s5 mixing rule is typically used when gas composition isavailable.

M��N

i�1

yi Mi , (3.50a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


Fig. 3.8—Heptanes-plus (pseudo)critical properties recom-mended for reservoir gases (from Standing,33 after Matthews etal.32).

Tpc��N

i�1

yi Tci , (3.50b). . . . . . . . . . . . . . . . . . . . . . . . . . . . .

and ppc��N

i�1

yi pci , (3.50c). . . . . . . . . . . . . . . . . . . . . . . . . . . .

where the pseudocritical properties of the C7+ fraction can be esti-mated from the Matthews et al.32 correlations (Fig. 3.8),3

Tc C7�� 608 � 364 log�MC7�

� 71.2�

� �2, 450 log MC7�� 3, 800� log �C7�

(3.51a). . . . . .

and pc C7�� 1, 188 � 431 log�MC7�

� 61.1�

� 2, 319� 852 log�MC7�� 53.7��C7�

� 0.8�.

(3.51b). . . . . . . . . . . . . . . . . . .

Kay’s mixing rule is usually adequate for lean natural gases thatcontain no nonhydrocarbons. Sutton suggests that pseudocriticalscalculated with Kay’s mixing rule are adequate up to �g� 0.85, butthat errors in calculated Z factors increase linearly at higher specificgravities, reaching 10 to 15% for �g� 1.5. This bias may be a resultof the C7+ critical-property correlations used by Sutton (not Eqs.3.51a and 3.51b).

When significant quantities of CO2 and H2S nonhydrocarbonsare present, Wichert and Aziz33,31 suggest corrections to arrive atpseudocritical properties that will yield reliable Z factors from theStanding-Katz chart. The Wichert and Aziz corrections are given by

Tpc� T *pc� , (3.52a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ppc�p*

pc�T *pc� �

T *pc� yH2S�1� yH2S�

, (3.52b). . . . . . . . . . . . . . . . .

and � 120 �yCO2� yH2S�0.9� �yCO2

� yH2S�1.6�

� 15�y0.5H2S� y4

H2S� , (3.52c). . . . . . . . . . . . . . . . . . . . . . .

where T *pc and p*

pc are mixture pseudocriticals based on Kay’s mix-ing rule. This method was developed from extensive data from natu-ral gases containing nonhydrocarbons, with CO2 molar concentra-tion ranging from 0 to 55% and H2S molar concentrations rangingfrom 0 to 74%.

If only gas gravity and nonhydrocarbon content are known, thehydrocarbon specific gravity is first calculated from

�gHC��g��yN2

MN2� yCO2

MCO2�yH2S MH2S��Mair

1� yN2� yCO2

� yH2S.

(3.53). . . . . . . . . . . . . . . . . . . .

Hydrocarbon pseudocriticals are then calculated from Eqs. 3.47aand 3.47b, and these values are adjusted for nonhydrocarbon con-tent on the basis of Kay’s5 mixing rule.

p*pc� �1� yN2

� yCO2� yH2S�ppcHC

� yN2pcN2� yCO2

pc CO2� yH2S pcH2S (3.54a). . . . . . . . . .

and T*pc� (1� yN2

� yCO2� yH2S)TpcHC

� yN2TcN2� yCO2

Tc CO2�yH2STcH2S . (3.54b). . . .

T *c and p*

c are used in the Wichert-Aziz equations with CO2 and H2Smole fractions to obtain mixture Tpc and ppc.

The Sutton7 correlations (Eqs. 3.47a and 3.47b) are recom-mended for hydrocarbon pseudocritical properties. If compositionis available, Kay’s mixing rule should be used with the Matthews etal.32 pseudocriticals for C7+. Gases containing significant amountsof CO2 and H2S nonhydrocarbons should always be corrected withthe Wichert-Aziz equations. Finally, for gas-condensate fluids thewellstream specific gravity, �w (discussed in the next section),should replace �g in the equations above.

3.3.4 Wellstream Specific Gravity. Gas mixtures that produce con-densate at surface conditions may exist as a single-phase gas in thereservoir and production tubing. This can be verified by determin-ing the dewpoint pressure at the prevailing temperature. If well-stream properties are desired at conditions where the mixture issingle-phase, surface-gas and -oil properties must be converted toa wellstream specific gravity, �w. This gravity should be usedinstead of �g to estimate pseudocritical properties.

Wellstream gravityrp represents the average molecular weight ofthe produced mixture (relative to air) and is readily calculated fromthe producing-oil (condensate)/gas ratio, rp; average surface-gasgravity �g ; surface-condensate gravity, �o ; and surface-condensatemolecular weight Mo .

�w��

g� 4, 580 rp �o

1� 133, 000 rp ��M�o, (3.55). . . . . . . . . . . . . . . . . . .

with rp in STB/scf. Average surface-gas gravity is given by

�g�

�Nsp

i�1

Rpi�gi

�Nsp

i�1

Rpi

, (3.56). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

where Rpi�GOR of Separator Stage i. Standing33 presents Eq.3.55 graphically in Fig. 3.9.

When Mo is not available, Standing gives the following correla-tion.

26 PHASE BEHAVIOR

Fig. 3.9—Wellstream gravity relative to surface average gas grav-ity as a function of solution oil/gas ratio and surface gravities.

Oil/Gas Ratio, STB/MMscf

Mo� 240� 2.22 �API . (3.57). . . . . . . . . . . . . . . . . . . . . . . .

This relation should not be extrapolated outside the range45� �API� 60. Eilerts34 gives a relation for (��M)o ,

��M�o� �1.892� 10�3� � �7.35� 10�5��API

� �4.52� 10�8�� 2API , (3.58). . . . . . . . . . . . . . . . . . .

which should be reliable for most condensates. When condensatemolecular weight is not available, the recommended correlation forMo is the Cragoe35 correlation,

Mo�6, 084

�API � 5.9, (3.59). . . . . . . . . . . . . . . . . . . . . . . . . .

which gives reasonable values for all surface condensates andstock-tank oils.

A typical problem that often arises in the engineering of gas-con-densate reservoirs is that all the data required to calculate well-stream gas volumes and wellstream specific gravity are not avail-able and must be estimated.36-38 In practice, we often report only thefirst-stage-separator GOR (relative to stock-tank-oil volume) andgas specific gravity, Rs1 and �g1, respectively; the stock-tank-oilgravity, �o ; and the primary-separator conditions, psp1 and Tsp1.

To calculate �w from Eq. 3.55 we need total producing OGR, rp,which equals the inverse of Rs1 plus the additional gas that will bereleased from the first-stage separator oil, Rs�,

rp�1

�Rs1 � Rs��. (3.60). . . . . . . . . . . . . . . . . . . . . . . . . . .

Rs� can be estimated from several correlations.37,39 Whitson38 pro-poses use of a bubblepoint pressure correlation (e.g., the Standing40

correlation),

Rs� � A1�g� (3.61a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

and A1� � psp1

18.2� 1.4�10�0.0125�API�0.00091Tsp1

��1.205

,

(3.61b). . . . . . . . . . . . . . . . . . .

with psp1 in psia, Tsp1 in °F, and Rs� in scf/STB. �g� is the gas grav-ity of the additional solution gas released from the separator oil. TheKatz41 correlation (Fig. 3.10) can be used to estimate �g�, where abest-fit representation of his graphical correlation is

�g� � A2� A3 Rs�, (3.62). . . . . . . . . . . . . . . . . . . . . . . . . .

where A2� 0.25� 0.02�API and A3�� (3.57� 10�6)�API .

Fig. 3.10—Correlation for separator-oil dissolved gas gravity asa function of stock-tank-oil gravity and separator-oil GOR (fromRef. 41).

Solution Gas/Oil Ratio, scf/STB

Solving Eqs. 3.61 and 3.62 for Rs� gives

Rs� �A1 A2

�1� A1 A3�

. (3.63). . . . . . . . . . . . . . . . . . . . . . . . . .

Average surface separator gas gravity, �g, is given by

�g�

�g1 Rs1� �g� Rs�

Rs1� Rs�. (3.64). . . . . . . . . . . . . . . . . . . . . .

Although the Katz correlation is only approximate, the impact of afew percent error in �g� is not of practical consequence to the cal-culation of �w because Rs� is usually much less than Rs1.

3.3.5 Gas Viscosity. Viscosity of reservoir gases generally rangesfrom 0.01 to 0.03 cp at standard and reservoir conditions, reachingup to 0.1 cp for near-critical gas condensates. Estimation of gas vis-cosities at elevated pressure and temperature is typically a two-stepprocedure: (1) calculating mixture low-pressure viscosity �gsc atpsc and T from Chapman-Enskog theory3,6 and (2) correcting thisvalue for the effect of pressure and temperature with a correspond-ing-states or dense-gas correlation. These correlations relate the ac-tual viscosity �g at p and T to low-pressure viscosity by use of theratio �g��gsc or difference ( �g� �gsc) as a function of pseudore-duced properties ppr and Tpr or as a function of pseudoreduced den-sity �pr.

Gas viscosities are rarely measured because most laboratories donot have the required equipment; thus, the prediction of gas viscos-ity is particularly important. Gas viscosity of reservoir systems isoften estimated from the graphical correlation �g��gsc� f(Tr, pr)proposed by Carr et al.42 (Fig. 3.11). Dempsey43 gives a polynomialapproximation of the Carr et al. correlation. With these correlations,gas viscosities can be estimated with an accuracy of about �3% formost applications. The Dempsey correlation is valid in the range1.2� Tr� 3 and 1� pr� 20.

The Lee-Gonzalez gas viscosity correlation (used by most PVTlaboratories when reporting gas viscosities) is given by44

�g� A1� 10�4 exp�A2�A3g� , (3.65a). . . . . . . . . . . . . . . . . .

where A1��9.379� 0.01607Mg�T1.5

209.2� 19.26Mg� T,

A2� 3.448� �986.4�T� � 0.01009Mg ,

and A3� 2.447� 0.2224A2 , (3.65b). . . . . . . . . . . . . . . . . . . .

with �g in cp, �g in g/cm3, and T in °R. McCain19 indicates the ac-curacy of this correlation is 2 to 4% for �g�1.0, with errors up to20% for rich gas condensates with �g� 1.5.


Fig. 3.11—Carr et al.42 gas-viscosity correlation.

Pseudoreduced Temperature, Tr

Molecular Weight

Gas Gravity (air�1)

H2S, mol%

N2, mol% CO2, mol%

go

Lucas45 proposes the following gas viscosity correlation, whichis valid in the range 1� Tr� 40 and 0� pr� 100 (Fig. 3.12)6:

�g��gsc� 1 �A1 p1.3088

pr

A2 pA5pr � �1 � A3 pA4

pr��1

, (3.66a). . . . . . .

where A1�(1.245� 10�3) exp�5.1726T�0.3286

pr�

Tpr,

A2� A1�1.6553Tpr� 1.2723� ,

A3�0.4489 exp�3.0578T�37.7332

pr�

Tpr,

A4�1.7368 exp�2.2310T�7.6351

pr�

Tpr,

and A5� 0.9425 exp�� 0.1853T 0.4489pr� , (3.66b). . . . . . . . . . .

where �gsc�� 0.807T 0.618pr � 0.357 exp�� 0.449Tpr�

� 0.340 exp�� 4.058Tpr� � 0.018� ,

�� 9, 490� Tpc

M3p4pc�

1�6

,

and ppc� RTpc

�N

i�1

yi Zci

�N

i�1

yivci

, (3.67). . . . . . . . . . . . . . . . . . . . . . . .

with � in cp�1, T and Tc in °R, and pc in psia. Special correctionsshould be applied to the Lucas correlation when polar compounds,such as H2S and water, are present in a gas mixture. The effect ofH2S is always �1% and can be neglected, and appropriate correc-tions can be made to treat water if necessary.

Given its wide range of applicability, the Lucas method is recom-mended for general use. When compositions are not available, cor-relations for pseudocritical properties in terms of specific gravitycan be used instead. Standing2 gives equations for �gsc in terms of�g, temperature, and nonhydrocarbon content,

�gsc� ��gsc�uncorrected� ��N2

� ��CO2� ��H2S ,

(3.68a). . . . . . . . . . . . . . . . . . . .

28 PHASE BEHAVIOR

Fig. 3.12—Lucas45 corresponding-states generalized viscositycorrelation (Ref. 6); ��dynamic viscosity and �p�micro-poise�10�6 poise�10�4 cp.

��

��

� � ��

��

where ��gsc�uncorrected� �8.188� 10�3� � �1.709� 10�5�

� �2.062� 10�6��g�T� �6.15� 10�3� log �g ,

��N2� yN2

�8.48� 10�3� log �g� �9.59� 10�3��,

��CO2� yCO2

�9.08� 10�3� log �g� �6.24� 10�3��,

and ��H2S� yH2S �8.49� 10�3� log �g� �3.73� 10�3��.

(3.68b). . . . . . . . . . . . . . . . . . .

Reid et al.6 review other gas viscosity correlations with accuracysimilar to that of the Lucas correlation.

3.3.6 Dewpoint Pressure. The prediction of retrograde dewpointpressure is not widely practiced. It is generally recognized that thecomplexity of retrograde phase behavior necessitates experimentaldetermination of the dewpoint condition. Sage and Olds’46 data areperhaps the most extensive tabular correlation of dewpoint pressur-es. Eilerts et al.47,48 also present dewpoint pressures for severallight-condensate systems.

Nemeth and Kennedy49 have proposed a dewpoint correlationbased on composition and C7+ properties.

ln pd�A1 zC2� zCO2

� zH2S�zC6�2(zC3

�zC4)� zC5

� 0.4zC1�0.2zN2� � A2�C7�

� A3��

zC1

�zC7�� 0.002��

� A4T��A5zC7�MC7�� A6�zC7�

MC7��2

� A7�zC7�MC7�� 3� A8��

MC7�

��C7�� 0.0001��

��

� A9��

MC7�

� �C7�� 0.0001��

��

2

� A10��

MC7�

��C7�� 0.0001��

��

3

� A11 , (3.69). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

where A1��2.0623054, A2�6.6259728, A3��4.4670559�10�3, A4�1.0448346�10�4, A5�3.2673714�10�2, A6��3.6453277�10�3, A7�7.4299951�10�5, A8��1.1381195�10�1, A9�6.2476497�10�4, A10��1.0716866�10�6, andA11�1.0746622�101.

The range of properties used to develop this correlation includesdewpoints from 1,000 to 10,000 psia, temperatures from 40 to 320°F,and a wide range of reservoir compositions. The correlation usuallycan be expected to predict dewpoints with an accuracy of �10% forcondensates that do not contain large amounts of nonhydrocarbons.This is acceptable in light of the fact that experimental dewpoint pres-sures are probably determined with an accuracy of only �5%. Thecorrelation is generally used only for preliminary reservoir studiesconducted before an experimental dewpoint is available.

Organick and Golding50 and Kurata and Katz51 present graphicalcorrelations for dewpoint pressure.

3.3.7 Total FVF. Total FVF,3,17,46 Bt, is defined as the volume ofa two-phase, gas-oil mixture (or sometimes a single-phase mixture)at elevated pressure and temperature divided by the stock-tank-oilvolume resulting when the two-phase mixture is brought to surfaceconditions,

Bt�Vo� Vg

(Vo)sc�

Vo� Vg

Vo. (3.70). . . . . . . . . . . . . . . . . . . . .

Bt is used for calculating the oil in place for gas-condensate reser-voirs, where Vo� 0 in Eq. 3.70. Assuming 1 res bbl of hydrocar-bon PV, the initial condensate in place is given by N� 1�Bt (inSTB) and the initial “dry” separator gas in place is G� N�rp,where rp�initial producing (solution) OGR.

For gas-condensate systems, Sage and Olds46 give a tabulatedcorrelation for Bt.

Bt�RpT

p Z*, (3.71). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

where Rp�producing GOR in scf/STB, Bt is in bbl/STB, T is in °R,and p is in psia. Z* varies with pressure and temperature, where thetabulated correlation for Z* is well represented by

Z*� A0� A1p� A2p1.5� A3

pT� A4

p1.5

T, (3.72). . . . . . . .

where A0�5.050�10�3, A1��2.740�10�6, A2�3.331�10�8,A3�2.198�10�3, and A4��2.675�10�5 with p in psia and Tin °R. Although the Sage and Olds data only cover the range600�p�3,000 psia and 100�T�250°F, Eq. 3.72 gives acceptableresults up to 10,000 psia and 350°F (when gas volume is much largerthan oil volume).

When reservoir hydrocarbon volume consists only of gas, the fol-lowing relations apply for total FVF.

Bt� Bgd Rp� Bgw �Rp� Cog� , (3.73a). . . . . . . . . . . . . . . . .

Cog� 133, 000 ��o�Mo

� , (3.73b). . . . . . . . . . . . . . . . . . . . . .

Mo� 6, 084��API� 5.9� , (3.73c). . . . . . . . . . . . . . . . . . . . .

and �API� 141.5�(131.5� �o) , (3.73d). . . . . . . . . . . . . . . . .

where Bgd�dry gas FVF in ft3/scf, Bgw�wet-gas FVF in ft3/scf(given by Eq. 3.38), Cog�gas equivalent conversion factor in scf/STB (see Chap. 7), and Rp�producing GOR in scf/STB.


Fig. 3.13—Effect of paraffinicity, Kw, on bubblepoint pressure.

C7+ Watson Characterization Factor, KwC7+ C7+ Watson Characterization Factor, KwC7+

PR EOSGlasø UncorrectedGlasø Corrected

Standing3 gives a graphical correlation for Bt using a correlationparameter A defined as

A� RpT 0.5

�0.3g

�ao , (3.74). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

where a� 2.9� 10�0.00027 Rp. Standing’s correlation is valid forboth oil and gas-condensate systems and can be represented with

log Bt�� 5.262� 47.4� 12.22� log A* , (3.75a). . . . . . . . .

where log A*� log A��10.1� 96.86.604� log p

� (3.75b). .

and A is given by Eq. 3.74. On the basis of data from North Sea oils,Glasø52 gives a correlation for Bt using the Standing correlation pa-rameter A (Eq. 3.74):

log Bt� �8.0135� 10�2� � 0.47257 log A*

� 0.17351�log A* �2 , (3.76). . . . . . . . . . . . . . . . . . . . .

where A*�Ap�1.1089.Either the Standing or the Glasø correlations for Bt can be used

with approximately the same accuracy. However, neither correla-tion is consistent with the limiting conditions

Bt� Bo for Vg� 0 (3.77a). . . . . . . . . . . . . . . . . . . . . . . . . .

and Bt� Bgd Rp for Vo� 0. (3.77b). . . . . . . . . . . . . . . . . . .

Bt correlations evaluated at a bubblepoint usually will underpredictthe actual Bob by �0.2.

��" �� !��

This section gives correlations for PVT properties of reservoir oils,including bubblepoint pressure and oil density, compressibility,FVF, and viscosity. With only a few exceptions, oil properties havebeen correlated in terms of surface-oil and -gas properties, includingsolution gas/oil ratio, oil gravity, average surface-gas gravity, andtemperature. A few correlations are also given in terms of composi-tion and component properties.

Reservoir oils typically contain dissolved gas consisting mainlyof methane and ethane, some intermediates (C3 through C6), andlesser quantities of nonhydrocarbons. The amount of dissolved gashas an important effect on oil properties. At the bubblepoint a dis-continuity in the system volumetric behavior is caused by gas com-ing out of solution, with the system compressibility changing dra-

matically.8 An accurate method is needed to correlate thebubblepoint pressure, temperature, and solution gas/oil ratio.

Oil properties can be grouped into two categories: saturated andundersaturated properties. Saturated properties apply at pressures ator below the bubblepoint, and undersaturated properties apply atpressures greater than the bubblepoint. For oils with initial GOR’sless than �500 scf/STB, assuming linear variation of undersaturat-ed-oil properties with pressure is usually acceptable.

3.4.1 Bubblepoint Pressure. The correlation of bubblepoint pres-sure has received more attention than any other oil-property correla-tion. Standing3,17,40 developed the first accurate bubblepoint cor-relation, which was based on California crude oils.

pb� 18.2�A� 1.4�, (3.78). . . . . . . . . . . . . . . . . . . . . . . . . .

where A� �Rs��g�0.83 10�0.00091T�0.0125�API�, with Rs in scf/STB, T

in °F, and pb in psia.Lasater53 used a somewhat different approach to correlate bub-

blepoint pressure, where mole fraction yg of solution gas in the res-ervoir oil is used as the main correlating parameter17:

pb� A T�g

, (3.79). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

with T in °R and pb in psia. The function A(yg) is given graphicallyby Lasater, and his correlation can be accurately described by

A� 0.83918� 101.17664yg y0.57246g ; yg� 0.6 (3.80a). . . . . . . .

and A� 0.83918� 101.08000yg y0.31109g ; yg� 0.6, (3.80b). . . . .

where yg� 1 � 133, 000 ��M�oRs��1

, (3.81). . . . . . . . . . . .

with Rs in scf/STB. In this correlation, the gas mole fraction is de-pendent mainly on solution gas/oil ratio, but also on the propertiesof the stock-tank oil. The Cragoe35 correlation given by Eq. 3.59 isrecommended for estimating Mo when stock-tank-oil molecularweight is not known.

Standing’s approach was used by Glasø52 for North Sea oils, re-sulting in the correlation

log pb� 1.7669� 1.7447 log A� 0.30218(log A)2,

(3.82). . . . . . . . . . . . . . . . . . . .

where A� �Rs��g�0.816 �T 0.172��0.989API� with pb in psia, T in °F and

Rs in scf/STB. Glasø’s corrections for nonhydrocarbon contentand stock-tank-oil paraffinicity are not widely used, primarily be-

30 PHASE BEHAVIOR

cause the necessary data are not available. Sutton and Farshad54

mention that the API correction for paraffinicity worsened bubble-point predictions for gulf coast fluids. Fig. 3.13 gives an explana-tion for this observation.

Fig. 3.13 shows the effect of paraffinicity (which is quantified bythe Watson characterization factor, Kw) on bubblepoint pressure;the figure is based on calculations with a tuned EOS for an Asian oil(solid circles). The oil composition is constant in the example cal-culation. The 12 C7+ fractions are each split into a paraffinic pseudo-component and an aromatic pseudocomponent (i.e., 24 C7+ pseudo-components). The paraffinic fraction was varied, and bubblepointcalculations were made. The variation in paraffinicity is expressedin terms of the overall C7+ Watson characterization factor. Alsoshown in the figure are the variation in solution gas/oil ratio and theoil specific gravity with KwC7�

.The actual reservoir oil has a KwC7�

� 11.55, where the EOSbubblepoint is close to the uncorrected Glasø bubblepoint predic-tion. When the correction for paraffinicity is applied, the correctiongives a poorer bubblepoint prediction (even though the overall trendin bubblepoints is improved by the Glasø paraffinicity corrections).

A quantitatively similar correction to the Glasø correction (buteasier to use) is based on the estimate for Whitson’s55,56 Watsoncharacterization factor, Kw, and yields

��o� corrected� ��o�measured�Kw�11.9�1.1824. (3.83). . . . . . . . . . .

The corrected specific gravity correlation is used in the Glasø bubble-point correlation instead of the measured specific gravity. An estimateof Kw for the stock-tank oil must be available to use this correction.

Vazquez and Beggs57 give the following correlations. For�API� 30,

pb��

27.64�Rs�gc�10��11.172 �API

T�460��

0.9143

, (3.84). . . . . . . . . .

and, for �API� 30,

pb� 56.06�Rs�gc� 10��10.393�API

T�460��

0.8425

, (3.85). . . . . . . . .

with pb in psia, T in °F and Rs in scf/STB. These equations are basedon a large number of data from commercial laboratories. Vazquezand Beggs correct for the effect of separator conditions using a mo-dified gas specific gravity, �gc, which is correlated with first-stage-separator pressure and temperature, and stock-tank-oil gravity.

�gc� �g 1� �0.5912� 10�4� �APITsp log� psp

114.7��,

(3.86). . . . . . . . . . . . . . . . . . . .

with Tsp in °F and psp in psia.Standing’s correlation can be used to develop field- or reservoir-

specific bubblepoint correlations. A linear relation is usually as-sumed between bubblepoint pressure and the Standing correlatingcoefficient. This is a standard approach used in the industry, and theStanding bubblepoint correlating parameter is well suited for devel-oping field-specific correlations.

Sometimes the solution gas/oil ratio is needed at a given pressure,and this is readily calculated by solving the bubblepoint correlationfor Rs. For the Standing correlation,

Rs� �g (0.055p� 1.4)100.0125�API

100.00091T�

1.205

; (3.87). . . . . . . . .

similar relations can be derived for the other bubblepoint correlations.

In summary, significant differences in predicted bubblepoint pres-sures should not be expected for most reservoir oils with most of theprevious correlations. The Lasater and Standing equations are recom-mended for general use and as a starting point for developing reser-voir-specific correlations. Correlations developed for a specific re-gion, such as Glasø’s correlation for the North Sea, should probablybe used in that region and, in the case of Glasø’s correlation, may beextended to other regions by use of the paraffinicity correction.

3.4.2 Oil Density. Density of reservoir oil varies from 30 lbm/ft3 forlight volatile oils to 60 lbm/ft3 for heavy crudes with little or no solu-tion gas. Oil compressibility may range from 3�10�6 psi�1 forheavy crude oils to 50�10�6 psi�1 for light oils. The variation ofoil compressibility with pressure is usually small, although for vola-tile oils the effect can be significant, particularly for material-balanceand reservoir-simulation calculations of highly undersaturated vola-tile oils. Several methods have been used successfully to correlate oilvolumetric properties, including extensions of ideal-solution mixing,EOS’s, corresponding-states correlations, and empirical correlations.

Oil density based on black-oil properties is given by

�o�62.4�o� 0.0136�g Rs

Bo, (3.88). . . . . . . . . . . . . . . . . . . .

with �o in lbm/ft3, Bo in bbl/STB, and Rs in scf/STB. Correlationscan be used to estimate Rs and Bo from �o, �g, p, and T.

Standing-Katz Method. Standing3,17 and Standing and Katz58

give an accurate method for estimating oil densities that uses an ex-tension of ideal-solution mixing.

�o� �po� ��p� ��T , (3.89). . . . . . . . . . . . . . . . . . . . . .

where �po is the pseudoliquid density at standard conditions and theterms ��T and ��p give corrections for temperature and pressure,respectively. Pseudoliquid density is calculated with ideal-solutionmixing and correlations for the apparent liquid densities of ethane

Fig. 3.14—Apparent liquid densities of methane and ethane(from Standing33).

System Density at 60°F and 14.7 psia, g/cm3


Fig. 3.15—Chart for calculating pseudoliquid density of reservoir oil (from Standing33).

g

and methane at standard conditions. Given oil composition xi, �po

is calculated from

�po�

�N

i�1

xi Mi

�N

i�1

�xi Mi��i�

, (3.90). . . . . . . . . . . . . . . . . . . . . . . . . . .

where Standing and Katz show that apparent liquid densities �i ofC2 and C1 are functions of the densities �2� and �po, respectively(Fig. 3.14).

�C2� 15.3� 0.3167 �C2�

�C1� 0.312� 0.45 �po , (3.91). . . . . . . . . . . . . . . . . . . . . . .

where �C2��

�C7�

i�C2

xi Mi

�C7�

i�C2

�xi Mi��i�

, (3.92). . . . . . . . . . . . . . . . . . . .

with � in lbm/ft3. Application of these correlations results in an ap-parent trial-and-error calculation for �po. Standing33 presents agraphical correlation (Fig. 3.15) based on these relations, where �po

is found from �C3� and weight fractions of C2 and C1 (wC2

and wC1,

respectively).Figs. 3.16 and 3.17 show the pressure and temperature correc-

tions, ��p and ��T

, graphically. ��p is a function of �po, and ��T

is a function of ( �po� ��p). Madrazo59 introduced modifiedcurves for ��p and ��

T that improve predictions at higher pressur-

es and temperatures. Standing3 gives best-fit equations for his origi-nal graphical correlations of ��p and ��

T (Eqs. 3.98 and 3.99),

which are not recommended at temperatures �240°F; instead, Ma-drazo’s graphical correlation can be used. The correction factors canalso be used to determine isothermal compressibility and oil FVF atundersaturated conditions.

The treatment of nonhydrocarbons in the Standing-Katz methodhas not received much attention, and the method is not recom-mended when concentrations of nonhydrocarbons exceed 10 mol%.Standing3 suggests that an apparent liquid density of 29.9 lbm/ft3

can be used for nitrogen but does not address how the nonhydrocar-bons should be considered in the calculation procedure (i.e., as partof the C3+ material or following the calculation of �C2

and �C1).

Madrazo indicates that the volume contribution of nonhydrocar-

32 PHASE BEHAVIOR

Fig. 3.16—Pressure correction to the pseudoliquid density at 14.7 psia and 60°F (from Ref. 59).

Density of System at 60°F and 14.7 psia, lbm/ft3

bons can be neglected completely if the total content is �6 mol%.Vogel and Yarborough60 suggest that the weight fraction of nitrogenshould be added to the weight fraction of ethane.

Using additive volumes and Eqs. 3.91 and 3.92, we can show that�C2�

and �po can be calculated explicitly. Thus, the following is themost direct procedure for calculating �o from the Standing-Katzmethod.

1. Calculate the mass of each component.

mi� xi Mi . (3.93). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2. Calculate VC3�.

VC3��

C7�

i�C3

mi�i

, (3.94). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

where �i are component densities at standard conditions (Appen-dix A).

3. Calculate �C2�.

�C2�� b� b2� 4ac�

2a, (3.95). . . . . . . . . . . . . . . . . . . .

where a� 0.3167VC3�, b� mC2

�0.3167mC2�� 15.3VC3�

,and c�� 15.3mC2�

.4. Calculate VC2�

.

VC2�� VC3�

�mC2�C2

� VC3��

mC2

15.3� 0.3167�C2�

. (3.96). . . . . . . . . . . .

5. Calculate �po.

�po��b � b2 � 4ac�

2a, (3.97). . . . . . . . . . . . . . . . . . . . .

where a� 0.45VC2�, b� mC1

� 0.45mC1�� 0.312VC2�

, andc�� 0.312mC1�

.6. Calculate the pressure effect on density.

��p� 10�3 0.167� �16.181� 10�0.0425�po�� p

� 10�8 0.299� �263� 10�0.0603�po�� p2. (3.98). . . . .


Fig. 3.17—Temperature correction to the pseudoliquid density at pressure and 60°F (from Ref. 59).

Density of System at Pressure and 60°F, lbm/ft3

7. Calculate the temperature effect on density.

��T� (T� 60) 0.0133� 152.4��po� ��p��2.45�

� (T� 60)2��8.1� 10�6�

� 0.0622� 10�0.0764(�po�� p)�� . (3.99). . . . . . . . . . .

8. Calculate mixture density from Eq. 3.89.In the absence of oil composition, Katz41 suggests calculating the

pseudoliquid density from stock-tank-oil gravity, �o, solution gas/oil ratio, Rs, and apparent liquid density of the surface gas, �ga, tak-en from a graphical correlation (Fig. 3.18),

�po�62.4�o� 0.0136 Rs �g

1� 0.0136�Rs �g��ga�. (3.100). . . . . . . . . . . . . . . . . .

Standing gives an equation for �ga.

�ga� 38.52� 10�0.00326 �API

� (94.75� 33.93 log �API) log �g , (3.101). . . . . . . . . . .

with �ga in lbm/ft3 and Rs in scf/STB.Alani-Kennedy61 Method. The Alani-Kennedy method for cal-

culating oil density is a modification of the original van der WaalsEOS, with constants a and b given as functions of temperature fornormal paraffins C1 to C10 and iso-butane (Table 3.1); two sets ofcoefficients are reported for methane (for temperatures from 70 to300°F and from 301 to 460°F) and two sets for ethane (for tempera-tures from 100 to 249°F and from 250 to 460°F). Lohrenz et al.62

give Alani-Kennedy temperature-dependent coefficients for non-hydrocarbons N2, CO2, and H2S (Table 3.1). The Alani-Kennedyequations are summarized next. Eqs. 3.102b and 3.102c are in theoriginal van der Waals EOS but are not used.

p� RTv� b

� av2 , (3.102a). . . . . . . . . . . . . . . . . . . . . . . . . . . .

34 PHASE BEHAVIOR

Fig. 3.18—Apparent pseudoliquid density of separator gas (fromStanding,33 after Katz41).

ai �2764

R2T 2ci

pci, (3.102b). . . . . . . . . . . . . . . . . . . . . . . . . . . .

bi �18

RTcipci

, (3.102c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

a��N

i�1

xi ai , (3.102d). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

b��N

i�1

xi bi , (3.102e). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ai�a1i

T� log a2i; i� C7�, (3.102f). . . . . . . . . . . . . . . . . .

and bi� b1iT� b2i ; i� C7�, (3.102g). . . . . . . . . . . . . . . . .

where log aC7�� 3.8405985� 10�3�MC7�

� �9.5638281� 10�4�MC7��C7�

� 261.80818T

� �7.3104464� 10�6�M2C7�

� 10.753517 (3.103a). . . . . . . . . . . . . . . . . . . . .

and bC7�� 3.499274� 10�2�MC7�

� 7.2725403�C7�

� �2.232395� 10�4�T� �1.6322572� 10�2�MC7��C7�

� 6.2256545, (3.103b). . . . . . . . . . . . . . . . . . . . . . .

with � in lbm/ft3, v in ft3/lbm mol, T in °R, p in psia, and R�univer-sal gas constant�10.73.

Solution of the cubic equation for volume is presented in Chap.4. Density is given by ��M/v, where M is the mixture molecularweight and v is the molar volume given by the solution to the cubicequation. The Alani-Kennedy method can also be used to estimateoil compressibilities.

Rackett,63 Hankinson and Thomson,64 and Hankinson et al.65

give accurate correlations for pure-component saturated-liquid den-sities, and although these correlations can be extended to mixtures,they have not been tested extensively for reservoir systems. Cullicket al.66 give a modified corresponding-states method for predictingdensity of reservoir fluids, The method has a better foundation andextrapolating capability than the methods discussed previously(particularly for systems with nonhydrocarbons); however, spacedoes not allow presentation of the method in its entirety.

Either the Standing-Katz or Alani-Kennedy method should esti-mate the densities of most reservoir oils with an accuracy of �2%.The Alani-Kennedy method is suggested for systems at tempera-tures �250°F and for systems containing appreciable amounts ofnonhydrocarbons (�5 mol%). Cubic EOS’s (e.g., Peng-Robinsonor Soave-Redlich-Kwong) that use volume translation also estimateliquid densities with an accuracy of a few percent (e.g., the recom-mended characterization procedures in Chap. 5 or other proposedcharacterizations67,68).

TABLE 3.1—CONSTANTS FOR ALANI-KENNEDY61 OIL DENSITY CORRELATION

Component a1 a2 b1�104 b2

N2 4,300 2.293 4.49 0.3853

CO2 8,166 126.0 0.1818 0.3872

H2S 13,200 0.0 17.9 0.3945

C1

At 70 to 300°F 9,160.6413 61.893223 �3.3162472 0.50874303

At 300 to 460°F 147.47333 3,247.4533 �14.072637 1.8326695

C2

At 100 to 250°F 46,709.573 �404.48844 5.1520981 0.52239654

At 250 to 460°F 17,495.343 34.163551 2.8201736 0.62309877

C3 20,247.757 190.24420 2.1586448 0.90832519

i-C4 32,204.420 131.63171 3.3862284 1.1013834

n-C4 33,016.212 146.15445 2.902157 1.1168144

i-C5 37,046.234 299.62630 2.1954785 1.4364289

n-C5 37,046.234 299.62630 2.1954785 1.4364289

n-C6 52,093.006 254.56097 3.6961858 1.5929406

n-C7 82,295.457 64.380112 5.2577968 1.7299902

n-C8 89,185.432 149.39026 5.9897530 1.9310993

n-C9 124,062.650 37.917238 6.7299934 2.1519973

n-C10 146,643.830 26.524103 7.8561789 2.3329874


3.4.3 Undersaturated-Oil Compressibility. With measured dataor an appropriate correlation for Bo or �o, Eq. 3.14 readily definesthe isothermal compressibility of an oil at pressures greater than thebubblepoint. “Instantaneous” undersaturated-oil compressibility,defined by Eq. 3.15 with the pressure derivative evaluated at a spe-cific pressure, is used in reservoir simulation and well-test inter-pretation. Another definition of oil compressibility may be used inmaterial-balance calculations (e.g., Craft and Hawkins69)—the“cumulative” or “average” compressibility defines the cumulativevolumetric change of oil from the initial reservoir pressure to cur-rent reservoir pressure.

co� p� �

Voi�pi

p

co� p� dp

Voi�pi� p�

� � � 1Voi� Voi� Vo�p�

pi� p �. (3.104). . . . . . . . . . . . . . . .

The cumulative compressibility is readily identified because it ismultiplied by the cumulative reservoir pressure drop, pi� pR. Usu-ally co is assumed constant; however, this assumption may not bejustified for high-pressure volatile oils.

Oil compressibility is used to calculate the variation in undersatu-rated density and FVF with pressure.

�o� �ob exp co�p� pb��

� �ob 1� co�pb� p�� (3.105a). . . . . . . . . . . . . . . . . . . . .

and Bo� Bob exp co�pb� p��

�Bob 1� co�p� pb

�� , (3.105b). . . . . . . . . . . . . . . . .

where consistent units must be used. These equations are derivedfrom the definition of isothermal compressibility assuming that co isconstant. When oil compressibility varies significantly with pressure,Eqs. 3.105a and 3.105b are not really valid. The approximations�o� �ob [1� co( pb� p)] and Bo�Bob [1� co( p� pb)] areused in many applications, and to predict volumetric behavior cor-rectly with these relations requires that co be defined by

co(p)�� 1Vob� Vo�p� � Vob

p� pb� . (3.106). . . . . . . . . . . . . . .

Strictly speaking, the compressibility of an oil mixture is definedonly for pressures greater than the bubblepoint pressure. If an oil isat its bubblepoint, the compressibility can be determined and de-fined only for a positive change in pressure. A reduction in pressurefrom the bubblepoint results in gas coming out of solution and, sub-sequently, a change in the mass of the original system for whichcompressibility is to be determined. Implicit in the definition ofcompressibility is that the system mass remains constant.

Vazquez and Beggs57 propose the following correlation forinstantaneous undersaturated-oil compressibility.

co� A�p, (3.107). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

whereA� 10�5(5Rsb� 17.2T� 1, 180�gc� 12.61�API� 1, 433), withco in psi�1, Rsb in scf/STB, T in °F, and p in psia. With this correla-tion for oil compressibility, undersaturated-oil FVF can be calcu-lated analytically from

Bo� Bob( pb�p) A. (3.108). . . . . . . . . . . . . . . . . . . . . . . . . . .

If measured pressure/volume data are available (see Sec. 6.4 inChap. 6), these data can be used to determine A (e.g., by plottingVo�Vob vs. p�pb on log-log paper). Constant A can then be used tocompute compressibilities from the simple relation co� A�p.

Fig. 3.19—Undersaturated-oil-compressibility correlation (fromStanding33).

Bubblepoint Oil Density, lbm/ft3

Compressibility atBubblepoint +

BubblepointPlus 1,000 psia

BubblepointPlus 2,000 psia

Constant A determined in this way is a useful correlating parameter,one that helps to identify erroneous undersaturated p-Vo data.

Standing33 gives a graphical correlation for undersaturated co

(Fig. 3.19) that can be represented by

co� 10�6 exp �ob� 0.004347 �p� pb� � 79.1

(7.141� 10�4)�p� pb� � 12.938�,

(3.109). . . . . . . . . . . . . . . . . . .

with co in psi�1, �ob in lbm/ft3, and p in psia.The Alani-Kennedy EOS can also be solved analytically for oil

compressibility, and Trube70 gives a corresponding-states methodfor determining oil compressibility with charts.

Any of the correlations mentioned here should yield reasonableestimates of co. However, we recommend that experimental data beused for volatile oils when co is greater than about 20�10�6 psi�1.A simple polynomial fit of the relative volume data, Vro� Vo�Vob,from a PVT report allows an accurate and explicit equation for un-dersaturated-oil compressibility.

Vro� A0� A1 p� A2 p2 (3.110a). . . . . . . . . . . . . . . . . . . . . . .

and co��1

Vro�Vro

p�

T

�� A1� 2A2 p�

A0� A1 p� A2 p2 , (3.110b). . . . . . . . . . . . . . . . . . . .

where A0, A1, and A2 are determined from experimental data. Alter-natively, measured data can be fit by use of Eq. 3.108.

3.4.4 Bubblepoint-Oil FVF. Oil FVF ranges from 1 bbl/STB foroils containing little solution gas to about 2.5 bbl/STB for volatileoils. Bob increases more or less linearly with the amount of gas insolution, a fact which explains why Bob correlations are similar tobubblepoint pressure correlations. For example, Standing’s3,17,40

correlation for California crude oils is

Bob� 0.9759� �12� 10�5�A1.2, (3.111). . . . . . . . . . . . . . .

where A� Rs��g��o� 0.5� 1.25T.

36 PHASE BEHAVIOR

Glasø’s52 correlation for North Sea crude oils is

log�Bob� 1� � � 6.585� 2.9133 log A� 0.2768�log A�2 ,

(3.112). . . . . . . . . . . . . . . . . . . .

where A� Rs ��g��o� 0.526� 0.968T.The Vazquez and Beggs57 correlation, based on data from com-

mercial laboratories, is

Bob� 1� �4.677� 10�4�Rs� �0.1751� 10�4�(T� 60)

��API�gc�� 1.8106� 10�8�Rs(T� 60)��API

�gc�

(3.113a). . . . . . . . . . . . . . . . . .

for �API� 30 and

Bob� 1� �4.67� 10�4�Rs� �0.11� 10�4�(T� 60)��API�gc�

� �0.1337� 10�8�Rs(T� 60)��API�gc� (3.113b). . . . . . . . .

for �API� 30, where the effect of separator conditions is includedby use of a corrected gas gravity �gc (Eq. 3.86).

The Standing and the Vazquez-Beggs correlations indicate that aplot of Bo vs. Rs should correlate almost linearly. This plot is usefulfor checking the consistency of reported PVT data from a differen-tial liberation plot. Eq. 3.114,71 which performs considerably betterfor Middle Eastern oils, also suggests a linear relationship betweenBob and Rs.

Bob� 1.0� �0.177342� 10�3� Rs� �0.220163� 10�3�

� Rs��g��o� ��4.292580� 10�6�Rs(T� 60)(1� �o)

��0.528707� 10�3�(T� 60). (3.114). . . . . . . . . . . . . . .

All three Bob correlations (Eqs. 3.113a, 3.113b, and 3.114)should give approximately the same accuracy. Sutton and Far-shad’s54 comparative study of gulf coast oils indicates that Stand-ing’s correlation is slightly better for Bob� 1.4 and that Glasø’scorrelation is best for Bob� 1.4.

3.4.5 Saturated-Oil Compressibility. Perrine8 introduces a defini-tion for the compressibility of a saturated oil that includes theshrinkage effect of saturated-oil FVF, Bo�p, and the expansioneffect of gas coming out of solution, Bg(Rs�p),

co��1Bo�Bo

p�

T

� 15.615

Bg

Bo�Rs

p�

T

. (3.115). . . . . . . . .

co is used in the definition of total system compressibility, ct .

ct� cf � cw Sw � co So � cg Sg , (3.116). . . . . . . . . . . . . .

where cf�rock compressibility. Bg has units ft3/scf. Rs is in scf/STB, and Bo in bbl/STB�saturated-oil FVF at the pressure of inter-est, at or below the original oil’s bubblepoint pressure (where bothgas and oil are present).

3.4.6 Oil Viscosity. Typical oil viscosities range from 0.1 cp fornear-critical oils to �100 cp for heavy crudes. Temperature, stock-tank-oil density, and dissolved gas are the key parameters determin-ing oil viscosity, where viscosity decreases with decreasing stock-tank-oil density (increasing oil gravity), increasing temperature,and increasing solution gas.

Oil viscosity is one of the most difficult properties to estimate,and most methods offer an accuracy of only about 10 to 20%. Twoapproaches are used to estimate oil viscosity: empirical and corre-sponding-states correlations. The empirical methods correlategas-saturated-oil viscosity in terms of dead-oil (residual or stock-tank-oil) viscosity and solution gas/oil ratio. Undersaturated-oilviscosity is related to bubblepoint viscosity and the ratio or differ-

Fig. 3.20—Beal dead-oil (stock-tank-oil) viscosity correlation in-cluding data in Frick (from Standing33).

4,000

2,000

600800

400

200

100

6080

40

20

10

68

4

2

1,000

1

0.60.8

0.4

0.2

0.10 10 20 30 40 50 60

120°F

100°F

140°F

160°F

200°F

180°F

220°F

240°F

Sources of DataBeal (1946)Frick (1962)

ence in actual and bubblepoint pressures. Corresponding-statesmethods use reduced density or reduced pressure and temperatureas correlating parameters.

3.4.7 Dead-Oil (Residual- or Stock-Tank-Oil) Viscosity. Severalcorrelations for dead-oil viscosity are given in terms of oil gravityand temperature. Standing,3 for example, gives best-fit equationsfor the original Beal72 graphical correlation,

�oD� �0.32� 1.8� 107

� 4.53API

�� 360T� 200�A, (3.117). . . . . . . . .

where A� 10 0.43��8.33��API��.

A somewhat modified version of the original correlation is givenin Fig. 3.20 by Standing.33 Beggs18 and Beggs and Robinson73 givethe following equation for the original Beal correlation,

�oD�� 1� 10 T�1.163 exp�6.9824�0.04658�API

�� . (3.118). . . . . .

Bergman* claims that the temperature dependence of the Beggs andRobinson correlation is not valid at lower temperatures (�70°F)and suggests the following correlation, based on viscosity data, forpure compounds and reservoir oils.

ln ln(�oD� 1)� A0 � A1 ln(T� 310), (3.119). . . . . . . . .

where A0�22.33�0.194�API� 0.00033 �2API and A1�� 3.20

� 0.0185 �API.Glasø52 gives a relation (used in the paraffinicity correction of his

bubblepoint pressure correlation) for oils with Kw� 11.9.

�oD� (3.141� 1010)T�3.444(log �API)[10.313(log T )�36.447].

(3.120). . . . . . . . . . . . . . . . . . .

Al-Khafaji et al.74 give the correlation

�oD�104.9563�0.00488T

��API � T�30� 14.29�2.709, (3.121). . . . . . . . . . .

with T in °F and �oD in cp for Eqs. 3.117 through 3.121.

*Personal communication with D.F. Bergman, Amoco Research, Tulsa, Oklahoma (1992).


Fig. 3.21—Dead-oil (stock-tank-oil) viscosities at 100°F for vary-ing paraffinicity (from Ref. 33).

Standing75 gives a relation for dead-oil viscosity in terms of dead-oil density, temperature, and the Watson characterization factor.

log(�oD ��o)�1

A3 Kw��8.24��o�� 1.639A2�1.059

�2.17,

(3.122a). . . . . . . . . . . . . . . . . . .

where A1� 1� 8.69 log T� 460560

, (3.122b). . . . . . . . . . . . .

A2� 1� 0.554 log T� 460560

, (3.122c). . . . . . . . . . . . . . . . .

A3�� 0.1285�2.87A1� 1��o

2.87A1� �o, (3.122d). . . . . . . . . . . . . . .

and �o��o

1� 0.000321(T� 60)100.00462�API, (3.122e). . . . . .

with � in cp, T in °F, and � in g/cm3 for Eqs. 3.117 through 3.122.Eqs. 3.122a through 3.122e represent a best fit of the nomograph forviscosity in terms of temperature, gravity, and characterization fac-tor. Eq. 3.122e (at standard pressure and temperature) is a best fit ofthermal expansion data for crude oils.

Dead-oil viscosity is one of the most unreliable properties to pre-dict with correlations primarily because of the large effect that oiltype (paraffinicity, aromaticity, and asphaltene content) has on vis-cosity. For example, the oil viscosity of a crude oil with Kw� 12may be 3 to 100 times the viscosity of a less paraffinic crude oil hav-ing the same gravity and Kw� 11. For this reason the Standing cor-relation based on the Watson characterization factor is recom-mended when Kw is known. Using an incorrectly estimated Kw,however, may lead to a potentially large error in dead-oil viscosity.

Fig. 3.22—Live-oil (saturated) viscosity as a function of dead-oilviscosity and solution gas/oil ratio (from Standing,33 after Beal72

correlation).

Solution gas/oil ratio, scf/STB

Fig. 3.21 shows dead-oil viscosities calculated at 100°F for a rangeof paraffinicities expressed in terms of Kw, together with the Berg-man* and Glasø48 correlations.

3.4.8 Bubblepoint-Oil Viscosity. The original approach by Chewand Connally76 for correlating saturated-oil viscosity in terms ofdead-oil viscosity and solution gas/oil ratio is still widely used.

�ob� A1��oD�A2. (3.123). . . . . . . . . . . . . . . . . . . . . . . . . . . .

Fig. 3.22 shows the variation in �ob with �oD as a function of Rs.The functional relations for A1 and A2 reported by various authorsdiffer somewhat, but most are best-fit equations of Chew and Con-nally’s tabulated results.

Beggs and Robinson.73

A1� 10.715(Rs� 100)�0.515 (3.124a). . . . . . . . . . . . . . . . . .

and A2� 5.44(Rs� 150)�0.338 . (3.124b). . . . . . . . . . . . . . . . .

Bergman.*

ln A1� 4.768 � 0.8359 ln(Rs� 300) (3.125a). . . . . . . . . .

and A2� 0.555 � 133.5Rs� 300

. (3.125b). . . . . . . . . . . . . . . . .

Standing.3

A1� 10��7.4�10�4�Rs��2.2�10�7�R2s (3.126a). . . . . . . . . . . . . . .

and A2�0.68

10�8.62�10�5�Rs� 0.25

10�1.1�10�3�Rs� 0.062

10�3.74�10�3�Rs.

(3.126b). . . . . . . . . . . . . . . . . .

Aziz et al.77

A1� 0.20 � �0.80� 10–0.00081 Rs� (3.127a). . . . . . . . . . . . . .

and A2� 0.43 � �0.57� 10–0.00072 Rs� . (3.127b). . . . . . . . . .


38 PHASE BEHAVIOR

Al-Khafaji et al.74 extend the Chew-Connally76 correlation to high-er GOR’s (up to 2,000 scf/STB).

A1� 0.247�0.2824A0� 0.5657A20

� 0.4065A30� 0.0631A4

0 (3.128a). . . . . . . . . . . . . . . . . . .

and A2� 0.894� 0.0546A0� 0.07667A20

� 0.0736A30� 0.01008A4

0 , (3.128b). . . . . . . . . . . . . .

where A0� log(Rs) and Rs� 0.1 yields A1� A2� 1. Rs isgiven in scf/STB for Eqs. 3.124 through 3.128. Chew and Connallyindicate that their correlation is based primarily on data with GOR’sof �1,000 scf/STB and that the scatter in A1 at higher GOR’s isprobably the result of insufficient data. Eqs. 3.128a and 3.128b arebased on additional data at higher GOR’s. Eqs. 3.127a and 3.128bappear to be the most well behaved.

An interesting observation by Abu-Khamsin and Al-Marhoun78

is that saturated-oil viscosity, �ob, correlates very well with satu-rated-oil density, �ob.

ln �ob�� 2.652294� 8.484462 �4ob , (3.129). . . . . . . . . .

with �ob in g/cm3. This behavior is expected from the Lohrenz etal.62 correlation discussed later. Although Abu-Khamsin and Al-Marhoun do not comment on the applicability of Eq. 3.129 to under-saturated oils, it would seem reasonable that their correlation shouldapply to both saturated and undersaturated oils. In fact, the correla-tion even appears to predict accurately dead-oil viscosities, �oD, ex-cept at low temperatures for heavy crudes. Simon and Graue givegraphical correlations for the viscosity of saturated CO2/oil systems(see Chap. 8).79

3.4.9 Undersaturated-Oil Viscosities. Beal72 gives the variation ofundersaturated-oil viscosity with pressure graphically where it hasbeen curve fit by Standing.2

�o� �ob

0.001( p� pb)� 0.024�1.6

ob � 0.038�0.56ob . (3.130). . . . . . . .

The Vazquez and Beggs57 correlation is

�o� �ob�p�pb

�A , (3.131). . . . . . . . . . . . . . . . . . . . . . . . . . .

where A� 2.6 p1.187 exp � 11.513� �8.98� 10�5�p�.A more recent correlation by Abdul-Majeed et al.80 is

�o� �ob � 10 A� 5.2106 � 1.11 log� p�pb��, (3.132a). . . . . . . .

where A� 1.9311� 0.89941 �ln Rs� � 0.001194 �2API

� 0.0092545�API�ln Rs�. (3.132b). . . . . . . . . . . . . . .

Eq. 3.132 is based on the observation that a plot of log(�o� �ob)vs. log(p� pb) plots as a straight line with slope of �1.11. Be-cause this observation appears to be fairly general, it can be usedto check reported undersaturated-oil viscosities and to developfield-specific correlations.

Sutton and Farshad54 and Khan et al.81 present results that indicatethat the Standing equation gives good results and that the Vaz-quez-Beggs correlation tends to overpredict viscosities somewhat.Abdul-Majeed et al.80 indicate that both the Standing and Vaz-quez-Beggs correlations overpredict viscosities of North Africanand Middle Eastern oils (253 data), and that their own correlation per-formed best for these data and for the data used by Vazquezand Beggs.

3.4.10 Compositional Correlation. In compositional reservoirsimulation of miscible-gas-injection processes and the depletion ofnear-critical reservoir fluids, the oil and gas compositions may bevery similar. A single viscosity relation consistent for both phases

is therefore desired. Several corresponding-states viscosity correla-tions can be used for both oil and high-pressure gas mixtures; theLohrenz et al.62 correlation has become a standard in compositionalreservoir simulation. Lohrenz et al. use the Jossi et al.82 correlationfor dense-gas mixtures ( �pr�0.1),6

�� o��T� 10�4�1�4� 0.10230� 0.023364�pr

� 0.058533�2pr� 0.040758�3

pr

� 0.0093324�4pr , (3.133a). . . . . . . . . . . . . . . . . . . . . .

where �T� 5.35� Tpc

M3p4pc�

1�6

, (3.133b). . . . . . . . . . . . . . . . . .

�pr��pc

��

Mvpc, (3.133c). . . . . . . . . . . . . . . . . . . . . . . .

and �o�

�N

i�1

zi�i Mi�

�N

i�1

zi Mi�

. (3.133d). . . . . . . . . . . . . . . . . . . . . . .

Pseudocritical properties Tpc, ppc, and vpc are calculated withKay’s mixing rule.

Component viscosities, �i, can be calculated from the Lucas45

low-pressure correlation Eq. 3.67 or from the Stiel and Thodos83

correlation (as suggested by Lohrenz et al.62).

�i�Ti� �34� 10�5�T 0.94ri

(3.134a). . . . . . . . . . . . . . . . . . . . . .

for Tri �1.5, and

�i�Ti� �17.78� 10�5�(4.58Tri� 1.67)5�8 (3.134b). . . . . .

for Tri� 1.5, where �Ti� 5.35�Tci M3i�p

4ci�

1�6.

Lohrenz et al.62 give a special relation for vc C7� of the C7+ fraction.

vcC7�� 21.573� 0.015122MC7�

� 27.656�C7�

� 0.070615MC 7��C7�

, (3.135). . . . . . . . . . . . . . . .

with � in cp, � in cp�1, � in lbm/ft3, v in ft3/lbm mol, T in �R, pin psia, and M in lbm/lbm mol. The Lohrenz et al. method is verysensitive to mixture density and to the critical volumes of heavycomponents. Adjustment of the critical volumes of heavy (andsometimes light) components to match experimental oil viscosi-ties is usually necessary.

��# �$% �� &��

3.5.1 IFT. Weinaug and Katz84 propose an extension of the Ma-cleod85 relationship for multicomponent mixtures.

�1�4go ��

N

i�1

Pi�xi�o

Mo� yi

�g

Mg� , (3.136). . . . . . . . . . . . . . . . .

with � in dynes/cm (mN/m) and � in g/cm3. Pi is the parachor ofComponent i, which can be calculated by group contributions, asshown in Table 3.2. For n-alkanes, the parachors can be expressed by

Pi� 25.2� 2.86Mi . (3.137). . . . . . . . . . . . . . . . . . . . . . . .

Several authors propose parachors for pure hydrocarbons that devi-ate from the group-contribution values. For example, PC1

�77 isoften cited for methane instead of the group-contribution value ofPC1�71. Likewise, PN2

�41 is often used for nitrogen instead ofthe group-contribution value of PN2

�35. Fig. 3.23 plots parachorsvs. molecular weight for pure components and petroleum fractions.


TABLE 3.2—PARACHORS FOR PURE COMPONENTS ANDCOMPOUND GROUPS

Pure Component

C1 71

C2 111

C3 151

C4 (also i-C4) 191

C5 (also i-C5) 231

C6 271

C7 311

C8 351

C9 391

C10 431

N2 35

CO2 49

H2S 80

Group

C 9.0

H 15.5

CH3 55.5

CH2 [in (CH2)n ] 40.0

N 17.5

O 20.0

S 49.1

Example: For methane, CH4. PC1=PC+4(PH)=9+4(15.5)=71.

Nokay86 gives a simple relation for parachors of pure hydrocarbons(paraffins, olefins, naphthenes, and aromatics) with a normal boilingpoint between 400 and 1,400°R and specific gravity �1,

log Pi�� 4.20895� 2.29319 log� Tbi

�0.5937i

�, (3.138). . . . .

with Tb in °R.Katz and Saltman87 and Katz et al.88 give parachor data for C7+

fractions measured by Standing and Katz,58,89 which are approxi-mately correlated by

Pi� 35� 2.40Mi . (3.139). . . . . . . . . . . . . . . . . . . . . . . . . .

The API recommended procedure for estimating petroleum fractionIFT’s is based on an unpublished correlation.27 The graphical cor-relation can be expressed by

�i�602(1� Tri)1.194

Kwi, (3.140). . . . . . . . . . . . . . . . . . . . . . .

where Kw� T1�3b��, with Tb in °R. The parachor can be estimated

with the Macleod relation,

Pi� �1�4i�Mi�sLi� , (3.141). . . . . . . . . . . . . . . . . . . . . . . . . . . .

where �sL� �sv is assumed. The saturated-liquid density can beestimated, for example, with the Rackett63 equation.

�sLi�Mi pci

RTciZ� 1��1�Tri

�2�7�Ri

, (3.142). . . . . . . . . . . . . . . . .

where ZRi� Zci� 0.291� 0.08 i (3.143). . . . . . . . . . . . . .

and R�universal gas constant. The parachors predicted from Eqs.3.140 through 3.143 are practically constant for a given petroleumfraction (i.e., the temperature effect cancels out).

Fig. 3.23—Hydrocarbon parachors.

n-paraffinsHeptanes plus of Ref. 4GasolinesCrude oil

Firoozabadi et al.90 give an equation that can be used to approxi-mate the parachor of pure hydrocarbons from C1 through C6 and forC7+ fractions,

Pi� 11.4 � 3.23Mi � 0.0022M2i . (3.144). . . . . . . . . . .

They also discuss the qualitative effect of asphaltenes on IFT andsuggest that the parachor of asphaltic substances generally will notfollow the relations of lighter C7+ fractions.

Ramey91 gives a method for estimating gas/oil IFT with black-oilPVT properties. We extend the method here to include the effect ofsolution oil/gas ratio, rs. Considering stock-tank oil and separatorgas as the “components” (o and g) making up reservoir oil and gas,respectively, the Weinaug-Katz84 relation can be written

��

go� Po xo��o

Mo�� yo�

�g

Mo�� Pg xg��o

Mo�� yg��g

Mg��,

(3.145a). . . . . . . . . . . . . . . . . . .

where xo �1

1� (7.52� 10�6)Rs�Mo��o�

, (3.145b). . . . . . . .

xg� 1� xo , (3.145c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

yo�1

1� (7.52� 10�6)�Mo��o�rs

, (3.145d). . . . . . . . . . .

yg� 1� yo , (3.145e). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

�o�62.4�o� 0.0136�g Rs

62.4Bo, (3.145f). . . . . . . . . . . . . . . . . .

�g� 0.0932�pMg�ZT� , (3.145g). . . . . . . . . . . . . . . . . . . . . .

Mo� xo Mo� xg Mg , (3.145h). . . . . . . . . . . . . . . . . . . . . . . .

40 PHASE BEHAVIOR

Mg� yo Mo� yg Mg ,�� (3.145i). . . . . . . . . . . . . . . . . . . . . . . .

Mo� 6, 084��API� 5.9 , (3.145j). . . . . . . . . . . . . . . . . . . . .

Mg� 28.97�g , (3.145k). . . . . . . . . . . . . . . . . . . . . . . . . . . .

Po� �2.376� 0.0102�API��Mo , (3.145l). . . . . . . . . . . . . . .

and Pg� 25.2� 2.86Mg , (3.145m). . . . . . . . . . . . . . . . . . . . .

with � in g/cm3, Rs in scf/STB, Bo in bbl/STB, T in °R, and p in psiaand where xo and xg�mole fractions of the surface-oil and -gascomponents, respectively, in the oil phase, and yo and yg�molefractions of the surface-oil and -gas components, respectively, in thegas phase. In the traditional black-oil approach rs� 0, simplifyingthe relations to those originally suggested by Ramey.91

Eq. 3.145 is useful in black-oil reservoir simulation and whencompositional data are not available. The black-oil approach gener-ally is not recommended for predicting gas/oil IFT’s unless the sur-face-oil parachor has been fit to experimental IFT data (or to IFT’scalculated with compositions and densities from an EOS character-ization by use of Eq. 3.136).

3.5.2 Diffusion Coefficients. Molecular diffusion in multicompon-ent mixtures is a complex problem. The standard engineering ap-proach uses an effective diffusion coefficient for Component i in amixture, Dim, where Dim can be calculated in one of two ways: (1)from binary diffusion coefficients and mixture composition or (2)from Component i properties and mixture viscosity. The first ap-proach uses the Wilke92 formula to calculate Dim.

Dim�1� yi

�N

j�1j�i

yj�Dij

, (3.146). . . . . . . . . . . . . . . . . . . . . . . . . . . .

where yi�mixture mole fraction and Dij� Dji is the binary diffu-sivity at the pressure and temperature of the mixture.

Sigmund93 correlates the effect of pressure and temperature ondiffusion coefficients using a corresponding-states approach withreduced density.

�M Dij

�oM

Doij� 0.99589� 0.096016�pr� 0.22035�2

pr

� 0.032874�3pr , (3.147). . . . . . . . . . . . . . . . . . . . . . .

where Dij�diffusion coefficient at pressure and temperature,�pr�pseudoreduced density� �M��Mpc� ��M�vpc, �M�mix-ture molar density, �o

M Doij�low-pressure density-diffusivity prod-

uct, and vpc�pseudocritical molar volume calculated with Kay’s5

mixing rule. Note that the ratio �M Dij��oDo

ij is the same for allbinary pairs in a mixture because �pr is a function of only mixturedensity and composition.

da Silva and Belery12 note that the Sigmund correlation does notwork well for liquid systems and propose the following extrapola-tion for �pr�3.

�MDij

�oM Do

ij

� 0.18839 exp(3� �pr) , (3.148). . . . . . . . . . . . . . . .

which avoids negative Dij for oils at �pr�3.7 as estimated by theSigmund correlation.

Low-pressure binary gas diffusion coefficients,6 Doij, can be esti-

mated from

Doij� 0.001883

T 3�2 �1�Mi� � �1�Mj

��0.5

po�2ij�ij

, (3.149a). . . . . . . .

where �ij�1.06036T 0.1561

ij

� 0.193exp�0.47635Tij

�

� 1.03587exp�1.52996Tij

�� 1.76474

exp�3.89411Tij�

,

(3.149b). . . . . . . . . . . . . . . . . .

Tij �T

(�k)ij, (3.149c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

��k� ij� ��k�i ��k�j�1�2 , (3.149d). . . . . . . . . . . . . . . . . . . .

��k�i� 65.3Tci Z18�5ci

, (3.149e). . . . . . . . . . . . . . . . . . . . . . . .

�ij� 0.5��i� �j� , (3.149f). . . . . . . . . . . . . . . . . . . . . . . . . . .

and �i� 0.1866v1�3

ci

Z6�5ci

, (3.149g). . . . . . . . . . . . . . . . . . . . . . . .

with the diffusion coefficient, Doij, in cm2/s; molecular weight, M,

in kg/kmol; temperature, T, in K, pressure; p, in bar; Lennard Jones12-6 potential parameter, �, in Å; Lennard-Jones 12-6 potentialparameter, �/k, in K; and critical volume, vc , in m3/kmol and whereZc�critical compressibility factor and i and j�diffusing and con-centrated species, respectively.

To obtain the low-pressure density-diffusivity product, we use theideal-gas law, �o

M� po�RT, to get

Doij�

oM� �2.2648� 10�5�

T1�2 �1�Mi� � �1�Mj

��1�2

�2ij �ij

,

(3.150). . . . . . . . . . . . . . . . . .

where � and �M have units g mol/cm3.The accuracy of the Sigmund correlation for liquids is not known,

but the extension proposed by da Silva and Belery (Eq. 148) forlarge reduced densities does avoid negative diffusivities calculatedby the Sigmund equation.94 Renner95 proposes a generalized cor-relation for effective diffusion coefficients of light hydrocarbonsand CO2 in reservoir liquids that can be used as an alternative to theSigmund-type correlation.

Dim� 10�9 ��0.4562o M�0.6898

i �1.706Mi p�1.831 T 4.524,

(3.151). . . . . . . . . . . . . . . . . . .

with D in cm2/s and where �o�oil viscosity in cp, Mi�molecularweight, �Mi�molar density of Component i at p and T in g mol/cm3,p�pressure in psia, and T�temperature in K. This correlation isbased on 141 experimental data with the following property ranges:0.2� �o�134 cp; 16�Mi�44; 0.04� �Mi�7 kmol/m3; 14.7�p�2,560 psia; and 273�T�333 K, where i�CO2, C1, C2, and C3.

Renner also gives a correlation for diffusivity of CO2 in water/brine systems.

DCO2�w� �6.392� 103� � 6.911CO2

� �0.1584w , (3.152). . . . . . . . .

with D in cm2/s and � in cp.

��' �()��

This section covers the estimation of equilibrium K values by cor-relations and the calculation of two-phase equilibrium when K val-ues are known. The K value is defined as the ratio of equilibrium gascomposition yi to the equilibrium liquid composition xi,

Ki� yi�xi . (3.153). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Ki is a function of pressure, temperature, and overall composition.K values can be estimated with empirical correlations or by satisfy-ing the equal-fugacity constraint with an EOS (see Chap. 4).

Although the increasing use of EOS’s has tended to lessen interestin empirical K-value correlations, empirical methods are still usefulfor such engineering calculations as (1) multistage surface separa-tion, (2) compositional reservoir material balance, and (3) checkingthe consistency of separator-oil and gas compositions.


Fig. 3.24—General behavior of a K value vs. pressure plot on log-log scale.

Several methods for correlating K values have appeared in thepast 50 years. Most rely on two limiting conditions for describingthe pressure dependence of K values. First, at low pressures,Raoult’s and Dalton’s laws3 can be used to show that

Ki� pvi�T ��p, (3.154). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

where pv�component vapor pressure at the system temperature.The limitations of this equation are that temperature must be less thanthe component critical temperature (because vapor pressure is not de-fined at supercritical temperatures) and that the component behavesas an ideal gas. Also, the equation implies that the K value is indepen-dent of overall composition. In fact, the pressure dependence of low-pressure K values is closely approximated by Eq. 3.154.

The second observation is that, at high pressures, the K values ofall components in a mixture tend to converge to unity at the samepressure. This pressure is called the convergence pressure96 and, forbinaries, represents the actual mixture critical pressure. For multi-component mixtures, the convergence pressure is a nonphysicalcondition unless the system temperature equals the mixture criticaltemperature.97,98 This is because a mixture becomes single phase atthe bubblepoint or dewpoint pressure before reaching the conver-gence pressure.

The log-log plot of Ki vs. pressure in Fig. 3.24 shows how theideal-gas and convergence-pressure conditions define the K-valuebehavior at limiting conditions. For light components (whereT� Tci ), K values decrease monotonically toward the conver-gence pressure. For heavier components (where T� Tci ), K valuesinitially decrease as a function of pressure at low pressures, passingthrough unity when system pressure equals the vapor pressure of aparticular component, reaching a minimum, and finally increasingtoward unity at the convergence pressure.

For reservoir fluids, the pressure where K values reach a mini-mum is usually �1,000 psia (Fig. 3.25), implying that K values aremore or less independent of convergence pressure (i.e., composi-tion) at pressures �1,000 psia. This observation has been used todevelop general “low-pressure” K-value correlations for surface-separator calculations.

3.6.1 Hoffman et al. Method. Hoffman et al.99 propose a methodfor correlating K values that has received widespread application.

Ki�10�A0 � A1 Fi

�

p

or log Ki p� A0 � A1 Fi , (3.155). . . . . . . . . . . . . . . . . . . . . .

where Fi�1�Tbi� 1�T

1�Tbi� 1�Tcilog�pci�psc�; (3.156). . . . . . . . . . . .

Tc�critical temperature; pc�pressure; Tb�normal boiling point;psc�pressure at standard conditions; and A1 and A0�slope and in-tercept, respectively, of the plot log(Ki p) vs. Fi.

Hoffman et al. show that measured K values for a reservoir gascondensate correlate well with the proposed equation. They foundthat trend of log(Ki p) vs. Fi is linear for components C1 through C6at all pressures, while the function turns downward for heavier com-ponents at low pressures. Interestingly, the trend becomes more lin-ear for all components at higher pressures.

As Fig. 3.26 shows, Slope A1 and Intercept A0 vary with pres-sure. For low pressures, Ki� pv�p. With the Clapeyron vapor pres-sure relation,5 log(pv)� a� b�T results in A0� log(psc) andA1� 1. These limiting values of A0 and A1 are close to the valuesfound when A0(p) and A1(p) are extrapolated to p� psc. Because

42 PHASE BEHAVIOR

Fig. 3.25—K values at 120°F for binary- and reservoir-fluid systems with convergence pressuresranging from 800 to 10,000 psia (from Standing3).

K values tend toward unity as pressure approaches the convergencepressure, pK , it is necessary that A0� log(pK) and A1 0. Severalauthors have noted that plots of log(Ki p) vs. Fi tend to converge ata common point. Brinkman and Sicking101 suggest that this “pivot”point represents the convergence pressure where Ki� 1 andp� pK. The value of Fi at the pivot point, FK, is easily shown toequal log(pK�psc).

It is interesting to note that the well-known Wilson102,103 equation,

Ki�exp 5.37(1� i)�1� T�1

ri�

pri, (3.157). . . . . . . . . . . . . .

is identical to the Hoffman et al.99 relation for A0� log(psc) andA1� 1 when the Edmister104 correlation for acentric factor equation,

i�37

Tbi�Tci

1� Tbi�Tcilog�pci�psc� � 1 , (3.158). . . . . . . . . . . .

is used in the Wilson equation. Note that 5.37�(7/3) ln (10).Whitson and Torp100 suggest a generalized form of the Hoffman

et al.99 equation in terms of convergence pressure and acentric factor.

Ki� �pcipK�

A1�1 exp 5.37 A1 (1� i)�1� T�1ri��

pri,

(3.159). . . . . . . . . . . . . . . . . . .

where A1�a function of pressure, with A1� 1 at p� psc andA1� 0 at p� pK. The key characteristics of K values vs. pressure


Fig. 3.26—Pressure dependence of slope, A1, and intercept, A0,in Hoffman et al. Kp-F relationship (Eq. 3.155) for a North Seagas condesate NS-1 (from Whitson and Torp100).

Pressure, psia

Intercept A0

Slope A1

–

log pK

and temperature are correctly predicted by Eq. 3.159, where the fol-lowing pressure dependence for A1 is suggested.

A1� 1� (p�pK)A2 , (3.160). . . . . . . . . . . . . . . . . . . . . . . . . .

where A2 ranges from 0.5 to 0.8 and pressures p and pK are givenin psig. Canfield105 also suggests a simple K-value correlationbased on convergence pressure.

3.6.2 Standing Low-Pressure K Values. Standing106 uses the Hof-fman et al.99 method to generate a low-pressure K-value equationfor surface-separator calculations ( psp� 1, 000 psia andTsp� 200�F). Standing fits A1 and A0 in Eq. 3.155 as a functionof pressure using K-value data from an Oklahoma City crude oil. Hetreats the C7� by correlating the behavior of KC7�

as a function of“effective” carbon number nC7�

. The Standing equations are

Ki�1

psp10�A0 � A1 Fi

�, (3.161a). . . . . . . . . . . . . . . . . . . . . . . .

Fi� bi�1�Tbi� 1�T�, (3.161b). . . . . . . . . . . . . . . . . . . . . . .

bi� log�pci�psc��1�Tbi� 1�Tci�, (3.161c). . . . . . . . . . . . . . .

A0�p� � 1.2� �4.5� 10�4�p� �15� 10�8�p2,

(3.161d). . . . . . . . . . . . . . . . . .

A1�p� � 0.890� �1.7� 10�4�p� �3.5� 10�8�p2,

(3.161e). . . . . . . . . . . . . . . . . . .

nC7�� 7.3� 0.0075T� 0.0016p, (3.161f). . . . . . . . . . . . .

bC7�� 1, 013� 324nC7�

� 4.256n2C7�

, (3.161g). . . . . . .

and TbC7�� 301� 59.85nC7�

� 0.971n2C7�

, (3.161h). . . . .

with T in °R except when calculating nC7� (for nC7�

, T is in °F) andp in psia. Standing suggests modified values of bi and Tbi for nonhy-drocarbons, methane, and ethane (Table 3.3). Glasø and Whitson107

show that these equations are accurate for separator flash calcula-tions of crude oils with GOR’s ranging from 300 to 1,500 scf/STBand oil gravity ranging from 26 to 48°API. Experience shows, how-ever, that significant errors in calculated GOR may result for leangas condensates, probably because of inaccurate C1 and

TABLE 3.3—VALUES OF b AND Tb FOR USE INSTANDING LOW-PRESSURE K-VALUE CORRELATION

Component, ibi

(cycle-°R)Tbi°R

N2 470 109

CO2 652 194

H2S 1,136 331

C1 300 94

C2 1,145 303

C3 1,799 416

i-C4 2,037 471

n-C4 2,153 491

i-C5 2,368 542

n-C5 2,480 557

C6 (lumped) 2,738 610

n-C6 2,780 616

n-C7 3,068 669

n-C8 3,335 718

n-C9 3,590 763

n-C10 3,828 805

For C7+ fractions, see Eqs. 3.161f through 3.161h

C7� K values. The Hoffman et al. method with Standing’s low-pres-sure correlations are particularly useful for checking the consisten-cy of separator-gas and -oil compositions.

3.6.3 Galimberti-Campbell Method. Galimberti and Camp-bell108,109 suggested another useful approach for correlating K val-ues where

log Ki� A0� A1T2ci

(3.162). . . . . . . . . . . . . . . . . . . . . . . . .

is shown to correlate K values for several simple mixtures contain-ing hydrocarbons C1 through C10 at pressures up to 3,000 psia andtemperatures from �60 to 300°F.

Whitson developed a low-pressure K-value correlation, based ondata from Roland,110 at pressures �1,000 psia and temperaturesfrom 40 to 200°F, for separator calculations of gas condensates.

A0� 4.276� �7.6� 10�4�T

� � 1.18� �5.675� 10�4�T� log p , (3.163a). . . . . . . .

A1� 10�6�� 4.9563� 0.00955T � � �1.9094� 10�3�

� �1.235� 10�5�T� �3.34� 10�8�T 2�p�, (3.163b). . .

TcC1� 343� 0.04p, (3.163c). . . . . . . . . . . . . . . . . . . . . . . . .

and TcC7� � 1, 052.5� 0.5125T� 0.00375T 2 , (3.163d). . . .

with p in psia, T in °F, and Tc in °R.

3.6.4 Nonhydrocarbon K Values. Lohrenz et al.111 reported non-hydrocarbon K values as a function of pressure, temperature, andconvergence pressure.

ln KH2S� �1� ppK�

0.8

6.3992127�1, 399.2204

T

� 0.76885112 ln p�18.215052 ln p

T

�1, 112, 446.2

T 2�, (3.164a). . . . . . . . . . . . . . . . . . . . .

44 PHASE BEHAVIOR

ln KN2� �1� p

pK�

0.4

�11.294748�1, 184.2409

T

� 0.90459907 ln p�, (3.164b). . . . . . . . . . . . . . . . . . .

ln KCO2� �1� p

pK�

0.6

�7.0201913� 152.7291T

� 1.8896974

� ln p �1, 719.2956 ln p

T�

644, 740.69 ln pT 2� ,(3.164c). . . . . . . . . . . . . . . . . . .

with p in psia and T in °R. For low-pressure K-value estimation, thefirst term in Eq. 3.164 simplifies to unity (assuming that1� p�pK� 1) and the K values become functions of pressure andtemperature only. However, these equations do not give the correctlow-pressure value of (ln Ki)�(ln p)�� 1

3.6.5 Convergence-Pressure Estimation. For correlation pur-poses, convergence pressure is used as a variable to define the com-position dependence of K values. Convergence pressure is a func-tion of overall composition and temperature. Whitson andMichelsen112 show that convergence pressure is a thermodynamicphenomenon, with the characteristics of a true mixture critical point,that can be predicted with EOS’s.

Rzasa et al.113 give an empirical correlation for convergencepressure as a function of temperature and the product (M�)C7�

.Standing2 suggests that convergence pressure of reservoir fluids va-ries almost linearly with C7� molecular weight.

Convergence pressure can also be calculated with a trial-and-er-ror procedure suggested by Rowe.97,98,114 This procedure involvesthe use of several empirical correlations for estimating mixture criti-cal pressure and temperature, pseudocomponent critical properties,and the K values of methane and octane. The Galimberti and Camp-bell108,109 K-value method is used to estimate K values of othercomponents by interpolation and extrapolation of the C1 and C8 Kvalues. This approach to convergence pressure is necessary if the Kvalues are used for processes that approach critical conditions orwhere K values change significantly because of overall compositioneffects. The method cannot, of course, be more accurate than thecorrelations it uses and therefore is expected to yield only qualita-tively correct results.

For reservoir calculations where convergence pressure can be as-sumed constant (e.g., pressure depletion), a more direct approach todetermining convergence pressure is suggested. With a K-value cor-relation of the form Ki� K(pK, p, T ) as in Eq. 3.159, the conver-gence pressure can be estimated from a single experimental satura-tion pressure. For a bubblepoint and a dewpoint, Eqs. 3.165 and3.166, respectively, must be satisfied.

F �pK� � 1��

N

i�1

zi Ki�pK, pb, T� � 0 (3.165). . . . . . . . . . . . .

and F�pK� � 1��

N

i�1

zi

Ki�pK, pd , T�� 0, (3.166). . . . . . . . . . .

where zi, pb, or pd and T are specified and pK is determined.The two-phase flash calculation, with K values given, is dis-

cussed in Chap. 4 in the Phase-Split Calculation section.

��

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84. Weinaug, C.F. and Katz, D.L.: “Surface Tensions of Methane-PropaneMixtures,” Ind. & Eng. Chem. (1943) 35, 239.

85. Macleod, D.B.: “On a Relation Between Surface Tension and Density,”Trans., Faraday Soc. (1923) 19, 38.

86. Nokay, R.: “Estimate Petrochemical Properties,” Chem. Eng. (23 Feb-ruary 1959) 147.

87. Katz, D.L. and Saltman, W.: “Surface Tension of Hydrocarbons,” Ind.& Eng. Chem. (January 1939) 31, 91.

88. Katz, D.L., Monroe, R.R., and Trainer, R.P.: “Surface Tension of CrudeOils Containing Dissolved Gases,” Pet. Tech. (September 1943).

89. Standing, M.B. and Katz, D.L.: “Vapor-Liquid Equilibria of NaturalGas-Crude Oil Systems,” Trans., AIME (1944) 155, 232.

90. Firoozabadi, A. et al.: “Surface Tension of Reservoir Crude-Oil/GasSystems Recognizing the Asphalt in the Heavy Fraction,” SPERE (Feb-ruary 1988) 265.

91. Ramey, H.J. Jr.: “Correlations of Surface and Interfacial Tensions ofReservoir Fluids,” paper SPE 4429 available from SPE, Richardson,Texas (1973).

92. Wilke, C.R.: “A Viscosity Equation for Gas Mixtures,” J. Chem. Phy.(1950) 18, 517.

93. Sigmund, P.M.: “Prediction of Molecular Diffusion at Reservoir Condi-tions. Part I—Measurement and Prediction of Binary Dense Gas Diffu-sion Coefficients,” J. Cdn. Pet. Tech. (April–June 1976) 48.

94. Christoffersen, K.: “High-Pressure Experiments with Application toNaturally Fractured Chalk Reservoirs. 1. Constant Volume Diffusion.2. Gas-Oil Capillary Pressure,” Dr.Ing. dissertation, U. Trondheim,Trondheim, Norway (1992).

95. Renner, T.A.: “Measurement and Correlation of Diffusion Coefficientsfor CO2 and Rich-Gas Applications,” SPERE (May 1988) 517; Trans.,AIME, 285.

96. Hadden, S.T.: “Convergence Pressure in Hydrocarbon Vapor-LiquidEquilibra,” Chem. Eng. Prog. (1953) 49, No. 7, 53.

97. Rowe, A.M. Jr.: “Applications of a New Convergence Pressure Con-cept to the Enriched Gas Drive Process,” PhD dissertation, U. of Texas,Austin, Texas (1964).

98. Rowe, A.M. Jr.: “The Critical Composition Method—A New Conver-gence Pressure Method,” SPEJ (March 1967) 54; Trans., AIME, 240.

99. Hoffmann, A.E., Crump, J.S., and Hocott, C.R.: “EquilibriumConstants for a Gas-Condensate System,” Trans., AIME (1953) 198, 1.

46 PHASE BEHAVIOR

100. Whitson, C.H. and Torp, S.B.: “Evaluating Constant Volume DepletionData,” JPT (March 1983) ; Trans., AIME, 275.

101. Brinkman, F.H. and Sicking, J.N.: “Equilibrium Ratios for ReservoirStudies,” Trans., AIME (1960) 219, 313.

102. Wilson, G.M.: “Calculation of Enthalpy Data From a Modified Re-dlich-Kwong EOS,” Advances in Cryogenic Eng. (1966) 11, 392.

103. Wilson, G.M.: “A Modified Redlich-Kwong EOS, Application to Gen-eral Physical Data Calculations,” paper 15c presented at the 1969AIChE Natl. Meeting, Cleveland, Ohio.

104. Edmister, W.C.: “Applied Hydrocarbon Thermodynamics, Part 4: Com-pressibility Factors and Equations of State,” Pet. Ref. (April 1958) 37, 173.

105. Canfield, F.B.: “Estimate K-Values with the Computer,” Hydro. Proc.(April 1971) 137.

106. Standing, M.B.: “A Set of Equations for Computing Equilibrium Ratiosof a Crude Oil/Natural Gas System at Pressures Below 1,000 psia,” JPT(September 1979) 1193.

107. Glasø, O. and Whitson, C.H.: “The Accuracy of PVT Parameters Cal-culated From Computer Flash Separation at Pressures Less Than 1,000psia,” JPT (August 1980) 1811.

108. Galimberti, M. and Campbell, J.M.: “Dependence of Equilibrium Va-porization Ratios (K-Values) on Critical Temperature,” Proc., 48thNGPA Annual Convention (1969) 68.

109. Galimberti, M. and Campbell, J.M.: “New Method Helps Correlate KValues for Behavior of Paraffin Hydrocarbons,” Oil & Gas J. (Novem-ber 1969) 64.

110. Roland, C.H.: “Vapor Liquid Equilibrium for Natural Gas-Crude OilMixtures,” Ind. & Eng. Chem. (1945) 37, 930.

111. Lohrenz, J., Clark, G.C., and Francis, R.J.: “A Compositional MaterialBalance for Combination Drive Reservoirs With Gas and Water Injec-tion,” JPT (November 1963) 1233; Trans., AIME, 228.

112. Whitson, C.H. and Michelsen, M.L.: “The Negative Flash,” FluidPhase Equilibria (1989) 53, 51.

113. Rzasa, M.J., Glass, E.D., and Opfell, J.B.: “Prediction of Critical Prop-erties and Equilibrium Vaporization Constants for Complex Hydrocar-bon Systems,” Chem. Eng. Prog. (1952) 2, 28.

114. Rowe, A.M. Jr.: “Internally Consistent Correlations for PredictingPhase Compositions for Use in Reservoir Composition Simulators,”paper SPE 7475 presented at the 1978 SPE Annual Technical Confer-ence and Exhibition, Houston, 1–3 October.

*� �� $��

Å�1.0* E�01�nm�� g/cm3

bar�1.0* E�05�Pabbl�1.589 873 E�01�m3

Btu/lbm mol�2.236 E�03�J/molcp�1.0* E�03�Pas

cSt�1.0* E�06�m2/sdyne/cm�1.0* E�00�mN/m

ft�3.048* E�01�mft2�9.290 304* E�02�m2

ft3�2.831 685 E�02�m3

ft3/lbm mol�6.242 796 E�02�m3/kmol°F (°F�32)/1.8 �°C°F (°F�459.67)/1.8 �K

in.2�6.451 6* E�00�cm2

lbm�4.535 924 E�01�kglbm mol�4.535 924 E�01�kmol

psi�6.894 757 E�00�kPapsi�1�1.450 377 E�01�kPa�1

°R�5/9 ��

*Conversion factor is exact.

EQUATION-OF-STATE CALCULATIONS 1

��

��

��

Cubic equations of state (EOS’s) are simple equations relating pres-sure, volume, and temperature (PVT). They accurately describe thevolumetric and phase behavior of pure compounds and mixtures, re-quiring only critical properties and acentric factor of each compo-nent. The same equation is used to calculate the properties of allphases, thereby ensuring consistency in reservoir processes that ap-proach critical conditions (e.g., miscible-gas injection and depletionof volatile-oil/gas-condensate reservoirs). Problems involving mul-tiphase behavior, such as low-temperature CO2 flooding, can betreated with an EOS, and even water-/hydrocarbon-phase behaviorcan be predicted accurately with a cubic EOS.

Volumetric behavior is calculated by solving a simple cubic equa-tion, usually expressed in terms of Z�pv/RT,

Z3 � A2 Z2 � A1 Z � A0 � 0, (4.1). . . . . . . . . . . . . . . . . . . .

where constants A0, A1, and A2 are functions of pressure, tempera-ture, and phase composition.

Phase equilibria are calculated with an EOS by satisfying thecondition of chemical equilibrium. For a two-phase system, thechemical potential of each component in the liquid phase �i(x) mustequal the chemical potential of each component in the vapor phase�i( y), �i(x)��i( y). Chemical potential is usually expressed interms of fugacity, fi, where �i�RT ln fi��i(T ) and �i(T ) areconstant terms that drop out in most problems.1-3 It is readily shownthat the condition �i(x)��i( y) is satisfied by the equal-fugacityconstraint, fLi�fvi, where fugacity is given by

ln �i � lnfi

yi p� 1

RT��

V

��p�ni

� RTV� dV � ln Z. (4.2). . . . . .

Other thermodynamic properties, such as Helmholz energy, enthal-py, and entropy, can be readily defined in terms of the fugacity coef-ficient. Michelsen4 gives a particularly compact and useful discus-sion of the relation between thermodynamic properties aimed atmaking efficient EOS calculations.

A component material balance is also required to solve vapor/liq-uid equilibrium problems: zi�Fv yi�(1�Fv)xi, where Fv�molefraction of the vapor phase�nv/(nv�nL). Integrating the componentbalance in the two-phase flash calculation is discussed in Sec. 4.3.1.

Solving phase equilibria with an EOS is a trial-and-error proce-dure, requiring considerable computations. With today’s comput-ers, however, the task is fast and reliable. The accuracy of EOS pre-dictions has also improved considerably during the past 15 years,

during which emphasis has been on improved liquid volumetric pre-dictions and treating the heptanes-plus fraction (Chap. 5).

This chapter provides the equations and algorithms necessary forcalculating phase and volumetric behavior of reservoir fluids witha cubic EOS. Sec. 4.2 reviews the most important cubic equations,starting with van der Waals’5 EOS from 1873 and concluding withthe method of volume translation, which has greatly improved thevolumetric capabilities of cubic EOS’s.

In Secs. 4.3 through 4.5, we present algorithms for solving vapor/liquid equilibrium (VLE) problems, including the two-phase flash,phase-stability-test, and saturation-pressure calculations. Refer-ence is also made to methods for solving three-phase and critical-point calculations. Sec. 4.6 deals specifically with compositionalgradients with depth caused by gravity and thermal diffusion. Final-ly, Sec. 4.7 covers how to “tune” an EOS to match experimentalPVT data (see also Appendix C).

��

Since the introduction of the van der Waals EOS, many cubic EOS’shave been proposed—e.g., the Redlich and Kwong6 EOS (RK EOS)in 1949, the Peng and Robinson7 EOS (PR EOS) in 1976, and theMartin8 EOS in 1979, to name only a few.9-15 Most of these equa-tions retain the original van der Waals repulsive term RT/(v�b),modifying only the denominator in the attractive term. The Redlich-Kwong equation has been the most popular basis for developingnew EOS’s. Another trend has been to propose generalized three-,four-, and five-constant cubic equations that can be simplified to thePR EOS, RK EOS, or other familiar forms. Kumar and Starling16,17

use the most general five-constant cubic EOS to fit volumetric andphase behavior of nonpolar compounds, although they do not applythe equation to mixtures.

Most petroleum engineering applications rely on the PR EOS or amodification of the RK EOS. Several modified Redlich-Kwongequations have found acceptance, with Soave’s18 modification (SRKEOS) being the simplest and most widely used. Unfortunately theSRK EOS yields poor liquid densities. Zudkevitch and Joffe19 pro-posed a modified RK EOS, the ZJRK EOS, where both EOSconstants are corrected by temperature-dependent functions, result-ing in improved volumetric predictions. Yarborough11 proposed ageneralized form of the ZJRK EOS for petroleum reservoir mixtures.

The PR EOS is comparable with the SRK EOS in simplicity andform. Peng and Robinson7 report that their equation predicts liquiddensities better than the SRK EOS, although PR EOS densities areusually inferior to those calculated by the ZJRK EOS. A distinct ad-vantage of the Peng-Robinson and Soave-Redlich-Kwong equa-

2 PHASE BEHAVIOR

Fig. 4.1—p-V relation of a pure component at subcritical, critical,and supercritical temperatures.

tions, where a simple temperature-dependent correction is used forEOS constant A, is reproducibility. The ZJRK EOS’s rely on tablesor complex functions to represent the highly nonlinear correctionterms for EOS constants A and B.

Peneloux et al.’s20 volume-translation method modifies a two-constant cubic equation by introducing a third EOS constant, c,without changing the equilibrium calculations of the original two-constant equation. The volume-translation constant c eliminates theinherent volumetric deficiency suffered by all two-constant equa-tions, and, for practical purposes, volume translation makes anytwo-constant EOS as accurate as any three-constant equation.12-15

4.2.1 van der Waals5 Equation. van der Waals proposed the first cu-bic EOS in 1873. The van der Waals EOS gives a simple, qualitativelyaccurate relation between pressure, temperature, and molar volume.

p � RTv � b

� av 2

, (4.3). . . . . . . . . . . . . . . . . . . . . . . . . . . . .

where a�“attraction” parameter, b�“repulsion” parameter, andR�universal gas constant. Comparing this equation with the idealgas law, p�RT/v, we see that the van der Waals equation offers twoimportant improvements. First, the prediction of liquid behavior ismore accurate because volume approaches a limiting value, b, athigh pressures,

limp �

v � p� � b , (4.4). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

where b is sometimes referred to as the “covolume” (effective mo-lecular volume). The term RT/(v�b) dictates liquid behavior andphysically represents the repulsive component of pressure on a mo-lecular scale.

The van der Waals equation also improves the description of non-ideal gas behavior, where the term RT/(v�b) approximates idealgas behavior ( pRT/v) and the term a/v2 accounts for nonideal be-havior. The a/v2 term reduces system pressure and traditionally isinterpreted as the attractive component of pressure.

van der Waals also stated the critical criteria that are used to definethe two EOS constants a and b—namely, that the first and secondderivatives of pressure with respect to volume equal zero at the criti-cal point of a pure component.

��p�v�

pc,Tc,vc� ��2p

�v2�

pc,Tc,vc� 0. (4.5). . . . . . . . . . . . . . . . . .

Martin and Hou21 show that this constraint is equivalent to the condi-tion (Z�Zc)3�0 at the critical point. Fig. 4.1 shows the p-v relationof a pure compound for T�Tc , T�Tc, and T�Tc, indicating the in-flection point on the critical isotherm that represents the van derWaals critical criteria. Imposing Eq. 4.5 on Eq. 4.3 and specifying pcand Tc (as opposed to specifying two of the other critical properties),the constants a and b in the van der Waals equation are given by

a � 2764

R2 T 2c

pc

and b � 18

RTcpc

. (4.6). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The critical volume is given by vc�(3/8)(RTc/pc), resulting in aconstant critical compressibility factor.

Zc �pc vc

RTc� 3

8� 0.375. (4.7). . . . . . . . . . . . . . . . . . . . . . .

The van der Waals equation also can be written in terms of the Z fac-tor (Z�pv/RT ).

Z3 � (B � 1)Z2 � AZ � AB � 0 , (4.8). . . . . . . . . . . . . . .

where A � ap

(RT) 2 �2764

pr

T 2r

and B � bp

RT� 1

8pr

Tr. (4.9). . . . . . . . . . . . . . . . . . . . . . . . . . .

Abbott22 gives an interesting historical review of the van derWaals EOS, its strengths and weaknesses, and its analogy to othercubic EOS’s.

4.2.2 Redlich-Kwong6 Equations. The RK EOS is

p � RTv � b

� av (v � b)

(4.10). . . . . . . . . . . . . . . . . . . . . . . .

or, in terms of Z factor,

Z 3 � Z 2 � �A � B � B 2 �Z � AB � 0

and Zc � 1 3 , (4.11). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

with EOS constants defined as

a � �oa

R2T 2c

pc�(Tr), (4.12a). . . . . . . . . . . . . . . . . . . . . . . . . . .

where �oa � 0.42748;

b � �ob

RTcpc

, (4.12b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

where �ob � 0.08664;

A � ap

(RT)2 � �oa

pr

T2r�(Tr), (4.12c). . . . . . . . . . . . . . . . . . . .

where �(Tr) � T�0.5r ;

and B � bp

RT� �

ob

pr

Tr. (4.12d). . . . . . . . . . . . . . . . . . . . . . . .

The fugacity expression for pure components is

lnfp � ln � � Z � 1 � ln(Z � B) � A

Bln�1 � B

Z�.

(4.13). . . . . . . . . . . . . . . . . . . .


TABLE 4.1—BIP’s FOR THE PR EOS AND SRK EOS

PR EOS* SRK EOS**

N2 CO2 H2S N2 CO2 H2S

N2

CO2

H2S

C1

C2

C3

i-C4

C4

i-C5

C5

C6

C7 +

—

0.000

0.130

0.025

0.010

0.090

0.095

0.095

0.100

0.110

0.110

0.110

—

—

0.135

0.105

0.130

0.125

0.120

0.115

0.115

0.115

0.115

0.115

—

—

—

0.070

0.085

0.080

0.075

0.075

0.070

0.070

0.055

0.050‡

—

0.000

0.120†

0.020

0.060

0.080

0.080

0.080

0.080

0.080

0.080

0.080

—

—

0.120

0.120

0.150

0.150

0.150

0.150

0.150

0.150

0.150

0.150

—

—

—

0.080

0.070

0.070

0.060

0.060

0.060

0.060

0.050

0.030‡

*Nonhydrocarbon BIP’s from Nagy and Shirkovskiy.24 Use for both the original PR EOS (Ref. 7) and modified PR EOS (Ref. 25).**Nonhydrocarbon BIP’s from Reid et al.3

†Not reported by Reid et al.3

‡Should decrease gradually with increasing carbon number.

The cubic Z-factor equation can readily be solved with an analyti-cal or a trial-and-error approach.1,2 One or three real roots may ex-ist, where the smallest root (assuming that it is greater than B) is typi-cally chosen for liquids and the largest root is chosen for vapors. Themiddle root is always discarded as a nonphysical value. For mix-tures, the choice of lower or upper root is not known a priori and thecorrect root is chosen as the one with the lowest normalized Gibbsenergy, g*,23

g*y ��

N

i�1

yi ln fi�y�

and g*x ��

N

i�1

xi ln fi� x� , (4.14). . . . . . . . . . . . . . . . . . . . . . . . .

where yi and xi�mole fractions of vapor and liquid, respectively,and fi�multicomponent fugacity given (for a vapor phase) by

lnfi

yi p� ln �i �

Bi

B(Z � 1) � ln(Z � B)

� AB �Bi

B� 2

A�

N

j�1

yj Aij� ln�1 � BZ� . (4.15). . . . . . .

The traditional quadratic mixing rule is used for A, and a linear mix-ing rule is used for B. For a vapor phase with composition yi, theseare given by

A ��N

i�1

�N

j�1

yi yj Aij ,

B ��N

i�1

yi Bi ,

and Aij � �1 � kij� Ai Aj� , (4.16). . . . . . . . . . . . . . . . . . . . . . .

where kij�binary-interaction parameters (BIP’s), where kii�0 andkij�kji. Usually, kij�0 for most hydrocarbon/hydrocarbon (HC/HC) pairs, except perhaps C1/C7� pairs. Nonhydrocarbon/HC kijare usually nonzero, where kij0.1 to 0.15 for N2/HC and CO2/HCpairs (Table 4.1).3,24,25

Many students of the RK EOS have been intrigued by its simplic-ity, accuracy, and the pleasure of deriving its thermodynamic prop-erties. This has led to innumerable attempts to improve and extendthe original Redlich-Kwong equation. Certainly hundreds, if notthousands, of technical papers and theses have been written aboutthe RK EOS. With the advent of digital computers, this “craze” de-veloped into what Abbott10 called the Redlich-Kwong decade(1967–77). Abbott claims that the remarkable success of the RKEOS results from its excellent prediction of the second virial coeffi-cient (securing good performance at low densities) and reliable pre-dictions at high densities in the supercritical region. This latter ob-servation results from the compromise fit of densities in thenear-critical region; all components have a critical compressibilityfactor of Zc�1/3, where, in fact, Zc ranges from 0.29 for methaneto �0.2 for heavy C7� fractions. The Redlich-Kwong value ofZc�1/3 is reasonable for lighter hydrocarbons but is unsatisfactoryfor heavier components.

4.2.3 Soave-Redlich-Kwong. Several attempts have been made toimprove VLE predictions of the RK EOS by introducing a compo-nent-dependent correction term � for EOS constant A. Soave18 usedvapor pressures to determine the functional relation for the correc-tion factor used in Eq. 4.12,

� � �1 � m�1 � T 0.5r

�� 2

and m � 0.480 � 1.574�� 0.176� 2. (4.17). . . . . . . . . . . . .

Acentric factor � is defined in Chap. 5, and values for pure compo-nents can be found in Appendix A. Table 4.1 gives nonhydrocarbonBIP’s for the SRK EOS as recommended by Reid et al.3; kij�0 isgenerally recommended for HC/HC pairs.

The Soave-Redlich-Kwong equation is the most widely used RKEOS proposed to date even though it grossly overestimates liquidvolumes (and underestimates liquid densities) of petroleum mix-tures. The present use of the SRK EOS results from historical andpractical reasons. It offers an excellent predictive tool for systemsrequiring accurate predictions of VLE and vapor properties. Volumetranslation (discussed in Sec. 4.2.6) is highly recommended, if notmandatory, when liquid densities are needed from the EOS. ThePedersen et al.26,27 C7� characterization method is recommendedwhen the SRK EOS is used.

