A COMPOSITIONAL SIMULATION MODEL FOR CARBON ...

A COMPOSITIONAL SIMULATION MODEL FOR

CARBON DIOXIDE FLOODING WITH

IMPROVED FLUID TRAPPING

by

Jeffrey S. Brown

c© Copyright by Jeffrey S. Brown, 2014

All Rights Reserved

A thesis submitted to the Faculty and the Board of Trustees of the Colorado School of Mines

in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Petroleum Engi-

neering).

Golden, Colorado

Date

Signed:Jeffrey S. Brown

Signed:Dr. Hossein Kazemi

Thesis Advisor

Golden, Colorado

Date

Signed:Dr. Erdal Ozkan

Professor and Interim HeadDepartment of Petroleum Engineering

ii

ABSTRACT

A new formulation for fluid trapping using a dual-media approach which includes compositional

trapping and interphase mass transfer was developed, coded, and validated. This formulation does

not exist in notable commercial reservoir simulators. The formulation was incorporated into a

three-dimensional, three-phase, parallel compositional simulator to simulate carbon dioxide (CO2)

water-alternating-gas (WAG) injection. Fluid phase trapping is both a channeling issue and a pore-

scale issue. Pore-scale phase trapping is strongly related to hysteresis in the relative permeability

and capillary pressure; the simulator incorporates them in a methodology consistent with these

issues. New algorithms were developed to implement the CO2 solubility in water and oil and CO2

phase trapping in a way that preserves the mass balance of the oil, water, and gas phases. The new

simulator was implemented using a parallel infrastructure to facilitate computationally intensive

fine grid systems.

For test examples, we focused on a mixed wet carbonate reservoir in the Middle East. These

tests were used to evaluate the significance of various trapping scenarios. Compositional trapping,

gas relative permeability hysteresis, CO2 solubility in water, and permeability heterogeneity were

found to have significant impacts on oil recovery and timing, as well as CO2 storage and utilization

during waterflood and CO2 WAG processes.

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TABLE OF CONTENTS

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxii

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxiv

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxvi

DEDICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxvii

CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

CHAPTER 2 LITERATURE REVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1 Enhanced Oil Recovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.1 Miscible Flooding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.2 Gas Injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.3 Other Enhanced Oil Recovery Methods . . . . . . . . . . . . . . . . . . . . . 5

2.2 CO2 Enhanced Recovery and Sequestration . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2.1 CO2 Enhanced Oil Recovery . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2.2 CO2 Flood Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.3 CO2 WAG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.4 CO2 Sequestration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.5 CO2 Simulation with TOUGH . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.6 CO2 Water Solubility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.7 CO2 Trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.8 Other Articles on CO2 Injection . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Reservoir Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

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2.3.1 Computation Approaches in Reservoir Simulation . . . . . . . . . . . . . . 10

2.3.2 Fractured Reservoir Simulation . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3.3 Compositional Reservoir Simulation . . . . . . . . . . . . . . . . . . . . . . 12

2.3.4 CO2 and Miscible Flood Simulation . . . . . . . . . . . . . . . . . . . . . . 13

2.3.5 Parallel Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.6 Simulation of Trapping and Bypassing . . . . . . . . . . . . . . . . . . . . 15

2.3.7 Simulation of Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.8 Additional Simulation Topics . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.9 SPE Comparative Solution Projects . . . . . . . . . . . . . . . . . . . . . . 16

2.4 Geologic Characterization in Middle East . . . . . . . . . . . . . . . . . . . . . . . . 17

2.5 Relative Permeability and Capillary Pressure . . . . . . . . . . . . . . . . . . . . . . 17

2.5.1 General Articles on Relative Permeability . . . . . . . . . . . . . . . . . . 17

2.5.2 General Articles on Capillary Pressure . . . . . . . . . . . . . . . . . . . . 18

2.5.3 Trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.5.4 Three-Phase Relative Permeability . . . . . . . . . . . . . . . . . . . . . . 18

2.5.5 Relative Permeability Hysteresis . . . . . . . . . . . . . . . . . . . . . . . . 21

2.5.6 Capillary Pressure Hysteresis . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.5.7 Combined Relative Permeability and Capillary Pressure Hysteresis . . . . 22

2.5.8 Non-zero Relative Permeability Derivative . . . . . . . . . . . . . . . . . . 23

2.5.9 Additional Relative Permeability Effects . . . . . . . . . . . . . . . . . . . 23

2.6 Equation of State Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.6.1 Calculation of Equation of State . . . . . . . . . . . . . . . . . . . . . . . . 24

2.6.2 Adjusting Equation of State Parameters . . . . . . . . . . . . . . . . . . . 25

2.6.3 Modifications to Equation of State Model when CO2 is Present . . . . . . 25

v

2.6.4 Phase Behavior Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.6.5 Other Equation of State References . . . . . . . . . . . . . . . . . . . . . . 26

2.7 Pore Scale Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.7.1 Network Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.7.2 Micro Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.7.3 Additional Pore Scale Simulation Discussion . . . . . . . . . . . . . . . . . 28

2.8 Interfacial Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.8.1 Interfacial Tension Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.8.2 Interfacial Tension and Relative Permeability . . . . . . . . . . . . . . . . 29

2.8.3 Spreading Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.8.4 Interfacial Tension Fit Gas-Oil . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.8.5 Water Interfacial Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.8.6 CO2-Brine Interfacial Tension . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.9 Liquid-Liquid-Vapor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.10 Asphaltenes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

CHAPTER 3 COMPOSITIONAL RESERVOIR SIMULATION OVERVIEW . . . . . . . . 36

3.1 Compositional Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2 Commercial Simulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.3 Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.4 Partially Implicit Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.4.1 IMPES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.4.2 IMPSEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.4.3 Fully Implicit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.4.4 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

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3.5 Thermodynamic Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.6 Typical Sizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.7 Off-Diagonal Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.8 Well Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.9 Right Hand Side . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.10 Total Rate Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.11 Accumulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.12 Accumulation Pressure Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.13 Accumulation Saturation Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.14 Accumulation Composition Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.15 Pressure Spatial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.16 Fugacity Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.17 Computation for Fixed Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.18 Computation for Fixed Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.19 Additional Implicit Decisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

CHAPTER 4 MATHEMATICAL FORMULATION OVERVIEW . . . . . . . . . . . . . . . 57

4.1 Primary Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.2 Secondary Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.2.1 Calculation of Secondary Variables . . . . . . . . . . . . . . . . . . . . . . 62

4.2.2 Storage of Secondary Variables . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.2.3 List of Secondary Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.3 Overview of Simulation Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.4 Assemble the Jacobian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.4.1 Single Medium (No Trapping) . . . . . . . . . . . . . . . . . . . . . . . . . 79

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4.4.2 Degenerate Case with Oil and Water Only . . . . . . . . . . . . . . . . . . 79

4.4.3 Degenerate Case with Gas and Water Only . . . . . . . . . . . . . . . . . . 79

4.4.4 Degenerate Case with Gas and Oil Only . . . . . . . . . . . . . . . . . . . 80

4.4.5 Degenerate Case with Water Only . . . . . . . . . . . . . . . . . . . . . . . 80

4.4.6 Degenerate Case with Oil Only . . . . . . . . . . . . . . . . . . . . . . . . 80

4.4.7 Degenerate Case with Gas Only . . . . . . . . . . . . . . . . . . . . . . . . 81

4.4.8 Three-Phase Degenerate Case with Fewer Components . . . . . . . . . . . 81

4.5 Rewrite Base Equations for Um Solve . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.6 Update Primary Variables at Each Nonlinear Iteration . . . . . . . . . . . . . . . . 85

4.7 Update Primary Variables at Each Nonlinear Iteration: Flash . . . . . . . . . . . . 87

4.8 Update WCO2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

CHAPTER 5 TRAPPING FORMULATION . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.1 Trapping Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.2 Initialize Trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.3 Update Trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.3.1 Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.3.2 Mass at Time n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.3.3 Transfer Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.3.4 Update Mole Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.3.5 Compute the Volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.3.6 Compute the Saturations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.4 Single Porosity Irreversible Trapping . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.5 Dual Porosity as Reversible Trapping . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.6 Dual Porosity Computation Options . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

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5.6.1 Implicit Pm2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.6.2 Explicit Pm2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.6.3 Implicit τ = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.6.4 Explicit τ = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.7 Computation of the Solution of a Dual Porosity System . . . . . . . . . . . . . . . . 106

CHAPTER 6 TIME DERIVATIVES FORMULATION . . . . . . . . . . . . . . . . . . . . . 108

6.1 Pressure Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.2 Saturation Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.3 Composition Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

CHAPTER 7 SPACE DERIVATIVES FORMULATION . . . . . . . . . . . . . . . . . . . . 111

7.1 Initial Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

7.2 Transmissibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

7.3 Expand Deltas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

7.4 Expand Terms on Left-Hand-Side . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

7.5 Rearrange Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

7.6 Combine Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

7.7 Upstream Weighting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

7.8 Time Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

CHAPTER 8 EQUATION OF STATE FORMULATION . . . . . . . . . . . . . . . . . . . . 119

8.0.1 Expand Fugacities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

8.1 Fugacity Equations - Above Bubble Point . . . . . . . . . . . . . . . . . . . . . . . . 120

8.2 Fugacity Equations - Below Dew Point . . . . . . . . . . . . . . . . . . . . . . . . . 121

8.3 Method for Peng-Robinson Flash Calculation . . . . . . . . . . . . . . . . . . . . . 122

8.3.1 Peneloux Volume Adjustment . . . . . . . . . . . . . . . . . . . . . . . . . 123

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8.3.2 Constants for This Formulation . . . . . . . . . . . . . . . . . . . . . . . . 123

8.3.3 Initial Values, Compute Km, (Full Flash Only) . . . . . . . . . . . . . . . 124

8.3.4 Flash to Calculate the Vapor Fraction, (Full Flash Only) . . . . . . . . . . 125

8.3.5 Calculate the Mixing Parameters . . . . . . . . . . . . . . . . . . . . . . . 126

8.3.6 Calculate the z-factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

8.3.7 Calculate the Fugacities f (Not if Only Computing z) . . . . . . . . . . . 127

8.3.8 Calculate the Tolerance (Full Flash Only) . . . . . . . . . . . . . . . . . . 127

8.3.9 Calculate the Densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

8.3.10 Calculate the saturations (Full Flash Only) . . . . . . . . . . . . . . . . . . 129

8.4 Evaluate Fugacity Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

8.5 Evaluate Peng-Robinson Pressure Derivatives . . . . . . . . . . . . . . . . . . . . . 130

8.5.1 Evaluate ∂ξ∂P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

8.5.2 Evaluate ∂z∂P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

8.5.3 Evaluate Derivatives of f(z) . . . . . . . . . . . . . . . . . . . . . . . . . . 131

8.5.4 Evaluate Derivatives of A and B . . . . . . . . . . . . . . . . . . . . . . . . 132

8.6 Evaluate Peng-Robinson Composition Derivatives . . . . . . . . . . . . . . . . . . . 132

8.6.1 Evaluate ∂ξ∂Xm′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

8.6.2 Evaluate ∂z∂Xm′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

8.6.3 Evaluate Derivatives of A and B . . . . . . . . . . . . . . . . . . . . . . . . 133

8.6.4 Evaluate ∂a∂Xm′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

8.6.5 Evaluate ∂b∂Xm′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

8.7 Check Fugacity Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

8.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

8.7.2 Fugacity Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

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8.7.3 Pressure Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

8.7.4 Composition Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

8.7.5 Consistency Check . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

8.8 Solving Cubic Equations Numerically . . . . . . . . . . . . . . . . . . . . . . . . . . 140

8.8.1 Initialize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

8.8.2 Three Distinct Real Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

8.8.3 One Real Root . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

8.8.4 Three Real Roots, Two or More Coincide . . . . . . . . . . . . . . . . . . . 142

8.8.5 Newton Raphson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

8.9 Fugacity Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

8.10 Flash Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

8.11 Flowchart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

CHAPTER 9 FORMULATION OF WELLS . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

9.1 Well Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

9.2 Flow from Node to Well . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

9.3 Well Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

9.4 Properties for Flow in Wellbore . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

9.5 Pressure in Wellbore . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

9.6 Compute the Moody Friction Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

9.7 Computation for Fixed Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

9.8 Computation for Fixed Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

9.9 Wells with Single Completions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

9.9.1 Fixed Pressure Producer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

9.9.2 Fixed Rate Producer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

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9.9.3 Fixed Mole Rate Producer . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

9.9.4 Fixed Pressure Injector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

9.9.5 Fixed Rate Injector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

9.9.6 Fixed Pressure Producer with Switch to Rate Control . . . . . . . . . . . . 165

9.9.7 Fixed Rate Producer with Switch to Pressure Control . . . . . . . . . . . . 166

9.9.8 Fixed Pressure Injector with Switch to Rate Control . . . . . . . . . . . . 167

9.9.9 Fixed Rate Injector with Switch to Pressure Control . . . . . . . . . . . . 168

CHAPTER 10 MASS BALANCE CALCULATIONS . . . . . . . . . . . . . . . . . . . . . . . 170

10.1 Calculate Surface Conditions of Well Fluids Using Separators . . . . . . . . . . . . 170

10.2 Calculate Surface Conditions of Original Oil in Place . . . . . . . . . . . . . . . . . 173

10.3 Mass Balance Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

CHAPTER 11 RELATIVE PERMEABILITY AND CAPILLARY PRESSURE . . . . . . . . 177

11.1 Three Phase Relative Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

11.2 Hysteresis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

11.2.1 Hysteresis Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

11.2.2 Hysteresis Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

11.2.3 Combined Three-Phase Relative Permeability and Hysteresis . . . . . . . . 180

11.2.4 Combined Analysis of Algorithms . . . . . . . . . . . . . . . . . . . . . . . 181

11.3 Trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

11.3.1 Composition of Trapped Phase . . . . . . . . . . . . . . . . . . . . . . . . 183

11.3.2 Simple Trapping Composition . . . . . . . . . . . . . . . . . . . . . . . . . 183

11.3.3 Complex Trapping Composition . . . . . . . . . . . . . . . . . . . . . . . . 184

11.3.4 Composition Trapping Formulation . . . . . . . . . . . . . . . . . . . . . . 184

11.4 Interfacial Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

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11.4.1 Interfacial Tension Literature . . . . . . . . . . . . . . . . . . . . . . . . . 185

11.5 Rock Type and Wettability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

11.5.1 Rock Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

11.5.2 Wettability Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

11.5.3 Static Wettability Changes . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

11.5.4 Dynamic Wettability Changes . . . . . . . . . . . . . . . . . . . . . . . . . 187

11.6 Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

11.7 Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

11.8 Flow Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

11.9 Brooks-Corey Properties for Mixed Wet Rock . . . . . . . . . . . . . . . . . . . . . 190

11.9.1 Simplified Three-Phase Relative Permeability . . . . . . . . . . . . . . . . 190

11.9.2 Derivatives of Simplified Three-Phase Relative Permeability . . . . . . . . 192

11.9.3 Two-Phase Relative Permeabilities . . . . . . . . . . . . . . . . . . . . . . 193

11.9.4 Water-Oil Capillary Pressure for Mixed-Wet Systems . . . . . . . . . . . . 194

11.9.5 Gas-Oil Capillary Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

11.9.6 Derivatives of Capillary Pressure . . . . . . . . . . . . . . . . . . . . . . . 196

11.10 Three Phase Relative Permeability References . . . . . . . . . . . . . . . . . . . . . 196

11.11 Hysteresis References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

11.12 Combined Three-Phase Relative Permeability and Hysteresis References . . . . . . 198

CHAPTER 12 VISCOSITY FORMULATION . . . . . . . . . . . . . . . . . . . . . . . . . . 200

12.1 Treatment of Viscosity by Commercial Applications . . . . . . . . . . . . . . . . . . 200

12.2 Other Viscosity Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

12.3 Lohrenz-Brae-Clark Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

12.3.1 Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

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12.3.2 Time-Dependent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

12.4 Jossi plus Lee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

12.5 Corresponding States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

12.5.1 Methane Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

12.5.2 Methane Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

12.5.3 Corresponding States Calculations . . . . . . . . . . . . . . . . . . . . . . . 210

12.5.4 Heavy oil adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

12.6 Extended Corresponding States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

12.6.1 n-Decane Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

12.6.2 n-Decane Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

12.6.3 Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

12.7 f -Theory Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

12.7.1 Dilute Gas Viscosity and General Properties . . . . . . . . . . . . . . . . . 217

12.7.2 f -Theory Friction Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 217

12.7.3 Mixing Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

CHAPTER 13 FORMULATION FOR PROPERTIES OF WATER CONTAINING CO2 . . . 220

13.1 CO2 Solubility in Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

13.2 Adjustments to Equation of State . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

13.3 Other Special Properties of CO2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

13.4 Properties of Water Containing CO2, Overview . . . . . . . . . . . . . . . . . . . . 222

13.5 Commercial Simulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

13.6 Properties of Water Containing CO2, CMG GEM . . . . . . . . . . . . . . . . . . . 223

13.7 Units of concentration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

13.8 Selection Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

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13.8.1 Rowe, Brine Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

13.8.2 Garcıa, CO2 Brine Density and Partial Molar Volume . . . . . . . . . . . . 227

13.8.3 Kestin, Brine Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

13.8.4 Duan, Henry’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

13.9 Correlations for this Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

13.10 Computational Forms of WCO2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

13.10.1 Option 0: WCO2 = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

13.10.2 Option C: Constant WCO2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

13.10.3 Option ZW0: Compute ξw using WCO2 = 0 . . . . . . . . . . . . . . . . . . 231

13.10.4 Option ZW1: Compute ξw using WCO2 . . . . . . . . . . . . . . . . . . . . 232

13.10.5 Option KP1: Use a simplified model for WCO2 using YCO2 [Pb] . . . . . . . 232

13.10.6 Option KP2: Use a simplified model for WCO2 using YCO2 below thebubble point and YCO2 = 0 above the bubble point . . . . . . . . . . . . . 233

13.10.7 Option KP3: Use a simplified model for WCO2 using YCO2 [Pb] . . . . . . . 234

13.10.8 Option 1: W n+1CO2

fully implicit . . . . . . . . . . . . . . . . . . . . . . . . . 234

13.10.9 Option 2: W n+1CO2

implicit pressure, explicit fugacity coefficient . . . . . . . 235

13.10.10 Option 3: W n+1CO2

implicit pressure, explicit fugacity . . . . . . . . . . . . . 235

13.10.11 Option 4: W n+1CO2

implicit pressure, fugacity at � . . . . . . . . . . . . . . . 236

13.10.12 Option 2Z: W n+1Z,CO2

implicit pressure, explicit fugacity coefficient . . . . . . 236


implicit pressure, explicit fugacity . . . . . . . . . . . . 237


implicit pressure, fugacity at � . . . . . . . . . . . . . . 237

13.10.15 Option 1XY: W n+1CO2

partially implicit, function of both Xm′ and Ym′ . . . 238

13.10.16 Option Y1: W n+1CO2

fully implicit . . . . . . . . . . . . . . . . . . . . . . . . 240

13.10.17 Option Y5: W nCO2

explicit . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

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13.10.18 Option P1: W n+1CO2

[P only] fully implicit . . . . . . . . . . . . . . . . . . . . 241

13.10.19 Option K1: W n+1CO2

[YCO2 only], evaluate Y at � . . . . . . . . . . . . . . . . 242

13.10.20 Option K2: W n+1CO2

[YCO2 only], evaluate Y at � . . . . . . . . . . . . . . . . 243

13.10.21 Using WCO2 as a Transfer Term . . . . . . . . . . . . . . . . . . . . . . . . 243

13.10.22 Rowe, Brine Density, Eclipse + VIP+CMG, H2O+NaCl, ρw + Cw . . . . 245

13.10.23 Garcıa, CMG, Brine Density, H2O+CO2 +NaCl, ρw + vCO2 . . . . . . . . 246

13.10.24 Kestin, Brine Viscosity, Eclipse+VIP, H2O+NaCl, μw . . . . . . . . . . . 250

13.10.25 Duan, Henry’s Law, H2O+CO2 +NaCl, WCO2 +HCO2 . . . . . . . . . . 251

13.11 Correlations Used to Evaluate Other Correlations . . . . . . . . . . . . . . . . . . . 253

13.11.1 Zeebe, Henry’s Law for Seawater, H2O+CO2 +NaCl, H . . . . . . . . . . 254

13.11.2 Duan, Fugacity, H2O+CO2 . . . . . . . . . . . . . . . . . . . . . . . . . . 254

13.12 Henry’s Law Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

13.12.1 Chang, Mole Fraction, Eclipse + VIP, H2O+CO2 +NaCl,WCO2 +Rsw +Bw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

13.12.2 CMG, Henry’s Law, H2O+CO2 +NaCl, WCO2 +HCO2 . . . . . . . . . . 257

13.12.3 Enick, Henry’s Law, H2O+CO2 +NaCl, H +Rsw +WCO2 + μw . . . . . 258

13.13 Adjustments to Peng-Robinson Equation of State . . . . . . . . . . . . . . . . . . . 259

13.13.1 Peng-Robinson Equation of State Paramters . . . . . . . . . . . . . . . . . 259

13.13.2 Soreide, EOS, Eclipse, H2O+CO2 +NaCl, WCO2 + ρaq . . . . . . . . . . . 260

13.13.3 Delshad, EOS and IFT, H2O+CO2 +NaCl, WCO2 + ρaq + σgw . . . . . . 261

13.13.4 Yan, EOS, H2O+CO2 +NaCl, WCO2 + ρaq . . . . . . . . . . . . . . . . . 262

13.13.5 Melhem, EOS, H2O+CO2, WCO2 + ρaq . . . . . . . . . . . . . . . . . . . 262

13.13.6 Spycher, EOS, Eclipse, H2O+CO2 +NaCl, WCO2 + ρaq . . . . . . . . . . 263

13.14 Models Considered But Not Used . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

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CHAPTER 14 COMPUTATION: ASSEMBLY OF JACOBIAN . . . . . . . . . . . . . . . . . 265

14.1 Diagonal Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

14.2 Diagonal Terms Above the Bubble Point . . . . . . . . . . . . . . . . . . . . . . . . 266

14.3 Diagonal Terms Below the Dew Point . . . . . . . . . . . . . . . . . . . . . . . . . . 267

14.4 Off-Diagonal Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

14.5 Well Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268

14.6 Right Hand Side . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

14.7 Total Rate Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

14.8 Accumulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

14.9 Accumulation Derivatives: Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

14.10 Accumulation Derivatives: Saturation . . . . . . . . . . . . . . . . . . . . . . . . . . 271

14.11 Accumulation Derivatives: Composition . . . . . . . . . . . . . . . . . . . . . . . . . 272

14.12 Spatial Derivatives: Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

14.13 Fugacity Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

14.14 Fugacity Equations - Above Bubble Point . . . . . . . . . . . . . . . . . . . . . . . . 274

14.15 Fugacity Equations - Below Dew Point . . . . . . . . . . . . . . . . . . . . . . . . . 275

14.16 Computation for Fixed Rate Wells . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

14.17 Computation for Fixed Pressure Wells . . . . . . . . . . . . . . . . . . . . . . . . . 276

14.18 Additional Comments on Computation . . . . . . . . . . . . . . . . . . . . . . . . . 277

14.19 Computational Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

14.20 Illustration of Solution Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278

14.20.1 Illustration of a 5× 5× 3 Model, Well Geometry . . . . . . . . . . . . . . 279

14.20.2 Illustration of a 5× 5× 3 Model, Block Values . . . . . . . . . . . . . . . . 280

14.20.3 Illustration of a 5× 5× 3 Model, Matrix Assembly . . . . . . . . . . . . . 282

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14.20.4 Illustration of a 5× 5× 3 Model, Local LU Decomposition . . . . . . . . . 282

14.20.5 Illustration of a 5× 5× 3 Reduced Model . . . . . . . . . . . . . . . . . . 290

14.20.6 Illustration of a 16× 16× 3 Model . . . . . . . . . . . . . . . . . . . . . . 290

CHAPTER 15 COMPUTATION: DESCRIPTION OF LINEAR SOLVERS . . . . . . . . . . 295

15.1 Serial Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

15.1.1 Dense Gaussian Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . 295

15.1.2 Band Gaussian Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . 296

15.1.3 Special Gaussian Elimination . . . . . . . . . . . . . . . . . . . . . . . . . 298

15.1.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

15.2 Parallel Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

15.2.1 Direct LU Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

15.2.2 Iterative Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

15.2.3 Parallel Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

CHAPTER 16 COMPUTATATION: PARALLEL COMPUTING . . . . . . . . . . . . . . . . 304

16.1 Computation Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

16.2 Solution Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308

16.3 Initialize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310

16.4 Scalability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

16.4.1 Computation Magnitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312

16.4.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314

CHAPTER 17 VALIDATION CASES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318

17.1 Validation Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319

17.2 Description of model 760E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319


xviii


17.5 Description of model 760F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322



17.8 Description of model 760G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326



17.11 Compare CMG Model with my Model 760E and 761E . . . . . . . . . . . . . . . . . 335

17.12 Compare CMG Model with my Model 762E . . . . . . . . . . . . . . . . . . . . . . 339

17.13 Compare CMG Model with my Model 760F, 761F, and 762F . . . . . . . . . . . . . 341

17.14 Compare CMG Model with my Model 760G . . . . . . . . . . . . . . . . . . . . . . 344



CHAPTER 18 CASE STUDIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351

18.1 Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351

18.2 Variations in Porosity and Permeability . . . . . . . . . . . . . . . . . . . . . . . . . 356

18.3 Relative Permeability Test Case Literature Review . . . . . . . . . . . . . . . . . . 359

18.3.1 Water-Oil Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

18.3.2 Gas-Oil Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

18.3.3 Gas-Water Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

18.3.4 Two-Phase Experiments with Different Phases . . . . . . . . . . . . . . . . 365

18.3.5 Three-Phase Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

18.3.6 Relative Permeability Formulations . . . . . . . . . . . . . . . . . . . . . . 366

18.3.7 Relative Permeability Observations . . . . . . . . . . . . . . . . . . . . . . 366

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18.4 Relative Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367

18.4.1 Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367

18.4.2 Oil/Water Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368

18.4.3 Gas/Oil Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368

18.4.4 Trapped Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370

18.4.5 Trapped Oil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370

18.4.6 Cycle Dependent Residual Oil Saturations . . . . . . . . . . . . . . . . . . 372

18.4.7 Water Relative Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . 372

18.4.8 Gas Relative Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . 373

18.4.9 Oil Relative Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377

18.5 Capillary Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383

18.6 Future Test Case Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389

CHAPTER 19 DISCUSSION OF RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . 392

19.1 Evaluation of Primary Production Performance . . . . . . . . . . . . . . . . . . . . 395

19.2 Evaluation of Waterflood Performance . . . . . . . . . . . . . . . . . . . . . . . . . 395

19.3 Evaluation of Continuous CO2 Injection . . . . . . . . . . . . . . . . . . . . . . . . . 398

19.4 Evaluation of CO2 WAG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404

19.5 Evaluation of Compositional Recovery Factor . . . . . . . . . . . . . . . . . . . . . 409

19.6 Evaluation of CO2 Storage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409

19.7 Evaluation of CO2 Utilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415

CHAPTER 20 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421

CHAPTER 21 RECOMMENDED FUTURE WORK . . . . . . . . . . . . . . . . . . . . . . 423

21.1 Use of This Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423

21.2 Formulation and Computation Enhancements . . . . . . . . . . . . . . . . . . . . . 423

xx

21.3 Phase Labeling and Relative Permeability Experiments . . . . . . . . . . . . . . . . 423

CHAPTER 22 NOMENCLATURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426

REFERENCES CITED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434

APPENDIX - RESULTS FOR SPECIFIC TEST CASES . . . . . . . . . . . . . . . . . . . . 480

A.1 Primary Production Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480

A.2 Waterflood Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489

A.3 Continuous CO2 Injection Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499

A.4 WAG Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509

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LIST OF FIGURES

Figure 2.1 Low temperature phase behavior of Wasson crude showing the presence oftwo liquid hydrocarbon phases . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Figure 2.2 Various miscibility regions for a CO2 flood, . . . . . . . . . . . . . . . . . . . 33

Figure 3.1 Block 2: block geometry for the off-block diagonal values with the IMPESformulation for a NC = 5 problem. . . . . . . . . . . . . . . . . . . . . . . . . . 44

Figure 3.2 Block 4: well terms for the component equations for a NC = 5 problem. . . . . 44

Figure 3.3 Block 6: right-hand-side terms for the component equations for a NC = 5problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Figure 3.4 Block 5: blocks for the well equations for a NC = 5 problem. . . . . . . . . . . 46

Figure 8.1 Illustration of amn. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

Figure 8.2 Regular flash calculation flow chart. . . . . . . . . . . . . . . . . . . . . . . . 149

Figure 8.3 Flash calculation flow chart for thermodynamic minimum miscibilitypressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