4 PHASE BEHAVIOR

Fig. 4.2—Temperature and component-dependent EOS terms for the ZJRK EOS (from Yarborough11).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Reduced Temperature

Reduced Temperature

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.20

0.30

0.35

0.40

0.25

0.45

�oa�(Tr,�) and�

oa�(Tr,�)

4.2.4 Zudkevitch-Joffe-Redlich-Kwong. Zudkevitch and Joffe19

proposed a novel procedure for improving the volumetric predic-tions of the RK EOS without sacrificing VLE capabilities of theoriginal equation. They suggest that the EOS constants A and Bshould be corrected as functions of temperature to match saturatedliquid densities and liquid fugacities. They show that vapor fugaci-ties and fugacity ratios (K values) remain essentially unaffected andthat their procedure does not greatly affect vapor densities. Shortlyafter the original modification appeared, Joffe et al.28 suggested thatvapor pressures should be used instead of liquid fugacities. This isthe approach used today in what is still referred to as the Zudke-vitch-Joffe modification, the ZJRK EOS.

Haman et al.29 proposed the correction terms � and � for EOSconstants A and B in equation form for pure paraffins. Yarborough11

proposed generalized � and � charts for petroleum reservoir fluidsthat include heavy petroleum fractions.

a � �oa

R2T 2c

pcT �0.5

r ��Tr ,��

and b � �ob

RTcpc

��Tr ,��. (4.18). . . . . . . . . . . . . . . . . . . . . . .

Unfortunately, the temperature-dependent functions are complexbecause they are represented by higher-order polynomials or cubicsplines (see Fig. 4.2). The behavior of these functions is highly non-linear near Tr�1, and a discontinuity is introduced by setting thecorrection factors ��1 at Tr�1. A single set of � and � correc-tions is not used in the industry, making reproducing results fromone version to another difficult. Preferably, a table of � and � correc-tion factors should be provided when reporting a fluid characteriza-tion based on a ZJRK EOS.

Two Redlich-Kwong modifications, the SRK EOS and ZJRKEOS, have found widespread application to petroleum reservoirfluids. The Soave equation is sometimes preferred because of its sim-

plicity and overall accuracy (particularly when used with volumetranslation). The ZJRK EOS is surprisingly accurate for both liquidand vapor property estimations, where its main disadvantage is thecomplexity of functions used to represent temperature-dependentcorrections for the EOS constants A and B.

4.2.5 Peng-Robinson.7 In 1976, Peng and Robinson proposed atwo-constant equation that created great expectations for improvedEOS predictions and improved liquid-density predictions in partic-ular. The PR EOS is given by

p � RTv � b

� av(v � b) � b(v � b)

(4.19). . . . . . . . . . . . . .

or, in terms of Z factor,

Z 3 � (1 � B)Z 2 � �A � 3B2 � 2B�Z

� �AB � B2 � B3� � 0

and Zc � 0.3074 . (4.20). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The EOS constants are given by

a � �oa

R2T 2c

pc�, (4.21a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

where �oa � 0.45724;

b � �ob

RTcpc

, (4.21b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

where �ob � 0.07780;

� � �1 � m�1 � Tr� �� 2

; (4.21c). . . . . . . . . . . . . . . . . . . . . .

and m � 0.37464 � 1.54226�� 0.26992�2 . (4.21d). . . . . .

In 1979, Robinson et al.25 and Robinson and Peng30 proposed amodified expression for m that is recommended for heavier compo-nents (��0.49).

m � 0.3796 � 1.485� � 0.1644�2 � 0.01667�3.

(4.22). . . . . . . . . . . . . . . . . . . .

Fugacity expressions are given by

lnfp�ln ��Z � 1 � ln(Z� B)

� A2 2� B

ln �Z � �1 � 2� �BZ � �1 � 2� �B

�and ln

fi

yi p� ln �i �

Bi

B(Z � 1) � ln(Z � B)

� A2 2� B�Bi

B� 2

A�

N

j�1

yj Aij� ln�Z � �1 � 2� �BZ � �1 � 2� �B

�,

(4.23). . . . . . . . . . . . . . . . . . . .

where traditional mixing rules (Eq. 4.16) are used in the derivationof the multicomponent fugacity expression.

The PR EOS does not calculate inferior VLE’s compared with theRK EOS equations, and the temperature-dependent correction termfor EOS constant A is very similar to the Soave correction. The larg-est improvement offered by the PR EOS is a universal critical com-pressibility factor of 0.307, which is somewhat lower than the Red-lich-Kwong value of one-third and closer to experimental values forheavier hydrocarbons. The difference between PR EOS and SRKEOS liquid volumetric predictions can be substantial, although, inmany cases, the error in oil densities is unacceptable from bothequations. Some evidence exists that the PR EOS underpredicts sat-


Fig. 4.3—p-V diagram of a pure component as calculated by a cu-bic EOS illustrating the van der Waals’s “loop” defining vaporpressure by the equal-area rule.

uration pressure of reservoir fluids compared with the SRK EOS,thereby requiring somewhat larger HC/HC (C1/C7�) BIP’s for thePR EOS.

In review, the Peng-Robinson and Soave-Redlich-Kwong equationsare the two most widely used cubic EOS’s. They provide the same ac-curacy for VLE predictions and satisfactory volumetric predictions forvapor and liquid phases when used with volume translation.

4.2.6 Volume Translation. In 1979, Martin8 proposed a new con-cept in cubic EOS’s, volume translation. His application was to easethe comparison of his proposed generalized EOS with previouslypublished equations. In an independent study, Peneloux et al.20 usedvolume translation to improve volumetric capabilities of the SRKEOS. Peneloux et al.’s key contribution was to show that the volumeshift does not affect equilibrium calculations for pure componentsor mixtures and therefore does not affect the original VLE capabili-ties of the SRK EOS. Volume translation works equally well withany two-constant EOS, as Jhaveri and Youngren31 show for thePeng-Robinson equation.

Volume translation solves the main problem with two-constantEOS’s, poor liquid volumetric predictions. A simple correction termis applied to the EOS-calculated molar volume.

v � vEOS � c, (4.24). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

where v�corrected molar volume, vEOS�EOS-calculated volume,and c�component-specific constant. The shift in volume is actuallyequivalent to adding a third constant to the EOS but is special be-cause equilibrium conditions are unaltered. This is readily seen fora pure component, where the van der Waals “loop” (Fig. 4.3) definesvapor pressure by making the areas above and below the p�pv lineon a p-v plot equal. Shifting the p-v plot to the left or right along thevolume axis does not change the equal-area (fugacity) balance, andit can be readily seen that vapor-pressure predictions are unalteredby introducing the volume-shift term c.

Peneloux et al.20 also show that multicomponent VLE is unal-tered by introducing the correction term as a mole-fraction average.

vL � v EOSL

��N

i�1

xi ci

and vv � v EOSv ��

N

i�1

yi ci , (4.25). . . . . . . . . . . . . . . . . . . . . . .

where vEOSL and vEOS

v �EOS-calculated liquid and vapor molar vol-umes, respectively; xi and yi�liquid and vapor compositions, re-spectively; and ci�component-dependent volume-shift parame-

TABLE 4.2—JHAVERI-YOUNGREN31

VOLUME-TRANSLATION CORRELATION FORC7� FRACTIONS WITH THE PR EOS

si � 1 � A0 MA1i

Hydrocarbon Family A0 A1

Paraffins

Naphthenes

Aromatics

2.258

3.004

2.516

0.1823

0.2324

0.2008

TABLE 4.3—VOLUME-TRANSLATION COEFFICIENTS(si�ci/bi) FOR PURE COMPOUNDS FOR THE

PR EOS AND SRK EOS

Component PR EOS SRK EOS

N2

CO2

H2S

C1

C2

C3

i-C4

n-C4

i-C5

n-C5

n-C6

n-C7

n-C8

n-C9

n-C10

�0.1927

�0.0817

�0.1288

�0.1595

�0.1134

�0.0863

�0.0844

�0.0675

�0.0608

�0.0390

�0.0080

0.0033

0.0314

0.0408

0.0655

�0.0079

0.0833

0.0466

0.0234

0.0605

0.0825

0.0830

0.0975

0.1022

0.1209

0.1467

0.1554

0.1794

0.1868

0.2080

ters. When the volume shift is introduced to the EOS for mixtures,the resulting expressions for fugacity are

� fvi�

modified� �fvi�original exp�� ci

pRT�

and � fLi�

modified� (fLi) original exp�� cip

RT� . (4.26). . . . . . . . . .

This implies that fugacity ratios are unaltered by the volume shift,

� fLi fvi�

modified� �fLi fvi�original . (4.27). . . . . . . . . . . . . . . . . .

Applications that require direct use of fugacity (e.g., compositional-gradient calculation and semisolid phase equilibrium) must includethe volume-translation coefficient in the fugacity expression. Also,the constant c can be temperature dependent but cannot includepressure or composition dependence without derivation of new fu-gacity expressions.

Peneloux et al. propose that ci be determined for each componentseparately by matching the saturated-liquid density at Tr�0.7. cican actually be determined by matching the EOS to any density val-ue at a specified pressure and temperature. Jhaveri and Youngren31

write ci as a ratio, si�ci/bi, suggesting the following equation forC7� fractions,

si � ci bi � 1 � A0 MA1i

. (4.28). . . . . . . . . . . . . . . . . . .

Table 4.2 gives A0 and A1 values, and Table 4.3 gives si values forselected pure components that have been determined by matching

6 PHASE BEHAVIOR

Fig. 4.4—Variation of volume-translation parameter si�ci /bi vs.molecular weight.

��

� C1 through C10paraffins fit at T�0.7

— Jhaveri-Youngrenfor paraffins

the saturated liquid density at Tr�0.7. Fig. 4.4 shows the variationof si with M.

Volume translation can be applied to any two-constant cubicequation, thereby eliminating the volumetric deficiency suffered byall two-constant equations. For practical purposes volume transla-tion makes any two-constant EOS as accurate as any three-constantequation12-15 (see Fig. 4.5).

�� !��

The isothermal two-phase flash calculation is the workhorse of mostEOS applications. The problem consists of defining the amounts andcompositions of equilibrium phases, usually liquid and vapor, giventhe pressure, temperature, and overall composition. An inherent ob-stacle to solving this problem is not knowing whether two equilibriumphases form at the specified pressure and temperature. The mixturemay exist as a single phase or may split into two or more phases.

The algorithms presented in this section assume that a mathemati-cal solution to the two-phase problem exists: either a solution yield-ing equilibrium phase compositions or a “trivial” solution. Evenwhen the results appear physically consistent, a rigorous check ofthe solution with the phase-stability test (discussed in Sec. 4.4) maybe required. Alternatively, defining the phase stability before a two-phase flash calculation is made improves the reliability of the flashresults but adds computations. Mathematically, the two-phase flashcalculation can be solved by either (1) satisfying the equal-fugacityand material-balance constraints with a successive-substitution orNewton-Raphson algorithm32,33 or (2) minimizing the mixtureGibbs free energy function.34

The first approach is used almost exclusively because it is readilyimplemented with one of several iterative algorithms. Gibbs energyminimization has received less attention, and it is unclear whetherit has any fundamental advantages over the simpler and more directequal-fugacity approach, at least for two-phase problems.

The usual constraint equations for solving the two-phase flashproblem are equal fugacities and a component/phase material bal-ance. Assuming that all other forces are negligible (e.g., gravity),the criterion of thermodynamic equilibrium is that the chemical po-tential of Component i in Phase 1 equals the chemical potential ofComponent i in Phase 2; this is true for all Components i�1, . . . , N(and all phases). Fugacity, fi, is a useful expression for the chemicalpotential, �i, where �i�RT ln fi��i(T), and the equal-chemical-po-tential constraint can be written as

fLi � fvi , i � 1, . . . , N. (4.29). . . . . . . . . . . . . . . . . . . . . . .

This constraint can be solved numerically by use of some measureof convergence, such as

Fig. 4.5—Comparison of measured and EOS-calculated satu-rated-liquid densities of the binary system C1/C10 systems at100°F; SW�Schmidt-Wenzel.14.

Methane Concentration, mol%

�N

i�1

�fLi

fvi� 1�

2

� � , (4.30). . . . . . . . . . . . . . . . . . . . . . . . . . .

where � is a convergence tolerance (e.g., 1�10�13).

4.3.1 Two-Phase Split Calculation (Rachford-Rice35 Procedure).The component and phase material-balance constraints state that n to-tal moles of feed with Composition zi distribute into nv moles of vaporwith Composition yi and nL moles of liquid with Composition xi with-out loss of matter or chemical alteration of the component species.The material-balance constraints can be written as

n � nv � nL

and nzi � nv yi � nL xi , i � 1, . . . , N. (4.31). . . . . . . . . . . . . . .

Introducing the vapor mole fraction Fv�nv/(nL�nv), Eq. 4.31 canbe written as

zi � Fv yi � (1 � Fv)xi . (4.32). . . . . . . . . . . . . . . . . . . . . . .

Additionally, the mole fractions of equilibrium phases and the over-all mixture must sum to unity.

�N

i�1

yi ��N

i�1

xi ��N

i�1

zi � 1. (4.33). . . . . . . . . . . . . . . . . . . .

This constraint can be expressed as

�N

i�1

�yi � xi� � 0. (4.34). . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Introducing the equilibrium ratio Ki,

Ki � yi xi , (4.35). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

the number of unknowns can be reduced from 2N�1 (yi, xi, and Fv)to N�1 (Ki and Fv). By use of the component material balance (Eq.4.31) and by replacing yi by Ki xi, Eq. 4.34 can be solved in termsof a single variable Fv .

h(Fv) ��N

i�1

�yi � xi� ��

N

i�1

zi (Ki � 1)1 � Fv (Ki � 1)

� 0. (4.36). . .

Eq. 4.36 is usually referred to as the Rachford-Rice35 equation. Fig.4.6 shows the function h(Fv) for a five-component mixture.

With feed composition and K values known, the only remainingunknown is Fv. h(Fv) has asymptotes at Fv�1/(1�Ki), where eachK value gives an asymptote.36,37 Mathematically, it can be shownthat the only physically meaningful solution of h(Fv)—i.e., where


Fig. 4.6—Rachford-Rice35 function h(FV) for a five-componentmixture (from Ref. 37).

all Compositions xi and yi are positive—lies in the regionFvmin�Fv�Fvmax, where

Fv min �1

1 � Kmax

and Fv max �1

1 � Kmin. (4.37). . . . . . . . . . . . . . . . . . . . . . . . .

It can be shown that Fvmin�0 and Fvmax�1 if at least one K valueis �1 and one K value is �1. This implies that the solution forh(Fv)�0 should always be limited to the region Fvmin�Fv�Fvmax.

Because h(Fv) is monotonic and the derivative h�(Fv)�dh/dFvcan be expressed analytically, the Newton-Raphson algorithm iscommonly used to solve for Fv .

Fn�1v � Fn

v �h�Fn

v�

h��Fnv�

h�(Fv) �dhdFv

� ��N

i�1

zi(Ki � 1)2

�Fv (Ki � 1) � 1�2 , (4.38). . . .

where n�iteration counter. The first guess for Fv can be chosen ar-bitrarily as 0.5.

In 1949, Muskat and McDowell38 proposed a solution to the two-phase split calculation that is basically the same as the one proposedby Rachford and Rice35 but numerically more efficient. Introducingthe quantity ci�1/(Ki�1), where ci�� for Ki�1, Muskat andMcDowell proposed the following form of the function h(Fv).

h(Fv) ��N

i�1

zi

Fv � ci� 0, (4.39). . . . . . . . . . . . . . . . . . . . . .

where dhdFv

� ��N

i�1

zi

�Fv � ci�2 . (4.40). . . . . . . . . . . . . . . . .

If a Newton estimate from Eq. 4.38 with either the Muskat-McDowell or Rachford-Rice equations for h gives an estimate of Fvoutside the range Fvmin�Fv�Fvmax, the Newton method should bereplaced by interval bisection or modified regula falsi until conver-gence is achieved. Severe round-off errors may cause any solutiontechnique to fail when both K and z of one component are very small(e.g., KN�1�10�12 and zN�1�10�20).*

*Personal communication with A.A. Zick, Zick Technologies Inc., Portland, Oregon (1991).

Phase compositions are calculated from the material-balanceequations

xi �zi

Fv (Ki � 1) � 1

and yi �zi Ki

Fv (Ki � 1) � 1� Ki xi . (4.41). . . . . . . . . . . . . . . .

4.3.2 EOS Two-Phase Flash Algorithm. The flash calculation isinitialized by estimating a set of K values; the Wilson39 equation iscommonly used.

Ki �exp�5.37�1 � �i

��1 � T�1r i��

pr i. (4.42). . . . . . . . . . . . .

K values from this equation are not accurate at high pressures, whichpotentially cause the two-phase flash to converge incorrectly to atrivial solution. Results from a phase-stability test provide the mostreliable K-value estimates for initializing the two-phase flash but arerelatively expensive to obtain. Reliable K-value estimates can betaken from a converged flash of the same mixture or a “related” mix-ture at a pressure and temperature not too far removed from theconditions of the present flash calculation. For example, in simulat-ing a depletion experiment with an EOS, the K values at the satura-tion pressure can be used as initial estimates for the flash at the firstdepletion stage, the converged K values from this flash can be usedfor the flash at the second stage, and so on at lower pressures.

With estimated K values, the Rachford-Rice35 equation is solvedfor Fv, with the search for Fv always bounded by Fvmin and Fvmax.

Fv min �1

1 � Kmax� 0

and Fv max �1

1 � Kmin� 1. (4.43). . . . . . . . . . . . . . . . . . . . . .

Phase compositions are calculated from the material-balanceequations. Having calculated xi and yi, phase Z factors ZL and Zv andcomponent fugacities fLi and fvi are calculated with the EOS.

ZL � FEOS� x, p, T�

and Zv � FEOS� y, p, T� (4.44). . . . . . . . . . . . . . . . . . . . . . . . . . .

and fLi � FEOS� x, ZL, p, T�

and fvi � FEOS� y, Zv, p, T�. (4.45). . . . . . . . . . . . . . . . . . . . . . .

The “normalized” Gibbs energy function, g*, of each phase is calcu-lated from

g*L ��

N

i�1

xi ln fLi

and g*v ��

N

i�1

yi ln fvi , (4.46). . . . . . . . . . . . . . . . . . . . . . . . . . .

and the normalized mixture Gibbs energy is given by

g*mix � Fv g*

v � (1 � Fv)g*L . (4.47). . . . . . . . . . . . . . . . . . . .

If multiple Z-factor roots are found for either phase, the root with thelowest Gibbs energy should be chosen.23 For example, if three liq-uid Z-factor roots were calculated (ZL1, ZL2, and ZL3 ), the middleroot, ZL2, is automatically discarded and the two Gibbs energy func-tions, g*

L1 and g*L3, are calculated; fL1i are calculated with ZL1, and

fL3i are calculated with ZL3 . If g*L3 � g*

L1, ZL3 should be chosen;otherwise, choose ZL1 forg*

L1 � g*L3 .

Zick* suggests that this method of choosing the Z-factor root is notfail-safe because, at early iterations in the flash calculation, the incor-


8 PHASE BEHAVIOR

TABLE 4.4—SEQUENCE OF FLASH CALCULATIONS TOENSURE CORRECT SOLUTION WITH MULTIPLE ROOTS

Possible Orderof Multiple Flash

Liquid ZLRoot Chosen

Vapor, ZvRoot Chosen

Calculations Smallest Largest Smallest Largest

1 � �

2 � �

3 � �

4 � �

rect root may have a lower Gibbs energy than the correct root. He pro-poses that the flash calculation be converged completely with a con-sistent choice of roots (e.g., the smallest root always chosen for theliquid phase and the largest root always chosen for the vapor phase).If multiple roots in either phase are detected during this flash calcula-tion, a second, third, and potentially fourth flash calculation must becompleted, as summarized in Table 4.4. The two-phase solution withthe lowest mixture Gibbs energy is chosen as the correct solution.

With fugacities calculated for each phase, the fugacity constraint(Eq. 4.30) is checked. The recommended convergence tolerance is1�10�13, although a less stringent value can be used in some ap-plications. If convergence is not achieved, the K values can be modi-fied with successive substitution.

K(n�1)i � K(n)

i

f (n)Li

f (n)vi

, (4.48). . . . . . . . . . . . . . . . . . . . . . . . . . . .

where the superscripts (n) and (n�1) indicate the iteration level.With new K values, the Rachford-Rice35 equation is solved again(with new values of Fvmin and Fvmax), phase compositions are calcu-lated with the converged Fv value, phase Z factors and componentfugacities are calculated from the EOS, and the fugacity constraintis rechecked. This iterative procedure is repeated until convergenceis achieved. Three types of converged solutions can be obtained.

1. A physically acceptable solution is found with 0�Fv�1,where Fv�0 corresponds to a bubblepoint condition, Fv�1 corre-sponds to a dewpoint condition, and 0�Fv�1 indicates a two-phase condition.

2. A physically unacceptable solution is found with Fv�0 orFv�1,37 where the calculated equilibrium compositions satisfy theequal-fugacity constraint and the mathematical material-balanceequation. This solution indicates that the mixture is thermodynami-cally stable as a single phase and will not split into two phases. Forthis solution, the calculated equilibrium compositions would co-exist in thermodynamic equilibrium at the given pressure and tem-perature if they were mixed together in a physically meaningful pro-portion (creating, of course, a different mixture composition).

3. A so-called trivial solution is found where the calculated phasecompositions are identical to the mixture composition and K valuesequal one (xi�yi�zi and Ki�1).

The first solution is usually a “correct” solution. However, if a poten-tial three-phase solution exists, the two-phase solution may representonly a local minimum in the mixture Gibbs energy surface and the mix-ture Gibbs energy may be reduced further by locating the three-phasesolution or another two-phase solution. Michelsen32 suggests that thisproblem is best dealt with by use of phase-stability analysis.

Whitson and Michelsen37 refer to the second solution to the flashas a “negative” flash because one of the phase mole fractions is neg-ative (and the other phase fraction is �1). Although this conditionis physically unacceptable, the solution still has practical applica-tion. For example, phase properties and compositions are continu-ous across the phase boundaries. Also, a nontrivial negative flashsolution indicates phase stability with the same certainty as thephase-stability test, although the negative flash calculation requiresbetter initial K-value estimates than does the phase-stability test.

A trivial solution to the flash calculation should always bechecked with the phase-stability test to verify that the mixture is infact single phase. Trivial solutions arise for several reasons, the most

Fig. 4.7—p-T phase envelope and envelopes indicating the limitof a nontrivial negative flash and a nontrival stability test for thebinary C2/n-C4 system (from Ref. 37).

common being poor initial K-value estimates (e.g., from the Wil-son39 equation). A “valid” trivial solution occurs when two-phasesolutions do not exist. This occurs outside the p-T envelope thatWhitson and Michelsen define as the convergence-pressure enve-lope, where Fv �� in the negative flash (Fig. 4.7). Along thephase boundary and near a critical point, the Newton-Raphson flashtechnique tends to converge to a trivial solution more readily thando successive-substitution methods. Finally, as Michelsen23 shows,the two-phase flash never converges to a trivial solution withsuccessive substitution under the following conditions.

1. The phase-stability test indicates that the mixture is unstable.2. The K values resulting from the stability test are used to initial-

ize the flash calculation.3. The mixture Gibbs energy g*1

mix at the first iteration is less thanthe mixture Gibbs energy g*

z.The flash calculation initialized by a successful phase-stability

test is the safest solution method available, albeit more expensivethan a direct two-phase flash calculation.

Successive substitution is the safest solution technique for thetwo-phase flash problem, but it becomes slow when fugacity coeffi-cients are strongly composition dependent. The method is particu-larly slow near phase boundaries and critical points, where manythousands of iterations may be required to reduce the convergencecriterion to an acceptable value. Successive substitution can be ac-celerated with one of several methods as described in Refs. 33 and40 through 43 among others. Michelsen32 recommends the generaldominant eigenvalue method44 (GDEM); he shows that this methodis particularly well suited for the two-phase flash problem becausetwo dominant eigenvalues are found near phase boundaries and thecritical point. He recommends preceding each GDEM promotion(acceleration) with five successive-substitution iterations, wherethe GDEM K-value correction is given by

ln K(n�1)i � ln K(n)

i ��u(n)

i� �2�u(n�1)

i

1 � �1 � �2, (4.49). . . . . . . . .

where �ui � ln � fLi fvi�

and �1 � �b02 b12 � b01 b22� �b11 b22 � b12 b12

� , (4.50a). . . . . . .

�2 � �b01 b12 � b02 b11� �b11 b22 � b12 b12

� , (4.50b). . . . .

and bjk ��N

i�1

�u�n�j �i

�u(n�k)i . (4.50c). . . . . . . . . . . . . . . . . . . .


�1 and �2 are coefficients reflecting the relative magnitudes ofdominant eigenvalues �1 and �2. Michelsen suggests that promo-tions be rejected (or reduced) if the mixture Gibbs energy increasesafter a promotion.

Zick* shows that the coefficients �1 and �2 calculated with Eqs.4.50a and 4.50b can be seriously affected by round-off error. Hesuggests that the substitution �jk�(bjk�b12)/b12 eliminates theround-off problem and that this transformation of variables resultsin promotion coefficients �1 and �2 that can be used even near a crit-ical point. Also, the Michelsen32 suggestion to switch to a Newton-Raphson method after two GDEM iterations is unnecessary with themodified GDEM coefficients. For most practical reservoir applica-tions, GDEM will converge in two to three promotions (11 to 16 it-erations), with near-critical problems requiring up to six promotions(31 iterations).

In summary, the two-phase flash calculation can be outlined withthe following step-by-step procedure.

1. Estimate K values.2. Calculate Kmin and Kmax.3. Solve the Rachford-Rice phase-split calculation (Eq. 4.36)

for Fv, limited between Fvmin and Fvmax (Eq. 4.43).4. Calculate phase compositions x and y (Eq. 4.41).5. Calculate phase Z factors ZL and Zv from the EOS.6. Calculate component fugacities fLi and fvi from the EOS.7. Calculate phase Gibbs energy functions g*

L and g*v (Eq. 4.46),

determine the correct Z-factor roots of each phase (if multiple rootsexist), and calculate the mixture Gibbs energy (Eq. 4.47).

8. Check the equal-fugacity constraint (Eq. 4.30).9. (a) If convergence is reached, stop. (b) If convergence is not

reached, update the K values with the fugacity ratios (Eq. 4.48) ora GDEM promotion (Eq. 4.49); alternatively, use another accelera-tion technique or a Newton-Raphson K-value update.

10. Check for convergence at a trivial solution (Ki �1) with thecondition

�N

i�1

�ln Ki�2 � 10�4. (4.51). . . . . . . . . . . . . . . . . . . . . . . . . .

11. If a trivial solution is not detected, return to Step 2. Otherwise,confirm the trivial solution with a stability test.

For reservoir simulation, a Newton-Raphson solution to the flashproblem can be used because initial K-value estimates (from earliertimesteps and neighboring gridblocks) should be reliable, and the re-duced computation time of a Newton method compared with an accel-erated successive-substitution method can be significant.45 Michel-sen’s23 implementation of the Newton-Raphson method is considereda very efficient algorithm and is cited here directly from his originalpublication (with the exception of some Nomenclature changes).

“The set of equations to be solved is

ei(K) � �ln�nvi nv� � ln�vi� � �ln�nLi nL

� � ln�Li� � 0,

(4.52). . . . . . . . . . . . . . . . . . . .

where nvi and nLi�number of moles of Component i in the vaporand liquid phases, respectively.

“The Jacobian matrix is given by

Jij ��ei

� ln Kj, (4.53). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

and the correction � with �i��ln Ki is found from

J� � � e. (4.54). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

“The Jacobian matrix is calculated from

Jij ��ei

� ln Kj��

N

k�1

�ei

�nL k

�nL k

� ln Kj, (4.55). . . . . . . . . . . . . . . . .


yielding

J � BA�1, (4.56). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

with Bij �zi

xi yi�ij � 1 �

nvnL

nv � nL�� ln�i

�nj�

v

�� ln�i

�nj�

L

�(4.57). . . . . . . . . . . . . . . . . . . .

and Aij �zi

x i yi�ij � 1. (4.58). . . . . . . . . . . . . . . . . . . . . . . . . .

“Since B is symmetric, we can use the decomposition

B � LDLT, (4.59). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

where L is unit lower triangular and D is diagonal with positive ele-ments for a positive definite B.

“Then,

� � � AL�TD�1L�1e, (4.60). . . . . . . . . . . . . . . . . . . . . . . .

where the cost of the decomposition and the subsequent backsub-stitution is only about half of that required for conventional solutionof Eq. 4.54 by Gaussian elimination.” Application of the Michael-sen Newton-Raphson algorithm, as proposed here and withoutproper precautions, will lead to convergence problems near phaseboundaries because both matrices become singular at phase bound-aries and the solution will be severely affected by round-off errors.

�� "

One of the most difficult aspects of making VLE calculations withan EOS is knowing whether a mixture will actually split into two (ormore) phases at the specified pressure and temperature. Traditional-ly, this problem has been solved either by conducting a two-phaseflash or by making a saturation-pressure calculation; both methodsare expensive and not entirely reliable.

In 1982, two papers32,46 showed how the Gibbs tangent-plane cri-terion could be used to establish the thermodynamic stability of aphase [i.e., whether a given composition has a lower energy remain-ing as a single phase (stable) or whether the mixture Gibbs energy willdecrease by splitting the mixture into two or more phases (unstable)].Ref. 46 shows graphically how the Gibbs tangent-plane criterion isused to establish phase stability of simple binary systems, and Ref. 32gives an algorithm to establish phase stability numerically. This sec-tion on phase stability follows these references closely.

Phase stability deals with the question of whether a mixture canattain a lower energy by splitting into two or more phases. The Gibbsenergy for n moles of mixture Composition zi considered as a homo-geneous phase is given by

Gz ��N

i�1

�ni�i�z

� n �N

i�1

zi�zi. (4.61). . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The mixture will split into two phases y and x if the mixture Gibbsenergy, Gmix, is less than Gz, where Gmix is given by

Gmix ��N

i�1

�ni�i�v � �ni�i�L ;�Li � �vi � �i

��N

i�1

�nvi � nLi��i

��N

i�1

n�Fvyi � (1 � Fv)xi��i . (4.62). . . . . . . . . . . . . .

10 PHASE BEHAVIOR

Fig. 4.8—Gibbs energy surface for a binary system.

The Gibbs tangent-plane criterion considers the energy surfacefor a homogenous phase. In terms of overall mole fractions zi�ni/nwith fugacities evaluated for z, the normalized Gibbs energy func-tion, g* � G RT, is given by

g*z ��

N

i�1

zi ln fi (z) . (4.63). . . . . . . . . . . . . . . . . . . . . . . . . . .

g*z is a normalized Gibbs energy for the mixture composition. For

a binary mixture, the energy surface g* represents a curve that canbe plotted vs. one of the mole fractions (Fig. 4.8). For a ternary sys-tem, the energy surface can be plotted in three dimensions (g* vs.two of the mole fractions), but a graphical representation is not pos-sible for systems with more than three components.

Graphically, the condition of equilibrium for a binary system is es-tablished on a g* plot by drawing a straight line that is tangent to the

TABLE 4.5—PHASES IN EACH PRESSURE INTERVAL

Region Phases Present

I Only single-phase vapor, V

II Single-phase liquid, L1

Two-phase vapor/liquid, V/L1

Single-phase vapor, V

III Single-phase liquid, L1

Three-phase vapor/liquid/liquid, V/L1/L2


IV Single-phase liquid, L1

Liquid/liquid, L1/L2

Single-phase liquid, L2

Vapor/liquid, V/L2


V Single-phase liquid, L1

Liquid/liquid, L1/L2

Single-phase liquid, L2

Fig. 4.9—p-x plot of a two-component mixture exhibiting varioustwo- and three-phase equilibrium conditions (Ref. 46).

Mole Fraction Component 1

curve at two (or more) compositions. A valid tangent plane cannot in-tersect the Gibbs energy surface anywhere except at the points of tan-gency. For example, the vapor/liquid tangent passes through the twopoints �x, g*

L� and �y, g*v� in Fig. 4.8. The compositions through which

the tangent passes are equilibrium phases that satisfy the equal-fugac-ity condition. A physically acceptable two-phase solution requiresthat the mixture composition lie between the two equilibrium com-positions, x�z�y. If z lies outside the compositions bounded by xand y (z�x or z�y), the material-balance constraint is violated andthe mixture is stable. Likewise, z�y and z�x indicate stable condi-tions for a mixture at its dewpoint and bubblepoint, respectively.

When the overall composition z lies between the equilibrium com-positions (x�z�y), the mixture is unstable and will split into the twoequilibrium phases with compositions y and x, having a mixtureGibbs energy given by g*

mix � Fvg*v � (1 � Fv)g*

L. with g*mix

� g*z. The value of g*

mix is read directly from the tangent line at themixture composition, and the vapor mole fraction Fv is given by thedistance from z to y, relative to the total distance between x and y�Fv � (z � y) (x � y)�.

Baker et al.46 discuss the mathematical conditions associatedwith the Gibbs tangent-plane criterion and illustrate the techniquefor a binary system that exhibits two- and three-phase behavior atvarious pressures and a fixed temperature. Fig. 4.9, a p-x diagramdivided into five pressure intervals, is adapted from their example.Depending on the mixture composition, various combinations ofthe three potential phases [vapor (V), lower liquid (L1), and upperliquid (L2)] can form in each pressure interval. Table 4.5 shows thephases for each interval.

Figs. 4.10A through 4.10G and 4.11A through 4.11F presentGibbs energy plots for Regions II, III, and IV together with the p-xdiagram (Fig. 4.9). Fig. 4.10A shows the g* curve for a low pressurein Region II where only two “valleys” exist, and thereby only one tan-gent can be drawn. Equilibrium compositions are located at the twopoints where the tangent touches the g* curve, y and xL1, each ofwhich is near the bottom of a valley. Figs. 4.10B through 4.10D showthe g* curve for a higher pressure in Region II, where a middle valleydevelops between the two valleys exhibited in Fig. 4.10A. Only onevalid tangent can be drawn, between the L1 and V valleys. This tan-gent is valid because it does not pass through the g* curve at composi-tions other than the points of tangency, xL1 and y. Two other tangentscan be drawn, one yielding a liquid/liquid (L1/L2) solution betweenthe left and middle valleys and the other yielding a liquid/vapor(L2/V) solution between the middle and right valleys. These two tan-gents are, however, invalid because they lie above the g* curve inviolation of the tangent-plane criterion. Such tangents represent falsetwo-phase solutions that satisfy the equal-fugacity constraint but


Fig. 4.10A—Gibbs energy plot for the Baker et al.46 binary exam-ple, Region II.

yield only a local minimum in the mixture Gibbs energy. False two-phase solutions are difficult to detect unless one has a priori knowl-edge of the actual equilibrium condition. Low-temperatures and high-CO2 concentrations are conditions associated with three-phasebehavior that may be susceptible to false two-phase solutions.

Fig. 4.10E shows the g* curve for the three-phase pressure (RegionIII). A single line can be drawn that is tangent to three compositions( y, xL1, and xL2). The three-phase solution is physically valid for anycomposition lying between the lower liquid (xL1) and the vapor (y)compositions, with the relative amounts of each phase in a two-phasemixture being determined by the overall composition. For z�xL1 andz�y, the mixture is stable and remains as a single phase.

Fig. 4.10F shows the g* curve for a pressure in Region IV wherethe middle valley decreases relative to the left and right valleys. Thiscreates a curve that has two valid tangents, one representing a L1/L2solution and the other representing a L�2 V solution. Valid two-phasesolutions are found for mixture compositions in either the L1/L2 inter-val, xL1�z�xL2, or the L�2 V interval x�L2�z�y. Mixture composi-tions outside these two intervals will remain as a stable single phase.The tangent that can be drawn between a lower liquid and vapor phase(dashed line) is not a valid two-phase solution because the tangent liesabove the g* curve in the middle region of compositions (Fig. 4.10G).However, this is a potential two-phase solution that could readily becalculated and mistaken for a valid solution.

In Figs. 4.10A through 4.10G, the tangent-plane solutions thatpass through compositions where g* is convex have been ignored.This follows from the observation that any mixture compositionwith the condition ��2g* �z2� � 0 is intrinsically unstable,37 andany search for a solution to the tangent-plane criterion will moveaway from such “convex” solutions. Also, these tangents violate thetangent-plane criterion because they lie above the energy surface(see Fig. 4.11A).

Baker et al.’s46 graphical interpretation of stability analysis isparticularly useful for describing the Gibbs tangent-plane criterionbut does not lend itself to being implemented as a numerical algo-rithm that can be used to calculate phase stability. Michelsen32 pro-poses an algorithm that determines whether a mixture will remain

Fig. 4.10B—Gibbs energy plot for the Baker et al.46 binary exam-ple: Region II, correct two-phase V/L1 solution.

Developing Second LiquidPhase “Valley”

single phase or split into multiple phases. Michelsen’s algorithm issimilar to a flash calculation but is faster and safer (accurate K-valueestimates are not needed for the stability test).

The Michelsen stability test is based on locating “second-phase”compositions that have tangent planes parallel to the tangent planeof the mixture composition. If any of the parallel tangent planes liebelow the tangent plane of the mixture composition, the mixture isunstable and will split into at least two phases. If all compositionshaving parallel tangent planes lie above the mixture tangent planeor no composition has a parallel tangent plane, the mixture is stableas a single phase. In addition, if a composition (not equal to the mix-ture composition) lies on the same tangent plane as the mixture, themixture is at a bubble- or a dewpoint and the second phase is an in-cipient equilibrium phase. Figs. 4.11B through 4.11F graphically il-lustrate the Michelsen stability-test criteria for stable and unstablemixture compositions.

The mathematical description of Michelsen’s stability test is notwithin the scope of this monograph, but his stability-test algorithmfollows. Actually, two tests are usually required; one test assumes thatthe second phase is vapor-like, and the other assumes that the secondphase is liquid-like. This corresponds to initializing the search for asecond phase with two compositions where each search is conductedseparately. The compositions used to initialize each search shouldrepresent “poor” guesses (i.e., very vapor-like and very liquid-likecompositions) to expand the composition space being searched. Onecould conceivably use N stability tests (N�number of components),each starting with a pure component as the initial composition esti-mate, but this would be unnecessarily time consuming.

Michelsen shows that locating a second-phase composition witha tangent plane parallel to the tangent plane of the mixture composi-tion is equivalent to locating a composition y with component fuga-cities fyi equal to mixture component fugacities fzi times a constant,

fzi

fyi� S � I, (4.64). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12 PHASE BEHAVIOR

Fig. 4.10C—Gibbs energy plot for the Baker et al.46 binary exam-ple: Region II, false two-phase V/L2 solution.

False V/L2 Two-PhaseEquilibrium Condition

where I�constant. A successive-substitution algorithm, summa-rized in the following procedure, can readily be used to solve the Mi-chelsen stability test. Note that each test is conducted separately(e.g., converging the vapor-like search first, then converging the liq-uid-like search).

1. Calculate the mixture fugacities, fzi ; with multiple Z-factorroots, choose the root with the lowest g*.

2. Use the Wilson equation (Eq. 4.42) to estimate initial K values.

K1i �

exp�5.37(1 � �i)�1 � T�1ri��

pri. (4.42). . . . . . . . . . . . .

3. Calculate second-phase mole numbers, Yi, using the mixturecomposition zi and the present K-value estimates.

(Yi)v � zi (Ki)v

or (Yi)L � zi (Ki)L . (4.65). . . . . . . . . . . . . . . . . . . . . . . . . . . .

4. Sum the mole numbers.

Sv ��N

j�1

�Yj�v

or SL ��N

j�1

�Yj�

L . (4.66). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5. Normalize the second-phase mole numbers to get mole frac-tions, yi.

(yi)v �(Yi)v

�N

j�1

�Yj�v

�(Yi)v

Sv

Fig. 4.10D—Gibbs energy plot for the Baker et al.46 binary exam-ple: Region II, false two-phase L1/L2 solution.

False L1/LTwo-PhaseEquilibriumCondition

Fig. 4.10E—Gibbs energy plot for the Baker et al.46 binary exam-ple: Region III, correct three-phase solution.


Fig. 4.10F—Gibbs energy plot for the Baker et al.46 binary exam-ple: Region IV, two possible correct two-phase solutions (L1/L2or V/L2).

or (yi) L�(Yi)L

�N

j�1

�Yj�

L

�(Yi)L

SL. (4.67). . . . . . . . . . . . . . . . . . . . .

6. Calculate the second-phase fugacities ( fyi)v or ( fyi)L from theEOS; with multiple Z-factor roots (for a given phase), choose theroot with the lowest Gibbs energy g*.

7. Calculate the fugacity-ratio corrections for successive-sub-stitution update of the K values.

(Ri)v �fzi

�fyi�v

1Sv

or (Ri)L ��fyi�

L

fziSL . (4.68). . . . . . . . . . . . . . . . . . . . . . . . . . . .

8. Check whether convergence is achieved (e.g., ��1�10�12).

�N

i�1

(Ri � 1)2 � �. (4.69). . . . . . . . . . . . . . . . . . . . . . . . . . . .

9. If convergence is not obtained, update the K values.

K(n�1)i � K(n)

i R(n)i . (4.70). . . . . . . . . . . . . . . . . . . . . . . . . . . .

10. Check whether a trivial solution is being approached usingthe criterion

�N

i�1

�ln Ki�2

� 1 � 10�4. (4.71). . . . . . . . . . . . . . . . . . . . . .

11. If a trivial solution is not indicated, go to Step 3 for anotheriteration.

Fig. 4.10G—Gibbs energy plot for the Baker et al.46 binary exam-ple: Region IV, false two-phase V/L1 solution.

False V-L1 Two-PhaseEquilibrium Condition

Michelsen suggests that Step 9 of the successive substitution canbe accelerated with the GDEM approach with one eigenvalue (onlyone eigenvalue approaches 1 near the critical point in a stabilitytest). He recommends that four successive-substitution iterationsprecede each promotion. The GDEM update is given by

K(n�1)i � K(n)

i�R(n)

i��,

� � � b11

b11 � b01� ,

b01 ��N

i�1

ln R(n)i

ln R(n�1)i

,

and b11 ��N

i�1

ln R(n�1)i

ln R(n�1)i

, (4.72). . . . . . . . . . . . . . . . .

where the superscript (n) is the iteration counter.Table 4.6 summarizes the interpretation of the two-part stability

test. The mixture (very likely) is stable if both tests yield S�1, ifboth tests converge to a trivial solution, or if one test yields S�1 andthe other converges to a trivial solution. Theoretically, it is impossi-ble to establish without a doubt that a mixture is stable until all com-positions have been tested. However, both solutions indicating sta-bility from the two-part Michelsen test usually ensures that amixture is in fact single phase.

On the other hand, only one test indicating S�1 is sufficient todetermine that a mixture is definitely unstable. For an unstable solu-tion, the resulting K values from the stability test can be used to ini-tialize the two-phase flash. Potentially both SL and Sv are �1, inwhich case the best initial K values for the flash are given byKi�(yi)v/(yi)L�(Ki)v(Ki)L , requiring that both tests be completed(even though the first test positively indicates an unstable mixture).

Fig. 4.12 shows Nghiem and Li’s47 EOS calculations identifyingthe phase boundary of a reservoir oil. Also shown is the envelopewithin the phase boundary (dashed line) where one of the stability

14 PHASE BEHAVIOR

Fig. 4.11A—Gibbs energy plot for a hypothetical binary systemshowing a graphical interpretation of Michelsen’s32 phase-sta-bility test for region of compositions where stability test con-verges nontrivial.

Region of Compositions WhereStability Test Converges Nontrivial

tests converges to a trivial solution. The lower dashed line (startingfrom the critical point) shows where the liquid-like stability testconverges to a trivial solution, and the upper dashed curve showswhere the vapor-like stability test converges to a trivial solution. In-side the dashed-curve envelope, both the liquid- and vapor-like sta-bility tests converge to a nontrivial unstable solution (both SL and

Fig. 4.11C—Gibbs energy plot for a hypothetical binary systemshowing a graphical interpretation of Michelsen’s32 phase-sta-bility test for vapor-like feed, z, with one stable condition, y, lo-cated.

Fig. 4.11B—Gibbs energy plot for a hypothetical binary systemshowing a graphical interpretation of Michelsen’s32 phase-sta-bility test for liquid-like feed, z, with one unstable condition, y,located.

Sv are �0). Fig. 4.13 illustrates the behavior of SL and Sv vs. pres-sure at a fixed temperature for this system.

Michelsen’s phase-stability test has many applications; the fol-lowing summarizes the most important ones.

1. Determining whether a mixture composition is thermodynami-cally stable as a single phase. If the test indicates stability (assuming

Fig. 4.11D—Gibbs energy plot for a hypothetical binary systemshowing a graphical interpretation of Michelsen’s32 phase-sta-bility test for liquid-like feed, z, with two unstable conditions, yLand yv, located.


Fig. 4.11E—Gibbs energy plot for a hypothetical binary systemshowing a graphical interpretation of Michelsen’s32 phase-sta-bility test for vapor-like feed, z, with one unstable condition, yL ,located.

that both liquid- and vapor-like second phases have been tested), itis very likely that a two-phase solution does not exist.

2. With at least one unstable solution, initializing the two-phaseflash calculation with K values determined from the unstable solu-tion(s) of the stability test. This is particularly useful if K valuesfrom a converged flash at nearby conditions are not available.

3. Initializing and limiting the pressure range in a saturation-pres-sure calculation (see Sec. 4.5).

4. Checking the stability of a converged two-phase flash whenthree-phase behavior is suspected (e.g., for low-temperature andhigh-CO2 systems). This requires, however, two modifications of thestability test: (a) choice of appropriate initial K-value estimates for the“third”-phase search and (b) use of the converged two-phase fugaci-

TABLE 4.6—SUMMARY OF POSSIBLEPHASE-STABILITY-TEST RESULTS

Second Phase

Vapor-Like Liquid-LikeProbable

�Ki�v��yi�v

zi

�Ki�l�

zi

�yi�l

ProbableNumber of

Valleys on g*

Stable TS TS 1

SL�1 TS 2

TS SL�1 2

Sv�1 SL�1 3

Unstable Sv�1 TS 2

TS SL�1 2

Sv�1 SL�1 2

Sv�1 SL�1 3

Sv�1 SL�1 3

TS�trivial solution.

Fig. 4.11F—Gibbs energy plot for a hypothetical binary systemshowing a graphical interpretation of Michelsen’s32 phase-sta-bility test for liquid-like feed, z, with one unstable condition, yv,located.

ties, feqi�fvi�fLi, instead of fzi in the new search (i.e., locate a thirdcomposition y so that feqi/fyi equals a constant S, with S�1 indicatingstability; S�1 would indicate an unstable condition for the two-phasesolution, thereby guaranteeing a multiphase solution).

Fig. 4.12—Phase and stability-limit envelopes for a reservoir oil;stability limit represents the condition when one of the stabilitytests first converges to a trivial solution (from Nghiem and Li47).

16 PHASE BEHAVIOR

Fig. 4.13—Behavior of mole number sums from stability test, SLand Sv, vs. pressure for a fixed temperature; TS�trivial solution.

LIQUID-lIKE SECOND PHASEVAPOR-LIKE SECOND PHASE

TS TS

��# ��

For a mixture composition z at fixed temperature T, the saturation-pressure calculation involves finding the pressure(s) where the mix-ture is in equilibrium with an infinitesimal amount of an incipientphase. In terms of a two-phase flash, the saturation pressure definesa pressure where the vapor mole fraction, Fv, equals zero or one(Fv�0 at bubblepoint and Fv�1 at dewpoint).

One way to locate the saturation pressure of a mixture would beto make a 1D search in pressure for Fv�0 or 1, where the two-phaseflash is converged at each pressure estimate during the search. Al-though this approach would be safe, it also would be very slow. Sev-eral alternative saturation-pressure algorithms are available that areboth efficient and reliable when used with stability analysis.

The two conditions defining a saturation pressure are that the fu-gacities of all components are equal in both phases,

fzi � fyi , (4.73). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

and that the mole fractions of the incipient phase, y, equal unity,

�N

i�1

yi � 1. (4.74). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Expressing the incipient-phase mole fractions in terms of K values(yi�ziKi for a bubblepoint and yi�zi/Ki for a dewpoint), the tradi-tional equations used to solve bubble- and dewpoint calculations,respectively, are

1 ��N

i�1

zi Ki � 0 (4.75a). . . . . . . . . . . . . . . . . . . . . . . . . . . . .

and 1 ��N

i�1

zi Ki � 0. (4.75b). . . . . . . . . . . . . . . . . . . . . . . . .

In terms of stability analysis, the saturation-pressure conditioncorresponds to finding a second phase with a tangent plane that isparallel to the mixture composition’s tangent plane, with zero dis-tance between the two tangent planes. This is equivalent to the sumof incipient-phase mole numbers equaling unity.

�N

i�1

Yi � 1. (4.76). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Nghiem et al.48 use the condition of zero tangent-plane distance,dTP�0, to solve for saturation pressure, psat, and incipient-phasecomposition y.

dTP� psat , y� � ln��N

i�1

Yi� � 0. (4.77). . . . . . . . . . . . . . . . .

The recommended approach for determining saturation pressureis based on a slightly different approach proposed by Michelsen49;he uses the condition

Q� psat , y� � 1 ��N

i�1

zi��i (z) �i� y�� 0

� 1 ��N

i�1

yi� fzi fyi

�

� 1 ��N

i�1

Yi , (4.78). . . . . . . . . . . . . . . . . . . . . . .

where incipient-phase mole fractions are defined by

yi �Yi

�N

j�1

Yj

. (4.79). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

An efficient method to solve this equation uses a Newton-Raphsonupdate for pressure and accelerated successive substitution (GDEM)for composition. The following procedure outlines this approach.

1. Guess a saturation type: bubble- or dewpoint. An incorrectguess will not affect convergence, but the final K values may be “up-side down.”

2. Guess a pressure p*.3. Perform Michelsen’s stability test at p*.4. (a) If the mixture is stable for the current value of p*, this pres-

sure represents p* the upper bound of the search for a saturationpressure on the upper curve of the phase envelope. Return to Step1 and try a lower pressure to look for an unstable condition. (b)With an unstable condition at p*, this pressure represents the lowerbound in the search for a saturation pressure on the upper curve ofthe phase envelope.

5. Having found an unstable solution, use the K values from thestability test to calculate incipient-phase mole numbers at bubble-and dewpoint with Eqs. 4.80a and 4.80b, respectively.

Yi � zi Ki (4.80a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

and Yi � zi Ki . (4.80b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

If two unstable solutions were found in the stability test, use the Kvalues for the test with the largest mole number sum S.

At this point, the initialization is complete and the iteration se-quence begins.

6. Calculate the normalized incipient-phase compositions.

yi �Yi

�N

j�1

Yj

. (4.81). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7. Calculate phase Z factors, Zz and Zy, and component fugacities,fzi and fyi, from the EOS at the present saturation-pressure estimate.When multiple Z-factor roots are found for a given phase, the rootgiving the lowest Gibbs energy should be chosen.

8. Calculate fugacity-ratio corrections.

Ri �fzi

fyi��N

j�1

Yj��1

. (4.82). . . . . . . . . . . . . . . . . . . . . . . . .


9. Update incipient-phase mole numbers with the fugacity-ratiocorrections,

Y (n�1)i

� Y (n)i�R(n)

i��, (4.83). . . . . . . . . . . . . . . . . . . . . . . . . .

where four iterations use successive substitution (��1) followed bya GDEM promotion with � given by

� � � b11

b11 � b01� , (4.84). . . . . . . . . . . . . . . . . . . . . . . . . . . . .

where b01 ��N

i�1

ln R(n)i ln R(n�1)

i

and b11 ��N

i�1

ln R(n�1)i ln R(n�1)

i .

10. Calculate a new estimate of saturation pressure using a New-ton-Raphson update.

p(n�1)sat � p(n)

sat �Q(n)

��Q�p�

(n), (4.85). . . . . . . . . . . . . . . . . . . . . .

where�Q�p ��

N

i�1

Yi Ri ��fyi

�p1fyi

��fzi

�p1fzi� (4.86). . . . . . . . . . . . .

is evaluated at Iteration (n).If searching for an upper saturation pressure, the new pressure es-

timate must be higher than p*. If the new estimate is lower than p*,go to Step 1 and use a new initial-pressure estimate higher than thepresent p* value.

11. Check for convergence. Zick* suggests the following twocriteria.

�1 ��N

i�1

Yi � � 10�13

and ��Ni�1

ln(Ri)

ln�Yi zi��

2

� 10�8. (4.87). . . . . . . . . . . . . . . . . . . .

In addition, check for a trivial solution using the criterion

�N

i�1

�lnYizi�

2

� 10�4. (4.88). . . . . . . . . . . . . . . . . . . . . . . . . .

12. (a) If convergence is not achieved, return to Step 6. (b) If con-vergence is achieved, determine the saturation type by comparingthe mole fraction of the heaviest component in the mixture with thatin the incipient phase, where yN�zN indicates a bubblepoint withKi�yi/zi and yN�zN indicates a dewpoint where Ki�zi/yi, or bycomparing the density of the incipient phase with that of the feed.

This algorithm can be modified to search for both lower and up-per saturation pressures as well as saturation temperature at a spe-cified pressure.

Michelsen50 also gives an efficient procedure for calculating theentire phase envelope, including calculations through the criticalpoint. More recently, he presented an approximate phase-envelope al-gorithm51 that is up to 10 times faster than his original algorithm us-ing a trial-and-error solution directly for pressure and temperature(component fugacities do not need to be converged at each point onthe phase envelope). Surprisingly, the results are extremely close tofully converged saturation conditions and provide excellent startingestimates for a rigorous saturation-point calculation. He also showsthat the approximate solution is always inside the phase envelope,thus representing an unstable thermodynamic condition.


Michelsen52 also has proposed a critical-point calculation algo-rithm that is, surprisingly, as fast or faster than a two-phase flash cal-culation. The critical point is determined by a simple 2D search (intemperature and volume) with a function that requires only evalua-tion of the mixture fugacities.

��$ ��% �� &��'��" !��(

� %� �� &��

Gibbs53 was the first to give the formula for calculating composi-tional variation under the force of gravity for an isothermal system.The condition of equilibrium is satisfied by the constraint

�i� pref , zref , T� � �i�p, z, T� � Mi g�h � href

� ,

i � 1, 2, . . . , N, (4.89). . . . . . . . . . . . . . . . . . . . . . . . . . . . .

where �i�chemical potential of Component i, zref�homogeneous(single-phase) mixture at pressure pref at a reference depth href, andp�pressure and z�mixture composition at depth h. The entire sys-tem is at constant temperature (dT/dh�0).

In 1930, Muskat54 provided exact solutions to Eq. 4.89 for a sim-plified EOS (ideal mixing). Numerical examples based on this over-simplified EOS led to the misleading conclusion that gravity hasnegligible effect on compositional variation in reservoir systems. In1938, Sage and Lacey55 evaluated Eq. 4.89 using a more realisticEOS model. They provide examples showing significant variationsof composition with depth for reservoir mixtures. Furthermore, theymade the key observation that significant compositional variationsshould be expected in systems in the vicinity of a critical condition.

From 1938 to 1980, the petroleum literature is apparently void ofpublications regarding calculation of compositional gradients. Sever-al references during this period do, however, mention reservoirs ex-hibiting compositional variation. Schulte56 cites most of these refer-ences. He appears to be the first to solve Eq. 4.89 with a cubic EOS.His classic paper illustrates that significant compositional variationcan result from gravity segregation in petroleum reservoirs. Schultegives examples showing the effect of oil type (aromatic content) andinteraction coefficients (used in the mixing rules of a cubic EOS) oncompositional gradients. He also compares gradients calculated withthe Peng-Robinson and Soave-Redlich-Kwong equations.

In 1980, significant compositional gradients were reported in theBrent field, North Sea.56-58 In the Brent formation of the Brent field,a significant composition gradient was observed, with the transitionfrom gas to oil occurring at a saturated gas/oil contact (GOC). Thesepapers also describe the unusual transition from gas to oil in the ab-sence of a saturated GOC. This transition occurs at a depth where thereservoir fluid is a critical mixture, with a critical temperature equalto the reservoir temperature and a critical pressure less than the reser-voir pressure. Apparently, the Statfjord formation in the Brent fieldis an example of a reservoir with such an “undersaturated GOC.”

In 1983, Holt et al.59 presented a formulation of the compositional-gradient problem that includes thermal diffusion. Example calcula-tions in this paper were, unfortunately, limited to binary systems. Nu-merous publications on the subject of compositional gradient werepresented in 1984 and 1985.60,61 Most of these were field case histo-ries; in fact, a special session of the 1985 SPE Annual Technical Con-ference and Exhibition was dedicated to this subject.62-64

Hirschberg60 discusses the influence of asphaltenes on composi-tional grading. He uses a simplified two-component model with onecomponent representing asphaltenes and the other representing theremaining deasphalted oil. He makes the observation that composi-tional grading in heavier oils (o�0.85 or API�35°API) can bestrongly influenced by both the amount and the properties of asphal-tenes, which implies that quantitatively accurate estimates of com-positional grading resulting from asphaltenes are extremely diffi-cult because of the strong dependence of calculated results onphysical properties of the oil and asphaltene(s). Finally, Hirschbergdiscusses two mechanisms for the development of a tar mat.

Riemens et al.61 present an interesting evaluation of the composi-tional grading in the Birba field, Oman. On the basis of isothermalgravity/chemical equilibrium (GCE) calculations and field measure-

18 PHASE BEHAVIOR

ments of PVT data, they show that a significant compositional gradientexists. The authors also evaluate the possibility of injecting gas into theundersaturated oil zone where multicontact miscibility can develop.

Montel and Gouel65 suggest an algorithm for solving the isothermalGCE problem. The procedure is only approximate because it calculatespressure with an incremental hydrostatic term instead of solving direct-ly for pressure. They discuss the effect of fluid characterization on com-positional grading and the effect of reservoir temperature and pressure.Finally, the authors suggest that including thermal diffusion may im-prove the reliability of calculated compositional gradients (althoughthey do not include this effect in their study).

Metcalfe et al.63 report measured variation of composition andphysical properties of reservoir fluids in the Anschutz Ranch Eastfield in the U.S. Overthrust Belt. These authors use an EOS to char-acterize the PVT behavior of the entire range of fluids sampled fromthe reservoir. However, instead of calculating the compositionalvariation using gravity/chemical equilibrium and the developedEOS characterization, they correlate compositional variation graph-ically on the basis of measured data.

Creek and Schrader62 report compositional grading data for anoth-er Overthrust Belt reservoir, the East Painter field. Considerable dataare presented together with comparison of measured compositionalgradients and those calculated with the isothermal GCE model. Theyreport difficulty in matching observed saturation-pressure and GORgradients. Finally, the authors indicate that most reservoirs along theOverthrust Belt have varying degrees of compositional grading.

Belery and da Silva66 present a formulation describing the com-bined effects of gravity and thermal diffusion for a system of zero netmass flux. After assessing various approaches for treating thermaldiffusion, they selected the Dougherty and Drickamer67 method.Belery and da Silva extend this formulation (originally valid only forbinary systems) to multicomponent systems. They use a field exam-ple with EOS characterization and measured gradient data from theNorth Sea Ekofisk field to illustrate the gravity/thermal model. Be-cause measured PVT gradients were very scattered (probably be-cause of sampling problems), the comparison is not quantitatively ac-curate (with or without thermal diffusion). However, the calculationsshow qualitatively the effect of thermal diffusion and are the first suchcalculations reported for multicomponent systems.

Wheaton68 discusses an isothermal GCE model that includes theinfluence of capillary pressure. The addition of capillary forces wasapparently justified in an effort to assist in the initialization of reser-voir simulators. Simulators use capillary pressure curves to initializesaturation and pressure distributions discretely in the vertical direc-tion. Results of the calculated examples in Wheaton’s paper suggestthat neglecting compositional variations in a gas-condensate reser-voir may result in large errors in the initial hydrocarbons in place. Ob-viously, these results are primarily a consequence of neglecting thecompositional variation resulting from gravity/chemical equilibrium.Quantitatively similar results would have been obtained with or with-out the inclusion of capillary pressures. Finally, his observation thatneglecting compositional gradients will lead to incorrect specifica-tion of initial oil and gas in place is equally applicable to gas-conden-sate and oil reservoirs (i.e., practically any petroleum reservoir).

In his discussion of Wheaton’s paper, Chaback69 makes the ob-servation that nonisothermal effects can be on the same order ofmagnitude as gravity effects. More importantly, he notes that a non-isothermal system will never reach equilibrium (zero energy flux)even though a stationary (steady-state) condition of zero net massflux is reached.

Montel70 discusses compositional grading, including comments ontreating thermal diffusion. He provides an equation for calculating theRayleigh-Darcy number that is used to indicate whether a fluid/rocksystem will experience convection (mechanical instability).

Bedrikovetsky71 gives an extensive discussion and formal math-ematical treatment of compositional grading, including gravity,thermal, and capillary forces. The treatment yields complicated ex-pressions, which, in a few cases, are solved for simple conditions(idealized EOS and binary systems). Many of the results are similarto those given by Muskat.54 No examples are given for multicom-ponent mixtures with a realistic thermodynamic model.

Recently, Faissat et al.72 gave a theoretical review of equilibriumformulations that include gravity and thermal diffusion. Belery andda Silva66 mention most of the formulations, but Faissat et al. formal-ize the thermal-diffusion term in a generic way. Unfortunately, cal-culations are not provided for comparing the different formulations.

4.6.1 Isothermal GCE. Eq. 4.89 gives the condition for isothermalGCE, which is sometimes written in differential form asd�i � Mi gdh � 0, i � 1, 2, . . . , N . This condition represents Nequations. Together with the constraint that the sum of mole fractionsz(h) must add to one,

�N

i�1

zi(h) � 1, (4.90). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

it is possible to solve for composition z(h) and pressure p(h) at a spe-cified depth h.

Because chemical potential can be expressed as �i�RT�ln fi��(T ), Eq. 4.89 can be expressed in terms of fugacity.

ln fi�pref , zref , T� � ln fi

� p, z, T � � 1RT

Mi g�h � href� ,

i � 1, 2, . . . , N. (4.91). . . . . . . . . . . . . . . . . . . . . . . . .

For convenience, we define fi(h)�fi[p(h),z(h),T ] and fi(href)�fi�( pref,zref,T ), yielding

fi (h) � fi�href

� exp�� Mi g�h � href�

RT�,

i � 1, 2, . . . , N. (4.92). . . . . . . . . . . . . . . . . . . . . . . . .

The volume-translation method is widely used for correcting volu-metric deficiencies of the original Soave-Redlich-Kwong and Peng-Robinson equations. The method involves calculating a linearly trans-lated volume, v�, by adding a constant c to the molar volume, v,calculated from the original EOS, v��v�c. Peneloux et al.20 showthat the volume shift modifies the component fugacity asfi exp[ci( p/RT)] (see Eqs. 4.26 and 4.96). This correction must be in-cluded in the fugacity expressions used for gradient calculations andalso must be included in the pressure derivative of fugacity used in therecommended algorithm for solving the isothermal GCE problem.

On the basis of the Gibbs-Duhem equation,53 combining the me-chanical-equilibrium condition, dp dh�� g , with the GCE condi-tion, Eq. 4.89, guarantees automatic satisfaction of the condition

p(h) � p�href� � �

h

href

(h)gdh . (4.93). . . . . . . . . . . . . . . . . .

Interestingly, the isothermal GCE equations are still valid and satisfythis condition when a saturated GOC is located between href and h [i.e.,even when (h) is not a continuous function].

4.6.2 Isothermal GCE Algorithm. Eqs. 4.89 and 4.90 representequations similar to those used to calculate saturation pressure.Michelsen51 gives an efficient method for solving the saturation-pressure calculation, which has been modified here to solve theGCE problem,

Q� p, z� � 1 ��N

i�1

zi� f~i�pref, zref

� fi� p, z��

� 1 ��N

i�1

Yi , (4.94). . . . . . . . . . . . . . . . . . . . . . . . . .

where Yi � zi� f~i

�pref, zref� fi

� p, z�� (4.95). . . . . . . . . . . . . . . . . .

and f~

i�pref, zref

� � fi�pref, zref

� exp�� Mi g�h � href�

RT�.

(4.96). . . . . . . . . . . . . . . . . . . .


An efficient algorithm for solving Eq. 4.94 uses a Newton-Raphsonupdate for pressure and accelerated successive substitution(GDEM44) for composition. The following outlines this approach.

1. Calculate fugacities of the reference feed f~

i (yref, zref) and thegravity-corrected fugacity f

~

i (pref, zref) from Eqs. 4.26 and 4.96.This calculation needs to be made only once. Initial estimates ofcomposition and pressure at h are simply values at the referencedepth, z1(h) � zref and p1(h) � pref .

2. Calculate fugacities of the composition estimate z at the pres-sure estimate p. Calculate mole numbers from Eq. 4.95. Calculatefugacity-ratio corrections with

Ri �f~

i�pref , zref�fi�p, z�

��Nj�1

Yj��1

. (4.97). . . . . . . . . . . . . . . . . .

3. Update mole numbers using Eqs. 4.83 and 4.84.4. Calculate z(n�1)

i from Y(n�1)i

using

zi � Yi ��Nj�1

Yj�. (4.98). . . . . . . . . . . . . . . . . . . . . . . . . .

5. Update the pressure estimate using a Newton-Raphson estimate.

p(n�1) � p(n) �Q(n)

��Q �p�(n) , (4.99). . . . . . . . . . . . . . . . . . . .

where�Q�p ��

N

i�1

Yi Ri

��fi �p�

fi� p, z�

. (4.100). . . . . . . . . . . . . . . . . .

6. Check for convergence using Eq. 4.87.7. Iterate until convergence is achieved.After finding the composition z(h) and pressure p(h) that satisfy

Eqs. 4.89 and 4.90, a phase-stability test32 must be made to establishwhether the solution is valid. A valid solution is single phase (ther-modynamically stable). An unstable solution indicates that the cal-culated z and p will split into two (or more) phases, thereby makingthe solution invalid.

If the gradient solution is unstable, then the stability-test com-position y should be used to reinitialize the gradient calculation. Thestarting pressure for the new gradient calculation can be pref or, pre-ferably, the converged pressure from the gradient calculation thatled to the unstable solution. Note that unstable gradient solutionsusually occur only a short distance beyond a saturated GOC.

Locating a potential GOC requires a trial-and-error search. For asaturated GOC, three approaches might be considered: (1) stabilitytests, (2) negative flash calculations,37 or (3) saturation-pressurecalculations. The first and second methods should be the fastest,with the negative flash probably being faster because informationfrom previous flash calculations can be used for initialization ofsubsequent flash calculations.

Unfortunately, an algorithm based on either the stability test ornegative flash results may suffer from the fact that only trivial solu-tions exist over a large part of the reservoir thickness. On the otherhand, either method can be used efficiently to determine the satu-rated GOC once a nontrivial stability condition is found.

If an undersaturated GOC exists (i.e., a transition from gas to oilthrough a critical mixture), only a search based on saturation-pres-sure calculations can be used. The following algorithm is recom-mended for locating both saturated and undersaturated GOC’s.

First, calculate the composition and pressure at the top (zT and pT)and the bottom (zB and pB) of the reservoir; then, calculate satura-tion pressures psT and psB . If the saturation types (bubblepoint/dew-point) are the same at the top and bottom, then no GOC exists.Otherwise, a search for the GOC, hGOC, is made.

A straightforward search algorithm would be interval halvingbased on the saturation type. At Iteration n, a solution with a dew-point at depth h(n) would replace the top depth h(n�1)

T � h(n) for thenext iteration, and a solution with a bubblepoint at a given depthwould replace the bottom depth h(n�1)

B � h(n). The depth estimatefor a given iteration is calculated from h(n) � 0.5�h(n)

B � h(n)T�. The

number of iterations required to meet a tolerance �h would be1.5 ln�(hT � hB) �h� . For example, only 13 gradient and satura-tion-pressure calculations would be needed to achieve �h�0.33 ftfor a total thickness (hT � hB)�1,640 ft.

More efficient algorithms for locating the GOC can probably bedeveloped, particularly if a nontrivial stability solution can be lo-cated. Alternatively, Michelsen’s52 critical-point algorithm or hisnew method for calculating accurate approximations for saturationpressure and temperature51 may provide a good starting point fordeveloping an improved algorithm.

Whitson and Belery73 give a detailed discussion of composition-al-gradient calculations, including the application of isothermal andnonisothermal compositional-gradient algorithms to reservoir fluidsystems ranging from a saturated low-GOR black-oil/dry-gas sys-tem to a near-critical system.

��) *��+ �� *�� ,��

Most EOS characterizations (see Chap. 5) are not truly predictive74,75

because errors in saturation pressure are commonly �10%, those indensities are �5%, and compositions may be off by several mole per-cent for key components. Also, the EOS may predict a dewpoint in-correctly when the measured saturation condition is a bubblepoint, orvice versa. This lack of predictive capability by the EOS can be be-cause of insufficient compositional data for the C7� fractions, inac-curate properties for the C7� fractions, inadequate BIP’s, or incorrectoverall composition.

The EOS characterization can be improved in a number of ways.First, however, the experimental data and fluid compositions shouldbe checked for consistency (see Chap. 6). If the PVT data appearconsistent and the fluid compositions are considered representativeof the material that was analyzed in the PVT laboratory, modifyingthe parameters in the EOS to improve the fluid characterization willbe necessary. Refs. 26 and 74 through 79 present methods for modi-fying the cubic EOS to fit experimental PVT data. Most of thesemethods modify the properties of fractions making up the C7� (Tc,pc, �, or direct multipliers on the EOS constants �a and �b) andBIP’s kij between methane and C7� fractions. When an injectiongas containing significant amounts of nonhydrocarbons is beingstudied, the kij between nonhydrocarbon and C7� fractions mayalso be modified.

Some methods use nonlinear regression to modify the EOS pa-rameters automatically.74,78,79 Others have tried simply to makemanual adjustments to the EOS parameters through a trial-and-errorapproach.75,77,80 The trend is now to automate the EOS modifica-tion procedure with nonlinear regression, including large amountsof measured PVT and compositional data.81

Coats and Smart74 recommend five standard EOS modifications:�a and �b of methane; �a and �b of the heaviest C7� fraction; andkij between methane and the heaviest C7� fraction. Additional pa-rameters (nonhydrocarbon �a and �b and kij) are used for systemswith significant amounts of nonhydrocarbon components. Their ap-proach differs from other methods in that they do not use volumetranslation. As a result, significant methane corrections had to be ap-plied to EOS constants �a and �b . Using the Coats and Smart ap-proach with the PR EOS typically results in multipliers of the EOSconstants �a and �b ranging from 1.2 to 1.5 for methane and from0.6 to 0.8 for the heaviest C7� fraction; kij of the methane/C7�heavy fraction varies from 0 to 0.3. The � corrections can be inter-preted as modifications of the critical properties.75

With a somewhat untraditional regression approach, Coats andSmart minimize a sum of weighted absolute deviations using linearprogramming. They suggest weighting factors of 40 for saturationpressures, 10 for saturation densities, and 1 for most other data.Their results are impressive, showing excellent matches of near-critical fluids, hydrocarbon and nonhydrocarbon gas injection inoils and retrograde condensate systems, and simple depletion data.

With a two-constant cubic EOS with volume translation, the mod-ifications of EOS parameters (or critical properties) is typically only5 to 10% compared with the �30 to 40% modifications requiredwith the Coats and Smart approach without volume translation. Thisis explained by the initial predictions being much better with vol-

20 PHASE BEHAVIOR

ume translation, thereby requiring fewer modifications to achievethe same quality fit of measured data.

Interestingly, the same five standard regression parameters origi-nally suggested by Coats and Smart can be used with an EOS thatuses volume translation. However, the result is usually that methanecorrections to �a and �b remain close to 1.0 and corrections to �aand �b for the heaviest C7� fraction range from 0.9 to 1.1. There-fore, it may be better to drop methane corrections to �a and �b anduse instead one set of corrections to the �a and �b for the heaviestC7� fraction, and another set of corrections to the �a and �b for thenext-to-heaviest C7� fraction. This approach is particularly helpfulwhen matching liquid-dropout curves with a “tail” (see AppendixC) or in multicontact vaporization experiments.

Finally, an alternative to use of corrections to �a and �b directlywould be to modify Tc and pc instead (modification of � is not rec-ommended). Be aware, however, that the sensitivity of the mini-mization problem to Tc and pc is probably less than to �a and �b ,thereby making the mathematical search for a minimum more diffi-cult. Appendix C gives a thorough discussion of how nonlinear re-gression can be used to adjust EOS parameters systematically to fitmeasured PVT data.

-��

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57. Bath, P.G.H., Fowler, W.N., and Russell, M.P.M.: “The Brent Field, AReservoir Engineering Review,” paper EUR 164 presented at the 1980SPE European Offshore Petroleum Conference and Exhibition, Lon-don, 21–24 October.

58. Bath, P.G.H., van der Burgh, J., and Ypma, J.G.J.: “Enhanced Oil Re-covery in the North Sea,” Proc., 11th World Pet. Cong. (1983).

59. Holt, T., Lindeberg, E., and Ratkje, S.K.: “The Effect of Gravity andTemperature Gradients on Methane Distribution in Oil Reservoirs,” pa-per SPE 11761 available from SPE, Richardson, Texas (1983).

60. Hirschberg, A.: “Role of Asphaltenes in Compositional Grading of aReservoir’s Fluid Column,” JPT (January 1988) 89.

61. Riemens, W.G., Schulte, A.M., and de Jong, L.N.J.: “Birba Field PVTVariations Along the Hydrocarbon Column and Confirmatory FieldTests,” JPT (January 1988) 83.

62. Creek, J.L. and Schrader, M.L.: “East Painter Reservoir: An Exampleof a Compositional Gradient From a Gravitational Field,” paper SPE14411 presented at the 1985 SPE Annual Technical Conference and Ex-hibition, Las Vegas, 22–25 September.

63. Metcalfe, R.S., Vogel, J.L., and Morris, R.W.: “Compositional Gradientin the Anschutz Ranch East Field,” paper SPE 14412 presented at the1985 SPE Annual Technical Conference and Exhibition, Las Vegas,22–25 September.

64. Montel, F. and Gouel, P.L.: “A New Lumping Scheme of AnalyticalData for Compositional Studies,” paper SPE 13119 presented at the1984 SPE Annual Technical Conference and Exhibition, Houston,16–19 September.

65. Montel, F. and Gouel, P.L.: “Prediction of Compositional Grading ina Reservoir Fluid Column,” paper SPE 14410 presented at the 1985SPE Annual Technical Conference and Exhibition, Las Vegas, 22–25September.

66. Belery, P. and da Silva, F.V.: “Gravity and Thermal Diffusion in Hydro-carbon Reservoirs,” paper presented at the 1990 Chalk Research Pro-gram, Copenhagen, 11–12 June.

67. Dougherty, E.L. Jr. and Drickamer, H.G.: “Thermal Diffusion and Mo-lecular Motion in Liquids,” J. Phys. Chem. (1955) 59, 443.

68. Wheaton, R.J.: “Treatment of Variation of Composition With Depth inGas-Condensate Reservoirs,” SPERE (May 1991) 239.

69. Chaback, J.J.: “Discussion of Treatment of Variations of Composi-tion With Depth in Gas-Condensate Reservoirs,” SPERE (February1992) 157.

70. Montel, F.: “Phase Equilibria Needs for Petroleum Exploration andProduction Industry,” Fluid Phase Equilibria (1993) 84, 343.

71. Bedrikovetsky, P.G.: Mathematical Theory of Oil and Gas Recovery,Petroleum Engineering & Development Studies, Cluwer Academic,Horthreht, Russia (1993) 4.

72. Faissat, B. et al.: “Fundamental Statements about Thermal Diffusionfor a Multicomponent Mixture in a Porous Medium,” Fluid Phase Equi-libria (1995) 100, 1.

73. Whitson, C.H. and Belery, P.: “Compositional Gradients in PetroleumReservoirs,” paper SPE 28000 presented at the 1994 U. of Tulsa/SPECentennial Petroleum Engineering Symposium, Tulsa, Oklahoma,29–31 August.

74. Coats, K.H. and Smart, G.T.: “Application of a Regression-Based EOSPVT Program to Laboratory Data,” SPERE (May 1986) 277.

75. Whitson, C.H.: “Effect of C7� Properties on Equation-of-State Predic-tions,” SPEJ (December 1984) 685; Trans., AIME, 277.

76. Coats, K.H.: “Simulation of Gas Condensate Reservoir Performance,”JPT (October 1985) 1870.

77. Pedersen, K.S., Thomassen, P., and Fredenslund, A.: “On the Dangersof Tuning Equation of State Parameters,” paper SPE 14487 availablefrom SPE, Richardson, Texas (1985).

78. Agarwal, R., Li, Y.K., and Nghiem, L.X.: “A Regression TechniqueWith Dynamic-Parameter Selection for Phase Behavior Matching,”SPERE (February 1990) 115.

79. Søreide, I.: “Improved Phase Behavior Predictions of Petroleum Reser-voir Fluids From a Cubic Equation of State,” Dr.Ing. dissertion, Norwe-gian Inst. of Technology, Trondheim, Norway (1989).

80. Turek, E.A. et al.: “Phase Equilibria in CO2-Multicomponent Hydro-carbon Systems: Experimental Data and an Improved Prediction Tech-nique,” SPEJ (June 1984) 308.


�� *�� '�� !��

�� g/cm3

bar �1.0* E�05�Paft �3.048* E�01�m�F (�F�32)/1.8 ��C�F (�F�459.67)/1.8 �Kpsi �6.894 757 E�00�kPa


HEPTANES-PLUS CHARACTERIZATION 1

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Some phase-behavior applications require the use of an equation ofstate (EOS) to predict properties of petroleum reservoir fluids. Thecritical properties, acentric factor, molecular weight, and binary-in-teraction parameters (BIP’s) of components in a mixture are requiredfor EOS calculations. With existing chemical-separation techniques,we usually cannot identify the many hundreds and thousands of com-ponents found in reservoir fluids. Even if accurate separation werepossible, the critical properties and other EOS parameters of com-pounds heavier than approximately C20 would not be known accu-rately. Practically speaking, we resolve this problem by making anapproximate characterization of the heavier compounds with exper-imental and mathematical methods. The characterization of heptanes-plus (C7�) fractions can be grouped into three main tasks.1–3

1. Dividing the C7� fraction into a number of fractions withknown molar compositions.

2. Defining the molecular weight, specific gravity, and boilingpoint of each C7� fraction.

3. Estimating the critical properties and acentric factor of each C7�fraction and the key BIP’s for the specific EOS being used.

This chapter presents methods for performing these tasks andgives guidelines on when each method can be used. A unique char-acterization does not exist for a given reservoir fluid. For example,different component properties are required for different EOS’s;therefore, the engineer must determine the quality of a given charac-terization by testing the predictions of reservoir-fluid behavioragainst measured pressure/volume/temperature (PVT) data.

The amount of C7� typically found in reservoir fluids varies from�50 mol% for heavy oils to �1 mol% for light reservoir fluids.4

Average C7� properties also vary widely. For example, C7� molec-ular weight can vary from 110 to �300 and specific gravity from0.7 to 1.0. Because the C7� fraction is a mixture of many hundredsof paraffinic, naphthenic, aromatic, and other organic compounds,5

the C7� fraction cannot be resolved into its individual componentswith any precision. We must therefore resort to approximate de-scriptions of the C7� fraction.

Sec. 5.2 discusses experimental methods available for quantify-ing C7� into discrete fractions. True-boiling-point (TBP) distilla-tion provides the necessary data for complete C7� characterization,including mass and molar quantities, and the key inspection data foreach fraction (specific gravity, molecular weight, and boiling point).Gas chromatography (GC) is a less-expensive, time-saving alterna-tive to TBP distillation. However, GC analysis quantifies only themass of C7� fractions; such properties as specific gravity and boil-ing point are not provided by GC analysis.

Typically, the practicing engineer is faced with how to character-ize a C7� fraction when onlyzC7� the mole fraction, ; molecularweight, MC7�

; and specific gravity, �C7�, are known. Sec. 5.3 re-

views methods for splitting C7� into an arbitrary number of sub-fractions. Most methods assume that mole fraction decreases expo-nentially as a function of molecular weight or carbon number. Amore general model based on the gamma distribution has been suc-cessfully applied to many oil and gas-condensate systems. Othersplitting schemes can also be found in the literature; we summarizethe available methods.

Sec. 5.4 discusses how to estimate inspection properties � and Tbfor C7� fractions determined by GC analysis or calculated from amathematical split. Katz and Firoozabadi’s6 generalized single car-bon number (SCN) properties are widely used. Other methods forestimating specific gravities of C7� subfractions are based on forc-ing the calculated �C7�

to match the measured value.Many empirical correlations are available for estimating critical

properties of pure compounds and C7� fractions. Critical propertiescan also be estimated by forcing the EOS to match the boiling point andspecific gravity of each C7� fraction separately. In Sec. 5.5, we reviewthe most commonly used methods for estimating critical properties.

Finally, Sec. 5.6 discusses methods for reducing the number ofcomponents describing a reservoir mixture and, in particular, theC7� fraction. Splitting the C7� into pseudocomponents is particu-larly important for EOS-based compositional reservoir simulation.A large part of the computing time during a compositional reservoirsimulation is used to solve the flash calculations; accordingly, mini-mizing the number of components without jeopardizing the qualityof the fluid characterization is necessary.

��

The most reliable basis for C7� characterization is experimentaldata obtained from high-temperature distillation or GC. Many ex-perimental procedures are available for performing these analyses;in the following discussion, we review the most commonly usedmethods. TBP distillation provides the key data for C7� character-ization, including mass and molar quantities, specific gravity, mo-lecular weight, and boiling point of each distillation cut. Other suchinspection data as kinematic viscosity and refractive index also maybe measured on distillation cuts.

Simulated distillation by GC requires smaller samples and lesstime than TBP distillation.7-9 However, GC analysis measures onlythe mass of carbon-number fractions. Simulated distillation resultscan be calibrated against TBP data, thus providing physical proper-ties for the individual fractions. For many oils, simulated distillation

2 PHASE BEHAVIOR

Fig. 5.1—Standard apparatus for conducting TBP analysis ofcrude-oil and condensate samples at atmospheric and subat-mospheric pressures (after Ref. 11).

��

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provides the necessary information for C7� characterization in farless the time and at far less cost than that required for a completeTBP analysis. We recommend, however, that at least one completeTBP analysis be measured for (1) oil reservoirs that may be candi-dates for gas injection and (2) most gas-condensate reservoirs.

5.2.1 TBP Distillation. In TBP distillation, a stock-tank liquid (oilor condensate) is separated into fractions or “cuts” by boiling-pointrange. TBP distillation differs from the Hempel and American Soc.for Testing Materials (ASTM) D-158 distillations10 because TBPanalysis requires a high degree of separation, which is usually con-trolled by the number of theoretical trays in the apparatus and thereflux ratio. TBP fractions are often treated as components havingunique boiling points, critical temperatures, critical pressures, andother properties identified for pure compounds. This treatment isobviously more valid for a cut with a narrow boiling-point range.

The ASTM D-289211 procedure is a useful standard for TBPanalysis of stock-tank liquids. ASTM D-2892 specifies the generalprocedure for TBP distillation, including equipment specifications(see Fig. 5.1), reflux ratio, sample size, and calculations necessaryto arrive at a plot of cumulative volume percent vs. normal boilingpoint. Normal boiling point implies that boiling point is measuredat normal or atmospheric pressure. In practice, to avoid thermal de-composition (cracking), distillation starts at atmospheric pressureand is changed to subatmospheric distillation after reaching a limit-ing temperature. Subatmospheric boiling-point temperatures areconverted to normal boiling-point temperatures by use of a vapor-pressure correlation that corrects for the amount of vacuum and thefraction’s chemical composition. The boiling-point range for frac-tions is not specified in the ASTM standard. Katz and Firoozabadi6

recommend use of paraffin normal boiling points (plus 0.5°C) asboundaries, a practice that has been widely accepted.

Fig. 5.2—TBP curve for a North Sea gas-condensate sample il-lustrating the midvolume-point method for calculating averageboiling point (after Austad et al.7).

Cutoff (n-paraffin) boiling point

Midvolume (“normal”) boiling point

Fig. 5.27 shows a plot of typical TBP data for a North Sea sample.Normal boiling point is plotted vs. cumulative volume percent.Table 5.1 gives the data, including measured specific gravities andmolecular weights. Average boiling point is usually taken as the val-ue found at the midvolume percent of a cut. For example, the thirdcut in Table 5.1 boils from 258.8 to 303.8°F, with an initial 27.49vol% and a final 37.56 vol%. The midvolume percent is(27.49�37.56)/2�32.5 vol%; from Fig. 5.2, the boiling point atthis volume is �282°F. For normal-paraffin boiling-point intervals,Katz and Firoozabadi’s6 average boiling points of SCN fractionscan be used (see Table 5.2).

The mass, mi, of each distillation cut is measured directly duringa TBP analysis. The cut is quantified in moles ni with molecularweight, Mi, and the measured mass mi, where ni � mi�Mi. Volumeof the fraction is calculated from the mass and the density, �i (or spe-cific gravity, �i), where Vi � mi��i . Mi is measured by a cryoscop-ic method based on freezing-point depression, and �i is measuredby a pycnometer or electronic densitometer. Table 5.1 gives cumula-tive weight, mole, and volume percents for the North Sea sample.Average C7� properties are given by

MC7��

�N

i�1

mi

�N

i�1

ni

and �C7��

�N

i�1

mi

�N

i�1

Vi

, (5.1). . . . . . . . . . . . . . . . . . . . . . . . . . . . .

where �C7�� C7�

�w with �w �pure water density at standardconditions. These calculated averages are compared with mea-sured values of the C7� sample, and discrepancies are reported as“lost” material.

Refs. 7 and 15 through 20 give procedures for calculating proper-ties from TBP analyses. Also, the ASTM D-289211 procedure givesdetails on experimental equipment and the procedure for conductingTBP analysis at atmospheric and subatmospheric conditions. Table5.3 gives an example TBP analysis from a commercial laboratory.


TABLE 5.1—EXPERIMENTAL TBP RESULTS FOR A NORTH SEA CONDENSATE

Fraction

UpperTbi(°F)

AverageTbi*(°F)

mi(g) �i **

Mi(g/mol)

Vi(cm3)

ni(mol)

wi(%)

xVi%

xi%

�wi%

�xVi% Kw

C7

C8

C9

C10

C11

C12

C13

C14

C15

C16

C17

C18

C19

C20

C21�

208.4

258.8

303.8

347.0

381.2

420.8

455.0

492.8

523.4

550.4

579.2

604.4

629.6

653.0

194.0

235.4

282.2

325.4

363.2

401.1

438.8

474.8

509.0

537.8

564.8

591.8

617.0

642.2

90.2

214.6

225.3

199.3

128.8

136.8

123.8

120.5

101.6

74.1

76.8

58.2

50.2

45.3

427.6

0.7283

0.7459

0.7658

0.7711

0.7830

0.7909

0.8047

0.8221

0.8236

0.8278

0.8290

0.8378

0.8466

0.8536

0.8708

96

110

122

137

151

161

181

193

212

230

245

259

266

280

370

123.9

287.7

294.2

258.5

164.5

173.0

153.8

146.6

123.4

89.5

92.6

69.5

59.3

53.1

491.1

0.940

1.951

1.847

1.455

0.853

0.850

0.684

0.624

0.479

0.322

0.313

0.225

0.189

0.162

1.156

4.35

10.35

10.87

9.61

6.21

6.60

5.97

5.81

4.90

3.57

3.70

2.81

2.42

2.19

20.63

4.80

11.15

11.40

10.02

6.37

6.70

5.96

5.68

4.78

3.47

3.59

2.69

2.30

2.06

19.03

7.80

16.19

15.33

12.07

7.08

7.05

5.68

5.18

3.98

2.67

2.60

1.87

1.57

1.34

9.59

4.35

14.70

25.57

35.18

41.40

48.00

53.97

59.78

64.68

68.26

71.96

74.77

77.19

79.37

100.00

4.80

15.95

27.35

37.37

43.74

50.44

56.41

62.09

66.87

70.33

73.92

76.62

78.91

80.97

100.00

11.92

11.88

11.82

11.96

11.97

12.03

11.99

11.89

12.01

12.07

12.16

12.14

12.11

12.10

Sum 2,073.1 2,580.5 12.049 100.00 100.00 100.00

Average 0.8034 172 11.98

Reflux ratio�1 : 5; reflux cycle�18 seconds; distillation at atmospheric pressure�201.2 to 347°F; distillation at 100 mm Hg�347 to 471.2°F; and distillation at 10 mm Hg�471.2to 653°F.

Vi�mi /�i /0.9991; ni�mi /Mi ; wi�100�mi /2073.1; xVi�100�Vi/2580.5; xi�100�ni /12.049; �wi��wi ; �xVi��xVi ; and Kw�(Tbi+460)1/3/�i .*Average taken at midvolume point.

**Water�1.

Boiling points are not reported because normal-paraffin boiling-pointintervals are used as a standard; accordingly, the average boilingpoints suggested by Katz and Firoozabadi6 (Table 5.2) can be used.

5.2.2 Chromatography. GC and, to a lesser extent, liquid chroma-tography are used to quantify the relative amount of compoundsfound in oil and gas systems. The most important application ofchromatography to C7� characterization is simulated distillation byGC techniques.

Fig. 5.3 shows an example gas chromatogram for the North Seasample considered earlier. The dominant peaks are for normal paraf-fins, which are identified up to n-C22. As a good approximation fora paraffinic sample, the GC response for carbon number Ci starts atthe bottom response of n-Ci1 and extends to the bottom responseof n-Ci. The mass of carbon number Ci is calculated as the area underthe curve from the baseline to the GC response in the n-Ci1 to n-Ciinterval (see the shaded area for fraction C9 in Fig. 5.3). As Fig. 5.47

shows schematically, the baseline should be determined before run-ning the actual chromatogram.

Because stock-tank samples cannot be separated completely bystandard GC analysis, an internal standard must be used to relate GCarea to mass fraction. Normal hexane was used as an internal stan-dard for the sample in Fig. 5.3. The internal standard’s response fac-tor may need to be adjusted to achieve consistency between simu-lated and TBP distillation results. This factor will probably beconstant for a given oil, and the factor should be determined on thebasis of TBP analysis of at least one sample from a given field. Fig.5.5 shows the simulated vs. TBP distillation curves for the Austadet al.7 sample. A 15% correction to the internal-standard responsefactor was used to match the two distillation curves.

As an alternative to correcting the internal standard, Maddox andErbar15 suggest that the reported chromatographic boiling points beadjusted by a correction factor that depends on the reported boiling

point and the “paraffinicity” of the composite sample. This correc-tion factor varies from 1 to 1.15 and is slightly larger for aromaticthan paraffinic samples.

Several laboratories have calibrated GC analysis to provide simu-lated-distillation results up to C40. However, checking the accuracyof simulated distillation for SCN fractions greater than approxi-mately C25 is difficult because C25 is usually the upper limit for reli-able TBP distillation. The main disadvantage of simulated distilla-tion is that inspection data are not determined directly for eachfraction and must therefore either be correlated from TBP data or es-timated from correlations (see Sec. 5.4).

Sophisticated analytical methods, such as mass spectroscopy,may provide detailed information on the compounds separated byGC. For example, mass spectroscopy GC can establish the relativeamounts of paraffins, naphthenes, and aromatics (PNA’s) for car-bon-number fractions distilled by TBP analysis. Detailed PNA in-formation should provide more accurate estimation of the criticalproperties of petroleum fractions, but the analysis is relatively cost-ly and time-consuming from a practical point of view. Recent workhas shown that PNA analysis3,19-23 may improve C7� characteriza-tion for modeling phase behavior with EOS’s. Our experience, how-ever, is that PNA data have limited usefulness for improving EOSfluid characterizations.

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Molar distribution is usually thought of as the relation between molefraction and molecular weight. In fact, this concept is misleading be-cause a unique relation does not exist between molecular weight andmole fraction unless the fractions are separated in a consistent man-ner. Consider for example a C7� sample distilled into 10 cuts sepa-rated by normal-paraffin boiling points. If the same C7� sample isdistilled with constant 10-vol% cuts, the two sets of data will not

4 PHASE BEHAVIOR

TABLE 5.2—SINGLE CARBON NUMBER PROPERTIES FOR HEPTANES-PLUS (after Katz and Firoozabadi6)

Katz-Firoozabadi Generalized Properties

Tb Interval*Lee-Kesler12/Kesler-Lee13

Correlations Riazi14 Defined

FractionNumber

Lower(°F)

Upper(°F)

Average Tb(°F) (°R) �� M

DefinedKw

Tc(°R)

pc(psia)

�

��

Vc(ft3/lbm mol) Zc

6 97.7 156.7 147.0 606.7 0.690 84 12.27 914 476 0.271 5.6 0.273

7 156.7 210.0 197.4 657.1 0.727 96 11.96 976 457 0.310 6.2 0.272

8 210.0 259.0 242.1 701.7 0.749 107 11.86 1,027 428 0.349 6.9 0.269

9 259.0 304.3 288.0 747.6 0.768 121 11.82 1,077 397 0.392 7.7 0.266

10 304.3 346.3 330.4 790.1 0.782 134 11.82 1,120 367 0.437 8.6 0.262

11 346.3 385.5 369.0 828.6 0.793 147 11.84 1,158 341 0.479 9.4 0.257

12 385.5 422.2 406.9 866.6 0.804 161 11.86 1,195 318 0.523 10.2 0.253

13 422.2 456.6 441.0 900.6 0.815 175 11.85 1,228 301 0.561 10.9 0.249

14 456.6 489.0 475.5 935.2 0.826 190 11.84 1,261 284 0.601 11.7 0.245

15 489.0 520.0 510.8 970.5 0.836 206 11.84 1,294 268 0.644 12.5 0.241

16 520.0 548.6 541.4 1,001.1 0.843 222 11.87 1,321 253 0.684 13.3 0.236

17 548.6 577.4 572.0 1,031.7 0.851 237 11.87 1,349 240 0.723 14.0 0.232

18 577.4 602.6 595.4 1,055.1 0.856 251 11.89 1,369 230 0.754 14.6 0.229

19 602.6 627.8 617.0 1,076.7 0.861 263 11.90 1,388 221 0.784 15.2 0.226

20 627.8 651.2 640.4 1,100.1 0.866 275 11.92 1,408 212 0.816 15.9 0.222

21 651.2 674.6 663.8 1,123.5 0.871 291 11.94 1,428 203 0.849 16.5 0.219

22 674.6 692.6 685.4 1,145.1 0.876 305 11.94 1,447 195 0.879 17.1 0.215

23 692.6 717.8 707.0 1,166.7 0.881 318 11.95 1,466 188 0.909 17.7 0.212

24 717.8 737.6 726.8 1,186.5 0.885 331 11.96 1,482 182 0.936 18.3 0.209

25 737.6 755.6 746.6 1,206.3 0.888 345 11.99 1,498 175 0.965 18.9 0.206

26 755.6 775.4 766.4 1,226.1 0.892 359 12.00 1,515 168 0.992 19.5 0.203

27 775.4 793.4 786.2 1,245.9 0.896 374 12.01 1,531 163 1.019 20.1 0.199

28 793.4 809.6 804.2 1,263.9 0.899 388 12.03 1,545 157 1.044 20.7 0.196

29 809.6 825.8 820.4 1,280.1 0.902 402 12.04 1,559 152 1.065 21.3 0.194

30 825.8 842.0 834.8 1,294.5 0.905 416 12.04 1,571 149 1.084 21.7 0.191

31 842.0 858.2 851.0 1,310.7 0.909 430 12.04 1,584 145 1.104 22.2 0.189

32 858.2 874.4 865.4 1,325.1 0.912 444 12.04 1,596 141 1.122 22.7 0.187

33 874.4 888.8 879.8 1,339.5 0.915 458 12.05 1,608 138 1.141 23.1 0.185

34 888.8 901.4 892.4 1,352.1 0.917 472 12.06 1,618 135 1.157 23.5 0.183

35 901.4 915.8 906.8 1,366.5 0.920 486 12.06 1,630 131 1.175 24.0 0.180

36 919.4 1,379.1 0.922 500 12.07 1,640 128 1.192 24.5 0.178

37 932.0 1,391.7 0.925 514 12.07 1,650 126 1.207 24.9 0.176

38 946.4 1,406.1 0.927 528 12.09 1,661 122 1.226 25.4 0.174

39 959.0 1,418.7 0.929 542 12.10 1,671 119 1.242 25.8 0.172

40 971.6 1,431.3 0.931 556 12.10 1,681 116 1.258 26.3 0.170

41 982.4 1,442.1 0.933 570 12.11 1,690 114 1.272 26.7 0.168

42 993.2 1,452.9 0.934 584 12.13 1,697 112 1.287 27.1 0.166

43 1,004.0 1,463.7 0.936 598 12.13 1,706 109 1.300 27.5 0.164

44 1,016.6 1,476.3 0.938 612 12.14 1,716 107 1.316 27.9 0.162

45 1,027.4 1,487.1 0.940 626 12.14 1,724 105 1.328 28.3 0.160*At 1 atmosphere.

**Water�1.

produce the same plot of mole fraction vs. molecular weight. How-ever, a plot of cumulative mole fraction,

Qzi �

�i

j�1

zj

�N

j�1

zj

,�� (5.2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vs. cumulative average molecular weight,

QMi �

�i

j�1

zj Mj

�i

j�1

zj

, (5.3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


TABLE 5.3—STANDARD TBP RESULTS FROM COMMERCIAL PVT LABORATORY

Component mol% wt%Density(g/cm3)

Gravity��API

MolecularWeight

HeptanesOctanesNonanesDecanesUndecanesDodecanesTridecanesTetradecanesPentadecanes plus

1.121.301.180.980.620.570.740.534.10

2.52 3.08 3.15 2.96 2.10 2.18 3.05 2.3931.61

0.72580.74700.76540.77510.78080.79710.81050.82350.8736

63.257.753.150.949.545.842.940.130.3

96101114129144163177192330

*At 60°F.Note: Katz and Firoozabadi6 average boiling points (Table 5.2) can be used when normal paraffin boiling-point intervals are used.

should produce a single curve. Strictly speaking, therefore, molardistribution is the relation between cumulative molar quantity andsome expression for cumulative molecular weight.

In this section, we review methods commonly used to describemolar distribution. Some methods use a consistent separation offractions (e.g., by SCN) so the molar distribution can be expresseddirectly as a relationship between mole fraction and molecularweight of individual cuts. Most methods in this category assume thatC7� mole fractions decrease exponentially. A more general ap-proach uses the continuous three-parameter gamma probabilityfunction to describe molar distribution.

5.3.1 Exponential Distributions. The Lohrenz-Bray-Clark24

(LBC) viscosity correlation is one of the earliest attempts to use anexponential-type distribution for splitting C7�. The LBC methodsplits C7� into normal paraffins C7 though C40 with the relation

zi � zC6exp[A1(i 6) � A2(i 6)2], (5.4). . . . . . . . . . . . .

where i�carbon number and zC6�measured C6 mole fraction.

Constants A1 and A2 are determined by trial and error so that

zC7��

40

i�7

zi (5.5). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Fig. 5.3—Simulated distillation by GC of the North Sea gas-con-densate sample in Fig. 5.2 (after Austad et al.7).

and zC7�MC7�

��40

i�7

zi Mi (5.6). . . . . . . . . . . . . . . . . . . . . . . .

are satisfied. Paraffin molecular weights (Mi�14i�2) are used inEq. 5.6. A Newton-Raphson algorithm can be used to solve Eqs. 5.5and 5.6. Note that the LBC model cannot be used when zC7�

� zC6and MC7�

� MC40. The LBC form of the exponential distribution

has not found widespread application.More commonly, a linear form of the exponential distribution is

used to split the C7� fraction. Writing the exponential distributionin a general form for any Cn� fraction (n�7 being a special case),

zi � zCnexp A[(i n)], (5.7). . . . . . . . . . . . . . . . . . . . . . . .

where i�carbon number, zCn�mole fraction of Cn , andA�constant indicating the slope on a plot of ln zi vs. i. The constantszCn and A can be determined explicitly. With the general expression

Mi � 14 i � h (5.8). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

for molecular weight of Ci and the assumption that the distributionis infinite, constants zCn and A are given by

zCn�

14MCn�

14(n 1) h(5.9). . . . . . . . . . . . . . . . . .

and A � ln1 zCn� (5.10). . . . . . . . . . . . . . . . . . . . . . . . . . .

so that��

i�n

zi � 1 (5.11). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Fig. 5.4—GC simulated distillation chromatograms (a) withoutany sample (used to determine the baseline), (b) for a crude oil,and (c) for a crude oil with internal standard (after MacAllisterand DeRuiter9).