Figure 11.1 Illustration of pore doublet effect in a water-wet rock; green is oil and blue iswater. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

Figure 11.2 Variation of relative permeability with wettability changes . . . . . . . . . . . 188

Figure 12.1 Compare the density correlation for methane to Pedersen Figure 10.3. . . . . . 207

Figure 12.2 Compare the viscosity correlation for methane to Pedersen Figure 10.4. . . . . 209

Figure 12.3 Compare the Hanley viscosity correlation for methane to Gonzalez Figure 2. . 210

Figure 13.1 Solubility of methane in water . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

Figure 13.2 Solubility of CO2 in water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

Figure 13.3 Change in z-factor as a function of pressure for CO2. . . . . . . . . . . . . . . 222

Figure 14.1 Block 1: block geometry for the main block diagonal of a NC = 5 problem. . . 265

Figure 14.2 Block 7: block geometry above the bubble point for the main block diagonalof a NC = 5 problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

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Figure 14.3 Block 7: block geometry below the dew point for the main block diagonal ofa NC = 5 problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267



Figure 14.6 Block 6: right-hand-side terms for the component equations for a NC = 5problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269


Figure 14.8 Geometry of three horizontal wells for a 5× 5× 3 problem. . . . . . . . . . . 280

Figure 14.9 Matrix 0: block banded matrix for a 5× 5× 3 problem. . . . . . . . . . . . . . 280

Figure 14.10 Block 1: block geometry for the main block diagonal of a NC = 5 problem. . . 281


Figure 14.12 Block 3: block geometry for the off-block diagonal values with the IMPSECformulation for a NC = 5 problem. . . . . . . . . . . . . . . . . . . . . . . . . . 281



Figure 14.15 Matrix 1: Spatial derivatives for a 5× 5× 3× 9 problem. . . . . . . . . . . . . 283

Figure 14.16 Matrix 2: Time derivatives for a 5× 5× 3× 9 problem. . . . . . . . . . . . . . 284

Figure 14.17 Matrix 3: Combined matrix for a 5× 5× 3× 9 problem. . . . . . . . . . . . . 285

Figure 14.18 Matrix 4: Well matrix for a 5× 5× 3× 9 problem with three horizontal wells. . 286

Figure 14.19 Matrix 5: Combined matrix with wells for a 5× 5× 3× 9 problem with threehorizontal wells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

Figure 14.20 Matrix 6: Eliminate the q�w or P �well well terms from the component

equations for a 5× 5× 3× 9 problem with three horizontal wells. . . . . . . . 288

Figure 14.21 Matrix 7: Eliminate the off-band well terms from the component equationsfor a 5× 5× 3× 9 problem with three horizontal wells. . . . . . . . . . . . . . 289

Figure 14.22 Row 1: An example of a row without well terms for a 5× 5× 3× 9 problem. . 289

xxiii

Figure 14.23 The non-zero blocks of Row 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

Figure 14.24 The non-zero columns of Row 1. . . . . . . . . . . . . . . . . . . . . . . . . . . 289

Figure 14.25 The non-zero columns from Row 1 are stored with the main block diagonalfirst. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290

Figure 14.26 The result of local LU decomposition on Row 1 in the order they are stored. . 290

Figure 14.27 The result of local LU decomposition on Row 1. . . . . . . . . . . . . . . . . . 290

Figure 14.28 Matrix 8: banded matrix for a 5× 5× 3 problem without eliminating wells. . 291

Figure 14.29 Matrix 9: banded matrix for a 5× 5× 3 problem. . . . . . . . . . . . . . . . . 291

Figure 14.30 Geometry of three horizontal wells for a 16 × 16 × 3 problem. . . . . . . . . . 292

Figure 14.31 Matrix 10: banded matrix for a 16× 16× 3 problem without eliminatingwells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

Figure 14.32 Upper left corner of Matrix 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

Figure 14.33 Matrix 11: banded matrix for a 16× 16× 3 problem. . . . . . . . . . . . . . . 294

Figure 15.1 Jacobian matrix for a 3× 1× 1 system with NC = 5 and Nblock = 9. . . . . . . 295

Figure 15.2 The test case after the first stage of Gaussian elimination. . . . . . . . . . . . 296

Figure 15.3 The test case after the second stage of Gaussian elimination. . . . . . . . . . . 296

Figure 15.4 The banded structure for the test case. . . . . . . . . . . . . . . . . . . . . . . 297

Figure 15.5 The test case after the first stage of Banded Gaussian elimination. . . . . . . . 297

Figure 15.6 The test case after the second stage of Banded Gaussian elimination. . . . . . 298

Figure 15.7 The sparse storage structure of the local LU solvers. . . . . . . . . . . . . . . . 299

Figure 15.8 The first step of the local LU solvers. . . . . . . . . . . . . . . . . . . . . . . . 299

Figure 15.9 The condensed matrix after the local LU decomposition. . . . . . . . . . . . . 300

Figure 15.10 The banded structure of the condensed matrix. . . . . . . . . . . . . . . . . . . 300

Figure 15.11 The condensed matrix after the first step of the band solve. . . . . . . . . . . . 300

Figure 15.12 The condensed matrix after the second step of the band solve. . . . . . . . . . 300

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Figure 15.13 Use the values from the condensed matrix solve to perform a backsubstitution on each grid cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

Figure 16.1 Illustration of a group of 9 nodes with 8 processor cores each. . . . . . . . . . 305

Figure 16.2 Illustration of computations with a hybrid MPI/openMP 3× 3× 8 processorgrid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

Figure 16.3 Illustration of computations with an MPI 9× 8 processor grid. . . . . . . . . . 306

Figure 16.4 Illustration of computations with a linear array of 72 processors. . . . . . . . . 306

Figure 16.5 Parallel boundary computations. . . . . . . . . . . . . . . . . . . . . . . . . . . 306

Figure 16.6 Parallel boundary computations for a 3× 3 processor grid. . . . . . . . . . . . 307

Figure 16.7 Parallel boundary computations for a 9× 8 processor grid. . . . . . . . . . . . 307

Figure 16.8 Parallel computations for load balancing. . . . . . . . . . . . . . . . . . . . . . 308

Figure 16.9 Ra bandwidth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314

Figure 16.10 Efficiency plot for Nx = 80, Ny = 80, and Nz = 15. . . . . . . . . . . . . . . . 315

Figure 16.11 Speedup plot for Nx = 80, Ny = 80, and Nz = 15. . . . . . . . . . . . . . . . . 315

Figure 16.12 Scalability plot for Nx = Ny, Nz = 15, and EP = 0.1. . . . . . . . . . . . . . . 316

Figure 16.13 Memory constrained scalability plot. . . . . . . . . . . . . . . . . . . . . . . . . 317

Figure 17.1 Production rates at reservoir conditions for model 760E. . . . . . . . . . . . . 320

Figure 17.2 Production pressure for model 760E. . . . . . . . . . . . . . . . . . . . . . . . 321

Figure 17.3 Saturation for equivalent one-cell model for model 760E. . . . . . . . . . . . . 321

Figure 17.4 Mole fraction for equivalent one-cell model for model 760E. . . . . . . . . . . . 321

Figure 17.5 Molar recovery factor for model 760E. . . . . . . . . . . . . . . . . . . . . . . . 322

Figure 17.6 Saturation for equivalent one-cell model for model 762E. . . . . . . . . . . . . 323

Figure 17.7 Injection rates at reservoir conditions for model 760F. . . . . . . . . . . . . . . 323

Figure 17.8 Injection pressures for model 760F. . . . . . . . . . . . . . . . . . . . . . . . . 323

Figure 17.9 Production rates at reservoir conditions for model 760F. . . . . . . . . . . . . 324

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Figure 17.10 Production pressure for model 760F. . . . . . . . . . . . . . . . . . . . . . . . 324

Figure 17.11 Saturation for equivalent one-cell model for model 760F. . . . . . . . . . . . . 325

Figure 17.12 Mole fraction for equivalent one-cell model for model 760F. . . . . . . . . . . . 325

Figure 17.13 Molar recovery factor for model 760F. . . . . . . . . . . . . . . . . . . . . . . . 325

Figure 17.14 Saturation for equivalent one-cell model for model 762F. . . . . . . . . . . . . 326

Figure 17.15 Injection rates at reservoir conditions for model 760G. . . . . . . . . . . . . . . 327

Figure 17.16 Injection pressures for model 760G. . . . . . . . . . . . . . . . . . . . . . . . . 327

Figure 17.17 Production rates at reservoir conditions for model 760G. . . . . . . . . . . . . 328

Figure 17.18 Production pressure for model 760G. . . . . . . . . . . . . . . . . . . . . . . . 328

Figure 17.19 Saturation for equivalent one-cell model for model 760G. . . . . . . . . . . . . 329

Figure 17.20 Mole fraction for equivalent one-cell model for model 760G. . . . . . . . . . . . 329

Figure 17.21 Molar recovery factor for model 760G. . . . . . . . . . . . . . . . . . . . . . . . 330













xxvi



Figure 17.36 Comparison of production rates for model 760E. . . . . . . . . . . . . . . . . . 337

Figure 17.37 Comparison of producer grid cell pressures for model 760E. . . . . . . . . . . . 337

Figure 17.38 Difference of producer grid cell pressures for model 760E. . . . . . . . . . . . . 338

Figure 17.39 Comparison of total molar rates for model 760E. . . . . . . . . . . . . . . . . . 338

Figure 17.40 Comparison of recovery factors for model 760E. . . . . . . . . . . . . . . . . . 338


Figure 17.42 Comparison of production rates for model 762E. . . . . . . . . . . . . . . . . . 339

Figure 17.43 Comparison of producer grid cell pressures for model 762E. . . . . . . . . . . . 340

Figure 17.44 Difference of producer grid cell pressures for model 762E. . . . . . . . . . . . . 340


Figure 17.46 Comparison of production rates for model 760F. . . . . . . . . . . . . . . . . . 341

Figure 17.47 Comparison of producer grid cell pressures for model 760F. . . . . . . . . . . . 341

Figure 17.48 Comparison of recovery factors for model 760F. . . . . . . . . . . . . . . . . . 342

Figure 17.49 Comparison of water saturation for model 760F at 500 days. . . . . . . . . . . 342



Figure 17.52 Comparison of production rates for model 760G. . . . . . . . . . . . . . . . . . 344

Figure 17.53 Comparison of producer grid cell pressures for model 760G. . . . . . . . . . . . 344

Figure 17.54 Comparison of recovery factors for model 760G. . . . . . . . . . . . . . . . . . 345

Figure 17.55 Comparison of gas saturation for model 760G at 500 days. . . . . . . . . . . . 345

Figure 17.56 Comparison of water saturation for model 760G at 1000 days. . . . . . . . . . 346


xxvii

Figure 17.58 Comparison of production rates for model 761G. . . . . . . . . . . . . . . . . . 347

Figure 17.59 Comparison of producer grid cell pressures for model 761G. . . . . . . . . . . . 347


Figure 17.61 Comparison of pressure profiles for model 761G at 1000 days. . . . . . . . . . 348



Figure 17.64 Comparison of pressure profiles for model 761G at 1500 days. . . . . . . . . . 349




Figure 18.1 Porosity distribution for Facies 5 of Jobe. . . . . . . . . . . . . . . . . . . . . . 357

Figure 18.2 Porosity-permeability correlation for Facies 5 of Jobe. . . . . . . . . . . . . . . 358

Figure 18.3 Porosity and permeability for Geostatistical Realization # 1. . . . . . . . . . . 359






Figure 18.9 Oil and water relative permeability curves including the data points. . . . . . 369

Figure 18.10 Gas and oil relative permeability curves including the data points. . . . . . . . 370

Figure 18.11 Trapped gas saturation as a function of maximum achieved gas saturation. . . 371

Figure 18.12 Trapped oil saturation as a function of maximum oil saturation achievedafter the initial oil saturation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 371

Figure 18.13 Water relative permeability based on a fit to the oil-water data. . . . . . . . . 373

Figure 18.14 Bounding scanning curves for gas relative permeability . . . . . . . . . . . . . 374

xxviii

Figure 18.15 A decreasing gas relative permeability scanning curve. . . . . . . . . . . . . . . 376

Figure 18.16 An increasing gas relative permeability scanning curve. . . . . . . . . . . . . . 378

Figure 18.17 Oil relative permeability based on from the oil/water SCAL . . . . . . . . . . 379

Figure 18.18 Oil relative permeability based on from the gas/oil SCAL . . . . . . . . . . . . 380

Figure 18.19 Compare the krow and the krog. For this data set the curves are very similar. . 380

Figure 18.20 The krow scanning curves have no hysteresis because Smaxot ≤ Sorg < Sorw. . . . 381

Figure 18.21 The krog scanning curves have no hysteresis because Smaxot ≤ Sorg < Sorw. . . . 381

Figure 18.22 A decreasing oil relative permeability scanning curve. . . . . . . . . . . . . . . 383

Figure 18.23 An increasing oil relative permeability scanning curve. . . . . . . . . . . . . . . 384

Figure 18.24 Oil-water capillary pressure curves including the data points. . . . . . . . . . . 386

Figure 18.25 Capillary pressure bounding curves and interpolated scanning curves. . . . . . 387

Figure 18.26 Decreasing capillary pressure scanning curve. . . . . . . . . . . . . . . . . . . . 388

Figure 18.27 Increasing capillary pressure scanning curve. . . . . . . . . . . . . . . . . . . . 390

Figure A.1 Primary Production Pressures. . . . . . . . . . . . . . . . . . . . . . . . . . . . 480

Figure A.2 Primary Production Rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480

Figure A.3 Primary Production Ratios. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481

Figure A.4 Primary nonlinear iteration convergence. . . . . . . . . . . . . . . . . . . . . . 481

Figure A.5 Primary time step criteria. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481

Figure A.6 Primary pressure for cells along diagonal between wells. . . . . . . . . . . . . . 482

Figure A.7 Primary total mass of CO2 for cells along diagonal between wells. . . . . . . . 482

Figure A.8 Primary total mass of hydrocarbons (no CO2) for cells along diagonalbetween wells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483

Figure A.9 PRIM saturation for equivalent one cell model. . . . . . . . . . . . . . . . . . . 483

Figure A.10 Primary total mole fraction in the reservoir. . . . . . . . . . . . . . . . . . . . 483

Figure A.11 Primary recovery factor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484

xxix

Figure A.12 Primary compositional recovery factor. . . . . . . . . . . . . . . . . . . . . . . 484

Figure A.13 Distribution of pressures at primary economic limit. . . . . . . . . . . . . . . 485

Figure A.14 2-D pressure distribution at primary economic limit. . . . . . . . . . . . . . . 485

Figure A.15 Distribution of oil saturation at primary economic limit. . . . . . . . . . . . . 486

Figure A.16 2-D oil saturation distribution at primary economic limit. . . . . . . . . . . . 486

Figure A.17 Distribution of gas saturation at primary economic limit. . . . . . . . . . . . 487

Figure A.18 2-D gas saturation distribution at primary economic limit. . . . . . . . . . . . 487

Figure A.19 Distribution of water saturation at primary economic limit. . . . . . . . . . . 488

Figure A.20 2-D water saturation distribution at primary economic limit. . . . . . . . . . 488

Figure A.21 Waterflood Injection Pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . 489

Figure A.22 Waterflood Injection Rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489

Figure A.23 Waterflood Production Pressures. . . . . . . . . . . . . . . . . . . . . . . . . . 490

Figure A.24 Waterflood Production Rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . 490

Figure A.25 Waterflood Production Ratios. . . . . . . . . . . . . . . . . . . . . . . . . . . . 490

Figure A.26 WF− Primary Oil Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491

Figure A.27 Waterflood nonlinear iteration convergence. . . . . . . . . . . . . . . . . . . . . 491

Figure A.28 Waterflood time step criteria. . . . . . . . . . . . . . . . . . . . . . . . . . . . 491

Figure A.29 Waterflood pressure for cells along diagonal between wells. . . . . . . . . . . . 492

Figure A.30 Waterflood total mass of CO2 for cells along diagonal between wells. . . . . . . 492

Figure A.31 Waterflood total mass of hydrocarbons (no CO2) for cells along diagonalbetween wells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493

Figure A.32 WF saturation for equivalent one cell model. . . . . . . . . . . . . . . . . . . . 493

Figure A.33 Waterflood total mole fraction in the reservoir. . . . . . . . . . . . . . . . . . 493

Figure A.34 Waterflood recovery factor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494

Figure A.35 Waterflood compositional recovery factor. . . . . . . . . . . . . . . . . . . . . 494

xxx

Figure A.36 Distribution of pressures at waterflood economic limit. . . . . . . . . . . . . . 495

Figure A.37 2-D pressure distribution at waterflood economic limit. . . . . . . . . . . . . . 495

Figure A.38 Distribution of oil saturation at waterflood economic limit. . . . . . . . . . . 496

Figure A.39 2-D oil saturation distribution at waterflood economic limit. . . . . . . . . . . 496

Figure A.40 Distribution of gas saturation at waterflood economic limit. . . . . . . . . . . 497

Figure A.41 2-D gas saturation distribution at waterflood economic limit. . . . . . . . . . 497

Figure A.42 Distribution of water saturation at waterflood economic limit. . . . . . . . . . 498

Figure A.43 2-D water saturation distribution at waterflood economic limit. . . . . . . . . 498

Figure A.44 Continuous CO2 Injection Pressure. . . . . . . . . . . . . . . . . . . . . . . . . 499

Figure A.45 Continuous CO2 Injection Rates. . . . . . . . . . . . . . . . . . . . . . . . . . . 499

Figure A.46 Continuous CO2 Production Pressures. . . . . . . . . . . . . . . . . . . . . . . 500

Figure A.47 Continuous CO2 Production Rates. . . . . . . . . . . . . . . . . . . . . . . . . 500

Figure A.48 Continuous CO2 Production Ratios. . . . . . . . . . . . . . . . . . . . . . . . . 500

Figure A.49 GF−WF Oil Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501

Figure A.50 Continuous CO2 nonlinear iteration convergence. . . . . . . . . . . . . . . . . 501

Figure A.51 Continuous CO2 time step criteria. . . . . . . . . . . . . . . . . . . . . . . . . 502

Figure A.52 Continuous CO2 pressure for cells along diagonal between wells. . . . . . . . . 502

Figure A.53 Continuous CO2 total mass of CO2 for cells along diagonal between wells. . . 503

Figure A.54 Continuous CO2 total mass of hydrocarbons (no CO2) for cells alongdiagonal between wells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503

Figure A.55 Continuous CO2 saturation for equivalent one cell model. . . . . . . . . . . . . 503

Figure A.56 Continuous CO2 total mole fraction in the reservoir. . . . . . . . . . . . . . . 504

Figure A.57 Continuous CO2 recovery factor. . . . . . . . . . . . . . . . . . . . . . . . . . 504

Figure A.58 Continuous CO2 compositional recovery factor. . . . . . . . . . . . . . . . . . 505

Figure A.59 Continuous CO2 storage of CO2. . . . . . . . . . . . . . . . . . . . . . . . . . 505

xxxi

Figure A.60 Continuous CO2 utilization of CO2. . . . . . . . . . . . . . . . . . . . . . . . 505

Figure A.61 Distribution of pressures at Continuous CO2 economic limit. . . . . . . . . . 506

Figure A.62 2-D pressure distribution at Continuous CO2 economic limit. . . . . . . . . . 506

Figure A.63 Distribution of oil saturation at Continuous CO2 economic limit. . . . . . . . 507

Figure A.64 2-D oil saturation distribution at Continuous CO2 economic limit. . . . . . . 507

Figure A.65 Distribution of gas saturation at Continuous CO2 economic limit. . . . . . . . 508

Figure A.66 2-D gas saturation distribution at Continuous CO2 economic limit. . . . . . . 508

Figure A.67 Distribution of water saturation at Continuous CO2 economic limit. . . . . . 509

Figure A.68 2-D water saturation distribution at Continuous CO2 economic limit. . . . . 509

Figure A.69 WAG Injection Pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510

Figure A.70 WAG Injection Rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510

Figure A.71 WAG Production Pressures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510

Figure A.72 WAG Production Rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511

Figure A.73 WAG Production Ratios. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511

Figure A.74 WAG−WF Oil Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512

Figure A.75 WAG nonlinear iteration convergence. . . . . . . . . . . . . . . . . . . . . . . . 512

Figure A.76 WAG time step criteria. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512

Figure A.77 WAG pressure for cells along diagonal between wells. . . . . . . . . . . . . . . 513

Figure A.78 WAG total mass of CO2 for cells along diagonal between wells. . . . . . . . . . 513

Figure A.79 WAG total mass of hydrocarbons (no CO2) for cells along diagonal betweenwells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514

Figure A.80 WAG saturation for equivalent one cell model. . . . . . . . . . . . . . . . . . . 514

Figure A.81 WAG total mole fraction in the reservoir. . . . . . . . . . . . . . . . . . . . . 515

Figure A.82 WAG recovery factor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515

Figure A.83 WAG compositional recovery factor. . . . . . . . . . . . . . . . . . . . . . . . 515

xxxii

Figure A.84 WAG storage of CO2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516

Figure A.85 WAG utilization of CO2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516

Figure A.86 Distribution of pressures at WAG economic limit. . . . . . . . . . . . . . . . . 516

Figure A.87 2-D pressure distribution at WAG economic limit. . . . . . . . . . . . . . . . 517

Figure A.88 Distribution of oil saturation at WAG economic limit. . . . . . . . . . . . . . 517

Figure A.89 2-D oil saturation distribution at WAG economic limit. . . . . . . . . . . . . 518

Figure A.90 Distribution of gas saturation at WAG economic limit. . . . . . . . . . . . . . 518

Figure A.91 2-D gas saturation distribution at WAG economic limit. . . . . . . . . . . . . 519

Figure A.92 Distribution of water saturation at WAG economic limit. . . . . . . . . . . . 519

Figure A.93 2-D water saturation distribution at WAG economic limit. . . . . . . . . . . . 520

xxxiii

LIST OF TABLES

Table 3.1 Distribution of components in phases for NC = 8 . . . . . . . . . . . . . . . . . 38

Table 4.1 Distribution of components in phases for NC = 5 . . . . . . . . . . . . . . . . . 61

Table 4.2 Primary variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Table 4.3 Secondary variables which do not vary with time, DA notime . . . . . . . . . . 67

Table 4.4 Secondary variables at n which are not needed for the transmissibilitycalculations, DA cell only n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

Table 4.5 Secondary variables at n which are needed for the transmissibilitycalculations, DA for TRANS n . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Table 4.6 Secondary variables at �, DA cell only ell . . . . . . . . . . . . . . . . . . . . . 70

Table 4.7 Well properties at �, stored for each well. . . . . . . . . . . . . . . . . . . . . . 75

Table 9.1 Superscripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

Table 9.2 Subscripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

Table 9.3 Well variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

Table 12.1 Units for (12.21). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

Table 15.1 Computation and memory requirement for 3 different solvers . . . . . . . . . . 299

Table 17.1 Validation cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320

Table 19.1 Description of test case scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . 394

Table 19.2 Primary production: recovery factor and time to economic limit . . . . . . . . 396

Table 19.3 Waterflood time to economic limit . . . . . . . . . . . . . . . . . . . . . . . . . 397

Table 19.4 Waterflood recovery factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399

Table 19.5 Continuous CO2 recovery factor . . . . . . . . . . . . . . . . . . . . . . . . . . 400

Table 19.6 Continuous CO2 response time . . . . . . . . . . . . . . . . . . . . . . . . . . . 402

xxxiv

Table 19.7 Continuous CO2 response duration . . . . . . . . . . . . . . . . . . . . . . . . . 403

Table 19.8 WAG recovery factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406

Table 19.9 WAG recovery factor versus continuous CO2 recovery factor . . . . . . . . . . 407

Table 19.10 WAG response duration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408

Table 19.11 Compositional recovery factor for waterflood . . . . . . . . . . . . . . . . . . . 410

Table 19.12 Compositional recovery factor for continuous CO2 injection . . . . . . . . . . . 411

Table 19.13 Compositional recovery factor for WAG . . . . . . . . . . . . . . . . . . . . . . 412

Table 19.14 CO2 storage for continuous CO2 injection . . . . . . . . . . . . . . . . . . . . . 413

Table 19.15 CO2 storage for WAG injection . . . . . . . . . . . . . . . . . . . . . . . . . . . 414

Table 19.16 CO2 storage difference for continuous vs WAG CO2 injection . . . . . . . . . . 416

Table 19.17 CO2 utilization for continuous CO2 injection . . . . . . . . . . . . . . . . . . . 417

Table 19.18 CO2 utilization for WAG injection . . . . . . . . . . . . . . . . . . . . . . . . . 419

Table 19.19 CO2 utilization difference for continuous vs WAG CO2 injection . . . . . . . . 420

Table 22.1 Subscripts and superscripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426

Table 22.2 Variables used in this document . . . . . . . . . . . . . . . . . . . . . . . . . . 427

xxxv

ACKNOWLEDGMENTS

I am very thankful to the organizations which have funded portions of this work. These include

Saudi Aramco, The Petroleum Institute of Abu Dhabi, Marathon Center of Excellence for Reservoir

Studies, and Colorado School of Mines. I am very grateful to the members of Marathon Center of

Excellence for Reservoir Studies and the CSM/PI Integrated Carbonate Reservoir Studies groups

for valuable discussions during my time at Mines. I thank the faculty at CSM who have offered a

wonderful integrated learning experience. I would like to thank my committee members, Dr. Mark

Lusk, Dr. Erdal Ozkan, Dr. J. Rick Sarg, and Dr. Yu-Shu Wu, for providing valuable advice

throughout the process. I am indebted to my advisor, Dr. Hossein Kazemi, for his guidance,

support, and the important insight into different formulation possibilities and their importance in

field applications.

xxxvi

This dissertation is dedicated to my wife Ana, my mother Barbara, and my daughter Aiva. I

could not have finished when I did without their support, assistance, and understanding.

xxxvii

CHAPTER 1

INTRODUCTION

Enhanced oil recovery (EOR) is a group of methods designed to increase the production of oil

in addition to waterflooding. These methods are described by Green and Willhite (1998) and Lake

(1989). They include miscible and immiscible gas injection, thermal recovery, mobility control,

and chemical flooding. Based on the 2012 Worldwide EOR Survey conducted by the Oil and Gas

Journal, (Koottungal, 2012), carbon dioxide (CO2) enhanced oil recovery is now 351 MBOPD and

thermal recovery is 323 MBOPD with 89 MBOPD for other gas injection and no reported volumes

for chemical methods, carbonated waterflood, or microbial EOR.

Injecting carbon dioxide in oil reservoirs has two advantages: increasing the production of oil

and sequestering CO2. CO2 may be injected without water or as a water-alternating-gas (WAG)

injection. In the USA, potential enhanced oil recovery from CO2 injection is approximately 80

billion barrels, corresponding to approximately 25 billion metric tons of sequestered CO2. In the

world, potential enhanced oil recovery from CO2 injection is approximately 880 billion barrels,

corresponding to approximately 260 billion metric tons of sequestered CO2 (Rychel, 2012).

Injecting gas or water into a reservoir helps maintain the pressure and helps displace the oil.

Gas injection lowers the residual oil saturation and enhances gravity drainage. Injecting CO2

causes oil to swell and lowers the residual oil saturation more than methane. Gas injection in the

second or third WAG cycle will continue to decrease the residual oil saturation. CO2 is soluble in

water so it can access trapped oil by traveling through a water block. Mixing CO2 with reservoir

oil changes the viscosity and density of the oil; these changes may make it easier to displace the

oil. If the CO2 is miscible with the oil, it reduces trapping and further decreases the residual oil

saturation. CO2 injection leads to the sequestration of approximately 50% of the injected CO2.

WAG injection controls the mobility more than continuous gas injection. WAG injection will cause

each CO2 injection cycle to follow different pathways in the reservoir, which leads to increased oil

recovery. WAG is often used for economic reasons because CO2 supply is often more limited than

water supply and CO2 injection is often more expensive than water injection.

1

If two or more phases are present in a pore space, one may become isolated or trapped as one

phase displaces another. Trapped or bypassed fluids consist of isolated pockets of fluids that are

not connected between the injector and producer. In a mixed-wet water-oil system, there may be

connected oil, connected water, connected gas, disconnected oil, disconnected water, and discon-

nected gas all present at the same time. In core flood experiments, the residual oil saturation is

determined. The residual oil saturation measured is the sum of the residual connected oil and the

residual disconnected oil, although when oil production stops most of the oil is probably discon-

nected.

Compositional variations of the trapped fluids have a significant impact on the volume of oil re-

covery; the timing of oil, water, and gas production; and the amount of CO2 storage and utilization.