(a)

(b)

(c)

6 PHASE BEHAVIOR

Fig. 5.5—Comparison of TBP and GC-simulated distillation fora North Sea gas-condensate sample (after Austad et al.7).

and ��

i�n

zi Mi � MCn�(5.12). . . . . . . . . . . . . . . . . . . . . . . . . . .

are satisfied.Eqs. 5.9 and 5.10 imply that once a molecular weight relation is cho-

sen (i.e., h is fixed), the distribution is uniquely defined by C7� molec-ular weight. Realistically, all reservoir fluids having a given C7� mo-lecular weight will not have the same molar distribution, which is onereason why more complicated models have been proposed.

5.3.2 Gamma-Distribution Model. The three-parameter gammadistribution is a more general model for describing molar distribu-tion. Whitson2,25,26 and Whitson et al.27 discuss the gamma dis-tribution and its application to molar distribution. They give resultsfor 44 oil and condensate C7� samples that were fit by the gammadistribution with data from complete TBP analyses. The absoluteaverage deviation in estimated cut molecular weight was 2.5 amu(molecular weight units) for the 44 samples.

The gamma probability density function is

p(M) �(M �)�1 exp �M ��

��(�), (5.13). . . . . . . .

where ��gamma function and is given by

�

MC7� �

�. (5.14). . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The three parameters in the gamma distribution are �, �, andMC7�

The key parameter � defines the form of the distribution, andits value usually ranges from 0.5 to 2.5 for reservoir fluids; ��1gives an exponential distribution. Application of the gamma dis-tribution to heavy oils, bitumen, and petroleum residues indicatesthat the upper limit for � is 25 to 30, which statistically is approach-ing a log-normal distribution (see Fig. 5.628).

The parameter � can be physically interpreted as the minimummolecular weight found in the C7� fraction. An approximate rela-tion between � and � is

� �110

1 1 � 4��0.7�(5.15). . . . . . . . . . . . . . . . . . . . . . . . .

Fig. 5.6—Gamma distributions for petroleum residue (afterBrulé et al.28).

700 to 1,000°F Distillate

1,000 to 1,250°F Distillate

1,200°F Residue

for reservoir-fluid C7� fractions. Practically, � should be consideredas a mathematical constant more than as a physical property, eithercalculated from Eq. 5.15 or determined by fitting measured TBP data.

Fig. 5.7 shows the function p(M) for the Hoffman et al.29 oil anda North Sea oil. Parameters for these two oils were determined by fit-ting experimental TBP data. The Hoffman et al. oil has a relativelylarge � of 2.27, a relatively small � of 75.7, with MC7��198; theNorth Sea oil is described by ��0.82, ��93.2, and MC7��227.

The continuous distribution p(M ) is applied to petroleum frac-tions by dividing the area under the p(M ) curve into sections (shownschematically in Fig. 5.8). By definition, total area under the p(M )curve from � to � is unity. The area of a section is defined asnormalized mole fraction zi�zC7�

for the range of molecularweights Mbi1 to Mbi. If the area from � to molecular-weightboundary Mb is defined as P0(Mb), then the area of Section i isP0(Mbi)P0(Mbi1), also shown schematically in Fig. 5.8. Molefraction zi can be written

zi � zC7��P0Mb i

� P0Mb i1�� . (5.16). . . . . . . . . . . . . . .

Average molecular weight in the same interval is given by

Mi � � � �

P1Mb i� P1Mb i1

�

P0Mb i� P0Mb i1

�, (5.17). . . . . . . . . . .

where P0 � QS, (5.18). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

and P1 � QS 1��, (5.19). . . . . . . . . . . . . . . . . . . . . . . . . . .


Fig. 5.7—Gamma density function for the Hoffman et al.29 oil(dashed line) and a North Sea volatile oil (solid line). After Whit-son et al.27

� � 2.273� � 75.7MC7�

� 198.4

� � 0.817� � 93.2MC7�

� 227

where Q � ey y��(�), (5.20). . . . . . . . . . . . . . . . . . . . . . . . .

S ��

j�0

y j��j

k�0

(�� k)�1

, (5.21). . . . . . . . . . . . . . . . . . .

and y �Mb �

. (5.22). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Note that P0(Mb0��)�P1(Mb0��)�0.The summation in Eq. 5.21 should be performed until the last

term is �1�108. The gamma function can be estimated by30

�x � 1� � 1 ��8

i�1

Ai xi , (5.23). . . . . . . . . . . . . . . . . . . . .

where A1�0.577191652, A2�0.988205891, A3�0.897056937,A4�0.918206857, A5�0.756704078, A6�0.482199394, A7�

0.193527818, and A8�0.035868343 for 0�x�1. The recurrenceformula, �(x�1)�x�(x), is used for x�1 and x�1; furthermore,�(1)�1.

The equations for calculating zi and Mi are summarized in a shortFORTRAN program GAMSPL found in Appendix A. In this simpleprogram, the boundary molecular weights are chosen arbitrarily atincrements of 14 for the first 19 fractions, starting with � as the firstlower boundary. The last fraction is calculated by setting the uppermolecular-weight boundary equal to 10,000. Table 5.4 gives threesample outputs from GAMSPL for ��0.5, 1, and 2 with ��90 andMC7�

�200 held constant. Fig. 5.9 plots the results as log zi vs. Mi.The amount and molecular weight of the C26� fraction varies foreach value of �.

The gamma distribution can be fit to experimental molar-distribu-tion data by use of a nonlinear least-squares algorithm to determine�, �, and . Experimental TBP data are required, including weightfraction and molecular weight for at least five C7� fractions (use ofmore than 10 fractions is recommended to ensure a unique fit of mod-el parameters). The sum-of-squares function can be defined as

F�, � , � � �N1

i�1

(�Mi)2, (5.24). . . . . . . . . . . . . . . . . . . . . . .

where �Mi �

(Mi)mod (Mi)exp

(Mi)exp. (5.25). . . . . . . . . . . . . . . . . .

Subscripts mod and exp�model and experimental, respectively. Thissum-of-squares function weights the lower molecular weights morethan higher molecular weights, in accordance with the expected accu-racy for measurement of molecular weight. Also, the sum-of-squaresfunction does not include the last molecular weight because this mo-lecular weight may be inaccurate or backcalculated to match the mea-sured average C7� molecular weight. If the last fraction is not in-cluded, the model average molecular weight, (MC7�

)mod � �� ,can be compared with the experimental value as an independentcheck of the fit.

A simple graphical procedure can be used to fit parameters � and� if experimental MC7�

is fixed and used to define . Fig. 5.10shows a plot of cumulative weight fraction,

Qwi ��i

j�1

wi , (5.26). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vs. the cumulative dimensionless molecular-weight variable,

Q*M i �

QM i �

MC7� �

. (5.27). . . . . . . . . . . . . . . . . . . . . . . . . . .

Table 5.5 and the following outline describe the procedure for deter-mining model parameters with Fig. 5.10 and TBP data.

1. Tabulate measured mole fractions zi and molecular weights Mifor each fraction.

2. Calculate experimental weight fractions, wi � (zi Mi)� (zC7�

MC7�), if they are not reported.

3. Normalize weight fractions and calculate cumulative normal-ized weight fraction Qw i .

4. Calculate cumulative molecular weight QM i from Eq. 5.3.5. Assume several values of � (e.g., from 75 to 100) and calculate

Q*M i for each value of the estimated �.6. For each estimate of �, plot Q*

M i vs. Qwi on a copy of Fig. 5.10and choose the curve that fits one of the model curves best. Read thevalue of � from Fig. 5.10.

7. Calculate molecular weights and mole fractions of Fractions iusing the best-fit curve in Fig. 5.10. Enter the curve at measured val-ues of Qwi , read Q*

M i , and calculate Mi from

Mi � �� MC7� ��

Qwi Qwi1

�Qwi�Q*M i� Qwi1�Q*

M i1��

.

(5.28). . . . . . . . . . . . . . . . . . .

Fig. 5.8—Schematic showing the graphical interpretation of areas under the gamma densityfunction p(M) that are proportional to normalized mole fraction; A�area.

�

� P0Mbi

� P0Mbi1

�

�MbiA � P0

Mbi� A � P0

Mbi1�

�Mbi1

p(M)

A � zi�zC7�

8 PHASE BEHAVIOR

TABLE 5.4—RESULTS OF GAMSPL PROGRAM FOR THREE DATA SETS WITH DIFFERENT

GAMMA-DISTRIBUTION PARAMETER �

��0.5 ��1.0 ��2.0

Fraction

Number

Mole

Fraction

Molecular

Weight

Mole

Fraction

Molecular

Weight

Mole

Fraction

Molecular

Weight

1 0.2787233 94.588 0.1195065 96.852 0.0273900 99.132

2 0.1073842 110.525 0.1052247 110.852 0.0655834 111.490

3 0.0772607 124.690 0.0926497 124.852 0.0852269 125.172

4 0.0610991 138.758 0.0815774 138.852 0.0927292 139.038

5 0.0505020 152.796 0.0718284 152.852 0.0925552 152.963

6 0.0428377 166.819 0.0632444 166.852 0.0877762 166.916

7 0.0369618 180.836 0.0556863 180.852 0.0804707 180.883

8 0.0322804 194.848 0.0490314 194.852 0.0720157 194.859

9 0.0284480 208.857 0.0431719 208.852 0.0632969 208.841

10 0.0252470 222.864 0.0380125 222.852 0.0548597 222.826

11 0.0225321 236.870 0.0334698 236.852 0.0470180 236.814

12 0.0202013 250.875 0.0294699 250.852 0.0399302 250.805

13 0.0181808 264.879 0.0259481 264.852 0.0336535 264.797

14 0.0164152 278.883 0.0228471 278.852 0.0281813 278.790

15 0.0148619 292.886 0.0201167 292.852 0.0234690 292.784

16 0.0134879 306.888 0.0177127 306.852 0.0194514 306.778

17 0.0122665 320.890 0.0155959 320.852 0.0160543 320.774

18 0.0111762 334.892 0.0137321 334.852 0.0132017 334.770

19 0.0101996 348.894 0.0120910 348.852 0.0108204 348.766

20 0.1199341 539.651 0.0890834 466.000 0.0463166 420.424

Total 1.0000000 1.0000000 1.0000000

Average 200 200 200

For all three cases � � 90 and MC7�� 200.

Mole fractions zi are given by

zi � zC7� Qw i

Q*M i

Qw i1

Q*M i1� . (5.29). . . . . . . . . . . . . . . . . . .

For computer applications, Qwi and Q*M i can be calculated exactly

from Eqs. 5.16 through 5.23 with little extra effort.

MC7�

Fig. 5.9—Three example molar distributions for an oil samplewith =200 and �=90, calculated with the GAMSPL program(Table A-4) in Table 5.4.

MC7�� 200

� � 90�Mb � 14� � 2.0

� � 0.5� � 1.0}�

�

Fig. 5.11 shows a Q*M i Qwi match for the Hoffman et al.29 oil

with ��70, 72.5, 75, and 80 and indicates that a best fit is achievedfor ��72.5 and ��2.5 (see Fig. 5.12).

Although the gamma-distribution model has the flexibility oftreating reservoir fluids from light condensates to bitumen, mostreservoir fluids can be characterized with an exponential molar dis-tribution (��1) without adversely affecting the quality of EOS pre-

MC7�

Fig. 5.10—Cumulative-distribution type curve for fitting exper-imental TBP data to the gamma-distribution model. Parameters� and � are determined with held constant.

1.0

0.0

0.8

0.6

0.4

0.2

0.0 0.2 0.4 0.6 0.8 1.0

Cumulative Normalized Mole Fraction, Qzi

(

)(

)


TABLE 5.5—CALCULATION OF CUMULATIVE WEIGHT FRACTION ANDCUMULATIVE MOLECULAR WEIGHT VARIABLE FOR HOFFMAN et al.29 OIL

Q*Mi

Componenti zi ��zi� Mi ziMi ��ziMi� Qwi QMi ��70 ��72.5 ��75 ��80 ��85

789

101112131415161718192021222324252627282930

0.02630.02340.02350.02240.02410.02460.02660.03260.03630.02290.01710.01430.01300.01080.00870.00720.00580.00480.00390.00340.00280.00250.00230.0091

0.02630.04970.07320.09560.11970.14430.17090.20350.23980.26270.27990.29410.30720.31800.32670.33380.33960.34440.34830.35170.35450.35700.35930.3684

99110121132145158172186203222238252266279290301315329343357371385399444

2.6042.5742.8442.9573.4973.8824.5706.0677.3715.0934.0793.5963.4663.0082.5262.1521.8111.5821.3511.1961.0390.9630.9264.049

2.6045.1788.021

10.97814.47518.35722.92828.99536.36641.45845.53849.13452.60055.60758.13360.28562.09763.67965.03166.22767.26568.22869.15473.203

0.0360.0710.1100.1500.1980.2510.3130.3960.4970.5660.6220.6710.7190.7600.7940.8240.8480.8700.8880.9050.9190.9320.9451.000

99.0104.2109.6114.8120.9127.2134.2142.5151.7157.8162.7167.0171.2174.9178.0180.6182.9184.9186.7188.3189.8191.1192.5198.7

0.2250.2660.3080.3480.3960.4450.4990.5630.6340.6820.7200.7540.7870.8150.8390.8590.8770.8930.9070.9190.9310.9410.9521.000

0.2100.2510.2940.3350.3840.4340.4890.5550.6270.6760.7150.7490.7820.8110.8360.8570.8750.8910.9050.9180.9290.9400.9511.000

0.1940.2360.2800.3220.3710.4220.4780.5460.6200.6690.7090.7440.7780.8080.8320.8540.8720.8890.9030.9160.9280.9390.9501.000

0.1600.2040.2490.2930.3450.3980.4570.5260.6040.6550.6970.7330.7690.7990.8250.8470.8670.8840.8990.9130.9250.9360.9481.000

0.1230.1690.2160.2620.3160.3710.4330.5060.5860.6400.6830.7220.7580.7910.8180.8410.8610.8790.8940.9090.9210.9330.9451.000

Total 0.3684 198.7 73.203

dictions. Whitson et al.27 proposed perhaps the most useful applica-tion of the gamma-distribution model. With Gaussian quadrature,their method allows multiple reservoir-fluid samples from a com-mon reservoir to be treated simultaneously with a single fluid char-acterization. Each fluid sample can have different C7� propertieswhen the split is made so that each split fraction has the same molec-ular weight (and other properties, such as �, Tb, Tc, pc, and �), while

Fig. 5.11—Graphical fit of the Hoffman et al.29 oil molar distribu-tion by use of the cumulative-distribution type curve. Best-fitmodel parameters are �=2.5 and �=72.5.

1.0

0.0

0.8

0.6

0.4

0.2

0.0 0.2 0.4 0.6 0.8 1.0

�

Cumulative Normalized Mole Fraction, Qzi

� � 70� � 75� � 80

� � 65�

�

X

the mole fractions are different for each fluid sample. Example ap-plications include the characterization of a gas cap and underlyingreservoir oil and a reservoir with compositional gradient.

The following outlines the procedure for applying Gaussianquadrature to the gamma-distribution function.

1. Determine the number of C7� fractions, N, and obtain thequadrature values Xi and Wi from Table 5.6 (values are given forN�3 and N�5).

2. Specify � and �. When TBP data are not available to determinethese parameters, recommended values are ��90 and ��1.

3. Specify the heaviest molecular weight of fraction N (recom-mended value is MN � 2.5MC7�

). Calculate a modified * term,

*� MN ��XN.

Fig. 5.12—Calculated normalized mole fraction vs. molecularweight of fractions for the Hoffman et al.29 oil based on the bestfit in Fig. 5.11 with �=2.5 and �=72.5.

10 PHASE BEHAVIOR

TABLE 5.6—GAUSSIAN QUADRATURE FUNCTIONVARIABLES, X, AND WEIGHT FACTORS, W

X W

Three Quadrature Points (plus fractions)

123

0.415 774 556 7832.294 280 360 2796.289 945 082 937

7.110 930 099 29�101

2.785 177 335 69�101

1.038 925 650 16�102

Five Quadrature Points (plus fractions)

12345

0.263 560 319 7181.413 403 059 1073.596 425 771 0417.085 810 005 859

12.640 800 844 276

5.217 556 105 83�101

3.986 668 110 83�101

7.594 244 968 17�102

3.611 758 679 92�103

2.336 997 238 58�105

Quadrature function values and weight factors can be found for other quadrature numbersin mathematical handbooks.30

4. Calculate the parameter .

� exp �*

MC7� �

1� . (5.30). . . . . . . . . . . . . . . . . . .

5. Calculate the C7� mole fraction zi and Mi for each fraction.

zi � zC7��Wi f (Xi)�,

Mi � � � * Xi ,

and f(X) �(X)�1

�(�)

1 � ln ��

X . (5.31). . . . . . . . . . . . . . . . . .

6. Check whether the calculated MC7� from Eq. 5.12 equals themeasured value used in Step 4 to define . Because Gaussian quad-rature is only approximate, the calculated MC7� may be slightly inerror. This can be corrected by (slightly) modifying the value of ,and repeating Steps 5 and 6 until a satisfactory match is achieved.

When characterizing multiple samples simultaneously, the valuesof MN , �, and * must be the same for all samples. Individual samplevalues of MC7�

and � can, however, be different. The result of thischaracterization is one set of molecular weights for the C7� frac-tions, while each sample has different mole fractions zi (so that theiraverage molecular weights MC7� are honored).

Specific gravities for the C7� fractions can be calculated withone of the correlations given in Sec. 5.4 (e.g., Eq. 5.44), where thecharacterization factor (e.g., Fc) must be the same for all mixtures.The specific gravities, �C7�

, of each sample will not be exactly re-produced with this procedure (calculated with Eq. 5.37), but the av-erage characterization factor can be chosen so that the differencesare very small (��0.0005). Having defined Mi and �i for the C7�fractions, a complete fluid characterization can be determined withcorrelations in Sec. 5.5.

��" ��

5.4.1 Generalized Properties. The molecular weight, specific grav-ity, and boiling point of C7� fractions must be estimated in the ab-sence of experimental TBP data. This situation arises when simulateddistillation is used or when no experimental analysis of C7� is avail-able and a synthetic split must be made by use of a molar-distributionmodel. For either situation, inspection data from TBP analysis of asample from the same field would be the most reliable source of M,�, and Tb for each C7� fraction. The next-best source would be mea-sured TBP data from a field producing similar oil or condensate fromthe same geological formation. Generalized properties from a pro-ducing region, such as the North Sea, have been proposed.31

Katz and Firoozabadi6 suggest a generalized set of SCN proper-ties for petroleum fractions C6 through C45. Table 5.2 gives an ex-tended version of the Katz-Firoozabadi property table. Molecular

weights can be used to convert weight fractions, wi, from simulateddistillation to mole fractions,

zi �wi�Mi

�N

j�7

wj �Mj

. (5.32). . . . . . . . . . . . . . . . . . . . . . . . . . . . .

However, the molecular weight of the heaviest fraction, CN, is notknown. From a mass balance, MN is given by

MN �wN

wC7��MC7�

� �N1

i�7

wi�Mi�

, (5.33). . . . . . . . . . . .

where Mi for i�7,…, N1 are taken from Table 5.2. Unfortunately,the calculated molecular weight MN is often unrealistic because ofmeasurement errors in MC7� or in the chromatographic analysis andbecause generalized molecular weights are only approximate. BothwN and MC7� can be adjusted to give a “reasonable” MN, but cautionis required to avoid nonphysical adjustments. The same problem isinherent with backcalculating MN with any set of generalized molec-ular weights used for SCN Fractions 7 to N1 (e.g., paraffin values).

During the remainder of this section, molecular weights and molefractions are assumed to be known for C7� fractions, either fromchromatographic analysis or from a synthetic split. The generalizedproperties for specific gravity and boiling point can be assigned toSCN fractions, but the heaviest specific gravity must be backcalcu-lated to match the measured C7� specific gravity. The calculated �Nalso may be unrealistic, requiring some adjustment to generalizedspecific gravities. Finally, the boiling point of the heaviest fractionmust be estimated. TbN can be estimated from a correlation relatingboiling point to specific gravity and molecular weight.

5.4.2 Characterization Factors. Inspection properties M, �, and Tbreflect the chemical makeup of petroleum fractions. Some methodsfor estimating specific gravity and boiling point assume that a par-ticular characterization factor is constant for all C7� fractions.These methods are only approximate but are widely used.

Watson or Universal Oil Products (UOP) Characterization Fac-tor. The Watson or UOP factor, Kw, is based on normal boiling point,Tb , in °R and specific gravity, �.32,33

Kw �

T1�3b�

. (5.34). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Kw varies roughly from 8.5 to 13.5. For paraffinic compounds,Kw�12.5 to 13.5; for naphthenic compounds, Kw�11.0 to 12.5;and for aromatic compounds, Kw�8.5 to 11.0. Some overlap in Kwexists among these three families of hydrocarbons, and a combina-tion of paraffins and aromatics will obviously “appear” naphthenic.However, the utility of this and other characterization factors is thatthey give a qualitative measure of the composition of a petroleumfraction. The Watson characterization factor has been found to beuseful for approximate characterization and is widely used as a pa-rameter for correlating petroleum-fraction properties, such as mo-lecular weight, viscosity, vapor pressure, and critical properties.

An approximate relation2 for the Watson factor, based on molecu-lar weight and specific gravity, is

Kw � 4.5579 M 0.15178�0.84573. (5.35). . . . . . . . . . . . . . . . . .

This relation is derived from the Riazi-Daubert14 correlation formolecular weight and is generally valid for petroleum fractions withnormal boiling points ranging from 560 to 1,310°R (C7 throughC30). Experience has shown, however, that Eq. 5.35 is not very ac-curate for fractions heavier than C20.

Kw calculated with MC7� and �C7� in Eq. 5.35 is often constant

for a given field. Figs. 5.13A and 5.13B7 plot molecular weight vs.specific gravity for C7� fractions from two North Sea fields. Datafor the gas condensate in Fig. 5.13A indicate an averageKwC7��11.99�0.01 for a range of molecular weights from 135 to150. The volatile oil shown in Fig. 5.13B has an averageKwC7��11.90�0.01 for a range of molecular weights from 220 to


Fig. 5.13A—Specific gravity vs. molecular weight for C7� frac-tions for a North Sea Gas-Condensate Field 2 (after Austad et al.7).

Molecular Weight, MC 7+

255. The high degree of correlation for these two fields suggests ac-curate molecular-weight measurements by the laboratory. In gener-al, the spread in KwC7� values will exceed �0.01 when measure-ments are performed by a commercial laboratory.

When the characterization factor for a field can be determined,Eq. 5.35 is useful for checking the consistency of C7� molecular-weight and specific-gravity measurements. Significant deviation inKwC7�

, such as �0.03 for the North Sea fields above, indicates pos-sible error in the measured data. Because molecular weight is moreprone to error than determination of specific gravity, an anomalousKwC7�

usually indicates an erroneous molecular-weight measure-ment. For the gas condensate in Fig. 5.13A, a C7� sample with spe-cific gravity of 0.775 would be expected to have a molecular weightof �141 (for KwC7�

� 11.99). If the measured value was 135, theWatson characterization factor would be 11.90, which is significant-ly lower than the field average of 11.99. In this case, the C7� molec-ular weight should be redetermined.

Eq. 5.35 can also be used to calculate specific gravity of C7� frac-tions determined by simulated distillation or a synthetic split (i.e.,when only mole fractions and molecular weights are known). As-suming a constant Kw for each fraction, specific gravity, �i, can becalculated from

�i � 6.0108 M 0.17947i

K 1.18241w . (5.36). . . . . . . . . . . . . . . . .

Kw must be chosen so that experimentally measured C7� specificgravity, (�C7�

)exp, is calculated correctly.

�C7��

exp�

zC7�MC7�

�N

i�1

zi Mi��i�

. (5.37). . . . . . . . . . . . . . . . . . . . .

The Watson factor satisfying Eq. 5.37 is given by

Kw � �0.16637�C7�

A0

zC7�MC7�

�0.84573

, (5.38). . . . . . . . . . . . . . .

where A0 ��N

i�1

zi M0.82053i

. (5.39). . . . . . . . . . . . . . . . . . . . . .

Fig. 5.13B—Specific gravity vs. molecular weight for C7� frac-tions for a North Sea Volatile-Oil Field 3B(after Austad et al.7).

Molecular Weight, MC 7+

Fig. 5.14—Specific gravity vs. molecular weight for constant val-ues of the Jacoby aromaticity factor (solid lines) and the Watsoncharacterization factor (dashed lines). After Whitson.25

Ja

Kw

Jacoby Correlation(Aromaticity Factor, Ja)

Present Correlation(Watson Factor, Kw)

Boiling points, Tbi, can be estimated from Eq. 5.36.

Tbi � (Kw�i)3. (5.40). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Unfortunately, Eqs. 5.36 through 5.40 overpredict � and Tb at mo-lecular weights greater than �250 (an original limitation of theRiazi-Daubert14 molecular-weight correlation).

Jacoby Aromaticity Factor. The Jacoby aromaticity factor, Ja , isan alternative characterization factor for describing the relativecomposition of petroleum fractions.34 Fig. 5.142 shows the originalJacoby relation between specific gravity and molecular weight forseveral values of Ja . The behavior of specific gravity as a functionof molecular weight is similar for the Jacoby factor and the relationfor a constant Kw. However, specific gravity calculated with theJacoby method increases more rapidly at low molecular weights,flattening at high molecular weights (a more physically consistentbehavior). A relation for the Jacoby factor is

Ja �� 0.8468 � 15.8�M�

0.2456 1.77�M�(5.41). . . . . . . . . . . . . . . . . .

12 PHASE BEHAVIOR

Fig. 5.15—Specific gravity vs. carbon number for constant val-ues of the Yarborough aromaticity factor (after Yarborough1).

or, in terms of specific gravity,

� � 0.8468 15.8M

� Ja 0.2456 1.77M�. (5.42). . . . . .

The first two terms in Eq. 5.42 (i.e., when Ja�0) express the relationbetween specific gravity and molecular weight for normal paraffins.

The Jacoby factor can also be used to estimate fraction specificgravities when mole fractions and molecular weights are availablefrom simulated distillation or a synthetic split. The Jacoby factorsatisfying measured C7� specific gravity (Eq. 5.37) must be calcu-lated by trial and error. We have found that this relation is particular-ly accurate for gas-condensate systems.27

Yarborough Aromaticity Factor. Yarborough1 modified theJacoby aromaticity factor specifically for estimating specific gravi-ties when mole fractions and molecular weights are known. Yarbo-rough tries to improve the original Jacoby relation by reflecting thechanging character of fractions up to C13 better and by representingthe larger naphthenic content of heavier fractions better. Fig. 5.15shows how the Yarborough aromaticity factor, Ya , is related to spe-cific gravity and carbon number. A simple relation representing Yais not available; however, Whitson26 has fit the seven aromaticitycurves originally presented by Yarborough using the equation

�i � exp�A0 � A1 i1� A2 i � A3 ln(i)� , (5.43). . . . . . . . . .

where i�carbon number. Table 5.7 gives the constants for Eq. 5.43.The aromaticity factor required to satisfy measured C7� specificgravity (Eq. 5.37) is determined by trial and error. Linear interpola-tion of specific gravity should be used to calculate specific gravityfor a Ya value falling between two values of Ya in Table 5.7.

Søreide35 Correlations. Søreide developed an accurate specific-gravity correlation based on the analysis of 843 TBP fractions from68 reservoir C7� samples.

�i � 0.2855 � Cf (Mi 66)0.13. (5.44). . . . . . . . . . . . . . .

Cf typically has a value between 0.27 and 0.31 and is determined fora specific C7� sample by satisfying Eq. 5.37.

5.4.3 Boiling-Point Estimation. Boiling point can be estimatedfrom molecular weight and specific gravity with one of several cor-relations. Søreide also developed a boiling-point correlation basedon 843 TBP fractions from 68 reservoir C7� samples,

Tb � 1928.3 1.695 � 105� M0.03522�

3.266

� exp � 4.922 � 103�M 4.7685�

� 3.462 � 103�M�� , (5.45). . . . . . . . . . . . . . . . . . . . .

with Tb in °R.Table 5.8 gives estimated specific gravities determined with the

methods just described for a C7� sample with the exponential splitgiven in Table 5.4 (��1, ��90, MC7�

�200) and �C7��0.832.

The following equations also relate molecular weight to boilingpoint and specific gravity; any of these correlations can be solvedfor boiling point in terms of M and �. We recommend, however, theSøreide correlation for estimating Tb from M and �.

Kesler and Lee.12

M � � 12, 272.6 � 9, 486.4�� (4.6523 3.3287�)Tb�

� 1 0.77084� 0.02058�2�

� �1.3437 720.79T1b� � 107�T1

b�

� 1 0.80882�� 0.02226�2�

� �1.8828 181.98T–1b� � 1012�T3

b� . (5.46). . . . . . . .

Riazi and Daubert.14

M � (4.5673 � 105)T 2.1962b

�1.0164. (5.47). . . . . . . . . . . .

American Petroleum Inst. (API).36

M � 2.0438 � 102�T 0.118b

�1.88 exp0.00218Tb 3.07�� .

(5.48). . . . . . . . . . . . . . . . . . . .

Rao and Bardon.37

ln M � (1.27 � 0.071Kw) ln1.8Tb

22.31 � 1.68Kw� .

(5.49). . . . . . . . . . . . . . . . . . . .

Riazi and Daubert.18

M � 581.96T 0.97476b

�6.51274 exp�5.43076 � 103�Tb

9.53384�� 1.11056 � 103�Tb�� . (5.50). . . . . . . . .

TABLE 5.7—COEFFICIENTS FOR YARBOROUGH AROMATICITY FACTOR CORRELATION1,26

Ya A0 A1 A2 A2

0.0 7.43855�1021.72341 1.38058�103 3.34169�102

0.1 4.25800�101 7.00017�101 3.30947�105 8.65465�102

0.2 4.47553�101 7.65111�101 1.77982�104 1.07746�101

0.3 4.39105�101 9.44068�101 4.93708�104 1.19267�101

0.4 2.73719�1011.39960 3.80564�103 5.92005�102

0.6 7.39412�1031.97063 5.87273�103 1.67141�102

0.8 3.17618�101 7.78432�101 2.58616�103 1.08382�103


TABLE 5.8—COMPARISON OF SPECIFIC GRAVITIES WITH CORRELATIONS BY USE OFDIFFERENT CHARACTERIZATION FACTORS

�C7�� 0.832

�i for Different Correlations With Constant CharacterizationFactor Chosen To Match

Fraction zi Mi Kw�12.080 Ja�0.2395 Ya�0.2794 Cf�0.2864

1 0.1195 96.8 0.7177 0.7472 0.7051 0.7327

2 0.1052 110.8 0.7353 0.7684 0.7286 0.7550

3 0.0926 124.8 0.7511 0.7849 0.7486 0.7719

4 0.0816 138.8 0.7656 0.7981 0.7660 0.7856

5 0.0718 152.8 0.7789 0.8088 0.7813 0.7972

6 0.0632 166.8 0.7913 0.8178 0.7951 0.8072

7 0.0557 180.8 0.8028 0.8253 0.8075 0.8161

8 0.0490 194.8 0.8136 0.8318 0.8189 0.8241

9 0.0432 208.8 0.8238 0.8374 0.8294 0.8314

10 0.0380 222.8 0.8335 0.8423 0.8391 0.8380

11 0.0335 236.8 0.8426 0.8466 0.8482 0.8442

12 0.0295 250.8 0.8514 0.8505 0.8567 0.8500

13 0.0259 264.8 0.8597 0.8539 0.8646 0.8554

14 0.0228 278.8 0.8677 0.8570 0.8722 0.8604

15 0.0201 292.8 0.8753 0.8598 0.8793 0.8652

16 0.0177 306.8 0.8827 0.8623 0.8861 0.8697

17 0.0156 320.8 0.8898 0.8646 0.8926 0.8740

18 0.0137 334.8 0.8966 0.8668 0.8988 0.8782

19 0.0121 348.8 0.9033 0.8687 0.9048 0.8821

20 0.0891 466.0 0.9514 0.8805 0.9468 0.9096

Total 1.0000 200.0 0.8320 0.8320 0.8320 0.8320

��

Thus far, we have discussed how to split the C7� fraction intopseudocomponents described by mole fraction, molecular weight,specific gravity, and boiling point. Now we must consider the prob-lem of assigning critical properties to each pseudocomponent. Criti-cal temperature, Tc; critical pressure, pc; and acentric factor, �, ofeach component in a mixture are required by most cubic EOS’s.Critical volume, vc, is used instead of critical pressure in the Bene-dict-Webb-Rubin38 (BWR) EOS, and critical molar volume isused with the LBC viscosity correlation.24 Critical compressibilityfactor has been introduced as a parameter in three- and four-constantcubic EOS’s.

Critical-property estimation of petroleum fractions has a long his-tory beginning as early as the 1930’s; several reviews22,25,26,39,40

are available. We present the most commonly used correlations anda graphical comparison (Figs. 5.16 through 5.18) that is intendedto highlight differences between the correlations. Finally, correla-tions based on perturbation expansion (a concept borrowed fromstatistical mechanics) are discussed separately.

The units for the remaining equations in this section are Tb in °R,TbF in °F�Tb459.67, Tc in °R, pc in psia, and vc in ft3/lbm mol.Oil gravity is denoted �API and is related to specific gravity by�API�141.5/�131.5.

5.5.1 Critical Temperature. Tc is perhaps the most reliably corre-lated critical property for petroleum fractions. The following criti-cal-temperature correlations can be used for petroleum fractions.

Roess.41 (modified by API36).

Tc � 645.83 � 1.6667��TbF � 100��

0.7127 � 103��TbF � 100��2. (5.51). . . . . . . . . . .

Kesler-Lee.12

Tc � 341.7 � 811�� (0.4244 � 0.1174�)Tb

� (0.4669 3.2623�) � 105T1b . (5.52). . . . . . . . . . . .

Cavett.42

Tc � 768.07121 � 1.7133693TbF

0.10834003 � 102�T 2bF

0.89212579 � 102��APITbF

� 0.38890584 � 106�T 3bF

� 0.5309492 � 105��APIT2bF

� 0.327116 � 107��2APIT

2bF

. (5.53). . . . . . . . . . . . . . .

Riazi-Daubert.14

Tc � 24.27871T 0.58848b

�0.3596. (5.54). . . . . . . . . . . . . . . . . .

Nokay.43

Tc � 19.078T 0.62164b

�0.2985 . (5.55). . . . . . . . . . . . . . . . . . . .

5.5.2 Critical Pressure. pc correlations are less reliable than Tc cor-relations. The following are pc correlations that can be used for pe-troleum fractions.

Kesler-Lee.12

ln pc � 8.3634 0.0566

�

�0.24244 �2.2898

��

0.11857�2�� 103�Tb

14 PHASE BEHAVIOR

Fig. 5.16—Comparison of critical-temperature correlations forboiling points from 600 to 1,500°R assuming a constant Watsoncharacterization factor of 12.

��1.4685 �3.648�

�0.47227

�2�� 107�T2

b

�0.42019 �1.6977�2�� 1010�T3

b . (5.56). . . . .

Cavett.42

log pc � 2.8290406 � 0.94120109 � 103�TbF

0.30474749 � 105�T 2bF

0.2087611 � 104��APITbF

� 0.15184103 � 108�T 3bF

� 0.11047899 � 107��APIT2bF

0.48271599 � 107��2APITbF

� 0.13949619 � 109��2APIT

2bF

. (5.57). . . . . . . . . . .

Riazi-Daubert.14

pc � 3.12281 � 109�T2.3125b �

2.3201 . (5.58). . . . . . . . . . . . .

5.5.3 Acentric Factor. Pitzer et al.44 defined acentric factor as

� � log p*v

pc� 1, (5.59). . . . . . . . . . . . . . . . . . . . . . . .

where p*v�vapor pressure at temperature T�0.7Tc (Tr�0.7).

Practically, acentric factor gives a measure of the steepness of thevapor-pressure curve from Tr�0.7 to Tr�1, where p*

v/pc�0.1 for��0 and p*

v/pc�0.01 for ��1. Numerically, ��0.01 for meth-ane, �0.25 for C5, and�0.5 for C8 (see Table A.1 for literature val-ues of acentric factor for pure compounds). � increases to �1.0 forpetroleum fractions heavier than approximately C25 (see Table 5.2).

The Kesler-Lee12 acentric factor correlation (for Tb/Tc�0.8) isdeveloped specifically for petroleum fractions, whereas the correla-tion for Tb/Tc�0.8 is based on an accurate vapor-pressure correla-tion for pure compounds. The Edmister45 correlation is limited topure hydrocarbons and should not be used for C7� fractions. Thethree correlations follow.

Fig. 5.17—Comparison of critical-pressure correlations for boil-ing points from 600 to 1,500°R assuming a constant Watsoncharacterization factor of 12.

Fig. 5.18—Comparison of acentric factor correlations for boilingpoints from 600 to 1500°R assuming a constant Watson charac-terization factor of 12.

Lee-Kesler.13 (Tbr�Tb/Tc�0.8).

� �

– lnpc�14.7� � A1 � A2 T1br

� A3 ln Tbr � A4 T 6br

A5 � A6 T1br � A7 ln Tbr � A8 T 6

br

,

(5.60). . . . . . . . . . . . . . . . . . . .

where A1�5.92714, A2� 6.09648, A3� 1.28862, A4�0.169347, A5� 15.2518, A6�15.6875, A7�13.4721,and A8� 0.43577.

Kesler-Lee.12 (Tbr�Tb/Tc�0.8).

� � 7.904 � 0.1352Kw 0.007465K2w

� 8.359Tbr � (1.408 0.01063Kw)T1br . (5.61). . . . . . .

Edmister.45

� �37

logpc�14.7�

�Tc�Tb� 1�

1. (5.62). . . . . . . . . . . . . . . . . . . . .


5.5.4 Critical Volume. The Hall-Yarborough46 critical-volumecorrelation is given in terms of molecular weight and specific grav-ity, whereas the Riazi-Daubert14 correlation uses normal boilingpoint and specific gravity.

Hall-Yarborough.46

vc � 0.025M 1.15�0.7935. (5.63). . . . . . . . . . . . . . . . . . . . . .

Riazi-Daubert.14

vc � 7.0434 � 107�T 2.3829b

�1.683. (5.64). . . . . . . . . . . . .

Critical compressibility factor, Zc, is defined as

Zc �pcvc

RTc, (5.65). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

where R�universal gas constant. Thus, Zc can be calculated directlyfrom critical pressure, critical volume, and critical temperature. Reidet al.40 and Pitzer et al.44 give an approximate relation for Zc.

Zc � 0.291 0.08�. (5.66). . . . . . . . . . . . . . . . . . . . . . . . .

Eq. 5.66 is not particularly accurate (grossly overestimating Zc forheavier compounds) and is used only for approximate calculations.

5.5.5 Correlations Based on Perturbation Expansions. Correla-tions for critical temperature, critical pressure, critical volume, andmolecular weight have been developed for petroleum fractions witha perturbation-expansion model with normal paraffins as the refer-ence system. To calculate critical pressure, for example, criticaltemperature, critical volume, and specific gravity of a paraffin withthe same boiling point as the petroleum fraction must be calculatedfirst. Kesler et al.47 first used the perturbation expansion (with n-al-kanes as the reference fluid) to develop a suite of critical-propertyand acentric-factor correlations.

Twu48 uses the same approach to develop a suite of critical-prop-erty correlations. We give his normal-paraffin correlations first,then the correlations for petroleum fractions.

Normal Paraffins (Alkanes).

TcP � Tb�0.533272 � 0.191017 � 103�Tb

� 0.779681 � 107�T 2b 0.284376 � 1010�T 3

b

�(0.959468 � 102)

0.01Tb�13�

1

, (5.67). . . . . . . . . . . . . . . . . .

pcP � (3.83354 � 1.19629�0.5� 34.8888�

� 36.1952�2� 104.193�4)2 , (5.68). . . . . . . . . . . . . . .

vcP � [ 1 (0.419869 0.505839� 1.56436�3

9481.7�14)]8 , (5.69). . . . . . . . . . . . . . . . . . . . . . . . .

�P � 0.843593 0.128624� 3.36159�3

13749.5�12 , (5.70). . . . . . . . . . . . . . . . . . . . . . . . . . . .

and Tb � exp(5.71419 � 2.71579� 0.28659�2

39.8544�1 0.122488�2)

24.7522�� 35.3155�2 , (5.71). . . . . . . . . . . . . . . . .

where � � 1 Tb

TcP(5.72). . . . . . . . . . . . . . . . . . . . . . . . . . . .

and � � ln MP . (5.73). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Paraffin molecular weight, MP, is not explicitly a function of Tb , andEqs. 5.67 through 5.73 must be solved iteratively; an initial guessis given by

MP �Tb

10.44 0.0052Tb. (5.74). . . . . . . . . . . . . . . . . . . . .

Twu claims that the normal-paraffin correlations are valid for C1through C100, although the properties at higher carbon numbers areonly approximate because experimental data for paraffins heavierthan approximately C20 do not exist. The following relations areused to calculate petroleum-fraction properties.

Critical Temperature.

Tc � TcP1 � 2fT

1 2fT�

2

,

fT � ��T� 0.362456T 0.5

b

�0.0398285 0.948125

T 0.5b

��T�,

and ��T � exp[5(�P �)] 1. (5.75). . . . . . . . . . . . . . . . . .

Critical Volume.

vc � vcP1 � 2fv

1 2fv�

2

,

fv � ��v�0.466590T 0 .5

b

� 0.182421 �3.01721

T 0.5b

��v�,

and ��v � exp�4�2P �

2�� 1. (5.76). . . . . . . . . . . . . . . . . .

Critical Pressure.

pc � pcPTc

TcP�VcP

Vc�1 � 2fp

1 2fp�

2

,

fp � ��p�2.53262 46.1955

T 0.5b

0.00127885Tb�

� 11.4277 �252.14

T 0.5b

� 0.00230535Tb��p�,

and ��p � exp[0.5(�P �)] 1. (5.77). . . . . . . . . . . . . . . . .

Molecular Weight.

ln M � ln MP1 � 2fM

1 2fM�

2

,

fM � ��M�|x| � 0.0175691 �0.193168

T 0.5b

��M�,

x � 0.012342 0.328086

T 0.5b

,

and ��M � exp[5(�P �)]. (5.78). . . . . . . . . . . . . . . . . . . . .

Figs. 5.16 through 5.18 compare the various critical-property cor-relations for a range of boiling points from 600 to 1,500°R.

5.5.6 Methods Based on an EOS. Fig. 5.1928 illustrates the impor-tant influence that critical properties have on EOS-calculated proper-ties of pure components. Vapor pressure is particularly sensitive tocritical temperature. For example, the Riazi-Daubert19 critical-tem-perature correlation for toluene overpredicts the experimental value

16 PHASE BEHAVIOR

Fig. 5.19—Effect of critical temperature on vapor-pressure pre-diction of toluene with the PR EOS; AAD�absolute average devi-ation (after Brulé et al.28).

Tc underpredicted← →Tc overpredicted

Deviation From Experimental Value, %

by only 1.7%. Even with this slight error in Tc, the average error invapor pressures predicted by the Peng-Robinson49 (PR) EOS is 16%.The effect of critical properties and acentric factor on EOS calcula-tions for reservoir-fluid mixtures is summarized by Whitson.26

In principle, the EOS used for mixtures should also predict the be-havior of individual components found in the mixture. For purecompounds, the vapor pressure is accurately predicted because allEOS’s force fit vapor-pressure data. Some EOS’s are also fit to satu-rated-liquid densities at subcritical temperatures. The measuredproperties of petroleum fractions, boiling point, and specific gravitycan also be fit by the EOS, as discussed later.

For each petroleum fraction separately, two of the EOS parame-ters (Tc; pc; �; volume-shift factor, s; or multipliers of EOS constantsA and B) can be chosen so that the EOS exactly reproduces exper-imental boiling point and specific gravity. Because only two inspec-tion properties are available (Tb and �), only two of the EOS parame-ters can be determined. Whitson50 suggests fixing the value of �with an empirical correlation and adjusting Tc and pc to match nor-mal boiling point and molar volume (M/�) at standard conditions.Critical properties satisfying these criteria are given for a wide rangeof petroleum fractions by the PR EOS and the Soave-Redlich-Kwong (SRK) EOS.22,23 A better (and recommended) approach forcubic EOS’s is to use the volume-shift factor s (see Chap. 4) to matchspecific gravity or a saturated liquid density and acentric factor tomatch normal boiling point.

Other methods for forcing the EOS to match boiling point andspecific gravity have also been devised. Brulé and Starling51 pro-posed a method that uses viscosity as an additional inspection prop-erty of the fraction for determining critical properties. This ap-proach proved particularly successful when applied to the BWREOS for residual-oil supercritical extraction (ROSE).28

��# $�� %� ��

We recommend the following C7� characterization procedure forcubic EOS’s.

1. Use the Twu48 (or Lee-Kesler12) critical property correlationfor Tc and pc .

2. Choose the acentric factor to match Tb; alternatively, use theLee-Kesler12/Kesler-Lee13 correlations.

3. Determine volume-translation coefficients, si, to match specificgravities; alternatively, use Peneloux et al.’s52 correlation for the SRKEOS22,23 or Jhaveri and Youngren’s53 correlation for the PR EOS.49

When measured TBP data are not available, a mathematical splitshould be made with either (1) the gamma distribution (default

��1, ��90) with Gaussian-quadrature or equal-mass fractions or(2) the exponential distribution (Eq. 5.7). Specific gravities shouldbe estimated with the Søreide35 correlation (Eq. 5.44), choosing Cfto match measured C7� specific gravity (Eq. 5.37). Boiling pointsshould be estimated from the Søreide correlation (Eq. 5.45).

For the PR EOS, we recommend the nonhydrocarbon BIP’s givenin Chap. 4 and the modified Chueh-Prausnitz54 equation for C1through C7� pairs,

kij � A��

�

1 2v1�6

civ1�6

cj

v1�3ci

� v1�3cj

�B

��

�

, (5.79). . . . . . . . . . . . . . . .

with A�0.18 and B�6.

5.6.1 SRK-Recommended Characterization. Alternatively, thePedersen et al.55 characterization procedure can be used with theSRK EOS.

1. Split the plus fraction Cn� (preferably n�10) into SCN frac-tions up to C80 using Eqs. 5.7 through 5.11 and h�4.

2. Calculate SCN densities �i (�i� �i /0.999) using the equation�i�A0�A1 ln(i), where A0 and A1 are determined by satisfying theexperimental-plus density, �n�, and measured (or assumed) densi-ty, �n1 ( �6�0.690 can be used for C7�).

3. Calculate critical properties of all C7� fractions (distillationcuts from C7 to Cn1 and split SCN fractions from Cn through C80)using the correlations

Tc � 163.12�� 86.052 ln M � 0.43475M 1877.4

M,

ln pc � 0.13408 � 2.5019��208.46

M

3987.2M2 ,

and mSRK � 0.48 � 1.574� 0.176�2

� 0.7431 � 0.0048122M � 0.0096707�

3.7184 � 106�M2. (5.80). . . . . . . . . . . . . . . . . .

Note that the use of acentric factor is circumvented by directly calcu-lating the term m used in the � correction term to EOS Constant A.

4. Group C7� into 3 to 12 fractions using equal-weight fractionsin each group; use weight-average mixing rules.

5. Calculate volume-translation parameters for C7� fractions tomatch specific gravities; pure component c values are taken fromPeneloux et al.52

6. All hydrocarbon/hydrocarbon BIP’s are set to zero. SRK BIP’sgiven in Chap. 4 are used for nonhydrocarbon/hydrocarbon pairs.

The two recommended C7� characterization procedures out-lined previously for the PR EOS and SRK EOS are probably the bestcurrently available (other EOS characterizations, such as the Re-dlich-Kwong EOS modified by Zudkevitch and Joffe,56 and somethree-constant characterizations should provide similar accuracybut are not significantly better). Practically, the two characterizationprocedures give the same results for almost all PVT properties (usu-ally within 1 to 2%). With these EOS-characterization procedures,we can expect reasonable predictions of densities and Z factors (�1to 5%), saturation pressures (�5 to 15%), gas/oil ratios and forma-tion volume factors (�2 to 5%), and condensate-liquid dropout(�5 to 10% for maximum dropout, with poorer prediction of tail-like behavior just below the dewpoint).

The recommended EOS methods are less reliable for predictionof minimum miscibility conditions, near-critical saturation pressureand saturation type (bubblepoint or dewpoint), and both retrogradeand near-critical liquid volumes. Improved predictions can be ob-tained only by tuning EOS parameters to accurate PVT data cover-ing a relatively wide range of pressures, temperatures, and composi-tions (see Sec. 4.7 and Appendix C).

��% &��' �� (��'�' ��

The cost and computer resources required for compositional reser-voir simulation increase substantially with the number of compo-


nents used to describe the reservoir fluid. A compromise betweenaccuracy and the number of components must be made accordingto the process being simulated (i.e., according to the expected effectthat phase behavior will have on simulated results). For example, adetailed fluid description with 12 to 15 components may be neededto simulate developed miscibility in a slim-tube experiment. Withcurrent computer technology, however, a full-field simulation withfluids exhibiting near-critical phase behavior is not feasible for a15-component mixture. The following are the main questions re-garding component grouping.

1. How many components should be used?2. How should the components be chosen from the original fluid

description?3. How should the properties of pseudocomponents be calculated?

5.7.1 How Many and Which Components To Group. The numberof components used to describe a reservoir fluid depends mainly onthe process being simulated. However, the following rule of thumbreduces the number of components for most systems: group N2 withmethane, CO2 with ethane, iso-butane with n-butane, and iso-pen-tane with n-pentane. Nonhydrocarbon content should be less thana few percent in both the reservoir fluid and the injection gas if anonhydrocarbon is to be grouped with a hydrocarbon.

Five- to eight-component fluid characterizations should be suffi-cient to simulate practically any reservoir process, including (1) reser-voir depletion of volatile-oil and gas-condensate reservoirs, (2) gascycling above and below the dewpoint of a gas-condensate reservoir,(3) retrograde condensation near the wellbore of a producing well,and (4) immiscible and miscible gas-injection. Coats57 discusses amethod for combining a modified black-oil formula with a simplifiedEOS representation of separator oil and gas streams. The “oil” and“gas” pseudocomponents in this model contain all the original fluidcomponents in contrast to the typical method of grouping where eachpseudocomponent is made up of only selected original components.

Lee et al.58 suggest that C7� fractions can be grouped into twopseudocomponents according to a characterization factor deter-mined by averaging the tangents of fraction properties M, �, and Japlotted vs. boiling point.

Whitson2 suggests that the C7� fraction can be grouped into NHpseudocomponents given by

NH � 1 � 3.3 log(N 7), (5.81). . . . . . . . . . . . . . . . . . . . .

where N�carbon number of the heaviest fraction in the originalfluid description. The groups are separated by molecular weights MIgiven by

MI � MC7MN�MC7

�1�NH

, (5.82). . . . . . . . . . . . . . . . . . .

where I�1,..., NH . Molecular weights, Mi, from the original fluiddescription (i�7,..., N) falling within boundaries MI1 to MI are in-cluded in Group I. This method should only be used when C7� frac-tions are originally separated on a carbon-number basis and for Ngreater than �20.

Li et al.59 suggest a method for grouping components of an origi-nal fluid description that uses K values from a flash at reservoir tem-perature and the “average” operating pressure. The original mixtureis divided arbitrarily into “light” components (H2S, N2, CO2, and C1through C6) and “heavy” components (C7�). Different criteria areused to determine the number of light and heavy pseudocompon-ents. Li et al. also suggest use of phase diagrams and compositionalsimulation to verify the grouped fluid description (a practice that wehighly recommend).

Still other pseudoization methods have been proposed60,61; Schlij-per’s61 method also treats the problem of retrieving detailed composi-tional information from pseudoized (grouped) components. Behrensand Sandler62 suggest a grouping method for C7� fractions basedon application of the Gaussian-quadrature method to continuousthermodynamics. Although a simple exponential distribution isused with only two quadrature points (i.e., the C7� fractions aregrouped into two pseudocomponents), Whitson et al.27 show that

the method is general and can be applied to any molar-distributionmodel and for any number of C7� groups.

In general, most authors have found that broader grouping of C7�as C7 through C10, C11 through C15, C16 through C20, and C21� issubstantially better than splitting only the first few carbon-numberfractions (e.g., C7, C8, C9, and C10�). Gaussian quadrature is recom-mended for choosing the pseudocomponents in a C7� fraction;equal-mass fractions or the Li et al.59 approach are valid alternatives.

5.7.2 Mixing Rules. Several methods have been proposed for calcu-lating critical properties of pseudocomponents. The simplest andmost common mixing rule is

�I �

�i�I

zi�i

�i�I

zi

, (5.83). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

where �i�any property (Tc, pc, �, or M) and zi�original mole frac-tion for components (i�1,..., I) making up Pseudocomponent I. Av-erage specific gravity should always be calculated with the assump-tion of ideal solution mixing.

�I �

�i�I

zi Mi

�i�I

zi Mi��i�. (5.84). . . . . . . . . . . . . . . . . . . . . . . . . . .

Pedersen et al.55 and others suggest use of weight fraction insteadof mole fraction. Wu and Batycky’s63 empirical mixing-rule ap-proach uses both the molar- and weight-average mixing rules anda proportioning factor, F, to calculate pcI, TcI, and �I.

�I ��i�I

�i�i , (5.85). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

where �I represents pcI, TcI, and �I and �i�average of the molar andweight fractions,

�i � F�izi � (1 F)�i wi

and wi �zi Mi

�N

j�1

zj Mj

, (5.86). . . . . . . . . . . . . . . . . . . . . . . . . . . . .

with 0�F�1.A generalized mixing rule for BIP’s can be written

kIJ ��i�I

�j�J

�i�j kij , (5.87). . . . . . . . . . . . . . . . . . . . . . . . .

where �i is also given by Eq. 5.86.On the basis of Chueh and Prausnitz’s54 arguments, Lee-Kesler13

proposed the mixing rules in Eqs. 5.88 through 5.92.

vcI � �18�i�I

�j�J

zi zjv1�3

ci� v1�3

cj�

3

��i�I

zi�2

, (5.88). . .

TcI � � 18vcI

�i�I

�j�J

zi zjTci Tcj

�1�2v1�3ci

� v1�3cj�

3

�

��i�I

zi�2

, (5.89). . . . . . . . . . . . . . . . . . . . . . . . . . . .

�I � �i�I

zi�i��i�I

zi�, (5.90). . . . . . . . . . . . . . . . . . .

ZcI � 0.2905 0.085�I , (5.91). . . . . . . . . . . . . . . . . . . . . .

and pcI �ZcI RTcI

vcI. (5.92). . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18 PHASE BEHAVIOR

TABLE 5.9—EXAMPLE STEPWISE-REGRESSION PROCEDURE FOR PSEUDOIZATIONTO FEWER COMPONENTS FOR A GAS CONDENSATE FLUID UNDERGOING DEPLETION

Original

Component

Number

Original

Component Step 1 Step 2 Step 3 Step 4 Step 5

1 N2 N2�C1* N2�C1 N2�C1 N2�C1�CO2�C2* N2�C1�CO2�C2

2 CO2 CO2�C2* CO2�C2 CO2�C2 C3�i-C4�n-C4

�i-C5�n-C5�C6*

C3�i-C4�n-C4

�i-C5�n-C5�C6

3 C1 C3 C3 C3�i-C4�n-C4* F1 F1

4 C2 i-C4 i-C4�n-C4* i-C5�n-C5�C6* F2 F2�F3*

5 C3 n-C4 i-C5�n-C5* F1 F3

6 i-C4 i-C5 C6 F2

7 n-C4 n-C5 F1 F3

8 i-C5 C6 F2

9 n-C5 F1 F3

10 C6 F2

11 F1 F3

12 F2

13 F3

Regression Parameters

kij 1, 9, 10, and 11 1, 7, 8, and 9 1, 5, 6, and 7 1, 3, 4, and 5 1, 3, and 4

�a 1 4 3 1 3

�b 1 4 3 1 3

�a 2 5 4 2 4

�b 2 5 4 2 4*Indicates the grouped pseudocomponents being regressed in a particular step.

Lee et al.58 and Whitson2 consider an alternative method for cal-culating C7� critical properties based on the specific gravities andboiling points of grouped pseudocomponents.

Coats57 presents a method of pseudoization that basically elimi-nates the effect of mixing rules on pseudocomponent properties.The approach is simple and accurate. Coats requires the pseudoizedcharacterization to reproduce exactly the volumetric behavior of theoriginal reservoir fluid at undersaturated conditions. This isachieved by ensuring that the mixture EOS constants A and B areidentical for the original and the pseudoized characterizations. First,pseudocritical properties ( pcI, TcI, and �I) are estimated with anymixing rule (e.g., Kay’s64 mixing rule). Then �aI and �bI are deter-mined to satisfy the following equations.

�aI �

��i�I

�j�J

zi zj aiaj1 kij

��i�I

zi�2

R2T 2cI�pcI��I(TrI,�I)

and �bI �

�i�I

zi bi��i�I

zi�

RTcI�pcI�I(TrI,�I)

, (5.93). . . . . . . . . . . . . . . .

where ai � �ai

R2T2ci

pci�i (Tri,�i)

and bi � �biRTcipci

i(Tri,�i) . (5.94). . . . . . . . . . . . . . . . . . . . .

�ai and �bi may include previously determined corrections to thenumerical constants �o

a and �ob. This approach to determining

pseudocomponent properties, together with Eq. 5.87 for kI J, is sur-prisingly accurate even for VLE calculations. Coats also gives an

analogous procedure for determining pseudocomponent vcI for theLBC24 viscosity correlation.

Coats’ approach is preferred to all the other proposed methods. Itensures accurate volumetric calculations that are consistent with theoriginal EOS characterization, and the method is easy to implement.

5.7.3 Stepwise Regression. A reduced-component characterizationshould strive to reproduce the original complete characterizationthat has been used to match measured PVT data. One approach toachieve this goal is stepwise regression, summarized in the follow-ing procedure.

1. Complete a comprehensive match of all existing PVT data witha characterization containing light and intermediate pure compo-nents and at least three to five C7� fractions.

2. Simulate a suite of depletion and multicontact gas-injectionPVT experiments that cover the expected range of compositions inthe particular application.

3. Use the simulated PVT data as “real” data for pseudoizationbased on regression.

4. Create two new pseudocomponents from the existing set ofcomponents. Use the pseudoization procedure of Coats to obtain�aI and �bI values, and use Eq. 5.87 for kI J.

5. Use regression to fine tune the �aI and �bI values estimatedin Step 4; also regress on key BIP’s, such as (N2�C1)C7�,(CO2�C2)C7�, and other nonzero BIP’s involving pseudocom-ponents from Step 4.

6. Repeat Steps 4 and 5 until the quality of the characterizationdeteriorates beyond an acceptable fluid description. Table 5.9shows an example five-step pseudoization procedure.

In summary, any grouping of a complete EOS characterizationinto a limited number of pseudocomponents should be checked toensure that predicted phase behavior (e.g., multicontact gas injec-tion data, saturation pressures, and densities) are reasonably closeto the predictions for the original (complete) characterization. Step-wise regression is the best approach to determine the number and


properties of pseudocomponents that can accurately describe a res-ervoir fluid’s phase behavior. If stepwise regression is not possible,standard grouping of the light and intermediates (N2�C1,CO2�C2, i-C4�n-C4, and i-C5�n-C5) and Gaussian quadraturefor C7� (or equal-mass fractions) is recommended; a valid alterna-tive is the Li et al.59 method. The Coats57 method (Eqs. 5.93and 5.94) is always recommended for calculating pseudocompon-ent properties.

$�)��

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9. MacAllister, D.J. and DeRuiter, R.A.: “Further Development and Ap-plication of Simulated Distillation for Enhanced Oil Recovery,” paperSPE 14335 presented at the 1985 SPE Annual Technical Conferenceand Exhibition, Las Vegas, Nevada, 22–25 September.

10. Designation D158, Saybolt Distillation of Crude Petroleum, AnnualBook of ASTM Standards, ASTM, Philadelphia, Pennsylvania (1984).

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18. Riazi, M.R. and Daubert, T.E.: “Analytical Correlations InterconvertDistillation-Curve Types,” Oil & Gas J. (August 1986) 50.

19. Riazi, M.R. and Daubert, T.E.: “Characterization Parameters for Petro-leum Fractions,” Ind. Eng. Chem. Res. (1987) 26, 755.

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21. Riazi, M.R. and Daubert, T.E.: “Prediction of the Composition of Petro-leum Fractions,” Ind. Eng. Chem. Proc. Des. Dev. (1980) 19, 289.

22. Pedersen, K.S., Thomassen, P., and Fredenslund, A.: “Thermodynam-ics of Petroleum Mixtures Containing Heavy Hydrocarbons. 1. PhaseEnvelope Calculations by Use of the Soave-Redlich-Kwong Equationof State,” Ind. Eng. Chem. Proc. Des. Dev. (1984) 23, 163.

23. Pedersen, K.S., Thomassen, P., and Fredenslund, A.: “Thermodynam-ics of Petroleum Mixtures Containing Heavy Hydrocarbons. 2. Flashand PVT Calculations with the SRK Equation of State,” Ind. Eng.Chem. Proc. Des. Dev. (1984) 23, 566.


25. Whitson, C.H.: “Effect of C7� Properties on Equation-of-State Predic-tions,” paper SPE 11200 presented at the 1982 SPE Annual TechnicalConference and Exhibition, New Orleans, 26–29 September.

26. Whitson, C.H.: “Effect of C7� Properties on Equation-of-State Predic-tions,” SPEJ (December 1984) 685; Trans., AIME, 277.

27. Whitson, C.H., Andersen, T.F., and Søreide, I.: “C7� Characteriza-tion of Related Equilibrium Fluids Using the Gamma Distribution,”C7� Fraction Characterization, L.G. Chorn and G.A. Mansoori(eds.), Advances in Thermodynamics, Taylor & Francis, New YorkCity (1989) 1, 35–56.

28. Brulé, M.R., Kumar, K.H., and Watansiri, S.: “CharacterizationMethods Improve Phase-Behavior Predictions,” Oil & Gas J. (11February 1985) 87.

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30. Abramowitz, M. and Stegun, I.A.: Handbook of Mathematical Func-tions, Dover Publications Inc., New York City (1970) 923.

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32. Watson, K.M., Nelson, E.F., and Murphy, G.B.: “Characterization ofPetroleum Fractions,” Ind. Eng. Chem. (1935) 27, 1460.

33. Watson, K.M. and Nelson, E.F.: “Improved Methods for Approximat-ing Critical and Thermal Properties of Petroleum,” Ind. Eng. Chem.(1933) 25, No. 8, 880.

34. Jacoby, R.H. and Rzasa, M.J.: “Equilibrium Vaporization Ratios for Ni-trogen, Methane, Carbon Dioxide, Ethane, and Hydrogen Sulfide inAbsorber Oil/Natural Gas and Crude Oil/Natural Gas Systems,” Trans.,AIME (1952) 195, 99.

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36. Technical Data Book—Petroleum Refining, third edition, API, NewYork City (1977).

37. Rao, V.K. and Bardon, M.F.: “Estimating the Molecular Weight of Pe-troleum Fractions,” Ind. Eng. Chem. Proc. Des. Dev. (1985) 24, 498.

38. Benedict, M., Webb, G.B., and Rubin, L.C.: “An Empirical Equationfor Thermodynamic Properties of Light Hydrocarbons and Their Mix-tures, I. Methane, Ethane, Propane, and n-Butane,” J. Chem. Phys.(1940) 8, 334.

39. Reid, R.C.: “Present, Past, and Future Property Estimation Tech-niques,” Chem. Eng. Prog. (1968) 64, No. 5, 1.

40. Reid, R.C., Prausnitz, J.M., and Polling, B.E.: The Properties of Gasesand Liquids, fourth edition, McGraw-Hill Book Co. Inc., New YorkCity (1987) 12–24.

41. Roess, L.C.: “Determination of Critical Temperature and Pressure ofPetroleum Fractions,” J. Inst. Pet. Tech. (October 1936) 22, 1270.

42. Cavett, R.H.: “Physical Data for Distillation Calculations-Vapor-Liq-uid Equilibria,” Proc., 27th API Meeting, San Francisco (1962) 351.

43. Nokay, R.: “Estimate Petrochemical Properties,” Chem. Eng. (23 Feb-ruary 1959) 147.

44. Pitzer, K.S. et al.: “The Volumetric and Thermodynamic Properties ofFluids, II. Compressibility Factor, Vapor Pressure, and Entropy of Va-porization,” J. Amer. Chem. Soc. (1955) 77, No. 13, 3433.

45. Edmister, W.C.: “Applied Hydrocarbon Thermodynamics, Part 4:Compressibility Factors and Equations of State,” Pet. Ref. (April1958) 37, 173.

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48. Twu, C.H.: “An Internally Consistent Correlation for Predicting theCritical Properties and Molecular Weights of Petroleum and Coal-TarLiquids,” Fluid Phase Equilibria (1984) No. 16, 137.

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20 PHASE BEHAVIOR

52. Peneloux, A., Rauzy, E., and Freze, R.: “A Consistent Correction forRedlich-Kwong-Soave Volumes,” Fluid Phase Equilibria (1982) 8, 7.

53. Jhaveri, B.S. and Youngren, G.K.: “Three-Parameter Modification ofthe Peng-Robinson Equation of State To Improve Volumetric Predic-tions,” SPERE (August 1988) 1033; Trans., AIME, 285.

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*� �� (�� +��

ft3/lbm mol �6.242 796 E02�m3/kmol�F (�F32)/1.8 ��C�F (�F�459.67)/1.8 �Kpsi �6.894 757 E�00�kPa�R 5/9 �K

88 PHASE BEHAVIOR

��

��

��

This chapter reviews the standard experiments performed by pres-sure/volume/temperature (PVT) laboratories on reservoir fluidsamples: compositional analysis, multistage surface separation,constant composition expansion (CCE), differential liberation ex-pansion (DLE), and constant volume depletion (CVD). We presentdata from actual laboratory reports and give methods for checkingthe consistency of reported data for each experiment. Chaps. 5 and8 discuss special laboratory studies, including true-boiling-point(TBP) distillation and multicontact gas-injection tests, respectively.

Table 6.1 summarizes experiments typically performed on oilsand gas condensates. From this table, we see that the DLE experi-ment is the only test never performed on gas-condensate systems.We begin by discussing standard analyses performed on oil and gas-condensate samples.

6.1.1 General Information Sheet. Most commercial laboratoriesreport general information on a cover sheet of the laboratory report,including formation and well characteristics and sampling condi-tions. Tables 6.2 and 6.31,2 show this information, which may beimportant for correct application and interpretation of the fluid anal-yses. This is particularly true for wells where separator samplesmust be recombined to give a representative wellstream composi-tion. Most of these data are supplied by the contractor of the fluidstudy and are recorded during sampling. Therefore, the representa-tive for the company contracting the fluid study is responsible forthe correctness and completeness of reported data.

We strongly recommend that the following data always be reportedin a general information sheet: (1) separator gas/oil ratio (GOR) instandard cubic feet/separator barrel, (2) separator conditions at sam-pling, (3) field shrinkage factor used (� Bosp), (4) flowing bottom-hole pressure (FBHP) at sampling, (5) static reservoir pressure, (6)minimum FBHP before and during sampling, (7) time and date ofsampling, (8) production rates during sampling, (9) dimensions ofsample container, (10) total number and types of samples collectedduring the drillstem test, and (11) perforation intervals.

6.1.2 Oil PVT Analyses. Standard PVT analyses performed on res-ervoir oils usually include (1) bottomhole wellstream compositionalanalysis through C7�, (2) CCE, (3) DLE, and (4) multistage-separa-tor tests. The CCE experiment determines the bubblepoint pressureand volumetric properties of the undersaturated oil. It also givestwo-phase volumetric behavior below the bubblepoint; however,these data are rarely used. The DLE experiment and separator testare used together to calculate traditional black-oil properties, Bo

and Rs, for reservoir-engineering calculations. Occasionally,

instead of a DLE study, a CVD experiment is run on a volatile oil.Also, the C7� fraction may be separated into single-carbon-numbercuts from C7 through approximately C20� by TBP analysis or simu-lated distillation (see Chap. 5).

6.1.3 Gas-Condensate PVT Analyses. The standard experimentalprogram for a gas-condensate fluid includes (1) recombined well-stream compositional analysis through C7�, (2) CCE, and (3) CVD.The CCE and CVD data are measured in a high-pressure visual cellwhere the dewpoint pressure is determined visually. Total volume/pressure and liquid-dropout behavior is measured in the CCE ex-periment. Phase volumes defining retrograde behavior are mea-sured in the CVD experiment together with Z factors andproduced-gas compositions through C7�. Optionally, a multistage-separator test can be performed as well as TBP analysis or simulateddistillation of the C7� into single-carbon-number cuts from C7 toabout C20� (see Chap. 5).

��

PVT studies usually are based on one or more samples taken duringa production test. Bottomhole samples can be obtained by wirelinewith a high-pressure container during either production testing or ashut-in period. Alternatively, separator samples can be taken duringa production test. Bottomhole sampling is the preferred method formost oil reservoirs, while recombined samples are traditionally usedfor gas-condensate reservoirs.3-8 Taking both bottomhole and sepa-rator samples in oil wells is not uncommon. The advantage of sepa-rator samples is that they can be recombined in varying proportionsto achieve a desired bubblepoint pressure (e.g., initial reservoirpressure); these larger samples are needed for special PVT tests(e.g., TBP and slim tube among others).

6.2.1 Bottomhole Sample. Table 6.4 shows the reported wellstreamcomposition of a reservoir oil where C7� specific gravity and molec-ular weight are also reported. In the example report, composition isgiven both as mole and weight percent although many laboratories re-port only molar composition. Experimentally, the composition of abottomhole sample is determined by the following (Fig. 6.1).

1. Flashing the sample to atmospheric conditions.2. Measuring the volumes of surface gas, Vg , and surface oil, Vo .3. Determining the normalized weight fractions, wgi and woi, of

surface samples by gas chromatography.4. Measuring surface-oil molecular weight, Mo , and specific

gravity, �o .

CONVENTIONAL PVT MEASUREMENTS 89

TABLE 6.1—LABORATORY ANALYSES PERFORMED ONRESERVOIR-OIL AND GAS-CONDENSATE SYSTEMS

Laboratory Analysis Oils Gas Condensates

Bottomhole sample � �

Recombined composition � �

C7+ TBP distillation � �

C7+ simulated distillation � �

Constant-composition expansion � �

Multistage surface separation � �

Differential liberation � N

CVD � �

Multicontact gas injection � �

��standard, ��can be performed, and N�not performed.

5. Converting wgi weight fractions to normalized mole fractionsyi and xi.

6. Recombining mathematically to the wellstream composition, zi.Eqs. 6.1 through 6.5 give Steps 1 through 6 mathematically.

zi � Fg yi � (1 � Fg)xi ; (6.1). . . . . . . . . . . . . . . . . . . . . . . .

Fg �1

1 � �133, 300��M�o�Rs�

, (6.2). . . . . . . . . . . . . . . . . .

where Rs �GOR Vg�Vo in scf/STB from the single-stage flash;

yi �wg i

�Mi

jC7�

�wg j�Mj

� � �wg C 7��Mg C 7�

�; (6.3). . . . . . . . .

xi �wo i

�Mi

jC7�

�wo j�Mj� � �wo C7�

�Mo C7�

�; (6.4). . . . . . . . . .

and Mo C7�

�

wo C7�

�1�Mo� �

jC7�

�woj�Mj�

. (6.5). . . . . . . . . . . . .

Surface gas usually contains less than 1 mol% C7� material con-sisting mainly of heptanes and octanes; Mg C 7�

� 105 is usually agood assumption. Surface oil contains less than 1 mol% of the lightconstituents C1, C2, and nonhydrocarbons. Low-temperature dis-tillation can be used to improve the accuracy of reported weightfractions for intermediate components in the surface oil ( C3 throughC6); however, gas chromatography is more widely used.

The most probable source of error in wellstream composition of abottomhole sample is the surface-oil molecular weight, Mo , whichappears in Eq. 6.2 for Fg and Eq. 6.4 for xi. Mo is usually accuratewithin �4 to 10%. In Chap. 5, we showed that the Watson character-ization factor, Kw, of surface oil (Eq. 5.35) should be constant (towithin �0.03 of the determined value) for a given reservoir. Once anaverage has been established for a reservoir (usually requiring threeseparate measurements), potential errors in Mo can be checked. Acalculated Kw that deviates from the field-average Kw by more than�0.03 may indicate an erroneous molecular-weight measurement.

Eqs. 6.1 through 6.4 show that all component compositions areaffected by Mo C 7�

, which is backcalculated from Mo with Eq.6.5. Fortunately, the amount of lighter components (particularly C1)in the surface oil are small, so the real effect on conversion fromweight to mole fractions of the surface oil usually is not significant.

6.2.2 Recombined Samples. Tables 6.5 and 6.6 present the separa-tor-oil and -gas compositional analyses of a gas-condensate fluidand recombined wellstream composition. The separator-oil com-position is obtained by use of the same procedure as that used forbottomhole oil samples (Eqs. 6.1 through 6.5). This involves bring-ing the separator oil to standard conditions, measuring properties

TABLE 6.2—EXAMPLE GENERAL INFORMATION SHEETFOR GOOD OIL CO. WELL 4 OIL SAMPLE

Formation Characteristics

Name Cretaceous

First well completed / /19 (m/d/y)

Original reservoir pressure at 8,692 ft, psig 4,100

Original produced GOR, scf/bbl 600

Production rate, B/D 300

Separator temperature, °F 75

Separator pressure, psig 200

Oil gravity at 60°F, °API

Datum 8,000

Original gas cap No

Well Characteristics

Elevation, ft 610

Total depth, ft 8,943

Producing interval, ft 8,684 to 8,700

Tubing size, in. 27/8

Tubing depth, ft 8,600

PI at 300 B/D, B-D/psi 1.1

Last reservoir pressure at 8,500 ft, psig 3,954*

Date / /19 (m/d/y)

Reservoir temperature at 8,500 ft, °F 217*

Well status Shut in 72 hours

Pressure gauge Amerada

Normal production rate, B/D 300

GOR, scf/bbl 600

Separator pressure, psig 200


Base pressure, psia 14.65

Well making water, % water cut 0

Sampling Conditions

Sample depth, ft 8,500

Well status Shut in 72 hours

GOR

Separator pressure, psig

Separator temperature, °F

Tubing pressure, psig 1,400

Casing pressure, psig

Sampled by

Sampler type Wofford

*Pressure and temperature extrapolated to the midpoint of the producing

interval�4,010 psig and 220°F.

and compositions of the resulting surface oil and gas, and recombin-ing these compositions to give the separator-oil composition; Tables6.5 and 6.6 report the results.

Separator gas is introduced directly into a gas chromatograph,which yields weight fractions, wg . These weight fractions are con-verted to mole fractions, yi, by use of appropriate molecularweights; Tables 6.5 and 6.6 show the results. C7� molecular weightis backcalculated with measured separator-gas specific gravity, �g .

Mg C 7�

� wg C 7�� 1

28.97�g�

iC7�

wg i

Mi�

�1

. (6.6). . . . . . .

90 PHASE BEHAVIOR

TABLE 6.3—EXAMPLE GENERAL INFORMATION SHEETFOR GOOD OIL CO. WELL 7 GAS CONDENSATE

Formation Characteristics

Formation name Pay sand

Date first well completed / /19 (m/d/y)

Original reservoir pressure at 10,148 ft, psig 5,713

Original produced-gas/liquid ratio, scf/bbl

Production rate, B/D

Separator pressure, psig

Separator temperature, °F

Liquid gravity at 60°F, °API

Datum, ft subsea 8,000

Well Characteristics

Elevation, ft KB 2,214

Total depth, ft 10,348

Producing interval, ft 10,124 to 10,176

Tubing size, in. 2

Tubing depth, ft 10,100

Open-flow potential, MMscf/D

Last reservoir pressure at 10,148 ft, psig 5,713

Date / /19 (m/d/y)

Reservoir temperature at 10,148 ft, °F 186

Status of well status Shut in

Pressure gauge Amerada

Sampling Conditions

Flowing tubing pressure, psig 3,375

FBHP, psig 5,500

Primary-separator pressure, psig 300

Primary-separator temperature, °F 62

Secondary-separator pressure, psig 20

Secondary-separator temperature, °F 60

Field stock-tank-liquid gravity at 60°F, °API 58.5

Primary-separator-gas production rate, Mscf/D 762.14

Pressure base, psia 14.696

Temperature base, °F 60

Compressibility factor, Fpv 1.043

Gas gravity (laboratory) 0.737

Gas-gravity factor, Fg 0.902

Stock-tank-liquid production rate at 60°F, B/D 127.3

Primary-separator-gas/stock-tank-liquid ratioIn scf/bblIn bbl/MMscf

5,987167.0

Sampled by

For the example PVT report (Tables 6.5 and 6.6), the separatorgas/oil ratio, Rsp, during sampling is reported as standard gas vol-ume per separator-oil volume (4,428 scf/bbl). In this report, the unitsare incorrectly labeled scf/bbl at 60°F, where in fact the separator-oilvolume is measured at separator pressure (300 psig) and tempera-ture (62°F). The separator-oil formation volume factor (FVF), Bosp,is 1.352 bbl/STB and represents the volume of separator oil requiredto yield 1 STB of oil (i.e., condensate).

The equation used to calculate wellstream composition, zi, is

zi � Fgsp yi � (1 � Fgsp)xi , (6.7). . . . . . . . . . . . . . . . . . . . .

where Fgsp �mole fraction of wellstream mixture that becomesseparator gas. In the laboratory report, Fgsp is reported as “primary-

separator gas/wellstream ratio” (801.66 Mscf/MMscf), which isequivalent to mole per mole ( Fgsp �0.80166 mol/mol). The re-ported value of Fgsp can be checked with

Fgsp � �1 �2, 130�osp

Mosp Rsp��1

, (6.8). . . . . . . . . . . . . . . . . . . . .

where Mosp �N

i�1

xi Mi . (6.9). . . . . . . . . . . . . . . . . . . . . . . . . . .

�osp in lbm/ft3 is calculated with a correlation (e.g., Standing-Katz9)or with the relation (62.4�o � 0.0136�g Rs)�Bo, where Rs andBo �separator-oil values in scf/STB and bbl/STB, respectively;


TABLE 6.4—WELLSTREAM (RESERVOIR-FLUID)COMPOSITION FOR GOOD OIL CO. WELL 4

BOTTOMHOLE OIL SAMPLE

Component mol% wt%Density*(g/cm3) °API*

MolecularWeight

H2S Nil Nil

CO2 0.91 0.43

N2 0.16 0.05

Methane 36.47 6.24

Ethane 9.67 3.10

Propane 6.95 3.27

i-butane 1.44 0.89

n-butane 3.93 2.44

i-pentane 1.44 1.11

n-pentane 1.41 1.09

Hexanes 4.33 3.97

Heptanes plus 33.29 77.41 0.8515 34.5 218

Total 100.00 100.00*At 60°F.

�o �stock-tank-oil density; and �g �gravity of gas released fromthe separator oil.

Finally, the value of stock-tank-liquid/wellstream ratio in bbl/MMscfrepresents the separator barrels produced per 1 MMscf of wellstream.In terms of Fgsp and separator properties, this value equals

bblMMscf

�

470(1�Fgsp)�Mosp��osp�

Bosp, (6.10). . . . . . . . . . . . . .

where 470�(1 million scf/MMscf)/[(379 scf/lbm mol)(5.615 ft3/bbl)].The separator-oil and -gas compositions can be checked for con-

sistency with the Hoffman et al.10 K-value method and Standing’s11

low-pressure K-value equations.

��

The multistage-separator test is performed on an oil sample primari-ly to provide a basis for converting differential-liberation data froma residual-oil to a stock-tank-oil basis. Occasionally, several separa-tor tests are run to help choose separator conditions that maximizestock-tank-oil production. Usually, two or three stages of separationare used, with the last stage at atmospheric pressure and near-ambi-ent temperature (60 to 80°F). The multistage-separator test can alsobe conducted for high-liquid-yield gas-condensate fluids.

Fig. 6.2 illustrates schematically how the separator test is per-formed. Initially, the reservoir sample is at saturation conditions andthe volume is measured ( Vob or Vgd). The sample is then brought tothe pressure and temperature of the first-stage separator. All the gasis removed, and the oil volume at the separator stage, Vosp, is notedtogether with the volume of removed gas, �Vg ; number of moles of

removed gas, ng ; and specific gravity of removed gas, �g. If re-quested, the gas samples can be analyzed chromatographically togive molar composition, y.

The oil remaining after gas removal is brought to the conditionsof the next separator stage. The gas is removed again and quantifiedby moles and specific gravity. Oil volume is noted, and the processis repeated until stock-tank conditions are reached. Final oil volume,Vo , and specific gravity, �o , are measured at 60°F.

Table 6.7 gives results from four separator tests, each consistingof two stages of separation. The first-stage-separator pressure is var-ied from 50 to 300 psig, and stock-tank conditions are held constantat 0 psig and 75°F. GOR’s are reported as standard gas volume perseparator-oil volume, Rsp, and as standard gas volume per stock-tank-oil volume, Rs, respectively.

�Rsp ��Vg

Vosp(6.11). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

and �Rs ��Vg

Vo. (6.12). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Total GOR is calculated by adding the stock-tank-oil-based GOR’sfrom each separator stage.

Rs �

Nsp

k�1

��Rs�k . (6.13). . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Separator-oil FVF’s, Bosp, are reported as the ratio of separator-oilvolume to stock-tank-oil volume.

Bosp �Vosp

V o. (6.14). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Accordingly, the relation between separator gas/oil ratio and stock-tank gas/oil ratio at a given stage is

�Rsp ��Rs

Bosp. (6.15). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Because Bosp � 1, it follows that Rsp Rs.Bubblepoint-oil FVF, Bob, is the ratio of bubblepoint-oil volume

to stock-tank-oil volume.

Bob �Vob

Vo. (6.16). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The average gas gravity, �g , is used in oil PVT correlations andto calculate reservoir densities on the basis of black-oil properties.The average gas gravity is calculated from

�g �

Nsp

k�1

��g�k��Rs�k

Nsp

k�1

��Rs�k

, (6.17). . . . . . . . . . . . . . . . . . . . . . . . . .

Fig. 6.1—Procedure for recombining single-stage separator samples to obtain wellstreamcomposition of a bottomhole sample; BHS�bottomhole sampler, GC�gas chromatograph,FDP� freezing-point depression, and DM�densitometer.

92 PHASE BEHAVIOR

TABLE 6.5—SEPARATOR AND RECOMBINED WELLSTREAM COMPOSITIONSFOR GOOD OIL CO. WELL 7 GAS CONDENSATE

Separator Products Hydrocarbon Analysis

Separator Liquid Separator Gas Wellstream

Component (mol%) (mol%) (gal/Mscf) (mol%) (gal/Mscf)

CO2 Trace 0.22 0.18

N2 Trace 0.16 0.13

Methane 7.78 75.31 61.92

Ethane 10.02 15.08 14.08

Propane 15.08 6.68 1.832 8.35 2.290

iso-butane 2.77 0.52 0.170 0.97 0.317

n-butane 11.39 1.44 0.453 3.41 1.073

iso-pentane 3.52 0.18 0.066 0.84 0.306

n-pentane 6.50 0.24 0.087 1.48 0.535

Hexanes 8.61 0.11 0.045 1.79 0.734

Heptanes plus 34.33 0.06 0.028 6.85 3.904

Total 100.00 100.00 2.681 100.00 9.159

Heptanes-Plus Properties

Oil gravity, °API 46.6

Specific gravityat 60/60°F

0.7946 0.795

Molecular weight 143 103 143

Parameters

Calculated separator gas gravity (air�1.000) 0.735

Calculated gross heating value for separator gas at 14.696 psia and60°F, BTU/ft3 dry gas

1,295

Primary-separator-gas*/-separator-liquid* ratio, scf/bbl at 60°F 4,428

Primary-separator-gas/stock-tank-liquid ratio at 60°F, bbl at 60°F/bbl 1.352

Primary-separator-gas/wellstream ratio, Mscf/MMscf 801.66

Stock-tank-liquid/wellstream ratio, bbl/MMscf 133.9

*Primary separator gas and liquid collected at 300 psig and 62°F.

TABLE 6.6—MATERIAL-BALANCE CALCULATIONS FORGOOD OIL CO. WELL 7 GAS-CONDENSATE SAMPLE

Liquid Composition at Specified Pressures(mol%)

Component At 3,500 psig At 2,900 psig At 2,100 psig At 1,300 psig At 605 psig

CO2 0.18 0.18 0.18 0.15 0.08

N2 0.13 0.08 0.06 0.03 0.01

C1 13.18 45.04 32.22 19.69 11.77

C2 8.12 14.05 13.99 12.32 7.44

C3 12.59 9.67 11.25 11.66 9.31

i-C4 3.44 1.14 1.59 1.85 1.64

n-C4 5.21 4.82 6.12 7.35 7.17

i-C5 2.67 1.25 1.77 2.43 2.79

n-C5 5.74 2.16 3.48 4.62 5.50

C6 8.47 3.11 4.55 6.40 8.37

C7+ 40.27 18.51 24.79 33.50 45.91

Total 100.00 100.00 100.00 100.00 100.00

Mo, g�mol 96.6 54.1 64.3 78.2 95.6

MoC7�, g�mol 168.8 160.1 152.1 149.9 150.3

�o, g�cm3 0.3235 0.2642 0.1625 0.0892 0.0398


Fig. 6.2—Schematic of a multistage-separator test.

pst�14.7 psiaTst�60°F

where ��g�

k �separator-gas gravity at Stage k. This relation is basedon the ideal gas law at standard conditions, where moles of gas are di-rectly proportional with standard gas volume (vg �379 scf/lbm mol).

Table 6.8 gives the composition of the first-stage-separator gasat 50 psig and 75°F. The gross heating value, Hg , of this gas is calcu-lated by Kay’s12 mixing rule and component heating values, Hi,given in Table A-1.

Hg �N

i�1

yi Hi . (6.18). . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Component liquid yields, Li, represent the liquid volumes of acomponent or group of components that can theoretically be pro-cessed from 1 Mscf of separator gas (gallons per million standardcubic feet). Li can be calculated from

Li � 19.73yi�Mi�i� , (6.19). . . . . . . . . . . . . . . . . . . . . . . . . . .

where Mi �molecular weight and �i � component liquid densityin lbm/ft3 at standard conditions (Table A-1). The C7� material inseparator gases is usually treated as normal heptane.

�� !"��

6.4.1 Oil Samples. For an oil sample, the CCE experiment is usedto determine bubblepoint pressure, undersaturated-oil density, iso-thermal oil compressibility, and two-phase volumetric behavior atpressures below the bubblepoint. Table 6.9 presents data from anexample CCE experiment for a reservoir oil.

Fig. 6.3 illustrates the procedure for the CCE experiment. A blindcell (i.e., a cell without a window) is filled with a known mass of reser-voir fluid. Reservoir temperature is held constant during the experi-ment. The sample initially is brought to a condition somewhat aboveinitial reservoir pressure, ensuring that the fluid is single phase. As thepressure is lowered, oil volume expands and is recorded.

The fluid is agitated at each pressure by rotating the cell. Thisavoids the phenomenon of supersaturation, or metastable equilibri-um, where a mixture remains as a single phase even though it shouldexist as two phases.13-15 Sometimes supersaturation occurs 50 to100 psi below actual bubblepoint pressure. By agitating the mixtureat each new pressure, the condition of supersaturation is avoided, al-lowing more accurate determination of the bubblepoint.

Just below the bubblepoint, the measured volume will increasemore rapidly because gas evolves from the oil, yielding a higher sys-tem compressibility. The total volume, Vt, is recorded after the two-phase mixture is brought to equilibrium. Pressure is lowered in stepsof 5 to 200 psi, where equilibrium is obtained at each pressure.When the lowest pressure is reached, total volume is three to fivetimes larger than the original bubblepoint volume.

The recorded cell volumes are plotted vs. pressure, and the result-ing curve should be similar to one of the curves in Fig. 6.4.16 For ablack oil (far from its critical temperature), the discontinuity in vol-ume at the bubblepoint is sharp and the bubblepoint pressure andvolume are easily read from the intersection of the p-V trends in thesingle- and two-phase regions.

Volatile oils do not exhibit the same clear discontinuity in volu-metric behavior at the bubblepoint pressure. Instead, the p-V curveis practically continuous in the region of the bubblepoint becausethe undersaturated-oil compressibility is similar to the effectivetwo-phase compressibility. This makes determining the bubble-point of volatile oils in a blind cell difficult. Instead, a windowed cell

TABLE 6.7—SEPARATOR TESTS (RESERVOIR-FLUID) OFGOOD OIL CO. WELL 4 OIL SAMPLE

SeparatorPressure

(psia)

SeparatorTemperature

(°F)GORb

(ft3/bbl)GORc

(ft3/bbl)

Stock-TankGravity(°API)

FVFd

(bbl/bbl)

SeparatorVolumeFactore(bbl/bbl)

Flashed-GasSpecificGravity

50to0

75

75

715

41

737

41 40.5 1.481

1.031

1.007

0.840

1.338

100to0

75

75

637

91

676

92 40.7 1.474

1.062

1.007

0.786

1.363

200to0

75

75

542

177

602

178 40.4 1.483

1.112

1.007

0.732

1.329

300to0

75

75

478

245

549

246 40.1 1.495

1.148

1.007

0.704

1.286aGauge.bIn cubic feet of gas at 60°F and 14.65 psi absolute per barrel of oil at indicated pressure and temperature.cIn cubic feet of gas at 60°F and 14.65 psi absolute per barrel of stock-tank oil at 60°F.dIn barrels of saturated oil at 2,620 psi gauge and 220°F per barrel of stock-tank oil at 60°F.eIn barrels of oil at indicated pressure and temperature per barrel of stock-tank oil at 60°F.

94 PHASE BEHAVIOR

TABLE 6.8—FIRST-STAGE SEPARATOR-GASCOMPOSITION AND GROSS HEATING VALUE FOR

GOOD OIL CO. WELL 4 OIL SAMPLE*

Component mol% gal/Mscf

H2S Nil

CO2 1.62

N2 0.30

C1 67.00

C2 16.04 4.265

C3 8.95 2.449

i-C4 1.29 0.420

n-C4 2.91 0.912

i-C5 0.53 0.193

n-C5 0.41 0.155

C6 0.44 0.178

C7+ 0.49 0.221

Total 100.00 8.793

Heating Value

Calculated gas gravity (air�1.000) 0.840

Calculated gross heating value, BTU/ft3

dry gas at 14.65 psia and 60°F1,405

*Collected at 50 psig and 75°F in the laboratory.

is used to observe visually the first bubble of gas and the liquid vol-umes below the bubblepoint.

Reported data from commercial laboratories usually include bub-blepoint pressure, pb; bubblepoint density, �ob, or specific volume,vob(v � 1��); and isothermal compressibility of the undersaturatedoil, co , at pressures above the bubblepoint (Table 6.9). The table alsoshows the oil’s thermal expansion, indicated by a ratio of undersatu-rated-oil volume at a specific pressure and reservoir temperature tothe oil volume at the same pressure and a lower temperature.

Total volumes are reported relative to the bubblepoint volume.

Vrt �Vt

Vob. (6.20). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Traditionally, isothermal compressibility data are reported for pres-sure intervals above the bubblepoint. In fact, the undersaturated-oilcompressibility varies continuously with pressure, and, becauseVt � Vo (Vrt � Vro) for p � pb, oil compressibility can be ex-pressed as

c �1

Vrt��Vrt

�p�

T

�1

Vro��Vro

�p�

T

; p � pb . (6.21). . . . . . . . .

Fig. 6.3—Schematic of a CCE experiment for an oil and a gascondensate.

TABLE 6.9—CCE DATA (RESERVOIR-FLUID)FOR GOOD OIL CO. WELL 4 OIL SAMPLE

Saturation (bubblepoint) pressure*, psig 2,620

Specific volume at saturationpressure*, ft3/lbm

0.02441

Thermal expansion of undersaturatedoil at 5,000 psi�V at 220°F/V at 76°F

1.08790

Compressibility of saturated oil atreservoir temperatureFrom 5,000 to 4,000 psi, vol/vol-psiFrom 4,000 to 3,000 psi, vol/vol-psiFrom 3,000 to 2,620 psi, vol/vol-psi

13.48x10–6

15.88x10–6

18.75x10–6

Pressure/Volume Relations*

Pressure(psig)

Relative volume(L)† Y function‡

5,000 0.9639

4,500 0.9703

4,000 0.9771

3,500 0.9846

3,000 0.9929

2,900 0.9946

2,800 0.9964

2,700 0.9983

2,620** 1.0000

2,605 1.0022 2.574

2,591 1.0041 2.688

2,516 1.0154 2.673

2,401 1.0350 2.593

2,253 1.0645 2.510

2,090 1.1040 2.422

1,897 1.1633 2.316

1,698 1.2426 2.219

1,477 1.3618 2.118

1,292 1.5012 2.028

1,040 1.7802 1.920

830 2.1623 1.823

640 2.7513 1.727

472 3.7226 1.621* At 220°F.

** Saturation pressure.1 Relative volume�V/Vsat in barrels at indicated pressure per barrel at saturation

pressure.‡ Y function�( psat�p)/(pabs)(V/Vsat�1).

The Vrt function at undersaturated conditions may be fit with a se-cond�degree polynomial, resulting in an explicit relation for under-saturated-oil compressibility (see Chap. 3).

Total volumes below the bubblepoint can be correlated by the Yfunction,16,17 defined as

Y �pb � p

p(Vrt � 1)�

pb � p

p��Vt�Vb� � 1�

, (6.22). . . . . . . . . . . . . .

where p and pb are given in absolute pressure units. As Fig. 6.5shows, Y vs. pressure should plot as a straight line and the lineartrend can be used to smooth Vrt data at pressures below the bubble-point. Standing16 and Clark17 discuss other smoothing techniquesand corrections that may be necessary when reservoir conditionsand laboratory PVT conditions are not the same.

6.4.2 Gas-Condensate Samples. The CCE data for a gas condensateusually include total relative volume, Vrt, defined as the volume ofgas or of gas-plus-oil mixture divided by the dewpoint volume. Z fac-


Fig. 6.4—Volume vs. pressure for an oil during a DLE test (after Standing16).

at 2

90 p

sia

tors are reported at pressures greater than and equal to the dewpointpressure. Table 6.10 gives these data for a gas-condensate example.

Reciprocal wet-gas FVF, bgw, is reported at dewpoint and initialreservoir pressures, where these values represent the gas equivalentor wet-gas volume at standard conditions produced from 1 bbl ofreservoir gas volume.

bgw � �5.615 � 10�3�Tscpsc

pZT

� 0.198p

ZT, (6.23). . . . . . . .

with bgw in Mscf/bbl, p in psia, and T in °R.Most CCE experiments are conducted in a visual cell for gas con-

densates, and relative oil (condensate) volumes, Vro, are reported atpressures below the dewpoint. Vro normally is defined as the oil vol-ume divided by the total volume of gas and oil, although some re-ports define it as the oil volume divided by the dewpoint volume.

��# $�%%�� &�'�� !"��

The DLE experiment is designed to approximate the depletion pro-cess of an oil reservoir18 and thereby provide suitable PVT data to

calculate reservoir performance.16,19-21 Fig. 6.6 illustrates the labo-ratory procedure of a DLE experiment. Figs. 6.7A through 6.7Cand Table 6.11 give DLE data for an oil sample.

A blind cell is filled with an oil sample, which is brought to asingle phase at reservoir temperature. Pressure is decreased until thefluid reaches its bubblepoint, where the oil volume, Vob, is recorded.Because the initial mass of the sample is known, bubblepoint densi-ty, �ob, can be calculated.

The pressure is decreased below the bubblepoint, and the cell isagitated until equilibrium is reached. All gas is removed at constantpressure. Then, the volume, �Vg; moles, �ng; and specific gravity,�g, of the removed gas are measured. The remaining oil volume, Vo,is also recorded. This procedure is repeated 10 to 15 times at de-creasing pressures and finally at atmospheric pressure. Residual-oilvolume, Vor, and specific gravity, �or , are measured at 60°F.

Other properties are calculated on the basis of measured data(�Vg , Vo , �ng , �g , Vor, and �or), including differential solutiongas/oil ratio, Rsd ; differential oil FVF, Bod ; oil density, �o; and gasZ factor, Z. For Stage k, these properties can be determined from

96 PHASE BEHAVIOR

Fig. 6.5—PVT relation and plot of Y function for an oil sample at pressures below the bubblepoint.

BubblepointTemperature

°5F80

163185205

Pressurepsia

1,9702,4372,5202,615

Volumecm3

82.3086.8887.9289.05

�Rsd�k �

k

j�1

379��ng�j

Vor, (6.24). . . . . . . . . . . . . . . . . . . . . . . .

�Bod�k �

�Vo�k

Vor, (6.25). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

��o�k �

Vor(62.4�or) �k

j�1

�28.97�5.615��ng�j��g�j

�Vo�k

�

350�or �k

j�1

0.0764��Rsd�j��g�j

5.615�Bod�k

,

(6.26). . . . . . . . . . . . . . . . . .

and (Z)k ��1�RT��p�Vg��ng�k

, (6.27). . . . . . . . . . . . . . . . . .

with Vor and Vo in bbl, Rsd in scf/bbl, Bod in bbl/bbl, �Vg in ft3, pin psia, �ng in lbm mol, �o in lbm/ft3, and T in °R. Note that the sub-script j�1 indicates the final DLE stage at atmospheric pressure andreservoir temperature. Reported oil densities are actually calculatedby material balance, not measured directly.

6.5.1 Converting From Differential to Stock-Tank Basis. Perhapsthe most important step in the application of oil PVT data for reservoircalculations is conversion of the differential solution gas/oil ratio,Rsd, and oil FVF, Bod, to a stock-tank-oil basis.16,20 For engineering

calculations, volume factors, Rs and Bo, are used to relate reservoir-oil volumes, Vo, to produced surface volumes, Vg and Vo; i.e.,

Rs �Vg

Vo(6.28). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

and Bo �Vo

Vo. (6.29). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Differential properties Rsd and Bod reported in the DLE report arerelative to residual-oil volume (i.e., the oil volume at the end of theDLE experiment, corrected from reservoir to standard temperature).

Rsd �Vg

Vor(6.30). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

and Bod �Vo

Vor. (6.31). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The equations commonly used to convert differential volume fac-tors to a stock-tank basis are

Rs � Rsb ��Rsdb � Rsd

��Bob

Bodb� (6.32). . . . . . . . . . . . . . . . . .

and Bo � Bod�Bob

Bodb� , (6.33). . . . . . . . . . . . . . . . . . . . . . . . . . .

where Bob �bubblepoint-oil FVF, Rsb �solution gas/oil ratiofrom a multistage-separator flash, and Rsdb and Bodb �differentialvolume factors at the bubblepoint pressure. The term ( Bob�Bodb),


TABLE 6.10—CCE DATA FOR GOOD OIL CO.WELL 7 GAS-CONDENSATE SAMPLE

Pressure(psig) Relative volume

Deviation FactorZ

6,000 0.8808 1.144

5,713* 0.8948 1.107**

5,300 0.9158 1.051

5,000 0.9317 1.009

4,800 0.9434 0.981

4,600 0.9559 0.953

4,400 0.9690 0.924

4,300 0.9758 0.909

4,200 0.9832 0.895

4,100 0.9914 0.881

4,000† 1.0000 0.867‡

3,905 1.0089

3,800 1.0194

3,710 1.0299

3,500 1.0559

3,300 1.0878

3,000 1.1496

2,705 1.2430

2,205 1.5246

1,605 2.1035

1,010 3.5665

Pressure/volume relations of reservoir fluid at 186°F.* Reservoir pressure.

** Gas FVF�1.591 Mscf/bbl.†Dewpoint pressure.‡Gas FVF�1.424 Mscf/bbl.

representing the volume ratio,Vor�Vo , is used to eliminate the resid-ual-oil volume, Vor, from the Rsd and Bod data. Note that the conver-sion from differential to “flash” data depends on the separatorconditions because Bob and Rsb depend on separator conditions.

Although, the conversions given by Eqs. 6.32 and 6.33 typicallyare used, they are only approximate. The preferred method, as origi-nally suggested by Dodson et al.,22 requires that some equilibriumoil be taken at each stage of the DLE experiment and flashed througha multistage separator to give the volume ratios, Rs and Bo. This lab-oratory procedure is costly and time-consuming and is seldom used.However, the method is readily incorporated into an equation-of-state (EOS) -based PVT program.

�� $��

The CVD experiment is designed to provide volumetric and com-positional data for gas-condensate and volatile-oil reservoirs pro-ducing by pressure depletion. Fig. 6.8 shows the stepwise procedureof a CVD experiment schematically, and Figs. 6.9A through 6.9Dand Table 6.12 give CVD data for an example gas-condensate fluid.

The CVD experiment provides data that can be used directly bythe reservoir engineer, including (1) a reservoir material balancethat gives average reservoir pressure vs. recovery of total well-stream (wet-gas recovery), sales gas, condensate, and natural gasliquids; (2) produced-wellstream composition and surface productsvs. reservoir pressure; and (3) average oil saturation in the reservoir(liquid dropout and revaporization) that occurs during pressuredepletion. For many gas-condensate reservoirs, the recoveries andoil saturation vs. pressure data from the CVD analysis closelyapproximate actual field performance for reservoirs producing bypressure depletion. When other recovery mechanisms, such as wa-terdrive and gas cycling, are considered, the basic data required forreservoir engineering are still taken mainly from a CVD report. Thissection provides a description of the data provided in a standard

Fig. 6.6—Schematic of DLE experiment.

CVD analysis, ways to check the data for consistency,23-25 and howto extract reservoir-engineering quantities from the data.23,26

Initially, the dewpoint, pd, or bubblepoint pressure, pb, of the res-ervoir sample is established visually and the cell volume, Vcell, atsaturated conditions is recorded. The pressure is then reduced by300 to 800 psi and usually by smaller amounts (50 to 250 psi) justbelow the saturation pressure of more-volatile systems. The cell isagitated until equilibrium is achieved, and volumes Vo and Vg aremeasured. At constant pressure, sufficient gas, �Vg, is removed toreturn the cell volume to the original saturated volume.

In the laboratory, the removed gas (wellstream) is brought to at-mospheric conditions, where the amount of surface gas and conden-sate are measured. Surface compositions yg and xo of the producedsurface volumes from the reservoir gas are measured, as are the vol-umes �Vo and �Vg , densities �o and �g and oil molecular weightMo . From these quantities, we can calculate the moles of gas re-moved, �ng.

�ng ��Vo�o

Mo�

�Vg

379. (6.34). . . . . . . . . . . . . . . . . . . . . . . .

These data are reported as cumulative wellstream produced, np, rel-ative to the initial moles n.

�np

n �k �1n

k

j�1

(�ng)j , (6.35). . . . . . . . . . . . . . . . . . . . . . . . .

where j�1 corresponds to saturation pressure and (�ng)1 � 0. Theinitial amount (in moles) of the saturated fluid is known when the cellis charged. The quantity np�n is usually reported as cumulative wetgas produced in MMscf/MMscf, which is equivalent to mol/mol.

Surface compositions yg and xo of the removed reservoir gas andproperties of the removed gas are not reported directly in the labora-tory report but are recombined to yield the equilibrium gas (well-stream) composition, yi, which also represents the equilibrium gasremaining in the cell. The C7� molecular weight of the wellstream,MgC7�, is backcalculated from measured specific gravity( �w � �g) and reservoir-gas composition, y. C7� specific gravity ofthe produced gas, �gC7�� is also reported, but this value is calculatedfrom a correlation.

Knowing the cumulative moles removed and its volume occupiedas a single-phase gas at the removal pressure, we can calculate theequilibrium gas Z factor from

Z �

p�Vg

�ng RT. (6.36). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

A “two-phase” Z factor is also reported that is calculated assum-ing that the gas-condensate reservoir depletes according to the ma-terial balance for a dry gas and that the initial condition of the reser-voir is at dewpoint pressure.

98 PHASE BEHAVIOR

Fig. 6.7A—DLE data for an oil sample from Good Oil Co. Well 4; differential solution gas/oilratio, Rsd .

pZ2

� �pd

Zd��1 �

Gpw

Gw�, (6.37). . . . . . . . . . . . . . . . . . . . . . . .

where Gpw �cumulative wellstream (wet gas) produced andGw �initial wet gas in place. As defined in Eq. 6.37, the term Gpw�Gw

equals np�n reported in the CVD report. From Eq. 6.37, the only un-known at a given pressure is Z2, and the two-phase Z factor is then giv-en by

Z2 �p

�pd�Zd��1 � �np�n��

. (6.38). . . . . . . . . . . . . . . . . . . . .

Theoretical liquid yields, Li, are also reported for C3� throughC5� groups in the produced wellstreams at each pressure-depletionstage. These values are calculated with

Li � 19.73yi�Mi�i� (6.39). . . . . . . . . . . . . . . . . . . . . . . . . . . . .

and by summing the yields of components in the particular “plus”group. For example, the liquid yield of C5� material at CVD Stagek is given by

�LC5��

k�

C7�

j�i C5

�Lj�

k � 19.73 C7�

j�i C5

�yj�

k�Mj

�j� . (6.40). . . . .

Table 6.13 gives various calculated cumulative recoveries basedon the reservoir initially being at its dewpoint. The basis for the cal-culations is 1 MMscf of dewpoint wet gas in place, Gw; the corre-sponding initial moles in place at dewpoint pressure is given by

n �Gwvg

�1 � 106 scf

379 scf�lbm mol� 2, 638 lbm mol. (6.41). . . . . . . . .

The first row of recoveries (wellstream) simply represents thecumulative moles produced, np�n, expressed as wet-gas volumes,Gpw, in Mscf.

Gpw � nvg�npn �

� (2, 638 lbm mol)�379 scf�lbm mol�

� �1 � 103 Mscf�scf��np

n �

� 1 � 103 �np

n �. (6.42). . . . . . . . . . . . . . . . . . . . . . . . .

Recoveries in Rows 2 through 4 (Normal Temperature Separa-tion, Total Plant Products in Primary-Separator Gas, and Total PlantProducts in Second-Stage-Separator Gas) refer to production whenthe reservoir is produced through a three-stage separator. Fig. 6.10


Fig. 6.7B—DLE data for an oil sample from Good Oil Co. Well 4; differential oil FVF (relativevolume), Bod .

illustrates the process schematically. The calculated recoveries arebased on multistage-separator calculations that use low-pressure Kvalues and a set of separator conditions chosen arbitrarily or speci-fied when the PVT study is requested.

6.6.1 Recoveries: “Normal Temperature Separation.” Column1: Initial in Place. In Column 1, Row 2a the stock-tank oil in solu-tion in the initial dewpoint fluid (N�135.7 STB) is calculated byflashing 1 MMscf of the original dewpoint fluid, Gw, through amultistage separator.

Rows 2b through 2d give the volumes of separator gas at eachstage of a three-stage flash of the initial dewpoint fluid: 757.87,96.68, and 24.23 Mscf, respectively. The mole fraction of well-stream resulting as a surface gas Fgg is given by

Fgg �Gd

Gw� �757.87 � 96.68 � 24.23 Mscf�lbm mol�

� �1 � 103 scf�Mscf��379 scf�lbm mol�

� 0.8788 lbm mol�lbm mol, (6.43). . . . . . . . . . . . . . .

where Gd �total separator “dry” gas and the corresponding molefraction of stock-tank oil is 0.1212 mol/mol. Fgg is used to calculatedry-gas FVF (see Eq. 3.41). For the dewpoint pressure, this gives

Bgd �Bgw

Fgg�

�psc�Tsc��ZT�p�

Fgg

�

�14.7�520��[0.867(186 � 460)]�4015�

0.8788

� 4.487 � 10�3 ft3�scf . (6.44). . . . . . . . . . . . . . . . . . .

The producing GOR of the dewpoint mixture for the specifiedseparator conditions can be calculated as

Rp �GN� ��757.87 � 96.68 � 24.23 Mscf�lbm mol�

� �1 � 103 scf�Mscf��135.7 STB�lbm mol

� 6, 476 scf�STB. (6.45). . . . . . . . . . . . . . . . . . . . . . . .

The dewpoint solution oil/gas ratio, rsd, is simply the inverse of Rp.

rsd � rp �1Rp

� 1.544 � 10�4 STB�scf � 154.4 STB�MMscf.

(6.46). . . . . . . . . . . . . . .

Note that specific gravities of stock-tank oil and separator gases arenot reported for the separator calculations.

100 PHASE BEHAVIOR

Fig. 6.7C—DLE data for an oil sample from Good Oil Co. Well 4; oil viscosity, �o .

Column 2 and Higher. On the basis of 1 MMscf of initial dew-point fluid, Rows 2a through 2d give cumulative volumes of separa-tor products at each depletion pressure ( Np, Gp1, Gp2, and Gp3).The producing GOR of the wellstream produced during a depletionstage is given by

�Rp�k�

�Gp1 � Gp2 � Gp3�

k� �Gp1 � Gp2 � Gp3

�k�1

�Np�k� �Np�k�1

.

(6.47). . . . . . . . . . . . . . . . . .

For 2,100 psig, this gives

Rp � �[(301.57 � 20.75 � 5.61) � (124.78 � 12.09 � 3.16)]

� �1 � 103��(24.0 � 15.4)

� 21, 850 scf�STB. (6.48). . . . . . . . . . . . . . . . . . . . . . . .

In terms of the solution oil/gas ratio,

rs � rp �1Rp

�1

21, 580 scf�STB� 4.58 � 10�5 STB�scf

� 45.8 STB�scf . (6.49). . . . . . . . . . . . . . . . . . . . . . . . . .

At a given pressure, the mole fraction of the removed CVD gaswellstream that becomes dry separator gas is given by

�Fgg�

k�

�Gp1 � Gp2 � Gp3�

k� �Gp1 � Gp2 � Gp3

�k�1

Gw��np�n�

k� �np�n�

k�1�

.

(6.50). . . . . . . . . . . . . . . . . .

For p�2,100 psig, this gives

Fgg � [(301.57 � 20.75 � 5.61)�(124.78 � 12.09

� 3.16)]�1 � 103��1 � 106�(0.35096 � 0.15438)

� 0.9558 . (6.51). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The dry-gas FVF at 2,100 psig is

Bgd �

�14.7�520��0.762(186 � 460)�2, 115�

0.9558

� 6.884 � 10�3 ft3�scf . (6.52). . . . . . . . . . . . . . . . . . .

In summary, the information provided in the rows labeled NormalTemperature Separation gives estimates of the condensate andsales-gas recoveries assuming a multistage surface separation. Forexample, at an abandonment pressure of 605 psig, the condensaterecovery is 35.1 STB of the 135.7 STB initially in place (in solutionin the dewpoint mixture), or 26% condensate recovery. Dry-gas re-covery is (685.02�37.79�10.40)�733.21 Mscf of the 878.78


TABLE 6.11—DLE DATA FOR GOOD OIL CO. WELL 4 OIL SAMPLE

Differential Vaporization

Pressure(psig)

SolutionGOR

(scf/bbl*)

RelativeOil Volume(RB/bbl*)

RelativeTotal Volume

(RB/bbl*)

OilDensity(g/cm3)

DeviationFactor

ZGas FVF(RB/bbl*)

IncrementalGas Gravity

2,620 854 1.600 1.600 0.6562

2,350 763 1.554 1.665 0.6655 0.846 0.00685 0.825

2,100 684 1.515 1.748 0.6731 0.851 0.00771 0.818

1.850 612 1.479 1.859 0.6808 0.859 0.00882 0.797

1,600 544 1.445 2.016 0.6889 0.872 0.01034 0.791

1,350 479 1.412 2.244 0.6969 0.887 0.01245 0.794

1,110 416 1.382 2.593 0.7044 0.903 0.01552 0.809

850 354 1.351 3.169 0.7121 0.922 0.02042 0.831

600 292 1.320 4.254 0.7198 0.941 0.02931 0.881

350 223 1.283 6.975 0.7291 0.965 0.05065 0.988

159 157 1.244 14.693 0.7382 0.984 0.10834 1.213

0 0 1.075 0.7892 2.039

1.000**

DLE Viscosity Data at 220°F

Pressure(psig)

Oil Viscosity(cp)

Calculated GasViscosity

(cp)

5,000 0.450

4,500 0.434

4,000 0.418

3,500 0.401

3,000 0.385

2,800 0.379

2,620 0.373

2,350 0.396 0.0191

2,100 0.417 0.0180

1,850 0.442 0.0169

1,600 0.469 0.0160

1,350 0.502 0.0151

1,100 0.542 0.0143

850 0.592 0.0135

600 0.654 0.0126

350 0.783 0.0121

159 0.855 0.0114

0 1.286 0.0093Gravity of residual oil�35.1°API at 60°F.

*Barrels of residual oil.**At 60°F.

Mscf dry gas originally in place, or 83.4%. These recoveries can becompared with the reported wet-gas (or molar) recovery of 76.787%at 605 psig. In addition to recoveries, the calculated results in thissection can be used to calculate solution oil/gas ratio, rs, and dry-gasFVF, Bgd, for modified black-oil applications.

6.6.2 Recovery: Plant Products. Rows 3 through 5 considertheoretical liquid recoveries for propane, butanes, and pentanes-plus assuming 100% plant efficiency. Recoveries in Rows 3 and 4are for the calculated separator gases from Stages 1 and 2 of thethree-stage surface separation. Recoveries in Row 5 are for the pro-duced wellstreams from the CVD experiment and represent the ab-solute maximum liquid recoveries that can be expected if the reser-voir is produced by pressure depletion. Fig. 6.10 illustrates therecovery calculations schematically. Liquid volumes (in gal/MMscfof initial dewpoint fluid) at CVD Stage k are calculated from

(Li)k � 19, 730�Mi�i��

k

j�1

��ng

n �j

�yi�j�, (6.53). . . . . . . . .

Fig. 6.8—Schematic of CVD experiment.

102 PHASE BEHAVIOR

Fig. 6.9A—CVD data for gas-condensate sample from Good Oil Co. Well 7; liquid-dropout curve, Vro .

where j � 1 represents the dewpoint, yi �compositions of well-stream entering the gas plant at various stages of depletion,Mi �component molecular weights, and �i � liquid componentdensities in lbm/ft3 at standard conditions (Table A-1).

Calculated liquid recoveries below the dewpoint use the moles ofwellstream produced (�ng�n) and the compositions yi from the sep-arator gas (Rows 3 and 4) or wellstream (Row 5) entering the plant.Column 1 (Initial in Place) gives the total recoveries assuming thatthe entire initial dewpoint fluid is taken to the surface and processed[i.e., k � 1 and (�ng�n)1 � 1 in Eq. 6.53].

Note that cumulative recovery of propanes from the first-stageseparator during depletion (1,276 gal) is larger than the liquid pro-pane produced in the first-stage-separator gas of the original dew-point mixture (1,198 gal). This means that the stock-tank oil fromthe separation of original dewpoint mixture contains more propanethan the cumulative stock-tank-oil volumes produced by depletionand three-stage separation.

The results given in Rows 3 and 4 cannot be calculated from re-ported data because surface separator compositions from the three-stage separation are not provided in the report. The results in Row5 can be checked. As an example, consider the C3 recoveries for theinitial-in-place fluid at 2,100 psig.

�LC3�

pd

� 19, 730 �44.09�31.66�� (1)(0.0837)�

� 2, 299 gal�MMscf (6.54a). . . . . . . . . . . . . . . . . . . .

and �LC3�

2100� 19, 730 �44.09�31.66� [0.0825(0.05374)

� 0.0810(0.15438 � 0.05374)

� 0.0757(0.35096 � 0.15438)]

� 754 gal�MMscf. (6.54b). . . . . . . . . . . . . . . .

For the C5� recoveries at the dewpoint,

�LC5��

pd

� 19, 730[(72.15�38.96) (0.0091)

� (72.15�39.36) (0.0152)

� (86.17�41.43) (0.0179) � (143�49.6) (0.0685)]

� 5, 513 gal�MMscf . (6.55). . . . . . . . . . . . . . . . .

6.6.3 Correcting Recoveries for Initial Pressure Greater ThanDewpoint Pressure. All recoveries given in Table 6.13 assume thatthe reservoir pressure is initially at dewpoint. This assumption ismade because initial reservoir pressure is not always known withcertainty when PVT calculations are made. However, adjusting re-ported recoveries is straightforward when initial pressure is greaterthan dewpoint pressure. With QTable as recoveries given in Columns2 and higher in Table 6.13, Qd as hydrocarbons in place in Column


Fig. 6.9B—CVD data for gas-condensate sample from Good Oil Co. Well 7; equilibrium gascompositions, yi .

Dew

po

int

Pre

ssu

re

1 at dewpoint pressure, and Q as actual cumulative recoveries basedon hydrocarbons in place at the initial pressure,

Q � Qd��p�Z�i�p�Z�d

�

�p�Z�

�p�Z�d�; p � pd , (6.56). . . . . . . . . . . .

Q � QTable � �Qd ; p pd , (6.57). . . . . . . . . . . . . . . . . . .

and �Qd � Qd�(p�Z)i

(p�Z)d� 1�, (6.58). . . . . . . . . . . . . . . . . . . .

where �Qd �additional recovery from initial to dewpoint pres-sure.

For the example report,

�Qd � ��5, 728�1.107�

�4, 015�0.867�� 1�Qd

� 0.1173Qd , (6.59). . . . . . . . . . . . . . . . . . . . . . . . . . .

recalling that moles of material at dewpoint is 2,638 lbm mol, molesof material at initial pressure of 5,728 psig is n�2, 638(1 � 0.1173)� 2, 947 lbm mol, and the basis of calculations is Gw � 1.173MMscf of wet gas in place at initial pressure of 5,728 psia.

The cumulative wellstream produced at the dewpoint pressure of4,000 psig is 0.1173(1, 000) � 117.3 Mscf. Recovery at 3,500 psigis 117.3 � 53.74 � 171.0 Mscf. Likewise, wet-gas recovery

should be increased by 117.3 Mscf for all depletion pressures in theCVD table.

For stock-tank-oil recovery, Qd � 135.7 STB, so �Qd � 15.9STB. Stock-tank-oil recovery at 4,000 psig is 15.9 � 0 � 15.9STB; at 3,500 psig the recovery should be 15.9 � 6.4 � 22.3 STB,and so on.

On the basis of 1 MMscf wet gas at the dewpoint or 1.1173 MMscfat initial reservoir pressure, the laboratory hydrocarbon pore vol-ume (HCPV), VpHClab, is the same.

VpHClab ��GwBgw�d

� �1 � 106��14.7520��0.867(186 � 460)

4, 015��

� 3, 943 ft3

� �Gw Bgw�i

� 1.1173 � 106��14.7520��1.107(186 � 460)

5728��

� 3, 943 ft3 . (6.60). . . . . . . . . . . . . . . . . . . . . . . . . .

The actual HCPV of a reservoir is much larger than VpHClab, and theconversion to obtain recoveries for the actual HCPV is simply

104 PHASE BEHAVIOR

Fig. 6.9C—CVD data for gas-condensate sample from Good Oil Co. Well 7; equilibrium gas Zfactor, Zg .

Qactual � Qlab

VpHCactual

VpHClab, (6.61). . . . . . . . . . . . . . . . . . . . . . .

where Qlab �laboratory value given by Eqs. 6.55 and 6.57. As an ex-ample, suppose geological data indicate a HCPV of 625,000 bbl(82.45 acre-ft), or 3.509�106 ft3. Then, original wet gas in place is

Gw � 1.1173 � 106 3.509 � 106

3, 943

� 994.3 MMscf (6.62). . . . . . . . . . . . . . . . . . . . . . . . . . .

and condensate in solution at initial pressure is given by

N � 135.7(1.1173) 3.509 � 106

3, 943

� 134, 900 STB . (6.63). . . . . . . . . . . . . . . . . . . . . . . . . .

6.6.4 Liquid-Dropout Curve. Table 6.11 and Figs. 6.9A through6.9D show relative oil volumes, Vro, measured in the example CVDexperiment. Vro is defined as the volume of oil, Vo, at a given pres-sure divided by the original saturation volume, Vs. This relative vol-ume is an excellent measure of the average reservoir-oil saturation(normalized) that will develop during depletion of a gas-condensate

reservoir. Correcting for water saturation, Sw, the reservoir-oil satu-ration can be calculated from Vro with

So � (1 � Sw)Vro . (6.64). . . . . . . . . . . . . . . . . . . . . . . . . . .

For most gas condensates, Vro shows a maximum near 2,000 to2,500 psia. Cho et al.27 give a correlation for maximum liquid drop-out as a function of temperature and C7� mole percent in the dew-point mixture.

�Vro�max � 93.404 � 4.799 zC7�� 19.73 ln T , (6.65). . . . . .

with zC7� in mole percent and T in °F. The correlation predicts

(Vro)max �23.2% for the example condensate fluid compared with24% measured experimentally (at 2,100 psig). Fig. 6.11 shows val-ues of (Vro)max vs. T and zC7�from Eq. 6.65.

Considerable attention usually is given to matching the liquid-dropout curve when an EOS is used. Some gas condensates have-what is referred to as a “tail,” where liquid drops out very slowly(sometimes for several thousand psi below the dewpoint) before fi-nally increasing toward a maximum. Matching this behavior withan EOS can prove difficult, and the question is whether matching thetail is really necessary (see Appendix C).

What really matters for reservoir calculations of a gas-condensatefluid is how much original stock-tank condensate is “lost” becauseof retrograde condensation in the reservoir. The shape and magni-


Fig. 6.9D—CVD data for gas-condensate sample from Good Oil Co. Well 7; wet-gas materialbalance.

tude of liquid dropout reflects the change in producing oil/gas ratio,rp � rs. A tail on a liquid-dropout curve implies that the producingwellstream is becoming only slightly leaner (i.e., rs is decreasingonly slightly). The cumulative condensate recovery is given by

Np � �

Gp

0

rs dGp , (6.66). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

where Gp �cumulative dry gas produced. Cumulative condensateproduction is readily evaluated from a plot of rs vs. Gp.

One of the most important checks of an EOS characterization forany gas condensate, particularly one with a tail, is Np calculatedfrom CVD data vs. Np calculated from the EOS characterization. Itis alarming how much the surface condensate recovery can be un-derestimated if the tail is not matched properly. We do not recom-mend matching the dewpoint exactly with a liquid-dropout curvethat is severely overpredicted in the region where measured resultsindicate little dropout. If the EOS characterization cannot be modi-fied to honor the tail of liquid-dropout curve, it is preferable tounderpredict the measured dewpoint pressure and match only thehigher liquid-dropout volumes.

In summary, oil relative volume, Vro, is not important per se; how-ever, the effect of liquid dropout on surface condensate production

should be emphasized. In fact, the effect of shape and magnitude ofliquid dropout on fluid flow in the reservoir is negligible, and anyEOS match will probably have the same effect on fluid flow from thereservoir into the wellbore (i.e., inflow performance).

6.6.5 Consistency Check of CVD Data. Reudelhuber and Hinds24

give a detailed procedure for checking CVD data consistency thatinvolves a material-balance check on components and phases andyields oil compositions, density, molecular weight, and MC7�. To-gether with reported data, these calculated properties allow K valuesto be calculated and checked for consistency with the Hoffman etal.10 method.11,28 Whitson and Torp’s23 material-balance equationsare summarized later. Similar equations can also be derived for aDLE experiment when equilibrium gas compositions and oil rela-tive volumes are reported. Reported CVD data include temperature,T ; dewpoint pressure, pd, or bubblepoint pressure, pb; dewpoint Zfactor, Zd, or bubblepoint-oil density, �ob . Additional data at eachDepletion Stage k include oil relative volume, Vro; initial fractionof cumulative moles produced, np�n; gas Z factor (not the two-phase Z factor), Z; equilibrium gas composition, yi; and equilibriumgas (wellstream) C7� molecular weight, Mg C7�.

The equilibrium gas density, �g; molecular weight, Mg; and well-stream gravity, �w � Mg�Mair , are readily calculated at each

106 PHASE BEHAVIOR

TABLE 6.12—CVD DATA FOR GOOD OIL CO. WELL 7 GAS-CONDENSATE SAMPLE 2*

Reservoir Pressure, psig

Component, mol% 5,713** 4,000† 3,500 2,900 2,100 1,300 605 0‡

CO2 0.18 0.18 0.18 0.18 0.18 0.19 0.21

N2 0.13 0.13 0.13 0.14 0.15 0.15 0.14

C1 61.72 61.72 63.10 65.21 69.79 70.77 66.59

C2 14.10 14.10 14.27 14.10 14.12 14.63 16.06

C3 8.37 8.37 8.26 8.10 7.57 7.73 9.11

i-C4 0.98 0.98 0.91 0.95 0.81 0.79 1.01

n-C4 3.45 3.45 3.40 3.16 2.71 2.59 3.31

i-C5 0.91 0.91 0.86 0.84 0.67 0.55 0.68

n-C5 1.52 1.52 1.40 1.39 0.97 0.81 1.02

C7 1.79 1.79 1.60 1.52 1.03 0.73 0.80

C7+ 6.85 6.85 5.90 4.41 2.00 1.06 1.07

Total 100.00 100.00 100.00 100.00 100.00 100.00 100.00

Properties

C7+ molecular weight 143 143 138 128 116 111 110

C7+ specific gravity 0.795 0.795 0.790 0.780 0.767 0.762 0.761

Equilibrium gas deviation factor, Z 1.107 0.867 0.799 0.748 0.762 0.819 0.902

Two-phase deviation factor, Z 1.107 0.867 0.802 0.744 0.704 0.671 0.576

Wellstream produced, cumulative% of initial

0.000 5.374 15.438 35.096 57.695 76.787 93.515

From smooth compositions

C3+, gal/Mscf 9.218 9.218 8.476 7.174 5.171 4.490 5.307

C4+, gal/Mscf 6.922 6.922 6.224 4.980 3.095 2.370 2.808

C5+, gal/Mscf 5.519 5.519 4.876 3.692 1.978 1.294 1.437

Retrograde Condensation During Gas Depletion

Retrograde liquid volume,

% hydrocarbon pore space

0.0 3.3 19.4 23.9 22.5 18.1 12.6

*Study conducted at 186°F.

** Original reservoir pressure.

† Dewpoint pressure.

‡0-psig residual-liquid properties: 47.5°API oil gravity at 60°; 0.7897 specific gravity at 60/60°F; and molecular weight of 140.

Depletion Stage k [and at the dewpoint ( k � 1) for a gas-conden-sate sample] from

�Mg�k�

N

i�1

(yi)k Mi , (6.67). . . . . . . . . . . . . . . . . . . . . . . . . .

��g�k�

p �Mg�k

(Z)k RT, (6.68). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

and ��g�k� ��w�k �

�Mg�k

28.97. (6.69). . . . . . . . . . . . . . . . . . . .

On a basis of 1 mol initial dewpoint fluid ( n � 1), the cell vol-ume is

Vcell �Zd RT

pd(6.70). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

for a gas condensate and

Vcell �Mob�ob

(6.71). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

for a volatile oil. Oil and gas volumes, respectively, at Stage k are

(Vo)k� Vcell (Vro)k

and �Vg�k � Vcell�1 � (Vro)k

� . (6.72). . . . . . . . . . . . . . . . . . . .

Moles and mass of the total material remaining in the cell at Stage kare given by

(nt)k � 1 � �np

n �k,

�ng�k�

�p�k�Vg�k

(Z)k RT,

and (no)k � (nt)k ��ng�k

, (6.73). . . . . . . . . . . . . . . . . . . . . . . .

and moles and mass of the individual phases remaining in the cell at

Stage k are given by

(mt)k � Ms �k

j�2

��ng

n �j

�Mg�j,

�mg�k� �ng�k

�Mg�k,

and (mo)k � (mt)k ��mg�k

. (6.74). . . . . . . . . . . . . . . . . . . . . .

In Eqs. 6.73 and 6.74,

��ng

n �j

� �np

n �j� �

np

n �j�1

, (6.75). . . . . . . . . . . . . . . . . . . .

Ms �saturated-fluid molecular weight, and (np�n)1 � 0.

Densities and molecular weights of the oil phase are calculated from


TABLE 6.13—CALCULATED RECOVERIES* FROM CVD REPORTFOR GOOD OIL CO. WELL 7 GAS-CONDENSATE SAMPLE

Reservoir Pressure (psig)

Initial in Place 4,000** 3,500 2,900 2,100 1,300 605 0

Wellstream, Mscf 1,000 0 53.74 154.38 350.96 576.95 767.87 935.15

Normal temperature separation†

Stock-tank liquid, bbl 135.7 0 6.4 15.4 24.0 29.7 35.1

Primary-separator gas, Mscf 757.87 0 41.95 124.78 301.57 512.32 658.02

Second-stage gas, Mscf 96.68 0 4.74 12.09 20.75 27.95 37.79

Stock-tank gas, Mscf 24.23 0 1.21 3.16 5.61 7.71 10.4

Total plant products in primary separator‡

Propane, gal 1,198 0 67 204 513 910 1,276

Butanes, gal 410 0 23 72 190 346 491

Pentanes, gal 180 0 10 31 81 144 192

Total plant products in second-stage

separator‡

Propane, gal 669 0 33 86 149 205 286

Butanes, gal 308 0 15 41 76 108 159

Pentanes, gal 138 0 7 19 35 49 69

Total plant products in wellstream‡

Propane, gal 2,296 0 121 342 750 1,229 1,706

Butanes, gal 1,403 0 73 202 422 665 927

Pentanes, gal 5,519 0 262 634 1,022 1,315 1,589* Cumulative recovery per MMscf of original fluid calculated during depletion.**Dewpoint pressure.†Recovery basis: primary separation at 500 psia and 70°F, second-stage separation at 50 psia and 70°F, and stock tank at 14.7 psia and 70°F.‡Recovery assumes 100% plant efficiency.

��o�k �(mo)k

(Vo)k(6.76). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

and (Mo)k �

(mo)k

(no)k, (6.77). . . . . . . . . . . . . . . . . . . . . . . . . . . . .

and the oil composition is given by

(xi)k �(nt)k(zi)k � �ng�k

�yi�k

(nt)k ��ng�k

. (6.78). . . . . . . . . . . . . . . . . . .

K values can be calculated from Ki � yi�xi, and zi �overall com-position of the mixture remaining in the cell at Stage k .

(zi)k �1

(nt)k�(zi)1 �

k

j�2

��ng

n �j

�yi�j� . (6.79). . . . . . . . . . .

C7� molecular weight of the oil phase can be calculated from

�Mo C7��

k�

(Mo)k � iC7�

(xi)k Mi

�xC7��k

. (6.80). . . . . . . . . . . . . .

Table 6.6 summarizes these calculations for the sample gas-conden-sate mixture.

Fig. 6.10—Schematic of method of calculating plant recoveries in a CVD report for a gascondensate.

(Separator Gas 1)

(Separator Gas 2)

108 PHASE BEHAVIOR

Fig. 6.11—Calculated maximum retrograde oil relative volumes from the Cho et al.27 correlation.

Heptanes Plus, mol%

Nonphysical

The oil composition at the last depletion state (605 psig for the ex-ample condensate) can be measured, but it must be requested specif-ically. Also, the residual-oil molecular weight, Mor, and specificgravity, �or, remaining after depletion at atmospheric pressure aretypically measured and reported as shown in Table 6.12. These val-ues can be compared with calculated values by use of the material-balance equations shown earlier.

The material-balance calculations are more accurate for rich gascondensates and volatile oils. In fact, obtaining reasonable material-balance oil properties for lean gas condensates is difficult. Some-times it is useful to modify the reported oil relative volumes (partic-ularly those close to the dewpoint) to monitor the effect oncalculated oil properties.

An alternative material-balance check that may be even moreuseful for determining data consistency (particularly for leaner gascondensates) involves starting with reported final-stage condensatecomposition, (xi)k�N, and adding back the removed gases, (yi)k, foreach stage from k � N to k � 1. This results in the original gascomposition, (zi)k�1, which can be compared quantitatively withthe laboratory-reported composition.

(�%��

1. “Core Laboratories Good Oil Company Oil Well No. 4 PVT Study,” CoreLaboratories, Houston.

2. “Core Laboratories Good Oil Company Condensate Well No. 7 PVTStudy,” Core Laboratories, Houston.

3. Flaitz, J.M. and Parks, A.S.: “Sampling Gas-Condensate Wells,” Trans.,AIME (1942) 146, 13.

4. Katz, D.L., Brown, G.G., and Parks, A.S.: “NGAA Report on SamplingTwo-Phase Gas Streams from High Pressure Condensate Wells,” (Sep-tember 1945).

5. Reudelhuber, F.O.: “Sampling Procedures for Oil Reservoir Fluids,” JPT(December 1957) 15.

6. Clark, N.J.: “Sampling and Testing Oil Reservoir Samples,” JPT (Jan.1962) 12.

7. Clark, N.J.: “Sampling and Testing Gas Reservoir Samples,” JPT(March 1962) 266.

8. Recommended Practice for Sampling Petroleum Reservoir Fluids, API,Dallas (1966) 44.

9. Standing, M.B. and Katz, D.L.: “Density of Natural Gases,” Trans.,AIME, (1942) 146, 140.

10. Hoffmann, A.E., Crump, J.S., and Hocott, C.R.: “Equilibrium Constantsfor a Gas-Condensate System,” Trans., AIME (1953) 198, 1.



13. Kennedy, H.T. and Olson, C.R.: “Bubble Formation in SupersaturatedHydrocarbon Mixtures,” Oil & Gas J. (October 1952) 271.

14. Silvey, F.C., Reamer, H.H., and Sage, B.H.: “Supersaturation in Hydrocar-bon Systems: Methane-n-Decane,” Ind. Eng. Chem. (1958) 3, No. 2, 181.

15. Tindy, R. and Raynal, M.: “Are Test-Cell Saturation Pressures AccurateEnough?,” Oil & Gas J. (December 1966) 126.

16. Standing, M.B.: Volumetric and Phase Behavior of Oil Field Hydrocar-bon Systems, eighth edition, SPE, Richardson, Texas (1977).

17. Clark, N.J.: “Adjusting Oil Sample Data for Reservoir Studies,” JPT(February 1962) 143.

18. Moses, P.L.: “Engineering Applications of Phase Behavior of Crude-Oiland Condensate Systems,” JPT (July 1986) 715.

19. Amyx, J.W., Bass, D.M. Jr., and Whiting, R.L.: Petroleum Reservoir En-gineering, McGraw-Hill Book Co. Inc., New York City (1960).

20. Craft, B.C. and Hawkins, M.: Applied Petroleum Reservoir Engineering,first edition, Prentice-Hall Inc., Englewood Cliffs, New Jersey (1959).

21. Dake, L.P.: Fundamentals of Reservoir Engineering, Elsevier ScientificPublishing Co., Amsterdam (1978).

22. Dodson, C.R., Goodwill, D., and Mayer, E.H.: “Application of Labora-tory PVT Data to Reservoir Engineering Problems,” Trans., AIME(1953) 198, 287.

23. Whitson, C.H. and Torp, S.B.: “Evaluating Constant-Volume-DepletionData,” JPT (March 1983) 610; Trans., AIME, 275.

24. Drohm, J.K., Goldthorpe, W.H., and Trengove, R.: “Enhancing the Eval-uation of PVT Data,” paper OSEA 88174 presented at the 1988 OffshoreSoutheast Asia Conference, Singapore, 2–5 February.

25. Drohm, J.K., Trengove, R., and Goldthorpe, W.H.: “On the Quality ofData From Standard Gas-Condensate PVT Experiments,” paper SPE17768 presented at the 1988 Gas Technology Symposium, Dallas,13–15 June.

26. Reudelhuber, F.O. and Hinds, R.F.: “Compositional Material BalanceMethod for Prediction of Recovery From Volatile-Oil Depletion-DriveReservoirs,” JPT (January 1957) 19; Trans., AIME, 210.

27. Cho, S.J., Civan, F., and Starling, K.E.: “A Correlation To Predict Maxi-mum Condensation for Retrograde Condensation Fluids and Its Use inPressure-Depletion Calculations,” paper SPE 14268 presented at the1985 SPE Annual Technical Conference and Exhibition, Las Vegas, Ne-vada, 22–25 September.

28. Clark, N.J.: “Theoretical Aspects of Oil and Gas Equilibrium Calcula-tions,” JPT (April 1962) 373.

�� )��

�� g/cm3

bbl �1.589 873 E�01�m3

Btu �1.055 056 E�00�kJcp �1.0* E�03�Pa�sft �3.048* E�01�m

ft3 �2.831 685 E�02�m3

�F (�F�32)/1.8 ��Cgal �3.785 412 E�03�m3

in. �2.54* E�00�cmlbm mol �4.535 924 E�01�kmol

psi �6.894 757 E�00�kPa


BLACK-OIL PVT FORMULATIONS 1

��

��

��

This chapter reviews black-oil pressure/volume/temperature (PVT)formulations, gives examples of their application, and providesguidelines for choosing the proper PVT formulation for a given reser-voir. Sec. 7.2 reviews the traditional black-oil PVT formulation. Thethree basic PVT properties are introduced: solution gas/oil ratio, Rs;oil formation volume factor (FVF), Bo; and gas FVF, Bg. These prop-erties define the PVT behavior of reservoir-oil and -gas mixtures byquantifying the volumetric behavior and the distribution of surface-gas and surface-oil “components” as functions of pressure.

Many reservoirs being discovered today are at great depths, witha higher percentage of these deep reservoirs containing gas-conden-sate and volatile-oil fluids. Treatment of these reservoirs requiresmodification of the standard PVT formulation, as Sec. 7.3 discusses.In particular, the additional property solution oil/gas ratio, rs, isintroduced, together with a modified gas FVF.

Sec. 7.4 covers the application of black-oil PVT properties towell-rate deliverability and material-balance calculations. Sec. 7.5discusses alternative black-oil PVT formulations, including the par-tial-density approach. And finally, Sec. 7.6 briefly reviews somelimited compositional formulations that are used in the simulationof gas-injection processes.

��

It was already clear in the 1920’s that the engineering of oil reservoirsrequired knowledge of how much gas was dissolved in the oil at reser-voir conditions and how much the oil would shrink when it wasbrought to the surface. It was also recognized that free gas at reservoirconditions would expand up to several hundred times when broughtto surface conditions. Engineering quantities were needed to relatesurface volumes to reservoir volumes and vice versa. Three proper-ties evolved to serve this purpose: solution gas/oil ratio, Rs; oil FVF,Bo; and gas FVF, Bg. These properties are defined, respectively, by

Rs �volume of surface gas dissolved in reservoir oil

volume of stock-tank oil from reservoir oil,

(7.1a). . . . . . . . . . . . . . . . . . . . .

Bo �volume of reservoir oil

volume of stock-tank oil from reservoir oil,

(7.1b). . . . . . . . . . . . . . . . . . . .

and Bg �volume of reservoir gas

volume of surface gas from reservoir gas.

(7.1c). . . . . . . . . . . . . . . . . . . . .

These three properties constitute the traditional black-oil PVTformulation, which has the following assumptions.

1. Reservoir oil consists of two surface “components,” stock-tankoil and surface (total separator) gas.

2. Reservoir gas does not yield liquids when brought to the surface.3. Surface gas released from the reservoir oil has the same proper-

ties as the reservoir gas.4. Properties of stock-tank oil and surface gas do not change dur-

ing depletion of a reservoir.Fig. 7.11 illustrates schematically the relation between reservoir

phases and surface components. This simplified PVT formulationis still the standard for most petroleum engineering applications.The traditional black-oil quantities, Rs, Bo, and Bg, can be esti-mated with the correlations in Chap. 3 or can be calculated from dif-ferential-liberation and multistage-separator data (Chap. 6).

The validity of the traditional black-oil PVT formulation dependsprimarily on the reservoir-oil volatility. Any reservoir oil with lessthan �750 scf/STB initial solution gas/oil ratio can probably betreated with the traditional PVT formulation. Also, any oil reservoirthat produces at higher than its bubblepoint pressure during most ofthe reservoir’s productive life can be treated with this formulation(e.g., strong waterdrive, gas-cap-drive, or waterflooded reservoirs).

Volatile oils usually have an initial gas/oil ratio (GOR) greaterthan �1,000 scf/STB and an initial stock-tank-oil gravity�35°API. The following are the two main depletion characteristicsof a volatile-oil reservoir: (1) varying surface gravity of producedstock-tank oil and (2) the percentage of produced stock-tank oilcoming from the flowing reservoir gas increases from zero initiallyto a significant percentage at depletion (potentially �90%).

For most petroleum engineering calculations, the variation instock-tank-oil gravity can be neglected. However, neglecting thesurface oil that is produced from flowing reservoir gas may causegross underestimation of the ultimate stock-tank-oil recovery. Fig.7.2 shows the actual depletion characteristics of a volatile-oil reser-voir, where reservoir pressure decreases from 5,000 to 1,800 psia,produced surface-oil gravity increases from 44 to 62°API, and pro-ducing GOR increases from 3,800 to 22,000 scf/STB.

A good check of the traditional black-oil formulation is tocompare reservoir material-balance performance determined on thebasis of standard black-oil PVT properties (e.g., a material bal-

2 PHASE BEHAVIOR

Fig. 7.1—Schematic of traditional black-oil formulation relatingreservoir phases to surface components.

ance2) with depletion characteristics calculated from a composi-tional material balance. The traditional black-oil formulationshould not be used if the stock-tank-oil recoveries differ significant-ly (see Figs. 7.3 and 7.4).

Fig. 7.5 is another plot that indicates the relative volatility of anoil. Differential-liberation relative oil volumes are plotted as shrink-age (1 � Bod�Bodb) vs. normalized pressure ( p�pb), which indi-cates whether the shrinkage is rapid or slow. A curve that drops rap-idly indicates a highly volatile oil. A “black” oil will tend to plotabove the solid “unit-slope” line shown in Fig. 7.5.

�� !��" ��

Several modifications of the traditional black-oil formulation havebeen introduced to account for the surface-liquid content in reser-voir gases. Most formulations introduce an additional PVT proper-ty, the solution oil/gas ratio, rs, and a modified definition of the gasFVF. Fig. 7.6 shows schematically the relation between reservoirphases and surface components in the MBO formulation.

Because this chapter gives a detailed description of the MBOPVT formulation, we have introduced a more concise nomenclaturethat distinguishes between reservoir and surface phases. Tradition-ally, we use the subscript o to represent both reservoir oil and stock-tank oil and g to represent both reservoir gas and surface separator

Fig. 7.3—Average reservoir pressure and producing GOR vs. cu-mulative oil for near-critical oil Reservoir NS-2; comparison oftraditional and MBO formulations.

Cumulative Surface Oil Produced, fraction

Fig. 7.2—Depletion characteristics of a volatile-oil reservoir(adapted from Ref. 1).

Fig. 7.4—GOR’s vs. pressure for near-critical Reservoir NS-2and volatile-oil Reservoir NS-3.

Pressure, psia


Fig. 7.5—Oil shrinkage plot used to evaluate volatility of a reser-voir oil (from Ref. 3).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

gas. In this chapter, we use the following subscripts to distinguishbetween reservoir and surface phases: o�reservoir-oil phase at pand T, g�reservoir gas phase at p and T, oo�stock-tank oil fromreservoir oil, go�surface gas from reservoir oil (“solution” gas),og�stock-tank oil (condensate) from reservoir gas, gg�surfacegas from reservoir gas, o�total stock-tank oil, and g�total surfacegas, where the overbar indicates a surface-phase (component). Toavoid confusion, the standard term �w is used to represent the well-stream gravity of a reservoir gas (instead of �g).

The four MBO PVT parameters, oil FVF, solution gas/oil ratio,dry-gas FVF, and solution oil/gas ratio are defined respectively as

Bo �Vo

Voo, (7.2a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Rs �Vgo

Voo, (7.2b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Bgd �Vg

Vgg, (7.2c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

and rs �Vog

Vgg, (7.2d). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Fig. 7.7—Schematic of the Whitson-Torp4 method for calculat-ing MBO properties on the basis of depletion experiments andmultistage separation.

Fig. 7.6—Schematic showing relation between reservoir phasesand surface phases (components) for MBO formulation.

where Vo �reservoir-oil volume, Voo �volume of stock-tank oilproduced from the reservoir oil, Vgo �volume of surface gas pro-duced from the reservoir oil, Vg �reservoir gas volume, Vgg �vol-ume of surface gas produced from the reservoir gas, andVog �stock-tank oil (condensate) produced from the reservoir gas.

Fig. 7.7 outlines one procedure for determining MBO properties.The equilibrium-gas and -oil phases from a depletion experiment[constant composition expansion, constant volume depletion(CVD), or differential liberation] are passed separately through amultistage separator. The MBO properties are calculated accordingto the definitions given in Eq. 7.2. Figs. 7.8 through 7.11 showMBO properties calculated with the Whitson-Torp4 method for thegas condensate, near-critical oil, and volatile oils in Table 7.1. Refs.5 through 11 provide alternative methods.

7.3.1 Surface Gravities. When a well produces both reservoir oiland gas, the composite surface gravities,�o and �g, will be an aver-age of the surface gravities of the two reservoir phases, �oo and �gofor the reservoir oil and �og and �gg for the reservoir gas. The aver-age gas gravity is given by

�g � Fgg�gg ��1 � Fgg

��go , (7.3). . . . . . . . . . . . . . . . . . . .

Fig. 7.8—Solution GOR, Rs , vs. pressure for volatile reservoirFluids NS-1, NS-2, and NS-3 calculated with the Whitson-Torp4

method.

4 PHASE BEHAVIOR

Fig. 7.9—Oil FVF, Bo , vs. pressure for volatile reservoir FluidsNS-1, NS-2, and NS-3 calculated with the Whitson-Torp4 method.

where Fgg �fraction of total surface gas produced from the res-ervoir gas.

Fgg �Vgg

Vgg � Vgo�

Vgg

Vg�

1 � Rs�Rp

1 � Rsrs. (7.4). . . . . . . . . . .

The average stock-tank-oil gravity is given by

�o � Foo�oo � (1 � Foo )�og , (7.5). . . . . . . . . . . . . . . . . . .

where Foo �fraction of total stock-tank oil that comes from the res-ervoir oil.

Foo �Voo

Voo � Vog�

Voo

Vo�

1 � rs Rp

1 � Rsrs, (7.6). . . . . . . . . . . .

with Rp and Rs in scf/STB and rs in STB/scf in Eqs. 7.4 and 7.6.Surface gravities �oo, �og, �go, and �gg are determined separate-

ly for the reservoir-oil and reservoir-gas phases from multistage-sepa-rator calculations. Because the compositions of reservoir oil and gaschange during pressure depletion, the surface gravities also vary withpressure. The variation in �og and �go in Figs. 7.12 and 7.13 is typicalof volatile-oil and gas-condensate mixtures. On the other hand, �ooand �gg usually do not vary significantly with pressure.

Although the variation in surface gravities should be consideredin engineering calculations, most MBO formulations assume that

�oo � �og � �o � constant

and �go � �gg � �g � constant. (7.7). . . . . . . . . . . . . . . . . . .

Fig. 7.10—Solution OGR, rs , vs. pressure for volatile reservoirFluids NS-1 and NS-3 calculated with the Whitson-Torp4 method.

Clearly, the assumption that �oo � �og � �o makes predicting thevariation in overall stock-tank-oil gravity during depletion impossi-ble. As Fig. 7.2 shows, this variation can be significant.

Fig. 7.11—Inverse dry-gas FVF, bgd (�1/Bgd), vs. pressure forGas-Condensate NS-1 calculated with the Whitson-Torp4 method.

TABLE 7.1—SOLUTION OIL/GAS RATIO CALCULATED FROM FIELD STOCK-TANK-OILGRAVITY COMPARED WITH EOS-CALCULATED VALUES

rs (STB/MMscf)

Test DatepR

(psia)Rp

(scf/STB)Rs

(scf/STB) �o �oo

EOS�og

EOSFrom �o

EOS

Bubblepoint

January 1979

June 1980

November 1983

May 1987

5,555

4,455

3,685

3,105

2,683

1,500

2,215

3,840

7,480

9,480

1,500

1,006

768

615

514

0.8430

0.8353

0.8289

0.8189

0.8146

0.843

0.843

0.843

0.843

0.843

0.7595

0.7467

0.7401

0.7356

0.7325

62

43

32

28

100

61

44

34

29

Note: Whitson-Torp4 method used to calculate in last column. does not change appreciably with pressure and is therefore assumed constant.Rs, �oo, �og, and rs �oo


Fig. 7.12—Surface-gas gravities vs. pressure during depletion.

Because the constant-gravity assumption is widely used, it shouldbe considered when determining the MBO properties Rs, Bo, Bgd,and rs. For example, Coats8 gives a procedure for determiningMBO properties of a gas condensate where the original mixture isfirst passed through a separator to determine the surface gravities;these gravities are assumed to be constant. A depletion experimentis then simulated with an equation of state (EOS), and the equilibri-um gas from each depletion stage is passed through a separator todetermine rs at the particular pressure. With constant surface gravi-ties and rs as a function of pressure, Bgd, Bo, and Rs, are deter-mined so that reservoir-oil and -gas densities and the oil relative vol-umes from the depletion experiment are honored.

Surface-oil and -gas gravities are used in reservoir simulators toconvert Bo, Rs, Bgd, and rs to reservoir-oil and -gas densities.

�o �

62.4�oo � 0.0136�go Rs

Bo

and �g �

0.0764�gg � 350�og rs

Bgd. (7.8). . . . . . . . . . . . . . .

Accurate phase densities can be important for reservoir processeswhere gravity affects the recovery mechanism (e.g., gravity drain-age in naturally fractured reservoirs). Therefore, manual checkingof MBO properties and surface gravities used as input for reservoirsimulation is recommended to ensure that the reservoir-oil and -gasdensities are calculated accurately.

7.3.2 Gas FVF. The traditional definition of gas FVF assumes thatthe number of moles of gas at the surface equals the number of molesof gas at reservoir conditions. This obviously is not correct if the res-ervoir gas yields condensate at the surface. The definition is stillused, however, for liquid-yielding reservoir gases and is called the“wet”-gas FVF, Bgw. The surface volume is a hypothetical wet-gasvolume consisting of the “dry”-surface-gas volume and the surfacecondensate converted to an equivalent surface-gas volume.

Bgw �

Vg

Vgw. (7.9). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

With Vg � ng ZRT�p and Vgw � ng RTsc�psc, Bgw is simply givenby the traditional equation for gas FVF.

Bgw �

psc

Tsc

ZTp � 0.02827 ZT

p , (7.10). . . . . . . . . . . . . . . . . .

where Bgw is in ft3/scf, T is in °R, and p is in psia.

Fig. 7.13—Surface-oil gravities vs. pressure during depletion.

A dry-gas FVF, Bgd (defined as the volume of reservoir gas di-vided by the volume of surface gas resulting after separation of thereservoir gas), is used for the MBO formulation.

Bgd �Vg

Vgg. (7.11). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

With Vg � ng ZRT�p and Vgg � ngg RTsc�psc, the dry-gas FVFcan be written

Bgd �psc

Tsc

ZTp (1 � Cog rs) � 0.02827

ZTp (1 � Cog rs)

� Bgw(1 � Cog rs), (7.12). . . . . . . . . . . . . . . . . . . . . . .

where rs is in STB/scf, Bgd and Bgw are in ft3/scf, T is in °R, and pis in psia. Cog is a conversion from surface-oil volume in STB to an“equivalent” surface gas in scf.

Cog � 379� scflbm mol

� 5.615� ft3

STB� 62.4

�og

Mog�lbm mol

ft3�

� 133, 000�og

Mog� scfSTB

�. (7.13). . . . . . . . . . . . . . . . . . . . . .

If condensate molecular weight, Mog, is not measured, it can be esti-mated with the Cragoe12 correlation,

Mo �6, 084

�API � 5.9. (7.14). . . . . . . . . . . . . . . . . . . . . . . . . . . .

The term (1 � Cog rs)�1 represents the mole fraction of reservoirgas that becomes dry surface gas after separation and usually rangesfrom 0.85 for rich gases to 1.0 for dry gases. Fig. 7.14 shows the be-havior of the ratio as a function of pressure during depletion of a gascondensate and a volatile oil.

7.3.3 Solution Oil/Gas Ratio. The following simplified relation canbe used to calculate rs in terms of reservoir-gas specific gravity, �w.

rs ��w � �gg

4, 600�og � Cog �w. (7.15). . . . . . . . . . . . . . . . . . . . . .

�w is reported as a function of pressure in the differential-liberationexperiment and is readily calculated from reported compositions ina CVD experiment. Assuming that �gg � �g and �og � �o , surfacegravities usually are taken from a multistage separation of the origi-nal reservoir mixture and assumed constant throughout depletion.

On the basis of field production data, rs can be calculated in termsof the actual measured stock-tank-oil gravity, �o.

6 PHASE BEHAVIOR

Fig. 7.14—Fraction of reservoir gas that becomes “dry” sur-face gas vs. pressure during depletion of a gas condensate anda volatile oil.

rs ��o � �oo

Rs��o � �og� � Rp ��oo � �og

�. (7.16). . . . . . . . . . . . . .

Table 7.1 compares rs values from this relation (determined with fielddata from a volatile-oil reservoir) with rs from EOS calculations.

7.3.4 Compositional Information. Engineering calculations basedon black-oil properties actually contain more compositional in-formation than is commonly used. Given the compositions of stock-tank oil and separator gases, we can calculate oil and gas composi-tions (and K values) at reservoir conditions using black-oil PVTproperties. Also, wellstream composition can be calculated from theproducing GOR.

To develop the compositional relations, we use a basis of Vo

stock-tank barrels of total stock-tank oil. Volume of reservoir-oiland -gas phases, respectively, is

Vo � 5.615 Vo Foo Bo

and Vg � Vo Bgd�Rp � Rs Foo�, (7.17). . . . . . . . . . . . . . . . . . .

with Vo and Vg in ft3, Rp and Rs in scf/STB, Bo in bbl/STB, and Bgd

in ft3/scf. Foo is the fraction of total stock-tank oil that comes fromthe reservoir oil (Eq. 7.4).

Mass of reservoir-oil and -gas phases, respectively, in lbm is

mo � moo � mgo

and mg � mog � mgg, (7.18). . . . . . . . . . . . . . . . . . . . . . . . . . .

where moo � 350Vo Foo�oo ,

mgo � 0.076Vo Foo Rs�go ,

mog � 350Vo�Rp � Rs Foo��og,

and mgg � 0.076Vo�Rp � Rs Foo��gg . (7.19). . . . . . . . . . . . . .

This yields

mo � Vo Foo �350�oo � 0.076Rs�go�

and mg � Vo�Rp � Rs Foo��350�og rs � 0.076�gg�. (7.20). . .

Moles of reservoir oil and gas, respectively, in lbm mol are

no � noo � ngo

and ng � nog � ngg , (7.21). . . . . . . . . . . . . . . . . . . . . . . . . . . .

where noo �Vo FooCoo

379,

ngo �

Vo Foo Rs

379,

nog �

Vo �Rp � Rs Foo�rsCog

379,

and ngg �

Vo �Rp � Rs Foo�

379. (7.22). . . . . . . . . . . . . . . . . . . .

This yields

no �Vo Foo�Coo � Rs�

379

and ng �

Vo�Rp � Rs Foo��1 � rs Cog�

379, (7.23). . . . . . . . . . . .

with Coo and Cog given by

Coo � 133, 000�oo

Moo

and Cog � 133, 000�og

Mog. (7.24). . . . . . . . . . . . . . . . . . . . . . . .

On the basis of these relations, the mole fractions of surface com-ponents in the reservoir oil are

xo �noo

no�

1�1 � Rs�Coo

�

and xg �ngo

no

� 1 � xo, (7.25). . . . . . . . . . . . . . . . . . . . . . . . .

and the mole fractions of surface components in the reservoir gas are

yo �nog

ng

�1

1 � �rs Cog��1

and yg �ngg

ng

� 1 � yo, (7.26). . . . . . . . . . . . . . . . . . . . . . . . .

with K values Ko � yo�xo and Kg � yg�xg. Strictly speaking,Components o and g are not the same “species” and K values can-not be interpreted physically unless (1) the properties of surface oilsfrom reservoir gas and oil are equal and constant and (2) the surfacegases from reservoir gas and oil are equal and constant.

The mole fraction of the wellstream that comes from the reservoirgas is Fg � ng�(ng � no); therefore,

Fg � 1 �Foo(Coo � Rs)

(1 � Foo)�Cog � r�1s��

�1

, (7.27). . . . . . . . . . .

with Coo, Cog, and Rs in scf/STB and rs in STB/scf.Compositions of reservoir oil, xi, and reservoir gas, yi, can be cal-

culated from black-oil properties Rs, rs, and surface properties by

yi �yggi � �Cogrs�xogi

1 � Cogrs

and xi �ygoi � �Coo�Rs�xooi

1 � Coo�Rs, (7.28). . . . . . . . . . . . . . . . . . . .

where yggi �average composition of surface gases produced fromthe reservoir gas; xogi �composition of surface oil produced fromthe reservoir gas; ygoi �average composition of surface gases pro-


TABLE 7.2—EOS-CALCULATED SEPARATOR-GAS AND –OIL COMPOSITIONS FROM THREE-STAGESEPARATION OF ORIGINAL DEWPOINT GAS AND EOS-CALCULATED EQUILIBRIUM OIL

Reservoir Gas Reservoir Oil

Component ysp1

ysp2

ysp3

ygg xog ygo xoo

CO2 0.026092 0.030059 0.036539 0.026388 0.000588 0.027475 0.000627

N2 0.003552 0.002154 0.000362 0.003460 5.9410�7 0.003265 6.1210�7

C1 0.827710 0.809814 0.389891 0.816791 0.002079 0.809754 0.002103

C2 0.083029 0.099069 0.209316 0.086288 0.006739 0.090387 0.006730

C3 0.033261 0.036388 0.183444 0.036976 0.022803 0.039307 0.022010

i-C4 0.005535 0.005410 0.039898 0.006376 0.013315 0.006609 0.012297

n-C4 0.010249 0.009582 0.077103 0.011882 0.036997 0.012158 0.033498

i-C5 0.003145 0.002559 0.022571 0.003616 0.030334 0.003486 0.026016

n-C5 0.002939 0.002287 0.020158 0.003355 0.036413 0.003183 0.030772

C6 0.002425 0.001577 0.012855 0.002673 0.081629 0.002398 0.066496

F1 0.001671 0.000953 0.007116 0.001798 0.135151 0.001612 0.111360

F2 0.000380 0.000141 0.000739 3.8710�4 0.252945 0.000353 0.221341

F3 6.3410�6 1.0310�6 2.6310�6 6.2010�6 0.223155 6.3010�6 0.230727

F4 7.6210�9 3.4010�10 2.7610�10 7.3710�9 0.120536 8.9110�9 0.162246

F5 4.1010�13 2.9010�15 4.5010�16 3.9310�13 0.037312 5.9010�13 0.073772

TABLE 7.3—RESERVOIR EQUILIBRIUM COMPOSITIONS CALCULATED FROM EOSAND FROM MBO PVT PROPERTIES WITH SURFACE-GAS AND -OIL COMPOSITIONS

Dewpoint* Reservoir Pressure**

y x y x

Component Feed EOS EOS MBO EOS MBO

CO2

N2

C1

C2

C3

i-C4

n-C4

i-C5

n-C5

C6

F1

F2

F3

F4

F5

C7+

0.0237

0.0031

0.7319

0.0780

0.0355

0.0071

0.0145

0.0064

0.0068

0.0109

0.0157

0.0267

0.0233

0.0126

0.0039

0.0821

0.0237

0.0031

0.7319

0.0780

0.0355

0.0071

0.0145

0.0064

0.0068

0.0109

0.0157

0.0267

0.0233

0.0126

0.0039

0.1302

0.0245

0.0034

0.7817

0.0791

0.0344

0.0066

0.0133

0.0056

0.0058

0.0088

0.0115

0.0158

0.0081

0.0015

0.0001

0.0369

0.0251

0.0033

0.7774

0.0824

0.0363

0.0067

0.0131

0.0049

0.0050

0.0065

0.0082

0.0126

0.0108

0.0058

0.0018

0.0393

0.0206

0.0018

0.5316

0.0737

0.0401

0.0090

0.0194

0.0097

0.0106

0.0194

0.0325

0.0704

0.0841

0.0573

0.0196

0.2639

0.0189

0.0022

0.5517

0.0637

0.0338

0.0084

0.0190

0.0107

0.0120

0.0229

0.0367

0.0709

0.0737

0.0518

0.0236

0.2567

*6,750 psia.**4,315 psia.

duced from the reservoir oil; xooi �composition of surface oil pro-duced from the reservoir oil; and Coo, Cog, and Rs are in scf/STBand rs is in STB/scf.

Average surface-gas compositions yggi and ygoi are calculatedseparately with the relations

ygoi �

�

Nsp

j�1

�ygoi�j�Rs�j

�

Nsp

j�1

�Rs�j

yggi �

�

Nsp

j�1

�yggi�j��rs�j

�

Nsp

j�1

�1� rs�j

, (7.29). . . . . . . . . . . . . . . . . . . . . . . . . .

where the subscript j indicates the separator stage. Stage GOR’sand OGR’s, (Rs)j and (rs)j , respectively, are based on stock-tankvolumes.

The four surface compositions (and gravities) are, in principle,functions of pressure. However, the average separator-gas composi-tions from reservoir oil and from reservoir gas may be similar, and

8 PHASE BEHAVIOR

Fig. 7.15—Calculated compositions for reservoir gas based onMBO properties and surface-component compositions; com-parison with EOS compositions.

yggi � ygoi � constant is a reasonable assumption (as is�gg � �go � constant). These compositions are readily determinedfrom a multistage flash of the original reservoir mixture (see Table7.2). Table 7.3 and Figs. 7.15 through 7.17 show calculated reser-voir-phase compositions based on Eq. 7.26 for a gas-condensatemixture. K values are also calculated (Ki � yi�xi) and comparedwith EOS results for a simulated CVD experiment (Fig. 7.18).

Wellstream composition, zi, can be calculated from reservoirphase compositions yi and xi.

zi � yi Fg � xi�1 � Fg�, (7.30). . . . . . . . . . . . . . . . . . . . . .

where Fg is given by Eq. 7.25 in terms of producing GOR, Rp

(through the quantity Foo). Note that values of Rs and rs used to cal-culate Fg , yi , and xi must be evaluated at the same pressure.

��# $��

The MBO PVT approach has been limited mainly to reservoir simu-lation, although some applications have been reported in well-test

Fig. 7.17—Calculated compositions for reservoir oil based onMBO properties and surface-component compositions; com-parison with EOS compositions.

Modified Black-Oil PropertiesEOSEOSEOS

Fig. 7.16—Calculated methane mole fractions for reservoir oiland gas based on MBO properties and surface-component com-positions; comparison with EOS compositions.

analysis, inflow performance, and reservoir material balance.Multiphase flow in pipe is another obvious application. To aid in theuse of MBO properties for volatile reservoir fluids, we present sev-eral engineering equations in terms of MBO properties.

7.4.1 Rate Equations (IPR)—Traditional Black-Oil PVT. In-flow-performance relations (IPR’s) give the relation between totalsurface rates, qo and qg ; wellbore flowing pressure, pwf ; and aver-age reservoir pressure, pR. For example, consider the radial-flowequation for an undersaturated oil well.13

qo � qoo �

kh�pR � pwf�

141.2�o Boln�re�rw� � 0.75 � s�, (7.31). . . . .

where qo is in STB/D, k �absolute permeability at irreducible wa-ter saturation, md; h �total reservoir thickness, ft; �o �oil viscos-ity, cp; Bo �oil FVF, bbl/STB; re �outer drainage radius, ft;rw �actual wellbore radius, ft; and s �total skin factor.

Fig. 7.18—Calculated K values for reservoir oil and gas based onMBO properties and surface component compositions; compar-ison with EOS compositions.


Fig. 7.19—Fraction of wellbore rate from reservoir oil, fraction ofsurface oil from reservoir oil, GOR, and pwf during depletion ofa volatile-oil reservoir.

For saturated-oil wells producing both reservoir oil and gas, theoil-rate equation can be written terms of traditional black-oil PVTproperties (rs � 0) as

qo � qoo �kh

141.2 ln�re�rw� � 0.75 � s�

pR

pwf

kro

�o Bodp.

(7.32). . . . . . . . . . . . . . . . . . . .

Total gas rate from a saturated-oil well is the product of the oil rateand total producing GOR.

qg � qo Rp , (7.33). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

where qg is in scf/D and Rp usually is available from material-bal-ance calculations.

The rate of the oil phase flowing anywhere in the tubing or reser-voir can be calculated as

qo � qo Bo , (7.34). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

with qo in B/D and Bo evaluated at a specific pressure and temperature.The flow rate of free gas at the same conditions is calculated from

qg � qo�Rp � Rs�

Bgd

5.615, (7.35). . . . . . . . . . . . . . . . . . . . . .

with qg in B/D, qo in STB/D, Rs and Rp in scf/STB, and Bgd in ft3/scf.Rs and Bgd are evaluated at the same pressure and temperature.

7.4.2 IPR—MBO PVT. Eqs. 7.32 and 7.33 are based on the tradi-tional black-oil PVT formulation where reservoir gas is assumed tohave no liquid content. For volatile reservoir fluids, the surface oilconsists of surface oil from the flowing liquid and condensed fromthe flowing vapor. Likewise, the surface-gas rate consists of surfacegas from the flowing vapor and released from the flowing liquid.

The appropriate equations to calculate rates in the production sys-tem are14,15

qo � qoo � qog �kh

141.2 ln�re�rw� � 0.75 � s�

pR

pwf

� kro

�o Bo� 5.615

krg rs

�g Bgd�dp

and qg � qgo � qgg �kh

141.2 ln�re�rw� � 0.75 � s�

pR

pwf

�kro Rs

�o Bo� 5.615

krg

�g Bgd�dp , (7.36). . . . . . . . . . . .

with qo in STB/D and qg in scf/D.The liquid and vapor rates in the tubing or reservoir are given by

qo � qo Foo Bo

and qg � qo�Rp � Rs Foo�Bgd , (7.37). . . . . . . . . . . . . . . . . . .

where Foo �fraction of total surface oil coming from the flowingliquid (Eq. 7.6).

Foo �qoo

qo�

1 � Rp rs

1 � Rs rs. (7.38). . . . . . . . . . . . . . . . . . . . . . .

PVT properties used to calculate qo and qg are evaluated at the pres-sure and temperature in the reservoir or the production tubing.

Evaluation of the integrals in Eq. 7.36 is not straightforward. In fact,using only one of the two rate equations would be logical, dependingon which phase was dominant. For a predominantly oil system, the oilrate in Eq. 7.36 should be used for qo and the gas rate could be calcu-lated from the total producing GOR. Likewise, for a predominantly gassystem, the gas rate in Eq. 7.36 should be used for qg and the oil ratecan be calculated from the total producing GOR. Producing GORwould be available from material-balance calculations.

The volumetric fraction of reservoir fluids flowing as an oil phaseat wellbore conditions is

qo

qo � qg�

qo Bo

qo Bo � qg Bgd�1 �

�Rp � Rs Foo�Bgd

5.615 FooBo�

�1

,

(7.39). . . . . . . . . . . . . . . . . . . .

where Bo, Rs, Bgd, and rs are evaluated at the wellbore flowing pres-sure, pwf . For a volatile-oil reservoir, the oil fraction will drop to lessthan 50% during depletion (see Fig. 7.19), marking the point whenthe gas phase becomes the dominant flowing phase. The relativeamounts of reservoir oil and gas flowing at the wellbore should beconsidered in the interpretation of well tests and application of IPR’s.

7.4.3 Reservoir Material Balance—MBO PVT. Reservoir mate-rial-balance relations for solution-gas-drive and dry-gas reservoirsare well known and widely used. Borthne16 presents a reservoir ma-terial balance based on MBO properties that can be used for blackoils, volatile oils, and gas condensates. Modifications to the materialbalance that account for connate water with dissolved gas, water in-flux, and other such factors can be included readily.

The basis of calculation is 1 bbl reservoir bulk volume. The con-servation-of-mass equations for a single-cell material balanceyields the following difference equations for reservoir-oil and -gasphases during a timestep �tk � tk � tk�1 with a change in averagepressure from (pR)k�1 to (pR)k .

�Ao�k ��Ao� k�1� �Np � 0

and �Ag�k ��Ag�k�1 � �Gp � 0, (7.40). . . . . . . . . . . . . . . .

10 PHASE BEHAVIOR

TABLE 7.4—MBO PROPERTIES FOR GAS CONDENSATE NS-1

Pressure(psia)

Bo(bbl/STB)

Rs(scf/STB) �oo

�goBgd

(ft3/scf)rs

(STB/MMscf)�og

�gg Fgg

6,748.2

6,514.7

6,014.7

5,514.7

4,314.7

3,114.7

2,114.7

1,214.7

714.7

2.6490

2.4693

2.2241

2.0495

1.7427

1.5116

1.3525

1.2277

1.1651

3,005

2,662

2,180

1,829

1,211

757

456

232

124

0.7837

0.7849

0.7859

0.7859

0.7845

0.7832

0.7829

0.7843

0.7864

0.7155

0.7171

0.7208

0.7251

0.7397

0.7629

0.7927

0.8324

0.8663

0.004244

0.004205

0.004226

0.004333

0.004940

0.006371

0.009179

0.016214

0.028276

181.0

158.2

125.7

102.4

64.0

39.3

26.2

21.2

24.7

0.7689

0.7647

0.7575

0.7516

0.7399

0.7298

0.7224

0.7151

0.7088

0.7114

0.7110

0.7107

0.7106

0.7114

0.7139

0.7181

0.7268

0.7386

0.8958

0.9051

0.9194

0.9306

0.9516

0.9677

0.9772

0.9808

0.9771

Water ��1.

where �Np and �Gp �incremental quantities of total surface oiland total surface gas, respectively, produced during the timestep;

Ao � � So

Bo�

5.615(1 � Sw � So)rs�*o

Bgd�

and Ag � � So Rs �

*g

Bo�

5.615(1 � Sw � So)Bgd

�; (7.41). . . . .

and �*o �

�og�oo

and �*g �

�go�gg

. (7.42). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

In Eqs. 7.40 through 7.42, �Np and Ao are in STB/bbl,�Gp and Ag are in scf/bbl, Rs is in scf/STB, Bo is in bbl/STB, rs isin STB/scf, and Bgd is in ft3/scf. Other quantities used in the materi-al-balance procedure are

Eo � 1 � 5.615rs�*o

krg�o Bo

kro�g Bgd,

Eg � Rs�*g � 5.615

krg�o Bo

kro�g Bgd,

Rp ��Gp

�Np,

andkrg

kro� f�So�, (7.43). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

with �o and �g in cp; Rs, Rp, and Eg in scf/STB; rs in STB/scf;Eo in STB/STB; Bo in bbl/STB; and Bgd in ft3/scf. PVT propertiesand porosity are(�*

g )k functions of pressure only. Application ofthese relations is outlined for an oil and a gas-condensate reservoir.

Oil Reservoir.1. Specify (�Np)k , the total surface oil produced in STB/bbl of

bulk volume.2. Assume (pR)k and calculate PVT properties and porosity:

(Bo)k, (Rs)k , (�o)k , (�*o )k , (Bgd)k , (rs)k , (�g)k , (�*

g )k, and (�)k .3. Calculate oil saturation (So)k from Eqs. 7.39 through 7.41.

�So�k �

�Ao�k�1 ��Np�k �

�(1 � Swi)rs�*o�Bgd�k

��1�Bo � rs�*o^�Bgd��

k

.

(7.44). . . . . . . . . . . . . . . . . . . .

4. Calculate (krg�kro)k from (So)k .5. Calculate (Ao)k , (Ag)k , (Eo)k , and (Eg)k .6. Calculate �Npo , incremental surface oil produced from reser-

voir oil, where �Npo � �Np�Eo and Eo � 0.5[(Eo)k � (Eo)k�1].

7. Calculate �Gp, incremental total surface gas produced, where�Gp � �Npo Eg and Eg � 0.5[(Eg)k � (Eg)k�1].

8. Calculate the material-balance error,

� � �Ag�k ��Ag� k�1� �Gp. (7.45). . . . . . . . . . . . . . . . . . . .

9. If � is not sufficiently small, assume a new pressure (pR)k andredo Steps 2 through 8.

Gas-Condensate Reservoir.1. Specify (�Gp)k, total surface gas produced in scf/bbl of bulk

volume.2. Assume (pR)k and calculate PVT properties and porosity:

(Bo)k, (Rs)k , (�o)k , (�*o )k, (Bgd)k , (rs)k , (�g)k , (�*

g )k, and (�)k .3. Calculate oil saturation (So)k from Eqs. 7.39 through 7.41.

�So�k �

�Ag�k�1 ��Gp�k �

��1 � Swi��Bgd

�k

��Rs �*g�Bo � 1�Bgd

��k

. (7.46). . . .

4. Calculate (krg�kro)k from (So)k .5. Calculate (Ao)k , (Ag)k , (Eo)k , and (Eg)k .6. Calculate �Npo, incremental surface oil produced from reser-

voir oil, where �Npo � �Gp�Eg and Eg � 0.5[(Eg)k � (Eg)k�1].7. Calculate �Np, incremental total surface oil produced, where

�Npo � �Np�Eo and Eo � 0.5[(Eo)k � (Eo)k�1].8. Calculate the material-balance error,

� � (Ao)k � (Ao)k�1 � �Np. (7.47). . . . . . . . . . . . . . . . . . .

9. If � is not sufficiently small, assume a new pressure (pR)k andredo Steps 2 through 8.

��% �� &��' ��

In 1965, Kniazeff and Naville7 presented the first approach to mod-eling gas-condensate and volatile-oil systems with a simplifiedcompositional PVT formulation. They introduced four “partial den-sities” as PVT parameters in a radial, 1D numerical model to studythe inflow performance of a Middle East gas–condensate field. Theflow and conservation equations were written in terms of mass,where surface volumes were not considered directly.

Partial densities, �p, are defined as

�pij �mij

Vj, (7.48). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

where mij �surface mass of Component i in Phase j ; Vj �reservoirvolume of Phase j ; i�g and o�surface gas and oil, respectively; andj�g and o�reservoir gas and oil, respectively. The four partial densi-ties, �p, can be expressed as composite terms of MBO properties.

�pgg�0.0763�gg

Bgd,

�pog�350� og rs

Bgd,


Fig. 7.20—Partial densities vs. pressure for Gas-CondensateNS-1.

�pgo�0.0136� go Rs

Bo,

and � poo�62.4� oo

Bo, (7.49). . . . . . . . . . . . . . . . . . . . . . . . . . . .

with �p in lbm/ft3, Bo in bbl/STB, Rs in scf/STB, Bgd in ft3/scf, andrs in STB/scf. Table 7.4 and Fig. 7.20 show the behavior of partialdensities and their relation to MBO properties.

From Eq. 7.47, we see that the variation in surface gravities withpressure is included directly in the definitions of the PVT properties.In fact, this is necessary to maintain an exact mass balance. Drohmand Goldthorpe9 and Drohm et al.10,11 indicate that a similar ap-proach can be used for reservoir simulators on the basis of the MBOapproach. They correct the MBO parameters with surface densities,which indicates that an exact mass balance can be maintained if thecorrected properties (B*

o, R*s, B*

gd, and r*s) are used instead of the

original parameters (Bo, Rs, Bgd, and rs).

B*o �

Bo

62.4� oo,

R*s � Rs�

� go

� oo

� ,

B*gd �

Bgd

62.4� gg,

and r*s � rs�

� og

� gg

� , (7.50). . . . . . . . . . . . . . . . . . . . . . . . . . . . .

with densities in lbm/ft3, Bo in bbl/STB, Rs in scf/STB, Bgd inft3/scf, and rs in STB/scf. Reservoir models based on the Drohm-Goldthorpe or the partial-density approach still do not yield a con-sistent surface-volume material balance unless surface gravities areconsidered pressure dependent.

��( �� )�� *��

Cook et al.5 extend the MBO formulation for vaporizing-gas-injec-tion processes, where a gas-injection parameter, Gi, is defined as thecumulative volume of injection gas entering a grid cell, divided by thegrid-cell volume. PVT properties Bo, Rs, Bgd, and rs are correlatedin tabular form vs. Gi (see Fig. 7.21). Lo and Youngren,17 Whitsonet al.,18 and others propose other extensions to the MBO formulation.

Fig. 7.21—Variation in black-oil PVT properties with gas-injec-tion parameter Gi (adapted from Ref. 5).

Gas Injection Parameter, Gi,Mscf/bbl Oil in Cell

Gas Injection Parameter, Gi,Mscf/bbl Oil in Cell

The complexity of some formulations is disturbing because so manynonphysical quantities are used to correlate compositional effects.

With the increasing speed of compositional simulators and the in-crease in available computing power, it is difficult to justify the effortto develop these highly empirical, pseudo-PVT formulations for gas-injection projects where compositional effects are important. If a sim-plified formulation is used, it should be checked with a compositionalformulation. Tang and Zick19 recently proposed and interesting andaccurate pseudocompositional model with the computational speedof a black-oil model and the accuracy of an EOS model that is of par-ticular interest in miscible-gas-injection simulations.

+� ��

1. Woods, R.W.: “Case History of Reservoir Performance of a Highly Vol-atile Type Oil Reservoir,” JPT (October 1955) 156; Trans., AIME, 204.

2. Dake, L.P.: Fundamentals of Reservoir Engineering, Elsevier Scientif-ic Publishing Co., Amsterdam (1978).

3. Cronquist, C.: “Dimensionless PVT Behavior of Gulf Coast ReservoirOils,” JPT (May 1973) 538.


5. Cook, R.E., Jacoby, R.H., and Ramesh, A.B.: “A Beta-Type ReservoirSimulator for Approximating Compositional Effects During Gas Injec-tion,” SPEJ (October 1974) 471.

6. Spivak, A. and Dixon, T.N.: “Simulation of Gas-Condensate Reser-voirs,” paper SPE 4271 presented at the 1973 SPE Annual Meeting,Houston, 10–12 January.

7. Kniazeff, V.J. and Naville, S.A.: “Two-Phase Flow of Volatile Hydro-carbons,” SPEJ (March 1965) 37; Trans., AIME, 234.

8. Coats, K.H.: “Simulation of Gas-Condensate Reservoir Performance,”JPT (October 1985) 1870.

12 PHASE BEHAVIOR

9. Drohm, J.K. and Goldthorpe, W.H.: “Black Oil PVT Revisited—Use ofPseudocomponent Mass for an Exact Material Balance,” paper SPE17081 available from SPE, Richardson, Texas (1987).

10. Drohm, J.K., Goldthorpe, W.H., and Trengove, R.: “Enhancing theEvaluation of PVT Data,” paper SPE 17685 presented at the 1988 SPEOffshore Southeast Asia Conference, Singapore, 2–5 February.

11. Drohm, J.K., Trengove, R., and Goldthorpe, W.H.: “On the Quality ofData From Standard Gas-Condensate PVT Experiments,” paper SPE17768 presented at the 1988 SPE Gas Technology Symposium, Dallas,13–15 June.

12. Cragoe, C.S.: “Thermodynamic Properties of Petroleum Products,”U.S. Dept. Commerce, Washington, DC (1929) 97.

13. Golan, M. and Whitson, C.H.: Well Performance, second edition, Pren-tice-Hall Inc., Englewood Cliffs, New Jersey (1986).

14. Fetkovich, M.D. et al.: “Oil and Gas Relative Permeabilities Deter-mined From Rate/Time Performance Data,” paper SPE 15431 pres-ented at the 1986 SPE Annual Technical Conference and Exhibition,New Orleans, 5–8 October.

15. Boe, A., Skjaeveland, S., and Whitson, C.H.: “Two-Phase Pressure TestAnalysis,” SPEFE (December 1989) 604; Trans., AIME, 287.

16. Borthne, G.: “Development of a Material Balance and Inflow Perfor-mance for Oil and Gas-Condensate Reservoirs,” MS thesis, U. Trond-heim, Norwegian Inst. Technology, Trondheim, Norway (1986).

17. Lo, T.S. and Youngren, G.K.: “A New Approach to Limited Composi-tional Simulation: Direct Solution of the Phase Equilibrium Equa-tions,” SPERE (November 1987) 703; Trans., AIME, 283.

18. Whitson, C.H., da Silva, F.V., and Søreide, I.: “Simplified Composi-tional Formulation for Modified Black-Oil Simulators,” paper SPE18315 presented at the 1988 SPE Annual Technical Conference and Ex-hibition, Houston, 2–5 October.

19. Tang, D.E. and Zick, A.A.: “A New Limited Compositional ReservoirSimulator,” paper SPE 25255 presented at the 1993 SPE Symposium onReservoir Simulation, New Orleans, 28 February–3 March.

,� �� -��

�� g/cm3

bbl 1.589 873 E�01�m3

ft3 2.831 685 E�02�m3

�F (�F�32)/1.8 ��Clbm 4.535 924 E�01�kg

lbm mol 4.535 924 E�01�kmolpsi 6.894 757 E�00�kPa

GAS-INJECTION PROCESSES 1

��

��

��

For the past 50 years, gas injection has been used successfully inboth oil and gas-condensate reservoirs. Hydrocarbon recoverieshave been increased over what could be obtained by primary drivemechanisms and waterflooding. It was recognized early that thephase and volumetric behavior of gas/oil systems during gas injec-tion had an important effect on ultimate recovery efficiency. Recov-ery efficiency is defined as the product of areal and vertical sweepefficiencies and the microscopic displacement efficiency of the con-tacted reservoir volume. Fluid properties influence all three compo-nents of overall recovery efficiency.

1. Viscosities are found in the definition of mobility ratio, which af-fects areal and vertical sweep efficiency, including viscous fingering.

2. Phase densities define the degree of gravity segregation, whichin turn affects vertical sweep efficiency by gravity bypassing(tonguing) in gravity-dominated processes.

3. Interfacial tensions, viscosities, interphase mass transfer (i.e.,vaporization and condensation), and miscibility affect the residualoil saturation (ROS) defining microscopic displacement efficiency.

Gas-injection processes are designed to enhance the recovery ofoil. The first application of gas injection was intended simply tomaintain reservoir pressure at a level that would sustain existingproduction rates. Another purpose for pressure maintenance in gas-condensate reservoirs was to avoid low liquids recovery resultingfrom retrograde condensation.

Injection of lean gas consisting mainly of methane or nitrogencan, by vaporization, recover significant quantities of light and in-termediate hydrocarbons (C5 through C12) from reservoir oil. Nitro-gen-rich-gas injection can theoretically recover most of the hydro-carbons making up solution gas (C1 through C7). In gas-condensatereservoirs, lean-gas injection can be miscible if reservoir pressureis above the dewpoint; otherwise, lean gas can revaporize liquidsthat drop out by retrograde condensation, which occurs when reser-voir pressure drops below the dewpoint. In oil reservoirs, vaporiza-tion at sufficiently high pressure may develop an in-situ gas that be-comes sufficiently enriched in intermediate components to displacethe reservoir-oil miscibility; this process is called the vaporizing-gas miscible drive.1

Miscibility also can be attained by injecting a gas that is enrichedwith liquefied petroleum gases (LPG’s)—mainly propane. Throughphase equilibrium, the injected gas transfers the LPG’s to the reser-voir oil, which is typically deficient in these intermediate compo-nents. Repeated contacts with enriched gas develops an oil that maybecome miscible with the injection gas; this process is traditionallycalled the enriched-gas, or condensing-gas, miscible drive.1

Zick2 shows that another mechanism may develop from injectionof enriched gas that results in miscible-like recoveries (�95%)without necessarily achieving a miscible condition. The combinedcondensing/vaporizing mechanism he describes is a process that ex-hibits a sharp near-miscible “front.” A condensing mechanism oc-curs just ahead of the front, and a vaporizing mechanism trails thefront. A practical consequence of this mechanism is that a lower en-richment level can be used for the injection gas than would be esti-mated from the traditional interpretation of the enriched-gas mis-cible drive process.

Miscible displacement also can be achieved by a miscible-slugdrive process, where a slug of propane-rich mixture is injected andmixes miscibly with the reservoir oil on first contact. After a suffi-cient volume of slug has been injected [5 to 20% of reservoir porevolume (PV)], a dry gas is injected to drive the slug. The dry gas maybe followed by continuous water injection or by a water-alternating-gas (WAG) injection sequence.

Since the 1970’s, CO2 flooding has been considered one of themost promising gas-injection processes in the U.S.3-5 Major invest-ments have been made to transport large quantities of CO2 in pipe-lines from CO2 reservoirs in Colorado and New Mexico to oil reser-voirs in west Texas and Oklahoma and from Mississippi toLouisiana. CO2 flooding has been used successfully in a wide vari-ety of oil reservoirs, with stock-tank-oil gravities ranging from 15to 45°API, reservoir temperatures from 80 to 300°F, reservoir pres-sures from less than 1,000 to more than 4,000 psia, and in both sand-stone and carbonate formations that vary in thickness from less thanten to more than several hundred feet.

Recovery mechanisms involved with CO2 flooding include oilswelling, oil-viscosity reduction, vaporization of intermediate to heavyhydrocarbons (C5 through C30), and development of multicontact mis-cibility. Other phase behavior exhibited by CO2/oil systems includesasphaltene deposition and three-phase [vapor/liquid/liquid (VLL)] be-havior in low-temperature systems. Phase and volumetric behavior areimportant in both miscible and immiscible CO2 processes.

All the gas-injection methods mentioned can be initiated as sec-ondary or tertiary projects (i.e., following, in conjunction with, oras a replacement for a waterflood). The occurrence of large watersaturations in tertiary and WAG processes does not appear to influ-ence the role of phase and volumetric behavior on these EOR pro-cesses. However, CO2 solubility in water may affect oil recovery ifthe loss of CO2 to connate and injected water is significant.

2 PHASE BEHAVIOR

Fig. 8.1—Phase behavior of ethane/heptane system, includingcritical locus defining MMP conditions (from Ref. 8).

��

Miscible gas displacement typically is characterized by high recover-ies in slim-tube displacement experiments. These recoveries are usu-ally greater than 90% and somewhat less than the 100% theoreticalrecovery expected for “first-contact”-miscible displacement. Thesmall ROS (2 to 10% of PV) is an immobile, highly viscous oil con-sisting mainly of heavy, nonvolatile hydrocarbons. Miscible gas dis-placement may also cause deposition of a solid asphaltene precipitatethat can alter wettability and water injectivity.6,7 As a thermodynamiccondition, miscibility is defined as the condition when two fluids aremixed in any proportion and the resulting mixture is a single phase.For example, gasoline and kerosene are miscible at room conditions,whereas stock-tank oil and water are clearly immiscible.

8.2.1 Binary Systems. For a binary system, the condition of misci-bility is readily defined on a pressure/temperature (p-T) diagram.The dashed line in Fig. 8.1 represents the locus of critical points forall mixtures of ethane and heptane. The critical-locus curve for abinary system will always enclose the two-phase region for all pos-sible mixtures of the two components. Thus, for a binary mixture ata specific temperature, the pressure on the critical-locus curve rep-resents the minimum pressure where miscibility can occur indepen-dently of overall composition. At all pressures greater than thisminimum miscibility pressure (MMP), any mixture of the binarywill form a single phase. Fig. 8.1 also shows that temperature in-creases MMP for a binary mixture at lower temperatures, but the ef-fect reverses at higher temperatures, where MMP decreases with in-creasing the temperature.

8.2.2 Ternary Systems. The condition of miscibility for a ternarysystem can also be depicted on a p-T diagram. Fig. 8.2 shows thecurves defining critical pressure vs. temperature for the ternary sys-tem methane/butane/decane (C1/n-C4/n-C10). For a specific com-position defined in terms of mole percent methane zC1

and the pa-

Fig. 8.2—Phase behavior of the methane/butane/decane ternary,including critical-pressure curves for mixtures of fixed com-position as functions of temperature (from Refs. 9 and 10).

rameter C � zC4�(zC4

� zC10), the locus of critical pressures

indirectly defines the condition of miscibility as a function of tem-perature. For a specific temperature, Fig. 8.2 gives the compositiondependence of critical pressure. Choosing, for example, 280°F and2,000 psia, the composition corresponding to this critical conditionis zC1

� 0.5 and C � 0.85 (zC4� 0.42, zC10

� 0.08). At 280°Fand 3,000 psia, the critical composition is zC1

� 0.68 andC � 0.65 (zC4

� 0.21, zC10� 0.11).

Knowing only the critical composition of a ternary system at a spe-cific temperature and pressure does not directly determine whethertwo mixtures of the three components will be miscible. Graphically,a ternary composition diagram can be used to determine whether twomixtures are first-contact miscible or whether the two mixtures candevelop miscibility. Fig. 8.3 shows the ternary diagram for the meth-

Fig. 8.3—Ternary composition diagram for C1/n-C4/n-C10 systemat 280°F and 2,000 psia (data from Ref. 10).


Fig. 8.4A—Path of developed miscibility by the vaporizing-gasmiscible drive process for C1/n-C4/n-C10 system.

ane/butane/decane system at 280°F and 2,000 psia. The critical com-position determined from Fig. 8.2 is shown as the critical point, C.Other compositional data for equilibrium systems at the same condi-tion are plotted on the ternary diagram. These data define the phaseenvelope enclosing all compositions that will split into two phaseswhen brought to this specific pressure and temperature. The upper

Fig. 8.5A—EOS calculated slim-tube profiles for the enriched-gas miscible drive process for C1/n-C4/n-C10 system (adaptedfrom Ref. 2).

Fig. 8.4B—EOS calculated slim-tube profiles for the vaporiz-ing-gas miscible drive process for C1/n-C4/n-C10 system(adapted from Ref. 2).

curve of the phase diagram defines the dewpoint curve, while the low-er curve defines the bubblepoint curve. The dewpoint and bubble-point curves join at the critical composition, C.

A tie-line is a straight line on a ternary diagram joining an equilib-rium-vapor composition with its equilibrium-liquid composition

Fig. 8.5B—Path of developed miscibility by the enriched-gasmiscible drive process for C1/n-C4/n-C10 system.

4 PHASE BEHAVIOR

(e.g., Line XY). Any system with an overall composition lying onthis tie-line will split into the same equilibrium-liquid and -vaporcompositions defined by X and Y (e.g., overall compositions ZA andZB). Fig. 8.3 shows three tie-lines inside the phase envelope. Everypoint on the two-phase envelope is connected to another point on theenvelope by a tie-line. A limiting tie-line can be drawn through thecritical composition (dashed line in Fig. 8.3). This critical tie-linedetermines whether two mixtures in a ternary system can developmiscibility by a multiple-contact process.

Strictly speaking, two fluids are first-contact miscible if the lineconnecting the two compositions does not pass through the two-phaseenvelope on a ternary diagram. In Fig. 8.3, the G1/G2, G2/O2, andO1/O2 mixtures are first-contact miscible and the G1/O1, G1/O2, andG2/O1 mixtures are not. Some systems can develop miscibility by amultiple-contact process. The criterion for developed multicontactmiscibility in a ternary system is that the two original mixtures lie onopposite sides of the critical tie-line. The following paragraphs de-scribe two methods of developing miscibility in a ternary system.

G1 and O2 can develop miscibility by the vaporizing-gas miscibledrive process. Here, intermediate and heavy components (C4 and C10)are vaporized from the original oil, O2, into the lean gas, G1, makinga richer gas that contacts O2 and develops an even richer gas that againcontacts O2. Finally, the gas composition approaches Critical Com-position C, which is miscible with O2. Fig. 8.4A shows the path of de-veloped miscibility for this process on a ternary diagram. Fig. 8.4Bshows simulated slim-tube profiles of oil saturation, phase densities,and K values for the vaporizing-gas miscible drive of the methane/bu-tane/decane system determined with the Peng-Robinson11 EOS.

G2 and O1 can develop miscibility by the traditional enriched-gasmiscible drive process. Here the intermediate component (C4) in theoriginal gas, G2, transfers to oil, O1. This enriched oil is made evenricher by new contacts with G2 until the oil is modified so that itscomposition approaches Critical Composition C. This developedcritical “oil” is miscible with G2. Fig. 8.5A shows the path of devel-oped miscibility for this process on a ternary diagram. Fig. 8.5Bshows simulated slim-tube profiles of oil saturation, phase densi-ties, and K values for enriched-gas miscible drive of the methane/butane/decane system determined with the Peng-Robinson EOS.

8.2.3 Pseudoternary Diagrams for Multicomponent Systems.For a true three-component system, first-contact and developedmiscibility can be determined uniquely from a ternary diagram at aspecific pressure and temperature. Pseudoternary diagrams are alsoused for multicomponent mixtures, where several components aregrouped together and represented at each apex on the ternary dia-gram. This method is used despite the inherent limitation that multi-component phase behavior cannot be represented uniquely with aternary diagram. Strictly speaking, a ternary representation of amulticomponent system is valid only if the relative amounts of allcomponents defining each pseudocomponent remain constant. Thiscondition cannot be satisfied for oil systems, but the graphical repre-sentation is still used.

Methane, N2, and CO2 are usually treated as the light pseudocom-ponent in a pseudoternary diagram, with ethane through hexanestreated as the intermediate pseudocomponent and heptanes-plus asthe heavy pseudocomponent. Sometimes CO2 is included with theintermediate components. The general characteristic of pseudoter-nary phase behavior described earlier (namely, that developed mis-cibility can be achieved if the injection gas and reservoir oil lie onopposite sides of the critical tie-line) are applied directly to multi-component systems.

Unlike the pseudoternary phase envelope for a three-componentsystem, the pseudoternary phase envelope for a multicomponentsystem is not unique. It must be developed from a sequence of multi-ple contacts, where the multicontact procedure starts with the origi-nal injection gas and the reservoir oil. Thereafter, the procedures forvaporizing- and condensing-gas drives are different. A forward-contact procedure is used for the vaporizing-gas drive, while a back-ward-contact procedure is used for the enriched-gas drive.

The forward-contact procedure starts by mixing the injection gaswith the reservoir oil to obtain a two-phase mixture. The equilibri-um-gas and -oil compositions provide two points and a tie-line on

the pseudoternary diagram. The gas from the two-phase mixture isthen removed and put into contact with original reservoir oil to forma new two-phase mixture, providing two more points and anothertie-line on the pseudoternary diagram. The process of removingequilibrium gas and mixing it with original reservoir oil is repeateduntil either (1) the enriched gas becomes miscible with the originalreservoir oil or (2) the compositions of the equilibrium gas and equi-librium oil no longer change. If Condition 1 is achieved, the processis multicontact miscible and most of the phase envelope is estab-lished up to the critical point. If Condition 2 is achieved, the processis not multicontact miscible and only part of the phase diagram isestablished. When miscibility is not achieved, the reservoir oil is lo-cated on an extension of a tie-line with the equilibrium mixtures andno further component exchange is achieved by mixing the equilibri-um gas with the original reservoir oil.

The pseudoternary diagram for the traditional enriched-gas driveprocess is developed by a backward-contact procedure. This starts bymixing the injection gas with reservoir oil to obtain a two-phase mix-ture. The equilibrium-gas and -oil compositions determine a pointand a tie-line on the pseudoternary diagram. The equilibrium oil isthen put into contact with the original injection gas to form a new two-phase mixture, yielding another point and tie-line on the pseudoterna-ry diagram. The process of mixing altered equilibrium oil with origi-nal injection gas is repeated until either Condition 1 or 2 (describedin the preceding paragraph) is achieved. Interpretation of the miscibil-ity condition is the same as that for the vaporizing-gas drive process.

Zick2 claims that the pseudoternary representation of enriched-gas injection may lead to erroneous interpretation of the actual re-covery mechanism. He further claims that the traditional enriched-gas miscible drive (developed by the multicontact process justdescribed) may rarely, if ever, occur in reservoir systems. His ob-servations are covered in more detail in the Sec. 8.4. On the otherhand, pseudoternary representation of the vaporizing-gas miscibledrive process probably gives a reasonable description of the actualdisplacement mechanism.

Quaternary diagrams have also been used to describe multicon-tact displacement in multicomponent systems; however, the addi-tional dimension added by the fourth component makes this graphi-cal representation more difficult to understand. Also, the“uniqueness” (i.e., oversimplification) of a single critical tie-line ona ternary diagram is no longer valid with a quaternary representa-tion. In their discussion of N2 in a vaporizing-gas miscible drive pro-cess (Fig. 8.6), Koch and Hutchinson12 give perhaps the most illus-trative use of a quaternary diagram.

8.2.4 Slim-Tube Displacements. A single definition of multicon-tact miscibility has not been accepted for multicomponent systems.Most definitions relate to recovery performance curves from labora-

Fig. 8.6—Illustration of phase relations for vaporizing-gas mis-cible drive process with N2 as injection gas (from Ref. 12).


Fig. 8.7—Schematic of a slim-tube displacement apparatus;sand-packed coil consists of 40-ft-long, 1/4-in.-OD stainless-steel tubing packed with 160/200-mesh Ottawa sand (adaptedfrom Ref. 13).

Capillary-TubeSight Glass

Sand-Packed Coil

CO2 Supply Cylinder

CO2

H2O FromPositive-DisplacementPump

Test OIl

Solvent

Well Test Meter

100-cm3 Burette

BackpressureRegulator

tory displacement tests. A slim-tube apparatus is used in the dis-placement experiments. Most slim tubes consist of 0.25-in.-outer-diameter coiled tubing, from 25 to 75 ft in length, packed withuniform sand or beads and housed in a constant-temperature con-tainer. Fig. 8.7 is a schematic of a slim tube. Orr et al.14 summarizeslim-tube characteristics described in various miscible studies.

Slim-tube results are interpreted by plotting cumulative oil recov-ery vs. PV of gas injected. Two recoveries are usually reported, atbreakthrough and after injection of 1.2 PV of gas. To determine theMMP, several slim-tube experiments are conducted at varying dis-placement pressures. Recovery at 1.2 PV of gas injected is thenplotted vs. displacement pressure. For immiscible displacements,where relative permeabilities and viscosities influence the recoveryprocess, recovery increases with pressure.

The recovery-pressure curve starts to flatten when the displace-ment becomes near miscible. Depending on the type of displace-ment process, temperature, injection gas, and other factors, the tran-sition from immiscible to miscible may be abrupt or gradual (Figs.8.8A and 8.8B). Table 8.1 gives reservoir-oil and -gas compositionsfor Fig. 8.8B. Choice of the “break point” defining MMP is some-what arbitrary. Some investigators use a specific recovery factor,such as 90% at 1.2 PV of gas injected, to define MMP. For CO2/oilsystems, Holm and Josendal16-19 use a definition of MMP that re-quires 80% recovery at breakthrough and 94% recovery at a produc-ing gas/oil ratio (GOR) of 40,000 scf/bbl (occurring at approximate-ly 1.1 to 1.3 PV injected gas).

Color change and lack of multiphase production from the slim-tubeapparatus have also been used to define the MMP. Yellig and Met-calfe13 give a particularly good discussion of criteria for definingMMP on the basis of slim-tube data for CO2 systems. Fig. 8.9 showsrecovery vs. PV of gas injected for a CO2 miscible displacement;changing colors of the produced oil are noted on the curve (from darkto red to orange to yellow to clear). Fig. 8.10 shows the qualitativecharacter of produced fluids from a series of slim-tube tests at immis-cible and miscible conditions for an enriched-gas displacement. Thesolid line indicates fluid density based on photoelectric-cell output,and shading represents two-phase production.

It is generally accepted that slim-tube displacements yield the mostreliable information for defining true multicontact miscibility. Al-though the slim-tube-determined miscibility condition is affected al-most exclusively by the phase behavior of the fluids being studied,this miscibility condition is not the same as “thermodynamic misci-

Fig. 8.8A—Experimental slim-tube for CO2 displacement of awest Texas 30°API oil showing effect of temperature on recov-ery-pressure behavior (adapted from Ref. 13).

Fig. 8.8B—Experimental slim-tube results for high-pressure dis-placement of a reservoir oil showing effect of injected lean-gascomposition on recovery-pressure behavior (from Ref. 15).