Disconnected or trapped oil will not automatically equilibrate with mobile oil and gas, especially if

it is isolated by a water phase. Disconnected or trapped gas will not automatically equilibrate with

mobile oil and gas, especially if it is isolated by a water phase. In a water-oil system, an increasing

water saturation, water displacing oil leads to trapped oil. In a water-gas system, an increasing

water saturation leads to trapped gas; this is sometimes called “water blocking”. In a water-oil-gas

system, any of the three phases can be trapped. Trapping can relate to a microscopic effect such as

snap-off or pore doublet trapping. Trapped oil, gas, and water can also be related to a bypassing

effect, where a preferential flow path leaves fluids behind. This effect occurs at scales from the pore

network through inter-well scales.

The goal of this research was to evaluate the effects of variations in the composition of trapped

fluids on CO2 WAG simulation. A three-dimensional, three-phase, parallel compositional simulator

was developed with a specialized formulation to handle compositional trapping and CO2 WAG

injection. This formulation tracks the compositional differences between the trapped oil, gas, and

water and the mobile oil, gas, and water using a dual porosity approach. The mobile oil, gas, and

water (m1 system) are analogous to the fracture system and the disconnected oil, gas, and water

(m2 system) are analogous to the matrix system. The approach differs from Coats, Thomas, and

Pierson (2004a) method for tracking bypassed oil because the gas, oil, and water may all be trapped

with compositional variations. The amount and composition of the trapped fluids changes with

time. Reservoir simulation allows us to predict future performance of CO2 enhanced oil recovery

and sequestration at different scales.

2

Test cases with properties based on mixed wet carbonate reservoirs were used to evaluate the

effects of compositional trapping, gas relative permeability hysteresis, the solubility of CO2 in water,

and other trapping effects on the volume of oil recovery; the timing of oil, water, and gas production;

and the amount of CO2 storage and utilization. Primary production, waterflood, continuous CO2

injection, and CO2 WAG production schemes were evaluated.

3

CHAPTER 2

LITERATURE REVIEW

This chapter presents a literature review of different topics related to compositional simulation

of CO2 enhanced oil recovery. First there is a general discussion of enhanced oil recovery, followed

by papers dealing specifically with CO2 enhanced oil recovery and CO2 sequestration. Next is

a discussion of numerical reservoir simulation; the simulator in this dissertation is an example

of a numerical reservoir simulator. Test cases for this dissertation were based on a field in the

Middle East, and there is a brief review of papers characterizing geology in the Middle East.

Relative permeability is an important property of multiphase fluid flow and is especially important

to understand the effects of trapping. Compositional simulation is based on the calculation of phase

properties from an equation of state model.

2.1 Enhanced Oil Recovery

Enhanced oil recovery (EOR) is a group of methods designed to increase the production of oil

in addition to waterflooding. These methods are described by Green and Willhite (1998) and Lake

(1989). They include miscible and immiscible gas injection, thermal recovery, mobility control,

and chemical flooding. Based on the 2012 Worldwide EOR Survey conducted by the Oil and Gas

Journal, (Koottungal, 2012), carbon dioxide (CO2) enhanced oil recovery is now 351 MBOPD and

thermal recovery is 323 MBOPD with 89 MBOPD for other gas injection and no reported volumes

for chemical methods, carbonated waterflood, or microbial EOR.

2.1.1 Miscible Flooding

Katz and Stalkup (1983) discusses some of the limitations of reservoir simulation of miscible

floods. Stalkup (1983) presents an overview of miscible displacement processes. Uleberg and Høier

(2002) describes a method for determining minimum miscibility pressure for a dual porosity system.

2.1.2 Gas Injection

van Vark, Masalmeh, van Dorp, Al Nasr, and Al-Khanbashi (2004) conducted compositional

simulations of an Abu Dhabi reservoir to evaluate different injection mixtures of CH4, CO2, and

4

H2S. H2S yielded even better miscibility than CO2. Changes in heterogeneity also had a significant

impact on recovery.

2.1.3 Other Enhanced Oil Recovery Methods

Agbalaka, Dandekar, Patil, Khataniar, and Hemsath (2008) summarizes the conclusion from the

literature that wettability has a significant impact on recovery during water injection, gas injection,

and WAG. Teletzke, Wattenbarger, and Wilkinson (2010) presents an overview of how to set up a

field pilot study for an EOR project.

2.2 CO2 Enhanced Recovery and Sequestration

Injecting carbon dioxide in oil reservoirs has two advantages: increasing the production of oil

and sequestering CO2. CO2 may be injected without water or as a water-alternating-gas (WAG)

injection. In the USA, potential enhanced oil recovery from CO2 injection is approximately 80

billion barrels, corresponding to approximately 25 billion metric tons of sequestered CO2. In the

world, potential enhanced oil recovery from CO2 injection is approximately 880 billion barrels,

corresponding to approximately 260 billion metric tons of sequestered CO2 (Rychel, 2012).

2.2.1 CO2 Enhanced Oil Recovery

Holm and Josendal (1974) presents the following summary of the benefits of CO2. These benefits

are still the primary reasons today for CO2 injection.

• CO2 promotes swelling.

• CO2 reduces oil viscosity.

• CO2 increases oil density.

• CO2 is soluble in water.

• CO2 exerts an acidic effect on rock.

• CO2 can vaporize portions of the oil.

• CO2 can be transported chromatographically through porous rock.

5

Injecting CO2 results in several displacement mechanisms, including solution gas drive, immis-

cible CO2, multi-contact miscible CO2, and miscible CO2.

Zekri, Shedid, and Almehaideb (2007) conducted core flood experiments related to CO2 EOR

in Abu Dhabi. Rawahi, Hafez, Al-Yafei, Al-Hammadi, Ghori, Putney, Matthews, and Harb (2012)

describes a CO2 EOR pilot design in Abu Dhabi. Yan and Stenby (2009) presents a study incorpo-

rating the effects of different CO2 solubilities in water on the oil recovery. Berenblyum, Calderon,

and Surguchev (2009) presents an overview of the mechanisms for CO2 enhanced oil recovery.

Ghedan (2009) presents an overview of laboratory experiments related to CO2 enhanced oil re-

covery. Al-Abri, Sidiq, and Amin (2009) describes experimental results for enhancing condensate

recovery by injecting CO2 and CH4. Riazi, Sohrabi, Jamiolahmady, Ireland, and Brown (2009)

describes micromodel experiments for carbonated water injection. Manrique, Thomas, Ravikiran,

Izadi, Lantz, Romero, and Alvarado (2010) presents an overview of enhanced oil recovery projects

based on Oil and Gas Journal reports and additional references. Prieditis, Wolle, and Notz (1991)

describes a CO2 WAG flood in west Texas San Andres formation.

2.2.2 CO2 Flood Simulation

Chase and Todd (1984) describes a compositional reservoir simulator which includes CO2 solu-

bility in brine. Chase and Todd (1984) also use a water blocking function based on Raimondi and

Torcaso (1964)

Stwb =Sorw

1 + β krokrw

(2.1)

(2.1) uses a parameter β to vary how strong the water blocking effect is; β = 1 would correspond to

a highly water wet sandstone; β = 5 is the much weaker blocking effect in a mixed wet west Texas

San Andres carbonate. Chase and Todd (1984) use a transition parameter α to vary the relative

permeabilities, viscosities, and densities between the oil and gas phases. Jackson, Andrews, and

Claridge (1985) presents simulation analysis of WAG ratio, using (2.1).

LaForce and Jessen (2007) presents an analysis of WAG simulations. Chang, Coats, and Nolen

(1998) describes a compositional reservoir simulator for CO2 flooding, including CO2 solubility in

water. Christensen, Stenby, and Skauge (1998) discusses the results of compositional simulation of

WAG using hysteresis options by Larsen and Skauge (1998). Christensen et al. (1998) concludes that

6

gas relative permeability hysteresis and slug size had very little effect on the results. Oil viscosity,

compositional simulation, and three-phase hysteresis were important for their simulations.

Hustad, Kløv, Lerdahl, Berge, Stensen, and Øren (2002) presents the results of 2D cross-section

simulation models of WAG with hysteresis. Nghiem, Sammon, Grabenstetter, and Ohkuma (2004)

describes modifications to CMG GEM which handle CO2 solubility in the aqueous phase and

aqueous geochemistry for simulation of CO2 sequestration in aquifers. Shtepani (2007) describes

experimental and modeling requirements for CO2 EOR. Shtepani (2007) recommends scaling the

residual oil saturation, gas-oil relative permeability, and gas-oil capillary pressure based on a ratio

of interfacial tensions:

α =σ − σmin

σmax − σmin(2.2)

2.2.3 CO2 WAG

Rogers and Grigg (2001) contains a thorough literature review of CO2 WAG processes. Injectiv-

ity of CO2 is sometimes higher and sometimes lower than waterflood injectivity. Awan, Teigland,

and Kleppe (2008) presents a review of gas injection and WAG injection projects in the North Sea.

Rogers and Grigg (2001) also contains a discussion of trapping/bypassing; the points summa-

rized here will be discussed with the papers cited by Rogers and Grigg (2001). WAG ratio should

be based on volume, not based on time, and should increase with time for the best results.

Surguchev, Korbol, and Krakstad (1992) discusses calculations of the optimum WAG ratio

which will vary for each field. Based on Surguchev et al. (1992) and Rogers and Grigg (2001),

technical factors include heterogeneity, wettability, fluid properties, miscibility conditions, injection

techniques, WAG parameters, flow geometry, and physical dispersion. Surguchev et al. (1992) uses

a North Sea reservoir example to conclude that optimization of WAG depends on stratification,

hysteresis, and three-phase flow effects.

Gorell (1988) determined the amount of trapped solvent, trapped oil, mobile solvent, mobile

oil, and water as a result of 1-D simulations which simulate WAG as simultaneous gas and water

injection.

Todd, Cobb, and McCarter (1982) presents results for simulation of a field case in west Texas

(Wasson San Andres field). Prieditis and Brugman (1993) presents data at reservoir temperature

7

showing hysteresis in the water relative permeability for West Texas carbonates, showing that the

residual water saturation after waterflood is higher than the conate water saturation. The presence

of a residual miscible oil or CO2 saturation significantly reduces the predicted oil recovery. The

experimental data was simulated using a Todd and Longstaff (1972) approach. Dria, Pope, and

Sephrnoori (1993) presents three-phase relative permeability data for dolomite cores. Schneider

and Owens (1976) present measurements of hysteresis for rich gas injection in oil-wet carbonates

in West Texas.

Rogers and Grigg (2001) based on Wegener and Harpole (1996) states that macroscopic bypass-

ing is related to heterogeneity and mobility differences; this is compounded by effects of varying

trapped gas saturations . Wegener and Harpole (1996) describes composite core flood experiments

of a West Texas carbonate. This study showed hysteresis in water relative permeability, with the

irreducible water saturation 15–20% saturation units higher than the connate water saturation.

Rampersad, Ogbe, Kamath, and Islam (1995) presents a good overview of the performance

effects of oil trapped by water during WAG. Fatemi, Sohrabi, Jamiolahmady, Ireland, and Robert-

son (2011) presents experimental results for multiple cycles of CO2 WAG in high permeability

water-wet and mixed-wet sandstones.

2.2.4 CO2 Sequestration

Haugen and Eide (1996) discusses CO2 sequestration options; the options have not changed since

1996. Flett, Gurton, and Taggart (2004) concludes that gas-water relative permeability hysteresis

and trapping has a significant effect on the amount of CO2 stored in an aquifer sequestration

case. Bachu and Bennion (2008) measured CO2-brine relative permeabilities, capillary pressures,

and interfacial tensions for several different reservoirs in Alberta. Flett, Gurton, and Weir (2007)

presents simulation results for CO2 sequestration in aquifers.

Burton and Bryant (2009) presents a method for injecting CO2 dissolved in brine rather than

pure CO2 for CO2 sequestration in aquifers. Noh, Lake, Bryant, and Araque-Martinez (2007)

discusses a fractional flow based analytical model for simulating CO2 sequestration in aquifers.

Thibeau, Nghiem, and Ohkuma (2007) evaluates the long-term effect of geochemical reactions on

CO2 sequestration in aquifers. Bryant (2007) presents an overview of geologic CO2 sequestration.

8

Nattwongasem and Jessen (2009) presents a study of CO2 sequestration using CMG GEM.

Economides and Ehlig-Economides (2009) provides an overview of the volume requirement for a

regional CO2 sequestration subject to injection pressure constraints. Nghiem, Yang, Shrivatava,

Kohse, Hassam, Chen, and Card (2009) optimizes the amount of gas trapped by residual gas

trapping and solubility trapping for an example saline aquifer. Esposito and Benson (2010) presents

a simulation of CO2 leakage from a sequestration site along with possible remediation efforts.

Javaheri and Jessen (2011) measured co-current and counter-current relative permeability curves

and used these to calculate the effect on CO2 sequestration in an aquifer. Altundas, Ramakrishnan,

Chugunov, and de Loubens (2011) presents a simulation study of CO2 trapping caused by capillary

pressure hysteresis.

2.2.5 CO2 Simulation with TOUGH

Battistelli, Calore, and Pruess (1997) describes the EWASG module in TOUGH2 for geothermal

brine plus gas. It includes variations in the salt content and an option for CO2-brine. Pruess,

Xu, Apps, and Garcia (2003) discusses CO2 injection in aquifers using the TOUGH2 suite. Zhang,

Doughty, Wu, and Pruess (2007) describes a parallel version of the TOUGH2 codes for use in CO2

sequestration studies. Battistelli and Marcolini (2009) presents the TMGAS module in TOUGH2 for

injection of gas into a brine aquifer; the gas may contain CO2, H2S, light hydrocarbons, nitrogen,

oxygen, and sulpher dioxide.

2.2.6 CO2 Water Solubility

Enick and Klara (1992) discusses the effect of CO2 solubility in brine on reservoir simulation

models. Do and Pinczewski (1993) discusses how CO2 solubility in water can diffuse through thin

layers of “water blocking” to get to trapped oil. Diffusion equilibrium is reached in approximately

100 hours. Takenouchi and Kennedy (1965) presents experimental work for water containing H2O,

CO2, and NaCl.

2.2.7 CO2 Trapping

Dai and Orr Jr. (1987) describes some trapping effects related to CO2 flooding. Dai and Orr Jr.

(1987) categorizes oil into flowing, dendritic, and trapped oil, including the effects of trapped oil

not mixing completely with the dendritic oil. Salter and Mohanty (1982) presents experimental

9

results for tracer floods in a four-foot long Berea sandstone core. Salter and Mohanty (1982) uses a

capacitance model to describe flowing, dendritic, and isolated oil fractions in a strongly water-wet

media, based on Coats and Smith (1964). Coats and Smith (1964) describes dead-end space using

a diffusion model.

2.2.8 Other Articles on CO2 Injection

Patton, Coats, and Spence (1982) describes a CO2 huff-and-puff process for increasing the

production of wells by altering the near-wellbore properties. It also provides data for eventual CO2

EOR operations. Morsi, Leslie, and Macdonald (2004) evaluates different methods for recovering

CO2 from flue gas for use as EOR in Abu Dhabi reservoirs. Seto, Jessen, and Orr (2007) evaluates

CO2 injection in a condensate reservoir. Plug and Bruining (2007) describes an experimental

procedure for measuring capillary pressure for a CO2-brine system in sand packs. Hassanzadeh,

Pooladi-Darvish, Elsharkawy, Keith, and Leonenko (2008) presents a good review of CO2 properties

in brine, including a diffusion coefficient for gaseous CO2 into brine and a diffusion coefficient for

aqueous CO2 in brine.

2.3 Reservoir Simulation

Numerical reservoir simulation provides a way to understand past performance and predict

future performance of fluid flow in reservoirs. It can also be used to understand the sensitivity of

different parameters. The reservoir simulator created in this dissertation was used to evaluate the

importance of compositional trapping and other trapping related phenomena.

Odeh (1969) provides an overview of reservoir simulation, including of 0-D, 1-D, 2-D, and 3-D

models. Coats (1982) provides an early review of reservoir simulation. Kazemi, Al-Kobaisi, Kur-

toglu, Heris, Charoenwongsa, Fakcharoenphol, and Akinboyewa (2012) provides a general discussion

of reservoir simulation in 2012.

Christensen, Larsen, and Nicolaisen (2000) presents a field test case using a WAG flood in a

North Sea reservoir. Masalmeh (2000) discusses oil recovery from transition zones.

2.3.1 Computation Approaches in Reservoir Simulation

Partially implicit methods are not mathematically stable for all combinations of grid cell size

and time step size. Courant, Friedrichs, and Lewy (1967) describes a way to calculate the stability

10

criteria, using a number now called the CFL number. Coats (2003a) describes a way to calculate

the CFL number for a compositional IMPES problem; Coats (2003b) presents related derivations.

In an overview of reservoir simulation methods Coats (1969) specifies that the implicit pressure

explicit saturation (IMPES) approach was first described in Stone and Garder (1961).

Christensen et al. (2000) describes saving a saturation value for use if the saturations oscillate

when calculating hysteresis.

Atan, Al-Matrook, Kazemi, Ozkan, and Gardner (2005) describes a method for reservoir simu-

lation using two different scales of grid cells.

The iterative approach described in Lu, Al-Shaalan, and Wheeler (2007) for a black oil system is

similar to the iterative approach used here, including the possibility of varying the solver tolerances

for pressure and saturation as a function of iteration number. Lu et al. (2007) refers to an older

paper by Dawson, Klıe, Wheeler, and Woodward (1997) that describes a related method.

Lu and Beckner (2011) describes a methodology for only solving for the grid cells which have

not yet converged.

2.3.2 Fractured Reservoir Simulation

Many hydrocarbon reservoirs are naturally fractured. To simulate naturally fractured reservoirs,

there are several approaches discussed in the literature. Dual porosity and dual permeability

systems partition the reservoir into two media: an interconnected fracture system which has a low

storage capacity but high flow capacity and a matrix system which provides high storage capacity

but low flow capacity. In a dual porosity representation, the matrix system connects to the fracture

system in the same grid cell but does not connect to adjacent fracture or matrix grid cells nor to

the wells. The fracture system connects to the matrix system, to adjacent grid cells, and to the

wells. In a dual permeability representation, the matrix blocks connect to the matrix system in

an adjacent grid cell. Multiple interacting continua (MINC) models provide connections between

several different levels of fracture and/or matrix systems (Wu and Pruess, 1988). Triple porosity

methods provide connections between two fracture systems and one matrix system or two matrix

systems and one fracture system. Fractures may be simulated using a discrete fracture network

where every fracture is represented individually or a network system where a collection of fractures

is represented by an interconnected network of fractures. A network representation often uses a

11

sugar cube model of the reservoir, where the matrix system is inside the sugar cubes and the

fracture system is between the sugar cubes (Warren and Root, 1963).

For this project, trapped fluids are represented using a dual porosity approach. The mobile

fluids are equivalent to the fractures in a dual porosity system: the m1 system is connected to wells,

neighboring grid cells, and to the trapped m2 system within the grid cell. The trapped fluids are

equivalent to the matrix in a dual porosity system: the m2 system is only connected to the m1

system in the same grid cell.

One of the earliest papers discussing naturally fractured reservoir systems is Warren and Root

(1963). Another early paper is Kazemi, Merrill, Porterfield, and Zeman (1976), which describes the

basic formulation used here. Gilman and Kazemi (1988) refines the formulation of Kazemi et al.

(1976), adding additional resolution to the gravity and capillary pressure in the fracture/matrix

transfer. Kazemi, Atan, Al-Matrook, Dreier, and Ozkan (2005) describes simulation of a system

with multiple levels of fractures; it uses an example with three fracture systems and one matrix

system. A slight modification of this approach would apply to a naturally fractured system with

a mobile m1 matrix system and a trapped m2 system. Balogun, Kazemi, Ozkan, Al-Kobaisi, and

Ramirez (2009), Ramirez, Kazemi, Al-Kobaisi, Ozkan, and Atan (2009), and Al-Kobaisi, Kazemi,

Ramirez, Ozkan, and Atan (2009) describe an updated formulation for calculating the water-oil-gas

transfer functions in dual porosity simulation.

Gouth, Moen-Maurel, Jeanjean, Soyeur, and Aziz (2007) describes a triple porosity simulation

in Abu Dhabi. Detwiler, Rajaram, and Glass (2005) describes a method of calculating the fracture

relative permeability using variable aperture fractures. Fung, Middya, and Dogru (2011) presents

results of a triple porosity simulation in Saudi Arabia.

2.3.3 Compositional Reservoir Simulation

The foundation of the compositional simulation formulation used here is described in Kazemi,

Vestal, and Shank (1978). According to Kazemi et al. (1978) the approach of using a separate flash

calculation was first described in Tsutsumi and Dixon (1972). The formulation used here has two

primary differences: the CO2 is soluble in the aqueous phase (WCO2 > 0), and trapping is accounted

for as in a dual porosity system. Coats (1980) describes a similar approach for compositional

simulation, although the approach used by Coats (1980) is fully implicit. Coats (1980) compares

12

their method to several other methods, including the iterative approach of Fussell and Fussell

(1979). Coats (1989) extends the approach of Coats (1980) for a dual porosity compositional

simulator. Acs, Doleschall, and Farkas (1985) describes a slightly different approach using pressure

and the component masses as primary variables. Kendall, Morrell, Peaceman, Silliman, and Watts

(1983) describes the development of the MARS simulator at Exxon. Young and Stephenson (1983)

describes a compositional reservoir simulator. Watts (1986) describes an approach for compositional

simulation; this approach first solves a pressure equation; using the pressure solution it solves for

velocities; using the velocities it solves for implicit saturations and relative permeabilities. This

approach requires the calculation of the derivatives of partial molar volumes. Nghiem and Li (1990)

describes a way to simplify flash calculations for use in a compositional simulator. Nghiem and

Sammon (1997) assumes that the fluids in a grid cell equilibrate based on diffusion rather than

instantaneously being in equilibrium. Haukas, Aavatsmark, Espedal, and Reiso (2007) describes

additional compositional approaches. Wang and Pope (2001) describes the state of the art in 2001

for compositional simulation using an equation of state.

Voskov and Tchelepi (2008) describes performing compositional simulations in using compo-

sitional space parameterization rather than simulating based on total number of moles or mole

fractions. Pan and Tchelepi (2011) describes another set of variables for compositional simulation

plus methods for bypassing the stability analysis of the compositional system.

Wong, Firoozabadi, and Aziz (1990) compares several of the previous methods of compositional

simulation, with the conclusion that the methods are more similar than it might appear under

a casual inspection. Coats (2000) compares different compositional formulations and finds them

similar. Nghiem, Fong, and Aziz (1981) describes the earliest version of the CMG methodology for

compositional simulation. It is similar to Kazemi et al. (1978) and discusses convergence issues.

2.3.4 CO2 and Miscible Flood Simulation

Todd and Longstaff (1972) describes a way to calculate miscible flood performance using a four

“component” system consisting of water, oil, gas, and solvent. Todd and Longstaff (1972) also

describes a way to calculate viscosity, density, and relative permeability of a miscible oil and gas

hydrocarbon system.

13

Chase and Todd (1984) presents an early simulation study of CO2 flooding in a San Andres

carbonate reservoir in west Texas). Several features are included that are specific to CO2 floods,

including dropout of heavy components, water blocking, viscous instability and fingering, misci-

ble/immiscible transition, and a non-zero WCO2 . Water blocking was calculated as Sblock = Sorw

1+β krokrw

;

this is used to adjust the saturations accessible to CO2.

Enick and Klara (1992) discusses the effect of including the CO2 solubility in brine; they con-

clude that it is frequently necesary for accurate compositional simulation results of CO2 flooding.

Enick and Klara (1992) also provides correlations for calculating WCO2 based on the total dis-

solved solids present, and a methodology for updating WCO2 in a compitional formulation similar

to Kazemi et al. (1978) and the approach used in this dissertation.

Xiao and Jones (2007) describes a reactive transport model for dolomitization.

Coats, Whitson, and Thomas (2004b) describes modeling of dispersion. Garmeh and Johns

(2010) discusses the importance of mixing within grid cells for reservoir simulation.

2.3.5 Parallel Simulation

Killough and Bhogeswara (1991) describes an early parallel compositional simulator. Domain

decomposition, communication, and load balancing described in this paper are still issues today.

Zhang, Wu, Ding, Pruess, and Elmroth (2001) describes a parallel formulation for TOUGH2. Atan,

Kazemi, and Caldwell (2006) describes a method to use multiscale multimesh reservoir simula-

tion for parallel openMP based computations. Tarman, Wang, Killough, and Sepehrnoori (2011)

describes a method for decomposing a reservoir simulation into rectangular grids for parallel com-

putations.

Dogru, Li, Sunaidi, Habiballah, Fung, Al-Zamil, Shin, McDonald, and Srivastava (1999) de-

scribes the initial development of the parallel compositional simulator POWERS (Parallel Oil Wa-

ter and Gas Reservoir Simulator) at Saudi Aramco. Dogru, Sunaidi, Fung, Habiballah, Al-Zamel,

and Li (2002) describes an update of the work on POWERS. Al-Shaalan, Fung, and Dogru (2003)

describes a dual permeability extension to POWERS using a hybrid MPI/openMP parallelization

scheme. Fung and Dogru (2007) and Fung and Dogru (2008) describes an update to POWERS

using a parallel unstructured solver. Dogru, Fung, Al-Shaalan, Middya, and Pita (2008) describes

the extension of POWERS from mega-cell models to giga-cell models. Al-Shaalan, Klie, Dogru, and

14

Wheeler (2009), Dogru, Fung, Middya, Al-Shaalan, Pita, HemanthKumar, Su, Tan, Hoy, Dreiman,

Hahn, Al-Harbi, Al-Youbi, Al-Zamel, Mezghani, and Al-Mani (2009), and Dogru, Fung, Middya,

Al-Shaalan, Byer, Hoy, Hahn, Al-Zamel, Pita, Hemanthkumar, Mezghani, Al-Mana, Tan, Dreiman,

Fugl, and Al-Baiz (2011) describe extensions to GigaPOWERS.

2.3.6 Simulation of Trapping and Bypassing

Coats et al. (2004a) describes a formulation for accounting for bypassed oil. The formulation

described in Coats et al. (2004a) is the closest to the formulation presented in this dissertation of

all the literature reviewed.

Barker, Prevost, and Pitrat (2005) describes a way to modify the mobile compositions.

2.3.7 Simulation of Diffusion

da Silva and Belery (1989) presents equations for calculating the diffusion coefficients in a

compositional system. Nghiem and Sammon (1997) presents correlations for calculating diffusion

coefficients in a compositional system. Hoteit and Firoozabadi (2006a) describes compositional

simulations of diffusion in naturally fractured reservoirs with gas injection. Bahar and Liu (2008)

measured the diffusion coefficient of gaseous CO2 into brine.

2.3.8 Additional Simulation Topics

Coats (1980) computes oil and gas relative permeabilities weighted by f(σ) =(

σσ0

) 1n1

to scale

krg and krog as the system becomes miscible. It uses Stone (1973) to calculate kro from krog, krow,

krw, and krg.

Kelly (2006) describes using an equation of state to calculate the density variations within an

injection well as a function of depth; it is important to use multiple depths in the calculation

because CO2 density varies a lot with temperature and pressure. Wu and Bai (2009) describes

a method for simulating low salinity water flooding. Zhang, Yin, Wu, and Winterfeld (2012)

describes a methodology for non-isothermal reactive transport modeling with application to CO2

sequestration using the TOUGH framework. Das, Mirzaei, and Widdows (2006) describes how

microscopic heterogeneities can effect the relative permeability and capillary pressure.

15

2.3.9 SPE Comparative Solution Projects

The SPE Comparative Solution Project is a series of ten articles which compare different reser-

voir simulators: Odeh (1981), Weinstein, Chappelear, and Nolen (1986), Kenyon and Behie (1987),

Aziz, Ramesh, and Woo (1987), Killough and Kossack (1987), Firoozabadi and Thomas (1990),

Nghiem, Collins, and Sharma (1991), Quandalle (1993), Killough (1995), and Christie and Blunt

(2001a). These articles present test cases which can be used to evaluate new reservoir simulators.

The following provides a brief description of each article:

1. Odeh (1981) presents a 3-D 2-phase black oil problem involving gas injection.

2. Weinstein et al. (1986) present a radial 2-D 3-phase black oil problem involving coning of

both water and gas.

3. Kenyon and Behie (1987) present a 3-D 3-phase compositional problem involving retrograde

gas cycling.

4. Aziz et al. (1987) present a 2-D 3-phase steam injection problem.

5. Killough and Kossack (1987) present a 3-D 3-phase compositional problem involving misci-

ble hydrocarbon gas injection. This could possibly be used as a test case of the simulator

developed here.

6. Firoozabadi and Thomas (1990) present a 2-D 3-phase black oil problem involving a naturally

fractured reservoir.