Displacement Pressure, MPa

Well-Designed Slim Tubes: Miscible Recoveries�95%

Low Miscible RecoveriesIndicate Slim-Tube

Equipment Design Problems

21 23 25 27 29 31 33 35 37 39 41 43

90

80

40

100

50

70

60

TABLE 8.1—RESERVOIR-OIL AND INJECTION-GAS

COMPOSITIONS FOR FIG. 8.8B

Component

Reservoir

Oil Gas 1 Gas 2 Gas 3

H2S 0.00 0.0 0.00 0.00

N2 0.06 1.2 0.35 0.31

CO2 2.71 0.0 0.00 0.00

C1 34.66 93.3 81.71 69.61

C2 6.96 3.0 9.03 12.18

C3 6.46 1.1 4.31 8.83

i-C4 1.54 0.0 0.84 1.63

n-C4 4.09 0.7 1.63 3.42

i-C5 1.87 0.0 0.54 1.07

n-C5 2.57 0.7 0.59 1.22

C6 3.58 0.55 0.00

C7 3.66 0.00 1.73

C8 3.46 0.43

C9 3.13

C10 2.61

C11+ 22.64

6 PHASE BEHAVIOR

Fig. 8.9—Experimental slim-tube recovery vs. PV injected CO2curve indicating change in color (as viewed in sight glass) ofproduced fluid after breakthrough (adapted from Ref. 13).

Clear and Yellow

YellowOrangeRedDark

bility.” Slim-tube experiments are relatively fast and simple to con-duct, they do not require expensive equipment, and the experimentalprocedure can be automated readily with standard data-acquisitiontools. The rising-bubble apparatus has also been suggested as a meth-od to arrive at an indication of true miscibility21-24; however, we areskeptical of this claim for the condensing/vaporizing drive mecha-nism. Zhou and Orr25 appear to share this skepticism.

8.2.5 Multiple-Contact Pressure/Volume/Temperature (PVT)Experiments. Although the slim-tube displacement experiment isthe preferred method for determining the MMP of an injection gas,it does not provide controlled measurements of the system phaseand volumetric behavior. Various PVT experiments can be used tosupplement slim-tube measurements for miscible displacementprojects. Also, PVT experiments provide the only means of obtain-ing important data, such as viscosities, densities, compositions, andK values. Multicontact PVT data are particularly useful for tuningan equation of state (EOS) or any other PVT model that may be usedin reservoir simulation.

PVT experiments designed for gas-injection processes involvemultiple contacts of injection or equilibrium gas with original reser-voir oil or previously contacted equilibrium oil. The swelling test(Fig. 8.11) is the most common multicontact PVT experiment. Inthis experiment, injection gas is mixed with original reservoir oil invarying proportions, with each mixture quantified in terms of a mo-lar percentage of injection gas (e.g., 20 mol% CO2 indicates that 0.2moles of CO2 has been mixed with 0.8 moles of reservoir oil). Thesaturation pressure and phase volumes at more than and less than thesaturation pressure are measured for each mixture. The data arepresented in a pressure/composition ( p-x) diagram, as in Figs. 8.12and 8.13. Pressure/volume plots are also used, usually as crossplotsto determine quality lines on a p-x diagram. Occasionally, composi-tions of equilibrium-oil and -gas phases are determined for somemixtures in a swelling test (usually those at pressures close to the ex-pected operating pressure of the injection project and those close tothe critical point on the p-x diagram).

The forward- and backward-contact PVT experiments (Fig. 8.14)also provide useful phase and volumetric data for gas-injection stud-ies. The forward-contact experiment follows the procedure describedearlier for the vaporizing-gas miscible displacement process. That is,the equilibrium gas from each contact is removed and mixed withmore of the original reservoir oil. The amount of gas mixed with origi-nal oil at each contact may vary, but the amount should not affect theresults significantly. The developed gas should eventually reach mis-cibility with the original reservoir oil if the experiment is conductedat a pressure greater than the MMP. Otherwise, the forward-contact

Fig. 8.10—Produced wellstream character indicated by solid linerepresenting density from photoelectric-cell output; shading in-dicates two-phase production (from Ref. 20).

experiment gives information about how efficiently the developedgas vaporizes the original oil without achieving miscibility.

The backward-contact experiment follows the procedure de-scribed for the enriched-gas miscible drive process. Here the equi-librium oil resulting from a given contact is mixed with more of theoriginal injection gas. According to the traditional interpretation ofthe enriched-gas miscible drive process, miscibility should developbetween the original injection gas and the altered reservoir oil. Ben-ham et al.27 present backward-contact PVT results that indicatemiscibility can be achieved by this process (Fig. 8.15). On the otherhand, Zick2 presents backward-contact PVT experiments and EOSsimulations that convincingly show that miscibility is not achievedby this process even at pressures considerably higher than the MMPdetermined by slim-tube experiments (Figs. 8.16 and 8.17).

The backward-contact experiment also can be used to investigaterevaporization of retrograde condensate by lean injection gas. Figs.8.18 and 8.19 show swelling and backward-contact experimentaldata reported by Vogel and Yarborough.28 Here, N2 was used in thebackward-contact experiment to revaporize retrograde liquid thatformed when a lean-gas condensate was brought into contact with50% N2 in a swelling test. Vogel and Yarborough also give exper-imental results for the effect of N2 on a reservoir oil. Nitrogen wasmixed with the original reservoir oil in varying proportions (0, 0.144,0.5, and 1.5 PV of N2 per PV of original reservoir oil). For a givenN2/oil mixture, the system was brought to equilibrium at a specifiedpressure. The equilibrium gas was completely removed and dis-carded. The equilibrium oil was analyzed chromatographically, anda differential liberation experiment was conducted on the oil to deter-mine solution gas/oil ratio and oil volumetric properties. Figs. 8.20through 8.24 present some of the results from these experiments.


Fig. 8.11—Schematic of swelling test.

8.2.6 Calculation Algorithms. Several methods have been pro-posed for calculating MMP by multicontact calculations with anEOS or K-value model.27,29,30 These methods typically involve ei-ther a forward- or backward-contact mixing procedure, with theintention of simulating either a vaporizing- or condensing-gas driveprocess, respectively.

Metcalfe et al.31 proposed a more rigorous calculation approachbased on Cook et al.’s32,33 multicell vaporization model. With thisapproach, fluid mixing along a series of connected “cells” is usedto simulate the development of miscible conditions with time (Fig.8.25). Initially, all cells are filled with reservoir-fluid composition.Then, a volume of injection gas (approximately 20% of a cell vol-ume) is mixed with the contents of the first cell and brought to equi-librium. Part of the resulting equilibrium gas and oil is mixed with

Fig. 8.12—Experimental p-x diagram for mixtures of lean naturalgas with a Block 31 Devonian reservoir oil (adapted from Ref. 1).

BUBBLEPOINTS

DEWPOINTS

,

the contents of Cell 2 and brought to equilibrium, part of the result-ing equilibrium gas and oil from Cell 2 is mixed with Cell 3, and soon. Finally, production is recorded from the last cell; typically,approximately 50 cells are used.

This series of calculations constitutes one “batch” or “timestep.”The calculations are repeated with a new volume of injected gas, andthe compositional changes in one or more cells are monitored withtime. Metcalfe et al. plot the results on a ternary diagram to applythe critical tie-line approach for establishing the condition of misci-bility. Injection-gas composition can change at each timestep, there-

Fig. 8.13—Experimental p-x diagram for mixtures of N2 with aPainter reservoir oil (adapted from Ref. 26).

Bubblepointcurve

vol%liquid

N2 in Painter Reservoir, mol%

8 PHASE BEHAVIOR

Fig. 8.14—Schematic of forward- and backward-contact experi-ments.

INJECTIONGAS

INJECTIONGAS

INJECTIONGAS

INJECTIONGAS

CRUDE

CRUDE CRUDE CRUDE

by allowing the study of miscible-slug displacement and the effectof driving an enriched gas with a cheaper lean gas.

Metcalfe et al. propose three methods for determining whichphases are passed from one cell to the next and the amount of eachphase passed. The first method, originally proposed by Cook etal.,32,33 passes all equilibrium gas from cell to cell, simulating thevaporizing-gas (forward-contact) process. The second methodpasses only enough gas and oil to the next cell to ensure that the re-maining mixture in the current cell fills the cell volume. The thirdpasses equilibrium gas and oil according to the mobility ratio(krg/kro)�(�o/�g), where the relative permeability ratio krg/kro isentered as a function of saturation. For any miscible displacementprocess, the second and third methods should give the same condi-tions of miscibility in the limit of small injection volumes and a largenumber of cells. The first method is valid only for a vaporizing-gasdrive mechanism.

Short of simulating a slim-tube displacement, the multicell cal-culation approach is probably the best generalized scheme for deter-mining the conditions required to develop miscibility. It should givethe same conditions of developed miscibility as slim-tube results ifthe multicell calculations are interpreted correctly. Calculationmethods based strictly on forward- or backward-contact proceduresare not recommended.27,29,30

Johns et al.34,35 present analytical simulation results based on themethod of characteristics for three- and four-component systemsthat verify the mechanisms of the condensing/vaporizing drivemechanism originally described by Zick.2 Practically, their ap-proach is limited to four-component systems and is more difficultto program than a one-dimensional (1D) slim-tube or batch-type ex-periment. Wang and Orr36 recently proposed a generalized algo-rithm for computing MMP on the basis of complex tie-line analysis

Fig. 8.16—Experimental slim-tube recoveries at 1.2 PV injectedgas vs. injection pressure for Reservoir Oil A displaced by Sol-vent A (adapted from Ref. 2).

Fig. 8.15—Experimental backward-contact PVT data for en-riched-gas/reservoir-oil system (adapted from Ref. 27).

Intermediates, mol%

founded in the theory fo the method of characterisitics; the CNmechanism can be the method of developed miscibility.

��

Lean-gas injection with methane- and N2-rich gases has been usedfor reservoir management during primary production, as an alterna-tive to waterflooding for secondary recovery, and in gravity-stabletertiary projects. Successful projects include (1) pressure mainte-nance in oil reservoirs to maintain productivity, sometimes by de-veloping an artificial gas cap; (2) gravity-stable displacement in dip-ping, high-permeability oil reservoirs; (3) reservoir-voidagereplacement to maintain the oil/water contact in a strong-waterdrivereservoir; (4) recovery of upstructure “attic” oil and gas in strong-waterdrive reservoirs; (5) high-pressure multicontact miscibility inoil reservoirs; and (6) partial and full pressure maintenance in gas-condensate reservoirs.

Some of the more important justifications for lean-gas injectioninclude gas availability, better injectivity in low-permeability reser-voirs, conservation or environmental constraints, and superior oilrecoveries compared with alternative EOR methods. Not all ap-plications of lean-gas injection require special treatment of phase

Fig. 8.17—EOS-calculated multiple backward-contact PVT ex-periment for Oil A and Solvent A at 900 psi higher than MMP pres-sure indicated from experimental and simulated slim-tube re-sults (adapted from Ref. 2).


Fig. 8.18—Effect of N2 on the phase behavior of two gas-condensate reservoir fluids: equilibrium flash volumetric expansion testsrun at 381K and no material removed from p-V cell (from Ref. 28).

Liquid, vol% Liquid, vol%

0% (gas only, no added N2)10% cumulative added N230% cumulative added N250% cumulative added N2

0% (gas only, no added N2)10% cumulative added N230% cumulative added N250% cumulative added N2

and volumetric behavior. However, high-pressure injection in oilreservoirs and gas cycling in partially depleted condensate reser-voirs do require detailed knowledge of how the injected gas behaveswith the reservoir fluids.

8.3.1 Vaporizing-Gas Miscible Drive. Several high-pressure, lean-gas miscible projects have been reported for light oils with stock-tank-oil gravities greater than 35°API and with operational floodingpressures greater than approximately 3,500 psia.1 Lean gases containmostly methane or N2, with methane-rich gases also containingsmaller quantities of ethane and C3+ components. Nitrogen-richgases include flue gas, consisting of approximately 88% N2 and 12%CO2, and pure N2 generated from cryogenic air separation.

Lean gases tend to vaporize intermediate hydrocarbons in the C5to C12 range, depending on the pressure and injection-gas composi-tion. Nitrogen also tends to “trade places” with the solution gas inan oil, thereby improving natural gas recovery. At sufficiently high

Fig. 8.19—Revaporization of retrograde condensate by multiplecontacts with N2 (adapted from Ref. 28).

N2 Injective, cumulative PV

pressure, lean gas can develop an in-situ gas that is sufficiently richin intermediate components (C2 through C4) to develop miscibilitywith the reservoir oil. Another condition for developed miscibilityby the vaporizing-gas drive process is that the reservoir should nothave an initial free-gas saturation. That is, in a vaporizing-gas drivemechanism the gas saturation must always be zero ahead of the mis-cible displacement front.

Pressure is usually the primary design parameter in a vaporizing-gas miscible drive project. Other considerations include slug size,WAG ratio, and producer/injector pattern. Methane-rich injectiongases tend to develop miscibility at slightly lower pressures than N2-rich gases, depending mainly on the methane content in the reser-voir oil. Also, the price differential between N2 and lean natural gascan be significant, so the methane/N2 ratio may be a valid design pa-rameter in some lean-gas miscible projects.

Fig. 8.20—Experimental oil formation volume factors (FVF’s) formixtures of N2 with a reservoir oil (adapted from Ref. 28).

Cumulative PVN2 Contacted

0.0 PV N2

0.144 PV N2

0.50 PV N2

1.50 PV N2

Experimental

Calculated

10 PHASE BEHAVIOR

Fig. 8.21—Experimental solution GOR data for mixtures of N2with reservoir oil (adapted from Ref. 28).


0.0 PV N2

0.144 PV N2

0.50 PV N2

1.50 PV N2

Experimental

Calculated

Stalkup1 reports only one MMP correlation for the vaporizing-gas drive miscible process. This correlation gives MMP as a func-tion of reservoir-oil bubblepoint pressure; reduced temperature ofthe reservoir oil; and mass fraction of three groups in the reservoiroil: (1) C2 through C6 plus CO2 and H2S, (2) C1 plus N2, and (3) C7+.

8.3.2 Gas Cycling. Gas cycling in condensate reservoirs has beenused for the past 50 years to minimize lossses in liquid recovery.When reservoir pressure drops below the dewpoint pressure in agas-condensate reservoir, liquids condense and remain primarily asan immobile phase. The produced wellstream becomes leaner (asreflected by a decreasing condensate yield), and overall condensaterecovery may be as low as 15 to 20%. Further depletion at pressuresless than approximately 2,000 psia may revaporize some of the“lost” retrograde condensate, but this additional recovery is usuallynot significant.

To maximize liquid recovery, reservoir pressure should be kepthigher than the dewpoint pressure to avoid retrograde condensation.Typically, this is achieved by reinjecting produced gas that has beenseparated and processed for condensate and natural gas liquids

Fig. 8.23—Experimental oil viscosity data for mixtures of N2 withreservoir oil (adapted from Ref. 28).


0.144 PV N2

0.0 PV N2

1.50 PV N2

0.50 PV N2

Experimental

Calculated

20 PV

Fig. 8.22—Experimental oil density data for mixtures of N2 withreservoir oil (adapted from Ref. 28).


0.144 PV N2

0.0 PV N2

1.50 PV N2

0.50 PV N2

Experimental

Calculated

(NGL’s). Because the produced gas is not sufficient to replace thereservoir voidage caused by production, makeup gas must be ob-tained to achieve full pressure maintenance. If the reservoir is ini-tially undersaturated (i.e., the initial pressure is greater than the dew-point pressure), reinjecting only the produced gas is acceptable untilreservoir pressure approaches the dewpoint.

The economics of delaying gas sales to increase condensate re-covery may be prohibitive. Alternatives to delayed gas sales includereinjection of only part of the produced gas, purchasing cheapermakeup gas for reinjection, and replacing produced-gas reinjection

Fig. 8.24—Change in heptanes-plus distribution for oil that hasbeen in contact with 20 PV of N2: effect of cycling, simulated trueboiling point analysis by temperature-programmed gas chroma-tography (adapted from Ref. 28).

Experimental DataBefore Cycling With N2

After Cycling With N2


Fig. 8.25—Schematic of multicell calculation method: (a) stagnantoil, (b) moving excess oil, and (c) oil and gas moved by phase mo-bilities (adapted from Ref. 31).

(a)

(b)

(c)

Gas

Oil

Gas

Oil

Oil

Oil

Gas to Next Cell

OriginalCell

Condition

OriginalCell

Condition

OriginalCell

Condition

FinalCell

Condition

FinalCell

Condition

FinalCell

Condition

Oil

Oil OilOil

Oil

Gas

Oil

Oil

Gas

GasGas

Gas (and Oil)

Cell 1 Cell 2 Cell NN

InjectionGas

Batch 1

InjectionGas

InjectionGas

Batch 2

Batch N

OriginalOil

OriginalOil

OriginalOil

Oil Oil Oil

OilOilOil

Gas (and Oil)

Gas (and Oil) Gas (and Oil) Gas (and Oil)

Gas (and Oil)

Gas (and Oil)Gas (and Oil)

Gas (and Oil)Gas (and Oil)

Gas (and Oil)

Gas (and Oil) to Next Cell

Gas (and Oil) to Next Cell

Oil

with injection of flue gas or N2. Generating large quantities of N2cryogenically on location has been demonstrated in several largegas-cycling projects.1 Nitrogen has also been used as makeup gasto ensure full pressure maintenance.

In the early 1980’s, studies27,32 showed that N2 caused substan-tial condensation of liquids when mixed with a gas-condensate mix-ture (Figs. 8.18 and 8.26). This behavior caused concern that N2might worsen the problem of retrograde condensation if used tomaintain pressure in condensate reservoirs. Subsequent displace-ment and multicontact tests showed that practically all the liquidcondensed by initial contact with N2 was revaporized by later con-tacts with the N2 (Fig. 8.19).28,37-39 Slim-tube recoveries with N2displacing a gas condensate showed behavior similar to that ofmethane-rich gas displacements, with both gases yielding practical-ly 100% total hydrocarbon recovery.

��! " �� #��

Miscible displacement projects with enriched injection gas are re-ported in the literature for reservoir oils with stock-tank-oil gravitiesranging from 30 to 45°API.1 Typical flooding pressures range from1,500 to 4,000 psia. Enriched gases usually contain methane,ethane, and varying quantities of LPG components C3 through C4.CO2 also may be found in the injection gas without significantly af-fecting the miscibility condition. Reservoir displacement pressureand the degree of LPG enrichment are the two main design parame-

Fig. 8.26—Effect of N2 and lean natural gas on the dewpointpressure of a gas-condensate reservoir fluid (from Ref. 37).

Cumulative Gas Injected, scf/RB

ters used to optimize recovery and other factors affecting an en-riched-gas drive project.

8.4.1 Traditional Mechanism. Some difference of opinion existsconcerning the actual displacement mechanism responsible for highrecoveries reported in slim-tube experiments with enriched gases.The traditional enriched-gas displacement mechanism is based onan interpretation of a pseudoternary diagram, where miscibility isdeveloped by repeated contacts of the injection gas with the oilfound at the point of injection. The following is Benham et al.’s27

description (based on their Fig. 3) of this traditional interpretationof the enriched-gas miscible displacement process (Fig. 8.27).

“Assuming that a phase diagram of this general shape is an ap-propriate representation, the mechanism for obtaining miscibilitymay be illustrated by reference to Fig. 3. This figure has been pre-

Fig. 8.27—Pseudoternary representation of the traditional en-riched-gas-miscible drive process (adapted from Ref. 27).

12 PHASE BEHAVIOR

MC5�

Fig. 8.28—Benham et al.26 chart for determining maximum meth-ane content in an injection gas for miscibility to develop accord-ing to traditional enriched-gas miscible oil with �240 at 3,000psia (adapted from Ref. 27).

pared to demonstrate the mechanism involved in obtaining miscibil-ity between reservoir fluid represented by Point R and an enriched gasrepresented by Point RG. The reservoir fluid is in the two-phase re-gion and has a liquid phase of composition (m) and a vapor phase ofcomposition (a). As gas is first injected, it will tend to move both liq-uid and vapor until eventually the gas velocity is greater than the liq-uid velocity. The first mixing will be between liquid (m) and rich gas(RG). The over-all composition of this mixture could be Point �. Thismixture separates into two phases represented by Points n and b. Asmore rich gas is injected, it displaces the gas (b) and mixes with theliquid (n). These may mix to an overall composition (�), which sepa-rates into liquid (o) and vapor (c). Injection of more rich gas will resultin displacement of the vapor (c) and mixing of the liquid (o) with theinjection fluid (RG) to form the mix (�). This continues until injectionfluid (RG) mixes with the liquid (t), at which time a miscible displace-ment begins. Injection fluid (RG) miscibly displaces the liquid (t),which miscibly displaces the liquid (s), which miscibly displaces r,etc. The gases will also be miscibly displaced by the rich gas; there-fore, a completely miscible displacement has been achieved. The liq-uids will gradually build up in saturation with displacement until acompletely single-phase miscible displacement is achieved.

“It may be shown that the leanest mixture that will give a miscibledisplacement is represented by a point on the extension of the limit-ing tie-line (A-B), which passes through the critical point (C).”

Benham et al. use this interpretation of the displacement mecha-nism to develop a series of working curves for estimating the degreeof enrichment required to attain MMP for a given reservoir oil. Theirgraphical correlations require (1) average molecular weight of thereservoir-oil C5+ mixture, (2) average molecular weight of the C2+components that will be used to enrich the injection gas, and (3) res-ervoir temperature. With these three data, the appropriate charts areentered to obtain the allowable methane concentration in the injec-tion gas. Each chart represents an MMP; charts are provided forMMP’s of 1,500, 2,000, 2,500, and 3,000 psia (Fig. 8.28). A plot ofLPG enrichment vs. MMP can then be made for design calculations.

Zick2 reports an MMP of 3,100 psia at 185°F for his ReservoirFluid A with an injection gas consisting of 39 mol% methane (20%methane mixed with 80% Solvent A containing 23.5 mol% C1). Fig.8.29 plots slim-tube recovery at 1.2 PV gas injected vs. dilution of thesolvent with pure methane. With MC2�

�40 for the reported solventand MC5�

�260 for the reservoir oil, the Benham et al. charts givea maximum methane content for the injection gas somewhat greaterthan 50 mol%. That is, the Benham et al. charts indicate that MMPcan be achieved at 3,000 psia with the solvent diluted 35% with meth-ane. Fig. 8.29 indicates that the experimental slim-tube recovery isonly 65% for this injection-gas composition. The Kuo16 MMP cor-relation predicts a similar overestimation of methane dilution.

8.4.2 Combined Condensing/Vaporizing Mechanism. Zick pro-poses an alternative mechanism to explain the miscible-like recoveriesthat can be achieved by displacing a reservoir oil with enriched gas. The

Fig. 8.29—Experimental slim-tube recoveries at 3,100 psig asfunctions of solvent dilution with methane for Reservoir Oil A (de-pleted to 3,000 psig) and Solvent A at 185°F (adapted from Ref. 2).

mechanism is a combination of (1) a leading front that enriches originaloil with light intermediates found in the original injection gas andmiddle intermediates (C5 through C30) that have been vaporized fromthe reservoir oil behind the front and (2) a trailing front of injection gasthat vaporizes middle-intermediate components. A sharp transitionzone separates condensing and vaporizing fronts. This transition zoneis near miscible, or perhaps miscible in the absence of dispersion, re-covering practically all the reservoir oil with only a small ROS.

Fig. 8.30—EOS-calculated slim-tube profiles for condensing-/vaporizing-gas drive of Reservoir Oil A by Solvent A (adaptedfrom Ref. 2).


Fig. 8.31—Multiphase behavior for mixture of 81.72 mol% (67.99vol%) enriched driving gas (32% C1, 37% C2, and 30% C3) and areservoir oil at p�2,000 psia and 105°F (adapted from Ref. 40).

Four-Phase

-

Injection Gas, mol%

Fig. 8.30 shows the profile of oil saturation, phase densities, andK values for an enriched-gas displacement calculated by an EOSslim-tube simulator. Five regions are readily identified in this figure.On the basis of the proposed condensing/vaporizing mechanism,these five regions can be summarized as follows.

1. Original oil zone.2. A leading two-phase front with net condensation of intermedi-

ate components. The gas contains light-intermediate componentsfound in the original injection gas and middle-intermediate compo-nents that have been vaporized from the reservoir oil.

3. A sharp transition zone with near-miscible behavior. The frontside of the transition zone (toward Zone B) shows dramatic con-densation of intermediate and heavy components. The back side ofthe transition zone (toward Zone D) shows highly efficient vapor-ization of intermediate and heavy components. Only a small ROSis left behind the transition zone.

4. A trailing front of enriched gas, which vaporizes middle-inter-mediate components found in the remaining residual oil.

5. A stripped ROS, in equilibrium with the injection gas, remainsbehind. Little if any mass transfer occurs here. The residual oil con-sists of a heavy, nonvolatile material and the components making upthe injection gas.

The net mass transfer of components between the gas and oilphases is reflected by the slope of the K values plotted vs. distance.Net condensation from the gas phase into the oil phase occurs wherethe slope dKi /dx is negative for middle-intermediate and heavycomponents (Zone B). Net vaporization from the oil phase into thegas phase occurs where the slope dKi /dx is positive for the middle-intermediate and heavy components (Zone D).

Zick2 gives a fairly detailed summary of the condensing/vaporizingmechanism. With experimental and simulation results, he shows thatthe traditional enriched-gas miscible drive mechanism cannot explainmiscible-like recoveries for three different reservoir-oil/enriched-gassystems. His arguments basically hinge on the observation that the oilthat should first become miscible with an enriched gas (i.e., the oil near-est the point of injection) does not become miscible in multicontactPVT experiments or in simulations of slim-tube displacements.

He writes, “When the enriched gas comes into contact with theoil, the light intermediates condense from the gas into the oil, mak-ing the oil lighter. The equilibrium gas is more mobile than the oil,so it moves on ahead and is replaced by fresh injection gas, fromwhich more light intermediates condense, making the oil even light-

Fig. 8.32—p-x diagram showing multiphase behavior for an en-riched gas (32% C1, 37% C2, and 30% C3) mixed with a reservoiroil at 105°F (adapted from Ref. 40).

Injection Gas, mol%

er. If this kept occurring until the oil was light enough to be misciblewith the injection gas, it would constitute the condensing-gas drivemechanism. However, this is unlikely to occur with a real reservoiroil. As the light intermediates are condensing from the injection gasinto the oil, the middle intermediates are being stripped from the oilinto the gas. Since the injection gas contains none of these middleintermediates, they cannot be replenished in the oil. After a few con-tacts between the oil and the injection gas, the oil becomes essential-ly saturated in the light intermediates, but it continues to lose middleintermediates, which are stripped out and carried on ahead by themobile gas phase. The light intermediates of the injection gas cannotsubstitute for the middle intermediates the oil is losing. So after thefirst few contacts make the oil lighter by net condensation of [light]intermediates, subsequent contacts make the oil heavier by net va-porization of [middle] intermediates. Once this begins to occur, theoil no longer has a chance of becoming miscible with the gas. Ulti-mately, all the middle intermediates are removed and the residual oilwill be very heavy, containing only the heaviest, nonvolatile frac-tion and the components present in the injection gas.”

Zick goes on to explain how high recoveries can be obtained withenriched-gas displacement without necessarily achieving true misci-bility. Regardless of whether true miscibility develops, he insists thatthe miscibility (or near miscibility) that does occur is not developedaccording to the traditional enriched-gas drive mechanism (i.e., be-tween the injection gas and the oil at the point of injection). Instead,he proposes the combined condensing/vaporizing mechanism. Heclaims that reaching miscibility by the traditional enriched-gas pro-cess requires higher displacement pressures (or higher enrichmentlevels) than the MMP (or minimum miscibility enrichment) deter-mined by slim-tube experiments (Figs. 8.17 and 8.20). A characteris-tic of the combined condensing/vaporizing mechanism is that afree-gas saturation always exists ahead of the front and that someROS is found behind the front.

Novosad and Costain21 and Novosad et al.22 describe a displace-ment mechanism for enriched-gas drive that differs from both the

14 PHASE BEHAVIOR

TABLE 8.2—CO2 PHYSICAL PROPERTIES

M (��1.52) 44.01

Tc , °F 88

pc , psia 1,070

�c, gm�cm3 (lbm�ft3) 0.469 (29.2)

Zc 0.274

� (Pitzer acentric factor) 0.239

Tb ,°F, “dry ice” at 1 atm �110

CO2 equivalent1 ton, Mscf1 lbm, scf

17.28.6

condensing-gas and the combined condensing-/vaporizing-gas drivemechanisms. On the basis of their interpretation, they propose a sim-ple rising-bubble apparatus to determine the enrichment level re-quired to develop miscibility for a given oil. This experimental tech-nique implies, however, a type of vaporizing-gas drive mechanismthat would not seem to apply for most enriched-gas displacements.Even so, the experimental results they provide seem to give reason-able conditions of developed miscibility for the highly undersaturatedoils used in their studies.

8.4.3 Multiphase Behavior. Enriched-gas injection at low tempera-tures may yield complex multiphase VLL/solid (VLLS) behavior.Shelton and Yarborough40 present a thorough study of multiphasebehavior for a reservoir oil in contact with a rich gas consisting of32% methane, 37% ethane, and 30% propane at 105°F. Figs. 8.31and 8.32 show some of their study results. The multiphase VLL be-havior and asphaltene/wax precipitation are strikingly similar tothose of CO2/oil systems at the same temperature.40,41

Although experimental evidence is lacking, multiphase behaviorprobably can be anticipated when the system temperature is not

Fig. 8.34—CO2 Z factor (from Refs. 42 and 43).

Fig. 8.33—CO2 density (from Refs. 42 and 43).

Fig. 8.35—CO2 viscosity (from Refs. 42 and 43).


TABLE 8.3—CO2 DENSITY* (from Ref. 42)

Pressure (bar)

Temperature(°F) 25 50 75 100 150 200 250 300

68 0.0527 0.1423 0.8100 0.8550 0.9010 0.9335 0.9600 0.9832

86 0.0499 0.1251 0.6550 0.7820 0.8500 0.8887 0.9190 0.9460

104 0.0476 0.1135 0.2305 0.6380 0.7850 0.8415 0.8771 0.9077

122 0.0456 0.1052 0.1932 0.3901 0.7050 0.7855 0.8347 0.8687

140 0.0437 0.0984 0.1726 0.2868 0.6040 0.7240 0.7889 0.8292

158 0.0421 0.0930 0.1584 0.2478 0.5040 0.6605 0.7379 0.7882

176 0.0406 0.0883 0.1469 0.2215 0.4300 0.5935 0.6872 0.7466

194 0.0391 0.0845 0.1381 0.2019 0.3730 0.5325 0.6359 0.7040

212 0.0378 0.0810 0.1305 0.1877 0.3330 0.4815 0.5880 0.6630

230 0.0366 0.0778 0.1239 0.1765 0.3040 0.4378 0.5443 0.6230

248 0.0354 0.0749 0.1187 0.1673 0.2800 0.4015 0.5053 0.5855

266 0.0344 0.0722 0.1141 0.1595 0.2620 0.3718 0.4718 0.5517

284 0.0334 0.0697 0.1094 0.1525 0.2465 0.3470 0.4419 0.5200

302 0.0325 0.0674 0.1054 0.1461 0.2337 0.3267 0.4151 0.4925

320 0.0316 0.0653 0.1018 0.1403 0.2229 0.3089 0.3918 0.4680

*In gm/cm3.

more than approximately 50°F higher than the critical temperatureof the injection gas. For example, the pseudocritical temperature ofthe enriched gas used by Shelton and Yarborough was 60°F and sig-nificant multiphase behavior was observed at 105°F. CO2 systemsexhibit multiphase behavior up to approximately 130°F, about 40°Fhigher than the critical temperature of CO2. Accordingly, an injec-tion gas rich in NGL’s probably experiences multiphase behavior

Fig. 8.36—CO2 phase diagram (from Refs. 42 and 43).

and asphaltene precipitation at higher reservoir temperatures thana less enriched gas does.

��$ �%� � ��

8.5.1 CO2 Physical Properties. CO2 is a stable, nontoxic com-pound found in a gaseous state at standard conditions. For petro-leum applications, CO2 exists either as a gas or as a liquid-like su-

Fig. 8.37—CO2 FVF (from Refs. 42 and 43).

16 PHASE BEHAVIOR

Fig. 8.38—Correlation for solubility of CO2 in dead stock-tankoils (adapted from Ref. 44).

XCO in Oil With UOP K=11.72

0 0.1 0.2 0.3 0.4 0.5 0.6

1.06

1.04

1.02

1.00

0.98

0.96

0.94

0.92

0.90

0.88

0.86

0.84

0.82

0.80

0.78

percritical fluid. Table 8.2 gives the key physical properties of CO2.Figs. 8.33 through 8.35, respectively, show density, Z factor, andviscosity of CO2 as functions of pressure and temperature. Table 8.3gives tabular data for the density of pure CO2.

Fig. 8.36 shows the phase diagram of CO2 with an extrapolationof the critical isochor. The critical isochor defines supercriticalconditions where phase density equals the critical density of 0.47 g/cm3. Later, we show that CO2 density at reservoir conditions is themain parameter that determines MMP of CO2 with reservoir oils. Infact, the critical isochor drawn in Fig. 8.36 gives a close approxima-tion of the Yellig-Metcalfe13 correlation for CO2 MMP.

Fig. 8.37 gives the reservoir barrels occupied by 1 Mscf of CO2as a function of pressure and temperature. For most CO2 projects,approximately 2 Mscf of CO2 is required to fill 1 res bbl PV. Typi-cally, approximately 5 to 10 Mscf of CO2 is the “gross utilization”required to recover an additional 1 bbl of stock-tank oil by theCO2-miscible flooding process; gross utilization is driven strong-ly by economics and may differ from these typical values. As muchas half of the injected CO2 may remain in the reservoir at the endof a CO2 flood.

CO2, when mixed with water, forms carbonic acid. This acidicbyproduct may affect injectivity in carbonate reservoirs, but the cor-rosive effect on steel tubulars and surface equipment may be signifi-

Fig. 8.39—Correlation for swelling of a dead stock-tank oil whensaturated with CO2 (adapted from Ref. 44).

Stock-Tank Oil Molecular Weight

Stock-Tank Oil Specific Gravity

cant. Corrosion in CO2 floods, particularly in WAG projects, re-quires special attention.

8.5.2 Immiscible CO2/Oil Behavior. CO2 flooding has been ap-plied successfully in viscous, heavy-oil reservoirs. Oil swelling andoil-viscosity reduction are the two primary mechanisms in immis-cible CO2 displacement. Low-pressure reservoirs and reservoirswith stock-tank-oil gravities less than approximately 30°API aretypical candidates for immiscible CO2 displacement. Gravity-stabledisplacement with CO2 also may be an efficient immiscible process.

Simon and Graue44 give generalized graphical correlations for sol-ubility (Fig.8.38), swelling (Fig. 8.39), and viscosity reduction for“dead” stock-tank oils saturated with CO2 (Fig. 8.40). Reported accu-racies for the solubility and swelling correlations are 2 and 0.5%, re-spectively, and 12% deviation is reported for the viscosity correlation.

Fig. 8.38 shows that CO2 solubility in crude oils increases withdecreasing temperature. Solution gas/oil ratio in scf/STB can be cal-culated from CO2 mole fraction, xCO2

, in a CO2/oil mixture from

Rs � 133, 000�o

Mo

xCO2

1 � xCO2

. (8.1). . . . . . . . . . . . . . . . . . .

At temperatures less than approximately 200°F, the correction to sol-ubility based on the universal oil products (Watson) characterizationfactor is less than 2% for most reservoir oils (11.4 � Kw � 12.4).

Fig. 8.39 shows the swelling factor, expressed as the ratio ofCO2-saturated stock-tank-oil volume divided by original stock-tank-oil volume. Swelling increases with increasing CO2 solubilityand with decreasing stock-tank-oil molar volume (Mo��o).

Oil-viscosity reduction (Fig. 8.40) is substantial for all API-grav-ity stock-tank oils at pressures up to approximately 750 psia; the ef-fect diminishes at higher pressures because of reduced CO2 solubili-ty. High-viscosity oils are affected the most by CO2 solubility; oilviscosity may be reduced by as much as two orders of magnitude.

Practically, Simon and Graue’s correlations are valid only forheavier oils (�API � 25 and �o � 5 cp) without solution gas andat temperatures greater than approximately 120°F. Fig. 8.39 showsthe effect of solution gas on oil swelling. The Simon-Graue cor-relations cannot be used to calculate solubility and swelling in res-ervoir oils containing solution gas and also do not predict the dra-


Fig. 8.40—Effect of CO2 on oil viscosity (adapted from Ref. 44).

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

matic change in solubility and swelling behavior exhibited bysome oils at lower temperatures.

8.5.3 Miscible CO2/Oil Behavior. Fig. 8.41 shows the swelling be-havior of a stock-tank oil reported by Holm,5,45 Holm and Josen-dal,16-19 and Holm and O’Brien.46 The experiment starts with aconstant-volume visual cell initially filled approximately one-thirdwith stock-tank oil. CO2 is added in increments, and the cell is rockedfor each mixture until equilibrium is reached. The final pressure andoil volume are noted, and the oil volume, relative to the initial oil vol-ume, is plotted vs. pressure. When a certain pressure is reached, theoil phase, which was being swollen by increasing amounts of dis-

Fig. 8.42—Volumetric behavior of Cabin Creek stock-tank oil asCO2 is added to a constant-volume visual cell (adapted fromRefs. 16 through 19).

Fig. 8.41—Effect of solution gas on swelling of a reservoir oil byCO2 (adapted from Refs. 16 through 19).

solved CO2, suddenly decreases in volume. Significant extraction ofintermediate and heavy components (C5 through C30) from the oilphase into the upper CO2-rich phase causes this dramatic change inoil volumetric behavior. At sufficiently high pressures, the CO2-richphase may even become heavier than the oil (hydrocarbon-rich liq-uid) phase, resulting in phase inversion (Fig. 8.42).

Holm and Josendal note that the observed discontinuity in swell-ing behavior is caused by a change in the behavior of CO2-richphase from vapor-like to liquid-like that is almost coincident withthe pressure required to develop miscibility in slim-tube measure-ments. Qualitatively, the change in behavior of the CO2-rich phaseis analogous to the volumetric change that occurs for a pure compo-nent when pressure passes through the vapor pressure. That is, theCO2-rich phase behaves like a vapor at pressures below the “vaporpressure” and like a liquid at higher pressures. Once the CO2-richphase attains liquid-like behavior, it extracts intermediate and heavycomponents from the oil, as would be expected with a liquid solvent.

The temperature required for the CO2-rich phase to exhibit sharp,discontinuous volumetric behavior depends on the oil but is usually

Fig. 8.43—Volumetric behavior of Mead-Strawn stock-tank oil asCO2 is added to a constant-volume visual cell at different tem-peratures (adapted from Refs. 16 through 19).

18 PHASE BEHAVIOR

Fig. 8.44—Volumetric behavior and slim-tube results for theMead-Strawn and Fansworth stock-tank oils at 135°F (adaptedfrom Refs. 16 through 19).

less than 150°F. Fig. 8.43 shows swelling/extraction experiments forthe 41°API Mead-Strawn crude oil at different temperatures. Holmand Josendal note that miscibility develops at a pressure only slightlyhigher than the pressure where the character of swelling changes (i.e.,where significant hydrocarbon extraction starts). They also point outthat the sharpness of change in the swelling behavior is more pro-nounced at lower temperatures and is coincident with the sharpnessin change from immiscible to miscible displacement indicated on arecovery-pressure curve from slim-tube experiments (Fig. 8.44).

Holm and Josendal correlate miscibility development with the den-sity of pure CO2. They indicate that light oils can develop miscibilityat conditions where CO2 has a density as low as 0.4 g/cm3 (criticalCO2 density is 0.47 g/cm3) and that most oils will develop miscibilityat conditions where CO2 density ranges from 0.5 to 0.7 g/cm3 (Fig.8.45). Their correlation for MMP shows that the CO2 density requiredto develop miscibility depends primarily on the amount of gasolineand gas/oil components (C5 through C30) found in the stock-tank oil.Their correlation uses weight percent (wC5

through wC30)�wC5�

asa correlating parameter, with typical values ranging from 70 to 80%requiring CO2 densities ranging from 0.65 to 0.55 g/cm3 to develop

Fig. 8.46—CO2 MMP correlation equals the pressure corre-sponding to the CO2 density from the chart at reservoir tempera-ture; MMP may be less than the oil bubblepoint for a C/V misciblemechanism.

C5-C30 Content of Oil, (C5�C30)/C5+, wt%

Fig. 8.45—Density of CO2 required to develop miscibility for vari-ous oils at temperatures from 90°F to 190°F (adapted from Refs.16 through 19).

miscibility. Fig. 8.46 shows the Holm-Josendal correlation for MMP.Stalkup1 covers other correlations for CO2 MMP.

The distribution of components in the C5 through C30 cut of anoil also affects the MMP, but Holm and Josendal do not include thiseffect directly in their correlation. They do show, however, that thefraction of gasolines (C5 through C12) in the C5 through C30 cut hasa measurable effect. Typically, gasolines make up 40 to 50 wt% ofthe C5 through C30 cut. Higher gasoline content will decrease theMMP, and lower gasoline content will increase the MMP. The typeof hydrocarbons (paraffinic vs. aromatic) making up the C5 throughC30 material in a crude oil has negligible effect on MMP. Aromaticoils appear to have slightly lower MMP’s than paraffinic oils, allother conditions being the same.

Nitrogen and light C1 through C4 hydrocarbons in the reservoiroil generally have a negligible effect on CO2 MMP if the MMP isless than the reservoir-oil bubblepoint pressure. The light compo-nents in the reservoir oil are extracted ahead of the miscible front ina CO2 process (Fig. 8.47). The bank of light components does notaffect the extraction process or developed miscibility. Yellig andMetcalfe13 point out that the MMP of a reservoir oil equals the bub-blepoint of that oil if the bubblepoint pressure is greater than theMMP determined for a low-GOR sample of the same stock-tank oil.However, this is true only when considering the traditional vaporiz-ing-gas drive mechanism. With the condensing/vaporizing mecha-nism, the MMP can be lower than the bubblepoint pressure.

Methane, N2 , and C2 through C4 hydrocarbons mixed with theCO2 injection gas affect MMP significantly. Methane and N2 tend

Fig. 8.47—Schematic of distribution of components in CO2 dis-placement at miscible and near-miscible conditions based onslim-tube simulation results (adapted from Refs. 16 through 19).

135°F and 1,800 psi—near miscible

2,500 psi—multicontact miscible

2,500 psi—first contact miscible

0.33PV

0.17PV

0.15PV

0.15PV

0.24PV


Fig. 8.48—Experimental recoveries from slim-tube displace-ments for a Wasson stock-tank crude oil displaced by a CO2 slugpushed by N2 at 1,250 psig and 107°F with no gas in solution and100-ft coil (adapted from Ref. 47).

to increase MMP, while NGL’s tend to decrease MMP. However, asufficiently large PV of injected CO2 can be followed by N2 or leangas without affecting MMP (Fig. 8.48).

8.5.4 Multiphase Behavior. CO2/oil systems exhibit multiphaseVLLS behavior similar to that described earlier for enriched-gas/oilsystems.48 Three-phase VLL behavior is limited to reservoir tem-peratures less than approximately 130°F, pressures from 1,000 to1,500 psia (somewhat less than the MMP), and CO2 concentrationsgreater than approximately 50 mol%. At other conditions, vapor/liquid or liquid/liquid behavior is expected, with the upper phasecontaining mainly CO2 and the lower phase containing mostly hy-drocarbons and some dissolved CO2. Asphaltene precipitation canoccur over a relatively large range of pressures and CO2 concentra-tions (Fig. 8.49), usually including the VLL region. The threephases in a CO2/oil VLL system include a CO2-rich vapor (the up-per phase), a CO2-rich liquid (the middle phase) containing somehydrocarbons, and a hydrocarbon-rich liquid (the lower phase) con-taining C5+ with some dissolved natural gas and CO2.

Consider a CO2/oil mixture in the three-phase region in Fig.8.50. Moving up in pressure through the lower two-phase region,the hydrocarbon-rich liquid is in equilibrium with a CO2-rich va-por phase. Near 1,000 psia (the dewpoint of the CO2-rich vaporphase), a CO2-rich liquid phase appears. As pressure increasesthrough the three-phase region, the volume of the CO2-rich liquidphase increases, mostly at the expense of the CO2-rich vaporphase. At few hundred psi higher than the onset of three-phase be-havior (the bubblepoint of the CO2-rich liquid phase), theCO2-rich vapor phase disappears.

The onset of three-phase behavior in a CO2/oil system is relatedto the upper CO2-rich phase behaving like a component at its vaporpressure (see the earlier discussion on miscibility). Because theCO2-rich phase is actually a mixture, the transition from vapor-liketo liquid-like behavior occurs over a narrow range of pressurescompared with the abrupt change experienced at the vapor pressureof a pure component.

The volume of hydrocarbon-rich liquid increases because ofswelling in the low-pressure, vapor/liquid region and through thethree-phase region. When the CO2-rich phase completes its transi-tion from vapor-like to liquid-like behavior at the top of the three-phase region, the oil phase stops swelling and starts shrinking as aresult of strong extraction of C5 through C30 components by theCO2-rich liquid phase.

Practically, the effect of three-phase behavior on the CO2 dis-placement process is small and can be ignored when modeling fieldperformance. The three-phase region usually is located in geologi-cal layers that have experienced CO2 breakthrough, some distance

Fig. 8.49—Experimental p-x diagram for west Texas reservoir oiland up to 95% CO2 injection gas showing large region of asphal-tene precipitation (from Ref. 4).

CO2 in Mixture, mol%

away from the producing wells, where reservoir pressure is between1,000 and 1,500 psia. The three-phase region may, however, causeserious problems for compositional simulators based on a two-phase vapor/liquid equilibrium (VLE) algorithm.50 The problem

Fig. 8.50—Experimental p-x diagram for Wasson crude oil andCO2 injection gas at 105°F (from Ref. 49).

LOWER LIQUID PHASE, vol%

CO2 in Mixture, mol%

20 PHASE BEHAVIOR

Fig. 8.51—Solubility of CO2 in pure water and NaCl brines at100°F (adapted from Ref. 4; data from Ref. 52).

arises because, thermodynamically, the flash algorithm is searchingfor an equilibrium condition with only two phases. If the mixture be-ing flashed actually exhibits three-phase behavior according to thethermodynamic model being used, the VLE algorithm must chooseone of several valid two-phase solutions. Any of these two-phasesolutions satisfies the equilibrium constraints, but the two-phasesolutions merely represent local minimums in the Gibbs free energy,while the three-phase solution represents a global minimum (seeChap. 4). Numerical instabilities arise when a gridblock oscillatesbetween one two-phase solution and another.

In CO2 flooding, asphaltene precipitation could be a more seriousmultiphase problem than three-phase VLL behavior. First, asphal-tene precipitation occurs over a wider range of pressures and CO2compositions, potentially causing reduced injectivity and produc-tivity. Several authors40,48 have provided laboratory measurementsshowing that asphaltene precipitation occurs over a wide range ofconditions. Unfortunately, few investigators have documented thequantitative effect of asphaltenes on reservoir performance. Christ-man and Gorell6 give results that indicate that reduced injectivitiesexperienced in many tertiary CO2 projects can be modeled withoutaccounting for reduced permeability and altered wettability causedby asphaltene precipitation. Still, serious operational problemsassociated with asphaltenes have been reported in field operations.Monger and Trujillo41 report on a comprehensive study of the de-position of organic solids during CO2 and rich-gas flooding.

Few thermodynamic models have been suggested for predictingasphaltene precipitation. Kawanaka et al.51 propose a technique forpredicting organic deposition of asphaltene, wax, and other solid-like materials that may precipitate from reservoir oils. The modeluses a continuous distribution for the solid phase, and the authorsprovide results that give reasonable predictions for miscible-solventprocesses. Finally, they give a comprehensive review of literatureon asphaltene precipitation, measurements, and thermodynamicmodels for prediction of VLS phase behavior.

8.5.5 CO2/Water Behavior. Chap. 9 covers methods for estimatingCO2/water behavior. The two primary design considerations in aCO2-injection project related to CO2/water phase behavior are thetreatment of corrosion resulting from the formation of carbonic acidwhen CO2 mixes with water and the loss of injected CO2 resultingfrom the saturation of connate and injected water with CO2.

Fig. 8.51 shows CO2 solubility in water and brines at 100°F. In fact,CO2 solubility in water is not very sensitive to temperature at temper-atures greater than 100°F. Also, solubility increases only slightly atpressures greater than approximately 3,000 psia. Salinity has a signif-icant effect on CO2 solubility, reducing the solubility in brine byapproximately 30% for every 100,000 ppm of total dissolved solids.Water density and viscosity change only slightly when saturated withCO2. Enick and Klara53 reported on the effect of CO2 solubility inbrine on compositional simulation of CO2 flooding.

��&��

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24. Sibbald, L.R., Novosad, Z., and Costain, T.G.: “Authors’ Reply to Dis-cussion of Methodology for the Specification of Solvent Blends forMiscible Enriched-Gas Drives,” SPERE (February 1992) 156.

25. Zhou, D. and Orr, F.M. Jr.: “Analysis of Rising-Bubble Experiments ToDetermine Minimum Miscibility Pressures,” SPE Journal (March1998) 19.

26. Peterson, A.V.: “Optimal Recovery Experiments With N2 and CO2,”Pet. Eng. Intl. (November 1978) 40.

27. Benham, A.L., Dowden, W.E., and Kunzman, W.J.: “Miscible Fluid Dis-placement—Prediction of Miscibility,” Trans., AIME (1960) 219, 229.

28. Vogel, J.L. and Yarborough, L.: “ The Effect of Nitrogen on the PhaseBehavior and Physical Properties of Reservoir Fluids,” paper SPE 8815presented at the 1980 SPE Annual Technical Conference and Exhibi-tion, Tulsa, Oklahoma, 20–23 April.

29. Jensen, F. and Michelsen, M.L.: “Calculation of First Contact and Mul-tiple Contact Miscibility Pressures,” In Situ (1990) 14, 1.

30. Luks, K.D., Turek, E.A., and Baker, L.E.: “Calculation of Minimum Mis-cibility Pressure,” SPERE (November 1987) 501; Trans., AIME, 283.

31. Metcalfe, R.S., Fussell, D.D., and Shelton, J.L.: “A Multicell Equilibri-um Separation Model for the Study of Multiple-Contact Miscibility inRich-Gas Drives,” SPEJ (June 1973) 147; Trans., AIME, 255.

32. Cook, A.B. et al.: “Effects of Pressure, Temperature, and Type of Oilon Vaporization of Oil During Gas Cycling,” Report RI 7278, U.S. Bu-reau of Mines, Washington, DC (1969).

33. Cook, A.B., Walter, C.J., and Spencer, G.C.: “Realistic K Values of C7+Hydrocarbons for Calculating Oil Vaporization During Gas Cycling atHigh Pressure,” JPT (July 1969) 901; Trans., AIME, 246.

34. Johns, R.T., Orr, F.M. Jr., and Dindoruk, B.: “Analytical Theory ofCombined Condensing/Vaporizing Gas Drives,” paper SPE 24112presented at the 1992 SPE/DOE Symposium on Enhanced Oil Recov-ery, Tulsa, Oklahoma, 22–24 April.

35. Johns, R.T., Fayers, J.F., and Orr, F.M. Jr.: “Effect of Gas Enrichmentand Dispersion on Nearly Miscible Displacement in Condensing/Va-porizing Drives,” paper SPE 24938 presented at the 1992 SPE AnnualTechnical Conference and Exhibition, Washington, DC, 4–7 October.

36. Wang, Y. and Orr, F.M. Jr.: “Analytical Calculation of Minimum Misci-bility Pressure,” Fluid Phase Equilibria (1997) 139, 101.

37. Moses, P.L. and Wilson, K.: “Phase Equilibrium Considerations in Us-ing Nitrogen for Improved Recovery From Retrograde CondensateReservoirs,” JPT (February 1981) 256; Trans., AIME, 271.

38. Donohoe, C.W. and Buchanan, R.D. Jr.: “Economic Evaluation ofCycling Gas-Condensate Reservoirs With Nitrogen,” JPT (February1981) 263; Trans., AIME, 271.

39. Renner, T.A. et al.: “Displacement of a Rich-Gas Condensate by Nitro-gen: Laboratory Corefloods and Numerical Simulations,” SPERE(February 1989) 52; Trans., AIME, 287.

40. Shelton, J.L. and Yarborough, L.: “Multiple-Phase Behavior in Po-rous Media During CO2 or Rich-Gas Flooding,” JPT (September1977) 1171.

41. Monger, T.G. and Trujillo, D.E.: “Organic Deposition During CO2 andRich-Gas Flooding,” SPERE (February 1991) 17; Trans., AIME, 291.

42. Kennedy, G.C.: “Pressure-Volume-Temperature Relations in CO2 atElevated Temperatures and Pressures,” Amer. J. Sci. (April 1954)252, 225.

43. Kennedy, J.T. and Thodos, G.: “ The Transport Properties of CO2,”AIChE J. (1961) 7, 625.

44. Simon, R. and Graue, D.J.: “Generalized Correlations for PredictingSolubility, Swelling, and Viscosity Behavior of CO2/Crude Oil Sys-tems,” JPT (January 1965) 102; Trans., AIME, 234.

45. Holm, L.W.: “CO2 Requirements in CO2 Slug and Carbonated WaterOil Recovery Processes,” Prod. Monthly (September 1963).

46. Holm, L.W. and O’Brien, L.J.: “CO2 Test at the Mead-Strawn Field,”JPT (April 1971) 431.

47. O’Leary et al.: Nitrogen-Driven CO2 Slugs Reduced Costs,” Pet. Eng.Intl. (May 1979) 130.

48. Orr, F.M. Jr., Yu, A.D., and Lein, C.L.: “Phase Behavior of CO2 andCrude Oil in Low-Temperature Reservoirs,” SPEJ (August 1981) 480.

49. Gardner, J.W., Orr, F.M. Jr., and Patel, P.D.: “ The Effect of Phase Be-havior on CO2-Flood Displacement Efficiency,” JPT (November1981) 2067.

50. Perschke, D.R., Pope, G.A., and Sepehrnoori, K.: “Phase IdentificationDuring Compositional Simulation,” paper SPE 19442 available fromSPE, Richardson, Texas (1989).

51. Kawanaka, S., Park, S.J., and Mansoori, G.A.: “Organic DepositionFrom Reservoir Fluids: A Thermodynamic Predictive Technique,”SPERE (May 1991)185.

52. McRee, B.C.: “How It Works, Where It Works,” Pet. Eng. Intl. (No-vember 1977) 52.

53. Enick, R.M. and Klara, S.M.: “Effects of CO2 Solubility in Brine on theCompositional Simulation of CO2 Floods,” SPERE (May 1992) 253.

'� �� (��

atm �1.013 250 E�05�Pa�� g/cm3

bar �1.0* E�05�Pabbl �1.589 873 E�01�m3

cp �1.0* E�03�Pa�sft �3.048* E�01�m

ft3 �2.831 685 E�02�m3

�F (�F�32)/1.8 ��Cin. �2.54* E�00�cm

lbm �4.535 924 E�01�kgpsi �6.894 757 E�00�kPaton �9.071 847 E�01�Mg


WATER/HYDROCARBON SYSTEMS 1

��

��

��

The connate or “original” water found in petroleum reservoirs usu-ally contains both dissolved salts (consisting mainly of NaCl) andsolution gas (consisting mainly of methane and ethane). Initial wa-ter saturation can range from 5 to 50% of the pore volume (PV) inthe net-pay intervals of a reservoir (where production is primarilyoil and gas). Higher water saturations are found in the aquifer andwhere water has swept oil or gas during a waterflood.

From a reservoir-depletion point of view, the amount of waterconnected with a reservoir is as important as the properties of thewater, particularly in material-balance calculations where water ex-pansion (compressibility times water volume) may contribute sig-nificantly to pressure support.1,2 From a production point of view,water mobility is important, requiring determination of water satu-rations, water viscosity, and formation volume factor (FVF). Forsurface-processing calculations, water composition, water contentin the produced wellstream, and conditions where water and hydro-carbons coexist must be defined.

The three most important aspects of phase behavior involving wa-ter/hydrocarbon systems are mutual solubilities of gas and water, vol-umetric behavior of reservoir brines, and hydrate formation and treat-ment. Sec. 9.2 presents pressure/volume/temperature (PVT)correlations for water/hydrocarbon systems. Standard PVT proper-ties—solution gas/water ratio, Rsw; isothermal water compressibility,cw; water FVF, Bw; water viscosity, �w; and water content in gas,rsw—are correlated in terms of pressure, temperature, and salinity byuse of graphical charts and empirical equations. Correlations for wa-ter/hydrocarbon interfacial tension (IFT), �wh, are also presented.

At very high temperatures and pressures, some correlations andthe existing water-property data base are not adequate. Equations ofstate (EOS’s) have been used with reasonable success in predictingmutual solubilities and phase properties of hydrocarbon/water sys-tems up to 400°F and greater than 10,000 psia,3-8 as discussed inSec. 9.3. The effect of salinity on gas/water phase behavior has alsobeen treated to some extent by the EOS methods.9

Sec. 9.4 covers the physical structure of hydrates and how to cal-culate conditions under which hydrates form. Hydrate formationcan have a significant effect on production and surface-facilitiesequipment and even on deep drilling. Water/hydrocarbon phase dia-grams give the conditions of initial hydrate formation. These dia-grams are particularly useful for designing a production system toavoid hydrate formation. The formation of hydrates can also be esti-mated with vapor/solid equilibrium ratios.

��

Like all reservoir fluids, formation-water properties depend oncomposition, temperature, and pressure. Reservoir water is seldompure and usually contains dissolved gases and salts. Total dissolvedsolids (TDS), usually consisting mainly of NaCl, ranges from10,000 to �300,000 ppm; seawater salinity is � 30,000 ppm.

Water is limited as to how much salt it can keep in solution. Thelimiting concentration for NaCl brine is10

C*sw � 262, 180 � 72T � 1.06T 2 , (9.1). . . . . . . . . . . . . . . .

with T in °C and Cw in ppm. If reservoir temperature is known buta water sample cannot be obtained, this relation gives the limitingsalinity of the reservoir brine. Salinity of a brine usually is less than80% of the value given by Eq. 9.1. Otherwise, the best estimate ofbrine salinity can be taken from a neighboring reservoir in the samegeological formation.

Scale buildup in tubing and surface equipment is caused by theprecipitation of salts in produced brine,11 usually calcium carbon-ate, calcium sulfate (e.g., gypsum), barium or strontium sulfates,and iron compounds. Temperature, pressure, total salinity, and saltcomposition are the primary variables determining the severity ofscaling. Note that Eq. 9.1 should not be used to detect conditions thatresult in scale buildup.

Dissolved gas in water is usually less than 30 scf/STB (approxi-mately 0.4 mol%) at normal reservoir conditions. The effect of saltand gas content on water properties can be important, and the follow-ing discussion gives methods to estimate fluid properties in terms oftemperature, pressure, dissolved gas, and salinity. Methods for esti-mating PVT properties of formation water usually are based on initialestimates of the pure-water properties at reservoir temperature andpressure that are then corrected for salinity and dissolved gas.

9.2.1 Salinity. The cations dissolved in formation waters usually in-clude Na+, K+, Ca++, and Mg++, and the anions include Cl�, SO4

��,and HCO��

3 . Most formation waters contain primarily NaCl. Sus-pended salts, entrained solids, and corrosion-causing bacteria mayalso be present in reservoir waters, but these constituents usually donot affect formation-water PVT properties. The geochemistry offormation waters can be useful in detecting foreign-water encroach-ment and in determining its source. Table 9.1 gives example com-positions of reservoir brines.

Salinity defines the concentration of salts in a saline solution(brine) and may be specified as one of several quantities: weightfraction, ws; mole fraction, xs; molality, csw; molarity, csv; parts permillion by weight, Csw; and parts per million by volume, Csv. Table

2 PHASE BEHAVIOR

TABLE 9.1—EXAMPLE COMPOSITIONS OF FORMATION BRINES

Dodson-Standing13

ComponentSeawater

(ppm)Brine A(ppm)

Brine B(ppm)

Arun Field(mg/L)

Gulf Coast Frioa

(mg/L)Kansas Wilcoxb

(mg/L)Kansas Wilcoxa

(mg/L)

Sodium (Na) 10,560 3,160 12,100 5,212 40,600 10,800 142,500

Calcium (Ca) 400 58 520 80 5,100 790 14,400

Magnesium (Mg) 1,270 40 380 5 1,000 5,560 68,500

Sulfate (SO4) 2,650 0 5 262 110 80 300

Chloride (Cl) 18,980 4,680 20,000 7,090 69,100 10,870 142,600

Bicarbonate (HCO3) 140 696 980 1,536 990 20 530

Iodide (I) 0 0 130 0 0 0 3

Bromide (Br) 65 0 0 0 0 80 350

Others 515 0 0 0 0 0 0

Total 34,580 8,630 34,110 14,190 116,900 28,200 369,180

Specific gravity 1.0243c 1.006d,e 1.024d,e 1.014d 1.086d,e 1.015d 1.140d

aMaximum salt-containing composition reported for field/formation.bMinimum salt-containing composition reported for field/formation.cAt 20°F.dAt 60°F.eEstimated with Eq. 9.3.

TABLE 9.2—DEFINITIONS OF SALT CONCENTRATIONS

Quantity Symbol Unit Definition

Weight fraction ws g/g ms��ms � mow�

Mole fraction xs g mol/g mol ns��ns � now�

Molality csw g mol/kg 103ns�mow

Molarity csv g mol/L 103ns�Vw

ppm, weight basis Csw mg/kg 106ms��ms � mow�

ppm, volume basis Csv mg/L 106ns�Vw

ms � mass salt, mow � mass pure water, ns � moles salt, no

w � moles purewater, and Vw � volume brine mixture.

9.2 formally defines these quantities; in the table, ms�mass of saltin grams, mo

w�mass of pure water in grams, ns�moles of salt ingram moles, no

w�moles of pure water in gram moles, and Vw�vol-ume of the brine mixture in cubic centimeters.

Some common conversions for the various concentrations are

Csv � �wCsw , (9.2a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Csw �Csv�w

� 106 ws , (9.2b). . . . . . . . . . . . . . . . . . . . . . . . .

csw �17.1

106 C�1sw � 1

, (9.2c). . . . . . . . . . . . . . . . . . . . . . . . . . .

and Csw �106

17.1c�1sw � 1

, (9.2d). . . . . . . . . . . . . . . . . . . . . . .

where the Eqs. 9.2c and 9.2d apply for NaCl brines. If brine density,�w, at standard conditions (14.7 psia and 60°F) is not reported, it can

be estimated from the Rowe-Chou12 density correlation for NaCl.

�w�psc, Tsc� � �1.0009 � 0.7114ws � 0.26055w2s��1 ,

(9.3). . . . . . . . . . . . . . . . . . . . .

with �w in g/cm3 and ws in weight fraction TDS. For many engi-neering applications, �w�1 g/cm3 is assumed and the mass of saltis considered negligible compared with the mass of pure water, re-sulting in the approximate relations

csv � csw � cs ,

Csw � Csv � Cs ,

and cs � �17.1 � 10�6�Cs, (9.4). . . . . . . . . . . . . . . . . . . . . . .

where the constant 17.1�10�6 applies for NaCl brines.

9.2.2 Gas Solubilities in Water/Brine.The solubility of natural gases in water is rather complicated to esti-mate from empirical correlations. However, the effect of gas solubili-ty usually is minor except at high temperatures. At temperatures lessthan approximately 300°F and pressures less than 5,000 psia, solubil-ity usually is less than 0.4 mol%, or approximately 30 scf/STB. Ac-cording to Dodson and Standing’s13 results, this amount of dissolvedgas causes an increase of approximately 25% in water compressibility(e.g., from 3.8�10�6 to 4.8�10�6 psi�1). Experimental gas solu-bilities for C1 through C4 hydrocarbons, nonhydrocarbons, naturalgas, and a few binaries and ternaries are available in the literature.Figs. 9.1 through 9.3 present some of these data.

Kobayashi and Katz17 give a method for estimating gas solubili-ties in pure water based on Henry’s law for dilute solutions.

ln xi � ln fi � ln Hi �v~i�p � pvw�

RT, (9.5). . . . . . . . . . . . .

where xi�solubility of gas Component i in water, fi�partial fugac-ity, Hi�Henry’s constant, and v~i�modified molar volume. Hi andv~i are nonlinear functions of temperature. Cramer18 uses a similarapproach to correlate gas solubilities for methane/water and meth-ane/NaCl-brine systems over a wide range of pressures, tempera-tures, and salinities.

At reservoir conditions, the solubility of methane in water and theeffect of salinity are the most important variables affecting waterproperties. The following empirical equation gives a reasonable fitof the Culberson and McKetta14,19 solubility data for methane inpure water at conditions 100T350°F and 0p10,000 psia,

xC1� 10�3�

3

i�0

��3

j�0

Ai jTj�pi� , (9.6). . . . . . . . . . . . . . .

where A00�0.299, A01��1.273�10�3, A02�0.000, A03�0.000,A10�2.283�10�3, A11��1.870�10�5, A12�7.494�10�8,A13��7.881�10�11, A20��2.850�10�7, A21�2.720�10�9,A22��1.123�10�11, A23�1.361�10�14, A30�1.181�10�11,A31��1.082�10�13, A32�4.275�10�16, and A33��4.846�10�19, with T in °F and p in psia. Gas solubility expressed as a solu-tion gas/water ratio, Rsw at standard conditions is

Rsw � 7, 370xg

1 � xg� 7, 370xg , (9.7). . . . . . . . . . . . . . . . .

with Rsw in scf/STB.


Fig. 9.1—Gas-solubility data for methane in pure water (adaptedfrom Ref. 14).

Amirijafari and Campbell20 give experimental component solu-bilities and an empirical method for calculating the total gas solubil-ity of the C1/C2/C3 ternary mixture. However, for most applicationsgas solubility can be estimated by assuming that the gas consistsonly of methane. A standard two-phase flash calculation with a cu-bic EOS gives a surprisingly accurate prediction of gas solubilities,as discussed in Sec. 9.3. This approach is the recommended proce-dure for estimating gas solubilities of hydrocarbon/water/brine mix-tures at high pressures and temperatures.

9.2.3 Salinity Correction for Solubilities. Refs. 9 and 21 give theSetchenow (sometimes written Secenov) relation for correcting hy-drocarbon solubility in pure water for salt content.

Fig. 9.2—Gas-solubility data for natural gas in pure water(adapted from Ref. 13).

Pressure,psia

ks � limcs 0c�1

s log��

i�w

��

i�o

w

�, (9.8). . . . . . . . . . . . . . . . .

where ks�Setchenow constant, cs�salt concentration, and (��

i )w

and (��

i )ow�fugacity coefficients of Component i at infinite dilu-

tion in the salt solution and in pure water, respectively. Both molal-ity and molarity have been used in the literature for defining Setche-now constants; however, molality, csw, is now considered to be thepreferred concentration. The unit for the Setchenow constant isM�1 (i.e., kg/g mol), where M�molarity.

The ratio of infinite-dilution fugacity coefficients is traditionallyassumed to give an accurate estimate of the ratio of solubilities,yielding the relation

Rsw

Rosw�

xg

xog� 10�ks cs

� 10��17.1�10�6�ks Cs, (9.9). . . . . . . . .

where Rosw�solubility of gas in pure water and Rsw�solubility of gas

in brine. For ks�0, the gas solubility is less in brines than in pure wa-ter, a fact that has led to the term “salting-out coefficient” for ks.

The Setchenow constant is more or less independent of pressurebut is a strong function of temperature. Cramer18 gives a detailedtreatment of Setchenow (and Henry’s) constants for the C1/NaCl sys-tem using data at temperatures up to 570°F and pressures up to 2,000psia. He proposes the temperature dependence of ks shown in Fig.9.4. This figure also shows values of ks reported elsewhere for theC1/NaCl system, illustrating the relatively large uncertainty in salt-ing-out coefficients, even for such a well-defined system. Søreide andWhitson9 give a best-fit relation for the Cramer correlation.

Fig. 9.3—Gas-solubility data for CO2 in pure water (adapted fromRefs. 15 and 16).

Pressure, psia�1,000

4 PHASE BEHAVIOR

Fig. 9.4—Temperature dependence of the Setchenow (salting-out) coefficient for light hydrocarbons (Ref. 9).

�� Methane�� Ethane

Propanen-butane

(ks)C1�NaCl � 0.1813 � �7.692 � 10�4�T

� �2.6614 � 10�6�T 2� �2.612 � 10�9�T 3,

(9.10). . . . . . . . . . . . . . . . . . . .

with ks in M�1 and T in °F. Using relations suggested by Pawli-kowski and Prausnitz21 relating ks of methane to ks of other hydro-carbons, Søreide and Whitson9 propose the following relation forHydrocarbon i.

ksi � (ks)C1�NaCl � 0.000445�Tbi � 111.6� , (9.11). . . . . . .

with ks in M�1 and the normal boiling point, Tbi, in K. Fig. 9.4shows the temperature dependence of ks for light hydrocarbons (C2through C4) based on Eqs. 9.10 and 9.11.

Clever and Holland22 give salting-out correlations for C1/NaCland CO2/NaCl systems. The correlation for CO2/NaCl is

(ks)CO2�NaCl � 0.257555 � �0.157492 � 10�3�T

� �0.253024 � 10�5�T 2� �0.438362 � 10�8�T 3 ,

(9.12). . . . . . . . . . . . . . . . . .

with T in K and ks in M�1. The temperature range for Eq. 9.12 is40T660°F. The Setchenow coefficient varies somewhat withpressure for the CO2/NaCl system, thereby making Eq. 9.12 less ac-curate than hydrocarbon/NaCl correlations. Fig. 9.5 illustrates theeffect of salts other than NaCl on low-pressure solubilities by use oflines of equal gas solubility vs. molality of the salt, where NaCl isthe reference salt.

9.2.4 Equilibrium Conditions in Oil/Gas/Water Systems. Allphases (oil, gas, and water) in a reservoir are initially in thermody-namic equilibrium. This implies that the water phase contains finitequantities of all hydrocarbon and nonhydrocarbon componentsfound in the hydrocarbon phases and that the hydrocarbon phasescontain a finite quantity of water. The amount of lighter compounds(C1, C2, N2, CO2, and H2S) in the water phase can be significant anddepends mainly on the amount of each component in the hydrocar-bon phase(s). The amount of C3+ hydrocarbons found in water isusually small and can be neglected.

The K value representing the ratio of the mole fraction of Compo-nent i in the hydrocarbon phase to the mole fraction of Componenti in the water phase (Ki � zi,HC�xi,aq) is approximately constant ata given pressure and temperature, independent of overall hydrocar-

Fig. 9.5—Lines of equal gas solubility for various salts with NaClas a reference (adapted from Ref. 23).

bon composition and whether the hydrocarbon is single phase ortwo phase. For example, the amount of methane dissolved in waterfor a methane-rich natural gas will be higher than the amount ofmethane dissolved in water for an oil (above its bubblepoint). Fur-thermore, the amount of methane dissolved in water for a gas/oilsystem with overall methane content of 40 mol% will probably beabout the same as for a single-phase oil with 40 mol% methane.

An oil that is undersaturated (with respect to gas) is still in equilibri-um with the water phase. When pressure is lowered, a new equilibriumstate is reached between the undersaturated oil and water. The result isthat some of the methane will move from the water to the oil (withoutfree gas forming); i.e., the solution gas/water ratio decreases. At somelower pressure, the oil will reach its bubblepoint and further reductionin pressure will yield two sources of free gas: gas coming out of solu-tion from the oil and gas coming out of solution from the water.

Therefore, for an undersaturated-oil reservoir, the solution gas/water ratio of reservoir brine will decrease continuously from theinitial reservoir pressure to the reservoir-oil bubblepoint pressureand even further at lower pressures. Correspondingly, the reservoir-oil solution gas/oil ratio will increase (albeit slightly) from initial tobubblepoint pressure and then decrease below the bubblepoint. AnEOS must be used to quantify the changing solution gas/water andsolution gas/oil ratios in this situation.

Fig. 9.6 shows calculations with an EOS that illustrate the relativegas solubility in a reservoir oil and a reservoir gas. The oil and gascompositions are in equilibrium at approximately 3,500 psia. Athigher pressures, the gas solubility in water is higher in the gas/wa-ter system than in the oil/water system. At less than 3,500 psia, threephases will exist in either system and the two-phase flash calcula-tion gives only approximate solubilities on the basis of treating thehydrocarbon as a single phase.

9.2.5 Water/Brine FVF and Compressibility. The FVF of reser-voir water, Bw, depends on pressure, temperature, salinity, and dis-solved gas. Fig. 9.7 gives Dodson and Standing’s13 results for purewater with and without solution gas. Contrary to saturated-oil volu-metric behavior, the liquid volume of a gas-saturated water in-creases with decreasing pressure. That is, the expansion caused byisothermal compressibility is larger than the shrinkage caused bygas coming out of solution.

The pressure dependence of Bw that Dodson and Standing givefor gas-saturated water/brine applies to all gas and oil reservoirs thathave appreciable solution gas. Even if the oil is undersaturated, asdiscussed earlier, the solution gas/water ratio decreases continuous-


Fig. 9.6—Gas dissolved in water for reservoir-oil/water and res-ervoir-gas/water systems, EOS two-phase calculations.

T�258°F

Gas/Water

Oil/Water

Gas/Oil Bubblepoint, 3,500 psia

ly from the initial pressure to the oil bubblepoint pressure and fur-ther thereafter. This precludes the pressure dependence of waterFVF shown in Fig. 9.8, where a discontinuity occurs at some bub-blepoint condition. The only way a reservoir brine could have thisbehavior is if the hydrocarbons that originally saturated the brinehad migrated away completely and the reservoir pressure subse-quently increased with further burial (creating an undersaturatedcondition for the brine with respect to hydrocarbon components).

The FVF of brine at atmospheric pressure, reservoir temperature,and without dissolved gas, Bo

w, is

Bow �

�w�psc, Tsc�

�ow�psc, T�

�v o

w�psc, T�

vw�psc, Tsc�. (9.13). . . . . . . . . . . . . . . .

Long and Chierici24,25 give experimental data and correlations for thedensity of pure water and NaCl-brine solutions, although the pro-posed correlations extrapolate poorly at temperatures greater thanapproximately 250°F. Kutasov26 gives several accurate correlationsfor FVF’s of pure water, but the equation for Bw results in a constantisothermal compressibility that is independent of pressure.

Rowe and Chou12 give the following correlation for water andNaCl-brine specific volume at zero pressure (also applicable at at-mospheric pressure).

v ow�psc, T� � 1

�ow�psc, T�

� A0 � A1ws � A2w2s ,

where A0 � 5.91635 � 0.01035794T

� �0.9270048 � 10�5�T 2

� 1, 127.522T�1� 100, 674.1T�2 ,

A1 � � 2.5166 � 0.0111766T � �0.170552 � 10�4�T 2 ,

and A2 � 2.84851 � 0.0154305T � �0.223982 � 10�4�T 2 ,

(9.14). . . . . . . . . . . . . . . . . . . .

with v ow in cm3/g, T in K, and ws in weight fraction of NaCl. The ef-

fect of pressure on FVF can be calculated by use of the definition ofwater compressibility,

c*w � �

1Bw��Bw

�p�Cs ,T, (9.15). . . . . . . . . . . . . . . . . . . . . . .

Fig. 9.7—FVF of pure water with and without natural gas(adapted from Ref. 13).

which, when integrated, gives

lnB*

w�p, T�

Bow�psc, T�

� � �

p

0

c*w�p, T� dp. (9.16). . . . . . . . . . . . . . . .

With the compressibility data reported by Rowe and Chou cover-ing the conditions 70T350°F, 150p4,500 psia, and0ws0.3, a general correlation for the compressibility of a brine(without solution gas), c*

w, is

c*w�p, T � � �A0 � A1 p�

�1,

Fig. 9.8—Effect of gas solubility on water FVF at saturated andundersaturated conditions, EOS two-phase calculations.

6 PHASE BEHAVIOR

Fig. 9.9—Water/NaCl-brine viscosity as a function of tempera-ture and salinity.

where A0 � 1060.314 � 0.58ws � �1.9 � 10�4�T

��1.45 � 10�6�T 2�

and A1 � 8 � 50ws � 0.125wsT, (9.17). . . . . . . . . . . . . . . . .

with c*w in psi�1, p in psia, T in °F, and ws in weight fraction of NaCl.

Solving Eq. 9.16 for the FVF of a brine without solution gas, B*w, gives

B*w�p, T� � Bo

w�psc, T��1 �A1

A0p�

�1�A1�

, (9.18). . . . . . . . . .

where A0 and A1 are given by Eq. 9.17. Eq. 9.18 results in water andbrine densities that are within 0.5% of values given by Rogers and Pit-zer’s27 highly accurate correlation for 60T400°F, 0p15,000psia, and 0Cs300,000 ppm. For the same range of conditions,Eq. 9.17 calculates isothermal compressibilities within approximate-ly 5% of Rogers and Pitzer’s values.

With Dodson and Standing’s13 data for pure water saturated witha natural gas, an approximate correction for dissolved gas on water/brine FVF at saturated conditions is

Bw�p, T, Rsw� � B*w�p, T��1 � 0.0001R 1.5

sw�, (9.19). . . . . . .

with Rsw in scf/STB. This relation fits the Dodson-Standing dataat 150, 200, and 250°F but overpredicts the effect of dissolved gasat 100°F.

Dodson and Standing also give a correction for the effect of dis-solved gas on water/brine compressibility.

cw�p, T, Rsw� � c*w�p, T� �1 � 0.00877Rsw� , (9.20). . . . . . . .

with Rsw in scf/STB. This relation is valid only for undersaturated-oil/water systems at higher than oil bubblepoint pressure. For gas/water systems and saturated-oil/water systems, the total compress-ibility effect is given by the Perrine formula,28

ctw � �1

Bw��Bw

�p�

T, Rsw

�1

5.615Bg

Bw��Rsw

�p�

T

. (9.21). . . .

Fig. 9.10—Effect of CO2 solubility (in terms of saturation pres-sure) on water viscosity.

9.4.6 Water/Brine Viscosity. Fig. 9.9 presents the viscosities ofpure water and NaCl brines as functions of temperature and salinity.The following equations (except for the pressure correction A0) arepresented by Kestin et al.,29 who report an accuracy of �0.5% inthe range 70T300°F, 0p5,000 psia, and 0Csw300,000ppm (0csw5 M).

�w � �1 � A0 p��*w ,

log�*

w

�ow� A1 � A2 log

�ow

�ow20

,

A0 � 10�3[0.8 � 0.01(T � 90) exp(� 0.25csw)],

A1 ��3

i�1

a1i cisw ,

A2 ��3

i�1

a2 icisw ,

log�o

w

�ow20

��4

i�1

a3i

�20 � T �i

96 � T,

and �ow20 � 1.002 cp, (9.22). . . . . . . . . . . . . . . . . . . . . . . . . .

where a11�3.324�10�2, a12�3.624�10�3, a13��1.879�10�4,a21��3.96�10�2, a22�1.02�10�2, a23��7.02�10�4, a31�1.2378, a32��1.303�10�3, a33�3.060�10�6, a34�2.550�10�8, with � in cp, T in °C, and p in MPa. Kestin et al.’s pressurecorrection A0 contains 13 constants and does not extrapolate well athigh temperatures. The pressure correction for A0 in Eq. 9.22 ismore well-behaved, with only small deviations from the originalKestin et al. correlation at low temperatures.

The effect of dissolved gas on water viscosity has not been re-ported. Intuitively, one might suspect that water viscosity decreaseswith increasing gas solubility, although Collins30 suggests that dis-solved gas may increase brine viscosity. As Fig. 9.10 shows, sys-tems saturated with CO2 show an increase in viscosity with increas-ing gas solubility.

9.2.7 Solubility of Water in Natural Gas. Fig. 9.11 shows the solu-bility of pure water in methane. McKetta and Wehe31 give two chartinserts for correcting pure-water solubilities for salinity and gasgravity (based mainly on Dodson and Standing’s13 values). A best-fit equation for these charts is

yw � yow Ag As, (9.23a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ln yow �

0.05227p � 142.3 ln p � 9, 625T � 460

� 1.117 ln p � 16.44, (9.23b). . . . . . . . . . . . . . . . .


Fig. 9.11—Water solubility in natural gases, including gas-composition and salinity effects(adapted from Ref. 31).

Dewpoint of Natural GasJ.J. McKetta and A.H. Wehe,

U. of Texas (1958)

Ag � 1 �

�g � 0.55(1.55 � 104)�g T�1.446 � (1.83 � 104)T �1.288

,

(9.23c). . . . . . . . . . . . . . . . . .

As � 1 � �2.222 � 10�6�Cs , (9.23d). . . . . . . . . . . . . . . . .

and As � 1 � �3.92 � 10�9 �C1.44s , (9.23e). . . . . . . . . . . . . . .

with T in °F, p in psia, and Cs in ppm or mg/L. Eq. 9.23 yields anabsolute average deviation of 2.5% for yo

w, with a maximum error

less than 10% for 100T460°F and 200p10,000 psia. Eq.9.23d is from the Dodson-Standing correlation and is not recom-mended. Eq. 9.23e is from the Katz et al.32 correlation and is recom-mended. Mole fraction of water in gas, yw, can be converted to a wa-ter/gas ratio, rsw, with

rsw � 135yw

1 � yw� 135yw , (9.24). . . . . . . . . . . . . . . . . . .

where rsw is in STB/MMscf. Replacing the constant 135 with47,300 yields rsw in lbm/MMscf.

8 PHASE BEHAVIOR

Fig. 9.12—Water/brine/oil IFT data correlated with the McLeodparameter (adapted from Ref. 33).

ow

The correction term for salinity that Dodson and Standing13 pro-posed is based on limited results for one low-salinity brine. The Katzet al.32 salinity correction is based on lowering of vapor pressure forbrine solutions at 100°C, where the assumption is made that

yw

yow�

pvw(100�C, Cs)po

vw(100�C), (9.25). . . . . . . . . . . . . . . . . . . . . . . . .

where pvw�brine vapor pressure and pw�pure-water vapor pres-sure, both measured at 100°C. Very little data are available to con-firm these two salinity corrections. However, EOS calculations in-dicate that the Katz et al. correlation is probably valid up to M�3;at higher molalities, the EOS-calculated ratio yw�y is less than thatpredicted by the Katz et al. correlation (see Sec. 9.3).

Finally, water dissolved in reservoir gas and oil mixtures will notcontain salts (i.e., it is fresh water), a fact that can help in identifyingwhere produced water comes from.

9.2.8 Water/Brine/Hydrocarbon IFT. The IFT of water/hydrocarbonsystems, ��wh, varies from approximately 72 dynes/cm for water/brine/gas systems at atmospheric conditions to 20 to 30 dynes/cm forwater/brine/stock-tank-oil systems at atmospheric conditions. The vari-ation in �wh is nearly linear with the density difference between waterand the hydrocarbon phase ��wh (i.e., ��wo or ��wg), where�wh�72 dynes/cm at ��wh��wg�1. This can be expressed inequation form as

�wh � �o� (72 � �

o)��wh , (9.26). . . . . . . . . . . . . . . . . . .

where �o�intercept at ��wh�0.

Ramey33 proposes a correlation for �wh based on the Macleod pa-rameter ��

��. This parameter was plotted vs. �� (Fig. 9.12) withdata for brines with stock-tank oil, saturated and undersaturated res-ervoir oils, and natural gases. Eq. 9.26, where �o

�15, representsRamey’s graphical correlation surprisingly well. A near-exact fit ofhis correlation is

�wg � 20 � 36��wh. (9.27). . . . . . . . . . . . . . . . . . . . . . .

Ramey’s data that do not lie on his general correlation are accuratelyrepresented by Eq. 9.26, with �o ranging from 5 to 30. Fig. 9.13shows a graphical correlation for �wg given by Standing34 for water/brine/methane systems (apparently based on Hocott’s35 natural-gas/brine data).

Firoozabadi and Ramey36 consider the IFT of water and hydro-carbons using data for distilled water and pure hydrocarbons. Theyarrive at a graphical relation similar to Ramey’s33 original correla-tion, with the addition of reduced temperature as a correlating pa-rameter. Unfortunately, their correlation does not predict water/brine/oil IFT’s with more accuracy than the original Rameycorrelation (or Eq. 9.27). As Fig. 9.14 shows, water/gas IFT’s re-

Fig. 9.13—Water/brine/oil IFT data correlation (adapted fromRef. 34).

Temperature, °F

Salt Concentration, ppm

ported by various authors show considerable scatter, and it seemsthat any correlation will give only approximate IFT values for suchsystems until consistent data become available. Mutual-solubilityeffects of gas dissolved in water and water dissolved in gas may af-fect IFT’s, perhaps explaining some of the difference in methane/brine and methane/water IFT’s in Fig. 9.14. Otherwise, the seeming-ly erratic behavior of some water/brine/oil IFT data may beexplained by aromatic compounds and asphaltenes. Also, crude-oilsamples exposed to atmospheric conditions for long periods of timemay experience oxidation that can affect IFT measurements.

Fig. 9.14—Methane/water and methane/brine IFT’s.


Fig. 9.15—Pure-water and NaCl-brine vapor-pressure curves.

��

Mutual solubilities and volumetric properties of water/hydrocarbonsystems can be predicted with reasonable accuracy with one of sever-al modifications to existing cubic EOS’s. Other types of EOS’s alsohave been applied to these systems but do not show a clearly superiorpredictive capability. Although cubic EOS’s are not widely used forreservoir water/hydrocarbon systems, this approach eventually is ex-pected to replace the empirical correlations currently being used.

To improve vapor-pressure predictions of water (and solubilities ofwater in the nonaqueous phase), Peng and Robinson37 proposed amodified correction term, � (applied to EOS Constant a), for water.

�H2O � 1.008568 � 0.8215�1 � T 0.5rw��

2, (9.28). . . . . . . .

which can be used for 0.44Tr0.72 (60T400°F). Alterna-tively, the Søreide-Whitson9 relation for �H2O can be used with thePeng-Robinson38 EOS (PR EOS).

�0.5H2O � 1 � 0.4531 � T

r H2O�1 � 0.0103c1.1

sw �

� 0.0034�T �3r H2O

� 1�� . (9.29). . . . . . . . . . . . . . . .

Eq. 9.29 predicts pure-water vapor pressures within 0.2% of steam-table values for 0.44Trw1 (i.e., T�60°F) and can be used to pre-dict vapor pressures of NaCl solutions with the same accuracy. Fig.9.15 shows vapor pressures of pure water and NaCl-brine solutions re-ported by Haas.10 With a correction for salinity in the � term, the pre-dicted water solubilities in nonaqueous phases are expected to improve.

The most important modification of existing cubic EOS’s for wa-ter/hydrocarbon systems is the introduction of alternative mixingrules for EOS Constant A, where different binary-interaction param-eters (BIP’s), kij, are used for the aqueous and nonaqueous (hydro-carbon) phases. Peng and Robinson37 propose a simple EOS modi-fication for hydrocarbon/water systems; namely, they define twosets of kij: kij,HC for the hydrocarbon phase(s) and kij,aq for theaqueous phase. EOS Constant A is therefore calculated differentlyfor the hydrocarbon and aqueous phases,

AHC ��N

i�1

�N

j�1

yi,HC yj,HC Ai Aj �1 � ki j,HC�

and Aaq ��N

i�1

�N

j�1

xi,aq xj,aq Ai Aj �1 � ki j,aq�, (9.30). . . . . . . . .

TABLE 9.3—RECOMMENDED BIP’s FOR THE PR EOS TO PREDICT SOLUBILITIESIN WATER/HYDROCARBON SYSTEMS*

Aqueous Phase

Hydrocarbons kij, aq � �1 � a0csw�A0 ��1 � a1csw�A1Tri �

�1 � a2csw�A2T2ri ,

where a0 � 0.017407, a1 � 0.033516, a2 � 0.011478

A0 � 1.112 � 1.7369��0.1i , A1 � 1.1001 � 0.83�i

A2��0.15742�1.0988�i , i�hydrocarbons, and j�water/brine.

N2 kij, aq � � 1.70235�1 � 0.025587c0.75sw � � 0.44338�1 � 0.08126c0.75

sw �Tri ,

where i�N2 and j�water/brine.

CO2 kij, aq � � 0.31092�1 � 0.15587c0.75sw �

� 0.2358�1 � 0.17837c0.98sw �Tri � 21.2566 exp(� 6.7222Tr � csw),

where i�CO2 and j�water/brine.

H2S kij, aq � � 0.20441 � 0.23426Tri , where i � H2S and j � water�brine.

Nonaqueous Phase

i kij ,HC, where j�water

C1 0.4850

C2 0.4920

C3 0.5525

C4 0.5091

C5� 0.5000

N2 0.4778

CO2 0.1896

H2S 0.19031�0.05965Tri

Acentric factors � used in developing hydrocarbon/water BIP’s are C1�0.0108, C2�0.0998, C3�0.1517, and C4�0.1931.*Modified Peng-Robinson � term for water/brine, Eq. 9.29.

10 PHASE BEHAVIOR

Fig. 9.16—Predicted gas-phase water solubilities for methane/NaCl-brine mixtures at 250°F determined with the general �wterm (Eq. 9.31).

�� 0�� 0.86

1.712.57

+++ 3.42�� 5.13

Brine Salinity, Csw

-

respectively, where yi,HC�hydrocarbon composition (gas or oil) andxi,aq�water-phase composition. Using two sets of kij has been ap-plied successfully to correlate mutual solubilities of hydrocarbon/water and nonhydrocarbon/water binary systems. Table 9.3 givesrecommended kij relations for aqueous and nonaqueous phases forthe PR EOS, where these interaction coefficients must be used withthe general �H2O relation (Eq. 9.29). The CO2/water/brine correla-tion gives the best results at pressures less than approximately 5,000psia because data in this region have been given more weight in de-velopment of the correlation.

Considerable data on solubilities of hydrocarbon and nonhydro-carbon gases in brine solutions were used in making the salinitycorrections for aqueous-phase kij. Similar data were not availablefor solubilities of water in the nonaqueous phase for mixtures con-taining brines. Until more data become available, it will be neces-sary to assume that the effect of salinity is adequately treated by themodified �H2O term (Eq. 9.30).

Fig. 9.16 shows predicted water solubilities for methane/NaCl-brine mixtures with varying salt concentration. The predicted reduc-tion in water solubility for mixtures containing brine, relative to sol-ubility for mixtures containing pure water, is more or lessindependent of pressure and temperature. Fig. 9.17 correlates the ra-tio yw�yo

w calculated by the modified PR EOS (with �H2O from Eq.9.30) vs. salinity. The effect of salinity is clearly less than that pre-dicted by the Dodson-Standing13 correlation (Eq. 9.23d), whereasthe Katz et al.32 correlation (Eq. 9.23e) appears to be consistent withthe EOS calculations up to M�3.

Simultaneous application of aqueous- and nonaqueous-phase in-teraction coefficients requires modification of the standard EOS im-plementation (which uses a single set of kij). Figs. 9.18 through9.22 show the accuracy of this approach for mutual-solubility pre-dictions of binaries and natural-gas/water/brine mixtures, suggest-ing that the required modification is probably warranted.

A standard implementation of the PR EOS can still be used withthe BIP’s in Table 9.3. If only gas solubility in the water phase isneeded, accurate gas solubilities can be predicted with the aqueous-phase kij,aq for both phases; however, calculated hydrocarbon-phasecomposition will not be accurate. Likewise, if only water solubilityin the hydrocarbon phase is needed, the hydrocarbon-phase kij,HCcan be used for both phases, but calculated aqueous-phase composi-tions will not be accurate in this case. Fig. 9.23 compares exper-imental solubilities for the methane/water system with results pre-dicted by the modified PR EOS (with two sets of kij) and by theoriginal PR EOS with a single set of kij.

Fig. 9.17—Effect of salinity on gas-phase water solubilityfor methane/NaCl-brine mixtures determined with the general�w term (Eq. 9.31).

�� Katz et al.32 Correlation ��Calculated at 100°F

Calculated at 250°F

-

Composition- and density-dependent mixing rules have also beenproposed for modifying cubic EOS’s for water/hydrocarbon sys-tems. Panagiotopoulos and Reid’s39 linear composition-dependentmixing rule has received considerable interest. Unfortunately, asKistenmacher and Michelsen40 point out, it violates several funda-mental thermodynamic conditions. Enick et al.8 propose tempera-ture-dependent correction terms for both EOS Constants A and B ofwater, together with a linear composition-dependent mixing rule forConstant A. With this approach, they successfully describe multi-phase equilibria for a multicomponent water/oil/CO2 system.

Several noncubic EOS’s3-5,41,42 have been proposed for water/hydrocarbon systems, including conventional activity-coefficientmodels that are limited to relatively low pressures and more generalelectrolyte EOS models. However, these models do not appear to bebetter than the simpler modifications of cubic EOS’s.

��

Gas hydrates are solutions of gases in crystalline solids called clath-rates. Gas molecules occupy the void spaces (cages) in the water-crystal lattice. Hydrates can form at temperatures considerably higherthan the freezing point of pure water. For example, in high-pressurewells (more than 15,000 psia), hydrates have been observed at tem-peratures much higher than 100°F. Hydrates resemble wet snow and,like ice, will float on water. In the oil field, hydrates look like a grayishsnow cone. When hydrate “snow” is tossed on the ground, the hydro-carbons escaping can be heard easily, giving the impression that thehydrocarbons were physically trapped in the snow. The distinctivecrackling sound is in fact caused by escaping natural-gas moleculesrupturing the crystal lattice of the hydrate molecules.

Hydrates were discovered in 1810 by Davy and were investigatedonly as curiosities of physical chemistry for many years thereafter.43 In1888, Villard became the first to determine the existence of hydrateswith typical components of natural gas, such as methane, ethane, andpropane.43 However, the real push to measure hydrate phase behaviordid not begin until the 1930’s when Hammerschmidt44 pointed out thathydrates were the culprits that were choking wellhead and productionequipment in gas fields. He also suggested ways to inhibit their forma-tion. Although hydrate inhibition has been practiced for more than 50years, the severe conditions encountered in arctic and deep drillinghave sparked a new wave of interest in measurement of hydrate forma-tion and inhibition at these conditions.

Although the kinetics and fluid mechanics of hydrate formationand dissociation are not covered here, they are nonetheless impor-tant in deepwater drilling operations. Because vast deposits of natu-


Fig. 9.18—EOS predictions of mutual solubilities for methane/water system determined with different sets of BIP’s for aqueousand nonaqueous phases.

ral-gas hydrates exist in the Arctic, a great deal of Russian researchhas been conducted on both the kinetics and thermodynamics of hy-drate formation and dissociation.43 Recovery of natural gas en-trapped in these vast hydrate deposits in permafrost regions (by hy-drate dissociation) has also been studied recently.45

The three most widely used calculation methods for predictinghydrate formation are (1) the vapor/solid K-value method of Katzand his coworkers46-51 and equations fitting the developed K-valuecharts; (2) methods of Campbell and his coworkers52-54; and (3)combined methods based on statistical thermodynamics (van derWaals and Platteeuw55) for the hydrate phase and EOS’s for the fluidphases. These methods are discussed later.

9.4.1 Crystallography of Hydrates. In the presence of a free-waterphase, hydrates will form below a certain temperature often referredto as the “hydrate temperature.” Hydrate crystals generally grow onlyin the presence of a free-liquid-water phase at typical oilfield condi-tions. Hydrates can also form in the presence of a dense-vapor-waterphase at temperatures sufficiently low to ensure hydrogen bonding.The general conditions under which hydrates form include gas at orbelow its water dewpoint (which can yield the free-water phase nec-essary for hydrate formation in the system) and conditions at moder-ately low temperature or high pressure. With respect to components

Fig. 9.19—EOS predictions of mutual solubilities for natural-gas/water system determined with different sets of BIP’s foraqueous and nonaqueous phases.

Mole Fraction Water in Vapor Phase, yw

Mole Fraction Natural Gas, xng

normally found in natural gas, hydrate formation has been observedand measured only for the light constituents found in natural gas: C1through C4 alkanes (including i-C4), N2, CO2, and H2S.

Fig. 9.24 shows a schematic of the natural-gas hydrate-crystallattice. Two common types of hydrate-crystal structures have beenproposed from interpretation of results of von Stackelberg andMüller’s56 X-ray diffraction studies of hydrates. Structure I is usual-ly a body-centered lattice, and Structure II has a diamond lattice.Structures I and II have different sized cages (i.e., void spaces). InStructure I hydrates, methane can fill the smaller cages, while thelarger cages can be filled only by larger hydrocarbon molecules,such as ethane. The cages in Structure II hydrates are larger, allow-ing entrapment of propane and i-butane in addition to methane andethane. Fig. 9.25 summarizes the components and correspondingsize ranges that fit into Structure I and II cavities. Light components,such as methane, ethane, and CO2, form Structure I hydrates; nitro-gen and the heavier alkanes, such as propane, n-butane, i-butane,and neopentane, form Structure II hydrates.

Enough cages must be filled with hydrocarbon molecules to sta-bilize the crystal lattice. Because all the cages do not have to be full,

12 PHASE BEHAVIOR

Fig. 9.20—EOS prediction of gas solubility for CO2/water/brinesystems at 302°F determined with different sets of BIP’s foraqueous and nonaqueous phases; symbols�experimental andlines�calculated.

Mole Fraction CO2 in Aqueous Phase, xCO2

the molecular weight of a clathrate hydrate is not fixed. The “vacan-cy” of the hydrate-crystal lattice depends on which “guest” natural-gas molecules happen to be available to occupy the void locationsbetween the interstices of the host water molecules and on the condi-tions under which the crystal lattice is formed. Thus, the presenceof methane and ethane leads only to the formation of Structure I hy-drates and the presence of methane, ethane, and propane leads to theformation of a mixture of Structure I and II hydrates.

The general trends of hydrate formation can be qualitatively pre-dicted for a particular natural-gas component. The two important fac-tors in formation of the two different structures of hydrates are sizeand solubility of the natural-gas molecules. The rate of clathration ispartially dependent on solubility because the more soluble a gaseouscomponent is in water, the higher the probability that it will be“caught” in a cage as the hydrate crystal is being formed. The size ofthe guest molecule not only determines the structure type but also therate of formation. For example, comparing the rate of clathration ofmethane with that of ethane, a higher pressure is required to form puremethane hydrates than pure ethane hydrates, even though methane isconsiderably more soluble in water than ethane. The reason is thatmethane is a smaller molecule that is more difficult to entrap as thecage of the crystal lattice closes. Furthermore, hydrates form morereadily from natural-gas mixtures than from pure components be-cause the range of molecular sizes in natural gas has a higher proba-bility of filling enough cavities to stabilize the hydrate-crystal lattice.

Researchers only recently proved that butanes are hydrate for-mers. McLeod and Campbell53 and others showed that the butanesare hydrate formers when methane is present to occupy the smallercavities in Structure II hydrates. They found that, like ethane andpropane and in contrast to n-pentane, the butanes lower the hydrate-forming pressure. Hydrates with n-butane are very unstable, and, atpressures higher than 10,000 psia, n-butane behavior reverts to thatof a nonhydrate former. Alkanes with a higher carbon number thann-butane are not believed to form hydrates.

9.4.2 Phase Diagrams for Hydrates. At cryogenic temperaturesand subatmospheric pressures, phase diagrams show a multitude ofhydrate forms. We cover only the simpler phase diagrams that repre-sent the most common conditions encountered in subsurface engi-neering and in surface facilities.

The temperature and pressure conditions for hydrate formation insurface gas-processing facilities are generally much lower thanthose considered in production and reservoir engineering. The

Fig. 9.21—EOS prediction of gas solubility for natural-gas/brinesystem determined with different sets of BIP’s for aqueous andnonaqueous phases.



conditions of initial hydrate formation are often given by simple p-Tphase diagrams for water/hydrocarbon systems. In 1885, Rooze-boom defined a lower hydrate quadruple point, Q1 ( I�Lw�H�V),and an upper quadruple point, Q2 (Lw�H�V�LHC), as on Fig. 9.26.43

His nomenclature for the phases is I�pure ice, Lw�liquid water,LHC�liquid hydrocarbon, V�vapor, and H�hydrate. The quadru-ple point defines the condition at which four phases are in equilibri-um. Because the Gibbs phase rule leads to zero degrees of freedomfor this system, the values of these quadruple points (Table 9.4) forthe eight natural-gas hydrate formers are unique and invariant andprovide a quantitative basis for classification of hydrate formers.

Each quadruple point is at the intersection of four three-phaselines. The lower quadruple point, Q1, represents the transition of Lw

to I. As temperature decreases to Point Q1, hydrates cease formingfrom vapor and liquid water and are forming from vapor and ice.The upper quadruple point, Q2, is the approximate intersection ofLine Lw�H�V with the vapor pressure of the hydrate former and rep-resents the upper temperature limit for hydrate formation for thatcomponent. Some of the lighter natural-gas components, such asmethane and nitrogen, do not have an upper quadruple point, so noupper temperature limit exists for hydrate formation. This is the rea-son that hydrates can still form at high temperatures (up to 120°F)in the surface facilities of high-pressure wells.


Fig. 9.22—EOS prediction of gas solubility for N2/NaCl-brinesystem at 217°F determined with different sets of BIP’s foraqueous and nonaqueous phases; symbols�experimental andlines�PR EOS predicted.

Mole Fraction Nitrogen in Aqueous Phase, xN 2

Fig. 9.27 shows the main area of hydrate formation in petroleum-engineering applications. Line FEG represents the natural-gas-mix-ture dewpoint curve. The dewpoint line is analogous to the vapor-pressure curves of the individual components in Fig. 9.26. Point Eis the maximum hydrate-forming temperature (analogous to thequadruple points, Q2, of the individual components in Fig. 9.26).The hydrate curve is Line BE. At the intersection of the dewpointand hydrate curves, the hydrate curve for many natural-gas systemsbecomes nearly vertical and establishes the maximum hydrate-forming temperature. For a natural-gas system with very high con-centrations of methane, such as encountered in the deep natural-gasplays in the Anadarko basin, the maximum hydrate-forming tem-perature may be essentially nonexistent (observe that no Q2 existsfor the methane curve in Fig. 9.26). The general approach to hydrateprediction in most engineering applications is to determine HydrateLine BE and the position of Dewpoint Line FEG on Line BE. Sec.9.4.3 discusses calculation methods.

Fig. 9.28 shows Deaton and Frost’s58 data for hydrate-formationconditions for methane/propane mixtures. These data show how hy-drate-formation conditions for natural gas are strongly dependenton the propane concentration. The general effect of increasing pro-pane concentration is to lower the hydrate-forming pressure and toincrease the hydrate-forming temperature. Katz et al.32 and Wilcoxet al.49 developed Fig. 9.29 to determine hydrate-forming condi-tions for natural gas at different specific gravities. Because Fig. 9.29is based on gas gravity, it is particularly useful as a quick guide toestimate the hydrate temperature for a natural gas. Fig. 9.29 shouldnot be used if CO2 or H2S is present at a combined concentration �1mol%. At pressures less than 12,000 psia, the Joule-Thompson ex-pansion of a natural gas, for example, across a separator choke, re-duces the temperature of the gas. Katz et al.32 present charts (Figs.9.30 through 9.32) that show the maximum permissible expansionof natural gases before hydrate formation occurs.

9.4.3 Calculation Method of Katz and Coworkers.32,47,49,51 Byapplying the analogy of vapor/liquid equilibrium K values to a solidsolution, Carson and Katz47 and Wilcox et al.49 developed the con-cept of a vapor/solid K value for predicting the temperature andpressure conditions under which hydrates form or dissociate.

Ki (v�s) �yi

xi (s), (9.31). . . . . . . . . . . . . . . . . . . . . . . . . . . .

Fig. 9.23—Predicted mutual solubilities of methane/water sys-tem at 100°F determined with the modified PR EOS with one andtwo sets of kij; all kij from Table 9.3.

Experimentalkij Different for Each Phase (modified �w),kij Same for Each Phase (modified �w),kij Same for Each Phase (original �w)

kij�0.485 (nonaqueous phase)kij��0.260 (nonaqueous phase)

Experimentalkij Different for Each Phase (modified �w),kij Same for Each Phase (modified �w),kij Same for Each Phase (original �w)

kij�0.485 (nonaqueous phase)kij��0.260 (nonaqueous phase)

where Ki(v�s)�vapor/solid equilibrium value of Component i,yi�gas composition, and xi(s)�mole fraction of Component i in thesolid on a water-free basis. Calculation of hydrate-formation tem-perature is analogous to calculation of a dewpoint temperature (dis-cussed in Chap. 3). A gas in the presence of a free-water phase willform a hydrate if

�N

i�1

yi

Ki (v�s)� 1. (9.32). . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Conversely, hydrate-dissociation temperature can be treated like abubblepoint calculation. A hydrate will dissociate if

�N

i�1

xi Ki (v�s) � 1. (9.33). . . . . . . . . . . . . . . . . . . . . . . . . . . .

Because Ki(v�s) is based on the mole fraction of a guest natural-gas component in the solid-phase hydrate mixture on a water-freebasis, the concept of Ki(v�s) is only an approximation of the original

14 PHASE BEHAVIOR

Fig. 9.24—Schematic of hydrate-crystal lattice; circles representwater molecules, lines represent hydrogen bonds (from Ref. 52).

definition of vapor/liquid equilibrium ratios, Ki. For example, theconcept of the vapor/solid K value cannot be used to calculate hy-drate-phase splits or equilibrium-phase compositions. The vapor/solid K value can be used only to predict the temperature or pressurewhere hydrates form or dissociate. However, on the basis of compo-nent Ki(v�s) values, where the natural-gas components will concen-trate can be determined qualitatively. If Ki(v�s) for a natural-gascomponent is greater than unity (nitrogen is a typical example), thecomponent will tend to concentrate in the gaseous phase rather thanin the hydrate phase. If Ki(v�s) is less than unity (for example, pro-pane), the component will tend to concentrate in the hydrate. Katzand his coworkers provide Ki(v�s) nomograms for several natural-gas components as functions of temperature and pressure.

Sloan57 developed the following polynomial-fit equation of theKatz-Carson charts, which can be used to estimate Ki(v�s).

ln Ki(v�s) � A0 � A1 T � A2 p � A3 T�1� A4�p � A5 pT

� A6 T 2� A7 p2

� A8�p�T� � A9 ln �p�T �

� A10 p�2� A11

�T�p� � A12�T 2

�p� � A13�p�T 2�

� A14�T�p3� � A15 T 3

� A16�p3

�T 2� � A17 T 4 .

(9.34). . . . . . . . . . . . . . . . . .

Table 9.5 gives the values of Constants A0 through A17.Ki(v�s) for nonhydrate formers are assumed to be infinity in the

calculation. The original work assumed that nitrogen and butaneswere not hydrate formers, which was subsequently shown to be in-correct. However, fairly reliable estimates can be obtained by as-suming that the Ki(v�s) for nitrogen and the butanes are also infinityas long as the pressure is less than approximately 1,000 psia. Thismethod becomes less reliable for pressures higher than 1,000 psia.

9.4.4 Calculation Methods of Campbell and Coworkers.52-54 Toaddress the pressure limitations of the Ki(v�s) method of Katz and hiscoworkers as well as the hydrate-temperature-depression effects ofmolecules too large to fit into the cavities of the hydrate crystal,Campbell and his coworkers52-54 developed additional empiricalprocedures. In general, these methods can be used for quick estimatesof hydrate-formation temperatures when pressures exceed the1,000-psia limitation of the Ki(v�s) method. The Trekell-Campbell54

Fig. 9.25—Summary of natural-gas components fitting into Struc-ture I and II (S-I and S-II, respectively) cavities (from Ref. 57).

No S-I or S-II Hydrates

CavitiesOccupied

Hydrate Former

512 + 51264

S-II

512 + 51262

S-I

51262

S-I

51264

S-II

No Hydrates

A

KrN2O2

CH4

Xe; H2S

CO2

C2H6

c-C3H6

C3H8

iso-C4H10

n-C4H10

17 H2O

72/3 H2O

53/4 H2O

52/3 H2O

(CH2)3O

3Å

4Å

5Å

6Å

7Å

8Å

method covers pressures from 1,000 to 6,000 psia, and theMcLeod-Campbell53 method covers pressures from 6,000 to10,000 psia.

The Trekell-Campbell method calculates additive effects of gasmolecules on the hydrate-forming temperature of methane. Theygive eight nomograms, six of which give positive displacements asfunctions of pressure for C3, n-C4, and i-C4 and two that give nega-tive corrections (depression) for nonhydrate formers, such as C5+.This method is strictly empirical and must be used with caution, butit is useful as a quick estimate at pressures up to 6,000 psia.

McLeod and Campbell developed another method to predict hy-drate-formation temperatures at very high pressures encountered indeep-gaswell drilling. They prepared a very simple correlationbased on a modified Clapeyron equation to describe the energy ofphase transition at pressures 6,000 to 10,000 psia.

T � 3.89 C�

and C ��N

i�1

yiCi , (9.35). . . . . . . . . . . . . . . . . . . . . . . . . . . . .

where T is in °R, yi�gas molar composition, and Table 9.6 givesthe hydrate-former constants Ci for C1 through C4 hydrocarbons.

The hydrate-prediction methods of Campbell and his coworkersare mostly empirical but do provide a reliable answer when computerprograms for the more theoretical models described in the next sec-


Fig. 9.26—Hydrate-formation conditions for natural-gas hydrateformers (from Ref. 57).

tion are not available. They can also be used as a check of the moresophisticated estimation methods (also described in the next section).

9.4.5 van der Waals and Platteeuw55 Model. Most modern com-puter-based methods of predicting hydrate formation are based on

van der Waals and Platteeuw’s statistical-mechanical solid-solutiontheory of clathrates. These authors developed an adsorption modelbased on statistical mechanics to derive a relation for the chemicalpotential of water in the hydrate phase. Their method is based on anequation that relates the chemical potential of water in the hydratestructure in much the same way that chemical potential of a compo-nent is related to the activity of a component in a mixture.

�i � �oi � RT ln ai , (9.36). . . . . . . . . . . . . . . . . . . . . . . . . .

where �i�chemical potential of pure Component i (see Chap. 4)and ai�activity of Component i in the mixture.

van der Waals and Platteeuw propose the following Langmuir ad-sorption-isotherm analogy that accounts for the microscopic hy-drate structure.

�wH � �wMT � RT�i

nci ln�1 ��j

yji�, (9.37). . . . . . . .

where �wH�chemical potential of water in the filled hydrate,MT �wMT�chemical potential of water in the empty hydrate,nci�number of Type i cavities per water molecule in hydrate-crys-tal lattice, and yji�fraction (probability) of Type j molecule occu-pying Type i cavity.

The Langmuir adsorption theory is applicable because “clathra-tion” and “declathration” are analogous to adsorption and desorption,respectively. The probability term, yji, depends on the interaction be-tween the guest gas molecule and its “cage” (the “site” by analogywith the original Langmuir theory). The term yji also depends on thefugacities of the components in the gas phase, which can be calcu-lated with an EOS. Parrish and Prausnitz59 were the first to extend thevan der Waals and Platteeuw statistical-mechanics model to multi-component systems. They used the Kihara potential to calculate theLangmuir constants. John et al.60 and Schroeter et al.61 also used theKihara potential to calculate the Langmuir constants.

Erickson and Sloan62 developed a calculation procedure using thevan der Waals and Platteeuw model. A computer program(CSMHYD) is included with Ref. 62, and a complete description ofthe calculation algorithm and computer-program flow chart are alsoprovided. Ref. 63 provides a description and algorithm for a similarapproach. These methods are fairly difficult to program from the liter-ature; therefore, Ref. 62 with the program diskettes is recommended.

Other researchers have developed calculation methods based onthe van der Waals and Platteeuw in combination with an EOS. Nget al.64 and Robinson and Mehta65 made predictions of hydrate-formation conditions using the PR EOS and developed a computer-based method that is available through the Gas Processors Assn.Schroeter et al.61 used the Benedict et al. EOS66 to model the fluidphase in hydrate calculations with sour-gas (including H2S) sys-tems. Munck et al.67 used the Soave-Redlich-Kwong EOS with thevan der Waals and Platteeuw adsorption model to calculate fugaci-ties of liquid and gaseous phases in equilibrium with hydrates. Theyused the Michelsen68 stability algorithms (see Chap. 4) to developa computer program that predicts hydrate-formation conditionswithout prior knowledge of the phases. To account for the effects ofnonelectrolyte inhibitors, Munck et al.67 used the UNIQUAC activ-ity-coefficient model. They obtained good agreement for hydratesin equilibrium with North Sea reservoir fluids.

9.4.6 Water Content of Vapor in Equilibrium With Hydrates. Theconcentration of water in the vapor phase in equilibrium with hydrateis usually very small, on the order of 0.001 mol% or less. Phase dia-grams and nomograms for determining the water content of vapor inequilibrium with hydrates are complicated by metastable equilibriumin the gas/ice region and are cumbersome to use for the many possiblecombinations of compositions. Song and Kobayashi69 present amathematical approach for determining the water content of gases inthe vapor/hydrate region. They studied methane-rich and CO2-richsystems, which are especially important in EOR operations (Fig.9.33). Sloan55 proposes a slight improvement to the Kobayashi etal.50 method and provides the necessary equations, along with an ex-tensive table of coefficients and an example of how to use the method.

16 PHASE BEHAVIOR

TABLE 9.4—QUADRUPLE POINTS FOR NATURAL-GAS HYDRATE FORMERS (from Ref. 57)

Lower Quadruple, Q1 Upper Quadruple, Q2

Natural-Gas

Component

Temperature

(°F)

Pressure

(psi)

Temperature

(°F)

Pressure

(psi)

C1 29.9 371.7 No Q2

C2 31.9 76.9 58.4 491.7

C3 31.9 24.9 42.2 80.6

i-C4 31.9 16.4 35.3 24.2

CO2 31.9 182.2 49.7 652.5

N2 29.8 2,079.5 No Q2

H2S 31.4 13.5 85.2 324.7

n-C4 does not form hydrate by itself; it requires the presence of a “help gas.”

Fig. 9.27—Characteristics of hydrate-forming natural-gas mix-ture at typical production conditions (from Ref. 52).

Algorithms for predicting hydrocarbon concentration in vapor inequilibrium with hydrate are also discussed.

Methods for predicting hydrate-formation conditions have im-proved, and the prediction of hydrate-formation conditions withthe van der Waals and Platteeuw49 method can be used reliably. Inextreme operating conditions, such as those encountered in deepdrilling, calculation methods for predicting hydrate formationmay not be reliable. In these situations, laboratory measurementsare recommended.

9.4.7 Hydrate Inhibition. Hammerschmidt70 presented a relationfor predicting the depression of the hydrate-forming temperature ofnatural gases in contact with dilute aqueous solutions of antifreezes,such as methanol and glycols (e.g., ethylene glycol). Hammersch-midt’s equation originates from the relationship for determining thecolligative properties (in this case, freezing or hydrate-formingpoint) of an ideal solution.

�T � 2, 335 w100M � wM

, (9.38). . . . . . . . . . . . . . . . . . .

Fig. 9.28—Hydrate-formation conditions for methane/propane/water mixtures (from Ref. 57).

with �T in °F, M�molecular weight of the antifreeze agent (e.g.,M�32 for methanol), and w�weight percent of the antifreezeagent in solution. Fig. 9.34 shows how the hydrate-formation tem-perature is depressed with the addition of methanol to water and atypical natural-gas component.

Use of the Hammerschmidt equation should be restricted to sweetnatural gases with antifreeze concentrations of less than 0.20 mol%.Campbell51 suggests that for glycols, the factor 2,335 should be re-placed by 4,000. For concentrated methanol solutions, like thoseused to free a plugged-up tubing string in a high-pressure well, Niel-sen and Bucklin71 propose the following modification of the Ham-merschmidt equation.

�T � � 129.6 ln�1 � xMeOH� , (9.39). . . . . . . . . . . . . . . . . .


Fig. 9.29—Temperature and pressure conditions of hydrateformation for natural gases (from Ref. 32).

where �T�depression of the hydrate-forming temperature in °Fand xMeOH�mole fraction of methanol inhibitor.

!�"��

1. Craft, B.C. and Hawkins, M.: Applied Petroleum Reservoir Engineering,first edition, Prentice-Hall Inc., Englewood Cliffs, New Jersey (1959).

2. Fetkovich, M.J., Reese, D.E., and Whitson, C.H.: “Application of a Gen-eral Material Balance for High-Pressure Gas Reservoirs,” SPE Journal(March 1998) 3.

3. Li, Y-.K. and Nghiem, L.X.: “Phase Equilibria of Oil, Gas, and Water/Brine Mixtures From a Cubic Equation of State and Henry’s Law,” Cdn.J. Chem. Eng. (June 1986) 64, 486.

4. Carroll, J.J. and Mather, A.E.: “Equilibrium in the System Water-Hydro-gen Sulfide: Modelling the Phase Behavior with an Equation of State,”Cdn. J. Chem. Eng. (1989) 67.

5. Michel, S., Hooper, H.H., and Prausnitz, J.M.: “Mutual Solubilities ofWater and Hydrocarbons From an Equation of State. Need for an Uncon-ventional Mixing Rule,” Fluid Phase Equilibria (1989) 45.

6. Firoozabadi, A. et al.: “EOS Predictions of Compressibility and Phase Be-havior in Systems Containing Water, Hydrocarbons, and CO2,” SPERE(May 1988) 673.

Fig. 9.30—Maximum permissible expansion of 0.6-gravity natu-ral gas without hydrate formation (from Ref. 32).

Initial Temperature, °F

7. Nutakki, R. et al.: “Calculation of Multiphase Equilibria for Water-Hy-drocarbon Systems at High Temperature,” paper SPE 17390 presentedat the 1988 SPE/DOE Enhanced Oil Recovery Symposium, Tulsa, Okla-homa, 17–20 April.

8. Enick, R.M., Holder, G.D., and Mohamed, R.: “Four-Phase Flash Equilib-rium Calculations Using the Peng-Robinson Equation of State and AMixing Rule for Asymmetric Systems,” SPERE (November 1987) 687.

9. Søreide, I. and Whitson, C.H.: “Peng-Robinson Predictions for Hydrocar-bons, CO2, N2 and H2S With Pure Water and NaCl-Brines,” Fluid PhaseEquilibria (1992).

10. Haas, J.L. Jr.: “Physical Properties of the Coexisting Phases and Ther-mochemical Properties of the H2O Component in Boiling NaCl Solu-tions,” Geological Survey Bulletin (1976) 1421-A and -B.

11. Patton, C.C.: Oil Field Water Systems, Campbell Petroleum Series, Nor-man, Oklahoma (1981).

18 PHASE BEHAVIOR



12. Rowe, A.M. Jr. and Chou, J.C.S.: “Pressure-Volume-Temperature-Con-

centration Relation of Aqueous NaCl Solutions,” J. Chem. Eng. Data

(1970) 15, 61.

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Solubility Relations for Natural Gas-Water Mixtures,” Drill. & Prod.

Prac. (1944) 173.

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Water Systems. III The Solubility of Methane in Water at Pressures to

10,000 psi,” Trans., AIME (1951) 192, 223.



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at Various Temperatures from 12 to 40°C and at Pressures to 500 Atmo-

spheres,” J. Amer. Chem. Soc. (1940) 62, 815.

16. Wiebe, R. and Gaddy, V.L.: “Vapor Phase Composition of Carbon Diox-

ide-Water Mixtures at Various Temperatures and at Pressures to 700 At-

mospheres,” J. Amer. Chem. Soc. (1941) 63, 475.

17. Kobayashi, R. and Katz, D.L.: “Vapor-Liquid Equilibria for Binary Hy-

drocarbon-Water Systems,” Ind. Eng. Chem. (1953) 45, No. 2, 440.

18. Cramer, S.D.: “Solubility of Methane in Brines From 0 to 300°C,” Ind.

Eng. Chem. Proc. Des. Dev. (1984) 23, No. 3, 533.


TABLE 9.5—VALUES OF COEFFICIENTS A0 THROUGH A17 IN EQ. 9.34

Coefficients

Component A0 A1 A2 A3 A4 A5

CH4 1.63636 0.0 0.0 31.6621 �49.3534 5.31x10�6

C2H6 6.41934 0.0 0.0 �290.283 2,629.10 0.0

C3H8 �7.8499 0.0 0.0 47.056 0.0 �1.17x10�6

i-C4H10 �2.17137 0.0 0.0 0.0 0.0 0.0

n-C4H10 �37.211 0.86564 0.0 732.20 0.0 0.0

N2 1.78857 0.0 �0.001356 �6.187 0.0 0.0

CO2 9.0242 0.0 0.0 �207.033 0.0 4.66x10�5

H2S �4.7071 0.06192 0.0 82.627 0.0 �7.39x10�6

A6 A7 A8 A9 A10 A11

CH4 0.0 0.0 0.128525 �0.78338 0.0 0.0

C2H6 0.0 9.0x10�8 0.129759 �1.19703 �8.46x104 �71.0352

C3H8 7.145x10�4 0.0 0.0 0.12348 1.669x104 0.0

i-C4H10 1.251x10�3 1.0x10�8 0.166097 �2.75945 0.0 0.0

n-C4H10 0.0 9.37x10�6 �1.07657 0.0 0.0 �66.221

N2 0.0 2.5x10�7 0.0 0.0 0.0 0.0

CO2 �6.992x10�3 2.89x10�6 �6.223x10�3 0.0 0.0 0.0

H2S 0.0 0.0 0.240869 �0.64405 0.0 0.0

A12 A13 A14 A15 A16 A17

CH4 0.0 �5.3569 0.0 �2.3x10�7 �2.0x10�8 0.0

C2H6 0.596404 �4.7437 7.82x104 0.0 0.0 0.0

C3H8 0.23319 0.0 �4.48x104 5.5x10�6 0.0 0.0

i-C4H10 0.0 0.0 �8.84x102 0.0 �5.7x10�7 �1.0x10�8

n-C4H10 0.0 0.0 9.17x105 0.0 4.98x10�6 �1.26x10�6

N2 0.0 0.0 5.87x105 0.0 1.0x10�8 1.1x10�7

CO2 0.27098 0.0 0.0 8.82x10�5 2.55x10�6 0.0

H2S 0.0 �12.704 0.0 �1.3x10�6 0.0 0.0

TABLE 9.6—COEFFICIENTS FOR EQ. 9.35AS FUNCTIONS OF PRESSURE

Hydrate-Former C Values

Pressure

(psia) C1 C2 C3 i-C4 n-C4

6,000 18,933 20,806 28,382 30,696 17,340

7,000 19,096 20,848 28,709 30,913 17,358

8,000 19,246 20,932 28,764 30,935 17,491

9,000 19,367 21,094 29,182 31,109 17,868

10,000 19,489 21,105 29,200 30,935 17,868

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20 PHASE BEHAVIOR

Fig. 9.33—Hydrate-formation conditions for CO2/water systems(adapted from Ref. 69).

39. Panagiotopoulos, A.Z. and Reid, R.C.: “New Mixing Rule for CubicEquations of State for Highly Polar, Asymmetric Systems,” Equationsof State: Theories and Applications, K.C. Chao and R.L. Robinson(eds.), ACS Symposium Series (1986) 571.

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Fig. 9.34—General effect of methanol added to water/ethanesystem (adapted from Ref. 71).

�T�

�T

59. Parrish, W.R. and Prausnitz, J.M.: “Dissociation Pressures of Gas Hy-drates Formed by Gas Mixtures,” Ind. Eng. Chem. Proc. Des. Dev.(1972) 11, No. 1, 26.

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61. Schroeter, J.P., Kobayashi, R., and Hildebrand, M.A.: “Hydrate Decom-position Conditions in the System H2S-Methane-Propane,” Ind. Eng.Chem. Fund. (1983) 22, 361.

62. Ericksen and Sloan, E.D.: “Calculation Procedure Using vdW-Plat-teeuw Model,” Clathrate Hydrates of Natural Gas, Marcel Dekker, NewYork City (1990).

63. Technical Data Book—Petroleum Refining, third edition, API, New YorkCity (1977).

64. Ng, H.-J., Chen, C.-J., and Saeterstad, T.: “Hydrate Formation and In-hibition in Gas Condensate and Hydrocarbon Liquid Systems,” FluidPhase Equilibria (1987) 36, 99.

65. Robinson, D.B. and Mehta, B.R.: “Hydrates in the Propane-CarbonDioxide-Water System,” J. Cdn. Pet. Tech. (January–March 1971) 33.

66. Starling, K.E. and Powers, J.E.: “Enthalpy of Mixtures by ModifiedBWR Equations,” Ind. & Eng. Chem. Fund. (1970) 9, 531.

67. Munck, J., Skjold-J¢rgensen, S., and Rasmussen, P.: “Computations of theFormation of Gas Hydrates,” Chem. Eng. Sci. (1988) 43, No. 10, 2661.

68. Michelsen, M.L.: “The Isothermal Flash Problem. Part I. Stability,”Fluid Phase Equilibria (1982) 9, 1.

69. Song, K.Y. and Kobayashi, R.: “Water Content of CO2 in EquilibriumWith Liquid Water and/or Hydrates,” SPEFE (December 1987) 500;Trans., AIME, 283.

70. Hammerschmidt, E.G.: “Formation of Gas Hydrates in Natural GasTransmission Lines,” Ind. & Eng. Chem. (August 1934) 26, No. 8, 851.

71. Nielsen, R.B. and Bucklin, R.W.: “Why Not Use Methanol for HydrateControl?” Hydro. Proc. (April 1983) 71.

�� #�� $�� %��

bar �1.0* E�05�Pabbl �1.589 873 E�01�m3

cp �1.0* E�03�Pa�sdyne/cm �1.0* E�00�mN/m

ft3 �2.831 685 E�02�m3

�F (�F�32)/1.8 ��C�F (�F�459.67)/1.8 �K

g mol �1.0* E�03�kmollbm �4.535 924 E�01�kgpsi �6.894 757 E�00�kPa

psi�1�1.450 377 E�01�kPa�1


PROPERTY TABLES AND UNITS 1

��

��

TABLE A-1A—COMPONENT PROPERTIES FOR CUSTOMARY UNITS

MolecularWeight Specific

LiquidDensity Critical Constants Acentric

NormalBoilingPoint

IdealLiquidYield

GrossHeatingValue

CompoundM

(lbm/lbm mol)Gravity*�� (lbm/ft3)

�sc pc

(psia)Tc

(°R)vc

(ft3/lbm mol) Zc

Factor��

Tb

(°R)L

(gal/Mscf)H

(Btu/scf)

Nitrogen N2 28.02 0.4700 29.31 493.0 227.3 1.443 0.2916 0.0450 139.3

Carbon dioxide CO2 44.01 0.5000 31.18 1,070.6 547.6 1.505 0.2742 0.2310 350.4

Hydrogen sulfide H2S 34.08 0.5000 31.18 1,306.0 672.4 1.564 0.2831 0.1000 383.1 672

Methane C1 16.04 0.3300 20.58 667.8 343.0 1.590 0.2884 0.0115 201.0 1,012

Ethane C2 30.07 0.4500 28.06 707.8 549.8 2.370 0.2843 0.0908 332.2 1,783

Propane C3 44.09 0.5077 31.66 616.3 665.7 3.250 0.2804 0.1454 416.0 27.4 2,557

iso-butane i-C4 58.12 0.5613 35.01 529.1 734.7 4.208 0.2824 0.1756 470.6 32.7 3,354

Butane n-C4 58.12 0.5844 36.45 550.7 765.3 4.080 0.2736 0.1928 490.8 31.4 3,369

iso-pentane i-C5 72.15 0.6274 39.13 490.4 828.8 4.899 0.2701 0.2273 541.8 36.3 4,001

Pentane n-C5 72.15 0.6301 39.30 488.6 845.4 4.870 0.2623 0.2510 556.6 36.2 4,009

Hexane n-C6 86.17 0.6604 41.19 436.9 913.4 5.929 0.2643 0.2957 615.4 41.2 4,756

Heptane n-C7 100.20 0.6828 42.58 396.8 972.5 6.924 0.2633 0.3506 668.8 46.3 5,503

Octane n-C8 114.20 0.7086 44.19 360.6 1,023.9 7.882 0.2587 0.3978 717.9 50.9 6,250

Nonane n-C9 128.30 0.7271 45.35 332.0 1,070.3 8.773 0.2536 0.4437 763.1 55.7 6,996

Decane n-C10 142.30 0.7324 45.68 304.0 1,111.8 9.661 0.2462 0.4902 805.2 61.4 7,743

Air 28.97 0.4700 29.31 547.0 239.0 1.364 0.2910 0.0400 141.9

Water H2O 18.02 1.0000 62.37 3,206.0 1,165.0 0.916 0.2350 0.3440 671.6

Oxygen O2 32.00 0.5000 31.18 732.0 278.0 1.174 0.2880 0.0250 162.2

*Water�1.

2 PHASE BEHAVIOR

TABLE A-1B—COMPONENT PROPERTIES IN SI METRIC UNITS

MolecularWeight Specific

LiquidDensity Critical Constants Acentric

NormalBoilingPoint

IdealLiquidYield

GrossHeatingValue

CompoundM

(kg/kmol)Gravity*�� (kg/m3)

�sc pc

(kPa)Tc

(K)vc

(m3/kmol) Zc

Factor��

Tb

(K)L

(m3/1000 m3)H

(MJ/std m3)

Nitrogen N2 28.02 0.4700 469.5 3 399 126.3 0.0901 0.2916 0.0450 77.39

Carbon dioxide CO2 44.01 0.5000 499.5 7 382 304.2 0.0940 0.2742 0.2310 194.67

Hydrogen sulfide H2S 34.08 0.5000 499.5 9 005 373.6 0.0976 0.2831 0.1000 212.83 25.04

Methane C1 16.04 0.3300 329.7 4 604 190.6 0.0993 0.2884 0.0115 111.67 37.71

Ethane C2 30.07 0.4500 449.6 4 880 305.4 0.1479 0.2843 0.0908 184.56 66.43

Propane C3 44.09 0.5077 507.2 4 249 369.8 0.2029 0.2804 0.1454 231.11 3.67 95.27

iso-butane i-C4 58.12 0.5613 560.7 3 648 408.2 0.2627 0.2824 0.1756 261.44 4.37 125.0

Butane n-C4 58.12 0.5844 583.8 3 797 425.2 0.2547 0.2736 0.1928 272.67 4.20 125.5

iso-pentane i-C5 72.15 0.6274 626.8 3 381 460.4 0.3058 0.2701 0.2273 301.00 4.86 149.1

Pentane n-C5 72.15 0.6301 629.5 3 369 469.7 0.3040 0.2623 0.2510 309.22 4.83 149.4

Hexane n-C6 86.17 0.6604 659.7 3 012 507.4 0.3701 0.2643 0.2957 341.89 5.51 177.2

Heptane n-C7 100.20 0.6828 682.1 2 736 540.3 0.4322 0.2633 0.3506 371.56 6.20 205.0

Octane n-C8 114.20 0.7086 707.9 2 486 568.8 0.4920 0.2587 0.3978 398.83 6.80 232.9

Nonane n-C9 128.30 0.7271 726.4 2 289 594.6 0.5477 0.2536 0.4437 423.94 7.45 260.7

Decane n-C10 142.30 0.7324 731.7 2 096 617.7 0.6031 0.2462 0.4902 447.33 8.20 288.5

Air 28.97 0.4700 469.5 3 771 132.8 0.0852 0.2910 0.0400 78.83

Water H2O 18.02 1.0000 999.0 22 105 647.2 0.0572 0.2350 0.3440 373.11

Oxygen O2 32.00 0.5000 499.5 5 047 154.4 0.0733 0.2880 0.0250 90.11

*Water�1.


TABLE A-2—UNIVERSAL GAS CONSTANT FOR DIFFERENT UNITS

Pressure

Unit

Volume

Unit

Temperature

Unit

Mass (mole)

Unit

Gas Constant

R

psia ft3 °R lbm 10.7315

psia cm3 °R lbm 303,880

psia cm3 °R g 669.94

bar ft3 °R lbm 0.73991

atm ft3 °R lbm 0.73023

atm cm3 °R g 45.586

Pa m3 K kg 8314.3

Pa m3 K g 8.3143

kPa m3 K kg 8.3143

kPa cm3 K g 8314.3

bar m3 K kg 0.083143

bar cm3 K g 83.143

atm m3 K kg 0.082055

atm cm3 K g 82.055

Energy Unit

Btu °R lbm 1.9858

Btu °R g 0.0043780

calorie °R lbm 500.76

calorie °R g 1.1040

kcal °R lbm 0.50076

kcal °R g 0.0011040

calorie K kg 1985.8

calorie K g 1.9858

erg K kg 8.3143�1010

erg K g 8.3143�107

J K kg 8314.3

J K g 8.3143

TABLE A-3—RECOMMENDED BIP’s

FOR PR EOS AND SRK EOS FOR

NONHYDROCARBON/HYDROCARBON COMPONENT PAIRS

PR EOS* SRK EOS**

N2 CO2 H2S N2 CO2 H2S

N2 — — — — — —

CO2 0.000 — — 0.000 — —

H2S 0.130 0.135 — 0.120† 0.120 —

C1 0.025 0.105 0.070 0.020 0.120 0.080

C2 0.010 0.130 0.085 0.060 0.150 0.070

C3 0.090 0.125 0.080 0.080 0.150 0.070

i-C4 0.095 0.120 0.075 0.080 0.150 0.060

C4 0.095 0.115 0.075 0.080 0.150 0.060

i-C5 0.100 0.115 0.070 0.080 0.150 0.060

C5 0.110 0.115 0.070 0.080 0.150 0.060

C6 0.110 0.115 0.055 0.080 0.150 0.050

C7+ 0.110 0.115 0.050‡ 0.080 0.150 0.030‡

*Nonhydrocarbon BIP’s from Ref. 1.**Nonhydrocarbon BIP’s from Ref. 2.†Not reported in Ref. 2.‡Should decrease gradually with increasing carbon number.

BIP�binary interaction parameter, PR EOS�Peng-Robinson equation of state, and

SRK EOS�Soave-Redlich-Kwong equation of state.

4 PHASE BEHAVIOR

TABLE A-4—FORTRAN PROGRAM FOR CALCULATING SPLIT OF C7+ WITH GAMMA DISTRIBUTION

CC–––– PROGRAM GAMSPLC

IMPLICIT DOUBLE PRECISION (A�H,O�Z)DOUBLE PRECISION MWBL,MWBU,MWAV,MW7POPEN(10,FILE�’GAMSPL.OUT’)WRITE(*,*) ’Input ALFA, ETA, M7+ >’READ (*,*) ALFA,ETA,MW7PBETA�(MW7P�ETA)/ALFAMWBU�ETAS1�0.0S2�0.0WRITE(10,2000) ALFA,ETA,MW7PDO 100 I�1,20

MWBL�MWBU

MWBU�MWBL�14.0IF (I.EQ.20) MWBU�10000.0CALL P0P1(ALFA,ETA,BETA,MWBL,P0L,P1L)CALL P0P1(ALFA,ETA,BETA,MWBU,P0U,P1U)Z�P0U�P0L

S1�S1�ZMWAV�ETA+ALFA*BETA*(P1U�P1L)/(P0U�P0L)

S2�S2�Z*MWAVWRITE(10,2100) I,Z,MWAV

100 CONTINUEWRITE(10,2200) S1,S2/S1

2000 FORMAT (/. ’ ALFA ........ :’,F10.3/. ’ ETA ......... :’,F10.3/. ’ MW7P ........ :’,F10.3/. ’ –––––––––––––––––––––––––––––’/. ’ Frac. Mole Molecular ’/. ’ No. Fraction Weight ’/. ’ ––––– –––––––––– –––––––––– ’)

2100 FORMAT (1X,I3,3X,F10.7,2X,F10.3)2200 FORMAT (’ ––––––––– ––––––– ’/7X,F10.7,2X,F10.3)

ENDSUBROUTINE P0P1 (ALFA,ETA,BETA,MWB,P0,P1)IMPLICIT DOUBLE PRECISION (A�H,O�Z)DOUBLE PRECISION MWBP0�0.0P1�0.0IF (MWB.LE.ETA) RETURNY�(MWB�ETA)/BETAQ�DEXP(�Y)*Y**ALFA/GAMA(ALFA)TERM�1.0/ALFAS�TERMDO 100 J�1,10000

TERM�TERM*Y/(ALFA�DFLOAT(J))

S�S�TERMIF (DABS(TERM).LE.1.0D�8) GOTO 200

100 CONTINUEWRITE (*,2000)

200 CONTINUEP0�Q*SP1�Q*(S�1.0/ALFA)

2000 FORMAT (1X,’*** PR : SUM DOES NOT CONVERGE’)RETURNEND

DOUBLE PRECISION FUNCTION GAMA (X)IMPLICIT DOUBLE PRECISION(A�H,O�Z)DIMENSION B(8)DATA B /�0.577191652, 0.988205891,�0.897056937,

. 0.918206857,�0.756704078, 0.482199394,

. �0.193527818, 0.035868343 /CONST�1.0XX�X

IF (X.LT.1.0) XX�X�1.0100 IF (XX.LE.2.0) GOTO 200

XX�XX�1.0CONST�XX*CONSTGOTO 100

200 XX�XX�1.0Y�1.0DO 300 I�1,8

Y�Y�B(I)*XX**I300 CONTINUE

GAMA�CONST*YIF (X.LT.1.0) GAMA�GAMA/XRETURNEND


TABLE A-5—GREEK ALPHABET

Upper Case Lower Case Name

� � Alpha

� � Beta

� � Gamma

� � Delta

� Epsilon

� Zeta

� � Eta

� � Theta

Iota

� Kappa

� � Lambda

� � Mu

� Nu

� � Xi

� � Omicron

� � Pi

� � Rho

� � Sigma

� � Tau

� � Upsilon

� � Phi

� � Chi

� � Psi

� � Omega

TABLE A-6—SI SYSTEM UNITS

Base SI Units Used in Phase Behavior

Quantity Unit Symbol

Length meter m

Time second s

Mass kilogram kg

Temperature kelvin K

Amount of substance mole mol

Quantity Unit Symbol Definition SI Term

Mass tonne Mg 1 Mg�103 kg Mg

Volume liter L 1 L�1 dm3 dm3

TABLE A-7—SI PREFIXES

Multiplication Factor Prefix Symbol*

1012 Tera T

109 Giga G

106 Mega M

103 Kilo k

102 Hecto h

10 Deka da

10�1 Deci d

10�2 Centi c

10�3 Milli m

10�6 Micro �

10�9 Nano n

10�12 Pico p

10�15 Femto f

10�18 Atto a*Only the symbols T (tera), G (giga), and M (mega) are capital letters. Compound

prefixes are not allowed; e.g., use nm (nanometer) rather than m�m (millimicrometer).

6 PHASE BEHAVIOR

TABLE A-8—PHYSICAL CONSTANTS AND VALUES (from Ref. 3)

Triple point of water 273.16 exactly K*

0.01 exactly °C491.688 exactly °R32.018 exactly °F

Absolute zero 0.00 exactly K*

�273.15 exactly °C0.00 exactly °R

�459.67 exactly °FGas constant, R 8.3143 J�mol�1

�K�1*

10.731 5 psia�ft3�(lbm-mol)�1�°R�1

Density of water at 60°F 999.014 kg�m�3*

[15.56°C, 288.71 K] 0.999 014 g�cm�3

62.366 4 lbm�ft�3

Standard atmosphere 1.013 2�105 Pa*

1.013 25 bar

14.696 0 psia

Density of air at 1 atm, 60°F 1.223 2 kg�m�3*

[15.56°C, 288.71 K] 1.223 2�10�3 g�cm�3

0.076 362 lbm�ft�3

Earth’s gravitational acceleration, g 9.806 650 m�s�2*

980.665 0 cm�s–2

32.174 05 ft�s�2

gc 1.000 000 kg�m�N�1�s�2*

1.000 000 g�cm�dyne�1�s�2

32.174 05 lbm�ft�lbf�1�s�2

� 3.141 593 …�API, °API [141.5/�(60°F)]�131.5

*SI values. All quantities are consistent with conversion factors for the current SI system.

TABLE A-9—TEMPERATURE SCALE CONVERSIONS (from Ref. 3)

To Convert To Solve

degree Fahrenheit, TF kelvin, TK TK = (TF + 459.67)/1.8

degree Rankine, TR kelvin, TK TK = TR /1.8

degree Fahrenheit, TF degree Rankine, TR TR = TF + 459.67

degree Fahrenheit, TF degree Celsius, TC TC = (TF�32)/1.8

degree Celsius, TC kelvin, TK TK = TC + 273.15The SI standard, the kelvin (K), is defined so that the triple point of water is 273.16 K exactly. The SI temperature symbol is written

K, without a degree symbol. The cgs (and common) temperature unit is degree Celsius, °C; the common oilfield unit is degreeFahrenheit, °F, or degree Rankine, °R.


TABLE A-10—CONVERSION FACTORS USEFUL IN PHASE BEHAVIOR (from Ref. 3)

To Convert From To Multiply By Inverse

Area

acre (acre) square meter (m2)*square foot (ft2)

4.046 856 E + 034.356 000** E + 04

2.471 054 E – 042.295 684 E – 05

darcy (darcy) square meter (m2)*square centimeter (cm2)square micrometer (�m2)millidarcy (md)cm2-cp�sec�1

�atm�1

9.869 23 E – 139.869 23 E – 099.869 23 E – 011.000 000** E + 031.000 000** E + 00

1.013 25 E + 121.013 25 E + 081.013 25 E + 001.000 000** E – 031.000 000** E + 00

square foot (ft2) square meter (m2)*

square centimeter (cm2)square inch (in.2)

9.290 304** E – 029.290 304** E + 021.440 000** E + 02

1.076 391 E + 011.076 391 E – 036.944 444 E – 03

hectare (ha) square meter (m2)*acre

1.000 000** E + 042.471 054 E + 00

1.000 000** E – 044.046 856 E – 01

square mile (sq mile) square meter (m2)*acre

2.589 988 E + 066.400 000** E + 02

3.861 022 E – 071.562 500** E – 03

Density

gram per cubic centimeter (g/cm3) kilogram/cubic meter (kg/m3)*pound-mass/cubic foot (lbm/ft3)pound-mass/gallon (lbm/gal)pound-mass/barrel (lbm/bbl)

1.000 000** E + 036.242 797 E + 018.345 405 E + 003.505 070 E + 02

1.000 000** E – 031.601 846 E – 021.198 264 E – 012.853 010 E – 03

pound-mass per cubic foot (lbm/ft3) kilogram/cubic meter (kg/m3)*pound-mass/gallon (lbm/gal)pound-mass/barrel (lbm/bbl)

1.601 846 E + 011.336 805 E – 015.614 583 E + 00

6.242 797 E – 027.480 520 E + 001.781 076 E – 01

pound-mass per gallon (lbm/gal) kilogram/cubic meter (kg/m3)*pound-mass/barrel (lbm/bbl)

1.198 264 E + 024.200 000 E + 01

8.345 406 E – 032.380 952 E – 02

Force

dyne (dyne) newton (N)*pound-force (lbf)

1.000 000** E – 052.248 089 E – 06

1.000 000** E + 054.448 222 E + 05

kilogram-force (kgf) newton (N)*pound-force (lbf)

9.806 650** E + 002.204 622 E + 00

1.019 716 E – 014.535 924 E – 01

pound-force (lbf) newton (N)* 4.448 222 E + 00 2.248 089 E – 01

Length

angstrom (Å) meter (m)* 1.000 000** E – 10 1.000 000** E + 10

centimeter (cm) meter (m)* 1.000 000** E – 02 1.000 000** E + 02

foot (ft) meter (m)*centimeter (cm)

3.048 000** E – 013.048 000** E + 01

3.280 840 E + 003.280 840 E – 02

inch (in.) meter (m)*centimeter (cm)

2.540 000** E – 022.540 000** E + 00

3.937 008 E + 013.937 008 E – 01

micron (�m) meter (m)* 1.000 000** E – 06 1.000 000** E + 06

mile (U.S. statute) meter (m)*foot

1.609 344** E + 035.280 000** E + 03

6.213 712 E – 041.893 939 E – 04

*SI conversions. All quantities are current to SI standards as of 1974.**Conversion factor is exact and all following digits are zero. All other factors have been rounded.

The notation E + 03 is used in place of 103, and so on.

8 PHASE BEHAVIOR

TABLE A-10 (continued)—CONVERSION FACTORS USEFUL IN PHASE BEHAVIOR (from Ref. 3)


Mass

gram-mass kilogram (kg)* 1.000 000** E – 03 1.000 000** E + 03

ounce-mass (avoirdupois) kilogram (kg)*gram (g)

2.834 952 E – 022.834 952 E + 01

3.527 397 E + 013.527 397 E – 02

pound-mass kilogram (kg)*ounce-mass

4.535 923 7** E – 011.600 000** E + 01

2.204 623 E + 006.250 000** E – 02

slug kilogram (kg)*pound-mass (lbm)

1.459 390 E + 013.217 405 E + 01

6.852 178 E – 023.108 095 E – 02

ton (U.S. short) kilogram (kg)*pound-mass (lbm)

9.071 847 E + 022.000 000** E + 03

1.102 311 E – 035.000 000** E – 04

ton (U.S. long) kilogram (kg)*pound-mass (lbm)

1.016 047 E + 032.240 000** E + 03

9.842 064 E – 044.464 286 E – 04

ton (metric) kilogram (kg)* 1.000 000** E + 03 1.000 000** E – 03

tonne kilogram (kg)* 1.000 000** E + 03 1.000 000** E – 03

Pressure

atmosphere (atm)(Normal is 760 mm Hg)

pascal (Pa)*mm Hg (0°C)feet water (4°C)pound-force/square inch (psi)bar

1.013 25 E + 057.600 000** E + 023.389 95 E + 011.469 60 E + 011.013 25 E + 00

9.869 23 E – 061.315 789 E – 032.949 90 E – 026.804 60 E – 029.869 23 E – 01

bar (bar) pascal (Pa)*pound-force/square inch (psi)

1.000 000** E + 051.450 377 E + 01

1.000 000** E – 056.894 757 E – 02

centimeter of Hg (32°F) pascal (Pa)*pound-force/square inch (psi)

1.333 22 E + 031.933 67 E – 01

7.500 64 E – 045.171 51 E + 00

dyne/square centimeter (dyne/cm2) pascal (Pa)*pound force/square inch (psi)

1.000 000** E – 011.450 377 E – 05

1.000 000** E + 016.894 757 E + 04

feet of water (39.2°F) pascal (Pa)*pound force/square inch (psi)

2.988 98 E + 034.335 15 E – 01

3.345 62 E – 042.306 73 E + 00

kilogram-force/square centimeter pascal (Pa)*barpound force/square inch (psi)

9.806 650** E + 049.806 650** E – 011.422 334 E + 01

1.019 716 E – 051.019 716 E + 007.030 695 E – 02

pound-force/inch2 (psi) pascal (Pa)* 6.894 757 E + 03 1.450 377 E – 04

Time

day (d) second (s)*minute (min)hour (h)

8.640 000** E + 041.440 000** E + 032.400 000** E + 01

1.157 407 E – 056.944 444 E – 044.166 667 E – 02

hour (h) second (s)*minute (min)

3.600 000** E + 036.000 000** E + 01

2.777 778 E – 041.666 667 E – 02

minute (min) second (s)* 6.000 000** E + 01 1.666 667 E – 02




TABLE A-10 (continued)—CONVERSION FACTORS USEFUL IN PHASE BEHAVIOR (from Ref. 3)


Viscosity

centipoise (cp) pascal-second (Pa�s)*dyne-second/square centimeter

(dyne-s/cm2)pound-mass/foot-second (lbm/ft-sec)pound-force-second/square foot

(lbf-sec/ft2)pound-mass/foot-hour (lbm/ft-hr)

1.000 000** E – 031.000 000** E – 026.719 689 E – 042.088 543 E – 052.419 088 E + 00

1.000 000** E + 031.000 000** E + 021.488 164 E + 034.788 026 E + 044.133 789 E– 01

centistoke (cSt) square meter/second (m2/s)*centipoise/gram-cubic centimeter

(cp/g-cm3)

1.000 000** E – 061.000 000** E + 00

1.000 000** E + 061.000 000** E + 00

poise pascal-second (Pa�s)* 1.000 000** E – 01 1.000 000** E + 01

pound-mass/foot-second (lbm/ft-sec) pascal-second (Pa�s)* 1.488 164 E + 00 6.719 689 E– 01

pound-mass/foot-hour (lbm/ft-hr) pascal-second (Pa�s)* 4.133 789 E – 04 2.419 088 E + 03

pound-force-second/square foot(lbf-sec/ft2)

pascal-second (Pa�s)* 4.788 026 E + 01 2.088 543 E– 02

Volume

acre-foot (acre-ft) cubic meter (m3)*cubic foot (ft3)barrel (bbl)

1.233 482 E + 034.356 000** E + 047.758 368 E + 03

8.107 131 E– 042.295 684 E– 051.288 931 E– 04

barrel (bbl) cubic meter (m3)*cubic foot (ft3)gallon (gal)

1.589 873 E – 015.614 583 E + 004.200 000** E + 01

6.289 811 E + 001.781 076 E– 012.380 952 E– 02

cubic foot (ft3) cubic meter (m3)*cubic inch (in.3)gallon (gal)

2.831 685 E – 021.728 000 E + 037.480 520 E + 00

3.531 466 E + 015.787 037 E– 041.336 805 E– 01

gallon (gal) cubic meter (m3)*cubic inch (in.3)

3.785 412 E – 032.310 001 E + 02

2.641 720 E + 024.329 003 E – 03

liter (L) cubic meter (m3)* 1.000 000** E – 03 1.000 000** E + 03

Volumetric rate

barrel/day (B/D) cubic meter/second (m3/s)*cubic meter/hour (m3/h)cubic meter/day (m3/d)cubic centimeter/second (cm3/s)cubic foot/minute (ft3/min)gallon/minute (gal/min)

1.840 131 E – 066.624 472 E – 031.589 873 E – 011.840 131 E + 003.899 016 E – 032.916 667 E – 02

5.434 396 E + 051.509 554 E + 026.289 810 E + 005.434 396 E – 012.564 750 E + 023.428 571 E + 01

cubic foot/minute (ft3/min) cubic meter/second (m3/s)* 4.719 474 E – 04 2.118 880 E + 03

cubic foot/second (ft3/sec) cubic meter/second (m3/s)* 2.831 685 E – 02 3.531 466 E + 01

gallon/minute (gal/min) cubic meter/second (m3/s)* 6.309 020 E – 05 1.585 032 E + 04



10 PHASE BEHAVIOR

TABLE A-11—ADDITIONAL CONVERSION FACTORS USEFUL IN PHASE BEHAVIOR


Amount of substance

mole (mol) pound-mass mole (lbm mol) 2.204 623 E + 03 4.535 923 E – 04

gram mole (gmol) 1.000 000* E + 00 1.000 000* E + 00

kilomole (kmol) 1.000 000* E – 03 1.000 000* E + 03

kilomole (kmol) mole (gmol) 1.000 000* E + 03 1.000 000* E – 03

gram mole (gmol) 1.000 000* E + 03 1.000 000* E – 03

pound-mass mole (lbm mol) 4.535 923 E – 01 2.204 623 E + 00

Diffusivity

square centimeter/second (cm2/s) square meter/second (m2/s) 1.000 000* E – 04 1.000 000* E + 04

square millimeter/second (mm2/s) 1.000 000* E + 02 1.000 000* E – 02

square foot/second (ft2/sec) 1.076 390 E – 03 9.290 304 E + 02

square foot/hour (ft2/hr) 3.875 000 E + 00 2.580 640 E – 01

Surface tension

milliNewton/meter (mN/m) dyne/centimeter (dyne/cm) 1.000 000* E + 00 1.000 000* E + 00

Energy

British thermal unit (Btu) kiloJoule (kJ) 1.055 056 E + 00 9.478 160 E – 01

calorie (cal) 2.521 640 E + 02 3.965 660 E – 03

kilocalorie (kcal) 2.521 640 E – 01 3.965 660 E + 00

erg 1.055 056 E + 10 9.478 160 E – 11*Conversion factor is exact.

��

1. Nagy, Z. and Shirkovskiy, A.I.: “Mathematical Simulation of Natural GasCondensation Processes Using the Peng-Robinson Equation of State,”paper SPE 10982 presented at the 1982 SPE Annual Technical Confer-ence and Exhibition, New Orleans, 26–29 September.

2. Reid, R.C., Prausnitz, J.M., and Polling, B.E.: The Properties of Gasesand Liquids, fourth edition, McGraw-Hill Book Co. Inc., New York City(1987).

3. Earlougher, R.C. Jr.: Advances in Well Test Analysis, SPE Monograph Se-ries, SPE, Richardson, Texas (1977) 5.

EXAMPLE PROBLEMS 1

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��

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Many of the problems presented here were introduced by Standingduring his 2 years as visiting professor at the Norwegian Inst. ofTechnology in Trondheim during 1973–74. Some of the problemshave been modified or expanded, and additional problems havebeen included to cover subjects presented in the monograph thatwere not necessarily covered in Standing’s problems.

��

Problem. A light-hydrocarbon gas has the compositional analysisgiven in Table B-1.

Calculate the following properties.a. Weight composition.b. Molecular weight.c. Specific gravity.d. Density in lbm/ft3 at 20 psia and 120°F, assuming ideal gas be-

havior.e. Density in kg/m3 at 3.1 atm and 50°C, assuming ideal gas be-

havior.

Solution. The problem is solved by calculating mass, mi�xiMi, andmass (weight) fractions, as shown in Table B-2. The followingequations have been used.

wi �mi

�Nj�1

mj

�ni Mi

�Nj�1

nj Mj

; (3.3). . . . . . . . . . . . . . . . . . . . . . .

�g ��g�sc

��air�sc

�Mg

Mair�

Mg

28.97

and Mg � 28.97 �g ; (3.28). . . . . . . . . . . . . . . . . . . . . . . . . . . .

�g � pMg�ZRT ; (3.34). . . . . . . . . . . . . . . . . . . . . . . . . . . . .

and M ��Ni�1

yi Mi , (3.50a). . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Tpc ��Ni�1

yi Tci , (3.50b). . . . . . . . . . . . . . . . . . . . . . . . . . . .

and ppc ��Ni�1

yi pci . (3.50c). . . . . . . . . . . . . . . . . . . . . . . . . . . .

a. Weight composition is given as wi in Eq. 3.3.b. The ratio of total mass, �mi, to total moles, �yi, gives the aver-

age molecular weight.

Mg � (24.97)�(1.00) � 24.97 lbm�lbm mol.

c. Gas specific gravity is given by

�g � (24.97)�(28.97) � 0.864 (air � 1).

d. Gas density is calculated with Eq. 3.35.

�g � (20)(24.97)�[(1)(10.732)(120 � 460)]

� 0.0801 lbm�ft3.

e. Gas density in SI units is also calculated with Eq. 3.35 with thecorrect gas constant, R, from Table A-2 in Appendix A.

�g � (3.1)(24.97)�[(1)(0.082055)(50 � 273)] � 2.92 kg�m3.

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Problem. Table B-3 gives the compositional analysis of a relative-ly-high-sulfur-content Canadian gas. If the gas-processing plantthat treats the gas removes 100% of the H2S and converts it to ele-mental sulfur, how many long tons (2,200 lbm) of sulfur will resultfrom processing 1,000 Mscf of field gas?

Solution. Total mass in lbm mol of 1,000 Mscf gas is calculatedfrom the real gas law,

pV � nZRT. (3.30). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

TABLE B-1—GAS COMPOSITIONAL ANALYSIS(PROBLEM 1)

Component Mole Fraction

Methane 0.49

Ethane 0.38

Propane 0.13

2 PHASE BEHAVIOR

TABLE B-2—MASS AND MASS (WEIGHT) FRACTIONS (PROBLEM 1)

Component i

Molecular WeightMi

(lbm/lbm mol)Mole Fraction

xi

Massmi =xiMi

(lbm)Weight Fraction

wi =mi/(�mj)

C1 16.04 0.49 7.84 0.314

C2 30.07 0.38 11.40 0.456

C3 44.09 0.13 5.73 0.230

Total 1.00 24.97 1.000

TABLE B-3—GAS COMPOSITIONAL ANALYSIS(PROBLEM 2)

Component iMole Fraction

zi

CO2 0.0112

H2S 0.2609

C1 0.5575

C2 0.0760

C3 0.0433

i-C4 0.0061

n-C4 0.0137

i-C5 0.0033

n-C5 0.0052

C6 0.0053

C7+ 0.0175

MC7�� 128 and �C7�

� 0.780.

Note: Canadian standard pressure base is 14.65 psia. Assume that Z=1 at standardconditions.

Solving for n,

n � pV�ZRT � (14.65)�1 � 106��[(1.0)(10.73)(60 � 460)]

� 2, 625 lbm mol.

Moles of H2S is calculated by multiplying the total moles by themole fraction of H2S.

nH2S � (0.2609)(2, 625) � 685 lbm mol.

There is one mole of sulfur (S) per mole of H2S, so

nS � (0.2609)(2, 625) � 685 lbm mol.

The mass of sulfur equals the moles of sulfur times the molecularweight of sulfur (MS�32),

mS � (685)(32)��2, 200 lbm�ton�

� 9.96 long tons�1, 000 Mscf produced gas.

TABLE B-4—LIQUID VOLUME COMPOSITION(PROBLEM 3)

Component iLiquid Volume Fraction

xVi

C3 0.05

n-C4 0.10

n-C5 0.15

n-C6 0.70

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Problem. At 15.56°C, a storage tank contains 1,000 m3 of gasolinewith the liquid volume composition given in Table B-4.

Calculate the following.a. Weight (mass) composition.b. Molar composition.c. Molecular weight.d. Specific gravity.e. Oil gravity (°API).f. Moles in kilogram moles (kmol) of nC6

in the tank.

g. Gallons of nC5 in the tank.

h. Pounds of nC4 in the tank.

Note: Use component properties from Appendix A and valuesfrom Table B-5.

Solution.a. Weight composition from Column 5, where wi�mi/(�mj).b. Mole composition from Column 8, where xi�ni/(�nj).c. Molecular weight from

M � (640.0 kg)�(8.248 kmol)

� 77.6 kg�kmol � 77.6 lbm�lbm mol.

d. Density from

�o � mo�Vo � (640.0 kg)��1.0 m3� � 640.0 kg�m3.

Specific gravity is calculated from �o � �o��w (Eq. 3.12), wheredensities are at standard conditions.

�o � �640.0 kg�m3��999.0 kg�m3� � 0.640 (water � 1).

TABLE B-5—COMPOSITION CONVERSIONS FOR MIXTURES (PROBLEM 3)

Column

1 2 3 4 5 6 7 8Liquid Volume Liquid Volume* Liquid Density Mass Weight Molecular Weight Moles Mole

Fraction Vi �i mi � �Vi Fraction Mi ni =mi /Mik FractionComponent i xVi (m3) (kg/m3) (kg) wi (kg/kmol) (kmol) xi

C3 0.05 0.05 507.2 25.36 0.040 44.09 0.575 0.070

n-C4 0.10 0.10 583.9 58.39 0.091 58.12 1.005 0.122

n-C5 0.15 0.15 629.5 94.43 0.148 72.15 1.309 0.159

n-C6 0.70 0.70 659.8 461.86 0.722 86.17 5.360 0.650

Total 1.00 640.04 1.000 8.248 1.000

*On the basis of 1 m3.

EXAMPLE PROBLEMS 3

TABLE B-6—SEPARATOR GAS AND SEPARATOR OILCOMPOSITIONS FOR WELLSTREAM RECOMBINATION

CALCULATION (PROBLEM 4)

Component iGas Mole Fraction

yi

Liquid Volume FractionxVi

C1 0.968 0.020

C2 0.010 0.006

C3 0.011 0.011

i-C4 0.003 0.009

n-C4 0.003 0.013

i-C5 0.002 0.016

n-C5 0.001 0.010

C6 0.002 0.038

C7+ 0.000 0.877

MC7�� 144 and �C7�

� 0.775.

e. �API � (141.5)�(0.640) � (131.5) � 89.4�API.

f. Moles of nC6� �1000 m3��5.36 kmol�m3� � 5360 kmol.

g. Volume of nC5� �1000 m3��0.15 m3�m3��6.289 bbl�m3�

� �42 gal�bbl� � 3.962 � 104 gal.

h. Mass of nC4� �1000 m3��58.39 kg�m3��2.205 lbm�kg�

� 1.2875 � 105 lbm.

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Problem. During a 24-hour test, a well produced 463 STB oil and5,783 Mscf of separator gas (these volumes are expressed at 14.4 psiaand 60°F). Table B-6 gives oil and gas compositions. Calculate thewell-effluent composition in mole fraction. Use apparent liquid den-sities for methane and ethane of 0.30 and 0.45 g/cm3, respectively.

Solution. From Eq. 3.18, the producing gas/oil ratio (GOR) is

Rp � qg�qo � �5.783 � 106��(463) � 12, 500 scf�STB,

or in terms of the producing oil/gas ratio (OGR) from Eq. 3.19,

rp � 1�Rp � �106 scf�MMscf��12, 500 scf�STB�

� 80 STB�MMscf.

On a basis of 1 STB, the moles of gas produced is given by solvingfor ng from the real gas law [ pV�nZRT (Eq. 3.30)], with Z�1,

ng � [(14.4)(12, 500)]�[(1.0)(10.73)(60 � 460)]

� 32.3 lbm mol.

Table B-7 calculates oil molar composition and recombined well-stream composition with 1 STB oil volume as a basis. Ideal solutionmixing is assumed for the stock-tank oil. Also, note that the compo-nent moles in the stock-tank oil are given by noi � 5.6146 Vi�i�Mi

(Eq. 3.4).

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Problem. A new well was completed with perforations in three sep-arate intervals. Initial pressure at midperforations (4,650 ft subsur-face) was 2,000 psig at 150°F. The first 24-hour production test gavethe information in Table B-8.

On the basis of these data, which of the following do you considerbest describes the well effluent.

a. Production of a single phase from a gas-condensate reservoir.b. Production of separate gas and liquid phases into the well.c. Production of undersaturated liquid into the well.Explain the basis for your decision.

Solution. The GOR of 19,000 might be descriptive of a gas-conden-sate system (Answer a). However, at the reservoir pressure of 2,000psi and 150°F, it would be unlikely that a 27°API liquid could dis-solve in the gas phase. The reservoir gas probably has been or cur-rently is in contact with a reservoir oil. At 2,000 psia, the K values(Ki�yi/xi) of the heavy components that make up a 27°API crudewould be extremely small (mostly 10�3) and the heaviest com-ponents would have the lowest K values. Even if the reservoir oilcontacting the reservoir gas is very heavy, the resulting amounts ofheavy components found in the equilibrium gas would be very smalland proportionally more of the lighter fractions would be found inthe reservoir gas. The condensate from such an equilibrium gaswould tend to have a lower gravity (e.g., �API50°API).

Answer c is also wrong because it is not possible to dissolve19,000 scf of gas in 1 STB of such a heavy crude oil.

Consequently, Answer b is the best answer. Both reservoir oil(with a gravity somewhat heavier than 27°API) and reservoir gas(with a much lighter condensate gravity) are both flowing into thewell simultaneously. Coning, leakage behind the casing, or multiplecompletion intervals are three situations that might cause the pro-duction characteristics seen in this well.

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Problem. Table B-9 gives the gas composition of the Sabine fieldin Texas. This is a typical composition of field gases produced fromprimary separators. Assuming that this gas is to be compressed andreinjected into a reservoir at 200°F, calculate the compressibilityfactor, Z; gas formation volume factor (FVF), Bg; and gas density,�g, at 2,000 psig and 160°F. Make the calculations using pseudo-critical properties calculated from the gas composition in Table B-9and from gas gravity.

TABLE B-7—OIL MOLAR COMPOSITION AND RECOMBINED WELLSTREAM COMPOSITION (PROBLEM 4)

Gas Gas Moles Oil Volume Liquid Density Molecular Weight Oil Moles Total Moles WellstreamComponent Mole Fraction ngi�ngyi Voi �i Mi noi ni�ngi +noi Mole Fraction

i yi (lbm mol) (STB) (lbm/ft3) (lbm/lbm mol) (lbm mol) (lbm mol) zi

C1 0.968 31.266 0.020 18.73 16.04 0.131 31.398 0.9123

C2 0.010 0.323 0.006 28.09 30.07 0.031 0.354 0.0103

C3 0.011 0.355 0.011 31.66 44.09 0.044 0.400 0.0116

i-C4 0.003 0.097 0.009 35.01 58.12 0.030 0.127 0.0037

C4 0.003 0.097 0.013 36.45 58.12 0.046 0.143 0.0041

i-C5 0.002 0.065 0.016 39.13 72.15 0.049 0.113 0.0033

C5 0.001 0.032 0.010 39.30 72.15 0.031 0.063 0.0018

C6 0.002 0.065 0.038 41.19 86.17 0.102 0.167 0.0048

C7+ 0.000 0.0000 0.877 48.33 144.00 1.653 1.653 0.0480

Total 1.000 1.000 2.117 34.417 1.0000Compostions are calculated on the basis of 1 STB oil volume.

4 PHASE BEHAVIOR

TABLE B-8—RESULTS OF FIRST 24-HOURPRODUCTION TEST (PROBLEM 5)

Oil produced, STB 65

Stock-tank-oil gravity, °API 27

Gas produced, MMscf 1.23

Gas/oil ratio, scf/bbl separator oil 19,000

Solution. Properties From Composition.

Mg � 28.97 �g , (3.28). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

M ��Ni�1

yi Mi , (3.50a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Tpc ��Ni�1

yi Tci , (3.50b). . . . . . . . . . . . . . . . . . . . . . . . . . . .

and ppc ��Ni�1

yi pci . (3.50c). . . . . . . . . . . . . . . . . . . . . . . . . . . .

With the pseudocritical properties in Table B-10, these equations give,

Tpc � 376�R,

ppc � 667 psia,

Mg � 18.83 (Kay�s mixing rule),

�g � (18.83)�(28.97) � 0.65 (air � 1),

Tpr � T�Tpc � (160 � 460)�376 � 1.65,

and ppr � p�ppc � 2, 015�667 � 3.02.

Gas Z factor is given by the Hall-Yarborough1,2 correlation.

Z � �ppr�y, (3.42). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

where � � 0.06125 t exp�� 1.2(1 � t)2 , where t � 1�Tpr. This

gives

t � 1�Tpr � 1�1.65 � 0.606,

� � (0.06125)(0.606) exp�(� 1.2)(1 � 0.606)2 � 0.0308,

y � 0.10996 �dF�dy � 0.79798�,

TABLE B-9—GAS COMPOSITION (PROBLEM 6)

ComponentMole Fraction

yi

C1 0.875

C2 0.083

C3 0.021

i-C4 0.006

n-C4 0.008

i-C5 0.003

n-C5 0.002

C6 0.001

C7+ 0.001

and Z � 0.846.

Gas density is given by

�g � pMg�ZRT , (3.34). . . . . . . . . . . . . . . . . . . . . . . . . . . . .

which yields

�g � (2, 015)(18.83)�[(0.846)(10.73)(160 � 460)]

� 6.74 lbm�ft3.

Properties From Specific Gravity Correlations.The Sutton3 correlations for pseudocritical properties are

TpcHC � 169.2 � 349.5�gHC � 74.0�2gHC (3.47a). . . . . . . . . .

and ppcHC � 756.8 � 131�gHC � 3.6�2gHC, (3.47b). . . . . . . . .

which give

Tpc � 169.2 � 349.5(0.65) � 74.0(0.65)2 � 365�R,

ppc � 756.8 � 131.0(0.65) � 3.6(0.65)2 � 670 psia ,

Tpr � T�Tpc � (160 � 460)�365 � 1.70,

ppr � p�ppc � 2, 015�670 � 3.01,

Z � 0.865,

and �g � 6.59 lbm�ft3.

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Problem. Calculate the viscosity of the Sabine field gas of Problem6 under reservoir conditions of 2,000 psig and 160°F. Use the Lu-cas4 and Lohrenz-Bray-Clark5 viscosity correlations based on gascomposition.

TABLE B-10—PSEUDOCRITICAL PROPERTIES (PROBLEM 6)

Component zi Mi

pci(psia)

Tci°R ziMi

zipci(psia)

ziTci°R

C1 0.8750 16.04 667.8 343.0 14.04 584.3 300.1

C2 0.0830 30.07 707.8 549.8 2.50 58.7 45.6

C3 0.0210 44.09 616.3 665.7 0.93 12.9 14.0

i-C4 0.0060 58.12 529.1 734.7 0.35 3.2 4.4

C4 0.0080 58.12 550.7 765.3 0.46 4.4 6.1

i-C5 0.0030 72.15 490.4 828.8 0.22 1.5 2.5

C5 0.0020 72.15 488.6 845.4 0.14 1.0 1.7

C6 0.0010 86.17 436.9 913.4 0.09 0.4 0.9

C7+* 0.0010 114.0 360.6 1,023.9 0.11 0.4 1.0

Total 1.0000 18.83 666.8 376.4

*Use properties for n-C8.

EXAMPLE PROBLEMS 5

TABLE B-11—LOHRENZ-BRAY-CLARK5 VISCOSITY CALCULATIONS (PROBLEM 7)

Component zi

vci(ft3/lbm mol) Zci

zivci(ft3/lbm mol) ziZci

C1 0.8750 1.590 0.2884 1.391 0.2524

C2 0.0830 2.370 0.2843 0.197 0.0236

C3 0.0210 3.250 0.2804 0.068 0.0059

i-C4 0.0060 4.208 0.2824 0.025 0.0017

C4 0.0080 4.080 0.2736 0.033 0.0022

i-C5 0.0030 4.899 0.2701 0.015 0.0008

C5 0.0020 4.870 0.2623 0.010 0.0005

C6 0.0010 5.929 0.2643 0.006 0.0003

C7+ 0.0010 7.882 0.2587 0.008 0.0003

Total 1.0000 1.752 0.2876

Solution. Lucas Correlation With Composition.

�g��gsc � 1 �A1 p1.3088

pr

A2 pA5pr � �1 � A3 pA4

pr��1 , (3.66a). . . . . .

where A1 �(1.245 � 10�3) exp�5.1726T�0.3286

pr�

Tpr,

A2 � A1�1.6553Tpr � 1.2723� ,

A3 �0.4489 exp�3.0578T�37.7332

pr�

Tpr,

A4 �1.7368 exp�2.2310T�7.6351

pr�

Tpr,

and A5 � 0.9425 exp�� 0.1853T 0.4489pr

�, (3.66b). . . . . . . . . . .

where �gsc� � �0.807T 0.618pr � 0.357 exp�� 0.449Tpr�

� 0.340 exp�� 4.058Tpr� � 0.018 ,

� � 9, 490� Tpc

M3p4pc�1�6

,

and ppc � RTpc

�Ni�1

yi Zci

�Ni�1

yivci

. (3.67). . . . . . . . . . . . . . . . . . . . . . . .

The Lucas correlation gives

Tpc � 376�R,

Zpc � 0.2876,

vpc � 1.752 ft3�lbm mol,

ppc � 663 psia,

M � 18.83 lbm�lbm mol,

Tpr � T�Tpc � (160 � 460)�376 � 1.65,

ppr � p�ppc � 2, 015�663 � 3.04,

� � 9, 490�(376)��(18.83)3(663)4 �1�6

� 77.3 cp�1

�gsc� � �0.807(1.65)0.618 � 0.357 exp[(� 0.449)(1.65)]

� 0.340 exp[(� 4.058)(1.65)] � 0.018� � 0.948,

�gsc � ��gsc�� 0.948�77.3 � 0.0123 cp,

A1 � 0.0607,

A2 � 0.0886,

A3 � 0.272,

A4 � 1.105,

A5 � 0.7473,

�g��gsc � 1.360,

and �g � 0.0167 cp.

Lohrenz-Bray-Clark Correlation. Eqs. 3.133 through 3.135 givethe Lohrenz-Bray-Clark correlation.

�� o��T � 10�4 1�4

� 0.10230 � 0.023364�pr

� 0.058533�2pr � 0.040758�3

pr � 0.0093324�4pr ,

where �T � 5.35� Tpc

M3p4pc�1�6

,

�pr ��

�pc�

�

Mvpc,

and �o �

�Ni�1

zi�i Mi�

�Ni�1

zi Mi�

. (3.133). . . . . . . . . . . . . . . . . . . . . . . .

�i�Ti � �34 � 10�5�T 0.94ri

(3.134a). . . . . . . . . . . . . . . . . . . . .

for Tri �1.5, and

�i�Ti � �17.78 � 10�5�(4.58Tri � 1.67)5�8

(3.134b). . . . . . .

for Tri1.5, where �Ti � 5.35�Tci M3i�p4

ci�1�6

.

vcC7�� 21.573 � 0.015122MC7�

� 27.656�C7�

� 0.070615MC 7��C7�

. (3.135). . . . . . . . . . . . . . .

On the basis of the data in Tables B-11 and B-12, this correlationyields

Tpc � 376�R,

Tpr � 1.65,

ppc � 663 psia,

6 PHASE BEHAVIOR

TABLE B-12—LOHRENZ-BRAY-CLARK VISCOSITY5 CALCULATIONS (PROBLEM 7)

Component zi Mi

pci(psia)

Tci(°R) Tri �i

�i(cp) zi�i M

�

i zi M�

i

C1 0.8750 16.04 667.8 343.0 1.81 0.0463 0.0125 0.0438 3.504

C2 0.0830 30.07 707.8 549.8 1.13 0.0352 0.0108 0.0049 0.455

C3 0.0210 44.09 616.3 665.7 0.93 0.0329 0.0097 0.0013 0.139

i-C4 0.0060 58.12 529.1 734.7 0.84 0.0322 0.0090 0.0004 0.046

C4 0.0080 58.12 550.7 765.3 0.81 0.0316 0.0088 0.0005 0.061

i-C5 0.0030 72.15 490.4 828.8 0.75 0.0310 0.0083 0.0002 0.025

C5 0.0020 72.15 488.6 845.4 0.73 0.0312 0.0081 0.0001 0.017

C6 0.0010 86.17 436.9 913.4 0.68 0.0312 0.0076 0.0001 0.009

C7+ 0.0010 114.00 360.6 1,023.9 0.61 0.0314 0.0068 0.0001 0.011

Total 1.0000 0.0516 4.268

TABLE B-13—ANALYSIS OF SOUR CANADIAN GAS(PROBLEM 8)

Componenti

Mole Fractionyi

CO2 0.0112

H2S 0.2609

C1 0.5575

C2 0.0760

C3 0.0433

i-C4 0.0061

n-C4 0.0137

i-C5 0.0033

n-C5 0.0052

C6 0.0053

C7+ 0.0175

MC7�� 128 and �C7�

� 0.780.

ppr � 3.04,

M � 18.83,

vMpc � 1.752 ft3�lbm mol,

�pr � �6.74�18.83�(1.752) � 0.627,

�T � 5.35�(376)��(18.83)3(663)4 �1�6

� 0.0436,

�gsc � 0.0516�4.268 � 0.0121 cp,

and �g � 0.0121 � �(0.131)4 � 10�4 �(0.0436) � 0.0166 cp.

��

Problem. Table B-13 gives the analysis of the sour Canadian gas ofProblem 2. Use the method developed by Wichert and Aziz6,7 andcalculate adjusted pseudocritical properties for use with the Stand-ing-Katz8 Z-factor chart. Then, calculate the gas FVF, Bg , at reser-voir conditions of 3,050 psig and 236°F. Note that Canadian stan-dard conditions are 14.65 psia and 60°F.

Solution. The Wichert-Aziz pseudocritical correction is given by

Tpc � T*pc � � , (3.52a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ppc �p*

pc�T*pc � ��

T *pc � yH2S�1 � yH2S��

, (3.52b). . . . . . . . . . . . . . . . .

and � � 120��yCO2� yH2S�

0.9

� �yCO2� yH2S�

1.6 � 15�y0.5

H2S � y4H2S� , (3.52c). . . . . . . . . . . . . . . . . . . . . . .

which (with the pseudocritical properties in Table B-14) gives

� � 29.8,

Tpc � 489.6 � 29.8 � 459.8�R,

ppc �(829.5)(489.6 � 29.8)

(489.6) � (0.2609)(1 � 0.2609)(29.8)� 770 psia,

Tpr � 696�459.8 � 1.51,

and ppr � 3, 065�770 � 3.98.

Where the Standing-Katz8 Z-factor chart is fit by the Hall-Yarbo-rough1,2 correlation,

Z � �ppr�y, (3.42). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

where � � 0.06125 t exp�� 1.2(1 � t)2 , where t � 1�Tpr ,

and F(y) � 0 � � �ppr �y � y2 � y3 � y4

(1 � y)3

� �14.76t � 9.76t 2 � 4.58t 3�y2

� �90.7t–242.2t2 � 42.4t3�y2.18�2.82t, (3.43). . . . . .

with t � 1�1.51 � 0.6622,

� � 0.06125(0.6622) exp�� 1.2(1 � 0.6622)2 � 0.03537,

y � 0.18088,

and Z � 0.778,

and gas FVF given by

Bg � �psc

Tsc� ZT

p (3.38). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

yields

Bg � �14.65�520��(0.778)(696)�(3, 065) � 0.00498 ft3�scf.

EXAMPLE PROBLEMS 7

TABLE B-14—PSEUDOCRITICAL-PROPERTY CALCULATIONSFOR A SOUR GAS (PROBLEM 8)

zi Mi

pci(psia)

Tci(°R) ziMi�

zipci(psia)

ziTci(°R)

CO2 0.0112 44.01 1,070.6 547.6 0.49 12.0 6.1

H2S 0.2609 34.08 1,306.0 672.4 8.89 340.7 175.4

C1 0.5575 16.04 667.8 343.0 8.94 372.3 191.2

C2 0.0760 30.07 707.8 549.8 2.29 53.8 41.8

C3 0.0433 44.09 616.3 665.7 1.91 26.7 28.8

i-C4 0.0061 58.12 529.1 734.7 0.35 3.2 4.5

C4 0.0137 58.12 550.7 765.3 0.80 7.5 10.5

i-C5 0.0033 72.15 490.4 828.8 0.24 1.6 2.7

C5 0.0052 72.15 488.6 845.4 0.38 2.5 4.4

C6 0.0053 86.17 436.9 913.4 0.46 2.3 4.8

C7+* 0.0175 128.0 386.7 1,099.5 2.24 6.8 19.2

Total 1.0000 26.98 829.5 489.6

*C7+ pseudocriticals from Eq. 3.51.

TABLE B-15—SURFACE PRODUCTION DATA(PROBLEM 9)

Reservoir pressure, psia 5,200

Reservoir temperature °F 250

Separator pressure, psia 950


Primary separator gas rate, Mscf/D 4,265

Primary separator gas gravity (air=1) 0.70

Tank-oil rate, STB/D 370

Tank-oil gravity, °API 45

��

Problem. Calculate the reservoir voidage, �VR , expressed as cubicfeet, resulting from 1 day of production from the gas-condensatereservoir with surface production data given in Table B-15.

Solution. On the basis of 1 day of production,

�VR � �Vg � ��Vg�� t�� tBgd � qg(1 day)Bgd � qg Bgd .

Surface-gas rate is qg � qo Rp � qo�R1 � Rs��, where R1 is theseparator gas/oil ratio (per stock-tank barrel of condensate) and Rs�is the solution gas/oil ratio of the separator oil.

Estimating the additional gas from the separator oil (Eqs. 3.61through 3.63),

Rs� � A1�gs1 (3.61a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

and A1 � �� psp1

18.2� 1.4�10�0.0125�API�0..00091Tsp1

� 1.205

;

(3.61b). . . . . . . . . . . . . . . . . . .

�gs1 � A2 � A3 Rs� , (3.62). . . . . . . . . . . . . . . . . . . . . . . . . .

where A2 � 0.25 � 0.2�API and A3 � � (3.57 � 10�6)�API ;

and Rs� �A1 A2

�1 � A1 A3� (3.63). . . . . . . . . . . . . . . . . . . . . . . . . .

gives A1 � ��95018.2

� 1.4�10(0.0125)(45)�(0.00091)(160) 1.205

� 385,

A2 � 0.25 � 0.02(45) � 1.15,

A3 � � 3.57 � 10�6(45) � � 1.607 � 10�4,

Rs� �(385)(1.15)

1 � (385)(� 1.607 � 10�4)� 417 scf�STB,

and �gs1 � 1.15 � 1.607 � 10�4(417) � 1.08 (air � 1).

The total GOR’s and OGR’s are given by

R1 � �4.265 � 106��(370) � 11, 527 scf�STB,

Rp � 11, 527 � 417 � 11, 944 scf�STB,

and rp � 1�Rp � 8.37 � 10�5 STB�scf � 83.7 STB�MMscf.

Total gas specific gravity is given by

�g ��g1 Rs1 � �gs1 Rs�

Rs1 � Rs�, (3.64). . . . . . . . . . . . . . . . . . . . . .

which yields

�g � [11, 527(0.70) � 417(1.08)]�(11, 527 � 417)

� 0.713 (air � 1).

The condensate stock-tank-oil molecular weight is estimatedfrom the Cragoe9 correlation (Eq. 3.59),

Mo �6, 084

�API � 5.9, (3.59). . . . . . . . . . . . . . . . . . . . . . . . . .

resulting in

Mo � 6, 084�(45 � 5.9) � 156,

which gives the wellstream specific gravity from Eq. 3.55.

�w ��g � 4, 580 rp �o

1 � 133, 000 rp ��M�o. (3.55). . . . . . . . . . . . . . . . . .

This yields

�w � 0.713 � (4, 580)(83.7 � 10�6)(0.8017)1 � (133, 000)(83.7 � 10�6)�0.8017�156�

� 0.963 (air � 1) .

The Sutton3 pseudocritical correlations

TpcHC � 169.2 � 349.5�gHC � 74.0�2gHC (3.47a). . . . . . . . . .

and ppcHC � 756.8 � 131�gHC � 3.6�2gHC (3.47b). . . . . . . . .

8 PHASE BEHAVIOR

give Tpc � 437�R and ppc � 627 psia, and reduced properties are

Tpr � T�Tpc � 710�437 � 1.625

and ppr � p�ppc � 5, 200�627 � 8.293.

The gas volumetric properties are given by Eqs. 3.42 and 3.43,

Z � �ppr�y, (3.42). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

where � � 0.06125 t exp�� (1.21 � t)2 , where t � 1�Tpr ,

and F(y) � 0 � � �ppr �y � y2 � y3 � y4

(1 � y)3

� �14.76t � 9.76t2 � 4.58t3�y2

� �90.7t � 242.2t2 � 42.4t3�y2.18�2.82t, (3.43). . . . .

giving Z�1.024. With Eq. 7.12,

Bgd � �psc

Tsc� ZT

p �1 � Cog rs� (7.12). . . . . . . . . . . . . . . . . . . .

and Cog given by

Cog � 133, 000� �og

Mog� (7.13). . . . . . . . . . . . . . . . . . . . . . . . .

� 133, 000�0.8017�156�

� 683 scf�STB.

So with rs � 1�Rp,

Bgd � ��14.7�520�(1.024)(160 � 460)�(5, 200)

� �1 � �683�11, 944�

� 0.00395 ft3�scf.

The initial daily reservoir voidage is then

�Vg � ��Vg�� t�� t��Bg�

� (370)(11, 944)(0.00395) � 17, 470 ft3 � 3, 110 bbl .

��

Problem. Table B-16 shows the composition of a reservoir oil in theKabob field, Canada. Bubblepoint pressure is 3,100 psia at 236°Freservoir temperature. Calculate the density in lbm/ft3 of the reser-voir oil at bubblepoint conditions using ideal-solution principles ac-cording to the Standing-Katz8 method.

Solution. Following the calculation procedure outlined in Chap .3,pseudoliquid density, �po, is calculated explicitly with Eqs. 3.94

through 3.97. From Table B-17 and Eqs. 3.93 and 3.94, volumes andmasses needed for the calculations are

VC3�� 1.385 ft3,

mC2� 3.40 lbm,

and mC2�� 69.97 lbm.

Recalling Eq. 3.95,

�C2�� b � b2 � 4ac�

2a, (3.95). . . . . . . . . . . . . . . . . . .

where a � 0.3167VC3�, b � mC2

� 0.3167mC2�� 15.3VC3�

,and c � � 15.3mC2�

, we calculate

a � 0.3167(1.385) � 0.439;

b � 3.40 � 0.3167(69.97) � 15.3(1.385) � 2.43;

c � � 15.3(69.97) � � 1, 071;

and �C2��

� (2.43) � (2.43)2 � 4(0.439)� (� 1, 071)2(0.439)

� 46.70 lbm�ft3 ,

the pseudoliquid density of the C2+ mixture at standard conditions.From Eq. 3.96,

VC2�� VC3�

�mC2�C2

� VC3��

mC2

15.3 � 0.3167�C2�

, (3.96). . . . . . . . . . .

TABLE B-16—OIL COMPOSITION (PROBLEM 10)


CO2 0.0111

C1 0.3950

C2 0.0969

C3 0.784

i-C4 0.0159

n-C4 0.0372

i-C5 0.0123

n-C5 0.0211

C6 0.0295

C7+ 0.3026

MC7�� 182 and �C7�

� 0.8275.

TABLE B-17—STANDING-KATZ8 DENSITY CALCULATION (PROBLEM 10)

�i mi � zi Mi Vi � mi��iComponent zi Mi (lbm/ft3) (lbm) (ft3)

C1 0.3950 16.04 6.34

C2 0.0969 30.07 2.91

CO2 0.0111 44.01 0.49

C3 0.0784 44.09 31.66 3.46 0.109

i-C4 0.0159 58.12 35.01 0.92 0.026

C4 0.0372 58.12 36.45 2.16 0.059

i-C5 0.0123 72.15 39.13 0.89 0.023

C5 0.0211 72.15 39.30 1.52 0.039

C6 0.0295 86.17 41.19 2.54 0.062

C7+ 0.3026 182.00 51.61 55.07 1.067

Total 1.0000 76.31Note: CO2 is treated as C2.

EXAMPLE PROBLEMS 9

which gives

VC2�� 1.385 � (3.40)�[15.3 � (0.3167)(46.70)]

� 1.50 ft3.

The mass of methane and of the total mixture (C1+) are taken fromTable B-17.

mC1 � 6.34 lbm and mC1�� 76.31 lbm.

From Eq. 3.97, the pseudoliquid density of the overall mixture iscalculated at standard conditions.

�po ��b � b2 � 4ac�

2a, (3.97). . . . . . . . . . . . . . . . . . . .

where a � 0.45(1.50) � 0.674,

b � 6.34 � 0.45(76.31) � 0.312(1.50) � � 27.53,

c � � 0.312(76.31) � � 23.81,

and �po �� (27.53) � (27.53)2 � 4(0.674)(� 23.81)�

2(0.674)

� 41.69 lbm�ft3.

The pressure correction is calculated with Eq. 3.98 and �po �41.69 lbm/ft3.

��p � 10�3 �0.167 � �16.181 � 10�0.0425�po� p

� 10�8 �0.299 � �263 � 10�0.0603�po� p2, (3.98). . . .

giving

��p � 10�3 �0.167 � �16.181 � 10�0.0425(41.69) � (3, 500)

� 10�8 �0.299 � �263 � 10�0.0603(41.69) � (3, 500)2

� �0.441 � 10�3�(3, 500) � �1.104 � 10�8�(3, 500)2

� 1.26 lbm�ft3.

The temperature correction is calculated with Eq. 3.99 and�po � ��p �41.69�1.26�42.95 lbm/ft3.

��T � (T � 60) �0.0133 � 152.4��po � ��p��2.45

� (T � 60)2��8.1 � 10�6�

� �0.0622 � 10�0.0764��po��p� �, (3.99). . . . . . . . . . .

giving

��T � (238 � 60)�0.0133 � 152.4(42.95)�2.45

� (238 � 60)2��8.1 � 10�6�

� �0.0622 � 10�0.0764(42.95) � � 5.85 lbm�ft3.

Eq. 3.89 gives the oil density at 3,100 psia and 238°F.

�o � �po � ��p � ��T , (3.89). . . . . . . . . . . . . . . . . . . . . .

resulting in

�o � 41.69 � 1.26 � 5.85 � 37.10 lbm�ft3 .

��

Problem. An oil well produces at a total GOR of 900 scf/STB. Totalgas gravity is 0.85 (air�1). Stock-tank-oil gravity is 36°API. Cal-culate, using ideal-solution principles and apparent liquid density ofthe gas, the density of the reservoir oil at 3,300 psia and 190°F. Ifreservoir pressure is 3,300 psia at 7,200 ft subsea, what would thereservoir pressure be at a datum level of 6,000 ft subsea?

Solution. From Eq. 3.100, pseudoliquid density, �po, can be calcu-lated from oil and gas surface gravities, �o and �g, respectively;solution gas/oil ratio, Rs; and apparent liquid density of separatorgas, �ga. Stock-tank-oil gravity is

�o � 141.5�(131.5 � 36) � 0.845 (water � 1).

Apparent gas pseudoliquid density is given by

�ga � 38.52 � 10�0.00326�API

� �94.75 � 33.93 log �API� log �g , (3.101). . . . . . . . . . .

which results in

�ga � 38.52 � 10�(0.00326)(36) � �94.75 � (33.93) log (36)

� log (0.85) � 26.4 lbm�ft3

Pseudoliquid oil density is given by

�po �62.4�o � 0.0136 Rs �g

1 � 0.0136�Rs �g��ga�, (3.100). . . . . . . . . . . . . . . . .

which gives

�po �62.4(0.845) � 0.0136(900)(0.85)

1 � (0.0136)�(900)(0.85)�(26.4) � 45.3 lbm�ft3.

The pressure correction to density, if given by Eq. 3.98 is��p � 1.1 lbm�ft3 and �po � ��p � 45.3 � 1.1 lbm�ft3 �46.4 lbm/ft3. On the basis of �po � ��p, the temperature correctionis given by Eq. 3.99 and is ��T � 3.5 lbm�ft3, yielding the reser-voir oil density from Eq. 3.89,

�o � �po � ��p � ��T , (3.89). . . . . . . . . . . . . . . . . . . . . .

as �o � 45.3 � 1.1 � 3.5 � 42.9 lbm�ft3.Pressure gradient with depth (dp/dh) in psi/ft is given by dp/dh

� �o�g�gc��144, where � is in lbm/ft3, g�32 ft/sec2, and gc�32lbm-ft/(lbf-sec2), giving

dp/dh�42.9(32/32)/144�0.298 psi/ft.

Assuming that this gradient is more or less constant from 7,200to 6,000 ft subsea, the oil pressure at a depth of 6,000 ft subsea is

�pR�6000 � 3, 300 � 0.298(7, 200 � 6, 000) � 2, 942 psia.

This result assumes that a continuous oil column exists from 6,000to 7,200 ft subsea.

��

Problem. For the reservoir considered in Problem 11, use the Stand-ing10 bubblepoint correlation to estimate bubblepoint pressure. Onthe basis of this estimate, is it possible that a gas cap might be foundbetween the test depth of 7,200 ft subsea and the structure top at6,000 ft subsea? If so, at what depth?

Solution. The Standing bubblepoint-pressure correlation, Eq. 3.78,

pb � 18.2�A � 1.4�, (3.78). . . . . . . . . . . . . . . . . . . . . . . . .

where A � �Rs��g�0.83 � 10�0.00091T�0.0125�API�, gives

A � �900�0.85�0.83

� 10[0.00091(190)�0.0125(36)] � 171.2

and pb � 18.2(171.2 � 1.4) � 3, 090 psia.

10 PHASE BEHAVIOR

If this bubblepoint-pressure estimate is accurate (even though thecorrelation accuracy is probably only �5%), a gas cap may be ex-pected at a subsea depth, calculated from

�pR�GOC � pb � 3, 090 � 3, 300 � 0.298�7, 200 � DGOC

� ,

where 0.298 psi/ft is the oil gradient calculated in Problem 11. Solv-ing this relation for DGOC gives DGOC � 6, 500 ft subsea.

��

Problem. If the hydrocarbon pore volume (HCPV) of the reservoirin Problem 11 is approximately 40�106 ft3/ft reservoir thickness,estimate the original oil in place, N, and original gas in place, G. Thewater/oil contact (WOC) is at 7,300 ft subsea, the gas/oil contact(GOC) depth is given in Problem 12 as 6,500 ft subsea, and the topof the structure is at 6,000 ft subsea.

Solution. To solve this problem, oil and gas FVF’s must be estimated.The oil FVF will vary throughout the 800-ft oil column. The oil is sat-urated at the GOC and undersaturated at depths down to the WOC.Several assumptions must be made because so little data are available.

1. Constant temperature is assumed throughout the reservoir, al-though a gradient of 1 to 2°F/100 ft probably exists.

2. Oil composition is assumed to be uniform in the oil column, al-though it would not be surprising if the GOR decreased somewhatfrom the GOC to the WOC (e.g., from 900 to 800 scf/STB).

3. Gas composition is assumed to be uniform in the gas cap (prob-ably a reasonable assumption).

4. A condensate yield must be assumed for the reservoir gas.From data in the literature (or from a similar reservoir in the samegeographical area), we can find a similar reservoir oil/gas system.An initial solution OGR, rsi, of 40 STB/MMscf is assumed here.

5. Surface condensate gravity of 60°API (�og � 0.739) is alsoassumed.

6. The surface-gas gravity for the reservoir gas is assumed to beslightly less than the surface-gas gravity for the reservoir oil,�gg � 0.80 (see, for example, Fig. 7.12).

The gas and oil column HCPV’s, VHCg and VHCo, respectively,are given by

VpHCg � �40 � 106�(6, 500 � 6, 000) � 20 � 109 ft3

and VpHCo � �40 � 106�(7, 300 � 6, 500) � 32 � 109 ft3

� 5.700 � 109 bbl.

Initial gas in place represents the free gas in place plus the gas insolution in the oil column. To calculate gas FVF, a wellstream grav-ity, �w, must be calculated first. With �og � 0.739, Eq. 3.59 givesan estimate of the condensate molecular weight.

Mo �6, 084

�API � 5.9, (3.59). . . . . . . . . . . . . . . . . . . . . . . . . .

giving

Mog � 6, 084�(60 � 5.9) � 112.

From Eq. 3.55,

�w ��g � 4, 580 rp �o

1 � 133, 000 rp ��M�o, (3.55). . . . . . . . . . . . . . . . . .

�w �0.8 � �4, 580 � (40 � 10�6)(0.739)

1 � 133, 000(40 � 10�6)�(0.739)�(112)

� 0.904 (air � 1).

With �gg � �gHC � 0.80, pseudocritical properties from the Stand-ing10 “wet-gas” correlations (Eq. 3.49),

TpcHC � 187 � 330�gHC � 71.5�2gHC (3.49a). . . . . . . . . . . . . .

and ppcHC � 706 � 51.7�gHC � 11.1�2gHC, (3.49b). . . . . . . . . .

are Tpc�426°R and ppc�650 psia.At the GOC, reduced properties are Tpr�(190�460)/426�1.526

and ppr�3,090/650�4.754.The Standing-Katz8 Z-factor correlation (Eq. 3.42),

Z � �ppr�y, (3.42). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

where �� 0.06125 t exp[� 1.2(1 � t)2], with t � 1�Tpr, givesZ�0.808.

Gas density is calculated at the GOC to obtain a gas gradient forestimating the average pressure in the gas cap.

�g �(3, 090)(28.97)(0.904)

(0.808)(10.73)(190 � 460)� 14.36 lbm�ft3 ,

�dp�dh�g � 14.36�144 � 0.0997 psi�ft

and �pR�g � [3, 090 � (0.0997)(6, 500 � 6, 000)]�2

� 3, 065 psia.

Pseudoreduced pressure at (pR)g is ppr � 3, 065�650 � 4.715,and the Z factor is 0.806. The wet-gas FVF at (pR)g is given by

Bg � �psc

Tsc� ZT

p , (3.38). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

resulting in

Bgw � (0.02827)(0.806)(190 � 460)�(3, 065)

� 0.00483 ft3�scf

and bgw � 1�Bgw � 207 scf�ft3.

However, the dry-gas FVF is needed to calculate dry surface gasfor the estimated VpHCg . Eqs. 7.12 and 7.11 are used to calculate Bgd.

Bgd �psc

Tsc

ZTp �1 � Cog rs� � 0.02827

ZTp �1 � Cog rs�

� Bgw�1 � Cog rs� , (7.12). . . . . . . . . . . . . . . . . . . . . . . .

where Cog � (133, 000)(0.739)(112) � 876 scf�STB,

and Bgd � Vg�Vgg , (7.11). . . . . . . . . . . . . . . . . . . . . . . . . . . . .

giving

Bgd � (0.00483)�1 � (876)�40 � 10�6� � 0.00500 ft3�scf

and bgd � 1�Bgd � 200 scf�ft3.

For the oil column, oil FVF must be estimated at an average oilpressure (pR)o.

�pR�

o� 3, 090 � (0.298)(7, 300 � 6, 500)�2 � 3, 209 psia.

Bubblepoint oil FVF is estimated from the Standing correlation(Eq. 3.111),

Bob � 0.9759 � �12 � 10�5�A1.2, (3.111). . . . . . . . . . . . . . .

where A � Rs��g��o� 0.5� 1.25T, giving

A � 900�0.85�0.845�0.5

� 1.25(190) � 1, 140

and Bob � 0.9759 � �12 � 10�5�(1, 140)1.2

� 1.535 bbl�STB.

Undersaturated oil FVF can be calculated with an estimate of theundersaturated oil compressibility with Eq. 3.107 for co at (pR)o ,and Eq. 3.105 for Bo.

co � A�p, (3.107). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

EXAMPLE PROBLEMS 11

giving

co � 10�5�(5)(900) � (17.2)(190) � (1, 180)(0.85)

� (12.61)(36) � 1, 433 �(3, 209) � 18.0 � 10�6 psi�1.

�o � �ob exp�co�p � pb�

� �ob�1 � co�pb � p� (3.105a). . . . . . . . . . . . . . . . . . . .

and Bo � Bob exp�co�pb � p�

�Bob�1 � co �p � pb

� , (3.105b). . . . . . . . . . . . . . . . .

which give

Bo � 1.535 exp��18.3 � 10�6�(3, 090 � 3.209)

� 1.532 bbl�STB.

However, a more exact approach uses Eq. 109, which properly ac-counts for the pressure dependence of oil compressibility.

co � 10�6 exp� �ob � 0.004347 �p � pb� � 79.1

(7.141 � 10�4)�p � pb� � 12.938

,(3.109). . . . . . . . . . . . . . . . . . .

resulting in

A � 10�5[(5)(900) � (17.2)(190) � (1, 180)(0.85)

� (12.61)(36) � 1, 433] � 0.05776

and Bo � 1.535�3, 090�3, 209�0.05776

� 1.532 bbl�STB.

The two approaches result in almost no difference in Bo for thisexample of slight undersaturation. However, for higher degrees ofundersaturation, the difference can be significant; therefore, in gen-eral, Eq. 3.109 is recommended.

Initial oil in place in the oil column is given by

(N)o � VpHCo�Boi � �5.700 � 109��(1.532)

� 3.720 � 109 STB.

Initial (dry) gas in place in the gas column is given by

(G)g � VpHCo bgd � �20 � 109�(200) � 4.000 � 1012 scf.

Initial condensate in place in solution in the gas column is given by

(N)g � Gd rsi � �4.000 � 1012��40 � 10�6�

� 160 � 106 STB.

Initial gas in solution in the oil column is given by

(G)o � NRsi � �3.720 � 109�(900) � 3.348 � 1012 scf .

Thus, the initial stock-tank oil plus condensate in place, N, and theinitial dry gas plus solution gas in place, G, are, respectively,

N � (N)o � (N)g � �3.720 � 109� � �0.160 � 109�

� 3.880 � 109 STB

and G � (G)g � (G)o � �4.000 � 1012� � �3.348 � 1012�

� 7.348 � 1012 scf.

Note that significant gas reserves are found as solution gas in thisoil reservoir. This is not uncommon for volatile and even moderate-ly volatile oil reservoirs (GOR750 scf/STB). In general, in largerfield developments, the economic value of solution gas cannot beignored as both production revenue for depletion drive and lost in-come in waterflooding projects.