7. Nghiem et al. (1991) present a 3-D 3-phase black oil problem involving horizontal wells.

8. Quandalle (1993) presents 3-D 3-phase black oil problem which compares different gridding

techniques.

9. Killough (1995) presents a 3-D 3-phase black oil problem with a 9000 grid cell geostatistically

populated grid.

10. Christie and Blunt (2001a), based on (Christie and Blunt, 2001b), present a 3-D 3-phase black

oil problem with a 1.1 million grid cell geostatistically populated grid. The paper focuses on

upscaling techniques.

16

2.4 Geologic Characterization in Middle East

Jobe (2013) presents a detailed review of the geologic characterization of the Abu Dhabi reser-

voirs studied by the CSM/PI Integrated Carbonate Reservoir Studies Group.

Al-Aruri, Ali, Ahmad, and Samad (1998) uses mercury injection capillary pressure data to

help group carbonate facies into petrophysical rock types in Abu Dhabi. Ghedan, Gunningham,

Ehmaid, and Azer (2002) describes a process to upscale a reservoir simulation model in Abu Dhabi.

Bushara, El Tawel, Borougha, Dabbouk, and Qotb (2002) describes a study by Zadco to charac-

terize the fracture permeability. Cantrell and Hagerty (2003) describes a way to characterize the

carbonate rocks in Ghawar. Ottinger, Kompanik, Al Suwaidi, Brantferger, and Edwards (2012)

describes geostatistical mapping of reservoir rock types conducted by Zadco. Yamamoto, Kom-

panik, Brantferger, Al-Zinati, Ottinger, Al-Ali, Dodge, and Edwards (2012) describes geostatistical

modeling of a dolomitized zone by Zadco.

Ghedan, Thiebot, and Boyd (2004) describes modeling a water-oil transition zone in Abu Dhabi,

including one author from Zadco. Ghedan (2007) uses dynamic reservoir rock types to assign

relative permeability and capillary pressure functions for grid cells in a reservoir simulation model

for Abu Dhabi. It’s important to account for the varying wettability if there is a transition zone

present. Ghedan, Canbaz, Boyd, Mani, and Haggag (2010) describes a new method for measuring

the wettability based on work done with Zadco.

2.5 Relative Permeability and Capillary Pressure

Relative permeability represents the reduced permeability when multiple fluids are present in a

reservoir.

2.5.1 General Articles on Relative Permeability

Mualem (1976) presents a two phase relative permeability model not including hysteresis.

Thomeer (1983) presents a two phase relative permeability model not including hysteresis. Chierici

(1984) presents a two phase relative permeability model not including hysteresis. Kamath, Meyer,

and Nakagawa (2001) presents two-phase oil/water relative permeability data for carbonate rocks.

Bennion and Bachu (2005) and Bennion and Bachu (2008b) present CO2/brine relative permeabil-

ity data for carbonate and sandstone cores in Canada. Byrnes and Bhattacharya (2006) presents

17

relative permeability data for carbonate reservoirs. Egermann, Laroche, Manceau, Delamaide, and

Bourbiaux (2007) presents gas/water relative permeability data for vuggy carbonates. Rustad,

Theting, and Held (2008) presents a simulation approach for assessing the uncertainty in rela-

tive permeabilities. Gawish and Al-Homadhi (2008) presents relative permeability experiments for

different temperatures, wettabilities, and overburden pressures.

2.5.2 General Articles on Capillary Pressure

Parker, Lenhard, and Kuppusamy (1987) presents a model for capillary pressure. Gray and Has-

sanizadeh (1991) presents a theoretical discussion of a capillary pressure model. Zhou and Blunt

(1997) presents a discussion of how the three-phase spreading coefficient effects capillary pressures.

Clerke (2009) presents a bimodal capillary pressure distribution. Lamy, Iglauer, Pentland, Blunt,

and Maitland (2010) presents capillary pressure data for carbonate cores. Iglauer, Wulling, Pent-

land, Al Mansoori, and Blunt (2009) presents a review of capillary trapping in sandstones along

with some new data.

2.5.3 Trapping

Land (1968) provides the trapped gas saturation as a function of the initial gas saturation. This

model is a very commonly used model and the base model for many comparisons. Keelan and Pugh

(1975) presents early experimental data for trapped gas saturations in carbonates. Torquato (1990)

presents a discussion of diffusion controlled trapping. Lin and Huang (1990) presents methods for

calculating trapping in an oil/water system in various wettabilities of Berea cores. Muller and

Lake (1991) presents a model of trapping using diffusion. All trapping amounts are presented as a

function of residence time. Bennion, Thomas, Bietz, and Bennion (1996) presents a discussion of

different trapping mechanisms. Pentland, Al-Mansoori, Iglauer, Bijeljic, and Blunt (2008) presents

measurements of trapping in sand packs.

2.5.4 Three-Phase Relative Permeability

Naar and Wygal (1961) presents an early model of three-phase relative permeability. Stone

(1970) and Stone (1973) present a three-phase relative permeability model. This model is a very

commonly used model and the base model for many comparisons. Dietrich and Bondor (1976)

presents a three-phase relative permeability model. Carlson (1981) presents a three-phase relative

18

permeability model using Killough and Kossack (1987) and Land (1968). This model is a very

commonly used model and the base model for many comparisons. Fayers and Matthews (1984)

analyzes three-phase relative permeability data from various literature sources. Thomas and Coats

(1992) rewrites Stone’s methods in terms of arbitrary permeabilities. Larsen and Skauge (1998)

presents a three-phase relative permeability formulation. Larsen and Skauge (1999) presents an

immiscible WAG simulation using Larsen and Skauge (1998). Blunt (2000) presents a three-phase

relative permeability formulation and a good summary of previous methods. van Dijke, Sorbie, and

McDougall (2000) and van Dijke, Sorbie, and McDougall (2001) present a formulation for three-

phase relative permeability. Oliveira and Demond (2003) presents a comparison of three-phase

relative permeability models. Juanes and Patzek (2004b) and Juanes and Patzek (2004a) present

a theoretical discussion under what conditions three-phase relative permeability models transition

between hyperbolic and elliptic regions. Yuen, Siu, Shenawi, Bukhamseen, Lyngra, and Al-Turki

(2008) presents a three-phase relative permeability model based on a curve fit to experimental data.

Yuan and Pope (2011) presents a three-phase relative permeability model including a new method

to transition between two phase gas-water and oil-water systems.

Delshad and Pope (1989) presents an analysis of seven different three-phase relative permeabil-

ity formulations. Baker (1988) presents an analysis of different three-phase relative permeability

formulations. Fayers (1989) presents an analysis of Stone’s methods for three-phase relative per-

meability formulations. Kokal and Maini (1990) presents analysis of several three-phase relative

permeability experiments and a modification of Stone’s method. Guzman, Giordano, Fayers, Aziz,

and Godi (1994) presents simulation results for WAG injections based on different three-phase

relative permeability models. Pope, Wu, Narayanaswamy, Delshad, Sharma, and Wang (1998)

presents an analysis of three-phase relative permeability data from various sources in terms of

trapping number. Whitson, Fevang, and Saevareid (1999) presents an analysis of three-phase rel-

ative permeability data using krg vs krg/kro and the capillary number for Berea sandstone and a

North Sea sandstone. This paper also discusses variations in the relative permeability curves as

a function of miscibility. Kossack (2000) presents a comparison three-phase relative permeabil-

ity models with hysteresis as implemented in Eclipse. Spiteri and Juanes (2004) and Spiteri and

Juanes (2006) present simulation of WAG injection with different three-phase relative permeability

models. Karkooti, Masoudi, Arif, Darman, and Othman (2011) presents a WAG case study using

19

three-phase relative permeability of a Malaysian field. Shahverdi, Sohrabi, Fatemi, Jamiolahmady,

Irelan, and Robertson (2011) presents a review of three-phase relative permeability formulations

and a simulation of experiments at Heriot-Watt.

Saraf, Batycky, Jackson, and Fisher (1982) presents three-phase relative permeability data for

Berea sandstone. Van Spronsen (1982) presents three-phase relative permeability data collected

using a centrifuge for both Berea and the Weeks Island sandstone. Ehrlich, Tracht, and Kaye (1984)

presents laboratory data for a dolomite reservoir subjected to a lab-based CO2 WAG flood. Oak

(1990) presents the results of very thorough experiments of three-phase relative permeability on

water-wet Berea sandstone. Kalaydjian, Moulu, Vizika, and Munkerud (1997) presents three-phase

relative permeability experiments for Fontainebleau sandstone and Clashach sandstone. Jerauld

(1997) presents three-phase relative permeability data and curve fits for mixed wet Prudhoe Bay

sandstone. Sahni, Burger, and Blunt (1998) presents three-phase relative permeability measure-

ments of packed sands and sandstones. Ebeltoft, Iversen, Vatne, Andersen, and Nordtvedt (1998)

presents three-phase relative permeability data for a chalk reservoir. Moulu, Vizika, Egermann,

and Kalaydjian (1999) presents three-phase relative permeability data for the Vosges sandstone

under different wettabilities. The paper uses a fractal correlation to match the experimental data.

Kralik, Manak, Jerauld, and Spence (2000) presents the results of three-phase relative permeabil-

ity experiments on an oil-wet sandstone. Egermann, Vizika, Dallet, Requin, and Sonier (2000)

presents simulations of three-phase relative permeability experiments on Estaillades limestone. El-

ement, Masters, Sargent, Jayasekera, and Goodyear (2003) presents WAG experiments that require

a three-phase relative permeability formulation with hysteresis in chalk. Dehghanpour, DiCarlo,

Aminzadeh, and Mirzaei (2010) presents WAG experiments using a water-wet sand pack. Cao and

Siddiqui (2011) presents three-phase relative permeability data for three immiscible fluids (not oil,

gas, and water, but interpreted as similar by the authors) in Berea sandstone. Fatemi and Sohrabi

(2012) presents a review of three-phase relative permeability models and experimental data for mul-

tiple WAG cycles. Fatemi, Sohrabi, Jamiolahmady, and Ireland (2012a) presents a history match

of experimental three-phase relative permeability data and a good literature review. Shahverdi and

Sohrabi (2012) presents an analysis of three-phase relative permeability data. Fatemi, Sohrabi,

Jamiolahmady, and Ireland (2012b) presents three-phase relative permeability data for water wet

and mixed wet cores.

20

2.5.5 Relative Permeability Hysteresis

Naar and Henderson (1961) and Naar and Wygal (1961) present an early model of relative

permeability hysteresis; better theories have been presented more recently. Philip (1964) has a

method for calculating hysteresis scanning curves based on the wetting and drying curves. Walsh,

Negahban, and Gupta (1989) uses the difference between the drainage and imbibition curves in

Berea sandstone to calculate the trapped saturation for a CO2 flood. Braun and Holland (1995)

presents experimental oil/water scanning curves for Berea sandstone and an Australian sandstone.

Chang, Mohanty, Huang, and Honarpour (1997) presents experimental measurements of mixed

wet oil/water relative permeability. Lenhard and Oostrom (1998) presents a discussion of two-

phase oil/water relative permeability with hysteresis. Bennion, Thomas, Jamaluddin, and Ma

(1998) presents a discussion of different kinds of hysteresis. Spiteri, Juanes, Blunt, and Orr (2005)

presents simulation models applied to CO2 injection with relative permeability hysteresis. Zhang,

Falcone, and Teodoriu (2010) presents the effects of relative permeability hysteresis on near-wellbore

pressures. Krause (2012) presents relative permeability data for Berea sandstone showing the 3D

variation in saturations in a core flood. Dernaika, Basioni, Dawoud, Kalam, and Skjaeveland (2012)

presents relative permeability data with hysteresis for various carbonate rocks.

Honarpour, Huang, and Dogru (1996) presents an experimental apparatus to simultaneously

measure relative permeability, capillary pressure, and electrical resistivity during a core flood. Hys-

teresis data is presented for Berea sandstone. Masalmeh (2001) presents a discussion of hysteresis

in water-wet, oil-wet, and mixed-wet porous media.

Behzadi (2010) presents a simulation of CO2 trapping in the Nugget formation. Altundas et al.

(2011) presents a simulation of CO2 trapping.

2.5.6 Capillary Pressure Hysteresis

Morrow and Harris (1965) provides data for capillary pressure hysteresis measured in a column

packed with glass beads. Morrow (1970) is an early summary of the thermodynamics of capillary

pressure hysteresis. Lenhard, Parker, and Kaluarachchi (1991) presents a two-phase gas/water cap-

illary pressure hysteresis model with experimental data. Kleppe, Delaplace, Lenormand, Hamon,

and Chaput (1997) presents measurements of gas/oil capillary pressure hysteresis and describes a

way to scale the scanning curves.

21

2.5.7 Combined Relative Permeability and Capillary Pressure Hysteresis

Killough and Kossack (1987) provides a method for computing capillary pressure and relative

permeability hysteresis. This model is a very commonly used model and the base model for many

comparisons. Parker and Lenhard (1987) and Lenhard and Parker (1987) present a model for

three-phase capillary pressure and relative permeability hysteresis. Bradford, Abriola, and Leij

(1997) presents a discussion of three-phase relative permeability and capillary pressure models.

Nordtvedt, Ebeltoft, lversen, Sylte, Urkedal, Vatne, and Watson (1997) presents three-phase capil-

lary pressure and relative permeability measurements. The three-phase data was fit using a product

of two splines. Hustad (2002) presents a three-phase capillary pressure and relative permeability

model with hysteresis. Fayers, Foakes, Lin, and Puckett (2000) presents a three-phase capillary

pressure and relative permeability model with hysteresis, including a method for weighting relative

permeabilities as miscibility is developed. Blunt (2000) presents an analysis of three-phase relative

permeability and capillary pressure experiments, including a discussion of trapped oil, spreading oil,

and mobile oil. Hustad et al. (2002) presents WAG simulation results and a description using the

IKU3P model for relative permeability and capillary pressure. Kjosavik, Ringen, and Skjaeveland

(2002) presents a relative permeability and capillary pressure formulation with hysteresis. Delshad,

Lenhard, Oostrom, and Pope (2003) presents a relative permeability and capillary pressure formu-

lation with hysteresis. Hustad and Browning (2009) presents a relative permeability and capillary

pressure formulation with hysteresis.

DiCarlo, Sahni, and Blunt (2000) presents three-phase capillary pressure and relative perme-

ability data for various wettability sandpacks. Masalmeh (2003) presents capillary pressure and

relative permeability data and their variations with wettability. Jackson, Valvatne, and Blunt

(2002) presents relative permeability and capillary pressures calculated from pore network simula-

tions of Berea sandstone. Masalmeh (2002) presents capillary pressure and relative permeability

data for mixed wet and oil wet Middle East carbonates. Masalmeh, Shiekah, and Jing (2007)

presents capillary pressure and relative permeability data and modeling for a carbonate reservoir.

Ghomian, Pope, and Sepehrnoori (2008) presents simulations of CO2 WAG for EOR and seques-

tration using different three-phase relative permeability and capillary pressure models. Olafuyi,

Cinar, Knackstedt, and Pinczewski (2008) presents experimental capillary pressure and relative

22

permeability data for Berea sandstone, Bentheim sandstone, and Mount Gambier carbonates.

Masalmeh and Wei (2010) presents a study of WAG options using three-phase relative perme-

ability and capillary pressure hysteresis. It uses a linear relative permeability option under miscible

conditions. Bhatti, Kalam, Hafez, and Kralik (2012) presents relative permeability and capillary

pressure data for Abu Dhabi carbonates.

2.5.8 Non-zero Relative Permeability Derivative

Bell, Trangenstein, and Shubin (1986) discusses reasons why the derivative of at least one of

the relative permeabilities with respect to saturation should be non-zero: it leads to a region of

the relative permeability space that has a non-hyperbolic solution; physically we would expect the

solution to be hyperbolic for all saturation values.

limSϕ→Sϕ

∂krϕ[Sϕ]

∂S�= 0 (2.3)

The following papers present a theoretical discussion under what conditions three-phase relative

permeability models transition between hyperbolic and elliptic regions: van Dijke et al. (2000),

van Dijke et al. (2001), Juanes and Patzek (2004b), and Juanes and Patzek (2004a). According to

Dr. Kazemi, if capillary pressure is appropriately calculated this is no longer required.

2.5.9 Additional Relative Permeability Effects

Wilson (1956) illustrates the effects of overburden pressure on oil/water relative permeability.

Overburden pressure causes Swr to increase and Sowr to decrease. Al-Quraishi and Khairy (2005)

presents experimental results showing changes in oil/water relative permeability as a function of

overburden pressure.

Coats and Smith (1964) describes diffusion-based mass transfer out of trapped pores. Sinnokrot,

Ramey, and Marsden (1971) describes the changes in capillary pressure with temperature. Swr

increases with increasing temperature for sandstone and decreases with increasing temperature for

carbonates.

Torabzadey (1984) and Kumar, Torabzadeh, and Handy (1985) present the experimental varia-

tions of water/oil relative permeability in Berea sandstone with temperature and interfacial tension.

The Sorw decreases with increasing temperature, but the change is much smaller for a low inter-

23

facial tension system. The Swr increases with increasing temperature for a low interfacial tension

system but is approximately constant with a high interfacial tension system.

Sorbie and van Dijke (2010) presents an analysis of near-miscible interfacial tension changes. It

also presents the results of some pore-scale and micro-model experiments of near-miscible WAG.

Middya and Dogru (2008) describes a method for calculating well drainage pressure as an average

of multiple grid cells rather than the value of a single cell.

2.6 Equation of State Literature

The phases and compositions are determined using “flash” calculations with an equation of

state. This section describes different methods for calculating an equation of state and methods

for adjusting equation of state parameters to fit experimental data.

2.6.1 Calculation of Equation of State

Rachford and Rice (1952) describes what is now considered the standard method of performing

hydrocarbon flash calculations.

Michelsen (1980) presents a method for calculating phase envelopes. Michelsen (1982a) and

Michelsen (1982b) present a methodology for flash calculations. Michelsen and Mollerup (1986)

specifies derivatives of thermodynamic properties. Whitson and Michelsen (1989) describes a flash

calculation using negative flash. Mollerup and Michelsen (1992) describe computations of ther-

modynamic derivatives that were used to check the flash calculations used in this dissertation.

Michelsen (1998) describes some ways to speed up flash calculations.

Li and Nghiem (1982) describes several different methods for flash calculations. Nghiem, Aziz,

and Li (1983) describes a flash calculation procedure.

Fussell and Yanosik (1978) presents a flash calculation procedure called Minimum Variable

Newton-Raphson (MVNR). Guehria, Thompson, and Reynolds (1990) describes a flash calcula-

tion procedure and ways to calculate derivatives of thermodynamic properties. Nagarajan, Cullick,

and Griewank (1991a) and Nagarajan, Cullick, and Griewank (1991b) describe a method for crit-

ical point calculations. Thomas, Bennion, and Bennion (1991) describes a method for calculating

pseudo-ternary diagrams. Firoozabadi and Pan (2002) describes improved stability analysis calcula-

tions for compositional modeling. Pan and Firoozabadi (2001) describes improved flash calculations

24

for compositional modeling. Li and Johns (2006) describes improved flash calculations for composi-

tional modeling. Rasmussen, Krejbjerg, Michelsen, and Bjurstrom (2006) describes improved flash

calculations for compositional modeling. Hoteit and Firoozabadi (2006b) provides a good overview

of flash calculations and suggests some improvements in stability testing.

Juanes (2008) describes a method to perform flash calculations by discretizing the tie lines.

Li and Firoozabadi (2010) describes a flash procedure, including liquid-vapor and liquid-liquid-

vapor systems. Voskov and Tchelepi (2008), Voskov and Tchelepi (2007), and Gasmi, Voskov, and

Tchelepi (2009) describe a flash procedure using a compositional space parameterization. Voskov,

Younis, and Tchelepi (2009) and Voskov (2011) describe several different parameterizations that

can be used for flash calculations and compares each of these methods.

2.6.2 Adjusting Equation of State Parameters

Rowe (1978) presents a methodology for using pseudo components in reservoir simulation.

Whitson (1984) discusses the importance of C7+ properties for EOS predictions. Whitson (1983)

describes methods for splitting the C7+ into pseudocomponents. Pedersen, Thomassen, and Fre-

denslund (1985) and Pedersen, Thomassen, and Fredenslund (1988) discuss appropriate ways for

fitting EOS parameters. Nishiumi, Arai, and Takeuchi (1988) discuss ways to calculate binary

interaction parameters for fitting an equation of state. Leibovici, Govel, and Piacentino (1993)

describes a method for calculating pseudo-component properties. Gasem, Gao, Pan, and Robinson

(2001) describes some changes for the Peng-Robinson EOS that may improve the fit to experimen-

tal data. Jaubert, Vitu, Mutelet, and Corriou (2005) describes modifications to the Peng-Robinson

EOS using experimental information about specific aromatic compounds. Ahmed (2007b) describes

modifications to the Peng-Robinson EOS that make the fits to experimental data better.

2.6.3 Modifications to Equation of State Model when CO2 is Present

The following papers describe modifications to the binary interaction coefficients of CO2 with

hydrocarbons using the Peng-Robinson EOS: Mulliken and Sandler (1980), Kato, Nagahama, and

Hirata (1981), Turek, Metcalfs, Yarborough, and Robinson (1984), Lin (1984), Nishiumi et al.

(1988), Kordas, Tsoutsouras, Stamataki, and Tassios (1994), Vitu, Privat, Jaubert, and Mutelet

(2008).

25

Coutinho, Kontogeorgis, and Stenby (1994) describe modifications to the binary interaction

coefficients of CO2 with hydrocarbons using the Soave-Redlich-Kwong EOS.

Metcalfe and Yarborough (1979) discusses the effects of mixing CO2 with oil on the phase

behavior of the system. Data is presented for a CO2-CH4-nC4-nC10 system, in addition to other

systems. Kuan, Kilpatrick, Sahimi, Scirven, and Davis (1986) presents CO2-water-hydrocarbon

phase behavior.

Turek et al. (1984) describes some methods for fitting the EOS properties for a system containing

CO2. Han and McPherson (2008) compares several different CO2-brine equations of state for CO2

sequestration applications.

2.6.4 Phase Behavior Illustration

Rowe Jr. and Silberberg (1965) describes the phase behavior for an enriched gas injection

process; it has an example of three-dimensional ternary diagrams with pressure along one axis. Rowe

(1967) presents four-dimensional plots of phase behavior. Kalippan and Rowe (1971) illustrates

additional ways to present phase behavior for more than three variables.

2.6.5 Other Equation of State References

Li and Nghiem (1986) describes a way to calculate solubility in the aqueous phase using Henry’s

Law and describes a three-phase oil-water-gas flash procedure. One of the examples is for a CO2-

brine system. Broad, Varotsis, and Pasadakis (2001) describes the effect of data quality on the

predictions of an EOS. Nagarajan, Honarpour, and Sampath (2007) describes the sampling pro-

cesses needed to accurately characterize reservoir fluids.

2.7 Pore Scale Simulation

Pore scale simulation is a specialized category of reservoir simulation devoted to simulating

devoted to simulating microscopic process to help understand macroscopic processes. Some of

these techniques are praising for future study of trapping.

2.7.1 Network Models

Ehrlich and Crane (1969) presents an overview of network models. One approach is to consider

a porous medium as a network of capillary tubes. Hysteresis can be explained by bypassing, Naar

26

and Henderson (1961). A simple pore doublet consists of one small and one large capillary tube.

Naar and Henderson (1961) describes a capillary tube based model for imbibition and drainage

relative permeabilities, including bypassed/blocked/trapped oil.

Blunt, Fenwick, and Zhou (1994) presents a discussion of spreading and non spreading oils, plus

a discussion of how individual pores or pore throats drain.

McDougall and Sorbie (1995) describes a cubic 20 × 20 × 20 network of nodes; each node has

an assigned capillary radius; each node is connected to six neighbors. After creating this network,

McDougall and Sorbie (1995) then conducted waterflood experiments for water wet and mixed wet

systems. The network simulations result in simulated capillary pressure and relative permeabilities.

Blunt (1997) describes network simulations where the contact angle varies between nodes.

Laroche, Vizika, and Kalaydjian (1999) describes another network model.

van Dijke and Sorbie (2003) discusses the results of network model simulations. They observe

that “displacement chains” can occur where an individual fluid is not in contact with the inlet and

outlet but is also not trapped; this is an extension of the “double displacement” concept.

Piri and Blunt (2002) describes a network model that consists of pores connected by pore

throats. Pores and pore throats have triangular, square, or circular cross sections. The network

used in Piri and Blunt (2002) is based on a 27 mm3 core of Berea with 12349 pores and 26146 throats;

condition number varies between 1 and 19, with an average of 4.19. Pores vary from 3.62 μm to

73.54 μm; the throats vary from 0.90 μm to 56.85 μm, with an absolute permeability of 2600 md.

Piri and Blunt (2005a) continue the discussion of how to conduct network simulations. Piri and

Blunt (2005b) discuss the results of mixed-wet network modeling, including relative permeability

predictions and the distribution of fluids in different pore sizes after primary drainage, water flood,

and tertiary gas injection. Oil moves into intermediate sized pores during gas injection as a result

of double displacement. Tertiary gas injection and secondary gas injection predict different relative

permeability curves.

Nguyen, Sheppard, Knackstedt, and Pinczewski (2006) compares the results of a Berea based

network model to the relative permeability measurements of Oak (1990). Suicmez, Piri, and Blunt

(2006) compares the results of a Berea based network model to experimental data of Oak (1990),

Egermann et al. (2000), and Element et al. (2003). Suicmez et al. (2006) hypothesizes that relative

permeability is independent of flow path if mobile saturations are used rather than mobile plus

27

trapped saturations.

Mahmud (2007) uses a cubic 64 × 64 × 64 network model based on Fontainebleau sandstone.

Suicmez, Piri, and Blunt (2008) describes the results of network model simulations, including

trapped oil and gas as a function of initial gas saturation. Suicmez, Piri, and Blunt (2007) presents

results of network simulations of WAG. Pentland, Tanino, Iglauer, and Blunt (2010) compares pore

network models to a new set of coreflood experiments.

Sheng, Thompson, Fredrich, and Salino (2011) compares different methods of network simula-

tion.

2.7.2 Micro Models

Campbell and Orr (1985) used micromodels to visualize CO2/oil displacements. Their 2D mod-

els were 88 mm × 63 mm with etched glass pores that are 750 μm × 140 μm. Sohrabi, Tehrani,

Danesh, and Henderson (2001) uses oil wet and mixed wet micromodels to visualize WAG. Sohrabi,

Tehrani, Danesh, and Henderson (2004) uses high pressure micromodels to visualize WAG processes.

Dong, Foraie, Huang, and Chatzis (2005) uses micromodels to illustrate pore scale effects in immis-

cible WAG. Chalbaud, Lombard, Martin, Robin, Bertin, and Egermann (2007) presents micromodel

experiments using CO2 and nitrogen. Bondino, McDougall, Ezeuko, and Hamon (2010) presents

the results of a re-pressurization micromodel experiment.

2.7.3 Additional Pore Scale Simulation Discussion

van Dijke, Sorbie, Sohrabi, Tehrani, and Danesh (2002) uses a combination of network sim-

ulation and micromodels to help understand WAG processes. Ajo-Franklin (2007) presents an

overview of different techniques for extracting a pore-network model from rocks; these include

2D optical microscope image analysis, micro-CT scans, scanning confocal microscopy, and ablation

combined with 2D imagery. Algive, Bekri, and Vizika (2009) uses a pore network model to evaluate

geochemical changes while injecting CO2.

Knackstedt, Dance, Kumar, Averdunk, and Paterson (2010) uses QEMscan images to construct

pore network model and compares the network simulation results to drainage and imbibition ex-

periments on the same cores. Youssef, Bauer, Bekri, Rosenberg, and Vizika (2010) uses microCT

scans to measure the in situ saturation during fluid flow experiments.

28

2.8 Interfacial Tension

Interfacial tension (IFT) represents the strength of the interface between two fluids. IFT is

often used to scale relative permeability as miscibility is developed based on a pressure change.

2.8.1 Interfacial Tension Methods

Weinaug and Katz (1943) describes an early approach for calculating the interfacial tension .

Lee and Chien (1984) describes a method for calculating interfacial tension. Danesh, Dandekar,

Todd, and Sarkar (1991) describes an adjustment to the calculation of interfacial tension. Zuo

and Stenby (1998) fits several different methods for calculating interfacial tension to experimental

data; these methods are based on Helmholtz free energies and chemical potential, which can be

calculated from an EOS.

Grigg and Schechter (1998) reviews various interfacial tension methods and concludes that an

exponent of 3.88 is best in the following equation:

σ =

(∑m

(ξnl P

#mXn

m − ξnvP#mY n

m

))3.88

(2.4)

Grigg and Schechter (1998) defines the following parachors, consistent with (2.4):

• PCO2 = 82.00

• PCH4 = 74.05

• PnC4 = 193.90

• PnC10 = 440.69

Schechter and Guo (1998) presents different ways to calculate parachors.

2.8.2 Interfacial Tension and Relative Permeability

Bardon and Longeron (1980) conducted gas-oil relative permeability measurements under dif-

ferent interfacial tensions. The krog changed curvature and endpoint saturations significantly over

interfacial tensions from σ = 12.6 × 10−3 N/m to σ = 0.001 × 10−3 N/m. The krg did not change

much above σ = 0.065 × 10−3 N/m, but changes significantly between σ = 0.065 × 10−3 N/m and

σ = 0.001 × 10−3 N/m

29

Harbert (1983) presents water-oil relative permeability data for several formations under inter-

facial tensions between σ = 0.1× 10−3 N/m and σ = 2.0× 10−3 N/m

Shen, Zhu, Li, and Wu (2006) evaluates changes in the krw and krow as the water-oil interfacial

tension changes σow. Al-Wahaibi, Grattoni, and Muggeridge (2006) presents changes in the gas-oil

relative permeability as the interfacial tension σgo changes. Fagerlund, Niemi, and Oden (2006)

scales the relative permeabilities based on interfacial tensions.

2.8.3 Spreading Coefficient

The spreading coefficient is defined as

Sow = σwg − σog − σow (2.5)

Oren and Pinczewski (1994) uses micromodels to study the effects of the spreading coefficient on

production mechanisms.

2.8.4 Interfacial Tension Fit Gas-Oil

Firoozabadi, Katz, Soroosh, and Sasjjadian (1988) presents interfacial tension fits for various

fluids. Pedersen, Lund, and Fredenslund (1989) presents interfacial tension fits for various fluids.

Rønningesen (1993) presents interfacial tension fits for North Sea fluids.

2.8.5 Water Interfacial Tension

Bahramian, Danesh, Gozalpour, Tohidi, and Todd (2007) presents fits of interfacial tension for a

water-methane-cyclohexane-decane system. Rushing, Newsham, Van Fraassen, Mehta, and Moore

(2008) presents gas-water interfacial tension data for several dry gas systems for temperatures

between 300◦F and 400◦F.

2.8.6 CO2-Brine Interfacial Tension

Bennion and Bachu (2006) presents data for brine-CO2 relative permeability curves, including

how they change with interfacial tension. Chalbaud, Robin, and Egermann (2006) and Chalbaud,

Robin, Lombard, Martin, Egermann, and Bertin (2009) present a correlation for brine-CO2 inter-

facial tension which includes the effects of different salinities, temperatures, and pressures.

30

σCO2,br[N

m] = 26×10−3[

N

m]+1.2550mNaCl[

mol

kg]+

(82

44.01

(ρbr

[ g

cm3

]− ρCO2

[ g

cm3

]))4.7180 (T [K]

Tc[K]

)1.0243

(2.6)

Bennion and Bachu (2008a) presents correlations for CO2-brine interfacial tension as a function

of pressure, temperature, and salinity. Delshad, Kong, and Wheeler (2011) presents a formulation

for CO2-brine interfacial tension, plus some adjustments for relative permeability.

Chun and Wilkinson (1995) presents a correlation for a CO2-H2O system; this correlation is

only applicable for this specific system. Ramey (1973) presents a general method for calculating

the oil-water interfacial tension, but the method requires reading one of the values from a graph.

Firoozabadi and Ramey (1988) presents several correlations for gas-water and oil-water interfacial

tensions. Firoozabadi and Ramey (1988) uses the Lee and Chien (1984) parachor for water of

52.0. Zuo and Stenby (1997) describes a way to calculate interfacial tension using gradient theory,

requiring the Helmholtz free energy and chemical potential. There are various adjustments for

pure compounds, including one for H2O and one for CO2. Shariat, Moore, Mehta, Van Fraassen,

Newsham, and Rushing (2011) presents a summary of gas-water interfacial tension data.

2.9 Liquid-Liquid-Vapor

At temperatures below 50◦C (for instance Nghiem and Li (1984)), it is possible for two liquid

hydrocarbon phases to form in addition to a liquid water phase, a gaseous phase, and a solid

asphaltene-rich phase. This is sometimes called the “LLV” region, corresponding to the two liquids

and vapor phase present for the hydrocarbon phases, or the “LLL” region when the aqueous phase

is included. The LLV region is typically relatively narrow in pressure and composition, Figure 2.1.

Jarell, Fox, Stein, and Webb (2002) and Lake (1989) discuss some of these effects. Figure 2.2 shows

the different possible displacement mechanisms for a CO2 flood; type III is the liquid-liquid vapor

region.

Several papers discuss experiments which show liquid-liquid-vapor portions of the phase dia-

gram. Gardner, Orr, and Patel (1981) present experiments using Wasson crude oil and CO2. Orr,

Yu, and Lien (1981) present experiments with Maljamar crude oil and mixtures of pure components.

Baker, Pierce, and Luks (1982) present experiments using CO2 plus various pure components and

31

Figure 2.1: Low temperature phase behavior of Wasson crude showing the presence of two liquidhydrocarbon phases (Lake, 1989).

32

Figure 2.2: Various miscibility regions for a CO2 flood, (Klins, 1984). Region I is immiscible. RegionII may develop miscilibility. Region III is a miscible region that may contain two hydrocarbon liquidphases. Region IV is the first contact miscible region. Region V involves liquid CO2.

some experiments using CO2 plus Levelland crude oil. Turek et al. (1984) present experiments

using a synthetic oil and CO2, plus an unnamed reservoir oil. Enick, Holder, and Morsi (1985)

present experimental data for a pure component system of CO2 and tridecane that displays LLV

behavior. Bryant and Monger (1988) present experimental data using Wasson crude oil plus CO2.

Godbole, Thele, and Reinbold (1995) and Wang and Strycker (2000) present experimental results

for fields in Alaska.

Other papers discuss methods for calculating LLV or LLLV equilibria. Fussell (1979) presents

an early discussion of the Minimum Variable Newton Raphson technique. Risnes and Dalen (1984)

describe a methodology for multi-phase flash. Nghiem and Li (1984) discuss the Quasi Newton

Successive Substitution method used in CMG. Nghiem and Li (1986) continue the discussion of

Nghiem and Li (1984) and illustrate the simulation of a slim tube experiment using Wasson crude

oil. Baker et al. (1982) provide the details of the computation of Gibbs Free Energy for determining

stability. Enick, Holder, and Mohamed (1987) provide a detailed description of the search strategy

for stable phases in a four-phase flash formulation. These are illustrated using Maljamar crude

33

oil mixed with CO2. Nagarajan et al. (1991a) provide a detailed description of four-phase flash

calculations. Lindeloff and Michelsen (2003) present the methods used by PVTsim and illustrates

these techniques using four different crude oils mixed with CO2. Li and Firoozabadi (2010) present

a good summary of several different methods for calculating stability analysis and three-phase flash

calculations, illustrated using a Maljamar crude oil mixed with CO2.

2.10 Asphaltenes

Asphaltenes are heavy hydrocarbon components that are not soluble in pentane, hexane, hep-

tane or CO2 but are soluble in benzene and toluene. Asphaltenes can alter the permeability and

wettability of rocks they are deposited on; this is often a concern for production or pipeline en-

gineers but is of lesser concern to reservoir engineers. Asphaltene literature was reviewed for this

project, but asphaltene deposition and simulation is being evaluated by another Ph.D. student at

Colorado School of Mines, Tadesse Teklu.

Pedersen and Christensen (2007) present a good review of different asphaltene deposition mech-

anisms. Leontaritis (1989) and Kokal and Sayegh (1995) present a good review of asphaltene liter-

ature including a description of different deposition mechanisms. The following additional articles

include good descriptions of specific asphaltene deposition models: Novosad and Costain (1990),

Nghiem, Hassam, Nutakki, and George (1993), Leontaritis, Amaefule, and Charles (1994), Man-

soori (1994), Nghiem, Coombe, and Farouq Ali (1998), Nghiem, Kohse, Farouq Ali, and Doan

(2000), Kohse, Nghiem, Maeda, and Ohno (2000), Nghiem, Sammon, and Kohse (2001), Kohse

and Nghiem (2004), and Fazelipour, Pope, and Sepehrnoori (2008).

Leontaritis and Mansoori (1988) and Mohammed, Arisaka, and Kumazaki (1998) provide re-

views of asphaltene issues in various fields. Kim, Boudh-Hir, and Mansoori (1990) provide a good

review of the role of asphaltenes in wettability alteration. Collins and Melrose (1983) and Yan,

Plancher, and Morrow (1997) describe experiments designed to measure wettability alteration with

asphaltene deposition. The following additional articles include the results of interesting experi-

ments related to asphaltene deposition in both clastic and carbonate rocks: Hirschberg, deJong,

Schipper, and Meijer (1984), Monger and Fu (1987), Monger and Trujillo (1991), Dubey and Wax-

man (1991), Minssieux (1997),Srivastava and Huang (1997), Srivastava, Huang, and Dong (1999),

Ali and Islam (1998), Nabzar, Aguilera, and Rajoub (2005), Broad, Al Binbrek, Neilson, and Gib-

34

son (2005), Oskui, Salman, Gholoum, Rashed, Al Matar, Al-Bahar, and Kahali (2006), Loahardjo,

Xie, and Morrow (2008), and Hashmi and Firoozabadi (2010).

35

CHAPTER 3

COMPOSITIONAL RESERVOIR SIMULATION OVERVIEW

The goal of this dissertation is to develop a practical, robust, computationally efficient compo-

sitional simulator with improved physics for CO2 flooding. This project consists of the following

three main topics.

1. Compositional simulation formulation

2. Formulation is amenable for parallel computing

3. Science of CO2 water-alternating-gas injection

• Evaluate generalized capillary pressure and relative permeability functional relationships

• Evaluate existing algorithms for three-phase relative permeability

• Evaluate existing and new algorithms for capillary pressure and relative permeability

hysteresis

• Evaluate existing and new algorithms for trapping of various phases, including compo-

sitional mixing associated with trapping

• Evaluate relative permeability and capillary pressure changes with wettability and in-

terfacial tension

• Account for CO2 phase behavior

– High solubility of CO2 in water phase

– Adjustments to equation of state

3.1 Compositional Simulation

This project involves the simulation of compositional fluid flow in porous media, as applied

to carbon dioxide (CO2) water-alternating-gas (WAG) injection. One of the earliest papers on

compositional simulation in the petroleum industry was Coats (1980). Chase and Todd (1984) was

an early paper on how to simulate CO2 injection. There are three main approaches for compositional

36

simulation formulations, as expressed by Wong et al. (1990), Acs et al. (1985), and Watts (1986).

Some of the details of the formulation used in this project are discussed in lecture notes from

Dr. Kazemi, (Kazemi, 2008a,b, 2009, 2010), and others were derived as part of my dissertation

work.

3.2 Commercial Simulators

There are several commercial reservoir simulators, but the commercial software with the largest

market share and which is most often used as a benchmark for simulation results is Eclipse, (Schlum-

berger, 2007a). The two primary manuals for Eclipse have a description of all of the available op-

tions in Schlumberger (2007a), and a more detailed technical description in Schlumberger (2007b).

Computer Modeling Group provides a suite of reservoir simulators; the CMG simulator applicable

to compositional simulation is “GEM”, CMG (2010). Landmark Graphics Corporation provides a

suite of reservoir simulators, including VIP core and VIP executive, Landmark (2000). In addition

to the commercial simulators, there are a number of proprietary reservoir simulators which have

been developed within large oil companies. Saudi Aramco’s “Gigapowers”, Dogru et al. (2008),

was designed form the beginning as a parallel reservoir simulator, with a specific goal to simulate

reservoirs with a large number of grid cells.

3.3 Mathematical Formulation

This project describes three-phase, compositional fluid flow in porous media, as applicable to

the oil and gas industry. The three phases considered are the oil phase, the gas phase, and the water

phase (also called the aqueous phase). Under normal conditions of pressure and temperature, all

three phases are immiscible with respect to each other, but under some conditions of temperature

and pressure the oil and gas phases become miscible.

The partial differential equations used to solve for compositional fluid flow are second order in

space and first order in time. This formulation uses Po, So, Sg, X1, . . . ,XNC−2, and Y1, . . . , YNC−2

as the primary variables1. NC is defined as the total number of components, including the H2O

component. This results in 2NC − 1 primary variables. There are NC component equations and

NC − 1 thermodynamic constraints.

1All variables are defined in Chapter 22.

37

Table 3.1 shows the distribution of the components in the three phases, illustrated with an

8-component system. The formulation used here accounts for the solubility of CO2 in the aque-

ous phase, but neglects the solubility of H2O in the oil and gas phases and the solubility of the

hydrocarbon components in the aqueous phase, since they are not expected to have a significant

effect in WAG injection problems. For Table 3.1, the primary variables are Po, So, Sg, and the pure

hydrocarbon components X1, . . . ,XNC−2, and Y1, . . . , YNC−2, or 2NC − 1 = 15 primary variables.

There are NC = 8 component equations, one for each of the composite hydrocarbon components,

one for CO2, and one for H2O. There are NC − 1 = 7 thermodynamic constraints, one for each of

the pure hydrocarbons and one for CO2. The Wm (the solubility of CO2 in water) are evaluated

explicitly for the spatial derivatives.

Table 3.1: Distribution of components in phases for NC = 8

component oil gas aqueous

C1 X1 Y1 0CI1 XI1 YI1 0CI2 XI2 YI2 0CH1 XH1 YH1 0CH2 XH2 YH2 0CH3 XH3 YH3 0

CO2 XCO2 YCO2 WCO2

H2O 0 0 WH2O

The following set of equations describes the differential equations used to solve for compositional

fluid flow in a porous medium, as used in the oil and gas industry. For each component (total NC),

the general partial differential equation for the component mass balance is in (3.1).

0.006328∇ ·(Xmξoλok(∇Po − γo∇D)

)+ 0.006328∇ ·

(Ymξgλgk(∇Po +∇Pcgo − γg∇D)

)+

0.006328∇ ·(Wmξwλwk(∇Po −∇Pcow − γw∇D)

)+(Xmξoqo + Ymξg qg +Wmξw qw

)=

∂

∂t

(φ(XmSoξo + YmSgξg +WmSwξw)

)(3.1)

The normalization constraints on each component are represented by (3.2)–(3.3). Because there

is no H2O in the hydrocarbon liquid or vapor phases, the upper limit of the sums are to NC − 1

not NC .

38

NC−1∑m

Xm = 1 =⇒ XNC−1 = 1−NC−2∑m′

Xm′ (3.2)

NC−1∑m

Ym = 1 =⇒ YNC−1 = 1−NC−2∑m′

Ym′ (3.3)

For the CO2 component, it is useful to recast (3.1) as (3.4). Use (3.2)–(3.3) to reduce the

degrees of freedom of the terms of (3.4).

0.006328∇ ·((1−

NC−2∑m′=1

Xm′)ξoλok(∇Po − γo∇D))+

0.006328∇ ·((1−

NC−2∑m′=1

Ym′)ξgλgk(∇Po +∇Pcgo − γg∇D))+


)+

((1 −

NC−2∑m′=1

Xm′)ξoqo + (1−NC−2∑m′=1

Ym′)ξg qg +Wmξw qw)=

∂

∂t

(φ((1 −

NC−2∑m′=1

Xm′)Soξo + (1 −NC−2∑m′=1

Ym′)Sgξg +WmSwξw))

(3.4)

For the H2O component, (3.1) simplifies to (3.5). It is useful to use WH2O +WCO2 = 1.

0.006328∇ ·((1−WCO2) ξwλwk(∇Po −∇Pcow − γw∇D)

)+((1−WCO2) ξw qw

)=

∂

∂t

(φ((1−WCO2)Swξw)

)(3.5)

3.4 Partially Implicit Formulation

Different primary variables can be evaluated at time n or iteration level �. The accumulation

terms are evaluated at time � for the new iteration and at n for the previous time step. In the

IMPES formulation, the pressure terms in the spatial derivatives are evaluated at � and all other

spatial and well variables are evaluated at n; some IMPES formulations evaluate the pressure in

the well terms at �. In the IMPSEC formulation, the pressure and saturation terms in the spatial

derivatives are evaluated at � and all other spatial and well variables are evaluated at n. In the

fully implicit formulation, all the primary variables in the spatial derivatives are evaluated at �.

39

3.4.1 IMPES

For the Implicit Pressure, Explicit Saturation (IMPES) formulation, the pressure is evaluated

at n+ 1 and the saturations and compositions are evaluated at time n. The finite difference form

of (3.1) using the IMPES formulation is as follows, (3.6).

0.006328∇ ·(Xn

mξnoμno

knrok#(∇Pn+1

o − γno∇D#)

)+

0.006328∇ ·(Y n

mξngμng

knrgk#(∇Pn+1

o +∇Pncgo − γn

g∇D#))+

0.006328∇ ·(Wn

mξnwμnw

knrwk#(∇Pn+1

o −∇Pncow − γn

w∇D#))+(Xn

mξno qno + Y n

mξng qng +Wn

mξnw qnw

)=

1

Δt

(φn+1(Xn+1

m Sn+1o ξn+1

o + Y n+1m Sn+1

g ξn+1g +Wn+1

m Sn+1w ξn+1

w ))−

1

Δt

(φn(Xn

mSno ξ

no + Y n

mSng ξ

ng +Wn

mSnwξ

nw))

(3.6)

3.4.2 IMPSEC

For the Implicit Pressure, Implicit Saturation, Explicit Composition (IMPSEC) formulation,

the pressure and saturations are evaluated at n+ 1 and the compositions are evaluated at time n.

The finite difference form of (3.1) using the IMPSEC formulation is as follows, (3.7).

0.006328∇ ·(Xn

mξnoμno

kn+1ro k#(∇Pn+1

o − γno∇D#)

)+

0.006328∇ ·(Y n

mξngμng

kn+1rg k#(∇Pn+1

o +∇Pn+1cgo − γn

g∇D#))+

0.006328∇ ·(Wn

mξnwμnw

kn+1rw k#(∇Pn+1

o −∇Pn+1cow − γn

w∇D#))+(Xn

mξno qno + Y n

mξng qng +Wn

mξnw qnw

)=

1

Δt

(φn+1(Xn+1

m Sn+1o ξn+1

o + Y n+1m Sn+1

g ξn+1g +Wn+1

m Sn+1w ξn+1

w ))−

1

Δt

(φn(Xn

mSno ξ

no + Y n

mSng ξ

ng +Wn

mSnwξ

nw))

(3.7)

3.4.3 Fully Implicit

For the Fully Implicit formulation, everything is evaluated at n+1. Our expectation is that the

Fully Implicit formulation will not be required for this project. The finite difference form of (3.1)

using the Fully Implicit formulation is as follows, (3.8).

40

0.006328∇ ·(Xn+1

m ξn+1o

μn+1o

kn+1ro k#(∇Pn+1

o − γn+1o ∇D#)

)+

0.006328∇ ·(Y n+1

m ξn+1g

μn+1g

kn+1rg k#(∇Pn+1

o +∇Pn+1cgo − γn+1

g ∇D#))+

0.006328∇ ·(Wn

mξnwμnw

kn+1rw k#(∇Pn+1

o −∇Pn+1cow − γn+1

w ∇D#))+

(Xn+1

m ξn+1o qn+1

o + Y n+1m ξn+1

g qn+1g +Wn+1

m ξn+1w qn+1

w

)=

1

Δt

(φn+1(Xn+1

m Sn+1o ξn+1

o + Y n+1m Sn+1

g ξn+1g +Wn+1

m Sn+1w ξn+1

w ))−

1

Δt

(φn(Xn

mSno ξ

no + Y n

mSng ξ

ng +Wn

mSnwξ

nw))

(3.8)

3.4.4 Comparison

The IMPES formulation is computationally very efficient. Using a banded solver with band-

width β = Nx × Nz and total number of grid cells Nxyz = Nx × Ny × Nz, the computational

order for the IMPES formulation is O [β2Nxyz

]. The IMPSEC formulation captures additional

variability in the saturations with the possibility of larger stable timesteps. The banded solver

for an IMPSEC algorithm has computational order O [(6β)2(3Nxyz)], or O [108× IMPES]. A fully

implicit algorithm is computationally inefficient. It would only be necessary if the compositional

gradient between grid cells were a significant driver for fluid flow between the grid cells. A fully

implicit algorithm has computational order O [(2 · (2NC − 1)β)2((2NC − 1)Nxyz)]. For NC = 8

components, this is O [(2)2(15)3β2Nxyz

], O [125 × IMPSEC], or O [13500 × IMPES].

3.5 Thermodynamic Constraints

There are NC − 1 thermodynamic constraints evaluated at time n + 1 which represent the

equilibrium conditions between the hydrocarbon liquid and vapor phases. The CO2 in the water

phase is evaluated explicitly using K-values.

fn+1om = fn+1

gm (3.9)

For the Peng-Robinson Equation of State (Peng and Robinson, 1976), this is defined as follows:

41

Xn+1m · Pn+1 · exp

[b#mbn+1l

(zn+1l − 1

)− ln(zn+1l −Bn+1

l

)+

− An+1l

2√2Bn+1

l

·(

2

an+1l

(∑n

Xn+1n a#mn

)− b#m

bn+1l

)· ln

(zn+1l +

(√2 + 1

)Bn+1

l

zn+1l − (√2− 1

)Bn+1

l

)]+

− Y n+1m · Pn+1 · exp

[b#mbn+1v

(zn+1v − 1

)− ln(zn+1v −Bn+1

v

)+

− An+1v

2√2Bn+1

v

·(

2

an+1v

(∑n

Y n+1n a#mn

)− b#m

bn+1v

)· ln

(zn+1v +

(√2 + 1

)Bn+1

v

zn+1v − (√2− 1

)Bn+1

v

)](3.10)

3.6 Typical Sizes

The formulation above involves three levels of iterations, assuming a direct matrix solve.

• Time loop n: time step sizes range from a few seconds to a maximum of roughly 30 days.

Total time ranges from a few years to about 150 years. Total time steps for a simulation are

typically between 100 and 1000.

• Time loop �: the progression to a new time step is an iterative process; typically this involves

between 3 and 15 iterations, with around 5 being typical.

• Flash loop iterations: this typically involves between 3 and 20 iterations, but there are some

cases near phase transitions which may require hundreds of iterations. Typical values are

probably around 5.

Multiplying out these typical values yields an expected value of 300×5×5 = 7, 500 solutions of

the matrix equation Aδ = R and 7500 ·Nxyz flash calculations. A starts out as a [(3NC) ·Nxyz]×[(3NC) · Nxyz] matrix. Some of the thermodynamic constraints have been evaluated explicitly in

this formulation, involving the simplification of A to a sparse [(2NC − 1) ·Nxyz]× [(2NC − 1) ·Nxyz]

matrix. This is further simplified into a [3Nxyz] × [3Nxyz] matrix for IMPSEC or a Nxyz × Nxyz

matrix for IMPES. The following are some typical values:

• NC , the total number of components, ranges from 4 to about 15, with 7–10 being typical.

Note that this is already simplified down from the hundreds of chemical components typically

present in a hydrocarbon system.

42

• Nxyz represents the total number of simulation grid cells. For a rectangular 3-D matrix,

Nxyz = Nx ∗Ny ∗Nz. The problem size is characterized by the following gradational scale for

Nxyz.

– Nxyz : (0, 104) most 1-D or 2-D problems and very small 3-D problems

– Nxyz : (104, 105) are considered small problems in industry. Problems through this size

are typically run in serial mode.

– Nxyz : (105, 106) are considered medium problems in industry. Problems through this

size are often run in serial mode, but sometimes run in parallel.

– Nxyz : (106, 107) are considered large or very large problems in industry, depending on

the hardware available. These are almost always run in parallel.

– Nxyz : (107, 109+) are considered very large problems. These are always run in parallel,

and only a few companies have simulators that can handle this size model. Saudi Aramco

ran their first billion cell model in the fall of 2008. They are actively developing software

to routinely run these billion cell models.

– Nxyz : 1012+: it is easy to define mathematically why models of 1012 or more grid cells

would be beneficial. For instance, if we have an oil field that is 10 km× 100 km× 100 m

and we split this into grid cells which are 1 m× 1 m× 0.1 m, this is 1012 grid cells. If we

consider basin modeling for a basin which is 1000 km× 1000 km× 10 km and simulate

this using a 100 m× 100 m× 1 m, this represents 1013 grid cells. Pore scale modeling of

a 10 cm × 10 cm × 10 cm block at a resolution of 1 μm× 1 μm × 1 μm represents 1012

grid cells.

If we use a medium sized problem with 105 grid cells and 8 components, a typical solution might

involve 7, 500 solves of the sparse matrix A of dimensions [(15) · (105)] × [(15) · (105)]. If we use a

naive dense matrix solution of O(N3), this represents approximately (1.5 ·106)3×7, 500 = 2.5 ·1022

FLOP. Fortunately, sparse matrix solves have a lower order than O(N3) (more details in final

report).

3.7 Off-Diagonal Terms

Off-diagonal terms have the following form, Figure 3.1.

43

Figure 3.1: Block 2: block geometry for the off-block diagonal values with the IMPES formulationfor a NC = 5 problem. Black represents non-zero values; gray represents zero values.

Po So Sg Xm′ Ym′

Cm X 0 0 0 0

Gm 0 0 0 0 0

(3.11)

Off-diagonal bands have the following form, here illustrated for i+ 1, j, k.


Cm DPmn[xyz

]t,i+1,jk

0 0 0 0

Gm 0 0 0 0 0

(3.12)

3.8 Well Terms

Figure 3.2: Block 4: well terms for the component equations for aNC = 5 problem. Black representsnon-zero values; gray represents zero values.

Well unknowns have the following form, Figure 3.2.

P �t,w|q�t,w

Cm X

Gm 0

(3.13)

Well unknowns have the following form.

44

P �t,w|q�t,w

Cm WDWmnijk

Gm 0

(3.14)

3.9 Right Hand Side

Figure 3.3: Block 6: right-hand-side terms for the component equations for a NC = 5 problem.Black represents non-zero values; gray represents zero values.

Right-hand-side, constant terms have the following form, Figure 3.3.

R

Cm X

Gm X

(3.15)

Right-hand-side, constant terms without well connections have the following form.

R

CmVRΔtAcc

m�ijk − VR

ΔtAccmnijk − DCmn�

xt,ijk −DCmn�yt,ijk −DCmn�

zt,ijk

Gm −fm�o,ijk + fm�

g,ijk

(3.16)

Right-hand-side, constant terms with well connections have the following form.

R

CmVRΔtAcc

m�ijk − VR


xt,ijk −DCmn�yt,ijk −DCmn�

zt,ijk +WCmn�ijk


g,ijk

(3.17)

Right-hand-side, constant terms above the bubble point or below the dew point without well

connections have the following form.

R

CmVRΔtAcc

m�ijk − VR


xt,ijk − DCmn�yt,ijk − DCmn�

zt,ijk


g,ijk

G′NC−1 −G�

NC−1,ijk

(3.18)

45

Right-hand-side, constant terms above the bubble point or below the dew point with well


R

CmVRΔtAcc

m�ijk − VR


xt,ijk − DCmn�yt,ijk − DCmn�

zt,ijk +WCmn�ijk


g,ijk

G′NC−1 −G�

NC−1,ijk

(3.19)

3.10 Total Rate Equations

Figure 3.4: Block 5: blocks for the well equations for a NC = 5 problem. Black represents non-zerovalues; gray represents zero values.

Total rate equations for each well have the following form, Figure 3.4.


Qw X 0 0 0 0(3.20)

Total rate equations for each well have the following form.


Qw QDPnijk 0 0 0 0

(3.21)

Diagonal terms for the total rate equations have the following form.

P �t,w|q�t,w

Qw X(3.22)


P �t,w|q�t,w

Qw QDWnijk

(3.23)

Right-hand-side, constant terms for the total rate equations have the following form.

R

Qw X(3.24)


46

R

Qw QCn�ijk

(3.25)

3.11 Accumulation

Define the accumulation term

Accmi =(φiξoiSoiXmi + φiξgiSgiYmi + φiξwiSwiWmi

)(3.26)

3.12 Accumulation Pressure Derivatives

For the normal hydrocarbon components,∂Acc�mi

∂P , for cell i and component m = 1 . . . NC − 2.

∂Acc�mi

∂P= ξ�oiS

�oiX

�mi

∂φ�i

∂P+ ξ�giS

�giY

�mi

∂φ�i

∂P+ φ�

iS�oiX

�mi

∂ξ�oi∂P

+ φ�iS

�giY

�mi

∂ξ�gi∂P

(3.27)

For the CO2 component,∂Acc�mi

∂P , for cell i and component m = NC − 1.