��

Problem. Estimate oil and gas viscosities at 2,500 psia and 190°Ffor the reservoir considered in Problems 11 through 13.

Solution. Gas viscosity can be estimated from the Lucas4 correla-tion (Eq. 3.66).

�g��gsc � 1 �A1 p1.3088

pr

A2 pA5pr � �1 � A3 pA4

pr��1 , (3.66a). . . . . . .

where A1 �(1.245 � 10�3) exp�5.1726T�0.3286

pr�

Tpr,

A2 � A1�1.6553Tpr � 1.2723� ,

A3 �0.4489 exp�3.0578T�37.7332

pr�

Tpr,

A4 �1.7368 exp�2.2310T�7.6351

pr�

Tpr,

and A5 � 0.9425 exp�� 0.1853T 0.4489pr

�. (3.66b). . . . . . . . . . .

Pseudocritical properties are estimated from reservoir gas (well-stream) gravity, �w. The initial wellstream gravity of 0.904 calculatedin Problem 13 is somewhat higher than would be expected for theequilibrium gas at 2,500 psia (see, for example, Table 6.11). We there-fore assume a current wellstream gravity of �w � �gHC � 0.85.With the Standing10 wet-gas correlations (Eq. 3.49) for pseudocriti-cal properties,

TpcHC � 187 � 330�gHC � 71.5�2gHC (3.49a). . . . . . . . . . . . . .

� 416�R

and ppcHC � 706 � 51.7�gHC � 11.1�2gHC, (3.49b). . . . . . . . .

� 654 psia,

giving pseudoreduced properties

Tpr � T�Tpc � (190 � 460)�416 � 1.562

and ppr � p�ppc � 2, 500�654 � 3.823.

The gas molecular weight is

Mg � (28.97)(0.85) � 24.62 lbm�lbm mol,

which is used to calculate �.

� � 9, 490�(416)�(24.62)3�(654)4 1�6

� 69.3 cp�1,

giving

�gsc � ��gsc�� 0.9046�69.3 � 0.0131 cp,

�g��gsc � 1.601,

and �g � 0.0210 cp.

Use the Lee-Gonzalez correlation (Eq. 3.65) to calculate gasviscosity.11

�g � A1 � 10�4 exp�A2�A3g� , (3.65a). . . . . . . . . . . . . . . . . .

where A1 ��9.379 � 0.01607Mg�T1.5

209.2 � 19.26Mg � T,

A2 � 3.448 � �986.4�T� � 0.01009Mg ,

and A3 � 2.447 � 0.2224A2 . (3.65b). . . . . . . . . . . . . . . . . . .

Gas density must be calculated first with

�g � pMg�ZRT. (3.34). . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12 PHASE BEHAVIOR

TABLE B-18—THREE-COMPONENT-SYSTEMCOMPOSITION (PROBLEM 15)


C1 0.20

C3 0.32

n-C5 0.48

TABLE B-19—COMPONENT PROPERTIES (PROBLEM 15)

Componentpci

(psia)Tci

(°R) �i Ki

C1 667.8 343.0 0.0115 9.208

C3 616.3 665.7 0.1454 1.439

n-C5 488.6 845.4 0.2510 0.358

TABLE B-20—CALCULATED RESULTSFROM ITERATIONS (PROBLEM 15)

Iteration Fv h(Fv) dh/dFv

1 0.5 �1.75�10�2 �0.98880

2 0.48227 1.48�10�4 �1.00606

3 0.48242 1.16�10�8 �1.00590

4 0.48242 7.07�10�17 �1.00590

This gives

�g �(2, 500)(24.62)

(0.803)(10.73)(190 � 460)

� 11.0 lbm�ft3 � 0.176 g�cm3,

where Z�0.803 is estimated from the Standing-Katz8 correlation(Eqs. 3.42 and 3.43). From Eq. 3.65b, the constants in the gas vis-cosity correlation are

A1 �[9.379 � 0.01607(24.62)](650)1.5

209.2 � 19.26(24.62) � 650� 121.5,

A2 � 3.448 � �986.4�650� � 0.01009(24.62) � 5.214,

and A3 � 2.447 � 0.2224(5.214) � 1.287 ,

giving

�g � �121.5 � 10�4� exp�(5.214)(0.176)1.287 � 0.0212 cp.

The oil viscosity is calculated by first estimating dead-oil viscos-ity, �oD. With the Bergman* correlation (Eq. 3.119),

ln ln��oD � 1� � A0 � A1 ln(T � 310), (3.119). . . . . . . . .

where A0�22.33�0.194(36)� 0.00033 (36)2 � 15.77,

A1 � � 3.20 � (0.0185)(36) � � 2.534,

and �oD � � 1 � exp� exp[15.77�2.534 ln(190 � 310)] �

� 1.78 cp.

The viscosity correction for a saturated live oil depends on theamount of gas in solution, Rs. We estimate the solution gas/oil ratiousing Standing’s10 bubblepoint pressure correlation (Eq. 3.78), set-ting the current pressure of 2,500 psia as the bubblepoint of the satu-rated oil and solving for Rs as given by Eq. 3.87,

Rs � (0.85)�[(0.055)(2, 500) � (1.4)] � 10(0.0125)(36)

10(0.00091)(190)�

� 700 scf�STB.


Using the Chew and Connally12 correlation (Eq. 3.123),

�ob � A1��oD

�A2, (3.123). . . . . . . . . . . . . . . . . . . . . . . . . . . .

and the Bergman equations for constants A1 and A2 (Eq. 3.125),

ln A1 � 4.768 � 0.8359 ln(Rs � 300) (3.125a). . . . . . . . . .

and A2 � 0.555 � 133.5Rs � 300

, (3.125b). . . . . . . . . . . . . . . . .

gives A1 � exp�4.768 � 0.8359 ln(700 � 300) � 0.3656,

A2 � 0.555 � (133.5)�(700 � 300) � 0.6885 ,

and �ob � (0.3656)(1.78)0.6885 � 0.544 cp.

��

Problem. Table B-18 gives the composition of a three-componentsystem of methane, propane, and normal pentane. Use the modifiedWilson13 K-value equation (Eq. 3.159) with a convergence pressureof 2,000 psia to estimate K values at 500 psia and 160°F. Make aflash calculation using the Muskat-McDowell14 (or Rachford-Rice15) algorithm given by Eqs. 4.36 through 4.40.

Solution. Table B-19 gives component properties taken from Ap-pendix A needed to calculate K values from the modified Wilson K-value equation. A0 � 0.7 is used in the modified Wilson K-valuecorrelation, where A � 1 � (p�pk )0.7 in Eq. 3.159. For example,the K value for methane is given by

Ki � �pcipk�A1�1 exp�5.37 A1 (1 � �i)�1 � T�1

ri�

pri,

(3.159). . . . . . . . . . . . . . . . . .

resulting in

A � 1 �� 500 � 14.72, 000 � 14.7

�0.7

� 0.627,

(Tr)C1� (160 � 460)�343 � 1.807,

and KC1� �667.8

2, 000�0.627�1

�exp�5.37(0.627)(1 � 0.0115)�1 � 1�1.807�

0.749� 9.21,

and the ci value for methane is

ci � 1�(Ki � 1) � 1�(9.21 � 1) � 0.122.

With these K values, four iterations are used to solve the Muskat-McDowell equation,

h(Fv) ��Ni�1

zi

Fv � ci� 0, (4.39). . . . . . . . . . . . . . . . . . . . . .

where ci � 1�(Ki � 1). Table B-20 summarizes the calculated re-sults from the iterations, and Table B-21 gives the final results forthe flash calculation, including equilibrium vapor and liquid com-positions.

��

Problem. Calculate the bubblepoint pressure for the ternary systemin Problem 15 at 160°F using the modified Wilson K-value equa-tion. Eq. 3.165 is used to solve for bubblepoint pressure given a K-value correlation based on convergence pressure.

F�pK� � 1 ��

N

i�1

zi Ki�pK, pb, T� � 0. (3.165). . . . . . . . . . . .

EXAMPLE PROBLEMS 13

TABLE B-21—FINAL FLASH-CALCULATION RESULTS (PROBLEM 15)

zi Ki ci zi /(Fv +ci) zi /(Fv +ci)2 xi yi

C1 0.20 9.208 0.122 0.331 0.548 0.0403 0.3713

C3 0.32 1.439 2.278 0.116 0.042 0.2641 0.3800

n-C5 0.48 0.358 �1.556 �0.447 0.416 0.6956 0.2487

Total 1.00 7.07�10�17 1.00590 1.0000 1.0000

h(Fv)�7.07�10�17 and h�(Fv)��1.00590.

TABLE B-22—PRESSURE-GUESS CALCULATIONS* (PROBLEM 16)

pK�1,300 psia pK�1,375 psia (correct) pK�1,500 psia

Component zi Ki yi�ziKi Ki yi�ziKi Ki yi�ziKi

C1 0.20 2.181 0.436 1.982 0.396 1.703 0.341

C3 0.32 1.003 0.321 0.996 0.319 0.988 0.316

C5 0.48 0.560 0.269 0.594 0.285 0.657 0.315

Total 1.00 1��yi��0.02591 1��yi��0.00002�0 1��yi�0.028024*At T�160°F.

TABLE B-23—OIL COMPOSITION (PROBLEM 17)


CO2 0.0111

C1 0.3950

C2 0.0969

C3 0.0784

i-C4 0.0159

n-C4 0.0372

i-C5 0.0123

n-C5 0.0211

C6 0.0295

C7+ 0.3026

MC 7�182

�C 7�0.8275

KwC 7�11.79

C7+ is split into three fractions; Table B-24 gives mole fractions and properties.

Solution. Although an iterative procedure, such as Newton-Raph-son, can be solved analytically with the modified Wilson K-valueequation, it takes only a few guesses to locate the pressure that satis-fies Eq. 3.165. Table B-22 summarizes the results of the calcula-tions for three guesses of pressure, where pK � 1, 375 psia givesa satisfactory result for bubblepoint pressure.

��

Problem. Tables B-23 and B-24 show the composition of a reser-voir oil in the Kabob field, Canada. Bubblepoint pressure is 3,100psia at 236°F reservoir temperature.

a. Calculate the convergence pressure, pK, that matches the mea-sured bubblepoint pressure. Use the modified Wilson K-value equa-tion (Eq. 3.159) with A0 � 0.7.

b. Use the K-value correlation developed in Part a to make asingle-stage separator flash calculation to 14.7 psia and 60°F. Re-port the stock-tank-gas and -oil compositions, GOR, oil gravity in°API, and gas specific gravity.

Solution. Table B-25 gives relevant component properties for thisproblem. The K values at reservoir conditions are calculated withT�236°F. The modified Wilson equation (Eq. 3.159) is

Ki � �pcipK�

A1�1 exp�5.37 A1 (1 � �i)�1 � T�1ri�

pri,

(3.159). . . . . . . . . . . . . . . . . .

where A � 1 � (p�pK)0.7 and A0 � 0.7 is assumed. By adjustingconvergence pressure, pK, the bubblepoint condition given by Eq.3.165 is satisfied with pK � 4, 052.8 psia.

Table B-26 gives the K values and incipient-phase gas composi-tion. The K-value correlation is then used to make a flash calculationat standard conditions p�14.7 psia and T�60°F. With K values atthese conditions, the Rachford-Rice equation, Eq. 4.36, is solved forgas-phase mole fraction(Fv � Fg) where Fg � 0.64241; stock-tank-oil and separator-gas compositions are given later.

On the basis of the surface-gas composition, specific gravity �g is

�g � 27.32�28.97 � 0.943 (air � 1)

and stock-tank oil properties are

Mo � ��mo i��no i

� � 167.8�1.0 � 167.8 lbm�lbm mol,

�o � ��mo i��Vo i

� � 167.8�3.323 � 50.48 lbm�ft3,

�o � �o��w � 0.8094 (water � 1),

and �API � 141.5��o � 131.5 � 43.3�API,

TABLE B-24—MOLE FRACTIONS AND PROPERTIES OF C7+ COMPONENT (PROBLEM 17)

C7+ Fraction zi Mi

Tci(°R)

pci(psia) ��i�� i*

Tbi(°R)

F1 0.1578 114.1 1065.5 409.6 0.3255 0.7674 727.0

F2 0.1243 223.1 1356.0 235.1 0.6538 0.8403 1,029.5

F3 0.0205 455.0 1689.1 134.6 1.1489 0.9254 1,410.1

Total 0.3026 182.0 0.8275*Water�1.

14 PHASE BEHAVIOR

TABLE B-25—COMPONENT PROPERTIES (PROBLEM 17)

�i pci TciComponent zi Mi (lbm/ft3) (psia) (°R) �i

CO2 0.0111 44.01 31.18 1,070.6 547.6 0.2310

C1 0.3950 16.04 20.58 667.8 343.0 0.0115

C2 0.0969 30.07 28.06 707.8 549.8 0.0908

C3 0.0784 44.09 31.66 616.3 665.7 0.1454

i-C4 0.0159 58.12 35.01 529.1 734.7 0.1756

C4 0.0372 58.12 36.45 550.7 765.3 0.1928

i-C5 0.0123 72.15 39.13 490.4 828.8 0.2273

C5 0.0211 72.15 39.30 488.6 845.4 0.2510

C6 0.0295 86.17 41.19 436.9 913.4 0.2957

F1 0.1578 114.10 47.86 409.6 1,065.5 0.3255

F2 0.1243 233.10 52.41 235.1 1,356.0 0.6538

F3 0.0205 455.00 57.72 134.6 1,689.1 1.1489

Total 1.0000

TABLE B-26—K VALUES AND INCIPIENT-PHASEGAS COMPOSITION (PROBLEM 17)

Component Tri pri

ModifiedWilson

Ki

IncipientPhaseyi�ziKi

CO2 1.270 2.90 1.324 0.0147

C1 2.028 4.64 1.539 0.6079

C2 1.265 4.38 1.196 0.1159

C3 1.045 5.03 0.990 0.0776

i-C4 0.947 5.86 0.867 0.0138

C4 0.909 5.63 0.831 0.0309

i-C5 0.839 6.32 0.733 0.0090

C5 0.823 6.34 0.709 0.0150

C6 0.762 7.10 0.614 0.0181

F1 0.653 7.57 0.461 0.0727

F2 0.513 13.19 0.189 0.0234

F3 0.412 23.03 0.04302 0.0009

Total 1.0000

where no i � xi, mo i � xi Mi, and Vo i � xi Mi��i.

On the basis of 1 mole of feed, the surface volumes are given by

Vg � 379Fg � 379(0.64241) � 243.5 scf

and Vo � �1 � Fg��Mo��o� � (1 � 0.64241)�167.8�50.48�

� 1.188 ft3 � 0.2117 STB

and the GOR is

Rgo � Vg�Vo � 243.5�0.2117 � 1, 150 scf�STB.

Table B-27 summarizes the results.

��

Problem. Make equation-of-state (EOS) calculations using thePeng-Robinson16 EOS (PR EOS) for the ternary system describedin Table B-28. Use the cubic m term (Eq. 4.22) for �0.4 (C10).a. Make a two-phase flash calculation at 500 psia and 280°F.b. Make a Michelsen phase-stability test followed by a two-phaseflash calculation at 1,500 psia and 280°F.

TABLE B-27—SEPARATOR FLASH CALCULATION (PROBLEM 17)

ModifiedWilson

Muskat-McDowell

Stock-TankOil

SeparatorGas moi � xi Mi Voi � xi Mi��i mgi � yi Mi

Component Tri pri Ki zi /(Fg+ci) xi yi (lbm) (ft3) (lbm)

CO2 0.949 0.014 51.05 0.0168 0.0003 0.0171 0.01 0.000 0.75

C1 1.515 0.022 287.94 06116 0.0021 0.6137 0.03 0.002 9.84

C2 0.945 0.021 34.28 0.1441 0.0043 0.1484 0.13 0.005 4.46

C3 0.781 0.024 7.44 0.0983 0.0153 0.1136 0.67 0.021 5.01

i-C4 0.707 0.028 2.64 0.0127 0.0077 0.0204 0.45 0.013 1.19

C4 0.679 0.027 1.81 0.0199 0.0244 0.0443 1.42 0.039 2.58

i-C5 0.627 0.030 0.662 �0.0053 0.0157 0.0104 1.13 0.029 0.75

C5 0.615 0.030 0.493 �0.0159 0.0313 0.0154 2.26 0.057 1.11

C6 0.569 0.034 0.153 �0.0549 0.0647 0.0099 5.58 0.135 0.85

F1 0.488 0.036 1.58x10�2 �0.4223 0.4291 0.0068 48.96 1.023 0.77

F2 0.383 0.063 9.93x10�6 �0.3476 0.3476 0.0000 81.02 1.546 0.00

F3 0.308 0.109 4.83x10�11 �0.0573 0.0573 0.0000 26.08 0.452 0.00

Total 0.0000 1.0000 1.0000 167.76 3.323 27.32

EXAMPLE PROBLEMS 15

TABLE B-28—TERNARY SYSTEM (PROBLEM 18)

Componenti zi Mi

Tci(°R)

pci(psia) �i si�ci/bi

C1 0.50 16.04 343.0 667.8 0.0115 �0.1595

C4 0.42 58.12 765.3 550.7 0.1928 �0.0675

C10 0.08 142.29 1,111.8 304.0 0.4902 0.0655

TABLE B-29—CHANGES DURING ITERATIONS(PROBLEM 18)

Iteration

ConvergenceTolerance

log[�(1–fLi /fvi)2]

Trivial SolutionIndicator�(ln Ki)2

Vapor-PhaseMole Fraction

Fv

1 0.708 30.73 0.852187

2 �2.230 15.24 0.853914

3 �4.380 14.66 0.853528

4 �6.454 14.61 0.853423

5 �8.457 14.61 0.853405

6 �15.236 14.61 0.853401

Solution.a. The flash calculation is made with five successive-substitution it-

erations followed by a general dominant eigenvalue method (GDEM)promotion. Tables B-29 and B-30 give the results of the calculationsfor the six iterations required to solve the flash problem. Table B-31shows the change in convergence tolerance, the trivial-solution indi-cator, and vapor-phase mole fraction during each iteration. The con-vergence tolerance indicates how close the phase fugacities of eachcomponent have come to one another. Convergence was specified as10�12 in this example. The trivial-solution indicator stabilizes afterthree iterations. Convergence toward a trivial solution is usually in-dicated for values �(ln Ki)

2 10�4. Details of the EOS calcula-tions for the first iteration are summarized later, step by step.

K-Value Estimate. The Wilson13 equation is used to estimate Kvalues.

Ki �exp�5.37�1 � �i

��1 � T�1r i�

pr i. (4.42). . . . . . . . . . . . .

For decane,

Tr � T�Tc � (280 � 640)�(1, 111.8) � 0.666,

and KC10�

exp�5.37(1 � 0.4902)�1 � 1�0.666� �500�304�

� 0.0108,

and for the other components, KC1� 24.58 and KC4

� 0.8820.

Phase Split. With K-value estimates and the feed compositionknown, a phase split is made with either the Rachford-Rice15 or Mus-kat-McDowell14 algorithms. This results in vapor-phase mole frac-tion, Fv � 0.852187, and the compositions given in Table B-30.

EOS Constants for Each Phase Separately. EOS Constants Aand B must now be calculated separately for the vapor and liquidphases on the basis Compositions yi and xi. For decane,

a � �oa

R2T 2c

pc� , where �

oa � 0.45724;

b � �ob

RTcpc

, where �ob � 0.07780;

� � �1 � m�1 � Tr� � 2

;

and m � 0.37464 � 1.54226�� 0.26992�2 . (4.21). . . . . . .

The modified relation for m (Eq. 4.22) is used for decane becauseits acentric factor is greater than 0.4,

m � 0.3796 � 1.485� � 0.1644�2 � 0.01667�3.

(4.22). . . . . . . . . . . . . . . . . . . .

This gives

(m)C10� 0.3796 � 1.485(0.4902) � 0.1644(0.4902)2

� 0.01667(0.4902)3 � 1.070,

(Tr)C10� (280 � 460)(1, 111.8) � 0.666,

(�)C10� �1 � (1.070)�1 � 0.666� �2 � 1.432,

(a)C10�

�a R2 T 2c

pc� � 0.45724

(10.73)2(1, 111.8)2

304.0(1.432)

� 306, 500 psia-ft3�lbm mol�1,

and (b)C10� �b

RTcpc

� 0.07780(10.73)(1, 111.8)

304.0

� 3.053 ft3�lbm mol.

From Eq. 4.9,

A � ap

(RT)2 �2764

pr

T 2r

and B � bp

RT� 1

8pr

Tr(4.9). . . . . . . . . . . . . . . . . . . . . . . . . . . .

at 500 psia and 280°F. EOS Constants A and B for decane are

(A)C10� (306, 500) 500

(10.73)2(280 � 460)2 � 2.435

and (B)C10� (3.053) 500

(10.73)(280 � 460)� 0.1922,

and for other components AC1� 0.04906, AC4

� 0.4544, BC1

� 0.02701, and BC4� 0. 07308.

The Ai and Bi constants are the same for both phases. To calcu-late A and B constants for the vapor phase (Av and Bv) and the liquidphase (AL and BL), traditional mixing rules are used (Eq. 4.16).

Av ��Ni�1

�Nj�1

yi yj Ai j ,

AL ��Ni�1

�Nj�1

xi xj Ai j ,

Ai j � �1 � ki j� Ai Aj�

Bv ��Ni�1

yi Bi ,

and BL ��Ni�1

xi Bi .

16 PHASE BEHAVIOR

TABLE B-30—FUGACITY CALCULATION RESULTS (PROBLEM 18)

Componenti yi xi Ki�yi /xi

fvi(psia)

fLi(psia) fLi /fvi

Iteration 1 (Wilson K–Value Estimate)

C1 0.58262 0.02370 24.5823 298.023 85.3847 0.28650

C4 0.41186 0.46695 0.882021 149.526 151.385 1.01243

C10 0.00553 0.50935 0.010854 1.06778 3.35554 3.14253

Iteration 2 (Successive Substitution)

C1 0.57165 0.08117 7.04293 294.517 279.596 0.94934

C4 0.41277 0.46224 0.892986 148.515 148.117 0.99732

C10 0.01557 0.45659 0.034107 2.89097 3.05736 1.05755


C1 0.57115 0.08542 6.6861 294.392 292.992 0.99524

C4 0.41258 0.46326 0.89059 148.363 148.288 0.99950

C10 0.01628 0.45132 0.03607 3.01459 3.02765 1.00433


C1 0.57114 0.08583 6.6543 294.394 294.253 0.99952

C4 0.41254 0.46345 0.890144 148.344 148.332 0.99992

C10 0.01633 0.45072 0.036227 3.02324 3.02426 1.00034


C1 0.57114 0.08587 6.6511 294.396 294.381 0.99995

C4 0.41253 0.46348 0.890073 148.342 148.34 0.99999

C10 0.01633 0.45065 0.036239 3.02377 3.02385 1.00003

Iteration 6 (GDEM Promotion)

C1 0.57114 0.08588 6.65071 294.397 294.397 1.00000

C4 0.41253 0.46349 0.890061 148.342 148.342 1.00000

C10 0.01633 0.45064 0.03624 3.02379 3.02379 1.00000

Recall that the compositions yi and xi result from the phase-splitcalculation based on feed composition zi and the current K-value es-timates. For the initial K-value estimates and resulting compositionsfrom the phase-split calculation, EOS Constants A and B are

AL � (0.02370)(0.02370)[(0.04906)(0.04906)]0.5(1 � 0)

� (0.02370)(0.46695)[(0.04906)(0.4544)]0.5(1 � 0)

� (0.02370)(0.50935)[(0.04906)(2.435)]0.5(1 � 0)

� (0.46695)(0.02370)[(0.4554)(0.04906)]0.5(1 � 0)

� (0.46695)(0.46695)[(0.4554)(0.4554)]0.5(1 � 0)

� (0.46695)(0.50935)[(0.4554)(2.435)]0.5(1 � 0)

� (0.50935)(0.02370)[(2.435)(0.04906)]0.5(1 � 0)

� (0.50935)(0.46695)[(2.435)(0.4544)]0.5(1 � 0)

� (0.50935)(0.50935)[(2.435)(2.435)]0.5(1 � 0)� 1.252,

BL � (0.02370)(0.02701) � (0.46695)(0.07308)� (0.50935)(0.1922) � 0.1327,

Av � 0.1725,

and Bv � 0.04690.

Z-Factor Calculation. With the EOS constants for each phase,the Z factor (i.e., volume solution to the cubic EOS) can be solved.Eq. 4.20 is used for each phase separately.

Z3L � �1 � BL

�Z2L � �AL � 3B2

L � 2BL�ZL

� �AL BL � B2L � B3

L� � 0

and Z3v � (1 � Bv)Z2

v � �Av � 3B2v � 2Bv�Zv

� �Av Bv � B2v � B3

v� � 0.

The solutions to these two equations with Constant A and B valuescalculated in the previous section are ZL � 0.1812 andZv � 0.8785.

We can check, for example, the liquid solution by substitutingZL � 0.1812 into Eq. 4.20 together with AL � 1.252 andBL � 0.1327.

�(0.1812)3 � (1 � 0.1327)(0.1812)2

� �1.252 � 3(0.1327)2 � 2(0.1327) (0.1812)

� �1.252(0.1327) � (0.1327)2 � (0.1327)3

� 0.0005 � 0.

Fugacity Calculations. Fugacity values of each component foreach phase are calculated with Eq. 4.23,

lnfp�ln �Z � 1 � ln(Z � B) � A

2 2� B

� ln�Z� �1 � 2� � B

Z� �1 � 2� � B

and lnfi

yi p� ln i �

Bi

B(Z � 1) � ln(Z � B)

� A2 2� B�Bi

B� 2

A�Nj�1

yj Aij� ln�Z� �1 � 2� � B

Z� �1 � 2� � B ,

(4.23). . . . . . . . . . . . . . . . . . . .

EXAMPLE PROBLEMS 17

TABLE B-31—PHASE STABILITY TEST RESULTS (PROBLEM 18)

Componenti yi zi Ki

fyi(psia)

fzi(psia) S�fzi /fyi

Vapor–Like Stability Test: Ki�yi /zi*

C1 0.66910 0.50 1.3540 1,053 1,066 1.0118

C4 0.30930 0.42 0.7450 194.7 197.0 1.0118

C10 0.02166 0.08 0.2740 2.712 2.744 1.0118

Liquid–Like Stability Test: Ki�zi /yi**

C1 0.31870 0.50 1.5430 1,048 1,066 1.0168

C4 0.47670 0.42 0.8664 193.7 197.0 1.0168

C10 0.20460 0.08 0.3846 2.699 2.744 1.0168

*Unstable; converged solution, SV=1.0118, 12 iterations.**Unstable; converged solution, SL=1.0168, 6 iterations.

TABLE B-32—CONVERGED FLASH SOLUTION (PROBLEM 18)

Componenti

Initial K Values From Stability TestKi�(yi)v /(yi)L yi xi Ki=yi /xi

fvi(psia)

fLi(psia)

C1 2.08907 0.629843 0.330082 1.90814 1,019.52 1,019.52

C4 0.645515 0.348699 0.513307 0.67932 210.076 210.076

C10 0.10537 0.021457 0.156611 0.13701 2.26859 2.26859

giving the results in Table B-30. Component fugacities are clearlynot equal within an acceptable tolerance; e.g., ( fv)C10

� 1.068 psiaand ( fL)C1

� 3.355 psia. K values are then updated with the fugac-ity ratio, fL�fv, as a correction term.

K (n�1)i

� K (n)i

f (n)Li

f (n)vi

. (4.48). . . . . . . . . . . . . . . . . . . . . . . . . . .

This type of simple K-value update is called successive substitution,and for decane the second K-value estimate is given by

K (2)C10

� K (1)C10

f (1)L ,C10

f (1)v ,C10

� (0.01085) 3.3551.068

� (0.01085)(3.142) � 0.0341.

After the first GDEM promotion, convergence was achieved, result-ing in vapor-phase mole fraction of Fv � 0.853401. K values wereKC1

� 6.65071, KC4� 0.890061, and KC10

� 0.03624. TableB-30 gives the phase compositions.

b. Table B-31 gives the phase-stability test results at 1,500 psia and280°F. Results from the converged solutions of the vapor- and liquid-like tests are shown. Both stability tests indicated that the feed com-position was unstable and would therefore split into two (or more)phases. The vapor-like test required 12 iterations to converge, includ-ing two GDEM promotions. The liquid-like stability test required sixiterations to converge, including one GDEM promotion.

Because two unstable solutions were found, the two-phase flashcalculation was initialized with K values based on the two incipient-phase compositions found in the stability tests; i.e., Ki � (yi)v�(yi)L.With these initial estimates, the two-phase flash calculation con-verged in eight iterations, including one GDEM promotion. The finalvapor-phase mole fraction was Fv � 0.566844. Note how close thefinal converged K values are to the initial estimates from the stabilitytest. Table B-32 gives the results.

��

Problem. The following are calculated phase properties from theflash calculation at 500 psia and 280°F in Problem 18.

ML � 111.7 lbm�lbm mol,

Mv � 35.46 lbm�lbm mol,

vL � 2.721 ft3�lbm mol,

and vv � 13.837 ft3�lbm mol.

These molar volumes include the effect of a slight shift in volumeby use of volume translation. What is the phase molar volume andliquid density without volume translation?

Solution. The volume shift, ci, for each component is calculated fromEOS constants bi and the volume translation ratios, si, given in Prob-lem 18. Eq. 4.21 gives the bi values for the PR EOS.16

a � �oa

R2T 2c

pc� , where �

oa � 0.45724;

b � �ob

RTcpc

, where �ob � 0.07780;

� � �1 � m�1 � Tr� � 2

;

and m � 0.37464 � 1.54226�� 0.26992�2 . (4.21). . . . . . .

This givesbC1

� 0. 07780(10.7315)(343.0)�(667.8)

� 0.4288 ft3�lbm mol,

bC4� 1.160 ft3�lbm mol,

bC10� 3.053 ft3�lbm mol,

cC1� (� 0.1595)(0.4288) � � 0.06840 ft3�lbm mol

cC4� (� 0.0675)(1.1603) � � 0.07832 ft3�lbm mol,

and cC10� (0.0655)(3.053) � 0.2000 ft3�lbm mol.

From Eq. 4.25,

vL � v EOSL

��N

i�1

xi ci

and vv � v EOSv ��

N

i�1

yi ci . (4.25). . . . . . . . . . . . . . . . . . . . . . .

18 PHASE BEHAVIOR

TABLE B-33—RECOMBINED SEPARATOR WELLSTREAM MOLAR COMPOSITION AND CONSISTENCY CHECK OFSEPARATOR K VALUES WITH THE STANDING18 LOW-PRESSURE K-VALUE CORRELATION (PROBLEM 20)

zi KiComponent yi xi Reported Calculated Fi Standing Reported

CO2 4.01 1.12 3.84 3.84 2.087 3.18 3.58

N2 0.85 0.03 0.80 0.80 3.394 37.0 28.3

C1 89.83 10.68 85.12 85.16 2.606 8.42 8.41

C2 2.88 2.56 2.86 2.86 1.543 1.14 1.13

C3 1.30 3.86 1.45 1.45 0.811 0.289 0.337

i-C4 0.32 2.60 0.46 0.45 0.346 0.121 0.123

n-C4 0.43 5.31 0.72 0.72 0.180 0.0884 0.0810

i-C5 0.15 3.88 0.37 0.37 �0.256 0.0390 0.0387

C5 0.11 4.16 0.35 0.35 �0.391 0.0303 0.0264

C6 0.07 7.58 0.52 0.51 �0.859 0.0126 0.0092

C7+ 0.05 58.22 3.51 3.48 �2.010 0.00145 0.00086

Total 100.00 100.00 100.00 100.00

�C 7�0.7783 0.778 0.778

MC 7�135 98 135 135

M 100.2 18.6 23.4 23.4

With liquid compositions calculated in Problem 18 at 500 psia and280°F, the molar volume without volume translation, v EOS

L, is given

by

v EOSL

� 2.721 � [(0.08588)(� 0.0684) � (0.46349)

� (� 0.07832) � (0.45064)(0.2000)]

� 2.721 � (� 0.006 � 0.0363 � 0.0901),

� 2.769 ft3�lbm mol.

The molecular weight of liquid is needed to convert from molarvolume to density.

ML � (0.08588)(16.04) � (0.46349)(58.12)

� (0.45064)(142.29) � 92.44 lbm�lbm mol,

which gives

�L � 94.44�2.769 � 33.38 lbm�ft3.

��

Problem. Separator samples were collected during a production testfrom the discovery well of a gas-condensate reservoir. Use the Hoff-mann-Crump-Hocott17 (HCH) K-value plot (Eqs. 3.155 and 3.156)to check the consistency of measured separator compositions. Plotthe data together with the low-pressure Standing18 K-value correla-tion line given by Eq. 3.161. Also recombine the separator samplesto check the reported wellstream composition (laboratory recom-bined values can be in error). Finally, calculate the Watson charac-terization factor of the C7� component.

Solution. Table B-33 gives reported separator compositions; calcu-lated K values from the ratio of separator-gas to separator-oil molarcompositions, Ki � yi�xi; and finally, the recombined wellstreamcomposition, zi.

Separator conditions are 390 psig and 52°F. The HCH variable Fi

is given by Eq. 3.156, with bi and Tbi values given in Table 3.3.Methane, for example, has an Fi value given by

bi � 300 cycle��R,

Tbi � 94�R,

and Fi � 300 �1�94 � 1(52 � 460) � 2.606,

where modified values of bi and Tbi are given by Standing (insteadof values given by Eq. 3.156).

The K-value pressure product for methane is given by

Ki psp � �89.83�10.68�(390 � 14.7) � (8.411)(404.7),

� 3, 404 psia.

which is plotted vs. Fi � 2.606 on semilog paper (Fig. B-1).The Standing low-pressure K-value correlation is plotted together

with the measured K-values on Fig. B-1. From Standing’s18 correla-tion, Slope A0 and Intercept A1 are

Ki �1

psp10�A0 � A1 Fi

�, (3.161a). . . . . . . . . . . . . . . . . . . . . . .

Fi � bi�1�Tbi � 1�T�, (3.161b). . . . . . . . . . . . . . . . . . . . . . .

bi � log�pci�psc��1�Tbi � 1�Tci�, (3.161c). . . . . . . . . . . . . .

A0(p) � 1.2 � �4.5 � 10�4�p � �15 � 10�8�p2,

(3.161d). . . . . . . . . . . . . . . . . .

A1�p� � 0.890 � �1.7 � 10�4�p � �3.5 � 10�8�p2,

(3.161e). . . . . . . . . . . . . . . . . . .

nC7�� 7.3 � 0.0075T � 0.0016p, (3.161f). . . . . . . . . . . . .

bC7�� 1, 013 � 324nC7�

� 4.256n2C7�

, (3.161g). . . . . . .

and TbC7�� 301 � 59.85nC7�

� 0.971n2C7�

, (3.161h). . . . .

giving

A0 � 1.2 � �4.5 � 10�4�(404.7) � �15 � 10�8�(404.7)2

� 1.407

and A1 � 0.890 � �1.7 � 10�4�(404.7)

� �3.5 � 10�8�(404.7)2 � 0.8155.

The methane K value from the Standing correlation is, for example,

KC1� �1�404.7�10[1.407�(0.8155)(2.606)] � 8.415,

which can be compared with the measured value of 8.411.

EXAMPLE PROBLEMS 19

The Standing C7� K value requires calculating bC7� and TbC7�

from separator conditions.

nC7�� 7.3 � 0.0075(52) � 0.0016(404.7)

� 8.34 (approximate carbon number),

bC7�� 1, 013 � (324)8.34 � (4.256)(8.34)2

� 3, 419 cycle-�R,

TbC7�� 301 � (59.85)(8.34) � (0.971)(8.34)2 � 732.6�R,

and FC7�� 3, 419�(1)�(732.6) � (1)�(52 � 460) � � 2.01.

This yields

KC7�� 1�404.7�10[1.407�(0.8155)(�2.01)] � 0.00145,

which can be compared with the measured value of KC7�� 0.050

� 58.22 � 0.00086.From Fig. B-1 the measured K-value data plot as a straight line

almost coincident with the Standing correlation. This indicates thatthe measured compositions are probably consistent.

Recombination is made on the basis of separator gas/oil ratio,Rspwith Eq. 6.8. Separator-oil density and molecular weight areboth required for the recombination calculation, and most laborato-ries use the Standing-Katz8 density correlation to estimate �osp onthe basis of separator-oil composition (oil molecular weight is cal-culated from Eq. 6.9). For this sample, the separator properties andgas mole fraction, Fgsp, are given by Mosp � 100.2 lbm/lbm moland �osp � 45.28 lbm/ft3.

Fgsp � �1 �2, 130�osp

Mosp Rsp��1

, (6.8). . . . . . . . . . . . . . . . . . . .

which yields

Fgsp � �1 �(2, 130)(45.28)

(100.2)�(15, 357) �1

� 0.94102 .

The wellstream composition is calculated from Eq. 6.7.

zi � Fgsp yi � �1 � Fgsp�xi . (6.7). . . . . . . . . . . . . . . . . . . . .

For methane, this is

zC1� (0.94102)(89.83) � (1 � 0.94102)(10.68) � 85.16%,

which is quite close to the reported value of 85.12% (as it should be).Occasionally, because of entry errors to the recombination comput-er program or possibly because of inconsistent recombination GORused in the laboratory, reported wellstream compositions may notbe the same as those calculated with Eqs. 6.7 through 6.9. In thesesituations, contact the laboratory about the inconsistency. It mayeven be worthwhile to request a preliminary report of the separatorand recombined compositions before completing pressure/volume/temperature study.

��

Problem. A refrigeration/expansion process is used to reduce waterand condensate content of a gas stream. The well effluent arrives atthe separator at 2,000 psia and 155°F. It is cooled in the separatorand heat exchanger. Separator pressure is 1,000 psia, separator-gasgravity is 0.70 (air�1), and separator-gas rate is 65 MMscf/D.

What is the minimum temperature upstream of the choke to pre-vent hydrate formation in the separator?

What is the water content of the separator gas?How much water in lbm/D must be removed from the separator

gas if sales specifications call for a maximum dewpoint of �15°Fat 1,000 psia?

Solution. From Fig. 9.29, the temperature for hydrate formation ofa 0.7-gravity gas at 1,000 psia is about 69°F. From Eq. 9.23, the wa-ter content in the gas at 1,000 psia and 69°F is

yw � yow Ag As , (9.23a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ln yow �

0.05227p � 142.3 ln p � 9, 625T � 460

� 1.117 ln p � 16.44 , (9.23b). . . . . . . . . . . . . . . . .

Ag � 1 ��g � 0.55

�1.55 � 104 ��g T�1.446 � �1.83 � 104 T ��1.288 ,

(9.23c). . . . . . . . . . . . . . . . . .

As � 1 � �2.22 � 10�6�Cs , (9.23d). . . . . . . . . . . . . . . . . .

and As � 1 � �3.92 � 10�9�C1.44s . (9.23e). . . . . . . . . . . . . . . .

This gives

Ag � 1 � �(0.7 � 0.55)��1.55 � 104�(0.7)(69)�1.446

� �1.83 � 104�(69)�1.288 � � 0.9972,

ln yow �

(0.05227)(1, 000) � 142.3 ln(1, 000) � 9, 62569 � 460

� 1.117 ln(1, 000) � 16.44,

yow � 0.000546,

and yw � (0.9972)(0.000546) � 0.000544.

From Eq. 9.24, the solution water/gas ratio is given by

rsw � 135yw

1 � yw� 135yw , (9.24). . . . . . . . . . . . . . . . . . .

which gives rsw � (47, 300)(0.000544)�(1 � 0.000544) � 25.7lbm/MMscf.

At 15°F and 1,000 psia, the water content is

Ag � 0.9996,

yow � 8.61 � 10�5,

yw � 8.61 � 10�5,

and rsw � 4.0 lbm�MMscf.

At a separator-gas rate of 65 MMscf/D, the water removal capacitymust then be (25.7�4.0)(65)�1,445 lbm/D.

��

Problem. Estimate gas solubility for the reservoir brine in Problem21 at reservoir conditions of 4,050 psia and 255°F. Also estimatebrine density, compressibility, and FVF. The reservoir gas yields 13STB/MMscf (MMscf of separator gas) of a 69°API stock-tank con-densate. Brine salinity is 36,200 ppm total dissolved salts. Assumeseparator conditions are 1,000 psia and 80°F.

Solution. The reservoir (wellstream hydrocarbon) specific gravityis given by

�w ��g � 4, 580 rp �o

1 � 133, 000 rp ��M�o. (3.55). . . . . . . . . . . . . . . . . .

However, we need to estimate the amount and specific gravity ofthe gas coming from separator condensate at 1,000 psia and 80°F us-ing Eqs. 3.62 through 3.64.

�g� � A2 � A3 Rs�, (3.62). . . . . . . . . . . . . . . . . . . . . . . . . .

20 PHASE BEHAVIOR

where A2 � 0.25 � 0.2�API and A3 � � �3.57 � 10�6��API ;

Rs� �A1 A2

�1 � A1 A3� ; (3.63). . . . . . . . . . . . . . . . . . . . . . . . . . .

and �g ��g1 Rs1 � �gs1 Rs�

Rs1 � Rs�. (3.64). . . . . . . . . . . . . . . . . .

This gives

A1 � 1, 152,

A2 � 1.63,

A3 � � 2.46 � 10�4,

�g� � 0.985 (air � 1),

Rs� � 2, 620 scf�STB,

rp � 1��1��13 � 10�6� � 2, 620 � 12.6 STB�MMscf,

and �g � 0.711 (air � 1) .

From Fig. 9.2, the gas solubility of a 0.65°API gravity gas in purewater at 4,000 psia and 250°F is about 19 scf/STB. Pure methanesolubility in pure water at reservoir conditions can be estimatedfrom Eq. 9.6.

xC1� 10�6��3

i�0

��3

j�0

Ai jTj�pi , (9.6). . . . . . . . . . . . . . . .

yielding

xC1� 10�3� � 0.0256 � (0.00107)(4, 050) � �9.59 � 10�8�

� (4, 050)2 � �3.98 � 10�12�(4, 050)3 � 2.73 � 10�3

or Rosw � 7, 370��2.73 � 10�3��1 � �2.73 � 10�3� �

� 20. 2 scf�STB.

This compares with 22 scf/STB from Fig. 9.1. We assume thatRo

sw � 17.5 for this gas in pure water (from a plot of 19 scf/STB at�g � 0.65 and 22 scf/STB at �g � 0.55).

Reduction in solubility resulting from salinity can be estimatedfrom the Setchenow correction (Eqs. 9.9 and 9.10).19 For methane,the Setchenow constant is

(ks)C1�NaCl � 0.1813 � �7.692 � 10�4�T

� �2.6614 � 10�6�T 2 � �2.612 � 10�9�T 3 ,

(9.10). . . . . . . . . . . . . . . . . .

which gives

ks � 0.1813 � �7.692 � 10�4�(255)�2.6614 � 10�6�(255)2

� �2.612 � 10�9�(255) 3 � 0.115.

This can be corrected for specific gravity with Eq. 9.11, but we ne-glect the correction for simplicity. The resulting gas solubility of thebrine is then

Rsw

Rosw�

xg

xog� 10�kscs � 10��17.1�10�6� ksCs , (9.9). . . . . . . . .

resulting in

Rsw � (17.5)10��17.1�10�6� (0.115)(36,000) .

Brine density at standard conditions is given by Eq. 9.14, withTsc � 60°F [289 K].

v ow�psc,T� � 1

�ow�psc,T�

� A0 � A1ws � A2w2s ,

where A0 � 5.916365 � 0.01035794T

� �0.9270048 � 10�5�T 2

� 1, 127.522T�1 � 1, 00674.1T�2 ,

A1 � � 2.5166 � 0.0111766T � �0.170552 � 10�4�T 2 ,

and A2 � 2.84851 � 0.0154305T � �0.223982 � 10�4�T 2 ,

(9.14). . . . . . . . . . . . . . . . . . . .

yielding

A0 � 1.00106,

A1 � � 0.7112,

A2 � 0.2601,

ws � 36, 200 � 10�6 � 0.0362,

vw � 0.9756 cm3�g,

and �w � 0.9756 g�cm3.

At reservoir temperature (397 K) and standard pressure, brinedensity, �o

w, is also given by Eq. 9.14, which results in

A0 � 1.0642,

A1 � � 0.768,

A2 � 0.253,

vow � 1.0367 cm3�g,

and �ow � 0.9646 g�cm3.

Compressibility of brine without solution gas is given by

c*w�p, T � � �A0 � A1 p�

�1,

where A � 106�0.314 � 0.58ws � �1.9 � 10�4�T

��1.45 � 10�6�T 2

and A1 � 8 � 50ws � 0.125wsT , (9.17). . . . . . . . . . . . . . . . .

yielding

A0 � 0.289 � 106,

A1 � 8.656,

and c*w � 3.09 � 10�6 psi�1.

The FVF of brine without dissolved gas at atmospheric pressureis given by

Bow �

�w�psc, Tsc�

�ow�psc, T�

�v o

w�psc, T�

vw�psc, Tsc�, (9.13). . . . . . . . . . . . . . . . .

yielding Bow � 1.025�0.9646 � 1.063 bbl/STB.

From Eq. 9.18, the FVF of brine at reservoir pressure and temper-ature without dissolved gas is

B*w�p, T� � Bo

w�psc, T��1 �A1

A0p�

�1�A1�, (9.18). . . . . . . . . .

EXAMPLE PROBLEMS 21

giving

B*w � 1.063�1 � (4, 050)(8.656)��0.289 � 106�

��1�8.656�

� 1.049 bbl�STB.

With the Dodson-Standing20 corrections for compressibility andFVF as a function of gas solubility (Eqs. 9.19 and 9.20, respectively),the brine volumetric properties including gas solubility effect are

Bw�p, T, Rsw� � B*w�p, T��1 � 0.0001R 1.5

sw� (9.19). . . . . . . .

and cw�p, T, Rsw� � c*w�p, T� �1 � 0.00877Rsw� , (9.20). . . . . . .

which give

Bw � (1.049)�1 � (0.0001)�17.5 1.5� � 1.057 bbl�STB

and cw � �3.09 � 10�6��1 � (0.00877)�17.5 �

� 3.55 � 10�6 psi�1.

�!��

1. Hall, K.R. and Yarborough, L.: “A New EOS for Z-factor Calculations,”Oil & Gas J. (18 June 1973) 82.

2. Yarborough, L. and Hall, K.R.: “How to Solve EOS for Z–factors,” Oil& Gas J. (18 February 1974) 86.

3. Sutton, R.P.: “Compressibility Factors for High-Molecular-Weight Res-ervoir Gases,” paper SPE 14265 presented at the 1985 SPE Annual Tech-nical Conference and Exhibition, Las Vegas, Nevada, 22–25 September.

4. Lucas, K.: Chem. Ing. Tech. (1981) 53, 959.5. Lohrenz, J., Bray, B.G., and Clark, C.R.: “Calculating Viscosities of Res-

ervoir Fluids From Their Compositions,” JPT (October 1964) 1171;Trans., AIME, 231.

6. Wichert, E. and Aziz, K.: “Compressibility Factor of Sour NaturalGases,” Cdn. J. Chem. Eng. (1971) 49, 267.

7. Wichert, E. and Aziz, K.: “Calculate Z’s for Sour Gases,” Hydro. Proc.(May 1972) 51, 119.


9. Cragoe, C.S.: “Thermodynamic Properties of Petroleum Products,” U.S.Dept. of Commerce, Washington, DC (1929) 97.


11. Lee, A.L., Gonzalez, M.H., and Eakin, B.E.: “The Viscosity of NaturalGases,” JPT (August 1966) 997; Trans., AIME, 237.

12. Chew, J.N. and Connally, C.A.: “A Viscosity Correlation for Gas-Satu-rated Crude Oils,” Trans., AIME (1959) 216, 23.

13. Wilson, G.M.: “A Modified Redlich-Kwong EOS, Application to Gener-al Physical Data Calculations,” paper 15c presented at the 1969 AIChENatl. Meeting, Cleveland, Ohio.


15. Rachford, H.H. and Rice, J.D.: “Procedure for Use of Electrical DigitalComputers in Calculating Flash Vaporization Hydrocarbon Equilibri-um,” JPT (October 1952) 19; Trans., AIME, 195.


17. Hoffmann, A.E., Crump, J.S., and Hocott, C.R.: “Equilibrium Constantsfor a Gas-Condensate System,” Trans., AIME (1953) 198, 1.


19. Pawlikowski, E.M. and Prausnitz, J.M.: “Estimation of SetchenowConstants for Nonpolar Gases in Common Salts at Moderate Tempera-tures,” Ind. Eng. Chem. Fund. (1983).

20. Dodson, C.R. and Standing, M.B.: “Pressure, Volume, Temperature andSolubility Relations for Natural Gas-Water Mixtures,” Drill. & Prod.Prac., API (1944) 173.

"� #�� $��%�� &��

�� g/cm3

atm �1.013 250* E�05�Pabbl �1.589 873 E�01�m3

cp �1.0* E�03�Pa�sft �3.048* E�01�m

ft3 �2.831 685 E�02�m3

�F (�F�32)/1.8 ��C�F (�F�459.67)/1.8 �Kgal �3.785 412 E�03�m3

lbm �4.535 924 E�01�kglbm mol �4.535 924 E�01�kmol

psi �6.894 757 E�00�kPapsi�1 �1.450 377 E�01�kPa�1

�R 5/9 �Kton �9.071 847 E�01�Mg


EQUATION-OF-STATE APPLICATIONS 1

��

��

This appendix presents two examples of fluid characterization withan equation of state (EOS). The examples treat the gas condensateand the oil discussed in Chap. 6, Good Oil Co. Wells 7 and 4, respec-tively. Details of developing a complete fluid characterization aregiven for the gas-condensate fluid, including the splitting of C7�into five fractions, determining volume-translation coefficients forthe C7� fractions, and estimating methane through C7� binary in-teraction parameters (BIP’s). The resulting characterization is thestarting point for EOS predictions and, particularly, the simulationof pressure/volume/temperature (PVT) experiments.

��

The characterization is developed for the Peng-Robinson1 EOS (PREOS) on the basis of the C7� characterization suggested in Chap.5 with five C7� fractions. First, predictions are made without modi-fying the EOS parameters. Then, the measured dewpoint is matchedby modifying the BIP between methane and all C7� fractions. Fi-nally, constant-volume-depletion (CVD) data are matched by modi-fying the characterization with three regression parameters.

C7 + Molar Distribution. The first step in the C7� characterizationis to split the heptanes-plus component into five fractions by use ofthe Gaussian quadrature model in Chap. 5. In the absence of exper-imental true-boiling-point data, the following parameters are as-sumed: ��1, ��90, and N�5, with MC7��143 and �C7��0.795.

The value selected for heaviest fraction molecular weight, MN, issomewhat higher than the recommended value of MN � 2.5MC7�� 2.5(143) � 358. Instead, we use MN � 500, which allows us todevelop a better characterization (particularly the tail-like behaviorof the liquid-dropout curve). The modified �* term is

� *� �MN � ��XN � (500 � 90)�(12.6408) � 32.435,

where X5 is taken from Table 5.6.The � parameter is calculated from Eq. 5.30.

� � exp� ��*

MC7��

� 1�, (5.30). . . . . . . . . . . . . . . . . . .

giving

� � exp�(1)(32.435)�[(143 � 90) � 1]� � 0.67840.

Table C-1 gives values of f(X) for each fraction, according toEq. 5.31, together with calculated mole fractions and molecularweights based on quadrature points and weighting factors,Xi and Wi , respectively.

zi � zC7�[Wi f(Xi)],

Mi � � � �* Xi ,

and f�X � ��X ��1

�(�)

�1 � ln ��

�X . (5.31). . . . . . . . . . . . . . . . .

For the first fraction,

X1 � 0.263560,

W1 � 0.52175561,

f�X1� �

(0.263560)(1�1)

�(1)[1 � ln(0.67840)]1

(0.67840)0.263560 � 0.677878,

z1 � 6.85(0.52175561)(0.677878) � 2.4228,

and M1 � 90 � 32.435(0.263560) � 98.55.

C7 + Specific Gravities and Boiling Points. Given mole fractionsand molecular weights of the fractions, specific gravities are esti-mated with the Søreide2 correlation.3

�i � 0.2855 � Cf�Mi � 66�

0.13, (5.44). . . . . . . . . . . . . . .

where the characterization factor, Cf, is modified to ensure that thecalculated C7� specific gravity equals the measured value of�C7��0.795.

��C7��

exp�

zC7�MC7�

Ni�1

�zi Mi��i�. (5.37). . . . . . . . . . . . . . . . . . . . .

By trial and error, Cf � 0.28927 is found to satisfy Eq. 5.37; TableC-2 gives the results. For the first fraction,

�F1� 0.2855 � 0.28927(98.55 � 66)0.13 � 0.7404.

Normal boiling points are calculated from the Søreide correlation.

Tb � 1, 928.3 � �1.695 105� M�0.03522 �3.266

exp � � �4.922 10�3�M � 4.7685�

� �3.462 10�3�M�� , (5.45). . . . . . . . . . . . . . . . . . . .

2 PHASE BEHAVIOR

TABLE C-1—GAUSSIAN QUADRATURE METHOD TO SPLIT C7+INTO FIVE FRACTIONS FOR RESERVOIR GAS CONDENSATE

C7+Fraction

i

QuadraturePoint

Xi

QuadratureWeight

Wi f(Xi)

MoleFraction

zi

MolecularWeight

Mi

Massmi�ziMi

1 0.263560 0.52175561 0.677878 2.4228 98.55 238.8

2 1.413403 0.39866681 1.059051 2.8921 135.84 392.9

3 3.596426 0.07594245 2.470516 1.2852 206.65 265.6

4 7.085810 0.00361176 9.567521 0.2367 319.83 75.7

5 12.640801 0.00002337 82.58395 0.0132 500.00 6.6

Total 6.8500 143.00* 979.5

*Equals 979.5/6.85.

TABLE C-2—PROPERTIES OF C7+ FRACTIONS FOR RESERVOIR GAS CONDENSATE

C7+Fraction

i

MolecularWeight

Mi

Massmi�ziMi

SpecificGravity

�i *

IdealVolume

V�ziMi /�i

Boiling PointTb°R

1 98.55 238.8 0.7407 322.5 674.1

2 135.84 392.9 0.7879 498.6 793.9

3 206.65 265.6 0.8358 317.8 972.7

4 319.83 75.7 0.8796 86.1 1,175.5

5 500.00 6.6 0.9226 7.2 1,386.3

143.00 979.5 0.7950 1,232.1

*Water�1.

TABLE C-3—TWU4 METHOD FOR CALCULATING CRITICAL PROPERTIES OF C7+ FRACTIONSFOR RESERVOIR GAS CONDENSATE

Componenti

Tb(°R) TcP � �P* �* ��T fT

Tc(°R)

1 674.1 978.7 0.3112 0.6908 0.7404 �0.2195 0.003224 1,004.3

2 793.9 1,102.9 0.2802 0.7304 0.7879 �0.2498 0.003599 1,135.1

3 972.7 1,268.7 0.2333 0.7705 0.8358 �0.2783 0.003965 1,309.6

4 1,175.5 1,434.6 0.1807 0.8005 0.8796 �0.3267 0.004754 1,490.2

5 1,386.3 1,589.5 0.1278 0.8201 0.9226 �0.4008 0.006209 1,670.5

vcP(ft3/lbm mol) ��v fv

vc(ft3/lbm mol)

pcP(psia) ��p fp

pc(psia)

1 6.90 �0.2471 �0.0085 6.4475 393.8 �0.0245 0.00256 441.4

2 9.33 �0.2947 �0.0114 8.5142 314.2 �0.0283 0.00294 362.7

3 14.15 �0.3424 �0.0152 12.5336 220.0 �0.0321 0.00504 266.9

4 21.69 �0.4124 �0.0217 18.2317 142.5 �0.0388 0.01028 191.2

5 32.14 �0.5103 �0.0328 24.7141 87.2 �0.0499 0.02037 140.4

*Water�1.

which, for the first fraction, gives

Tb � 1, 928.3 � �1.695 105�(98.55)�0.03522 (0.7404)3.266

exp � � �4.922 10�3�(98.55) � 4.7685(0.7404)

� �3.462 10�3�(98.55)(0.7404)� � 674.1�R .

C7 + Critical Properties. Critical properties Tc and pc are calcu-lated from the Twu4 correlations (Eqs. 5.68 through 5.78). TableC-3 shows the calculations from left to right, in the order requiredto solve the rather tedious Twu correlations.

Acentric factor is calculated from the Lee-Kesler5 correlation.

� �� ln�pc�14.7� � A1 � A2 T�1

br� A3 ln Tbr � A4 T 6

br

A5 � A6 T�1br � A7 ln Tbr � A8 T 6

br

(5.60). . . . . . . . . . . . . . . . . . . .

for reduced normal boiling points Tbr � Tb�Tc 0.8. The Kesler-Lee6 correlation,

� � � 7.904 � 0.1352Kw � 0.007465K2w

� 8.359Tbr � �1.408 � 0.01063Kw�T�1br , (5.61). . . . . . .

is used to calculate higher reduced boiling points [making use of theWatson characterization factor defined by Eq. 5.34, (Kw � T1�3

b/�)].

Table C-4 shows the results.


TABLE C-4—CALCULATION OF ACENTRIC FACTOR FORC7+ FRACTIONS OF RESERVOIR GAS CONDENSATE

Componenti Tb/Tc

Kw[(°R)1/3] �

1 0.671 11.842 0.2864

2 0.699 11.752 0.3881

3 0.743 11.855 0.5754

4 0.789 11.998 0.8313

5 0.830 12.086 1.1185

Volume-Translation Parameters. Volume-translation parameters,si, for pure components through C6 are taken from Table 4.3. Valuesof si for the C7� fractions are determined to ensure that the EOScharacterization for each separate C7� fraction correctly calculatesa density at standard conditions that is consistent with the specificgravity of that fraction. The actual molar volume at standard condi-tions, v � M�(62.37�) in ft3/lbm mol, is equal to the EOS-calcu-lated molar volume, vEOS (without volume translation), less the vol-ume-translation parameter, c,

v � vEOS � c � (98.55)�[(62.37)(0.7404)]

� 2.1340 ft3�lbm mol.

The correct c is determined when vEOS and EOS Constants a and bin the PR EOS,

p � RTv � b

� av(v � b) � b(v � b)

, (4.19). . . . . . . . . . . .

calculate a pressure of 14.7 psia at T � Tsc . The EOS constants arecalculated from Eqs. 4.20 through 4.22.

Z3 � (1 � B)Z2 � �A � 3B2 � 2B�

Z � �AB � B2 � B3� � 0

and Zc � 0.3074 . (4.20). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

a � �oa

R2T 2c

pc� , where �

oa � 0.45724;

b � �ob

RTcpc

, where �ob � 0.07780;

� � �1 � m�1 � Tr� �� 2

;

and m � 0.37464 � 1.54226�� 0.26992�2 . (4.21). . . . . . .

m � 0.3796 � 1.485� � 0.1644�2 � 0.01667�3.

(4.22). . . . . . . . . . . . . . . . . . . .

for ��0.49. This results in

Tr � T�Tc � (60 � 460)�(1, 004.3) � 0.5174,

m � 0.7941,

� � 1.4955,

a � 1.7995 105 psia-ft3�lbm mol

and b � 1.8997 ft3�lbm mol.

By trial and error, the value of c that gives p�14.7 psia from Eq.4.19 is

c � 0.06151 ft3�lbm mol

or s � c�b � (0.06151)�(1.8997) � 0.0324 .

Table C-5 gives results for the other fractions.

BIP’s. The BIP’s between nonhydrocarbons and hydrocarbons aretaken from Table 4.1. The modified Chueh-Prausnitz7 equation,

kij � A��

1 �� 2v1�6ci

v1�6cj

v1�3ci

� v1�3cj

�B

��

, (5.79). . . . . . . . . . . . . . . .

is used for methane/C7� pairs with A�0.18 and B�6. For use withthis correlation, hydrocarbon critical volumes should be estimatedwith the following approximate correlation.

vci � 0.4804 � 0.06011Mi � 0.00001076M2i ,

with vc in ft3/lbm mol. For the first fraction, vcF1� 6.508 ft3/lbm mol.

By use of the same approximate relation for methane, vc � 1.447ft3/lbm mol and kij for this pair is

kC1�F1� 0.18 � 0.18� 2(1.447)1�6(6.508)1�6

(1.447)1�3 � (6.508)1�3�6

� 0.0301.

Table C-6 gives other methane/C7� BIP’s, and Tables C-7 and C-8summarize the PR EOS fluid characterization.

��

With the PR EOS characterization given in Tables C-7 and C-8, adewpoint pressure of 3,535 psia is predicted at reservoir tempera-ture of 186°F; this is approximately 500 psi lower than the measuredvalue. Figs. C-1 through C-4 show calculated EOS results. The liq-uid-dropout data are seriously overpredicted at pressures from2,500 to 3,500 psia. Otherwise, the predictions are quite reasonable.Wellstream compositions are acceptable, being somewhat too richat 3,500 psia and somewhat too lean at 2,900 psia.

��

Multiplying the BIP’s between methane and all C7� fractions by afactor of 2.09 matches the measured dewpoint pressure of 4,015 psia.Figs. C-1 through C-4 present calculated CVD results. The predictedPVT data are not very good; in particular, the liquid-dropout curve at3,515 psia is overpredicted (21.2% vs. the measured value of 3.3%)and equilibrium-gas C7� compositions are severely underpredicted.

TABLE C-5—CALCULATION (CHECK) OF VOLUME-TRANSLATION PARAMETERS, s, FOR C7+FRACTIONS IN RESERVOIR GAS CONDENSATE

i v � M�� * Tr�Tsc /Tc m � a b Guess c vEOS�v+cpcalc(psia) s�c/b

1 2.1340 0.5174 0.7941 1.4955 1.7995105 1.8997 0.06151 2.1955 14.7 0.0324

2 2.7644 0.4578 0.9325 1.6941 3.1686105 2.6127 0.14435 2.9087 14.4 0.0552

3 3.9644 0.3968 1.1829 2.0671 6.9940105 4.0964 0.44038 4.4048 14.5 0.1075

4 5.8295 0.3487 1.5100 2.6189 1.6022106 6.5089 1.00387 6.8334 14.8 0.1542

5 8.6897 0.3111 1.8582 3.3189 3.4744106 9.9361 1.58485 10.2745 14.5 0.1595

* � � 62.37�.

4 PHASE BEHAVIOR

TABLE C-6—CALCULATION OF METHANE/C7+ BIP’s FORRESERVOIR GAS CONDENSATE

ComponentApproximate vc

(ft3/lbm mol) Methane kij

Methane 1.447 —

Fraction 1 6.509 0.0301

Fraction 2 8.844 0.0416

Fraction 3 13.362 0.0582

Fraction 4 20.806 0.0763

Fraction 5 33.225 0.0945

�� !� ��

The measured CVD data, including dewpoint pressure, were thenmatched by modifying parameters in the original EOS (the charac-terization with predicted dewpoint of 3,535 psia). Three regression

parameters were chosen, and a sum-of-squares (SSQ) function wasminimized with a nonlinear regression algorithm. The SSQ functionis defined as

FSSQ �Mi

r 2i ,

where M�total number of measured data included in the regres-sion. The residuals, ri, are defined in terms of experimental data,dxi; calculated data, dci; and weight factors, wi. For dewpoint pres-sure and Z factors,

ri � �dxi � dci

dxi�wi .

For relative oil volumes, Vro, and cumulative gas produced, np�n,

ri � (dxi � dci)wi .

All weight factors, wi, are set to unity.

TABLE C-7—FINAL PR EOS CHARACTERIZATION FOR RESERVOIR GAS CONDENSATE

Component z MTc

(°R)pc

(psia) �

vc(ft3/lbm mol) Zc �*

Tb(°R) s�c/b

N2 0.0018 44.01 547.6 1,070.6 0.2310 1.505 0.2742 0.5072 350.4 �0.0577

CO2 0.0013 28.01 227.3 493.0 0.0450 1.443 0.2916 0.4700 139.3 �0.1752

C1 0.6192 16.04 343.0 667.8 0.0115 1.590 0.2884 0.3300 201.0 �0.1651

C2 0.1408 30.07 549.8 707.8 0.0908 2.370 0.2843 0.4500 332.2 �0.1070

C3 0.0835 44.10 665.7 616.3 0.1454 3.250 0.2804 0.5077 416.0 �0.0848

i-C4 0.0097 58.12 734.7 529.1 0.1756 4.208 0.2824 0.5631 470.6 �0.0686

C4 0.0341 58.12 765.3 550.7 0.1928 4.080 0.2736 0.5844 490.8 �0.0686

i-C5 0.0084 72.15 828.8 490.4 0.2273 4.899 0.2701 0.6247 541.8 �0.0410

C5 0.0148 72.15 845.4 488.6 0.2510 4.870 0.2623 0.6310 556.6 �0.0410

C6 0.0179 86.18 913.4 436.9 0.2957 5.929 0.2643 0.6640 615.4 �0.0154

F1 0.024227 98.55 1,004.4 441.5 0.2864 6.447 0.2640 0.7405 674.1 0.0322

F2 0.028921 135.84 1,135.1 362.7 0.3882 8.514 0.2535 0.7879 793.9 0.0552

F3 0.012852 206.65 1,309.6 266.9 0.5756 12.535 0.2380 0.8357 972.7 0.1075

F4 0.002367 319.83 1,490.2 191.1 0.8316 18.236 0.2179 0.8796 1,175.5 0.1544

F5 0.000132 500.00 1,670.4 140.3 1.1188 24.725 0.1935 0.9224 1,386.4 0.1599

*Water�1.

TABLE C-8—BIP’s FOR FINAL PR EOS CHARACTERIZATION OFRESERVOIR GAS CONDENSATE

N2 CO2 C1 C2 C3 i-C4 C4 i-C5 C5 C6 F1 F2 F3 F4 F5

N2 0

CO2 0 0

C1 0.025 0.105 0

C2 0.010 0.130 0 0

C3 0.090 0.125 0 0 0

i-C4 0.095 0.120 0 0 0 0

C4 0.095 0.115 0 0 0 0 0

i-C5 0.100 0.115 0 0 0 0 0 0

C5 0.110 0.115 0 0 0 0 0 0 0

C6 0.110 0.115 0 0 0 0 0 0 0 0

F1 0.110 0.115 0.030 0 0 0 0 0 0 0 0

F2 0.110 0.115 0.042 0 0 0 0 0 0 0 0 0

F3 0.110 0.115 0.058 0 0 0 0 0 0 0 0 0 0

F4 0.110 0.115 0.076 0 0 0 0 0 0 0 0 0 0 0

F5 0.110 0.115 0.095 0 0 0 0 0 0 0 0 0 0 0 0


Fig. C-1—CVD liquid-dropout behavior for gas-condensate ex-ample comparing measured, EOS-predicted, and dewpoint-matched calculations.

Fig. C-3—CVD equilibrium-gas C7+ behavior for gas-condensateexample comparing measured, EOS-predicted, and dewpoint-matched calculations.

The total number of data is 17 and includes one saturation pres-sure, six Z factors, five relative oil volumes, and five cumulative gasproductions. Because the number of data is somewhat limited, onlythree regression parameters are used. Initially, before parametershave been changed, three data contribute most to the SSQ function:pd , Zd , and Vro at 2,915 psia. Each is approximately 25% of thetotal SSQ. The initial SSQ is approximately (FSSQ)i � 0.05.

Regression I. The first regression uses the following three regres-sion parameters: P1, the multiplier to BIP’s between methane andall C7� fractions; P2, the multiplier to Tc for all C7� fractions; andP3, the multiplier to pc for all C7� fractions. Fig. C-5 shows the re-duction in the SSQ function at each iteration. The final SSQ valueis approximately 4% of the initial value (0.002/0.05). Six iterationswere required to find a minimum. Practically, however, the mini-mum was located after four iterations, with only small parameter ad-justments made during the last two iterations. The final parametersare P1 � 4.34, P2 � 0.910, and P3 � 0.849.

Figs. C-6 and C-7 show the change in the multipliers at each it-eration. The BIP multipliers increase monotonically to a value ofapproximately 4.3, resulting in C1 through C7� BIP’s ranging from0.13 to 0.40. The large BIP values are outside the range of what isprobably acceptable because they generally should not exceedapproximately 0.3 for the PR EOS.

C7� critical temperatures decreased almost monotonically toapproximately 10% less than the initial values. C7� critical pressur-es increased during the first iterations, then finally decreased to

Fig. C-2—CVD Z-factor behavior for gas-condensate examplecomparing measured, EOS-predicted, and dewpoint-matchedcalculations.

Fig. C-4—CVD equilibrium-gas C7+ molecular-weight behaviorfor gas-condensate example comparing measured, EOS-pre-dicted, and dewpoint-matched calculations.

Fig. C-5—Reduction in SSQ function for regression cases withthree different sets of parameters to match measured gas-con-densate PVT data.

approximately 15% below the starting values. At Iteration 3, theminimum SSQ was almost reached, but the multiplier to criticalpressures was approximately 1.0. During the last three iterations,the multiplier was reduced to 0.85 without any significant reduction

6 PHASE BEHAVIOR

Fig. C-6—Variation in C1 through C7+ BIP multipliers used in Re-gressions I and II to match measured gas-condensate PVT data.

Fig. C-8—CVD liquid-dropout behavior for gas-condensate exam-ple comparing measured and EOS Regression I calculations.

Fig. C-10—CVD equilibrium-gas C7+ behavior for gas-conden-sate example comparing measured and EOS Regression I cal-culations.

in the SSQ function. This indicates that C7� critical pressures areprobably not very important when matching PVT data and thatanother parameter could be chosen instead.

Figs. C-8 through C-11 show calculated results for the CVD ex-periment. Dewpoint pressure was overpredicted by only 8 psi

Fig. C-7—Variation in C7+ critical-property multipliers used inRegression I to match measured gas-condensate PVT data.

pc

Tc

Fig. C-9—CVD Z-factor behavior for gas-condensate examplecomparing measured and EOS Regression I calculations.

Fig. C-11—CVD equilibrium-gas C7+ molecular-weight behaviorfor gas-condensate example comparing measured and EOS Re-gression I calculations.

(0.2%) despite the relative low weight factor used (a factor of 10 ormore is commonly used). Also, the experimental liquid dropout of3.3% at 3,515 psia was 4.9% with the modified characterization, avery good match.


Fig. C-12—Variation in the two C7+ critical-temperature multipli-ers used in Regression II to match measured gas-condensatePVT data.

Fig. C-14—CVD Z-factor behavior for gas-condensate examplecomparing measured and EOS Regression II calculations.

Regression II. The second regression uses the following threeregression parameters: P1, the multiplier to BIP’s between meth-ane and all C7� fractions; P2, the multiplier to Tc for C7� frac-tions F1 through F3; and P3, the multiplier to Tc for C7� frac-tions F4 and F5. Fig. C-5 shows the reduction in the SSQfunction at each iteration. The final SSQ function value isapproximately 3% of the initial value (0.017/0.05). Four itera-tions were required to find a minimum. The final parameters areP1 � 2.29, P2 � 0.932, and P3 � 1.047.

Figs. C-6 and C-12 show the change in the multipliers at each it-eration. The BIP multipliers converged to a value of approximately2.3, resulting in C1 through C7� BIP’s ranging from 0.07 to 0.22.These BIP values are reasonable for the PR EOS.

C7� critical temperatures for fractions F1 through F3 de-creased to approximately 7% less than the initial values. C7� criti-cal temperatures for fractions F4 and F5 increased, fluctuatingfrom 2 to 10% above the initial values, finally converging to a 5%increase. Figs. C-13 through C-16 show calculated results for theCVD experiment.

This regression gives an excellent match of almost all measuredPVT data, including the data used in the regression and equilibrium-gas compositions and properties that were not included in the re-gression. Dewpoint pressure was overpredicted by 8 psi (0.2%),which is sufficiently close, although a larger weight factor (e.g., 10)would force the calculated dewpoint to match the measured valuealmost exactly. On the other hand, the experimental accuracy ofdewpoint pressure is less than 0.2% and further refinement with a

Fig. C-13—CVD liquid-dropout behavior for gas-condensate ex-ample comparing measured and EOS Regression II calculations.

Fig. C-15—CVD equilibrium-gas C7+ behavior for gas-conden-sate example comparing measured and EOS Regression II cal-culations.

Fig. C-16—CVD equilibrium-gas C7+ molecular-weight behaviorfor gas-condensate example comparing measured and EOS Re-gression II calculations.

larger weight factor is probably not justified. Finally, the measuredliquid dropout of 3.3% at 3,515 psia was calculated to be 4.9%, alsoa very good match.

8 PHASE BEHAVIOR

Fig. C-17—Variation in the three C7+ critical-temperature multi-pliers used in Regression III to match measured gas-condensatePVT data.

Fig. C-19—CVD Z-factor behavior for gas-condensate examplecomparing measured and EOS Regression III calculations.

MeasuredCalculated

Regression III. Results almost as good as those for Regression II areachieved by fitting only critical temperatures of the C7+ fractions,namely multipliers to Tc(F1, F2), Tc(F3, F4), and Tc(F5), withthe final parameters being P1 � 0.915, P2 � 1.023, andP3 � 1.239 (Fig. C-17). The converged FSSQ � 0.0026 is 5% of theinitial value. The C1 through C7� BIP’s are the same as those usedin the prediction, ranging from 0.03 to 0.095. Calculated dewpoint is4,044 psia (0.7%), and liquid dropout at 3,515 psia is 4.9% comparedwith the measured value of 3.3%. Figs. C-18 through C-21 comparecalculated and measured results for the CVD experiment.

Comparing Different Fluid Characterizations. More analysis isneeded to determine whether any real difference exists between thefluid characterizations determined in Regressions II and III. Fordepletion calculations, the results are almost identical. For gas cycl-ing, however, they may provide quite different results.

When limited PVT data are available to tune an EOS (as in thisexample), it usually is good practice to evaluate two or three “equal-ly good” characterizations. As in our example, different modifyingparameters might be used. Alternative EOS’s can also be tried [e.g.,the Soave-Redlich-Kwong EOS8 (SRK EOS) with the Pedersen etal.9 fluid characterization as a starting point]. Each fluid character-ization can then be evaluated with the results from compositionalsimulation of the reservoir process being studied.

Fig. C-18—CVD liquid-dropout behavior for gas-condensateexample comparing measured and EOS Regression III calcula-tions.

Fig. C-20—CVD equilibrium-gas C7+ behavior for gas-conden-sate example comparing measured and EOS Regression III cal-culations.

Fig. C-21—CVD equilibrium-gas C7+ molecular-weight behaviorfor gas-condensate example comparing measured and EOS Re-gression III calculations.

Generating Modified Black-Oil PVT Data. Figs. C-22 throughC-24 present modified black-oil PVT properties calculated with thevarious characterizations discussed earlier. Figs. C-22 and C-23give oil properties Rs (solution gas/oil ratio) and Bo [oil formation


Fig. C-22—Modified black-oil PVT property solution gas/oil ratio,Rs , vs. pressure for gas-condensate example for three EOSmodels: dewpoint-match only and Regressions I and II.

volume factor (FVF)]. Note that these oil properties do not increasemonotonically, as is usually exhibited by reservoir oils. The reasonis that the first condensate that drops out just below the dewpoint isrelatively “heavy” compared with the condensate that drops out atlower pressures.

For example, the fluid characterization from Regression II yieldsa stock-tank-oil (STO) gravity of 45°API produced from the reser-voir condensate at the dewpoint, a 50°API STO produced from thereservoir condensate at 3,515 psia, and a 53°API STO producedfrom the reservoir condensate at 3,000 psia. The correspondingsolution gas/oil ratios at dewpoint, 3,515, and 3,000 psia are 1,500,1,880, and 2,100 scf/STB, respectively, and oil FVF’s are 1.835,2.109, and 2.319 bbl/STB, respectively.

This behavior is typical for gas condensates with a “tail” on theliquid-dropout curve; i.e., the retrograde condensation is small in apressure interval just below the dewpoint (approximately 500 psi inthis example), with the start of a more rapid increase in retrogradecondensation occurring at some lower pressure (at approximately3,500 psia in this example). In the region of the tail retrograde be-havior, produced reservoir gas has only slight changes in composi-tion during depletion because only small amounts of the heaviestcomponents are being lost from the original reservoir gas. Thisshould be reflected in the EOS characterization by only slight de-crease in C7� composition. The behavior should also be reflectedby modified black-oil PVT property rs (solution oil/gas ratio) of thereservoir gas. Solution oil/gas ratio should decrease only slightly inthe region of the tail-like retrograde condensation.

Fig. C-24—Modified black-oil PVT property solution oil/gas ra-tio, rs , vs. pressure for gas-condensate example for three EOSmodels: dewpoint-match only and Regressions I and II.

Fig. C-23—Modified black-oil PVT property saturated-oil FVF,Bo , vs. pressure for gas-condensate example for three EOSmodels: dewpoint-match only and Regressions I and II.

Referring again to the fluid characterization from Regression II,calculated rs decreases only slightly from 136 STB/MMscf at thedewpoint to 122 STB/MMscf at 3,515 psia. Compared with largerdecreases in rs at lower pressures (e.g., to 88 STB/MMscf at 3,015psia), the slight decrease in rs predicted from dewpoint to 3,515 psiaappears very reasonable. Calculations from Regressions I and IIIalso show similar rs behavior. Fig. C-25 summarizes the effect oftreating the tail-like retrograde behavior properly with an EOS fluidcharacterization. The figure plots cumulative stock-tank condensateproduced during depletion on the basis of modified black-oil PVTdata (rs) generated with the fluid characterizations discussed pre-viously. In particular, the characterization based only on fitting thedewpoint pressure is compared with the the fluid characterizationsdetermined in Regressions I and II. The effect on condensate recov-ery is clear from the comparison.

��"��

The second example treats the oil in Chap. 6, Good Oil Co. Well 4.This is a slightly volatile oil with a bubblepoint of 2,600 psi at 220°F,an initial solution gas/oil ratio of 750 scf/STB, and a bubblepoint oilFVF of 1.45 RB/STB. In this example, we look at two EOS charac-terizations. The first characterization uses the PR EOS with theSøreide2 and Whitson10 methods for developing three C7� frac-tions. This approach is basically the same as that used for the gascondensate presented earlier. The second characterization uses theSRK EOS with the Pedersen et al.9 method for characterizing the

Fig. C-25—CVD-based cumulative condensate recovery vs.pressure for gas-condensate example for three EOS models:dewpoint-match only and Regressions I and II.

10 PHASE BEHAVIOR

TABLE C-9—COMPOSITIONS OF RESERVOIR OIL AND EQUILIBRIUM GAS AND K VALUESAT 2,600-psia BUBBLEPOINT PRESSURE AND 220°F

Bubblepoint-Oil Composition Equilibrium-Gas Composition K Values at Bubblepoint

Component PR EOS SRK EOS PR EOS SRK EOS PR EOS SRK EOS

N2 0.16 0.16 0.52 0.59 3.28 3.66

CO2 0.91 0.91 1.31 1.43 1.44 1.57

C1 36.47 36.47 77.13 76.97 2.11 2.11

C2 9.67 9.67 10.16 10.57 1.05 1.09

C3 6.95 6.95 4.87 4.95 0.70 0.71

i-C4 1.44 1.44 0.77 0.78 0.54 0.54

C4 3.93 3.93 1.85 1.82 0.47 0.46

i-C5 1.44 1.44 0.51 0.50 0.36 0.35

C5 1.41 1.41 0.46 0.44 0.33 0.31

C6 4.33 4.33 1.00 0.94 0.23 0.22

F1 15.91 19.07 1.35 0.97 0.085 0.051

F2 14.28 9.31 0.0623 0.0358 0.0044 0.0038

F3 3.11 4.91 0.000050 0.000110 0.000016 0.000022

TABLE C-10—PR EOS CHARACTERIZATION OF RESERVOIR OIL WITH SØREIDE-WHITSON C7+ METHOD2,3,10

Component MTc

(°R)pc

(psia) � s�c/b ��

Tb(°R)

vc(ft3/lbm mol) Zc

N2 28.01 227.3 493.0 0.0450 �0.1930 0.4700 139.3 1.443 0.2916

CO2 44.01 547.6 1,070.6 0.2310 �0.0820 0.5072 350.4 1.505 0.2742

C1 16.04 343.0 667.8 0.0115 �0.1590 0.3300 201.0 1.590 0.2884

C2 30.07 549.8 707.8 0.0908 �0.1130 0.4500 332.2 2.370 0.2843

C3 44.10 665.7 616.3 0.1454 �0.0860 0.5077 416.0 3.250 0.2804

i-C4 58.12 734.7 529.1 0.1756 �0.0840 0.5631 470.6 4.208 0.2824

C4 58.12 765.3 550.7 0.1928 �0.0670 0.5844 490.8 4.080 0.2736

i-C5 72.15 828.8 490.4 0.2273 �0.0610 0.6247 541.8 4.899 0.2701

C5 72.15 845.4 488.6 0.2510 �0.0390 0.6310 556.6 4.870 0.2623

C6 86.18 913.4 436.9 0.2957 �0.0080 0.6640 615.4 5.929 0.2643

F1 120.08 1,086.6 397.1 0.3419 0.0403 0.7750 746.2 8.333 0.2838

F2 255.96 1,401.5 230.0 0.6866 0.1255 0.8618 1,070.9 17.562 0.2685

F3 545.00 1,707.3 137.0 1.2213 0.1326 0.9354 1,424.3 28.250 0.2113

*Water�1.

three C7� fractions. Both EOS characterizations predict the mea-sured PVT data reported in Chap. 6 (Tables 6.2 through 6.7) reason-ably well. The characterizations are not modified by regression inthis example (possibly an interesting exercise for the reader). Thetwo characterizations are presented first. Calculated results are thencompared with measured data reported in Chap. 6. Finally, a studyof modified black-oil PVT properties is given.

Peng-Robinson1 Characterization.The methods presented for thegas condensate in the Gas-Condensate-Fluid Characterization sec-tion (see also Sec. 5.6) were used to develop a fluid characterizationfor this reservoir oil. Three C7� fractions, determined with theGaussian quadrature approach, were used. Table C-9 gives molefractions of the reservoir oil, Table C-10 gives component proper-ties, and Table C-11 provides BIP’s. Volume translation was usedto ensure accurate volumetric predictions.

Soave-Redlich-Kwong Characterization.8 For comparison, thePedersen et al.9 characterization procedure (Sec. 5.6) was used todevelop an EOS description of the same reservoir oil. The split ofthe C7� fraction is made by use of an exponential distribution toC80, then regrouping in subfractions with approximately equal mass

fractions. Tables C-9, C-12, and C-13 give the resulting composi-tion and properties.

TABLE C-11—BIP’s FOR PR EOS CHARACTERIZATION OFRESERVOIR OIL

Component N2 CO2 C1

N2 0.000

CO2 0.000 0.000

C1 0.025 0.105 0.000

C2 0.010 0.130 0.000

C3 0.090 0.125 0.000

i-C4 0.095 0.120 0.000

C4 0.095 0.115 0.000

i-C5 0.100 0.115 0.000

C5 0.110 0.115 0.000

C6 0.110 0.115 0.000

F1 0.110 0.115 0.035

F2 0.110 0.115 0.063

F3 0.110 0.115 0.092


TABLE C-12—SRK EOS CHARACTERIZATION WITH PEDERSEN et al.9 C7+ METHOD FOR RESERVOIR OIL

Component MTc

(°R)pc

(psia) � s�c/b ��*Tc

(°R)vc

(ft3/lbm mol) Zc

N2 28.01 227.3 493.0 0.0450 �0.0080 0.4700 139.3 1.443 0.2916

CO2 44.01 547.6 1,070.6 0.2310 0.0830 0.5072 350.4 1.505 0.2742

C1 16.04 343.0 667.8 0.0115 0.0230 0.3300 201.0 1.590 0.2884

C2 30.07 549.8 707.8 0.0908 0.0600 0.4500 332.2 2.370 0.2843

C3 44.10 665.7 616.3 0.1454 0.0820 0.5077 416.0 3.250 0.2804

i-C4 58.12 734.7 529.1 0.1756 0.0830 0.5631 470.6 4.208 0.2824

C4 58.12 765.3 550.7 0.1928 0.0970 0.5844 490.8 4.080 0.2736

i-C5 72.15 828.8 490.4 0.2273 0.1020 0.6247 541.8 4.899 0.2701

C5 72.15 845.4 488.6 0.2510 0.1210 0.6310 556.6 4.870 0.2623

C6 86.18 913.4 436.9 0.2957 0.1470 0.6640 615.4 5.929 0.2643

F1 133.98 1,079.5 354.8 0.5935 0.1535 0.7899 802.3 9.561 0.2928

F2 258.05 1,307.1 232.5 0.9030 0.1422 0.8577 1,075.7 16.734 0.2774

F3 468.63 1,615.2 199.4 1.2322 �0.0422 0.9247 1,369.0 26.756 0.3078

TABLE C-13—BIP’s FOR SRK EOS CHARACTERIZATIONOF RESERVOIR OIL

Component N2 CO2 C1

N2 0.000

CO2 0.000 0.000 0.000

C1 0.020 0.150 0.000

C2 0.060 0.150 0.000

C3 0.080 0.150 0.000

i-C4 0.080 0.150 0.000

C4 0.080 0.150 0.000

i-C5 0.080 0.150 0.000

C5 0.080 0.150 0.000

C6 0.080 0.150 0.000

F1 0.080 0.150 0.000

F2 0.080 0.150 0.000

F3 0.080 0.150 0.000

Analyzing EOS Results. Fig. C-26 plots oil density vs. pressure.EOS predictions are accurate at the bubblepoint, somewhat too highat undersaturated conditions, and significantly overpredicted at lowpressures. Overall, the predictions are quite good, particularly in theimportant pressure regions. Fig. C-27 shows the differential oil vol-ume factor, Bod. The EOS predictions are similar, slightly overpre-dicting the undersaturated oil compressibility and overpredictingthe shrinkage of oil at lower pressures.

A useful graphical analysis for undersaturated oil behavior is alog-log plot of oil relative volume, Bod�Bod, b, vs. the pressure ratiop�pb (Fig. C-28). The slope of this plot yields Constant A(A��slope), where instantaneous oil compressibility is given by

co � A�p , (3.107). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

and cumulative oil compressibility, co (used in material-balanceequations), is given by

co � Api � p ln�pi�p� . (C-1). . . . . . . . . . . . . . . . . . . . . . . . . .

The constant A is a characteristic value for an oil reservoir withconstant bubblepoint pressure. With Eqs. 3.107 and C-1, theconstant allows easy calculation of undersaturated oil compressibil-ity at any reservoir pressure. For reservoirs with bubblepoint varia-tion, A correlates well with bubblepoint pressure (approximatelylinear and increasing with bubblepoint).

For this example, the plot in Fig. C-28 indicates that measureddata intercept the relative oil volume ratio, Vro � Bod�Bod,b, at a

Fig. C-26—DLE oil-density behavior for the reservoir-oil exam-ple comparing measured and EOS predictive models based onthe PR and SRK EOS’s that use the Søreide-Whitson meth-od2,3,10 and the Pedersen et al.9 method, respectively, for char-acterizing C7+ fractions.

value of log�p�pb� � 0.015, corresponding to p�pb � 1.03. Such

an intercept indicates that the reported bubblepoint pressure isapproximately 3% too low. Forcing measured data throughlog(Vro) � 0 should be done only if it results in a linear trendthrough all the data. For this example, it is very difficult to honor thereported bubblepoint pressure and still have a linear plot that passesthrough most of the reported oil relative volume data.

EOS results confirm that a linear trend with a zero interceptshould be expected. The PR EOS has a slope of 0.070, slightly lessthan the SRK EOS slope of 0.074. The measured data have a slope(with nonzero intercept) of 0.059. Then, resulting oil compressibili-ties at initial pressure of 5,000 psia are

(co)meas � 0.059�5, 000 � 11.8 10�6 psi�1,

(co)PREOS � 0.070�5, 000 � 14.0 10�6 psi�1,

and (co)SRKEOS � 0.074�5, 000 � 14.8 10�6 psi�1.

Cumulative oil compressibilities are given by

(co)meas � �0.059�(5, 000 � 2, 650)� ln(5, 000 � 2, 650)

12 PHASE BEHAVIOR

Fig. C-27—DLE differential-oil FVF behavior for the reservoir-oilexample comparing measured and EOS predictive modelsbased on the PR/Søreide-Whitson method2,3,10 and theSRK/Pedersen et al.9 method.

� 15.9 10�6 psi�1,

(co)PREOS � 18.9 10�6 psi�1,

and (co)SRKEOS � 19.9 10�6 psi�1.

Returning to Fig. C-27, the SRK EOS seems to underpredict dif-ferential oil volume factors more than the PR EOS. Practically, how-ever, the two characterizations predict nearly identical oil shrink-age. This is seen in Fig. C-29, which shows the oil volume ratio,Vro � Bod�Bod, b, vs. pressure. This ratio gives a true measure of thereservoir-oil shrinkage during depletion, whereas the ratio Bod ismisleadingly related to the “meaningless” residual oil volume. Wehighly recommend that the ratio Bod�Bod, b be used as “data” in re-gression (instead of Bod directly) to ensure accurate oil shrinkagefrom the EOS without also having to fit the residual oil volume atstandard conditions. [The residual oil is of no practical interest be-cause it will never be produced to the surface and probably never ex-

Fig. C-29—DLE oil-shrinkage behavior for reservoir-oil exam-ple comparing measured and EOS predictive models based onthe PR/Søreide2 method and the SRK/Pedersen et al.9 method.

Fig. C-28—DLE undersaturated-oil-volume (compressibility) be-havior for reservoir-oil example comparing measured and EOSpredictive models based on the PR/Søreide-Whitson meth-od2,3,10 method and the SRK/Pedersen et al.9 method.

Measured intercept implies reported pb too low.

isted in the reservoir. Furthermore, the experimental procedure usedin reducing the pressure from the last stage of depletion (approxi-mately 150 psi) to standard pressure and reservoir temperature in-volves bleeding the system down slowly. This bleeding is a non-equilibrium process that cannot really be simulated with a PVTpackage (it can be estimated by a series of 5 to 10 additional deple-tion stages, starting at the lowest reported depletion stage)].

Fig. C-30 shows the differential solution gas/oil ratio vs. pres-sure and indicates that both EOS characterizations overpredict themeasured data by 5 to 10%. Correcting this deviation from mea-sured data may lead to unnecessary and severe changes in the EOScharacterization. What is really important to predict are (1) theseparator flash gas/oil ratio (GOR) and (2) the cumulative gascoming out of solution during depletion.

Table C-14 shows the calculated and measured separator data.Interestingly, calculated separator gas/oil ratios are 1 to 2% lowerthan measured data. That is, the differential GOR’s are consider-

Fig. C-30—DLE solution gas/oil ratio behavior for reservoir-oilexample comparing measured and EOS predictive modelsbased on the PR/Søreide2 and SRK/Pedersen et al.9 methods.


TABLE C-14—MEASURED AND CALCULATED TWO-STAGE SEPARATOR TEST RESULTSFOR RESERVOIR OIL

GOR(scf/STB) ��g*

Bo(bbl/STB)

�API(°API)

MeasuredStage 1 (315 psia and 75°F)Stage 2 (14.7 psia and 60°F)Total or at bubblepoint

549246795

0.7041.2860.884

1.1481.0071.495 40.1

PR EOS CharacterizationStage 1Stage 2Total or at bubblepointPercent deviation

559219778

�2.1

0.7071.2720.866�2.0

1.1291.0061.483�0.8

40.10.0

SRK EOS CharacterizationStage 1Stage 2Total or at bubblepointPercent deviation

569216785

�1.2

0.7121.2700.865�2.1

1.1241.0061.494�0.1

39.0�2.6

*Air�1.

ably overpredicted, while the separator gas/oil ratios are onlyslightly underpredicted. Clearly, the separator gas/oil ratio predic-tions are accurate enough, satisfying the first requirement givenpreviously. However, the question is how well the EOS character-izations estimate cumulative gas coming out of solution. Fig.C-31, which plots �Rsd,b � Rsd

��Bod, b vs. pressure, shows this. Thefigure indicates that the measured data are somewhat overpre-dicted by both EOS’s (the two characterizations give very similarresults). Although the overprediction is not excessive, these data[ �Rsd,b � Rsd

��Bod, b] could be used in regression (together with oilshrinkage data Bod�Bod,b) to improve the EOS characterization.

Fig. C-32 shows the gas specific gravity of equilibrium gas re-leased during the differential-liberation experiment (DLE). The EOScharacterizations predict the measured data accurately, with slight un-derestimation at the two highest pressures. Laboratory gas specificgravities may be difficult to measure accurately because of samplingprocedures that can result in loss of liquids during transfer from thePVT cell to the sampler. Such errors would tend to result in specificgravities that are too low, the opposite of what Fig. C-32 shows.

A problem that may arise in fitting reservoir gas specific gravitieswith an EOS is the choice and number of components used to de-scribe the C7� fraction. Often the lightest EOS C7� fraction consti-

Fig. C-31—DLE cumulative-released-gas behavior for reservoir-oil example comparing measured and EOS predictive modelsbased on the PR/Søreide2 and SRK/Pedersen et al.9 methods.

tutes most of the total C7� material in the calculated reservoir gasphase (in certain pressure regions). If this component is too heavyor too light compared with the actual C7� material of the reservoirgas, it will cause the EOS-calculated gas specific gravity to be tooheavy or too light. For this example, the Soave-Redlich-Kwongcharacterization with 12 C7� fractions gave basically the same gasspecific gravities (within 1%) for all pressures down to 200 psia.

Gas specific gravity usually is not important in reservoir engi-neering calculations of oil reservoirs, particularly if gas Z factors arepredicted accurately. However, because the equilibrium-gas specif-ic gravity indirectly reflects the gas composition (and thus the liquidyield from the reservoir gas), it may be important for gas-conden-sate and volatile-oil reservoirs where a significant amount of stock-tank-liquid production comes from the reservoir gas phase.

Fig. C-33 shows the equilibrium-gas-phase Z factor. At pressuresjust below the bubblepoint the PR EOS predicts the measured data ac-curately, while the SRK EOS predicts the data somewhat better at in-termediate and lower pressures. Neither EOS predicts the generalshape of the measured Z-factor curve. As an independent check of theEOS Z factors, the Standing-Katz11 correlation (Eq. 3.42) was usedwith specific gravities from the PR EOS results and with the Sutton12

Fig. C-32—DLE released (equilibrium) -gas specific-gravity be-havior for reservoir-oil example comparing measured and EOSpredictive models based on the PR/Søreide2 and SRK/Pedersenet al.9 methods.

14 PHASE BEHAVIOR

Fig. C-33—DLE released (equilibrium) -gas Z-factor behavior forreservoir-oil example comparing measured and EOS predictivemodels based on the PR/Søreide2 and SRK/Pedersen et al.9methods; Standing-Katz11 Z factors calculated on the basis ofPR gas compositions also shown.

pseudocritical properties (Eq. 3.47). Fig. C-33 presents the Standing-Katz Z factors as open circles. The results are closest to the SRK EOSZ factors, which is not surprising. The SRK EOS usually gives bettergas volumetric properties than the PR EOS for methane-rich systems.

Fig. C-34 presents the oil viscosities. Measured values arecompared with calculated values by use of the Lohrenz-Bray-Clark13 correlation, with compositions and densities from EOS re-sults. Experimental oil viscosities are difficult to obtain with an ac-curacy of more than approximately 5 to 10%, so the resultspresented here are acceptable. To obtain these calculated results,the critical volumes of C7� fractions were modified by regressingon measured oil viscosities and reported (calculated) gas viscosi-ties. The default critical volumes were increased 10 to 20% to ob-tain the match. The modifications to C7� critical volumes differfor every reservoir system, mainly because the Lohrenz-Bray-Clark correlation is strongly dependent on both critical volumes

Fig. C-35—DLE oil-viscosity vs. -density behavior for reservoir-oilexample comparing measured and EOS predictive models basedon PR/Søreide2 and SRK/Pedersen et al.9 methods.

Oil Density, lbm/ft3

Fig. C-34—DLE oil-viscosity behavior for reservoir-oil examplecomparing measured and Lohrenz-Bray-Clark13 (LBC) viscositymodel (regressed Vc)/predictive PR EOS/Søreide2 and SRKEOS/Pedersen et al.9 methods.

and oil densities. The modifications are usually less when oil den-sities are accurately predicted by the EOS.

A useful plot for correlating oil viscosities measured at differentlaboratories is oil viscosity vs. density (Fig. C-35). Reservoir oilsfrom the same reservoir should have a unique viscosity/density rela-tionship. (One exception would be if a reservoir exhibited composi-tional gradients characterized by variation in relative oil paraffinic-ity/aromaticity.) Because most laboratories measure oil densityaccurately (i.e., consistently from one laboratory to another), erro-neous viscosity data from a laboratory will plot parallel to the reser-voir’s correct viscosity/density relation, shifted by a more or lessconstant amount.

Reported gas viscosities, even though they are calculated with acorrelation (on the basis of measured specific gravities), should beaccurate within 5% or less. Therefore, including gas viscosities inthe viscosity regression ensures that critical volumes of the C7�

Fig. C-36—DLE oil- and gas-viscosity behavior for reservoir-oilexample comparing measured and LBC viscosity model13 (re-gressed Vc )/predictive PR EOS/Søreide2 and SRK EOS/Peder-sen et al.9 methods.


Fig. C-37—Modified black-oil PVT property solution gas/oil ratio,Rs , for reservoir-oil example comparing measured/convertedand EOS predictive models based on the PR/Søreide2 and SRK/Pedersen et al.9 methods.

fractions are not modified unrealistically (i.e., to the point where gasviscosities are no longer predicted accurately). Fig. C-36 shows gasand oil viscosities together for this reservoir system.

Generating Black-Oil PVT Data. In this section, we consider cal-culation of modified black-oil PVT properties (Chap. 7). We lookat the problems involved in generating consistent black-oil PVTproperties for a reservoir with a gas cap in equilibrium with an un-derlying reservoir oil and try to determine whether black-oil PVTproperties are the same for the gas cap and reservoir oil.

Several other questions are also raised. How accurate are reser-voir phase densities calculated from black-oil PVT data? What sur-face gravities should be chosen? How do differential data correctedwith separator flash data (Eqs. 6.32 and 6.33) compare with resultsfrom the Whitson-Torp14 method? And finally, how should modi-fied black-oil PVT data be extrapolated for saturation conditionsabove the original saturation condition?

Fig. C-39—Modified black-oil PVT property solution oil/gas ratio,rs, for reservoir-oil example comparing EOS predictive modelsbased on the PR/Søreide2 and SRK/Pedersen et al.9 methods.

Fig. C-38—Modified black-oil PVT property saturated-oil FVF,Bo , for reservoir-oil example comparing measured/convertedand EOS predictive models based on the PR/Søreide2 and SRK/Pedersen et al.9 methods.

Gas Cap and Reservoir-Oil PVT. The Whitson-Torp method wasused to develop modified black-oil PVT for the reservoir oil with thetwo EOS characterizations presented previously. A DLE was simu-lated where the equilibrium oil and equilibrium gas from each stageof depletion was passed separately through a two-stage separator(300 psia at 75°F and 14.7 psia at 60°F). Figs. C-37 through C-40present the results for the reservoir oil for the two characterizationsas solid lines.

The reservoir was then considered to have a gas cap. The gas-capcomposition was taken from the bubblepoint calculation (Table C-9).This gas was depleted by a CVD experiment, where the equilibriumgas and equilibrium oil from each stage of depletion was passed sepa-rately through a two-stage separator under the same conditions as inthe previous paragraph. Figs. C-37 through C-40 present the resultsfor the two characterizations for the reservoir gas as dashed lines.

As Figs. C-37 through C-39 show, significant differences in mo-dified black-oil PVT data exist for the two characterizations. Signif-

Fig. C-40—Modified black-oil PVT property dry gas FVF, Bgd ,for reservoir-oil example comparing EOS predictive modelsbased on the PR/Søreide2 and SRK/Pedersen et al.9 methods.