∂Acc�mi

∂P= ξ�oiS

�oiX

�mi

∂φ�i

∂P+ ξ�giS

�giY

�mi

∂φ�i

∂P+ ξ�wiS

�wiW

�mi

∂φ�i

∂P+

φ�iS

�oiX

�mi

∂ξ�oi∂P

+ φ�iS

�giY

�mi

∂ξ�gi∂P

+ φ�iS

�wiW

�mi

∂ξ�wi

∂P+ φ�

iξ�wiS

�wi

∂W �CO2,i

∂P(3.28)

For the H2O component,∂Acc�mi

∂P , for cell i and component m = NC .

∂Acc�mi

∂P= ξ�wiS

�wiW

�mi

∂φ�i

∂P+ φ�

iS�wiW

�mi

∂ξ�wi

∂P− φ�

iξ�wiS

�wi

∂W �CO2,i

∂P(3.29)

3.13 Accumulation Saturation Derivatives

Evaluate∂Acc�mi∂So

.

∂Acc�mi

∂So= φ�

iξ�oiX

�mi − φ�

iξ�wiW

�mi (3.30)

Evaluate∂Acc�mi∂Sg

.

∂Acc�mi

∂Sg= φ�

iξ�giY

�mi − φ�

iξ�wiW

�mi (3.31)

47

Above the bubble point, Sg = 0 and Sg → Pb becomes a new primary variable and∂Acc�mi∂Pb

= 0.

Below the dew point, So = 0 and So → Pd becomes a new primary variable and∂Acc�mi∂Pd

= 0

3.14 Accumulation Composition Derivatives

For the normal hydrocarbon component equations Cm = 1 . . . NC − 2 and m′ = 1 . . . NC − 2,

evaluate∂Acc�mi∂X′

m.

∂Acc�mi

∂Xm′= φ�

iS�oiX

�mi

∂ξ�oi∂Xm′

+ φ�iξ

�oiS

�oiδm,m′ (3.32)


evaluate∂Acc�mi∂Y ′

m.

∂Acc�mi

∂Ym′= φ�

iS�giY

�mi

∂ξ�gi∂Ym′

+ φ�iξ

�giS

�giδm,m′ (3.33)

For the CO2 component equation Cm = CNC−1 and m′ = 1 . . . NC − 2, evaluate∂Acc�mi∂X′

m.

∂Acc�CO2,i

∂Xm′= φ�

iS�oiX

�CO2,i

∂ξ�oi∂Xm′

− φ�iS

�oiξ

�oi (3.34)

For the CO2 component equation Cm = CNC−1 and m′ = 1 . . . NC − 2, evaluate∂Acc�mi∂Y ′

m.

∂Acc�mi

∂Ym′= φ�

iS�giY

�CO2

∂ξ�gi∂Ym′

− φ�iS

�giξ

�gi + φ�

iS�wiWCO2

∂ξ�wi

∂Ym′+ φ�

iS�wiξ

�wi

∂WCO2

∂Ym′(3.35)

For the water component equation Cm = CNCand m′ = 1 . . . NC − 2, evaluate

∂Acc�mi∂X′

m.

∂Acc�H2O,i

∂Xm′= 0 (3.36)


∂Acc�mi∂Y ′

m.

∂Acc�H2O,i

∂Ym′= φ�

iS�wiWH2O

∂ξ�wi

∂Ym′− φ�

iS�wiξ

�wi

∂WCO2

∂Ym′(3.37)

48

3.15 Pressure Spatial Derivatives

The following derivatives are written in terms of x and i± 1. The same approach applies to y

and j ± 1 and z and k ± 1.

The following are the multiples of δPi±1. All ± are either positive or negative for this equation.

DPmnxt,i±1 =

(Tmnxo,i± 1

2

+ Tmnxg,i± 1

2

+ Tmnxw,i± 1

2

)(3.38)

The following are the multiples of δPi.

DPmnxt,i = −

(DPmn

xt,i+1 + DPmnxt,i−1

)=

−(Tmnxo,i+ 1

2+ Tmn

xo,i− 12+ Tmn

xg,i+ 12+ Tmn

xg,i− 12+ Tmn

xw,i+ 12+ Tmn

xw,i− 12

)(3.39)

The following do not multiply deltas. All ± are either positive or negative for this equation.

DCmn�xt,i±1 = Tmn

xo,i± 12

·(P �i±1 − γn

o,i± 12

Di±1

)+ Tmn

xg,i± 12

·(P �i±1 − γn

g,i± 12

Di±1 + Pncgo,i±1

)+

Tmnxw,i± 1

2

·(P �i±1 − γn

w,i± 12

Di±1 − Pncow,i±1

)(3.40)

The following do not multiply deltas.

DCmn�xt,i = −Tmn

xo,i+ 12

·(P �i − γn

o,i+ 12

Di

)− Tmn

xo,i− 12

·(P �i − γn

o,i− 12

Di

)+

− Tmnxg,i+ 1

2

·(P �i − γn

g,i+ 12

Di + Pncgo,i

)− Tmn

xg,i− 12

·(P �i − γn

g,i− 12

Di + Pncgo,i

)+

− Tmnxw,i+ 1

2

·(P �i − γn

w,i+ 12

Di − Pncow,i

)− Tmn

xw,i− 12

·(P �i − γn

w,i− 12

Di − Pncow,i

)(3.41)

3.16 Fugacity Equations

The fugacities are defined by

fmo = Φmo XmP fmg = Φm

g YmP (3.42)

Evaluate∂fm�

oi∂P , m = 1 . . . NC − 1:

∂fm�oi

∂P= fm�

oi

(1

Φm�oi

∂Φm�oi

∂P

)+Φm�

oi Xm (3.43)

49

Evaluate∂fm�

gi

∂P , m = 1 . . . NC − 1:

∂fm�gi

∂P= fm�

gi

(1

Φm�gi

∂Φm�gi

∂P

)+Φm�

gi Y�mi (3.44)

For the normal hydrocarbon equations m = 1 . . . NC − 2, evaluate∂fo�gi∂Xm′ for m′ = 1 . . . NC − 2:

∂fm�oi

∂Xm′= fm�

oi

(1

Φm�oi

∂Φm�oi

∂Xm′

)+Φm�

oi Pδm,m′ (3.45)

For the normal hydrocarbon equations m = 1 . . . NC − 2, evaluate∂fm�

gi

∂P for m′ = 1 . . . NC − 2:

∂fm�gi

∂Ym′= fm�

gi

(1

Φm�gi

∂Φm�gi

∂Ym′

)+Φm�

gi Pδm,m′ (3.46)

For the CO2 equations m = NC − 1, evaluate∂fl�mi∂Xm′ for m′ = 1 . . . NC − 2:

∂fm�oi

∂Xm′= fm�

oi

(1

Φm�oi

∂Φm�oi

∂Xm′

)− Φm�

oi P (3.47)

For the CO2 equations m = 1 . . . NC − 1, evaluate∂fm�

gi

∂P for m′ = 1 . . . NC − 2:

∂fm�gi

∂Ym′= fm�

gi

(1

Φm�gi

∂Φm�gi

∂Ym′

)− Φm�

gi P (3.48)

3.17 Computation for Fixed Rate

Each component equation Cw,α,m has a source term. The coefficient of δP is

WDPmnw,α = −WI#w,α ·

(Xn

m,w,αξno,w,αλ

no,w,α + Y n

m,w,αξng,w,αλ

ng,w,α +W n

m,w,αξnw,w,αλ

nw,w,α

)(3.49)

The coefficient of δP �w is

WDWmnw,α = WI#w,α ·

(Xn

m,w,αξno,w,αλ

no,w,α + Y n

m,w,αξng,w,αλ

ng,w,α +W n

m,w,αξnw,w,αλ

nw,w,α

)(3.50)

The constant terms associated with the well are

50

WCmn�w,α = WI#w,α ·

(Xn

m,w,αξno,w,αλ

no,w,α + Y n

m,w,αξng,w,αλ

ng,w,α +W n

m,w,αξnw,w,αλ

nw,w,α

) ·((P �w,α

)−(P �,�w + Pw′,n

w,α

))(3.51)

Each well has a total rate equation. This equation has the following form for a fixed rate well.

The coefficient of δP is

QDPnw,α = −

(WI#w,α

)·(qemaxo,w,αξ

no,w,αλ

no,w,α

ξw,emaxo,w,αmax

+qemaxg,w,αξ

ng,w,αλ

ng,w,α

ξw,emaxg,w,αmax

+qemaxw,w,αξ

nw,w,αλ

nw,w,α

ξw,emaxw,w,αmax

)(3.52)


QDWnw,α =

αmax∑α′=1

(WI#w,α′

)×

(qemaxo,w,α′ξno,w,α′λn

o,w,α′

ξw,emaxo,w,αmax

+qemaxg,w,α′ξng,w,α′λn

g,w,α′

ξw,emaxg,w,αmax

+qemaxw,w,α′ξnw,w,α′λn

w,w,α′

ξw,emaxw,w,αmax

)(3.53)

The constant terms associated with the constant rate equation

QCn�w,α =

RHS︷︸︸︷q�t,w +

αmax∑α′=1

(WI#w,α′

)×((

P �w,α′

)−(P �,�w + Pw,const @ n

w,α′

))×


o,w,α′

ξw,emaxo,w,αmax


g,w,α′

ξw,emaxg,w,αmax


w,w,α′

ξw,emaxw,w,αmax

)(3.54)

3.18 Computation for Fixed Pressure

Each component equation Cw,α,m has a source term. This term has the following form for a

fixed pressure well. The coefficient of δP is 0.

WDPmnw,α = 0 (3.55)

The coefficient of δq�t,w is

WDWmnw,α =

(WI#w,α∑αmax

α′=1 WI#w,α′λnt,w,α′

)×

(Xn

m,w,αξno,w,αλ

no,w,α + Y n

m,w,αξng,w,αλ

ng,w,α +W n

m,w,αξnw,w,αλ

nw,w,α

)(3.56)

51


WCmn�w,α = −

(WI#w,α∑αmax


)×

(Xn

m,w,αξno,w,αλ

no,w,α + Y n

m,w,αξng,w,αλ

ng,w,α +W n

m,w,αξnw,w,αλ

nw,w,α

)(3.57)

Each well has a total rate equation. This equation has the following form for a fixed pressure

well. The coefficient of δP is

QDPnw,α =

(WI#w,α

)·(qemaxo,w,αξ

no,w,αλ

no,w,α

ξw,emaxo,w,αmax

+qemaxg,w,αξ

ng,w,αλ

ng,w,α

ξw,emaxg,w,αmax

+qemaxw,w,αξ

nw,w,αλ

nw,w,α

ξw,emaxw,w,αmax

)(3.58)


QDWnw,α = 1 (3.59)


QCn�w,α = −q�,�t,w −

αmax∑α′=1

(WI#w,α′

)·(P �w,α′ − Pw,n

w,α′

)·


o,w,α′

ξw,emaxo,w,αmax


g,w,α′

ξw,emaxg,w,αmax


w,w,α′

ξw,emaxw,w,αmax

)(3.60)

3.19 Additional Implicit Decisions

• IMPES Primary variables 1

– implicit: Po is evaluated at time n+ 1.

– δPo are computed directly from the matrix equation.

• IMPES Primary variables 2

– mixed: So, Sg are evaluated at time n for the spatial derivatives and at time n + 1 for

the time derivatives.

– mixed: Sw = 1− So − Sg is evaluated at time n for the spatial derivatives and at time

n+ 1 for the time derivatives.

52

– mixed: Xm, Ym are evaluated at time n for the spatial derivatives and at time n+1 for


– mixed: Wm is evaluated at time n for the spatial derivatives and at time n + 1 for the

time derivatives. Evaluate thermodynamics at time �.

– Zm = Wm +Xm + Ym

–∑

mXm = 1,∑

m Ym = 1,∑

mWm = 1

– Xm, Ym, Wm are computed using the local LU decomposition of matrix A

• IMPES Secondary variables 1

– explicit: kro(So, Sw, Sg), krg(So, Sw, Sg), krw(So, Sw, Sg), saturations are evaluated at

time n. Functional dependence on wettability, interfacial tension, trapping, and hystere-

sis are evaluated less often.

– explicit: Pcgo(So, Sw, Sg), Pcow(So, Sw, Sg), Pcgw(So, Sw, Sg), saturations are evaluated

at time n. Functional dependence on wettability, interfacial tension, trapping, and

hysteresis are evaluated less often. Assume time derivatives of pressure refer to Po.

– mixed: ξo(P,Xm), ξg(P, Ym), ξw(P,Wm) are evaluated at time n for the spatial deriva-

tives and time n+ 1 for the time derivatives.

– mixed: Co(P ), Cg(P ) are evaluated at time n for the spatial derivatives and time n+ 1

for the time derivatives.

– All kr and all Pc and their derivatives are computed using their own functions

– All ξ, Co, and Cg are computed from the flash computation.

• IMPSEC Primary variables 1

– implicit: Po is evaluated at time n+ 1.

– implicit: So, Sg are evaluated at time n+ 1.

– implicit: Sw = 1− So − Sg is evaluated at time n+ 1.

– δPo, δSo, and δSg are computed directly from the matrix equation.

• IMPSEC Primary variables 2

53

– mixed: Xm, Ym are evaluated at time n for the spatial derivatives and at time n+1 for


– mixed: Wm is evaluated at time n for the spatial derivatives and at time n + 1 for the

time derivatives. Evaluate thermodynamics at time �.

– Zm = Wm +Xm + Ym

–∑

mXm = 1,∑

m Ym = 1,∑

mWm = 1

– Xm, Ym, Wm are computed using the local LU decomposition of matrix A

• IMPSEC Secondary variables 1

– implicit: kro(So, Sw, Sg), krg(So, Sw, Sg), krw(So, Sw, Sg), saturations are evaluated at

time n + 1. Functional dependence on wettability, interfacial tension, trapping, and

hysteresis are evaluated less often.

– implicit: Pcgo(So, Sw, Sg), Pcow(So, Sw, Sg), Pcgw(So, Sw, Sg), saturations are evaluated

at time n + 1. Functional dependence on wettability, interfacial tension, trapping, and

hysteresis are evaluated less often. Assume time derivatives of pressure refer to Po.

– mixed: ξo(P,Xm), ξg(P, Ym), ξw(P,Wm) are evaluated at time n for the spatial deriva-

tives and time n+ 1 for the time derivatives.

– mixed: Co(P ), Cg(P ) are evaluated at time n for the spatial derivatives and time n+ 1

for the time derivatives.

– All kr and all Pc and their derivatives are computed using their own functions.

– All ξ, Co, and Cg are computed from the flash computation.

• Secondary variables 2

– explicit, once per time step: μo(P,Xm), μg(P, Ym), μw(P,Wm) are evaluated at time n

since the viscosity does not change rapidly for small pressure changes.

– explicit, once per time step: γo(P,Xm, ξo,MWo), γg(P, Ym, ξg,MWg), γw(P,Wm, ξw,MWw)

are evaluated at time n since the specific gravity does not change rapidly for small pres-

sure changes.

54

– explicit, once per time step: Upstream weighting is evaluated at time n. The cell which

is upstream of another cell is used for many of the fluid properties, but the determination

of which cells are upstream is computed at most once per time step for every cell.

– explicit, once per time step: Source and sink terms are evaluated at time n.

– γ and μ are evaluated using their own functions after completion of a flash computation.

– Upstream weighting and source and sink terms are evaluated in their own functions.

• Tertiary variables - constant in this formulation

– constant: k, φ: permeability and porosity may be time dependent at n with asphaltene

deposition, but are constant in this formulation.

– constant: Cw, Cφ are not defined as functions of pressure for this formulation.

– constant: D - gravity does not vary with time

• Other considerations

– explicit, at most once per time step: Swt, Sgt, Sot are evaluated at time n or less

frequently if possible. Changes in the trapped oil, water, and gas are evaluated at most

once per time step for every cell.

– explicit, at most once per time step: k, φ are evaluated at time n. Changes in k and φ

are a result of solid deposition, adsorption, or dissolution. These reactions occur at most

once per time step per cell, but may be less frequent. Note that the compressibility of

the matrix Cφ is handled separately.

– explicit, at most once per time step: Relative permeability hysteresis is evaluated at

time n or less frequently if possible. This means that whether to use the increasing or

decreasing curve is determined at most once per time step for a cell.

– explicit, at most once per time step: Capillary pressure hysteresis is evaluated at time n

or less frequently if possible. This means that whether to use the increasing or decreasing

curve is determined at most once per time step for a cell.

– explicit, at most once per time step: Wettability changes are evaluated at time n or less

frequently if possible.

55

– explicit, at most once per time step: Pressure dependence of relative permeability (re-

lated to wettability and miscibility changes) is evaluated at time n or less frequently if

possible. These are treated in a similar way to hysteresis curves.

– explicit, at most once per time step: Pressure dependence of capillary pressure (related

to wettability and miscibility changes) is evaluated at time n or less frequently if possible.

These are treated in a similar way to hysteresis curves.

– explicit, at most once per time step: Adsorption is evaluated at time n or less frequently

if possible.

56

CHAPTER 4

MATHEMATICAL FORMULATION OVERVIEW

The basic equation for each component is:

Cm=1...NC ,m1 : 0.006328 VR ∇ ·(Xn

mm1ξnom1

knrom1

μnom1

k#m1(∇P �+1

om1− γn

om1∇D#)

)+

0.006328 VR ∇ ·(Y n

mm1ξngm1

knrgm1

μngm1

k#m1(∇P �+1

om1+∇Pn

cgom1− γn

gm1∇D#)

)+

0.006328 VR ∇ ·(Wn

mm1ξnwm1

knrwm1

μnwm1

k#m1(∇P �+1

om1−∇Pn

cowm1− γn

wm1∇D#)

)+(

Xnmm1

ξnom1q�+1om1

+ Y nmm1

ξngm1q�+1gm1

+Wnmm1

ξnwm1q�+1wm1

)− τ �+1mm1/m2

=

VR

Δt

(φ�+1m1

X�+1mm1

S�+1om1

ξ�+1om1

+ φ�+1m1

Y �+1mm1

S�+1gm1

ξ�+1gm1

+ φ�+1m1

W �+1mm1

S�+1wm1

ξ�+1wm1

)−

VR

Δt

(φnm1Xn

mm1Snom1

ξnom1+ φn

m1Y nmm1

Sngm1

ξngm1+ φn

m1Wn

mm1Snwm1

ξnwm1

)(4.1)

(5.40) represents the m2 pore system.

Cm=1...NC ,m2 : τ �+1mm1/m2

=

VR

Δt

(φ�+1m2

X�+1mm2

S�+1om2

ξ�+1om2

+ φ�+1m2

Y �+1mm2

S�+1gm2

ξ�+1gm2

+ φ�+1m2

W �+1mm2

S�+1wm2

ξ�+1wm2

)−

VR

Δt

(φnm2Xn

mm2Snom2

ξnom2+ φn

m2Y nmm2

Sngm2

ξngm2+ φn

m2Wn

mm2Snwm2

ξnwm2

)(4.2)

The m1/m2 transfer function is defined by:

τ �+1mm1/m2

= 0.006328 VR σ#m1/m2

k#m1/m2

(P �+1om1− P �+1

om2

)×(Xup,n

mm1/m2ξup,nom1/m2

kup,nrom1/m2

μup,nom1/m2

+Y up,nmm1/m2

ξup,ngm1/m2kup,nrgm1/m2

μup,ngm1/m2

+W up,n

mm1/m2ξup,nwm1/m2

kup,nrwm1/m2

μup,nwm1/m2

)(4.3)

And the thermodynamic constraints are:

Gm=1...NC−1,m1 : f�+1om,m1

− f�+1gm,m1

= 0 (4.4)

Gm=1...NC−1,m2 : f�+1om,m2

− f�+1gm,m2

= 0 (4.5)

57

4.1 Primary Variables

The formulation used here is an isothermal formulation; this means the temperature is constant

for the simulation run. The temperature T is measured in ◦F and converted as appropriate to

◦C, K, or R. The phase behavior, viscosity, density, solubility, capillary pressure, and relative

permeability all change with temperature.

The formulation used here assumes that the salinity remains constant within a simulation

run. The aqueous density and CO2 solubility change with the salinity of the water. Salinity is

represented as an equivalent mole fraction of NaCl, WNaCl, and converted as needed to a mass

fraction, molarity, or molality. Most of the experiments with brines first measure the properties

with a certain salt concentration and then measure the solubility or density changes of an additional

component separately. Both reservoir brines and seawater are dominated by Na and Cl; variations

in the composition of the salts only causes a small change in the solubility. As a result, the salinity

specified is relative to an equivalent system with H2O and NaCl only.

The pressures (measured in psia) in the oil, gas, and water phases change as a function of time

and space. After discretization, the mobile oil phase pressure Pnom1,ijk

or P �om1,ijk

and the trapped

oil phase pressure Pnom2,ijk

or P �om2,ijk

are stored for each grid cell, for the current time step n, and

for the current nonlinear iteration �. The gas phase pressure Pg,m1 is expanded using the gas-oil

capillary pressure:

Pgm1 − Pom1 = Pcgo[Sot, Sgt, Swt] (4.6)

The gas phase pressure Pg,m2 is expanded using the gas-oil capillary pressure:

Pgm2 − Pom2 = Pcgo[Sot, Sgt, Swt] (4.7)

The water phase pressure Pw,m1 is expanded using the oil-water capillary pressure:

Pom1 − Pwm1 = Pcow[Sot, Sgt, Swt] (4.8)

The water phase pressure Pw,m2 is expanded using the oil-water capillary pressure:

Pom2 − Pwm2 = Pcow[Sot, Sgt, Swt] (4.9)

58

The saturations (measured as a volume fraction) in the oil, gas, and water phases change as a

function of time and space. After discretization, the water saturation in the mobile system Snwm1,ijk

or S�wm1,ijk

and the water saturation in the trapped system Snwm2,ijk

or S�wm2,ijk

are stored for each

grid cell, for the current time step n, and for the current nonlinear iteration �. The oil saturation

in the mobile system Snom1,ijk

or S�om1,ijk

and the oil saturation in the trapped system Snom2,ijk

or

S�om2,ijk

are stored for each grid cell, for the current time step n, and for the current nonlinear

iteration �. The sum of the mobile saturations is equal to 1:

So,m1 + Sg,m1 + Sw,m1 = 1 (4.10)

The sum of the trapped saturations is equal to 1:

So,m2 + Sg,m2 + Sw,m2 = 1 (4.11)

There are several degenerate cases that need to be considered.

1. All phases are present; store Sw and So and calculate Sg = 1− Sw − So as needed.

2. The gas saturation Sg = 0; store Sw and calculate So = 1− Sw as needed.

3. The oil saturation So = 0; store Sw and calculate Sg = 1− Sw as needed.

4. The water saturation Sw = 1, the oil saturation So = 0, and the gas saturation Sg = 0.

5. The water saturation Sw = 0; store the oil saturation So and calculate the gas saturation

Sg = 1− So as needed.



The mole fractions (measured as a fraction of lbmol) of each component in the oil, gas, and water

phases change as a function of time and space. For this work, only the CO2 and H2O components

are present in the water phase; the oil and gas phases do not contain any H2O. For a system with

NC = 5 components, there are three hydrocarbon components, one component for CO2, and one

component for H2O; refer to Table 4.1.

59

The mole fractions of each component in the mobile oil phase sum to 1. The mole fractions

X1,m1 , X2,m1 , and X3,m1 are stored and XCO2,m1 is calculated when needed.

XCO2,m1 = 1−X1,m1 −X2,m1 −X3,m1 (4.12)

The mole fractions of each component in the trapped oil phase sum to 1. The mole fractions X1,m2 ,

X2,m2 , and X3,m2 are stored and XCO2,m2 is calculated when needed.

XCO2,m2 = 1−X1,m2 −X2,m2 −X3,m2 (4.13)

The mole fractions of each component in the mobile gas phase sum to 1. The mole fractions

Y1,m1 , Y2,m1 , and Y3,m1 are stored and YCO2,m1 is calculated when needed.

YCO2,m1 = 1− Y1,m1 − Y2,m1 − Y3,m1 (4.14)

The mole fractions of each component in the trapped gas phase sum to 1. The mole fractions Y1,m2 ,

Y2,m2 , and Y3,m2 are stored and YCO2,m2 is calculated when needed.

YCO2,m2 = 1− Y1,m2 − Y2,m2 − Y3,m2 (4.15)

The mole fractions of each component in the mobile water phase sum to 1. The mole fraction

WCO2,m1 is stored and WH2O,m1 is calculated when needed.

WH2O,m1 = 1−WCO2,m1 (4.16)

The mole fractions of each component in the trapped water phase sum to 1. The mole fraction

WCO2,m1 is stored and WH2O,m2 is calculated when needed.

WH2O,m2 = 1−WCO2,m2 (4.17)

For the formulations used here, WCO2 is calculated as a function of P , T , WNaCl, XCO2 , and YCO2

as needed.All of the primary variables after simplification are listed in Table 4.2. The two primary variables

which do not depend on the spatial location, T and WNaCl are stored on each processor. Thevariables which depend on the spatial location are stored in PetSc distributed arrays, includingboth the local values and the “ghost” values for the adjacent grid cells. PetSc automatically handles

60

Table 4.1: Distribution of components in phases for NC = 5

component oil gas aqueous

C{L}ight or C1 or CH4 X1 Y1 0

C{I}ntermediate or C2 or nC4 X2 Y2 0

C{H}eavy or C3 or nC10 X3 Y3 0

C4 or CO2 X4 Y4 W4

C5 or H2O 0 0 W5

the communication of the ghost properties. There are three arrays of primary variables; one fortime n (DA primary n), one for time � (DA primary ell), and one for the best iteration value from1..� (DA primary best ell) in case the nonlinear iterations fail to converge. Each primary variabledistributed array includes Pom1 , Swm1 , Som1 , WCO2,m1 , Xm′,m1 , Ym′,m1 , Pom2 , Swm2 , Som2 , WCO2,m2 ,Xm′,m2 , and Ym′,m2 .

Table 4.2: Primary variables

variable units name

T# ◦F Constant temperature.

W#NaCl lbmol/lbmol Constant salinity.

Pno,m1,ijk

psia Pressure in mobile oil phase at time n for every grid cell.

P �o,m1,ijk

psia Pressure in mobile oil phase at nonlinear iteration � for everygrid cell.

Pno,m2,ijk

psia Pressure in trapped oil phase at time n for every grid cell.

P �o,m2,ijk

psia Pressure in trapped oil phase at nonlinear iteration � for everygrid cell.

Snw,m1,ijk

ft3/ft3 Saturation in mobile water phase at time n for every grid cell.

S�w,m1,ijk

ft3/ft3 Saturation in mobile water phase at nonlinear iteration � forevery grid cell.

Snw,m2,ijk

ft3/ft3 Saturation in trapped water phase at time n for every grid cell.

S�w,m2,ijk

ft3/ft3 Saturation in trapped water phase at nonlinear iteration � forevery grid cell.

Sno,m1,ijk

ft3/ft3 Saturation in mobile oil phase at time n for every grid cell.

S�o,m1,ijk

ft3/ft3 Saturation in mobile oil phase at nonlinear iteration � for everygrid cell.

Sno,m2,ijk

ft3/ft3 Saturation in trapped oil phase at time n for every grid cell.

S�o,m2,ijk

ft3/ft3 Saturation in trapped oil phase at nonlinear iteration � forevery grid cell.

Xnm′,m1,ijk

lbmol/lbmol Mole fraction in mobile oil phase for componentm′ = 1 . . . NC − 2 at time n for every grid cell.

X�m′,m1,ijk

lbmol/lbmol Mole fraction in mobile oil phase for componentm′ = 1 . . . NC − 2 at nonlinear iteration � for every grid cell.

Continued.

61

Table 4.2: Continued.

Table 4.2: Primary variables (continued)

variable units name

Xnm′,m2,ijk

lbmol/lbmol Mole fraction in trapped oil phase for componentm′ = 1 . . . NC − 2 at time n for every grid cell.

X�m′,m2,ijk

lbmol/lbmol Mole fraction in trapped oil phase for componentm′ = 1 . . . NC − 2 at nonlinear iteration � for every grid cell.

Y nm′,m1,ijk

lbmol/lbmol Mole fraction in mobile gas phase for componentm′ = 1 . . . NC − 2 at time n for every grid cell.

Y �m′,m1,ijk

lbmol/lbmol Mole fraction in mobile gas phase for componentm′ = 1 . . . NC − 2 at nonlinear iteration � for every grid cell.

Y nm′,m2,ijk

lbmol/lbmol Mole fraction in trapped gas phase for componentm′ = 1 . . . NC − 2 at time n for every grid cell.

Y �m′,m2,ijk

lbmol/lbmol Mole fraction in trapped gas phase for componentm′ = 1 . . . NC − 2 at nonlinear iteration � for every grid cell.

W nCO2,m1,ijk

lbmol/lbmol Mole fraction of CO2 in the mobile aqueous phase at time n forevery grid cell.

W �CO2,m1,ijk

lbmol/lbmol Mole fraction of CO2 in the mobile aqueous phase at nonlineariteration � for every grid cell.

W nCO2,m2,ijk

lbmol/lbmol Mole fraction of CO2 in the trapped aqueous phase at time nfor every grid cell.

W �CO2,m2,ijk

lbmol/lbmol Mole fraction of CO2 in the trapped aqueous phase atnonlinear iteration � for every grid cell.

4.2 Secondary Variables

Secondary variables are calculated as a function of the primary variables. The following sec-

ondary variables appear directly in the partial differential equations or the IMPES finite difference

expansion of the partial differential equations.

4.2.1 Calculation of Secondary Variables

WCO2 may be calculated as a primary variable or a secondary variable, but here WCO2 is

calculated as a secondary variable as a function of T , WNaCl, Po,m1,ijk, YCO2 , and possibly XCO2 .

The derivatives∂WCO2

∂P ,∂WCO2∂Xm′ , and

∂WCO2∂Ym′ are evaluated analytically from the derivatives of the

correlations. In a three-phase system, the CO2 may partition between the water, oil, and gas

phases. The gas-oil partitioning is handled by a normal two-phase flash calculation. The gas-water

partitioning is handled by a CO2 solubility computation; the gas-water solubility used here is the

model by Duan and Sun (2003).

62

For a water-oil-gas system, several different possibilities are available to use with the Duan and

Sun (2003) correlation; option (4.18) is used here.

WCO2 = F [P, T,WNaCl, YCO2 ] (4.18)

WCO2 = αF [P, T,WNaCl, YCO2 ] + (1− α)F [P, T,WNaCl,XCO2 ] (4.19)

WCO2 = WCO2 [P, T,WNaCl, ZCO2 ] (4.20)

For a two-phase oil-water system, several different possibilities are available to use in the Duan and

Sun (2003) correlation; option (4.21) was found to work best.

WCO2 = F [Pb, T,WNaCl, YCO2 [Pb]] (4.21)

WCO2 = F [P, T,WNaCl, YCO2 [Pb]] (4.22)

WCO2 = αF [P, T,WNaCl, YCO2 [Pb]] + (1− α)F [P, T,WNaCl,XCO2 ] (4.23)

WCO2 = F [P, T,WNaCl, ZCO2 ] (4.24)

WCO2 = F [P, T,WNaCl, YCO2 = 0] =⇒ WCO2 = 0 (4.25)

Unfortunately, for both the oil-water system and the water-oil-gas system insufficient experimen-

tal data is available to decide between the different choices of CO2 models. A three-phase flash

calculation based on an equation of state like Peng-Robinson could also be used to represent the

CO2 partitioning in any of these systems, but the three-phase flash would also require additional

experimental data to calibrate the model.

The molar density in the oil and gas phases, ξo and ξg, are calculated as part of the Peng-

Robinson equation of state flash for a gas-oil system. The oil density ξo is a function of P , T , and

Xm. The gas density ξg is a function of P , T , and Ym. The derivatives ∂ξo∂P , ∂ξo

∂Xm′ ,∂ξg∂P , and

∂ξg∂Ym′

are evaluated using analytical derivatives of the Peng-Robinson equation of state. The oil specific

gravity γo[psi/ft] is calculated from ξo:

γo =ξoMWo

144(4.26)

The gas specific gravity γg[psi/ft] is calculated from ξg:

γg =ξgMWg

144(4.27)

63

The molar density in the aqueous phase, ξw, is calculated as a function of P , T , WNaCl, and

WCO2 . If WCO2 is a primary variable, the derivatives ∂ξw∂P and ∂ξw

∂WCO2are evaluated using analytical

derivatives of the correlations. If WCO2 is a secondary variable, the derivatives ∂ξw∂P , ∂ξw

∂Xm′ ,∂ξw∂Ym′ are

evaluated using analytical derivatives of the correlations. The water specific gravity γw[psi/ft] is

calculated from ξw:

γw =ξwMWw

144(4.28)

The total porosity φ or φt changes as a function of Pom1 . The mobile porosity and trapped

porosity φm1 and φm2 change as the trapping changes. The ratios φm1/φt and φm2/φt remain

constant except when the trapping changes. The derivatives ∂φ∂P ,

∂φm1∂P , and

∂φm2∂P are evaluated

using analytical derivatives. The total saturations are defined as follows:

Sot = (Som1φm1 + Som2φm2) /φt (4.29)

Sgt = (Sgm1φm1 + Sgm2φm2) /φt (4.30)

Swt = (Swm1φm1 + Swm2φm2) /φt (4.31)

The relative permeabilities and capillary pressures in both the mobile and trapped systems are

calculated as a function of the total saturations, Swt, Sot, Sgt, not the mobile or trapped saturation

only. This means that

krw = krwm1 = krwm2 (4.32)

kro = krom1 = krom2 (4.33)

krg = krgm1 = krgm2 (4.34)

Pcgo = Pcgom1 = Pcgom2 (4.35)

Pcow = Pcowm1 = Pcowm2 (4.36)

The relative permeability and capillary pressures are assumed to be representative of the initial

reservoir pressure and temperature. The water relative permeability krw is a function of Swt, Sot,

Sgt, and the saturation history of the grid cell. The oil relative permeability kro is a function of Swt,

Sot, Sgt, and the saturation history of the grid cell. The gas relative permeability krg is a function

of Swt, Sot, Sgt, and the saturation history of the grid cell. The gas-oil relative permeability Pcgo

is a function of Swt, Sot, Sgt, and the saturation history of the grid cell. The oil-water relative

64

permeability Pcow is a function of Swt, Sot, Sgt, and the saturation history of the grid cell.

The fugacity in the oil phase fom and the derivatives of fugacity ∂fom∂P and ∂fom

∂Xm′ are calculated

using the Peng-Robinson equation of state. The fugacity and fugacity derivatives are functions of

P and Xm′ . The fugacity in the gas phase fgm and the derivatives of fugacity∂fgm∂P and ∂fgm

∂Ym′ are

calculated using the Peng-Robinson equation of state. The fugacity and fugacity derivatives are

functions of P and Ym′ .

4.2.2 Storage of Secondary Variables

Secondary variables which are different in each grid cell are stored as PetSc distributed arrays.

There are different arrays for values stored at time n, nonlinear iteration �, and values which

don’t change with time. Arrays are also divided based on whether they contain “ghost” cells for

neighboring processors and whether they use ghost cells from other arrays. The arrays are also

split based on when the values need to be computed and used. The following list describes the

different distributed arrays in calculation order.

1. Initialization: values which do not change with time

1.1. DA notime, with ghost cells; these properties change with the grid cell but do not change

with time. Includes k, km1/mtwo, σm1/m2, D, cφ, Δx, Δy, Δz, and the constant portion

of the transmissibilities TC. The different hysteresis curves for relative permeability and

capillary pressure are also defined. See Table 4.3.

2. Update at time n

2.1. DA primary n, with ghost cells; the primary variables at n.

2.2. DA before TRANS n, no ghost cells; these values change when when the water, oil, or gas

saturation direction of individual grid cells changes to increasing or decreasing. Includes

the properties needed to calculate the relative permeability and capillary pressure curves

such as the saturation direction, endpoint saturations, maximum historical saturations,

and curvature. These values are calculated based on DA primary n and DA notime.

2.3. DA cell only n, no ghost cells; these properties are calculated at time n for every grid cell.

They are required by the jacobian calculation but are not required by the transmissibility

65

calculation. Includes the mobile and trapped φ, fom, and fgm. These properties are

calculated based on DA primary n and DA notime. See Table 4.4.

2.4. DA for TRANS n, with ghost cells; these values are needed for the transmissibility cal-

culations. Includes the mobile and trapped ξo, ξg, ξw, γo, γg, γw, μo, μg, μw, kro, krg,

krw, Pcow, and Pcgo. These properties are calculated based on DA primary n, DA notime,

DA cell only n, and DA before TRANS n. After the local properties are calculated, the

ghost values are communicated to the neighboring processors. See Table 4.5.

2.5. DA TRANS n, no ghost cells; the transmissibilities themselves are local but depend on

the local and ghost values of DA primary n, DA notime, and DA for TRANS n. Includes

the upstream potential Ψup, the upstream weighted specific gravities γup, the inter-grid

transmissibility Tm1/m1, and the intra-grid transmissibility Tm1/m2

.

2.6. DA jacobian n, no ghost cells; these are the jacobian values at time n. The portion of

the jacobian for each grid cell is a two-dimensional array. The 4NC rows of the array

represent each of the component equations and each of the thermodynamic equations for

the mobile system m1 and the trapped system m2. For a 7-point finite difference stencil

with single completion wells, the 4NC + 7 columns of the array represent the primary

variables, the mobile pressures at the adjacent grid cells, and the right-hand-side of the

jacobian equation corresponding to the grid cell. In degenerate cases or cases without

trapping, a portion of the local jacobian matrix is the identity matrix. DA jacobian n

is calculated based on DA primary n, DA notime, DA for TRANS n, DA TRANS n, and

DA cell only n.

2.7. DA after TRANS n, no ghost cells; the values used in the flash computation for the

primary variables. Includes the mobile and trapped values of Um, α, β, and Z2ph,m.

These properties are calculated based on DA primary n, DA notime, DA for TRANS n,

DA TRANS n, DA cell only n, and DA jacobian n.

3. Update at nonlinear iteration �

3.1. DA primary ell, with ghost cells; the primary variables at �.

66

3.2. DA cell only ell, no ghost cells; the secondary variables evaluated at �. This includes: the

mobile and trapped water saturation at the previous nonlinear iteration S�−1w ; the mobile

and trapped values of φ, GCO2 , Gmax, ξo, ξg, ξw, fom, fgm; and the derivatives of ξo, ξg,

ξw, WCO2 , fom, fgm, and Gmax. These properties are calculated based on DA primary ell

and DA notime. See Table 4.6.

3.3. DA jacobian ell, no ghost cells; these are the jacobian values at time �. The portion of

the jacobian for each grid cell is a two-dimensional array. The 4NC rows of the array

represent each of the component equations and each of the thermodynamic equations for

the mobile system m1 and the trapped system m2. For a 7-point finite difference stencil

with single completion wells, the 4NC + 7 columns of the array represent the primary

variables, the mobile pressures at the adjacent grid cells, and the right-hand-side of the

jacobian equation corresponding to the grid cell. In degenerate cases or cases without

trapping, a portion of the local jacobian matrix is the identity matrix. DA jacobian ell

is calculated based on DA primary ell, DA notime, DA jacobian n, and DA cell only ell.

3.4. DA solution ell, no ghost cells; the solution vector at �. Although the solution vector

contains all the primary variables as a result of LU-decomposition, some of them may

be degenerate and some are calculated by flash. When not degenerate, the pressures

Pom1 and Pom2 , water saturations Swm1 and Swm2 , and primary WCO2,m1 and WCO2,m2

are calculated from the solution vector. DA solution ell is calculated by the solver based

on DA jacobian ell.

3.5. DA primary best ell, with ghost cells; the value of the primary variables at the best of

the nonlinear iterations 1 . . . �.

4.2.3 List of Secondary Variables

Table 4.3: Secondary variables which do not vary with time, DA notime

variable units name

Δtn day Time step size for time n.

VR#ijk ft3 Rock volume for each grid cell; does not change with time.

Continued.

67


Table 4.3: Secondary variables which do not vary with time, DA notime (continued)

variable units name

D#ijk ft Depth to midpoint of each grid cell; does not change with

time.

k#ijk md Permeability of each grid cell; does not change with time.

k#m1,ijkmd Permeability of mobile system for each grid cell; does not

change with time.

k#m1/m2,ijkmd Permeability of transfer from trapped system to mobile

system for each grid cell; does not change with time.

k#xx,ijk md Permeability in the x or i direction of each grid cell; doesnot change with time.

k#yy,ijk md Permeability in the y or j direction of each grid cell; doesnot change with time.

k#zz,ijk md Permeability in the z or k direction of each grid cell; doesnot change with time.

σ#m1/m2,ijk

1/ft2 Shape factor for transfer between mobile and trappedsystem for each grid cell; does not change with time.

WI#wft3/daypsia/cp Well index for each well; does not change with time.

Table 4.4: Secondary variables at n which are not needed for the transmissibility calculations,DA cell only n

variable units name

φnt,ijk ft3 pore/ft3 rock Porosity at time n for every grid cell. It is a function of

Pno,m1,ijk

.

φnm1,ijk

ft3 mobile/ft3 rock Mobile pore fraction at time n for every grid cell. It isa function of Pn

o,m1,ijk.

φnm2,ijk

ft3 trapped/ft3 rock Trapped pore fraction at time n for every grid cell. It isa function of Pn

o,m1,ijk.

Snwt,ijk ft3 water/ft3 pore Total water saturation at time n for every grid cell. It

is a function of φnt,ijk, φ

nm1,ijk

, φnm2,ijk

, Snwm1,ijk

, andSnwm2,ijk

.

Snot,ijk ft3 oil/ft3 pore Total oil saturation at time n for every grid cell. It is a

function of φnt,ijk, φ

nm1,ijk

, φnm2,ijk

, Snom1,ijk

, and Snom2,ijk

.

Sngt,ijk ft3 gas/ft3 pore Total gas saturation at time n for every grid cell. It is a

function of φnt,ijk, φ

nm1,ijk

, φnm2,ijk

, Sngm1,ijk

, and Sngm2,ijk

.

MWno,m1,ijk lbm/lbmol Molecular weight of mobile oil phase at time n for

every grid cell. It is a function of Xnm,m1,ijk

and MW#m.

MWng,m1,ijk lbm/lbmol Molecular weight of mobile oil phase at time n for

every grid cell. It is a function of Y nm,m1,ijk

and MW#m.

Continued.

68


Table 4.4: Secondary variables at n which are not needed for the transmissibility calculations,DA cell only n (continued)

variable units name

MWnw,m1,ijk lbm/lbmol Molecular weight of mobile oil phase at time n for every

grid cell. It is a function of W#NaCl W

nm,m1,ijk

and MW#m.

fnom,m1,ijkpsi Mobile oil phase fugacity for component m at time n

for every grid cell. It is a function of T , Pno,m1,ijk

andXn

1...NC−1,m1,ijk.

fnom,m2,ijkpsi Trapped oil phase fugacity for component m at time n


andXn

1...NC−1,m2,ijk.

fngm,m1,ijkpsi Mobile gas phase fugacity for component m at time n


andY n1...NC−1,m1,ijk

.

fngm,m2,ijkpsi Trapped gas phase fugacity for component m at time n


andY n1...NC−1,m2,ijk

.

Table 4.5: Secondary variables at n which are needed for the transmissibility calculations, DA forTRANS n

variable units name

ξno,m1,ijklbmol/ft3 Molar density of mobile oil phase at time n for every grid

cell. It is a function of T , Pno,m1,ijk

and Xn1...NC−1,m1,ijk

.

ξno,m2,ijklbmol/ft3 Molar density of trapped oil phase at time n for every grid


and Xn1...NC−1,m2,ijk

.

ξng,m1,ijklbmol/ft3 Molar density of mobile gas phase at time n for every grid


and Y n1...NC−1,m1,ijk

.

ξng,m2,ijklbmol/ft3 Molar density of trapped gas phase at time n for every grid


and Y n1...NC−1,m2,ijk

.

ξnw,m1,ijklbmol/ft3 Molar density of mobile water phase at time n for every grid

cell. It is a function of T , WNaCl, Pno,m1,ijk

, and W nCO2,m1,ijk

.

ξnw,m2,ijklbmol/ft3 Molar density of trapped water phase at time n for every

grid cell. It is a function of T , WNaCl, Pno,m2,ijk

, andW n

CO2,m2,ijk.

μno,m1,ijk

cp Viscosity of mobile oil phase at time n for every grid cell. Itis a function of T , Pn

o,m1,ijkand Xn

1...NC−1,m1,ijk.

μno,m2,ijk

cp Viscosity of trapped oil phase at time n for every grid cell.It is a function of T , Pn

o,m2,ijkand Xn

1...NC−1,m2,ijk.

μng,m1,ijk

cp Viscosity of mobile gas phase at time n for every grid cell.It is a function of T , Pn

o,m1,ijkand Y n

1...NC−1,m1,ijk.

Continued.

69


Table 4.5: Secondary variables at n which are needed for the transmissibility calculations, DA forTRANS n (continued)

variable units name

μng,m2,ijk

cp Viscosity of trapped gas phase at time n for every grid cell.It is a function of T , Pn

o,m2,ijkand Y n

1...NC−1,m2,ijk.

μnw,m1,ijk

cp Viscosity of mobile water phase at time n for every grid cell.It is a function of T , WNaCl, P

no,m1,ijk

, and W nCO2,m1,ijk

.

μnw,m2,ijk

cp Viscosity of trapped water phase at time n for every gridcell. It is a function of T , WNaCl, P

no,m2,ijk

, and W nCO2,m2,ijk

.

γno,m1,ijkpsi/ft Specific gravity of mobile oil phase at time n for every grid

cell. It is a function of ξno,m1,ijkand MWn

o,m1,ijk.

γng,m1,ijkpsi/ft Specific gravity of mobile gas phase at time n for every grid

cell. It is a function of ξng,m1,ijkand MWn

g,m1,ijk.

γnw,m1,ijkpsi/ft Specific gravity of mobile water phase at time n for every

grid cell. It is a function of ξnw,m1,ijkand MWn

w,m1,ijk.

knrw,ijk md/md Relative permeability to water at time n for every grid cell.It is a function of Sn

w,t,ijk, Sno,t,ijk, and Sn

g,t,ijk.

knro,ijk md/md Relative permeability to oil at time n for every grid cell. Itis a function of Sn


g,t,ijk.

knrg,ijk md/md Relative permeability to gas at time n for every grid cell. Itis a function of Sn


g,t,ijk.

Pncgo,ijk psia Gas-oil capillary pressure at time n for every grid cell. It is

a function of Snw,t,ijk, S

no,t,ijk, and Sn

g,t,ijk.

Pncow,ijk psia Oil-water capillary pressure at time n for every grid cell. It

is a function of Snw,t,ijk, S

no,t,ijk, and Sn

g,t,ijk.

Table 4.6: Secondary variables at �, DA cell only ell

variable units name

φ�t,ijk ft3 pore/ft3 rock Porosity at nonlinear iteration � for every grid cell. It is

a function of P �o,m1,ijk

.

φ�m1,ijk

ft3 mobile/ft3 rock Mobile pore fraction at nonlinear iteration � for everygrid cell. It is a function of P �

o,m1,ijk.

φ�m2,ijk

ft3 trapped/ft3 rock Trapped pore fraction at nonlinear iteration � for everygrid cell. It is a function of P �

o,m1,ijk.

ξ�o,m1,ijklbmol/ft3 Molar density of mobile oil phase at nonlinear iteration

� for every grid cell. It is a function of T , P �o,m1,ijk

and

X�1...NC−1,m1,ijk

.

Continued.

70


Table 4.6: Secondary variables at �, DA cell only ell (continued)

variable units name

ξ�o,m2,ijklbmol/ft3 Molar density of trapped oil phase at nonlinear

iteration � for every grid cell. It is a function of T ,P �o,m2,ijk

and X�1...NC−1,m2,ijk

.

ξ�g,m1,ijklbmol/ft3 Molar density of mobile gas phase at nonlinear


and Y �1...NC−1,m1,ijk

.

ξ�g,m2,ijklbmol/ft3 Molar density of trapped gas phase at nonlinear


and Y �1...NC−1,m2,ijk

.

ξ�w,m1,ijklbmol/ft3 Molar density of mobile water phase at nonlinear

iteration � for every grid cell. It is a function of T ,WNaCl, P

�o,m1,ijk

, and W �CO2,m1,ijk

.

ξ�w,m2,ijklbmol/ft3 Molar density of trapped water phase at nonlinear

iteration � for every grid cell. It is a function of T ,WNaCl, P

�o,m2,ijk

, and W �CO2,m2,ijk

.

f�om,m1,ijkpsi Mobile oil phase fugacity for component m at nonlinear


and X�1...NC−1,m1,ijk

.

f�om,m2,ijkpsi Trapped oil phase fugacity for component m at

nonlinear iteration � for every grid cell. It is a functionof T , P �

o,m2,ijkand X�

1...NC−1,m2,ijk.

f�gm,m1,ijkpsi Mobile gas phase fugacity for component m at


o,m1,ijkand Y �

1...NC−1,m1,ijk.

f�gm,m2,ijkpsi Trapped gas phase fugacity for component m at


o,m2,ijkand Y �

1...NC−1,m2,ijk.

∂φ�m1,ijk

∂Pom1,ijk

ft3/ft3

psia Derivative of mobile pore fraction with respect topressure at nonlinear iteration � for every grid cell. It isa function of P �

o,m1,ijk.

∂φ�m2,ijk

∂Pom2,ijk

ft3/ft3

psia Derivative of trapped pore fraction with respect topressure at nonlinear iteration � for every grid cell. It isa function of P �

o,m2,ijk.

∂ξ�o,m1,ijk∂Po,m1,ijk

lbmol/ft3

psia Derivative of molar density of mobile oil phase withrespect to pressure at nonlinear iteration � for everygrid cell. It is a function of T , P �

o,m1,ijkand


.

Continued.

71



variable units name∂ξ�o,m2,ijk∂Po,m2,ijk

lbmol/ft3

psia Derivative of molar density of trapped oil phase withrespect to pressure at nonlinear iteration � for everygrid cell. It is a function of T , P �

o,m2,ijkand


.∂ξ�o,m1,ijk

∂Xm′,m1,ijklbmol/ft3

lbmol/lbmol Derivative of molar density of mobile oil phase withrespect to each component mole fraction Xm′,m1 atnonlinear iteration � for every grid cell. They are afunction of T , P �

o,m1,ijkand all the X�

1...NC−1,m1,ijk.

∂ξ�o,m2,ijk∂Xm′,m2,ijk

lbmol/ft3

lbmol/lbmol Derivative of molar density of trapped oil phase withrespect to each component mole fraction Xm′,m2 atnonlinear iteration � for every grid cell. They are afunction of T , P �


1...NC−1,m2,ijk.

∂ξ�g,m1,ijk∂Po,m1,ijk

lbmol/ft3

psia Derivative of molar density of mobile gas phase withrespect to pressure at nonlinear iteration � for everygrid cell. It is a function of T , P �

o,m1,ijkand

Y �1...NC−1,m1,ijk

.∂ξ�g,m2,ijk∂Po,m2,ijk

lbmol/ft3

psia Derivative of molar density of trapped gas phase withrespect to pressure at nonlinear iteration � for everygrid cell. It is a function of T , P �

o,m2,ijkand

Y �1...NC−1,m2,ijk

.∂ξ�g,m1,ijk∂Ym′,m1,ijk

lbmol/ft3

lbmol/lbmol Derivative of molar density of mobile gas phase withrespect to each component mole fraction Ym′,m1 atnonlinear iteration � for every grid cell. They are afunction of T , P �

o,m1,ijkand all the Y �

1...NC−1,m1,ijk.

∂ξ�g,m2,ijk∂Ym′,m2,ijk

lbmol/ft3

lbmol/lbmol Derivative of molar density of trapped gas phase withrespect to each component mole fraction Ym′,m2

atnonlinear iteration � for every grid cell. They are afunction of T , P �


1...NC−1,m2,ijk.

∂W �CO2,m1,ijk

∂Pom1,ijk

lbmol/lbmolpsia Derivative of mobile WCO2 with respect to pressure at

nonlinear iteration � for every grid cell. It is a functionof P �

o,m1,ijk, T , WNaCl, X

�CO2,m1,ijk

, and Y �CO2,m1,ijk

. It isevaluated only when WCO2 is a secondary variable.

∂W �CO2,m2,ijk

∂Pom2,ijk

lbmol/lbmolpsia Derivative of trapped WCO2 with respect to pressure at



�CO2,m2,ijk



Continued.

72



variable units name∂W �

CO2,m1,ijk

∂XCO2,m1,ijk

lbmol/lbmollbmol/lbmol Derivative of mobile WCO2 with respect to XCO2 at



�CO2,m1,ijk



∂W �CO2,m2,ijk

∂XCO2,m2,ijk

lbmol/lbmollbmol/lbmol Derivative of trapped WCO2 with respect to XCO2 at



�CO2,m2,ijk



∂W �CO2,m1,ijk

∂YCO2,m1,ijk

lbmol/lbmollbmol/lbmol Derivative of mobile WCO2 with respect to YCO2 at



�CO2,m1,ijk



∂W �CO2,m2,ijk

∂YCO2,m2,ijk

lbmol/lbmollbmol/lbmol Derivative of trapped WCO2 with respect to YCO2 at



�CO2,m2,ijk



∂ξ�w,m1,ijk

∂Po,m1,ijk

lbmol/ft3

psia Derivative of molar density of mobile water phase withrespect to pressure at nonlinear iteration � for everygrid cell. It is a function of T , WNaCl, P

�o,m1,ijk

, and

W �CO2,m1,ijk

.∂ξ�w,m2,ijk

∂Po,m2,ijk

lbmol/ft3

psia Derivative of molar density of trapped water phasewith respect to pressure at nonlinear iteration � forevery grid cell. It is a function of T , WNaCl, P

�o,m2,ijk

,

and W �CO2,m2,ijk

.∂ξ�w,m1,ijk

∂WCO2,m1,ijk

lbmol/ft3

lbmol/lbmol Derivative of molar density of mobile water phase withrespect to WCO2,m1 at nonlinear iteration � for everygrid cell. It is a function of T , WNaCl, P

�o,m1,ijk

, and

W �CO2,m1,ijk

. It is evaluated only when WCO2 is aprimary variable.

∂ξ�w,m2,ijk

∂WCO2,m2,ijk

lbmol/ft3

lbmol/lbmol Derivative of molar density of trapped water phasewith respect to WCO2,m2 at nonlinear iteration � forevery grid cell. It is a function of T , WNaCl, P

�o,m2,ijk

,

and W �CO2,m2,ijk

. It is evaluated only when WCO2 is aprimary variable.

Continued.

73



variable units name∂ξ�w,m1,ijk

∂XCO2,m1,ijk

lbmol/ft3

lbmol/lbmol Derivative of molar density of mobile water phase withrespect to XCO2,m1 at nonlinear iteration � for everygrid cell. It is a function of T , WNaCl, P

�o,m1,ijk

,

X�CO2,m1,ijk


. It is evaluated only whenWCO2 is a secondary variable.

∂ξ�w,m2,ijk

∂XCO2,m2,ijk

lbmol/ft3

lbmol/lbmol Derivative of molar density of trapped water phasewith respect to XCO2,m2 at nonlinear iteration � forevery grid cell. It is a function of T , WNaCl, P

�o,m2,ijk

,

X�CO2,m2,ijk



∂ξ�w,m1,ijk

∂YCO2,m1,ijk

lbmol/ft3

lbmol/lbmol Derivative of molar density of mobile water phase withrespect to YCO2,m1 at nonlinear iteration � for everygrid cell. It is a function of T , WNaCl, P

�o,m1,ijk

,

X�CO2,m1,ijk



∂ξ�w,m2,ijk

∂YCO2,m2,ijk

lbmol/ft3

lbmol/lbmol Derivative of molar density of trapped water phasewith respect to YCO2,m2 at nonlinear iteration � forevery grid cell. It is a function of T , WNaCl, P

�o,m2,ijk

,

X�CO2,m2,ijk



∂f�om,m1,ijk

∂Po,m1,ijk

psiapsia Derivative of mobile oil phase fugacity for component

m with respect to pressure at nonlinear iteration � forevery grid cell. They are a function of T , P �

o,m1,ijkand


.∂f�om,m2,ijk

∂Po,m2,ijk

psiapsia Derivative of trapped oil phase fugacity for component


o,m2,ijkand


.∂f�gm,m1,ijk

∂Po,m1,ijk

psiapsia Derivative of mobile gas phase fugacity for component


o,m1,ijkand

Y �1...NC−1,m1,ijk

.∂f�gm,m2,ijk

∂Po,m2,ijk

psiapsia Derivative of trapped gas phase fugacity for component


o,m2,ijkand

Y �1...NC−1,m2,ijk

.

Continued.

74



variable units name∂f�om,m1,ijk

∂Xm′,m1,ijkpsia

lbmol/lbmol Derivative of mobile oil phase fugacity for componentm with respect to each component mole fraction Xm′,m1

at nonlinear iteration � for every grid cell. They are afunction of T , P �


1...NC−1,m1,ijk.