16 PHASE BEHAVIOR

icant differences are also seen between the PVT properties gener-ated from the reservoir oil and the reservoir gas. The difference inPVT properties calculated for the two EOS characterizations seemsconsiderably larger than the differences in predictions of measuredPVT data. This is because the differences in black-oil PVT data liemainly in the gas-phase properties, which are not well-defined bythe experimental PVT data. Comparison of equilibrium-gas com-positions (Table C-9) supports the differences seen in Figs. C-37through C-39. Table C-9 shows that more C7� material is predictedby the PR EOS for the bubblepoint equilibrium gas.

The significant differences in black-oil PVT properties calculatedfrom the reservoir oil and gas cause a real dilemma. First, most res-ervoir simulators require that saturated Rs and Bo data increasemonotonically with pressure. From Figs. C-37 and C-38, we see thatonly the reservoir-oil PVT data satisfy this requirement. This leadsto the question of how use of the reservoir-oil PVT properties in thegas cap would affect reservoir performance? The answer can onlybe found by comparing black-oil with compositional simulations.

Another concern is choosing the surface oil and gas gravities.These gravities are used together with the pressure-dependent black-oil properties to calculate reservoir phase densities (Eq. 7.6). Typical-ly only one oil gravity and one gas gravity can be provided to a reser-voir simulator. If phase densities are important, then care must betaken to choose the surface gravities that give the best reservoir phasedensities, particularly in the range of pressures most important to thereservoir recovery mechanisms. For this example, the oil specific gra-vities range from 0.715 (from the reservoir gas) to 0.825 (from thereservoir oil) and the gas specific gravities range from 0.88 to 0.91.

Figs. C-37 and C-38 show the black-oil properties Bo and Rs cal-culated with Eqs. 6.32 and 6.33 on the basis of conversion of differ-ential-liberation data by use of separator test results. For this partic-ular oil, the traditional conversions are not bad, somewhatoverpredicting Bo and Rs. For more volatile oils, the difference canbe much more significant.

For a reservoir system that is initially undersaturated, the fluid canbecome saturated at a pressure greater than the initial saturationcondition. For example, the reservoir oil in this example might pro-duce at a low flowing bottomhole pressure that results in a high gassaturation near the wellbore. During a shut-in period, the pressureincrease near the wellbore will saturate the free gas developed dur-ing production. If the Rs vs. pressure curve increases only to the ini-tial bubblepoint and remains constant at higher pressures, the gaswould stop dissolving in the oil at the initial bubblepoint. To ensurethat free gas continues to dissolve into the oil at higher pressures, theRs curve must be extrapolated to higher pressures.

One approach to developing an extension to the Rs curve is toadd a small amount of equilibrium gas (evolved at the original bub-blepoint pressure) to the original oil. A new bubblepoint is deter-mined for the new mixture. The separator gas/oil ratio is also deter-mined, thereby providing a new point on the Rs vs. (bubblepoint)pressure curve. This procedure can be continued at increasing bub-blepoint pressures until the initial reservoir pressure is reached.Alternatively, equilibrium gas from each new bubblepoint can beused to generate the next mixture. This approach is often used toestimate the PVT properties for a reservoir that exhibits bubble-point variation with depth.

Two problems may arise when generating an extension of the Rs

curve. Either a maximum in bubblepoint pressure may be reachedthat is less than the initial reservoir pressure or a dewpoint insteadof a bubblepoint may be calculated, indicating that the procedurehas passed through a critical condition. In either of these situa-tions, completing the extension of the Rs to the initial pressure isnot possible. If successful, this method generates an extension tothe original Rs curve that may become flat or even exhibit a de-creasing slope at higher pressures.

Immiscible gas injection into an undersaturated oil reservoir de-fines a second situation that requires an extrapolated Rs curve. Thedevelopment of the extrapolated Rs for this situation is somewhatdifferent. Here, the injection gas should be used to determine mix-tures with increasing bubblepoints and GOR’s. A swelling test withthe injection gas can be simulated to obtain the necessary mixturesfor extending the Rs curve.

The same two problems that can occur with the equilibrium-gasprocedure also can occur with this method. Namely, that a maxi-mum can be reached below the initial pressure and that transitionthrough a critical mixture can result in a dewpoint condition. A rich-er injection gas tends to cause both problems, whereas leaner injec-tion gas may avoid the problems (depending somewhat on the de-gree of undersaturation). Extension of the Rs curve with this methodusually results in a relatively steep increase in Rs at increasing pres-sures. Leaner injection gas results in steeper curves.

Caution should be used in modeling a gas-injection process withmodified black-oil PVT properties, particularly when significantphase behavior effects are expected (e.g., vaporization and swelling).The Cook et al.15 method (Chap. 7) for modifying black-oil PVTproperties for vaporizing immiscible gas injection processes andcompositional simulation are alternatives that can be considered.

��


2. Søreide, I.: “Improved Phase Behavior Predictions of Petroleum Reser-voir Fluids From a Cubic Equation of State,” Dr.Ing. dissertation, Nor-wegian Inst. of Technology, Trondheim, Norway (1989).

3. Whitson, C.H., Andersen, T.F., and Søreide, I.: “C7� Characteriza-tion of Related Equilibrium Fluids Using the Gamma Distribution,”C7� Fraction Characterization, L.G. Chorn and G.A. Mansoori(eds.), Advances in Thermodynamics, Taylor & Francis, New YorkCity (1989) 1, 35–56.

4. Twu, C.H.: “An Internally Consistent Correlation for Predicting theCritical Properties and Molecular Weights of Petroleum and Coal-TarLiquids,” Fluid Phase Equilibria (1984) No. 16, 137.


6. Kesler, M.G. and Lee, B.I.: “Improve Predictions of Enthalpy of Frac-tions,” Hydro. Proc. (March 1976) 55, 153.

7. Chueh, P.L. and Prausnitz, J.M.: “Calculation of High-Pressure Vapor–Liquid Equilibria,” Ind. Eng. Chem. (1968) 60, No. 13.

8. Soave, G.: “Equilibrium Constants from a Modified Redlich-KwongEquation of State,” Chem. Eng. Sci. (1972) 27, No. 6, 1197.




12. Sutton, R.P.: “Compressibility Factors for High-Molecular Weight Reser-voir Gases,” paper SPE 14265 presented at the 1985 SPE Annual Techni-cal Conference and Exhibition, Las Vegas, Nevada, 22–25 September.



15. Cook, R.E., Jacoby, R.H., and Ramesh, A.B.: “A Beta-Type ReservoirSimulator for Approximating Compositional Effects During Gas Injec-tion,” SPEJ (October 1974) 471.

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bbl 1.589 873 E�01�m3

ft3 2.831 685 E�02�m3

�F (�F�32)/1.8 ��Clbm mol 4.535 924 E�01�kmol

psi 6.894 757 E�00�kPapsi�1 1.450 377 E�01�kPa–1

�R 5/9 �K

UNDERSTANDING LABORATORY OIL PVT REPORTS 1

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The subject of how to read and make proper use of information con-tained in laboratory pressure/volume/temperature (PVT) reportshas not been treated adequately in course texts. This is borne out bycomments of students in a basic petroleum engineering course, whofind the subject one of the most difficult to understand in the wholecourse. I hope the following discussion of the why and whereforeof a typical PVT report will be helpful. The discussion pertains toReport RFL 10641 on the Raleigh field contained in this section.Sample pages of this report are given at the end of this appendix.

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First, the form of data presentation in the report developed becauseof its use in material-balance calculations. Some of the tabular in-formation is set up to satisfy that need. Second, the report shouldcover all past, present, and future situations that might require cal-culations. To do this with a minimum of tables and curves, the dataare normalized to a reference state and only data for the referencestate are given. The petroleum engineer must then “work back”from the reference state to a particular situation.

Third, the laboratory tests are carried out on the basis of two differ-ent thermodynamic processes being under way at the same time.These are (1) flash equilibrium separation of gas and oil in the surfacetraps during production and (2) differential equilibrium separation ofgas and oil in the reservoir during pressure decline. As a consequence,the report gives both flash and differential data and it becomes neces-sary to be able to shift between the two sets of data. Finally, the reportgives data on the particular sample obtained. This may not be theproper “average” of all the fluid in the reservoir, and slight adjustmentof the data may be necessary at a later time. Therefore, some detailis given to the manner of obtaining the sample and the conditions thatexist at the sampling time. Also, the compositional analysis of thesample is given so that equilibrium calculations can be made forconditions other than those studied in the laboratory.

With these generalities in mind, we now consider specific datapresented in the Raleigh report. The surface flash separation data areconsidered first, followed by the reservoir differential data. We thenconsider how to convert certain differential data to equivalent flashdata. Page numbers refer to the pages of report.

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As we show later, one form of the material-balance equation is anequality between the expansion of the original reservoir oil (be-tween the initial pressure and any subsequent pressure) and the vol-ume voidage that has occurred down to the subsequent pressure.The separator test data on Page 5 of the report, which shows thequantity of surface gases and stock-tank oil (STO) that results when1 bbl of bubblepoint oil is flashed through certain surface trap se-quence, allows computation of voidages. The tabulation also givesthe oil gravity (°API) of the STO and, in some instances, the gravityof gas coming from the primary trap.

Cols. 1 and 2 give the pressure/temperature condition of the sur-face trap tests that were investigated. These should be specified bythe reservoir engineer at the time the test is planned so that they willapply to future field operations. Referring to the bottom line of data,the surface situation modeled here is a two-stage separation [i.e., aprimary trap operating at 200 psig and 73°F, followed by a stocktank operating at 14.7 psia (0 psig) and 73°F].

When 1 bbl of bubblepoint oil (defined in Footnote 2 as oil satu-rated at 3,236 psig and 258°F) is flashed (processed) through thistrap arrangement, the STO amounts to 0.5974 bbl and has a qualityof 48.5°API (Cols. 6 and 5). The formation volume factor (FVF)of the bubblepoint oil, Bob, is therefore 1/0.5974�1.674 bbl/bblSTO (Col. 7).

Cols. 3 and 4 show the surface gas/oil ratio from the trap and tank.The primary trap ratio is 875 ft3/bbl STO, and the tank vaporsamount to 134 ft3/bbl STO. The solution gas/oil ratio at bubblepointconditions (3,236 psig and 258°F), Rsb, is 875�134�1,009 ft3/bblSTO when flashed through this surface trap arrangement. As thistable shows, Rsb, Bob, and oil gravity all vary with the trap pressure/temperature situation. Surface-gas gravity does also, but usually isreported only for the single-stage atmosphere flash. To calculatereservoir voidage properly, the measured STO and the produced gashave to be handled according to the information in this table. How-ever, note that these data always refer only to the bubblepoint oil asthe reference fluid. Determination of FVF’s for other reservoirfluids requires additional information.

2 PHASE BEHAVIOR

FV

F, v

olu

me/

resi

du

al v

olu

me

Fig. D-1—FVF for oil and gas and for total (oil plus gas) system as a function of pressure aboveand below bubblepoint.

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We now consider situations at reservoir pressures greater than thebubblepoint pressure. We first look at the FVF, then at the fluid den-sity, and then the compressibility of the fluid. Cols. 1 and 2 of theReservoir Fluid Sample Tabular Data on Page 3 give the pressure/vol-ume relations of the original fluid at 258°F. Note that the data are pres-ented in terms of a unit volume at the bubblepoint condition. Col. 2gives the volume of the system at pressure p per unit system volumeat 3,236 psig and 258°F. These are listed as relative volumes (i.e., rel-ative to the bubblepoint).

Consider the FVF of the original oil in the reservoir. On Page 1,we see that the original pressure (listed as last reservoir pressure un-der well characteristics) was 5,783 psig at �12,650 ft. Thus, if wewant the oil FVF at 5,783 psig, we obtain it by multiplying the FVFat the bubblepoint by the relative volume (to the bubblepoint),Vrel � VR�Vob . We multiply because

Bo �VR

Vo�

Vob

Vo

VR

Vob(D-1). . . . . . . . . . . . . . . . . . . . . . . . . .

and the reference bubblepoint oil volume cancel out. Therefore, Boi,the initial FVF, is 1.674�0.9424�1.577 when the 200-psig prima-ry trap is involved. It will be different if another trap pressure is used.

Reservoir oil density at pressures greater than 3,236 psig alsomake use of the relative-volume data of Col. 2, Page 3. The addedinformation is the density of the bubblepoint oil. This is always giv-en in the summary data on Page 2 of the report. We see here that thespecific volume at the bubblepoint vôb � 0.02772 ft3/lbm. Thiscomes from direct weight/volume measurements on the sample inthe PVT cell. If now we wish the density, �oi, of the initial reservoiroil, we have

�o �1

vôi�

1vôb

1Vrel

(D-2). . . . . . . . . . . . . . . . . . . . . . . . . . . .

and �oi �1

0.02772 � 0.9424� 38.3 lbm�ft3 . (D-3). . . . . . .

Compressibility of reservoir oil at pressures higher than the bub-blepoint is also obtainable from the relative-volume data. Recallthat the definition of compressibility is

co �1V�dV

dp�

T

. (D-4). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

It makes no difference whether the volume units in the equation arerelative volumes to the bubblepoint, to FVF’s, or to specific volumevalues. To evaluate co at pressure p, it is only necessary to differenti-ate the p � Vrel data in Cols. 1 and 2 graphically to get dV/dp at the

pressure and divide by Vrel. A less accurate value can be obtainedby the assumption

co �1

Vrel

��Vrel

�p�

T

. (D-5). . . . . . . . . . . . . . . . . . . . . . . . . . . .

For example, to get co at 4,500 psig by use of relative-volume val-ues of 500 psi on each side of 4,500 psig

co �1

(0.9562 � 0.9781)�2(0.9562 � 0.9781)(5, 000 � 4, 000)

�1

0.96710.02191, 000

� 22.7 � 10�6 vol�vol-psi.

Note that Page 2 of the report lists some compressibility numbers.These are not the same as those indicated earlier because they arechanges in volume (in the pressure interval indicated) per unit vol-ume at the lower pressure. For example, the value of 22.33�10�6

for the 5,000- to 4,000-psi interval is obtained as

10.9781

(0.9562 � 0.9781)(5, 000 � 4, 000)

� 22.39 � 10�6.

The compressibility data on Page 2 are set up in this manner becauseof the way they are used in one form of material balance.

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We have seen that, to calculate the FVF of the oil at pressures higherthan bubblepoint, we multiply the bubblepoint FVF times the rela-tive volume given in Col. 2, Page 3. Obviously, if we multiplyBob by Vrel at pressures less than pb, we also get an FVF. In fact, weget the total FVF, Bt, of the original system. That is, at p � pb, wewill have two phases and Bt is the volume relation of both gas andliquid phases in equilibrium at pressure p (Fig. D-1).

We mentioned earlier that one form of the material balance makesuse of the expansion of the original oil between the initial systempressure and any subsequent pressure. This expansion is given by

Eo � N(Bt � Boi) , (D-6). . . . . . . . . . . . . . . . . . . . . . . . . . . .

where N�initial stock-tank barrels in the reservoir and(Bt � Boi) �the expansion per unit STO; therefore, Eo �expan-sion (in barrels) of the original oil system. Sometimes the expansionequation is written

Eo � N(Bt � Bti) . (D-7). . . . . . . . . . . . . . . . . . . . . . . . . . . . .

At p pb, whether the FVF is considered to be total FVF or an oilFVF makes no difference, it is the same thing. For example, see Eq.4.4 of Ref. 2 or Eq. 8.17 of Ref. 3.


Initialbubblepoint

oil

Fig. D-2—PVT cell volume vs. pressure during differential vaporization test (showing oil shrink-age) and incremental liberated gas volumes (b–b and c–c) at pressures below bubblepoint.

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Up to this point, we have considered what happens when reservoirfluid comes to the surface and is separated into surface gas and oilproducts. We modeled flash equilibrium conditions because we be-lieve that the action going on in the trap is essentially one where thewhole system entering the trap immediately separates into two com-ponents, trap gas and trap liquid. This constitutes the elements of aflash separation.

The standard PVT report includes data referred to as “differentialdata.” These are gas-solubility and phase-volume data taken in amanner to model what some people believe happens to the oil phasein the reservoir during pressure decline. The argument that differen-tial-liberation tests model the subsurface behavior comes primarilyfrom two things.

1. Reservoir pressure changes are not as violent or as large as thepressure changes that occur when entering surface traps. The subsur-face changes are more gradual and might be considered to be a seriesof infinitesimal changes.

2. Because of the relative permeability characteristics of reservoirrock/fluid systems, the gas phase moves toward the well at a fasterrate than the liquid phase. As a result, the overall composition of theentire reservoir system is changing.

These two ideas promote the idea that a test procedure modeledon a differential process should be used to study subsurface behav-ior. Because of experimental limitations and time/cost consider-ations, a laboratory cannot perform a true differential procedure.Instead, it performs a series of stepwise flashes at the reservoir tem-perature (usually about 10) beginning at the bubblepoint. Of course,the greater the number of steps, the more closely the true differentialprocess is modeled.

The differential data are reported in the last three columns of Page3. Supplementary differential-release data are given on Page 4. Notethat the three columns are headed “Differential Liberation at258°F.” The best way to understand these data is to explain how thevalues are obtained. The laboratory starts with a known volume ofthe original system in the PVT cell, which may be of the order of 100to 200 cm3. The volume at the bubblepoint pressure (3,236 psig inthis instance) is determined accurately because it is a reference forall subsequent measurements.

Referring to Page 3, we see that the first pressure step was to 2,938psig. At this pressure, the original system will be in two phases. Itsvolume would be at b on Fig. D-2. The first step in altering the over-all system composition is made at 2,938 psig by removing the gasphase from the PVT cell while maintaining constant pressure. Thequantity of gas removed is determined by collecting it in a calibratedcontainer. The volume that the gas phase occupied in the cell is de-

termined by the amount of mercury injected during the removal pro-cess. Also, the gas gravity is measured on the sample bleedoff. Thevolume of liquid remaining in the cell is at Point b on Fig. D-2.

This procedure is repeated by taking the 2,938-psig saturatedliquid to 2,607 psig (Point c) and removing a second batch of gasat that pressure. Again the volume of the displaced gas in the cellat 2,607 psig is determined along with the gravity of the removalgas. The volume of liquid phase remaining after the second gas-re-moval step is illustrated by Point c in Fig. D-2. This process of re-moving batches of equilibrium gas continues until the cell pressureat the last displacement is 0 psig. The differential data on Page 3show 11 equilibrium removals, all at 258°F. The final volume ofliquid phase remaining in the cell at 0 psig and 258°F is correctedby thermal-expansion tables (or by cooling the cell) to 0 psig and60°F. This 0-psig/60°F liquid is called residual oil. Note that resid-ual oil and STO are not the same thing. They are both products ofthe original oil in the system but are developed by different pres-sure/temperature routes.

Once residual oil has been reached, the data obtained are recalcu-lated and presented on the basis of a unit barrel of residual oil. Thecumulative amount of gas removed from the cell (liberated fromsolution) at each pressure step is given as a gas/oil ratio(GOR). Col.4 shows that 183 ft3/bbl residual oil was liberated between 3,236 and2,938 psig and 362 ft3/bbl residual oil was liberated between 2,938and 2,607 psig. By the time 0 psig and 258°F had been reached, theoriginal system had liberated 1,518 ft3/bbl residual oil. Col. 5 showsthe amount of gas in solution at the various pressures. This is the dif-ference of the 1,518 ft3 total liberated and the amount liberated be-tween the original bubblepoint pressure and that pressure. For ex-ample, the solution gas/oil ratio at 2,938 psig is 1,518�183�1,335ft3/bbl residual oil.

At this point, be sure that you understand why the solution gas/oilratio determined from surface flash and from differential removalwill be different. It is because the processes for obtaining residualoil and STO from bubblepoint oil are different. The first is a multipleseries of flashes at the elevated reservoir temperature, and the sec-ond is generally a one– or two-stage flash at low pressures and lowtemperature. The quantity of gas released will be different, and thequantity of final liquid will be different. Also, the quality (gravity)of the products will be different (compare °API of residual oil with°API of STO). The only thing that will be the same for the two pro-cesses is the total weight of the end products.

Col. 6 gives the relative volumes of the liquid phase measured dur-ing the differential liberation of gas. Note that these are volumes atpressure p per unit volume of residual oil. Again, these relative vol-umes must not be confused with FVF’s because FVF’s are specifiedper barrel of STO. Note on Page 3 that relative volumes start at 1.000

4 PHASE BEHAVIOR

Fig. D-3—Differential vaporization and flash-corrected solution gas/oil ratio vs. pressure aboveand below bubblepoint pressure.

at 0 psig/60°F and that the value of 1.109 at 0 psig/258°F is the ther-mal expansion of 42.2°API residual oil from 60 to 258°F. At pressur-es higher than 3,236 psig (the original bubblepoint), the system com-position remained constant. Therefore, the relation of the relative oilvolume at ppb to the bubblepoint value, 2.075, must be the sameas the relative-volume numbers in Col. 2 (e.g., 1.948/2.075�0.9387at 6,000 psig).

The data on Page 4 are differential liberation data that refer to theoil and gas phases in the reservoir at 258°F. Col. 2 shows that thegravity of the 183 ft3/bbl residual oil liberated between 3,236 and2,938 psig was 0.870. The next batch between 2,938 and 2,607 psig(362�183�179 ft3/bbl residual oil) was 0.846. The gas deviation(compressibility) factor of the first liberated gas was 0.886 at 2,938psig. The oil density at 2,938 psig/258°F was 0.5905 g/cm3.

Once you understand the basic difference between flash and dif-ferential data as given in the standard PVT report, proceed to cal-culation of flash solubilities and oil FVF’s at less than the bubble-point from the differential data.

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#�$ ��

The laboratory report requires calculation of flash solubility datarather than providing it because the laboratory does not know whattrap pressures will be used in the field during its producing life.Instead, the laboratory concentrates on providing sufficient data tohandle any normal situation by simple data conversions.

First, consider the solubility data we have.1. Differential solubility data at the bubblepoint state (3,236

psig/258°F) and at 11 pressures less than the bubblepoint pressure.The bubblepoint value is 1,518 ft3/bbl residual oil. All fluids at pres-sures greater than pb have this amount of gas.

2. Flash solubility of the bubblepoint oil for four different surfacetrap situations. These vary from 1,206 ft3/bbl STO for a single flashto atmospheric pressure to 1,009 ft3/bbl STO for a 200-psig prima-ry-trap-tank situation. Fig. D-3 shows these.

We now wish to determine the “flash-converted” values (i.e., theamount of gas obtained at the surface when a unit of saturated reser-voir oil at less than 3,236 psig is flashed through a surface trap set-up). To illustrate, we use the 200-psig-primary/0-psig-tank situationat the reservoir pressure of 2,301 psig. Looking at the differential-liberation data in Col. 4, Page 3, we see that 506 ft3 of gas has comeout of solution per barrel of residual oil when the pressure declinedfrom 3,236 to 2,301 psig. In other words, we can say that the2,301-psig saturated oil contains less gas by this amount. If this liq-uid were taken to the surface and processed through the traps, itwould also show somewhat less gas solubility than the 1,009 ft3/bblSTO that the bubblepoint oil shows; however, it would not be 506ft3 less because we are on a different oil base.

If we let (�Rs)diff be the liberated gas/oil ratio by differential va-porization, (�Rs)diff � (Rsb)diff � (Rs)diff , we can convert this to a(�Rs)flash as follows.

Vg

Vor

Vor

Vob�

Vg

Vob,

Vg

Vob

Vob

Vo�

Vg

Vo,

Vg

Vo� (�Rs)flash,

Vor

Vob�

12.075

,

and Vob

Vo� 1.674 RB�STB , (D-8). . . . . . . . . . . . . . . . . . . . . .

where Vg is in cubic feet and Vo , Vob, and Vor are in barrels.Therefore,

(�Rs)flash � (�Rs)diff1.6742.075

and (Rs)2301flash � 1, 009 � (�Rs)flash � 1009 � 506 1.674

2.075

� 1, 009 � 408 � 601 scf�STB. (D-9). . . . . . . .

This can be generalized as

(Rs)flash ��Rsb

�flash �

(�Rs)diff

Bob

�Vob�Vor �. (D-10). . . . . . . . .

At times, this relationship will yield negative values of (�Rs)flash atlow pressures. This is not inconsistent with the physics of the situa-tion because a saturated oil at a high reservoir temperature but a lowpressure may give off no gas when processed through the cooler sur-face traps. We would expect then to get a (�Rs)flash of zero at somefinite value of p (Fig. D-3).

* �� #�#+�

#�$ �� "� ��$��

We now consider FVF’s at pressures lower than bubblepoint. We havethe full relative-volume curve of the saturated-liquid phase in termsof residual oil. Fig. D-4 shows this. The bubblepoint state has a rela-tive oil volume of 2.075 bbl/bbl residual oil. We also have the FVFat the bubblepoint state, Bob, with a value of 1.674 bbl/bbl STO. Wecan see that the relative oil volume and the FVF at pressure p can berelated by transferring to the common point, the bubblepoint.

Let Vo�Vor �relative volume of saturated oil at pressure p in bbl/bbl residual oil. Then,


Nonphysical:neglect

Fig. D-4—Differential vaporization and flash-corrected oil FVF vs. pressure above and belowbubblepoint pressure.

Vo

Vor

Vor

Vob�

Vo

Vob,

Vo

Vob

Vob

Vo�

Vo

Vo,

and Vo

Vo� Bo ,

with all volumes in barrels. Therefore,

Bo � Vo�VorBob

�Vo�Vor�b

.

At 2,301 psig we would have

Bo � 1.787 1.6742.075

� 1.442 RB�STB.

As in the previous instance, we must give special considerationto the low range of pressure. If we apply the above equation to the0 psig/258°F point, we get Bo � 1.109(1.674�2.75) � 0.895. Thisis an absurdity because any oil at 258°F must have a Bo of close to1.1 owing to thermal expansion. Therefore, as Fig. D-4 shows, wedraw the oil FVF curve into the ordinate at the value of thermal ex-pansion shown by the differential curve. Alternatively, the Rs curve

vs. pressure can be used to the point where Rs � 0, with Rs � 0 atlower pressures. The corresponding Bo at Rs � 0 is taken from thelinear trend of flash-corrected Bo vs. flash-corrected Rs (Fig. D-5).

��

1. “Raleigh Field PVT Report,” Report RFL 1064, Core Laboratories Inc.,Houston (1958).

2. Amyx, J.W., Bass, D.M. Jr., and Whiting, R.L.: Petroluem Reservoir En-gineering: Physical Properties, McGraw-Hill Book Co. Inc. New YorkCity (1960).

3. Craft, B.C. and Hawkins, M.: Applied Petroleum Reservoir Engineering,Prentice-Hall Inc., Englewood Cliffs, New Jersey (1959).

�� *��"�� # ��

�� g/cm3

bbl �1.589 873 E�01�m3

cp �1.0* E�03�Pa�sft �3.048* E�01�m

ft3 �2.831 685 E�02�m3

�F (�F�32)/1.8 ��Clbm �4.535 924 E�01�kgpsi �6.894 757 E�00�kPa


Nonphysicalextrapolation

Fig. D-5—Flash-corrected oil FVF vs. flash-corrected solution gas/oil ratio showing “normal”linear behavior (in particular, the nonphysical behavior at low pressures because of the approx-imate nature of the traditional differential-to-flash correction.

6 PHASE BEHAVIOR

The California Co.Box 713Brookhaven, Mississippi

Attention: Mr. O.H. Fennell

Subject: Reservoir Fluid StudyCentral Oil Co. No. 5-2 WellRaleigh FieldSmith County, MississippiOur File Number: RFL 1064

Gentlemen:

Subsurface fluid samples were collected from the Central Oil Co. No. 5-2 well on March 14, 1958, by a representative of Core Labora-tories Inc. The results of fluid studies performed with these samples are transmitted to you in the following report.

The saturation pressure of the fluid was determined to be 3,236 psig at the reservoir temperature of 258°F. This value is considerablyless than the static reservoir pressure measured immediately before sampling and indicates that the reservoir exists in a highly under-saturated condition. The presence of a column of water in the bottom of the tubing necessitated obtaining the samples approximately900 ft above the producing formation. Because the reservoir was highly undersaturated, the pressure in the tubing at the point of sam-pling was still well above the measured saturation pressure. The data presented in this report are felt to be representative of the reser-voir fluid and may be applied to calculations without adjustment.

Differential pressure depletion of the fluid at the reservoir temperature of 258°F evolved 1,518 scf gas/bbl residual oil with an accom-panying FVF of 2.075 bbl saturated fluid/bbl residual oil. Under similar depletion conditions, the viscosity of the fluid was measuredfrom pressures exceeding reservoir pressure to atmospheric pressure. The viscosity of the fluid decreased to 0.093 cp at saturationpressure, then increased to a maximum of 0.700 cp at atmospheric pressure.

To determine the effects of changes in surface separation pressure on the produced fluid, flash vaporization tests were performed atfour operating pressures and atmospheric temperature. The tests indicate the optimum separator pressure to be approximately 150 psigwith near optimum conditions as low as 100 psig.

Again it was a pleasure to cooperate with you by performing this study. Should any questions arise or if we may assist you further,please do not hesitate to call.

Very truly yours,

P.L. Moses,Operations SupervisorCore Laboratories Inc.Reservoir Fluid Div.

PLM:ds

3 cc—Addressee

3 cc—Mr. C. L. PickettThe California Co.Natchez, Mississippi

3 cc—Mr. E. J. Deu Pree The California Co.New Orleans, Louisiana


Formation Name HosstonDate First Well Completed , , 19Original Reservoir Pressure psi at ftOriginal Produced Gas/Oil Ratio 1,100 ft3/bbl

Production Rate B/DSeparator Pressure and Temperature psi, °FOil Gravity at 60°F 49 °API

Datum ft subseaOriginal Gas Cap None

Elevation 438 ft DFTotal Depth 12,770 PBD ftCompletion Date 12,732–12, 752,12, 758–12,765 ftTubing Size and Depth 2 in. to 12,704 ftProductivity Index B/D-psi at B/DLast Reservoir Pressure 5,783 psi at 12,650 ft

Date March 14 ,1958Reservoir Temperature 256* °F at 2,650 ftStatus of Well Shut-In 27 HoursPressure Gauge Amerada (DO)

Normal Production Rate B/DGas/Oil Ratio 1,100 ft3/bblSeparator Pressure and Temperature psi, °FBase Pressure psi Abs.

Well Making Water % Cut

Sampled at 11,800 ftStatus of Well Shut-In 27 Hours**

Gas/Oil Ratio ft3/bblSeparator Pressure and Temperature psi, °FTubing Pressure 2,128 psiCasing Pressure psi

Core Laboratories Engineer LBBType Sampler Wofford

SAMPLING CONDITIONS

WELL CHARACTERISTICS

FORMATION CHARACTERISTICS

Company The California Co. Data Sampled March 14, 1958

Well Central Oil Co. No. 5-2 County Smith

Field State Raleigh State Mississippi

Page 1 of 11

File RFL 1064

REMARKS:*Temperature extrapolated to midpoint of perforations, 258°F.

**Before sampling, well was flowed at successive rates of 127, 115, and 103 B/D. The well was then shut infor 24 hours.

8 PHASE BEHAVIOR

1. Saturation pressure (bubblepoint pressure) 3,236 psi at 258°F.

2. Thermal expansion of saturated oil at 6,000 psi�

3. Compressibility of saturated oil at reservoir temperature: vol/vol-psi.From 6,000 psi to 5,000 psi�18.32�10�6

From 5,000 psi to 4,000 psi�22.33�10�6

From 4,000 psi to 3,236 psi�28.64�10�6

4. Specific volume at saturation pressure: ft3/lbm 0.02772 at 258°F.

VOLUMETRIC DATA OF Reservoir Fluid SAMPLE

Page 2 of 11

File RFL 1064

Well Central Oil Co. No. 5-2

V at 258°FV at 73°F � 1.13094 .


Page 3 of 11

File RFL 1064


Reservoir Fluid SAMPLE TABULAR DATA

Differential Liberation at 258°F

Gas/Oil Ratio

GaugePressure

(psi)

Pressure/VolumeRelation at 258 °F

Relative Volume of Oiland Gas

V/Vob

Viscosity*of Oil

at 258 °F(cp)

Liberated/bblResidual Oil

In Solution/bblResidual Oil

Relative OilVolumeVo /Vor

6,0005,5005,3005,0004,5904,5004,1004,0003,8003,7203,6003,5003,4003,3903,3003,2363,2003,1413,1103,0943,0392,9692,9382,8822,8002,7922,6402,6072,4482,3012,3002,2372,0241,9031,8251,8001,6651,5051,5011,3001,2611,0921,078

900800761686656518346310200970

0.93870.9471

0.9562

0.9666

0.97810.9833

0.98880.99180.9948

0.99791.00001.00471.0128

1.01921.02731.0387

1.0534

1.06971.1025

1.1517

1.21771.3003

1.3997

1.4994

1.6244

1.8717

2.15402.5475

2.9926

3.47414.3966

0.119

0.113

0.107

0.102

0.099

0.096

0.093

0.095

0.104

0.118

0.134

0.155

0.179

0.220

0.700

0

183

362

506

670

815

957

1,089

1,209

1,2621,3281,518

1,518

1,335

1,156

1,012

848

703

561

429

309

256190

0at 60°F=

1.9481.965

1.984

2.006

2.0302.040

2.0522.0582.064

2.0712.075

1.970

1 .867

1.787

1.698

1.624

1.544

1.472

1.399

1.3671.3111.1091.000

*Viscosity measurement made with differential-liberation procedure that is a separate experiment from the differential-liberation test used to measure PVT data.V�volume at given pressure; Vob�volume at bubblepoint pressure at the specified temperature, and Vor�residual oil volume at 14.7 psi absolute pressure and 60°F.Gravity of residual oil�42.2°API at 60°F.

10 PHASE BEHAVIOR

Page 4 of 11

File RFL 1064


Supplementary Differential-Liberation Data

Pressure(psig) Gas Gravity

Oil Density(g/cm3)

Deviation FactorZ

3,2362,9382,6072,3011,9031,505

0

0.8700.8460.8330.8300.8351.532

0.57730.59050.60550.61790.63260.64550.7340

0.8860.8790.8780.8840.897

Page 5 of 11

File RFL 1064


SEPARATOR TESTS OF Reservoir Fluid SAMPLE

Separator GOR1

Pressure(psi gauge)

Temperature(°F) Separator Stock Tank

Stock-TankGravity

(°API at 60°F)

ShrinkageFactor

Vor/Vob2

FormationVolume Factor

Vob /Vor3

Flashed GasSpecificGravity

050

100200

75747573

1,2061,011

950875

03568

134

45.648.148.548.5

0.54560.58720.59490.5974

1.8331.7031.6811.674

0.942

1Separator and stock-tank gas/oil ratio in cubic feet of gas at 60°F and 14.7 psi absolute per barrel of STO at 60°F.2Shrinkage factor, Vor /Vob , is barrels of STO at 60°F per barrel of saturated oil at 3,236 psi gauge and 258 °F.3FVF, Vob /Vor, is barrels of saturated oil at 3,236 psi gauge and 258°F per barrel of STO at 60°F.

This table provides results of four separate two-stage separator tests. The first two columns of data give the primary-separator conditions. In all tests, the second(final) separator is at standard (stock-tank) conditions. For example, conditions for the first two-stage separator test are (1) psp1�0 psig and Tsp1�75°F and (2)psp2�0 psig and Tsp2�60°F, with total Rsb�1,206�35�1,241 scf/STB, Bob�1.833, �API�45.6°API, and � �0.942.g


Company The California Co. Data Sampled March 14, 1958

Well Central Oil Co. No. 5-2 County Smith

Field State Raleigh State Mississippi

Page 6 of 11*

File RFL 1064

HYDROCARBON ANALYSIS OF RESERVOIR FLUID SAMPLE

Component wt% mol%Density at 60°F

(g/cm3)°API

at 60°FMolecular

Weight

Nitrogen

Methane

Ethane

Propane

iso–butane

n–butane

iso–pentane

n–pentane

Hexanes

Heavier

Carbon Dioxide

0.18

9.54

2.80

2.67

1.29

2.15

1.47

1.91

5.01

72.30

0.68

100.00

0.51

45.21

7.09

4.61

1.69

2.81

1.55

2.01

4.42

28.91

1.19

100.00 0.8142 42.1 190

Core Laboratories Inc.Reservoir Fluid Div.

P. L. Moses,Operations Supervisor

*Pages 7 through 11 of the original report are graphical representations of the tabular data in Pages 3 and 4.

NOMENCLATURE 1

��

a� numerical constant(s) used in equations;dimensional equation-of-state (EOS) constantdescribing molecular attractive forces,psia/(ft3-lbm mol)2

ai� EOS constant of Component iA� numerical constant(s) used in equations;

dimensionless EOS constant describing molecularattractive forces

Aaq� dimensionless EOS constant for aqueous phase inhydrocarbon/water system

AH� intermediate variable used for selectingpseudocomponents defined by the logarithm of C7

K valueAHC� EOS dimensionless constant for hydrocarbon phase

in hydrocarbon/water systemAij� intermediate terms in Newton-Raphson solution of

the Michelsen two-phase isothermal flash (Eq. 4.58)AL� intermediate variable used for selecting

pseudocomponents defined by the logarithm ofmaximum K value in a mixture

b� inverse FVF, b=1/B, L3/L3, scf/ft3 or STB/RB;dimensional EOS constant describing molecularrepulsive forces, L3/n, ft3/lbm mol

bg� inverse gas FVF, L3/L3, scf/ft3

bgd� inverse dry-gas FVF, L3/L3, scf/ft3

bgw� inverse wet-gas FVF, L3/L3, scf/ft3

bi� Hoffmann et al. K-value correlation parameter (Eq.3.156) for Component i; EOS constant ofComponent i

bo� inverse oil FVF, L3/L3, STB/RBB� FVF, L3/L3, RB/STB or ft3/scf; dimensionless EOS

constant describing molecular repulsive forces� wet-gas FVF gas, L3/L3, ft3/scf

Bgd� dry-gas FVF, L3/L3, ft3/scfB*

gd� modified dry-gas FVF, L3/L3, ft3/scfBgw� Bg�wet-gas FVF, L3/L3, ft3/scfBij� intermediate terms in Newton-Raphson solution of

the Michelsen two-phase isothermal flash (Eq. 4.57)Bo� oil FVF, L3/L3, RB/STB

B*o� modified oil FVF, L3/L3, RB/STB

Bob� oil FVF at bubblepoint (saturated) conditions,L3/L3, RB/STB

Bod� differential oil FVF, L3/L3, RB/residual bblBosp� separator-oil FVF, L3/L3, RB/STB bbl

Bt� total (gas plus oil) FVF of gas/oil system, L3/L3,RB/STB

Bti� Bt at initial reservoir pressure, L3/L3, RB/STBBtw� total (gas plus water) FVF of gas/water system,

L3/L3, RB/STBBw� gas-saturated brine FVF, L3/L3, RB/STBBo

w� brine FVF at atmospheric pressure and reservoirtemperature without solution gas, L3/L3, RB/STB

B*w� brine FVF at reservoir pressure and temperature

without solution gas, L3/L3, RB/STBc� isothermal compressibility, Lt2/m, psi�1;

dimensionless EOS volume-translation constant(volume shift), L3/n, ft3/lbm mol

c� cumulative (average) compressibility, Lt2/m, psi�1

cg� gas isothermal compressibility, Lt2/m, psi�1

cgw� total (gas plus water) isothermal compressibility ofgas/water system, Lt2/m, psi�1

ci� EOS volume-translation (“shift”) constant, ft3/lbmmol; ci�1/(Ki � 1) in Muskat-McDowellphase-split algorithm (Eq. 4.39)

co� oil isothermal compressibility, Lt2/m, psi�1

csw� salt concentration, molalitycw� saturated-brine isothermal compressibility,

Lt2/m, psi�1

c*w� brine isothermal compressibility without solution

gas, Lt2/m, psi�1

cwv� salt concentration, molarityCf� Søreide specific gravity correlation characterization

factor (Eq. 5.44)Ci� molar concentration, n/L3, lbm mol/ft3;

hydrate-former constantCog� conversion from stock-tank condensate (condensed

from a reservoir gas) to equivalent surface gas,L3/L3, scf/STB

Coo� conversion from stock-tank oil (produced froma reservoir oil) to equivalent surface gas,L3/L3, scf/STB

Csv� salt concentration in water, ppm by volumeCsw� salt concentration in water, ppm by weightdci� calculated Data i used in least-squares regressiondTP� tangent-plane distance, Ldxi� experimental Data i used in least-squares

regressionDCO2w� CO2/water binary-diffusion coefficient, L2/t, ft2/sec

2 PHASE BEHAVIOR

Dij� binary-diffusion coefficient, L2/t, ft2/secDo

ij� low-pressure binary-diffusion coefficient,L2/t, ft2/sec

Dim� effective diffusion coefficient of Component i in amixture, L2/t, ft2/sec

ei� intermediate terms in Newton-Raphson solution ofthe Michelsen two-phase isothermal flash (Eq. 4.52)

Eg� gas expansion term used in generalized gas/oilmaterial balance, L3/L3, scf/STB

Eg� average expansion term used in generalized gas/oilmaterial balance, L3/L3, scf/STB

Eo� oil expansion term used in generalized gas/oilmaterial balance, L3/L3, STB/STB

Eo� average oil expansion term used in generalizedgas/oil material balance, L3/L3, STB/STB

f� generic function; pure-component fugacity,m/Lt2, psia

f� pure-component fugacity, m/Lt2, psiafeqi� final converged-solution equilibrium fugacities in a

two-phase flash, m/Lt2, psiafi� fugacity of Component i in a mixture, m/Lt2, psiafi� fugacity of Component i in a mixture, including

gravity potential, L/mt2, psiafLi� fugacity of Component i in the liquid phase,

m/Lt2, psiafM� parameter in Twu correlation for molecular weightfpc� Twu correlation parameter for critical pressurefTc� Twu correlation parameter for critical temperaturefvc� Twu correlation parameter for critical volumefvi� Component i fugacity in the vapor phase,

m/Lt2, psiafyi� Component i fugacity in an incipient

(saturation-pressure calculation) or (phase-stabilitytest) trial phase, m/Lt2, psia

fzi� Component i fugacity in the overall (feed) mixture,m/Lt2, psia

F� sum-of-squares functionF� proportioning factor

FEOS� generic representation of an EOS functionFi� characterization factor in Hoffman et al. K-value

correlationFg� fv�mole fraction of wellstream or overall mixture

in the gas phaseFgg� mole fraction of reservoir gas that remains gas at

surface conditionsFgsp� mole fraction of wellstream that is gas in the

primary separatorFoo� volume fraction of total stock-tank oil that comes

from the reservoir oilFosp� mole fraction of wellstream that is oil in the

primary separatorg*� normalized Gibbs energy

gc� mass-to-force conversion factorg*

mix� overall-mixture normalized Gibbs energyg*

x� liquid-phase normalized Gibbs energyg*

y� vapor- or incipient-phase normalized Gibbs energyg*

z� feed-composition normalized Gibbs energy(considered as a single phase)

G� original gas in place, L3, scfGd� original dry gas in place, L3, scf

Gmix� mixture Gibbs energyGp� cumulative gas produced, L3, scf

Gpd� cumulative dry gas produced, L3, scfGpw� cumulative wet gas produced, L3, scfGw� original wet gas in place, L3, scfGz� overall-composition Gibbs energy (considered as a

single phase)

h� depth, L, ft; Rachford-Rice function in phase-splitcalculation

href� reservoir reference depth, L, ftHg� surface-gas gross-heating value, Btu/scfHi� component gross-heating value, Btu/scf;

Henry’s constanti� carbon numberI� constant in Eq. 4.64

Ja� Jacoby aromaticity factor, fractionJij� Jacobian terms in Newton-Raphson solution of

Michelsen two-phase isothermal flash (Eq. 4.55)k� permeability, L2, md

kij� EOS binary-interaction parameter betweenComponent Pair i-j

kijaq� binary-interaction parameter for Component Pair i-jin aqueous phase in a water/hydrocarbon system

kijHC� binary-interaction parameter for Component Pair i-jin nonaqeous phase in a water/hydrocarbon system

krg� gas relative permeabilitykro� oil relative permeabilityks� Setchenow constant, molarity (mol/kg)Ki� yi�xi�equilibrium ratio (K value) of Component i

Ki(vs)� equilibrium ratio of Component i in a vapor/solidsystem

Kw� Watson characterization factor, T1/3, °R1/3

L� total liquid yield, L3/L3, gal/MscfLi� liquid yield of Component i, L3/L3, gal/Mscfm� mass, m, lbm or g; correlating function in

correction term � for EOS Constant Amg� gas mass, m, lbmmo� oil mass, m, lbm

mSRK� function in correction term � for Constant A in theSoave-Redlich-Kwong EOS (Pedersen et al.charaterization procedure (Eq. 5.80)

ms� salt mass, m, gmt� total-system mass, m, lbm

mow� pure-water mass, m, g

M� molecular weight, m/n, lbm/lbm molMair� air molecular weight, m/n, lbm/lbm molMb� boundary molecular weight in gamma distribution

model, m/n, lbm/lbm molMCn�

� Cn� molecular weight, m/n, lbm/lbm molMC7

� molecular weight of C7, m/n, lbm/lbm molMC7�

� C7� molecular weight, m/n, lbm/lbm molMg� gas molecular weight, m/n, lbm/lbm molMg� surface-gas molecular weight, m/n, lbm/lbm molMN� heaviest C7� fraction molecular weight,

m/n, lbm/lbm molMo� oil molecular weight, m/n, lbm/lbm molMo� stock-tank oil molecular weight, m/n, lbm/lbm mol

Mosp� molecular weight of separator oil, m/n, lbm/lbmmol

MP� molecular weight of paraffin hydrocarbons, m/n,lbm/lbm mol

n� moles, n, lbm molnc� number of types of cavities per water molecule in

hydrate crystal lattice, nng� moles of gas, n, lbm molng� moles of surface gas, n, lbm molnL� moles of liquid phase, n, lbm molno� moles of oil, n, lbm molno� moles of stock-tank oil, n, lbm molno

w� moles of pure water, n, molenv� moles of vapor phase, n, lbm molN� original oil in place, L3

, STB; total number ofcomponents, n; last component in a mixture

NC7�� C7� approximate carbon number in Standing’s

low-pressure K-value correlation, n

NOMENCLATURE 3

NH� number of heavy (C7�) pseudocomponents, nNL� number of light pseudocomponents, nNp� cumulative oil produced, L3, STBNsp� number of separator stages, n

p� pressure, m/Lt2, psiapb� bubblepoint pressure, m/Lt2, psia

pcP� critical pressure of paraffin hydrocarbons,m/Lt2, psia

pc� critical pressure, m/Lt2, psiapc C7�

� critical pressure of C7�, m/Lt2, psiapd� dewpoint pressure, m/Lt2, psiapi� initial pressure, m/Lt2, psiapK� convergence pressure, m/Lt2, psiappc� pseudocritical pressure, m/Lt2, psiappc� pseudocritical pressure adjusted for

nonhydrocarbon content, m/Lt2, psiappcHC� pseudocritical pressure of hydrocarbon components

only in a gas, m/Lt2, psiappr� pseudoreduced pressure, dimensionlesspr� reduced pressure, dimensionless

pref� reference pressure, m/Lt2, psiapR� average reservoir pressure, m/Lt2, psiaps� saturation pressure, m/Lt2, psiapsc� pressure at standard conditions, m/Lt2, psiapsp� separator pressure, m/Lt2, psia

psp1, psp2� primary- and secondary-separator pressure,m/Lt2, psia

pst� stock-tank pressure, m/Lt2, psiapv� vapor pressure, m/Lt2, psia

pvw� water/brine vapor pressure, m/Lt2, psiapvpw� pure-water vapor pressure, m/Lt2, psiapwf� wellbore flowing pressure, m/Lt2, psia

p(M)= density function of the gamma probability molardistribution

P� parachorPc� capillary pressure, m/Lt2, psiPg� surface-gas-“component” parachorPo� stock-tank-oil-“component” parachorP0� integral (area) of p(M) from � to the

molecular-weight boundary Mb

P1� integral (area) of Mp(M) from � to themolecular-weight boundary Mb

qg, qg� total surface-gas production rate, L3/t, scf/Dqgg� production rate of surface gas from reservoir gas,

L3/t, scf/Dqgo� production rate of surface gas from reservoir oil,

L3/t, scf/Dqo, qo� total stock-tank-oil production rate, L3/t, STB/D

qog� production rate of stock-tank condensate fromreservoir gas, L3/t, STB/D

qoo� production rate of stock-tank oil from reservoir oil,L3/t, STB/D

Q� generic for cumulative production in theconstant-volume-depletion experiment; variable insaturation pressure algorithm; parameter in gammadistribution model

Qcum� cumulative production quantity fromconstant-volume-depletion table (produced fromdewpoint pressure)

Qd� cumulative production quantity from initial todewpoint pressure

QMi� cumulative molecular weight, m/n, lbm/lbm molQ*

Mi� normalized cumulative molecular weight variable,m/n, lbm/lbm mol

QWi� cumulative weight fractionQzi� cumulative mole fraction

r� radius, Lr� Residual i used in least-squares regression

r� average pore radius, Lre� well external drainage radius, L, ft

rog� oil/gas ratio, L3/L3, STB/scf or STB/MMscfrp� total producing oil/gas ratio, L3/L3, STB/scf or

STB/MMscfrs� solution oil/gas ratio, STB/scf or STB/MMscfr*

s� modified solution oil/gas ratio, L3/L3, STB/scf orSTB/MMscf

rsd� solution oil/gas ratio at dewpoint pressure, STB/scfrw� wellbore radius of a well, L, ftR� universal gas constant�10.73146 psia-ft3/

°R-lbm molRgo� GOR, L3/L3, scf/STBRi� fugacity ratio variable for Component iRp� total producing GOR, L3/L3, scf/STBRs� solution gas/oil ratio, L3/L3, scf/STBR*

s� modified solution gas/oil ratio, L3/L3, scf/STBRsd� differential solution gas/oil ratio, L3/L3,

scf/residual bblRsdb� differential solution gas/oil ratio at bubblepoint,

L3/L3, scf/residual bblRsp� separator-gas/oil ratio, L3/L3, scf/separator bbl

Rspw� solution gas/water (pure) ratio, L3/L3, scf/STBRsp1� GOR of first-stage separator, L3/L3,

scf/separator bblRsw� solution gas/water (brine) ratio, L3/L3, scf/STB

Rswg� solution water/gas ratio, L3/L3, STB/scf orSTB/MMscf

Rs1� GOR from first-stage separator, L3/L3, scf/STBRs�� solution gas/oil ratio of first-stage separator oil,

L3/L3, scf/STBR�g� surface-gas specific-gravity ratioR�o� stock-tank-oil specific-gravity ratio

s� skin factor, dimensionlesssi� ci�bi�dimensionless volume-translation (“shift”)

variable used in EOSS� sum of mole numbers (fugacity ratio) in

phase-stability test; gamma distributionmodel variable

Sg� gas saturation, fractionSL� sum of mole numbers in liquid phase

(phase-stability test)So� oil saturation, fractionSv� sum of mole numbers in vapor phase

(phase-stability test)Sw� water saturation, fractionS0� specific-gravity correlation variableT� temperature, T, °F or °R

�T� hydrate-forming point, T, °FTb� normal boiling point at 1 atm, T, °R

TbF� normal boiling point at 1 atm, T, °FTbr� reduced normal boiling pointTc� critical temperature, T, °R

TcP� critical temperature of paraffin hydrocarbons, T, °RTc C7�

� C7+ critical temperature, T, °RTij� low-pressure diffusion-coefficient-equation

parameter between Component Pair i-jTpc� pseudocritical temperature, T, °RT*

pc� pseudocritical temperature adjusted fornonhydrocarbon content, T, °R

TpcHC� hydrocarbon-component pseudocritical temperaturein a gas, T, °R

Tpr� pseudoreduced temperatureTr� reduced temperature

Trpw� reduced temperature of pure waterTsc� standard condition temperature, T, °F or °RTsp� separator temperature, T, °F

4 PHASE BEHAVIOR

Tsp1, Tsp2� primary- and secondary-separatortemperature,T, °F

Tst� stock-tank temperature, T, °Fui� component molar velocity, n/t, lbm mol/sec

�ui� logarithm of fugacity ratios used in GDEMpromotion algorithm

v� molar volume, L3/n, ft3/lbm molvc� critical molar volume, L3/n, ft3/lbm mol

vcP� critical molar volume of paraffin hydrocarbons,L3/n, ft3/lbm mol

vg� gas molar volume, L3/n, ft3/lbm molvg� gas molar volume at standard conditions, L3/n,

vg^�379 scf/lbm molv*

Mi� modified molar volume, L3/n, ft3/lbm molvpc� pseudocritical molar volume, L3/n, ft3/lbm molvpr� pseudoreduced molar volumevr� reduced molar volume�Vrv^~� specific volume, L3/m, ft3/lbm

v^~

w� specific volume of brine, L3/m, cm3/gv^*

w� brine specific volume at reservoir pressure andtemperature without solution gas, L3/m, cm3/g

v^~

wsc� brine specific volume at standard pressure andreservoir temperature without solution gas,L3/m, cm3/g

V� volume, L3, ft3 or bblV� average volume, L3, ft3 or bbl

Vc� critical volume, C2�L3, ft3 or bblV C2�

� ideal-solution liquid volume of C2�componentsV C3�

� ideal-solution liquid volume of C3�componentsVcell� original cell volume at saturation pressure in a PVT

experiment, L3, ft3

Vg� gas volume, L3, ft3 or bblVg� surface-gas volume, L3, scfVo� oil volume, L3, ft3 or bblVo� stock-tank oil volume, L3, STB

Vob� bubblepoint oil volume, L3, ft3 or bblVoi� initial oil volume, L3, ft3 or bblVor� residual oil volume, L3, ft3 or bblVor� residual oil volume at reservoir temperature from

differential-liberation experiment, L3, residual bblVosp� separator-oil volume, L3, bbl

VpHC� hydrocarbon pore volume (HCPV), L3,ft3 or res bbl

Vr� reduced volume, L3, ft3 or bblVR� reservoir oil volume, L3, ft3 or bblVro� oil volume/oil volume at saturation pressureVrt� total (gas�oil) volume relative to

saturation volumeVs� reservoir oil volume at saturation pressure,

L3, ft3 or bblVt� total (gas�oil) volume, ft3 or bblVw� water volume, L3, ft3 or bblwi� weight fractionwg� surface-gas weight fractionwo� stock-tank-oil weight fractionWi� Gaussian quadrature weight factor

x� coordinate directionxg, xg� surface-gas-“component” mole fraction in

reservoir oilxi� Component i mole fraction in oil phase

xMEOH� methanol inhibitor mole fractionx

o, x

o� stock-tank-oil-“component” mole fraction in

reservoir oilxvi� Component i volume fractionXi� Gaussian quadrature pointy� Hall-Yarborough Z-factor correlation

reduced-density parameter

yg, yg� surface-gas-“component” mole fraction in

reservoir gasyi� Component i mole fraction in gas phase or

incipient phaseyj i� fraction (probability) of Type j molecule occupying

Type i cavityy

o, y

o� stock-tank-condensate-“component” mole fraction

in reservoir gasyw� water mole fraction in reservoir gas

ypw� pure-water mole fraction in gas phaseY� function for smoothing two-phase (gas/oil)

volumetric data below bubblepoint duringconstant-composition-expansion experiment

Ya� Yarborough aromaticity factor, fractionYi� Component i mole number

z Cn� mole fraction of first carbon number component in

a Cn� fractionz C6

� C6 mole fraction in overall mixturez C7

� C7 mole fraction in overall mixturez C7�

� C7� mole fraction in overall mixturezi� Component i mole fraction in overall mixture

zref� reservoir mole fraction at reference depthZ� compressibility, or “deviation,” factor

Zc� critical Z factorZd� dewpoint pressure Z factorZL� liquid-phase Z factorZR� Rackett Z factor for calculating saturated

liquid densitiesZv� vapor phase Z factorZ2� two-phase Z factor�� correction term to Constant A in EOS’s;gamma

distribution model parameter; Hall-Yarboroughequation parameter for the Standing-Katz Z-factorchart; Twu property correlation parameter

�w� Constant A correction term in Peng-Robinson EOSfor water/brine

�� Constant B correction term in theZudkevitch-Joffe-Redlich-Kwong EOS; parameterin the gamma distribution model; solution vector inNewton-Raphson solution of the Michelsentwo-phase isothermal flash (Eq. 4.54)

�*� parameter in the modified gamma distribution

model used with Gaussian quadrature�� specific gravity, air�1 or water�1

�API� (141.5/�o)�131.5, oil gravity, °API� C7�

� C7� specific gravity, water�1�g , �g� gas specific gravity, air�1�

g, �

g� total average gas specific gravity, air�1

�gc� corrected separator gas specific gravity for Vazquezcorrelations, air�1

�gg� specific gravity of surface gas from reservoirgas, air�1

�gHC� gas specific gravity of hydrocarbon components ina gas mixture, air�1

�go� specific gravity of surface gas from reservoir oil, air�1

�g1� first-stage separator-gas specific gravity, air�1�g�� specific gravity of gas released from first-stage

separator oil, air�1�o , �o� stock-tank oil specific gravity, water�1

�og� specific gravity of stock-tank condensate fromreservoir gas, water�1

�oo� specific gravity of stock-tank oil from reservoiroil, water�1

�P� specific gravity of paraffin hydrocarbons, water�1�w� wellstream (reservoir gas) specific gravity, air�1;

brine specific gravity, water�1��M� parameter in the Twu molecular-weight correlation

NOMENCLATURE 5

��P� parameter in the Twu critical-pressure correlation��T� parameter in the Twu critical-temperature

correlation��v� parameter in the Twu critical-volume correlation�� gamma function�� parameter in the modified gamma distribution

model used with Gaussian quadrature�� deviation�� parameter used in the Wichert-Aziz

nonhydrocarbon correction method forpseudocritical properties

�/k� Leonard-Jones 12-6 potential parameter, K�� gamma distribution model parameter (minimum

molecular weight), m/n, lbm/lbm mol�� generic symbol for any component property; Twu

property correlation parameter�� generic property of “grouped” Pseudocomponent I,

where I contains “original” Components i (i�I);e.g., molecular weight MI (Eqs. 5.82 through 5.94)

�1, �2� eigenvalues� dynamic viscosity, m/Lt2, cpg� gas viscosity, m/Lt2, cp

gsc� low-pressure gas viscosity at specified temperature,m/Lt2, cp

i� low-pressure gas viscosity of Component i atspecified temperature, m/Lt2, cp

o� oil viscosity, m/Lt2, cpob� bubblepoint (saturated) oil viscosity, m/Lt2, cpoD� dead (degassed) oil viscosity at standard pressure

and specified temperature, m/Lt2, cpw� water viscosity, m/Lt2, cppw� pure-water viscosity at standard pressure and

specified temperature, m/Lt2, cp(pw)20�C� pure-water viscosity at standard pressure and 20°C,

m/Lt2, cp

*w� water/brine viscosity at standard pressure and

specified temperature, m/Lt2, cpcwH� water chemical potential of water in filled hydrate,

m/Lt2, psiacwMT�water chemical potential in empty hydrate,

m/Lt2, psia1, 2� GDEM-promotion eigenvalue parameters�CO2

� low-pressure gas-viscosity correction for CO2�H2S� low-pressure gas-viscosity correction for H2S�N2

� low-pressure gas-viscosity correction for N2� kinematic viscosity, L2/t, cSt�� Lucas gas-viscosity correlation parameter, cp�1

�T� Thodos (Lohrenz-Bray-Clark) gas viscositycorrelation parameter, cp�1

�� mass density, m/L3, lbm/ft3 or g/cm3

�air� air density, m/L3, lbm/ft3

�C1� C1 apparent pseudoliquid density at standard

conditions, m/L3, lbm/ft3

�C2� C2 apparent pseudoliquid density at standard

conditions, m/L3, lbm/ft3

�C2�� C2� pseudoliquid density at standard conditions,

m/L3, lbm/ft3

�g� gas density, m/L3, lbm/ft3

�g� surface-gas density, m/L3, lbm/ft3

�ga� separator-gas apparent pseudoliquid density,m/L3, lbm/ft3

�i� liquid density of Component i at standardconditions, m/L3, lbm/ft3

�o� oil density, m/L3, lbm/ft3

�o� stock-tank oil density, m/L3, lbm/ft3

�ob� bubblepoint oil density, m/L3, lbm/ft3

�osp� separator-oil density, m/L3, lbm/ft3

�M� molar density, n/L3, lbm mol/ft3

�Mc� critical molar density, n/L3, lbm mol/ft3

�Msc� low-pressure molar density, n/L3, lbm mol/ft3

�pij� partial density of surface Phase i produced fromreservoir Phase j, m/L3, lbm/ft3

�po� pseudoliquid density, m/L3, lbm/ft3

�pr� pseudoreduced density�r� reduced density

�ref� reference density (air or water), m/L3, lbm/ft3

�sL� saturated-liquid density, m/L3, lbm/ft3

�w� saturated-brine density, m/L3, g/cm3

�*w� water/brine density at reservoir pressure and

temperature without solution gas, m/L3, g/cm3

�wsc� brine density at standard pressure and reservoirtemperature without solution gas, m/L3, g/cm3

��p� density/pressure correction for Standing-Katz oildensity correlation, m/L3, lbm/ft3

��T� density/temperature correction for Standing-Katzoil density correlation, m/L3, lbm/ft3

��wH� density difference between water/brine and thehydrocarbon phase, m/L3, g/cm3

� interfacial tension (IFT), m/t2, dynes/cm lim� limiting hydrocarbon/water IFT at ��wH�0,

m/t2, dynes/cm go� gas/oil IFT, m/t2, dynes/cm i j� Leonard-Jones 12-6 potential parameter, Å wH� water/hydrocarbon IFT, m/t2, dynes/cm

�� sheer stress, m/Lt2, psi�� porosity�i� fugacity coefficient for Component i; generalized

weighting factor for mixing rule(�i)w� fugacity coefficient for Component i in brine

(�i)pw� fugacity coefficient for Component i in pure water�� acentric factor

�a,�b� constants in cubic EOS’s�

oa,�o

b� numerical constants in cubic EOS’s�i j� low-pressure diffusion-coefficient-correlation

parameter

Superscriptso� low pressure

AUTHOR INDEX 1

��

A

Abbott, M.M., 48, 49, 66Abdul-Majeed, G.H., 38, 45Abou-Kassem, J.H., 24, 44Abramowitz, M., 86Abu-Khamsin, S.A., 38, 45Agarwal, R., 67Ahmed, T., 45Al-Khafaji, A.H., 36, 38, 45Al-Marhoun, M.A., 38, 45Alani, G.H., 3, 33–35, 45Amirijafari, B., 144, 160Amyx, J.W., 44, 108Andersen, T.F., 86, 208Austad, T., 69, 70, 72, 73, 78, 86Auxiette, G., 140Aziz, K., 24, 25, 37, 44, 45, 66, 177, 192, 224

B

Baker, L.E., 56–59, 66, 141Bardon, M.F., 79, 86Bass, D.M. Jr., 44, 108Bath, P.G.H., 67Batycky, J.P., 84, 87Beal, C., 36–38, 45Bedrikovetsky, P.G., 64, 67Beggs, H.D., 24, 30, 35–38, 44, 45Behrens, R.A., 84, 87Belery, P., 40, 44, 64, 65, 67Benedict, M., 4, 80, 86Benham, A.L., 126, 131, 132, 141Bergman, D.F., 36, 37Beu, K.L., 3Bhagia, N.S., 3Bicher, L.B. Jr., 1, 3Boe, A., 120Borthne, G., 117, 120Bray, B.G., 45, 72, 86, 175–77, 192, 206, 208, 224

Brill, J.P., 24, 44Brinkman, F.H., 42, 46Brown, G.G., 3, 13, 17, 108

Brulé, M.R., 73, 83, 86Buchanan, R.D. Jr., 141Bucklin, R.W., 157, 161Burrows, D.B., 45

C

Campbell, J.M., 43, 44, 46, 144, 152, 153, 155, 157, 160, 161Canfield, F.B., 43, 46Carlson, H.A., 3Carr, N.L., 26, 27, 45Carroll, J.J., 158Carson, D.B., 2, 154, 155, 161Cavett, R.H., 80, 81, 86Chaback, J.J., 64, 67Chaperon, I., 140Chen, C.-J., 161

Chew, J.N., 37, 38, 45, 183, 192Chien, M.C.H., 45Chierici, G.L., 146, 160Cho, S.J., 104, 108Chorn, L.G., 86Chou, J.C.S., 143, 146, 159Christman, P.G., 140Christoffersen, K., 44, 45Chueh, P.L., 83, 84, 87, 195, 208Civan, F., 108Clark, C.R., 45, 72, 86, 175–77, 192, 206, 208, 224Clark, G.C., 17, 46

Clark, N.J., 94, 108Clever, H.L., 145, 160Coats, K.H., 4, 44, 65, 66, 67, 84, 85, 86, 87, 113, 119Collins, A.G., 147, 160Connally, C.A., 37, 38, 45, 183, 192Cook, A.B., 127, 128, 141Cook, R.E., 119, 208Correia, R.J., 160Costain, T.G., 133, 140, 141Craft, B.C., 35, 45, 86, 108, 158, 213Cragoe, C.S., 26, 29, 44, 113, 120, 178, 192Creek, J.L., 64, 67

Cronquist, C., 16, 17, 119Crowe, A.M., 66

2 PHASE BEHAVIOR

Crump, J.S., 45, 86, 108, 189, 192Culberson, O.L., 143, 159, 160Cullick, A.S., 34, 45

D

da Silva, F.V., 40, 44, 64, 67, 120Dake, L.P., 108, 119Dalen, V., 66Daubert, T.E., 77–82, 86David, R.A., 2, 17de Jong, L.N.J., 67Deaton, W.M., 154, 161Delclaud, J., 44Dempsey, J.R., 26, 45DeRuiter, R.A., 72, 86Dindoruk, B., 141Dixon, T.N., 119Dodson, C.R., 97, 108, 143, 145, 147–49, 151, 159, 192Donnelly, H.C., 3Donohoe, C.W., 141Dougherty, E.L. Jr., 64, 67Dowden, W.E., 141Dranchuk, P.M., 24, 44Drickamer, H.G., 64, 67Drohm, J.K., 108, 119, 120

E

Eakin, B.E., 45, 192Earlougher, R.C. Jr., 171Edmister, W.C., 12, 17, 42, 46, 66, 81, 86Eilerts, C.K., 1, 3, 11, 14, 17, 26, 28, 44, 45Ely, J.F., 44Enick, R.M., 151, 158Erbar, J.H., 70, 86Ericksen, 161

F

Faissat, B., 64, 67Farshad, F.F., 30, 36, 38, 45Fayers, J.F., 141Fetkovich, M.D., 120Fetkovich, M.J., 44, 158Fevang, Ø., 17Fick, A., 21, 44Firoozabadi, A., 39, 45, 68–72, 77, 86, 149, 158, 160Fiskin, J.M., 3Flaitz, J.M., 108Forgarasi, M., 45Fowler, W.N., 67Francis, R.J., 17, 46Fredenslund, A., 66, 67, 86, 87, 208Freze, R., 66, 87Frost, E.M., 154, 161Fuller, G.G., 66Fussell, D.D., 141

G

Gaddy, V.L., 159Galimberti, M., 43, 44, 46Gardner, J.W., 141

Gibbs, J.W., 1, 2, 49, 52–64, 67Glasø, O., 29, 30, 36, 37, 43, 45, 46Glass, E.D., 46Golan, M., 120Gold, D.K., 44, 45Golding, B.H., 28, 45, 86Goldthorpe, W.H., 108, 119, 120Gonzalez, M.H., 26, 45, 182, 192Goodrich, J.H., 140Goodwill, D., 108Gorell, S.B., 140Gouel, P.L., 64, 67Govier, G.W., 45Graue, D.J., 136, 141Griewank, A.K., 45

H

Haaland, S., 86Haas, J.L. Jr., 150, 158Hachmuth, K.K., 2Hadden, S.T., 45Hall, K.R., 23, 24, 44, 82, 86, 175, 177, 192, 223Haman, S.E.M., 50, 66Hammerschmidt, E.G., 151, 157, 161Hankinson, R.W., 34, 45Hanley, H.J.M., 44Harvey, A.H., 161Harvey, M.T., 140Hassoon, S.F., 45Hawkins, M., 35, 45, 86, 108, 158, 213Heidemann, R.A., 66Hicks, B.L., 3Hildebrand, M.A., 161Hinds, R.F., 17, 105, 108Hirschberg, A., 63, 67Hocott, C.R., 45, 86, 108, 149, 160, 189, 192Hoffmann, A.E., 45, 86, 108, 189, 192, 220Holder, G.D., 158, 161Holland, C.J., 145, 160Holm, L.W., 125, 137, 138, 140, 141Holt, T., 63, 67Hooper, H.H., 158Hou, Y.C., 48, 66Hutchinson, C.A. Jr., 124, 140

J

Jacoby, R.H., 78, 79, 86, 119, 208, 221Jennings, J.W., 44, 45Jensen, F., 141Jensen, J.I., 66Jhaveri, B.S., 51, 52, 66, 83, 87Joffe, J., 47, 50, 66, 83, 87, 223John, V.T., 156, 161Johns, R.T., 128, 141Josendal, V.A., 125, 137, 138, 140Jossi, J.A., 38, 45

K

Kattan, R.R., 45Katz, D.L., 1–3, 5, 9–11, 13, 14, 16, 17, 23–26, 28, 30–34, 38, 39, 44, 45,

68–72, 77, 86, 90, 108, 140, 143, 148, 149, 151, 152, 154, 155,159–61, 177, 179, 181, 183, 190, 192, 205, 206, 208, 223, 224

AUTHOR INDEX 3

Kawanaka, S., 140, 141

Kay, W.B., 11, 13, 17, 19, 24, 25, 38, 40, 44, 85, 87, 93, 108, 140Kelm, C.H., 140

Kennedy, G.C., 3, 141Kennedy, H.T., 1, 3, 28, 33–35, 45, 108

Kennedy, J.T., 141Kesler, M.G., 24, 44, 71, 79, 80–84, 86, 194, 208

Kestin, J., 147, 160Khalifa, H.E., 160

Khan, S.A., 38, 45Kistenmacher, H., 151, 161

Klins, M.A., 140Kniazeff, V.J., 118, 119

Kobayashi, R., 3, 45, 143, 156, 159, 161Kobe, K.A., 160

Koch, H.A. Jr., 124, 140Kuenen, J.P., 11, 17

Kumar, K.H., 47, 66Kunzman, W.J., 141

Kuo, S.S., 132, 140Kurata, F., 2, 10, 11, 17, 28, 45

Kutasov, I.M., 146, 160Kwong, J.N.S., 1, 4, 47, 48–51, 63, 64, 66, 221, 223

L

Lacey, W.N., 3, 63, 67, 140Lasater, J.A., 29, 30, 45

Lawsa, W.F., 161Lee, A.L., 26, 45, 182, 192

Lee, B.I., 12, 17, 24, 44, 66, 71, 79–81, 83, 84, 86, 194, 208Lee, R.L., 161

Lee, S.T., 84, 85, 87Lein, C.L., 141

Leshikar, A.G., 45Li, Y.-K., 59, 61, 66, 67, 84, 86, 87, 158

Lindeberg, E., 67Little, J.E., 3

Lo, T.S., 119, 120Lohrenz, J., 17, 33, 38, 43, 45, 46, 72, 86, 175–77, 192, 206, 208, 224

Long, G., 146, 160Lucas, K., 27, 28, 38, 45, 175, 176, 182, 192, 224

Ludecke, D., 161Luks, K.D., 66, 141

M

MacAllister, D.J., 72, 86

Macleod, D.B., 38, 39, 45Maddox, R.N., 70, 86

Madrazo, A., 31, 45Makogon, Y.F., 161

Malone, R.D., 161Mannan, M., 44

Mansoori, G.A., 141Markham, A.E., 160

Martin, J.J., 47, 48, 51, 66Mather, A.E., 158

Matthews, T.A., 2, 25, 44Mayer, E.H., 108

McAuliffe, J.C., 66McCain, W.D. Jr., 26, 44, 45, 86

McDowell, J.M., 1, 4, 53, 66, 183, 185, 186, 192, 220McKetta, J.J. Jr., 143, 147, 148, 159, 160

McLeod, H.D. Jr., 149, 153, 155, 161McRee, B.C., 141

Mehra, R.K., 66Mehta, B.R., 156, 161Merrill, R.C. Jr., 87Metcalfe, R.S., 64, 67, 125, 127, 128, 136, 138, 140, 141

Michel, S., 158Michelsen, M.L., 44, 46, 47, 54, 55, 57–67, 141, 151, 156, 161,

220, 221, 223

Mohamed, R., 158Monger, T.G., 140, 141Monroe, R.R., 3, 45Monroy, M.R., 45

Montel, F., 64, 67Morris, R.W., 67Moses, P.L., 17, 108, 141Muckleroy, J.A., 1, 2

Mullen, N.B., 3Muller, H.G., 152, 161Munck, J., 156, 161Murphy, G.B., 86

Muskat, M., 1, 4, 53, 63, 64, 66, 67, 183, 185, 186, 192, 220

N

Nagy, Z., 49, 66, 171

Naville, S.A., 118, 119Nectoux, A., 44Nelson, E.F., 86Nemeth, L.K., 3, 28, 45

Newley, T.M.J., 87Ng. H.-J., 156, 161Ng, H.-Y., 66Nghiem, L.X., 61, 62, 66, 67, 87, 158

Nielsen, R.B., 157, 161Nishio, M., 66Nokay, R., 39, 45, 80, 86Novosad, Z., 133, 140, 141

Nuttaki, R., 158

O

O’Brian, L.J., 137, 141

O’Leary, 141Olds, R.H., 3, 28, 45Olson, C.R., 108Opfell, J.B., 46

Organick, E.I., 28, 45, 86Orr, F.M. Jr., 125, 126, 128, 140, 141

P

Panagiotopoulos, A.Z., 151, 161Papadopoulos, K.D., 161Park, S.J., 141Parks, A.S., 3, 108

Parrish, W.R., 156, 161Patel, P.D., 141Patel, V.C., 66Patton, C.C., 158

Pawlikowski, E.M., 145, 160, 192Pebdani, F.N., 45Pedersen, K.S., 49, 66, 67, 83, 84, 86, 87, 200–08, 221Peneloux, A., 48, 51, 64, 66, 83, 87

4 PHASE BEHAVIOR

Peng, D.Y., 1, 4, 47, 50, 51, 63, 64, 66, 83, 86, 124, 140, 150, 160, 185,192, 193, 202, 208, 223

Perschke, D.R., 141Peterson, A.V., 141Pierce, A.C., 66Pitzer, K.S., 81, 82, 86, 147, 160Poettman, F.H., 3Polling, B.E., 44, 66, 171Pope, G.A., 141Powers, J.E., 161Prausnitz, J.M., 44, 66, 83, 84, 86, 87, 145, 156, 158, 160, 161, 171, 192,

195, 208

R

Rachford, H.H., 52–55, 66, 183, 184, 186, 192, 221Rackett, H.G., 34, 39, 45, 223Ramesh, A.B., 119, 208Ramey, H.J. Jr., 39, 40, 44, 45, 149, 160Rao, V.K., 79, 86Rasmussen, P., 161Ratkje, S.K., 67Rauzy, E., 66, 87Raynal, M., 108Razsa, M.J., 86Reamer, H.H., 3, 108Redlich, O., 1, 4, 47, 48, 49, 50, 51, 63, 64, 66, 221, 223Reese, D.E., 44, 158Reid, R.C., 28, 44, 49, 66, 82, 86, 151, 161, 171Renner, T.A., 40, 45, 141Reudelhuber, F.O., 17, 105, 108Riazi, M.R., 71, 77, 78, 79, 80, 81, 82, 86Rice, J.D., 52, 53, 54, 55, 66, 183, 184, 186, 192, 221Riemens, W.G., 63, 67Risnes, R., 66Robinson, D.B., 1, 4, 47, 50, 51, 63, 64, 66, 83, 86, 124, 140, 150, 156,

160, 161, 185, 192, 193, 202, 208, 223Robinson, J.R., 36, 37, 45Rochon, J., 44Roess, L.C., 80, 86Rogers, P.S.Z., 147, 160Roland, C.H., 2, 43, 44, 46Rosman, A., 45Rowe, A.M. Jr., 44, 45, 46, 143, 146, 159Rubin, L.C., 4, 80, 86Russell, M.P.M., 67Rzasa, M.J., 1–3, 44, 46

S

Saeterstad, T., 161Sage, B.H., 1, 3, 28, 45, 63, 67, 108, 140Salman, N.H., 45Saltman, W., 2, 39, 45Sandler, S.I., 84, 86, 87Savidge, J.L., 44Schaafsma, J.G., 3, 140Schlijper, A.G., 84, 87Schmidt, G., 52, 66Schrader, 64, 67Schroeder, G.M., 66Schroeter, J.P., 156, 161Schulte, A.M., 4, 63, 67Sepehrnoori, K., 141Shelton, J.L., 134, 135, 140, 141

Shirkovskiy, A.I., 49, 66, 171Sibbald, L.R., 140, 141Sicking, J.N., 42, 46Sigmund, P.M., 40, 45Silvey, F.C., 108Simon, R., 38, 45, 136, 141Singleterry, C.C., 2, 17Siu, A., 87Skjaeveland, S., 120Skjold-Jorgensen, S., 161Sloan, E.D., 155, 156, 161Smart, G.T., 65, 66, 67Soave, G., 1, 4, 47, 49, 50, 51, 63, 64, 66, 200, 202, 205, 208, 221Song, K.Y., 156, 161Søreide, I., 67, 79, 83, 86, 120, 144, 145, 150, 158, 193, 201–08Spencer, G.C., 141Spivak, A., 119Stalkup, F.I. Jr., 130, 138, 140Standing, M.B., 1–3, 23–27, 29–39, 42–46, 90, 91, 94, 95, 108, 143, 145,

147–49, 151, 159, 160, 172, 177, 179–83, 189, 190, 192, 205, 206,208, 221, 223, 224

Starling, K.E., 1, 4, 24, 44, 47, 66, 83, 86, 108, 161Stegun, I.A., 86Stephenson, R.E., 4Stiel, L.I., 38, 45Sutton, R.P., 24, 25, 30, 36, 38, 44, 45, 160, 175, 178, 192, 205, 208

T

Takacs, G., 24, 44Tang, D.E., 119, 120Teja, A.S., 66Terry, R.E., 86Thodos, G., 38, 45, 141, 224Thomassen, P., 66, 67, 86, 87, 208Thomson, G.H., 34, 45Tindy, R., 108Torp, S.B., 42–44, 46, 105, 108, 111, 112, 119, 207, 208Trainer, R.P., 3, 45Trangenstein, J.A., 66Trekell, R.E., 155, 161Trengove, R., 108, 120Trube, A.S., 35, 45Trujillo, D.E., 140, 141Turek, E.A., 67, 141Twu, C.H., 82, 83, 86, 194, 208, 221, 223, 224

U

Unruh, C.H., 3, 161Usdin, E., 66

V

van der Burgh, J., 67van der Waals, J.D., 1, 2, 24, 33, 44, 47, 48, 51, 66Vazquez, M., 30, 35, 36, 38, 45Vink, D.J., 2, 17Vogel, J.L., 32, 45, 67, 126, 141von Stackelberg, M., 152, 161

W

Walter, C.J., 141

AUTHOR INDEX 5

Wang, Y., 128, 141Watson, K.M., 77, 78, 81, 86Webb, G.B., 4, 80, 86Webster, D.C., 3Wehe, A.H., 147, 148, 160Weinaug, C.F., 3, 38, 39, 45Wenzel, H., 52, 66Wheaton, R.J., 64, 67Whiting, R.L., 44, 108Whitson, C.H., 4, 17, 26, 30, 42–46, 54, 65–67, 73, 74, 76, 78, 79, 83–86,

105, 108, 111, 112, 119, 120, 144, 145, 150, 158, 201–04, 207, 208Wichert, E., 24, 25, 44, 177, 192, 224Wiebe, R., 159Wilcox, W.I., 2, 154, 161Wilke, C.R., 40, 45Williams, B., 3Wilson, G.M., 42, 46, 53, 54, 58, 66, 183–87, 192Wilson, K., 141Woods, R.W., 119Wu, R.S., 84, 87

Y

Yale, W.D., 3

Yarborough, L., 4, 23, 24, 32, 44, 45, 47, 50, 66, 79, 82, 86, 126, 134, 135,

141, 175, 177, 192, 223

Yellig, W.F., 125, 136, 138, 140

Young, L., 66

Young, L.C., 4

Youngren, G.K., 51, 52, 66, 83, 87, 119, 120

Ypma, J.G.J., 67

Yu, A.D., 141

Z

Zana, E., 45

Zhou, D., 126, 141

Zick, A.A., 4, 53, 55, 63, 67, 119–21, 124, 126, 128, 132, 133, 140

Zudkevitch, D., 47, 50, 66, 83, 87, 223

SUBJECT INDEX 1

��

AAbsolute zero, 167Acentric factor, 81, 162, 163, 194Air density, 167Alani-Kennedy method, 33Algorithms, 53

Flash calculations, 53Gravity/chemical equilibrium, 64Michelsen stability test, 57Minimum miscibility pressure, 127Newton-Raphson, 53, 72Vapor/liquid equilibrium (VLE), 47, 139

Alkanes, 7, 38, 82American Soc. for Testing Materials

(ASTM), 69Antifreezes, 157API Research Projects, 6Aromaticity factor, 78, 79Asphaltene

Chemical structure, 9Precipitation, 134, 139

Atomic mass, 18Aziz correlation, 37

BBeggs-Robinson correlation, 37Bergman correlation, 37Binary-interaction parameters (BIP’s), 49, 150,

164, 193, 195, 202Binary mixtures

Critical locus, 122Gibbs energy surfaces, 56–61Phase equilibria, 56p-T phase envelope, 54

Black oil, 13, 15Composition, 6PVT formulations

Modified, 110, 116, 200, 207Traditional, 109, 116

PVT properties, 20, 109Boiling points, 162, 163

Correlation, 81Heptanes-plus fractions, 79, 193n-alkanes, 7, 8North Sea condensate, 69

Bottomhole oil, 88Brines

Composition, 143Gas/oil ratio, 21Properties (Problem 22), 190

Bubblepoint curves, 11, 13, 15Bubblepoint oil

FVF, 35Viscosity, 37

Bubblepoint pressure, 29, 210Calculation (Problems 12 and 16), 180, 183

CCampbell’s calculation methods, 155Carbon dioxide, 25

Diffusivity, 40Flooding, 121Hydrate-formation conditions, 161Injection, 135MMP correlation, 138Physical properties, 134, 135Slim-tube displacement, 125

Carbon-12 standard, 18Carr correlation, 27Chemical compounds, 18Chemical potential, 47Clathrates, 151, 156Color change, 125Component fractions, 19Compositional correlation, 38Compositional gradients, 63Compositional relations, 114Compressibility

Brine, 145, 190Gas, 23Gas (Problem 6), 174Isothermal, 20, 210Saturated oil, 36, 94Undersaturated oil, 35

Computer programsCSMHYD, 156GAMSPL, 74, 75, 165UNIQUAC, 156

Constant composition expansion (CCE), 88Gas condensate, 94Oil, 93

Constant volume depletion (CVD), 97Consistency check, 105Gas condensate, 102–07, 195–200

Convergence pressure, 44, 184Calculation (Problem 17), 184

Correlations, 18Corresponding states theory, 18Cricondenbar, 11Cricondentherm, 11Critical constants, 162, 163

Critical pressure, 80, 82Critical properties, 18

Rules for calculation, 84Critical temperature, 80, 82Critical volume, 82Crude oil

California, 29, 35Composition and properties, 6h series, 9Hydrocarbon classes, 8Simulated distillation, 72

Crystallography, of hydrates, 152

DDead stock-tank oil

Saturation with CO2, 136Viscosity, 36

Density, 19Air, 167Brines, 190Carbon dioxide, 134, 135, 138Conversion factors, 168Gas, 22Liquid, 162Oil, 30, 34, 113, 130, 179Partial, 118Water, 167

Depletion reservoirs, 12, 15, 97, 110Dewpoint curves, 11, 13, 15Dewpoint pressure, 28, 102, 131, 195Diatomic compounds, 18Differential liberation expansion (DLE), 88

Laboratory procedure, 95Oil sample, 98–101Oil volumetric properties, 126Raleigh report, 211Reservoir oil, 203–07

Diffusion coefficients, 21, 38, 40Conversion factors, 171

Distillation, 68Dry gas, 13

Composition, 6FVF, 111, 113

EEarth’s gravitational acceleration, 167Elements, chemical, 18Equations of state

Applications, 193Composition calculations, 115

2 PHASE BEHAVIOR

Critical-properties estimation, 82Cubic, 47, 151Matching to measured data, 65Multiple-contact PVT experiments, 126Oil-gravity calculations, 112Peng-Robinson, 50, 150, 156, 164, 185, 193,

196, 202Predictions, 195Redlich-Kwong, 48Slim-tube profiles, 123, 132Soave-Redlich-Kwong, 49, 83, 164, 200, 202Solubility predictions, 150–54Ternary system (problem 18), 185Two-phase flash algorithm, 53van der Waals, 48Water/hydrocarbon systems, 146Zudkevitch-Joffe-Redlich-Kwong, 50

EthaneDensity, 30p-V diagram, 11

Ethane/n-heptane system, 11, 13, 122

FFick’s law, 21Field shrinkage factor, 88Flash calculations, 52, 53, 89, 184, 212

Problem 17, 184Formation volume factor (FVF)

At less than bubblepoint pressure, 210, 212Brine, 146, 191Bubblepoint oil, 35, 91Carbon dioxide, 135Dry gas, 111, 113Gas, 20, 109Nitrogen/oil mixture, 129Oil, 20, 99, 109, 111Separator oil, 90Total, 28Water, 20Water/brine, 145Wet gas, 113

Formation-water properties, 142Fugacity, 49

Calculation (Problem 18), 187

GGalimberti-Campbell method, 43Gamma-distribution model, 73Gas

Composition, 114Density, 22, 113FVF, 20, 109, 113Gravity, 22, 111High-sulfur-content (Problem 2), 172Phase behavior, 5Properties, pseudocritical, 24

Problem 8, 177Properties and correlations, 18Properties (Problem 1), 172Volumetric properties, 5, 22

Gas cap, 207Gas chromatography, 68, 70, 89, 130Gas-condensate

Boiling points, 69Composition, 6Constant composition expansion, 94, 97Constant volume depletion, 102–07Effect of nitrogen, 129Fluid characterization, 193Isotherms, 14Material-balance calculations, 92MBO properties, 118p-T diagrams, 13

PVT analysis, 88Retrograde region, 14Stepwise-regression procedure, 85

Gas constant, 167Gas cycling, 130Gas injection

Methods, 121Modifications, 119

Gas mixtures, 22Gas/oil ratio (GOR), 13, 21, 109

CO2/oil system, 136Separator test, 88, 91, 100

Gasoline properties (Problem 3), 173Gas solubility, 143

Calculation (Problem 22), 190Gas viscosity, 26

CalculationProblem 7, 175Problem 14, 182

Gaussian quadrature functions, 77, 193General dominant eigenvalue method

(GDEM), 54Gibbs energy surfaces, 56–61Gibbs free energy, 52Gibbs phase rule, 8Gravity/chemical equilibrium (GCE), 63Greek alphabet, 166Gross heating value, 162, 163

HHammerschmidt’s equation, 157Henry’s law, 143Heptanes-plus (C7+) fractions

Acentric factor, 195Boiling points, 193Characterization, 68Critical properties, 194Gas cycling, 130Liquid-dropout curve, 108Molar distribution, 193Pseudocritical properties, 25Single carbon number, 71Specific gravity, 193

Hoffman method, 41Hydrate formation

Calculation methods, 154–56, 190Calculation (Problem 21), 190

Hydrates, 151Crystallography, 152Inhibition, 157Phase diagrams, 153

HydrocarbonsComponent properties, 162–63Crude oil, 8Heavy, 121, 138Hydrate-former constants, 155, 160Intermediate, 121, 129, 132Light, 121, 138, 145, 172/nonhydrocarbon component pairs, 164Parachors, 39p-T diagrams, 9, 13Ternary system, 122–24/water systems, 142

Hydrogen sulfide, 25, 172

IIdeal gas law, 22Ideal liquid yield, 162, 163Immiscible CO2/oil behavior, 136Inflow-performance relations (IPR’s), 116, 117Inspection-properties estimation, 77Interfacial tension (IFT), 21, 38

Methane/water system, 149

Water/brine/hydrocarbon systems, 149Isothermal gravity/chemical equilibrium, 64

JJacobian matrix, 55Jacoby aromaticity factor, 78Joule-Thompson expansion, 154, 158, 159

KKatz-Carson charts, 154, 155Kay’s mixing rule, 19, 24, 25, 93K values

Black oil, 114Calculation (Problem 15), 183Correlations, 40Hydrate formation, 154MMP calculations, 127Nonhydrocarbon, 43Reservoir oil/gas, 116, 202Standing low-pressure, 43

LLaboratory experiments

Differential liberation expansion, 95Slim-tube displacements, 124True-boiling-point, 68–70, 73

Laboratory reportsGeneral information sheet, 88Oil PVT, 209

Langmuir adsorption theory, 156Lasater equation, 29Lean-gas injection, 128Lee-Gonzalez correlation, 26Liquefied petroleum gases (LPG’s), 121, 131Liquid chromatography, 70Liquid density, 162, 163Liquid-dropout curves, 12, 15, 102, 104Lucas correlation, 28

MMacleod relation, 39Mass, 18

Conversion factors, 169Mass fractions, 19Mass spectroscopy, 70Material-balance relations, 53, 92,

108, 117, 209Methane

/brine system, 149/butane/decane system, 122/C7+ BIP’s, 196Density, 30/hydrocarbon mixtures, 11Maximum content determination, 132/NaCl brine mixtures, 151/propane/water mixtures, 157Solubility in water, 143/water system, 149, 154

Methane-rich injection gases, 128, 129Methanol, 157, 161Michelsen stability test, 57, 62, 185Midvolume-point method, 69Minimum miscibility pressure (MMP),

122, 125, 127Miscibility, 122

CO2/oil behavior, 137Temperature range, 138

Miscible displacement projectsEnriched-gas miscible drive, 131

SUBJECT INDEX 3

Vaporizing-gas miscible drive, 129Mixing rules, 19, 84Molality, 143Molar density, 19Molar distribution, 70

Exponential distributions, 72Gamma-distribution model, 73

Molarity, 143Molar mass, 18Molar volume, 19, 22, 188

Calculation (Problem 19), 188Mole, 18

Conversion factors, 171Molecular mass, 18Molecular weights, 18, 162, 163

Correlations, 82Cumulative, 76Gas-condensate example, 197Heptanes-plus fractions, 70, 73

Mole fractions, 114, 148Multicell vaporization model, 127Multicomponent mixtures

Pseudoternary diagrams, 124Rachford-Rice function, 53

Multiphase behaviorCO2/oil, 139Enriched-gas injection, 134

Multistage separation, 91, 111

Nn-alkanes

Boiling point, 7, 8Parachors, 38

Natural gas, 147Composition, 6, 155Correlations for PVT properties, 22Hydrate-formation conditions, 156Joule-Thompson expansion, 154, 158, 159Quadruple points, 157

Natural gas liquids (NGL’s), 21, 130Newton-Raphson algorithm, 53, 72Nitrogen

Effect on dewpoint pressure, 131Injection gas, 124/NaCl-brine system, 154/oil mixture, 127, 129, 130

Nitrogen-rich injection gases, 129Nomenclature, 2, 18, 220North Sea gas condensate

K-value correlation, 43Simulated distillation, 72, 73Specific gravity, 78TBP distillation, 69, 70

North Sea oils, 29Gamma density function, 74

OOil

Composition, 114Constant composition expansion, 94Differential liberation expansion, 95, 98–101FVF, 111General information sheet, 89, 90Gravity, 20Gross heating value, 94Near-critical, 6Phase behavior, 5Properties and correlations, 18PVT analysis, 88Separator test, 93Volumetric behavior, 5

Oil compressibilitySaturated oil, 36

Undersaturated oil, 35, 94Oil density, 30, 113

Alani-Kennedy method, 34Differential liberation expansion, 203Nitrogen/oil mixture, 130

Oil/gas ratio (OGR), 13, 21Oil/gas/water systems, 145Oil mixtures, 29Oil viscosity, 36, 100, 101, 130

Calculation (problem 14), 182Effect of CO2, 137Reservoir-oil example, 206

PParachors, 38Paraffinicity, 29Paraffins, normal, 82Paraffins/naphthalenes/aromatics (PNA’s), 70Partial-density formulation, 118Peng-Robinson equation, 50, 150, 156, 164,

185, 193, 196, 202Perturbation expansions, 82Petroleum compounds, 5Petroleum-refinery products, 6Petroleum residue, 73Phase behavior

Conversion factors, 168–71Gas systems, 5Historical review, 1Oil systems, 5

Phase diagramsCarbon dioxide, 135Hydrates, 153Multicomponent systems, 11Simple systems, 8Single-component systems, 9Two-component systems, 10

Phase envelope, 12, 15Phase equilibria, 47

Binary mixtures, 56In gravity field, 63

Phase stability, 55Physical constants, 167Pressure, conversion factors for, 169Pressure/temperature diagrams

Depletion experiments, 15Ethane/n-heptane system, 13Gas-cap fluid, 16Gas-condensate system, 13Hydrocarbon binaries, 13NaCl brine, 150Phase envelope, 54Pure fluids and mixtures, 12Pure water, 150Single-component system, 9

Pressure/volume diagramsEthane, 11Pure component, 48Pure fluids and mixtures, 12

Pressure/volume/temperature (PVT) diagramsBelow bubblepoint, 96Black oil, 20, 109, 116, 200, 207Conventional measurements, 88Gas cap, 207Laboratory reports, 209Multicontact experiments, 126Pure compound, 10Reservoir oil, 207

Problems, example, 172Pseudocomponents, 84, 124Pseudocritical properties, 24, 175, 177Pseudoization, 85Pseudoliquid density

Chart for calculating, 31Oil, 180

Pressure correction, 32Separator gas, 34Temperature correction, 33

Pseudoternary diagrams, 124, 131

QQuadruple points, 153, 157Quaternary diagrams, 124

RRachford-Rice equation, 52Radial-flow equation, 116Raleigh field report, 209–19Rate equations (IPR’s), 116, 117Recommendations

Heptane-plus characterization, 83Laboratory report, 88

Recoveries, 97Calculated, 98, 107Corrections, 102Gas injection, 121Normal temperature separation, 99Plant products, 101Slim-tube, 132Stock-tank oil, 103

Recovery-pressure curves, 125Redlich-Kwong equation, 48Reduced properties, 18Regression parameters, 196Reservoir fluids

At less than bubblepoint pressure, 210Characterization, 201Classification, 12Composition, 5Compressibility, 20FVF, 20Grouping and averaging properties, 83

Reservoir gas, 110Reservoir mixtures, 19Reservoir oil, 109

Density calculation (Problems 10 and 11),179, 180

/gas mixtures, 127Slim-tube displacement, 125

Reservoir voidage (Problem 9), 178Reservoir water, 142Residual oil saturation (ROS), 121, 133Retrograde condensation, 11, 108, 126, 129

SSalinity, 142

Correction, 144Salts

Concentrations, 143Gas solubility, 145

Sample analysisBottomhole oil, 88Gas-condensate, 92, 102–07, 195Oil, 93Subsurface fluid, 215

Saturated oil, 16Compressibility, 94Rate equation, 117

Saturation-pressure calculation, 62Separator gas

Composition, 43, 89Pseudoliquid density, 34Water content, 190

Separator-oil composition, 89, 174Separator test, 91, 189, 205

Raleigh report, 209

4 PHASE BEHAVIOR

Well-effluent composition (Problem 4), 174Setchenow relation, 144Simulated distillation, 70, 72Single carbon number (SCN), 69–72, 77SI standards, 18SI system units, 163, 166Slim-tube displacements, 122, 124, 138Soave-Redlich-Kwong equation, 49, 83Sodium chloride brine, 142Solubility

Carbon dioxide, 136, 140Differential, 212Gas in water/brine system, 143, 153Methane in water, 154Natural gas in brines, 153Salinity correction, 144Water in natural gas, 147

Solution gas, 142Solution gas/oil ratio, 21, 111, 143Solution oil/gas ratio, 112, 113Søreide correlations, 79, 193, 202Specific gravity, 19

Components, 162, 163Gas, 22, 111Heptanes-plus fractions, 70, 79, 193Oil, 20, 78Reservoir-oil example, 205Stock-tank oil, 112Wellstream, 25

Specific volume, 19Standard atmosphere, 167Standing-Katz method, 30, 179Standing-Katz Z-factor chart, 23, 177Standing’s correlations, 24, 29, 37, 43, 90, 180Stepwise regression, 85Stock-tank oil, 109

Cabin Creek, 137Gravity, 26PVT data, 96Recovery, 103Slim-tube displacements, 137–39True-boiling-point distillation, 69Viscosity, 36/wellstream ratio, 26

Subsea oil and gas (Problem 13), 181Subsurface sampling, 214Sulfur-rich gas (Problem 2), 172Sum-of-squares (SSQ) function, 196Surface gravity, 26, 111Surface-separator calculations, 40, 43Surface-separator gas, 109, 110Surface tension, 171Sutton correlations, 24, 175Swelling

Dead stock-tank oil, 136Reservoir oil by CO2, 137

Test, 126

TTemperature correction, 33Temperature scale conversions, 167Ternary systems, 122, 183, 185Thermodynamic properties, 47Toluene, 83Total dissolved solids (TDS), 142True-boiling-point (TBP) analysis, 68–70, 73Two-phase flash calculation, 52

UUndersaturated oil

Compressibility, 35, 94, 204Radial-flow equation, 116Viscosity, 38

Units, 2, 18, 162Universal gas constant, 22, 164Universal oil products (UOP) factor, 77

Vvan der Waals EOS, 48van der Waals-Platteeuw model, 156Vapor/liquid equilibrium (VLE) algorithms,

47, 139Vapor/liquid/liquid (VLL) behavior, 121, 139Vazguez-Beggs correlations, 30, 35Viscosity, 21

Bubblepoint oil, 37Carbon dioxide, 134Conversion factors, 170Correlation, 72Gas, 26, 175, 182Gas-free oil, 37Oil, 36, 100, 101, 182Oil/nitrogen mixture, 130Undersaturated oil, 38Water/brine, 147

Volatile oil, 13, 15, 109–11Composition, 6Gamma density function, 74

Volume fractions, 19Volume-translation parameters, 51, 195Volumetric behavior

Calculation, 47CO2-rich stock-tank oil, 137Gas systems, 5Oil systems, 5Two-phase systems, 93

Volumetric properties, 19, 22, 216

Conversion factors, 170

WWater

Content of separator gas (Problem 21), 190Density, 167FVF, 146/hydrocarbon systems, 142Reservoir, 142Solubility in methane/NaCl-brine mixture,

151Solubility in natural gas, 147Triple point, 167Vapor in equilibrium with hydrates, 156

Water-alternating-gas (WAG) ratio, 121, 129Water/brine

Compressibility, 145FVF, 145Viscosity, 147

Water/brine/hydrocarbon systems, 149Water/ethane system, 157, 161Water/hydrocarbon systems

EOS predictions, 150Watson characterization factor, 29, 77, 81, 189

Calculation (Problem 20), 189Weight factors, 77Weight fractions, 19, 76, 143Well production test (Problem 5), 174Wellstream composition, 88, 91, 116, 174Wellstream specific gravity, 25Wet gas, 13, 16

Composition, 6CVD data, 105FVF, 113

Whitson-Torp method, 111, 112Wichert-Aziz correlations, 25Wilson equation, 42, 53

YYarborough aromaticity factor, 79

ZZ factor, 22

Calculation (problem 18), 97, 187Carbon dioxide, 134Correlations, 23, 177Gas-condensate example, 197Reservoir-oil example, 206van der Waals equation, 48

Zudkevitch-Joffe-Redlich-Kwong equation, 50

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Whitson PVT SPE Phase Behaviour Monograph

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