∂f�om,m2,ijk

∂Xm′,m2,ijkpsia

lbmol/lbmol Derivative of trapped oil phase fugacity for componentm with respect to each component mole fraction Xm′,m2



1...NC−1,m2,ijk.

∂f�gm,m1,ijk

∂Ym′,m1,ijkpsia

lbmol/lbmol Derivative of mobile gas phase fugacity for componentm with respect to each component mole fraction Ym′,m1



1...NC−1,m1,ijk.

∂f�gm,m2,ijk

∂Ym′,m2,ijkpsia

lbmol/lbmol Derivative of trapped gas phase fugacity for componentm with respect to each component mole fraction Ym′,m2



1...NC−1,m2,ijk.

Table 4.7: Well properties at �, stored for each well.

variable units name

q�w,w ft3/day Water production or injection rate at reservoir conditionsfor nonlinear iteration � for every well. It is a function ofWI#w, P

�om1,ijk

, knrw,ijk, knro,ijk, k

nrg,ijk, μ

nw,ijk, μ

no,ijk, and

μng,ijk.

q�o,w ft3/day Oil production or injection rate at reservoir conditions fornonlinear iteration � for every well. It is a function of WI#w,P �om1,ijk


nrg,ijk, μ

nw,ijk, μ

no,ijk, and μn

g,ijk.

q�g,w ft3/day Gas production or injection rate at reservoir conditions fornonlinear iteration � for every well. It is a function of WI#w,P �om1,ijk


nrg,ijk, μ

nw,ijk, μ

no,ijk, and μn

g,ijk.

4.3 Overview of Simulation Process

The following steps are involved for a complete simulation run.

1. Initialize

1.1. Load properties for a specific simulation run from standard input and file(s).

75

1.2. Allocate memory: allocate global EOS, allocate local eos, allocate temp EOS, psim initialize

data sizes, psim initialize data 2spot, psim allocate DA variables, psim allocate other variables,

psim allocate init variables.

1.3. Initialize temperature dependent constants: initialize eos X4, initialize temperature constants,

initialize LBC viscosity constants, initialize aqueous constants,

1.4. Initialize grid properties for a specific simulation run: psim initialize 1D 0020, psim init

trap 1D 0020, psim initialize 1D 0001 flash, psim initialize 1D 0001 vector, psim init trap

1D 0001 vector

1.5. Initialize well properties for a specific simulation run: psim initialize 1D 0020 well

1.6. Initialize solver: psim solver init

2. For each time step n: psim solve iterate ell, psim solve n

2.1. Copy final iteration of previous timestep (n − 1, � + 1) to new timestep (n, � = 0):

psim COPY ell to n

2.2. Calculate interior and ghost cell properties at n; communicate ghost properties needed

for transmissibilities to neighbors at n: psim calculate local n.

2.3. Calculate transmissibilities at n: psim all TRANS nonly

2.4. Calculate and communicate wells at n: psim COPY well n

2.5. Calculate time step size at n: psim local timestep n

2.6. Calculate jacobian and other properties which depend on the transmissibilities at n:

psim calculate all jacobian n, psim calc after TRANS n

2.7. Copy properties for � = 0: psim COPY n to ell

3. For each nonlinear iteration �, before convergence or before maximum number of iterations

is exceeded: psim solve ell

3.1. When � > 0, calculate interior properties at � (transmissibilities depend on n not on �,

so no ghost properties are needed): psim calculate local ell.

3.2. Calculate wells at �: psim COPY well ell, sim well single completion ell

76

3.3. Calculate and communicate jacobian at �: psim calculate all jacobian ell

3.4. Solve matrix equation and communicate solution at �: psim solver solve

3.5. Update and communicate primary variables and determine if the solution has converged

at �: psim convergence update primary ell

3.6. If not converged and the maximum number of nonlinear iterations has not been exceeded,

return to Step 3.

4. After the final nonlinear iteration � of each timestep n: psim solve iterate ell only

4.1. If the maximum number of nonlinear iterations was exceeded, update the primary vari-

ables using the best solution at �.

4.2. Write out selected grid properties: psim calculate local ell, psim converged local ell

4.3. Write out selected well properties: psim COPY well ell, sim well single completion ell

4.4. If necessary, WAG to gas or WAG to water or transfer additional mass to the trapped

system: psim update trap properties n, psim 1D 0020 well WAG gas, psim 1D 0020 well

WAG water.

4.5. If not the final time step, return to Step 2.

5. Finalize

5.1. Finalize solver

5.2. Deallocate memory

4.4 Assemble the Jacobian

In a three-phase dual medium system with NC = 5 and no degeneracies illustrated in (4.37),

there are 2 × (2 × NC − 1) primary variables: Pm1 , Swm1 , Som1 , X1,m1 , X2,m1 , X3,m1 , Y1,m1 , Y2,m1 ,

Y3,m1 , Pm2 , Swm2 , Som2 , X1,m2 , X2,m2 , X3,m2 , Y1,m2 , Y2,m2 , and Y3,m2 . The primary variables are

reordered to facilitate the following main steps of the solution.

1. Pm1 is solved for first using a sparse matrix solve of a reduced set of pressure equations,

one per grid cell. The reduced set of equations is obtained by LU decomposition of the full

system of equations for each grid cell. This simplification is possible because only the Pm1

terms appear for off-block-diagonal grid cells.

77

2. Pm2 , Sw,m1 , and Sw,m2 are solved for next by back substitution local to the grid cell.

3. Som1 , X1,m1 , X2,m1 , X3,m1 , Y1,m1 , Y2,m1 , and Y3,m1 are solved using flash calculations local to

the grid cell.

4. Som2 , X1,m2 , X2,m2 , X3,m2 , Y1,m2 , Y2,m2 , and Y3,m2 are solved using flash calculations local to

the grid cell.

There are also 2 × (2 × NC − 1) equations: C1,m1 , C2,m1 , C3,m1 , C4,m14, C5,m1 , G1,m1 , G2,m1 ,

G3,m1 , G4,m1 , C1,m2 , C2,m2 , C3,m2 , C4,m24, C5,m2 , G1,m2 , G2,m2 , G3,m2 , and G4,m2 . These are ordered

in the following way to facilitate the LU decomposition:

1. CH2O,m1 , CH2O,m2 , CCO2,m1 , and CCO2,m2 . These component equations come first because they

typically have large non-zero coefficients of Pm1 , Pm2 , Swm1 , and Swm2 .

2. C1...NC−2,m1 and G1...NC−1,m1 ; these are associated with the flash variables Som1 , X1...NC−2,m1 ,

and Y1...NC−2,m1

3. C1...NC−2,m2 and G1...NC−1,m2 ; these are associated with the flash variables Som2 , X1...NC−2,m2 ,

and Y1...NC−2,m2

Pom

1

Pom

2

Swm

1

Swm

2

Som

1

Xm

=1,m

1

Xm

=2,m

1

Xm

=3,m

1

Ym

=1,m

1

Ym

=2,m

1

Ym

=3,m

1

Som

2

Xm

=1,m

2

Xm

=2,m

2

Xm

=3,m

2

Ym

=1,m

2

Ym

=2,m

2

Ym

=3,m

2

CH2O,m1 # X X 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

CH2O,m2 X # 0 X 0 0 0 0 0 0 0 0 0 0 0 0 0 0

CCO2,m1 X X # 0 X X X X X X X 0 0 0 0 0 0 0

CCO2,m2 X X 0 # 0 0 0 0 0 0 0 X X X X X X X

Cm=1,m1 X X X 0 # X X X X X X 0 0 0 0 0 0 0

Cm=2,m1 X X X 0 X # X X X X X 0 0 0 0 0 0 0

Cm=3,m1 X X X 0 X X # X X X X 0 0 0 0 0 0 0

Gg/o,m=1,m1X 0 0 0 0 X X # X X X 0 0 0 0 0 0 0

Gg/o,m=2,m1X 0 0 0 0 X X X # X X 0 0 0 0 0 0 0

Gg/o,m=3,m1X 0 0 0 0 X X X X # X 0 0 0 0 0 0 0

Gg/o,m=4,m1X 0 0 0 0 X X X X X # 0 0 0 0 0 0 0

Cm=1,m2 X X 0 X 0 0 0 0 0 0 0 # X X X X X X

Cm=2,m2 X X 0 X 0 0 0 0 0 0 0 X # X X X X X

Cm=3,m2 X X 0 X 0 0 0 0 0 0 0 X X # X X X X

Gg/o,m=1,m20 X 0 0 0 0 0 0 0 0 0 0 X X # X X X

Gg/o,m=2,m20 X 0 0 0 0 0 0 0 0 0 0 X X X # X X

Gg/o,m=3,m20 X 0 0 0 0 0 0 0 0 0 0 X X X X # X

Gg/o,m=4,m20 X 0 0 0 0 0 0 0 0 0 0 X X X X X #

(4.37)

78

4.4.1 Single Medium (No Trapping)

In a single medium system, all of the m2 variables and equations can be eliminated. These

variables and equations are eliminated in the order of their occurrence from the reordered lists.

The nine remaining variables are the following: Pm1 , Sw,m1 , Som1 , X1,m1 , X2,m1 , X3,m1 , Y1,m1 , Y2,m1 ,

and Y3,m1 . The nine remaining equations are the following: CH2O,m1 , CCO2,m1 , C1...NC−2,m1 , and

G1...NC−1,m1 .

For degenerate cases where one of the phases or one of the components is not present, the

same process is used. The variables and equations are eliminated in the order of their occurrence

from the reordered lists. Different phases or components can be present in the mobile and trapped

medium, leaving many potential options. To simplify the discussion the single medium case will be

used in the remainder of the discussion.

4.4.2 Degenerate Case with Oil and Water Only

When a three-phase gas-oil-water system degenerates into a two-phase oil-water system, instead

of three saturations Sw, So, Sg = 1−So−Sw, there are now only two saturations, Sw and So = 1−Sw.

Although the gas compositions Y1, Y2, Y3, Y4 can be defined, they are not relevant to a problem

with only oil and water. The five remaining primary variables are the following: Pm1 , Sw,m1 , X1,m1 ,

X2,m1 , andX3,m1 . All of the component equations are still relevant, but note that the terms referring

to gas saturations or gas relative permeabilities are zero. The thermodynamic constraints can be

defined but are not relevant to a problem without both oil and gas. The five remaining equations

are the following: CH2O,m1 , CCO2,m1 , and C1...NC−2,m1 .

4.4.3 Degenerate Case with Gas and Water Only

When a three-phase gas-oil-water system degenerates into a two-phase gas-water system, instead

of three saturations Sw, So, Sg = 1−So−Sw, there are now only two saturations, Sw and Sg = 1−Sw.

Although the oil compositions X1, X2, X3, X4 can be defined, they are not relevant to a problem

with only gas and water. The five remaining primary variables are the following: Pm1 , Sw,m1 , Y1,m1 ,

Y2,m1 , and Y3,m1 . All of the component equations are still relevant, but note that the terms referring

to oil saturations or oil relative permeabilities are zero. The thermodynamic constraints can be

defined but are not relevant to a problem without both oil and gas. The five remaining equations

79

are the following: CH2O,m1 , CCO2,m1 , and C1...NC−2,m1 .

4.4.4 Degenerate Case with Gas and Oil Only

The degenerate case where a three-phase gas-oil-water system degenerates into a two-phase

oil-gas system is unusual, but can occur in several different ways. In a steam injection scenario

(not considered in this dissertation), the water may all be vaporized into gas. For a system with

trapping, the trapped system may not contain any trapped water. A gas-only system or an oil-only

system can also evolve into a two-phase oil-gas system.

Instead of three saturations Sw, So, Sg = 1−So−Sw, there are now only two saturations, So and

Sg = 1 − So. The water component equation does not apply here. The eight remaining variables

are the following: Pm1 , Som1 , X1,m1 , X2,m1 , X3,m1 , Y1,m1 , Y2,m1 , and Y3,m1 . The eight remaining

equations are the following: CCO2,m1 , C1...NC−2,m1 , and G1...NC−1,m1 .

4.4.5 Degenerate Case with Water Only

The degenerate case where a three-phase gas-oil-water system degenerates into a single-phase

water system is unusual, but can occur in several different ways. In a steam injection scenario

(not considered in this dissertation), all the steam may condense into liquid water. For a system

with trapping, the trapped system may only contain water. A gas-water system may turn into a

water-only system with water injection or the dissolution of all the gas into the water.

Instead of three saturations Sw, So, Sg = 1 − So − Sw, there is now only one saturation, Sw;

but because Sw = 1, this saturation can be eliminated too. The X1,m1 , X2,m1 , X3,m1 , Y1,m1 , Y2,m1 ,

and Y3,m1 can be defined but are not applicable. This leaves only one primary variable, Pm1 and

one equation, CH2O,m1 .

4.4.6 Degenerate Case with Oil Only


oil system is unusual, but can occur in several different ways. For a system with trapping, the

trapped system may only contain oil. A gas-oil system may turn into an oil-only system based on

a phase transition.

Instead of three saturations Sw, So, Sg = 1 − So − Sw, there is now only one saturation, So;

but because So = 1, this saturation can be eliminated too. Although the gas compositions Y1,

80

Y2, Y3, Y4 can be defined, they are not relevant to a problem with oil only. This leaves only four

primary variables: Pm1 , X1,m1 , X2,m1 , and X3,m1 . The water component equation is not applicable

to a system without water. The thermodynamic constraints can be defined but are not relevant

to a problem without both oil and gas. This leaves the following four equations: CCO2,m1 and

C1...NC−2,m1 .

4.4.7 Degenerate Case with Gas Only


gas system is unusual, but can occur in several different ways. For a system with trapping, the

trapped system may only contain gas. A gas-oil system may turn into an oil-only system based on

a phase transition.

Instead of three saturations Sw, So, Sg = 1−So−Sw, there is now only one saturation, Sg; but

because Sg = 1, this saturation can be eliminated too. Although the oil compositions X1, X2, X3,

X4 can be defined, they are not relevant to a problem with gas only. This leaves only four primary

variables: Pm1 , Y1,m1 , Y2,m1 , and Y3,m1 . The water component equation is not applicable to a system

without water. The thermodynamic constraints can be defined but are not relevant to a problem

without both oil and gas. This leaves the following four equations: CCO2,m1 and C1...NC−2,m1 .

4.4.8 Three-Phase Degenerate Case with Fewer Components

When one of the components is zero, this eliminates two primary variables (Xm and Ym) and

two equations (Cm and Gm). These primary variables and equations are eliminated and the rest

of the order is preserved.

For instance, if Z1,m1 = 0, the seven remaining variables are the following: Pm1 , Sw,m1 , Som1 ,

X2,m1 , X3,m1 , Y2,m1 , and Y3,m1 . The seven remaining equations are the following: CH2O,m1 , CCO2,m1 ,

C2...NC−2,m1 , and G2...NC−1,m1 .

If ZCO2 = 0, then the NC − 2 = 3 component is eliminated from the variables since now

X3,m1 = 1−X1,m1−X2,m1 and Y3,m1 = 1−Y1,m1−Y2,m1 . The equations for CO2 are still eliminated.

The seven remaining variables are the following: Pm1 , Sw,m1 , Som1 , X1,m1 , X2,m1 , Y1,m1 , and Y2,m1 .

The seven remaining equations are the following: CH2O,m1 , C1...NC−2,m1 , and G1...NC−2,m1 .

A similar process applies if more than one component is zero.

81

4.5 Rewrite Base Equations for Um Solve

The solution approach used for each nonlinear iteration uses a careful labeling of the terms in

the component equations to solve for So, Sg, Xm, Ym, Wm based on a solution of P and Sw. This

section identifies the new variable definitions of the base equations. Section 4.6 identifies the steps

in the solution.

The component equations for the m1 system, (4.1) are rewritten as follows. The [� + 1] terms

are based on the pressure P �+1m1

and P �+1m2

which have already been calculated during the pressure

solve.

Cm=1...NC ,m1 :

U�+1m,m1︷︸︸︷

0.006328 VR ∇ ·(Xn

mm1ξnom1

knrom1

μnom1

k#m1(∇P [�+1]

om1− γn

om1∇D#)

)+

0.006328 VR ∇ ·(Y n

mm1ξngm1

knrgm1

μngm1

k#m1(∇P [�+1]

om1+∇Pn

cgom1− γn

gm1∇D#)

)+

0.006328 VR ∇ ·(Wn

mm1ξnwm1

knrwm1

μnwm1

k#m1(∇P [�+1]

om1−∇Pn

cowm1− γn

wm1∇D#)

)+(

Xnmm1

ξnom1q[�+1]om1

+ Y nmm1

ξngm1q[�+1]gm1

+Wnmm1

ξnwm1q[�+1]wm1

)−

0.006328 VR σ#m1/m2

k#m1/m2

(P

[�+1]om1

− P[�+1]om2

)×(Xup,n

mm1/m2ξup,nom1/m2

kup,nrom1/m2

μup,nom1/m2

+Y up,nmm1/m2


μup,ngm1/m2

+W up,n

mm1/m2ξup,nwm1/m2

kup,nrwm1/m2

μup,nwm1/m2

)

=

(Z�+1m,2ph,m1

α�+1m1︷︸︸︷

VR

Δtφ�+1m1

(S�+1om1

ξ�+1om1

+ S�+1gm1

ξ�+1gm1

) +W �+1mm1

β�+1m1︷︸︸︷

VR

Δtφ�+1m1

S�+1wm1

ξ�+1wm1

)−

(Znm,2ph,m1

αnm1︷︸︸︷

VR

Δtφnm1(Sn

om1ξnom1

+ Sngm1

ξngm1) +Wn

mm1

βnm1︷︸︸︷

VR

Δtφnm1Snwm1

ξnwm1

)(4.38)

The component equations for the m2 system, (4.2) are rewritten as follows. The [� + 1] terms

are based on the pressure P �+1m1

and P �+1m2

which have already been calculated during the pressure

solve.

82

Cm=1...NC ,m2 :

U�+1m,m2︷︸︸︷

0.006328 VR σ#m1/m2

k#m1/m2

(P

[�+1]om1

− P[�+1]om2

)×(Xup,n

mm1/m2ξup,nom1/m2

kup,nrom1/m2

μup,nom1/m2

+Y up,nmm1/m2


μup,ngm1/m2

+W up,n

mm1/m2ξup,nwm1/m2

kup,nrwm1/m2

μup,nwm1/m2

)=

(Z�+1m,2ph,m2

α�+1m2︷︸︸︷

VR

Δtφ�+1m2

(S�+1om2

ξ�+1om2

+ S�+1gm2

ξ�+1gm2

) +W �+1mm2

β�+1m2︷︸︸︷

VR

Δtφ�+1m2

S�+1wm2

ξ�+1wm2

)−

(Znm,2ph,m2

αnm2︷︸︸︷

VR

Δtφnm2(Sn

om2ξnom2

+ Sngm2

ξngm2) +Wn

mm2

βnm2︷︸︸︷

VR

Δtφnm2Snwm2

ξnwm2

)(4.39)

Add the NC equations (4.38) to obtain Ct,m1 . This eliminates Xmm1 , Ymm1 , and Wmm1 because∑mXmm1 = 1,

∑m Ymm1 = 1, and

∑mWmm1 = 1.

Ct,m1 :

U�+1t,m1︷︸︸︷

0.006328 VR ∇ ·(ξnom1

knrom1

μnom1

k#m1(∇P [�+1]

om1− γn

om1∇D#)

)+

0.006328 VR ∇ ·(ξngm1

knrgm1

μngm1

k#m1(∇P [�+1]

om1+∇Pn

cgom1− γn

gm1∇D#)

)+

0.006328 VR ∇ ·(ξnwm1

knrwm1

μnwm1

k#m1(∇P [�+1]

om1−∇Pn

cowm1− γn

wm1∇D#)

)+(

ξnom1q[�+1]om1

+ ξngm1q[�+1]gm1

+ ξnwm1q[�+1]wm1

)−

0.006328 VR σ#m1/m2

k#m1/m2

(P

[�+1]om1

− P[�+1]om2

)×(ξup,nom1/m2

kup,nrom1/m2

μup,nom1/m2

+ξup,ngm1/m2

kup,nrgm1/m2

μup,ngm1/m2

+ξup,nwm1/m2

kup,nrwm1/m2

μup,nwm1/m2

)

=

(α�+1

m1︷︸︸︷VR

Δtφ�+1m1

(S�+1om1

ξ�+1om1

+ S�+1gm1

ξ�+1gm1

) +

β�+1m1︷︸︸︷

VR

Δtφ�+1m1

S�+1wm1

ξ�+1wm1

)−

(αn

m1︷︸︸︷VR

Δtφnm1(Sn

om1ξnom1

+ Sngm1

ξngm1) +

βnm1︷︸︸︷

VR

Δtφnm1Snwm1

ξnwm1

)(4.40)

Add the NC equations (4.39) to obtain Ct,m2 . This eliminates Xmm2 , Ymm2 , and Wmm2 because∑mXmm2 = 1,

∑m Ymm2 = 1, and

∑mWmm2 = 1.

83

Ct,m2 :

U�+1t,m2︷︸︸︷

0.006328 VR σ#m1/m2

k#m1/m2

(P

[�+1]om1

− P[�+1]om2

)×(ξup,nom1/m2

kup,nrom1/m2

μup,nom1/m2

+ξup,ngm1/m2

kup,nrgm1/m2

μup,ngm1/m2

+ξup,nwm1/m2

kup,nrwm1/m2

μup,nwm1/m2

)=

(α�+1

m2︷︸︸︷VR

Δtφ�+1m2

(S�+1om2

ξ�+1om2

+ S�+1gm2

ξ�+1gm2

) +

β�+1m2︷︸︸︷

VR

Δtφ�+1m2

S�+1wm2

ξ�+1wm2

)−

(αn

m2︷︸︸︷VR

Δtφnm2(Sn

om2ξnom2

+ Sngm2

ξngm2) +

βnm2︷︸︸︷

VR

Δtφnm2Snwm2

ξnwm2

)(4.41)

Write the water equation for m1.

CWAT,m1 :

U�+1WAT,m1︷︸︸︷

0.006328 VR ∇ ·(ξnwm1

knrwm1

μnwm1

k#m1(∇P [�+1]

om1−∇Pn

cowm1− γn

wm1∇D#)

)+(

ξnwm1q[�+1]wm1

)−

0.006328 VR σ#m1/m2

k#m1/m2

(P

[�+1]om1

− P[�+1]om2

)×(ξup,nwm1/m2

kup,nrwm1/m2

μup,nwm1/m2

) =

(β�+1m1︷︸︸︷

VR

Δtφ�+1m1

S�+1wm1

ξ�+1wm1

)−(

βnm1︷︸︸︷

VR

Δtφnm1Snwm1

ξnwm1

)(4.42)

Write the water equation for m2.

CWAT,m2 :

U�+1WAT,m2︷︸︸︷

0.006328 VR σ#m1/m2

k#m1/m2

(P

[�+1]om1

− P[�+1]om2

)×(ξup,nwm1/m2

kup,nrwm1/m2

μup,nwm1/m2

)=

(β�+1m2︷︸︸︷

VR

Δtφ�+1m2

S�+1wm2

ξ�+1wm2

)−(

βnm2︷︸︸︷

VR

Δtφnm2Snwm2

ξnwm2

)(4.43)

84

4.6 Update Primary Variables at Each Nonlinear Iteration

The process involves the following steps:

1. Assemble the Jacobian using (4.1)–(4.5). The ordering of the primary variables and equations

listed here is for the gas/oil/water system with WCO2 as a secondary variable.

• The primary variables are ordered as follows: Pom1 , Pom2 , Swm1 , Swm2 , Som1 , Xm=1,m1 ,

Xm=2,m1 , Xm=3,m1 (up to Xm=NC−2,m1), Ym=1,m1 , Ym=2,m1 , Ym=3,m1 (up to Ym=NC−2,m1),

Som2 , Xm=1,m2 , Xm=2,m2 , Xm=3,m2 (up to Xm=NC−2,m2), Ym=1,m2 , Ym=2,m2 , Ym=3,m2 (up

to Ym=NC−2,m2)

• The equations are ordered as follows, and are reordered using partial pivoting. CH2O,m1 ,

CH2O,m2 , CCO2,m1 , CCO2,m2 , Cm=1,m1 , Cm=2,m1 , Cm=3,m1 (up to CNC−2,m1), Gm=1,m1 ,

Gm=2,m1 , Gm=3,m1 , Gm=4,m1 (up toGNC−1,m1), Cm=1,m2 , Cm=2,m2 , Cm=3,m2 (up to CNC−2,m2),

Gm=1,m2 , Gm=2,m2 , Gm=3,m2 , Gm=4,m2 (up to GNC−1,m2)

2. Set up and solve pressure equation.

2.1. Perform an LU decomposition for each grid cell.

2.2. The upper left corner of each block forms the LU-pressure equation.

2.3. Solve the system of LU-pressure equations for δP �+1om1

using a sparse matrix solver.

3. In psim convergence update primary ell, calculate primary variables at �+ 1.

3.1. Calculate P �+1om1

and P �+1om2

3.1.1. Calculate P �+1om1

= P �om1

+δP �+1om1

, with additional checks to keep Pmin < P �+1om1

< Pmax.

Test for convergence using

maxijk

∣∣∣δP �+1om1,ijk

∣∣∣ < εP (4.44)

3.1.2. Calculate δP �+1om2

by back substitution. Calculate P �+1om2

= P �om2

+ δP �+1om2

, with addi-

tional checks to ensure∣∣∣P �+1

om2− P �+1

om1

∣∣∣ < Pmax diff and Pmin < P �+1om2

< Pmax.

3.2. Calculate S�+1wm1

and S�+1wm2

85

3.2.1. If the previous iteration εSw < S�wm1

< 1−εSw , calculate δS�+1wm1

by back substitution.

Calculate S�+1wm1

= S�wm1

+ δS�+1wm1

, with additional checks to ensure 0 ≤ S�+1wm1≤ 1.

3.2.2. If the previous iteration S�wm1

< εSw or S�wm1

> 1− εSw , then

3.2.2.1. Calculate an approximate value for φ�+ 1

2m1

.

φ�+ 1

2m1

= φ�m1

+∂φ

∂PδP (4.45)

3.2.2.2. Calculate an approximate value for ξ�+ 1

2w,m1

. Accumulate pressure derivative at

this point.

ξ�+ 1

2w,m1

= ξ�wm1+

∂ξwm1

∂PδP +

∂ξwm1

∂Ym′δYm′ +

∂ξwm1

∂Xm′δXm′ (4.46)

3.2.3. If the previous iteration εSw < S�wm2

< 1−εSw , calculate δS�+1wm2

by back substitution.

Calculate S�+1wm2

= S�wm2

+ δS�+1wm2

, with additional checks to ensure 0 ≤ S�+1wm2≤ 1.

3.2.4. If the previous iteration S�wm2

< εSw or S�wm2

> 1− εSw , then

3.2.4.1. Calculate an approximate value for φ�+ 1

2m2

φ�+ 1

2m2

= φ�m2

+∂φ

∂PδP (4.47)

3.2.4.2. Calculate an approximate value for ξ�+ 1

2w,m2

. Accumulate pressure derivative at

this point.

ξ�+ 1

2w,m2

= ξ�wm2+

∂ξwm2

∂PδP +

∂ξwm2

∂Ym′δYm′ +

∂ξwm2

∂Xm′δXm′ (4.48)

3.3. Calculate the well properties q�+1om1

, q�+1gm1

, and q�+1wm1

by calling psim COPY well ell and

sim well single completion converged ell. as a function of P �+1om1

and properties at n. See

Chapter 9 for a discussion of the calculation of well rates using fixed rate, fixed pressure,

and mixed pressure and rate constraints.

4. In psim convergence update primary ell, call psim new primary from flash ell. Calculate the pri-

mary variables at �+ 1 which depend on flash calculations. See Section 4.7 for more details.

86

• The primary variables calculated by psim new primary from flash ell: Som1 , Som2 ,Xm′=1...NC−2,m1,

Ym′=1...NC−2,m1 , Xm′=1...NC−2,m2 , Ym′=1...NC−2,m2

• If the previous iteration S�wm1

< εSw or S�wm1

> 1− εSw , calculate Swm1


< εSw or S�wm2


• The secondary variables calculated by psim new primary from flash ell: WCO2m1 , ξgm1 ,

ξom1 , ξwm1 , WCO2m2 , ξgm2 , ξom2 , and ξwm2 .

5. In psim convergence update primary ell, calculate mass balance for grid cells in psim converged

local ell.

Mnt =

∑m

(Mn

omm1+Mn

gmm1+Mn

wmm1

)+∑m

(Mn

omm2+Mn

gmm2+Mn

wmm2

)(4.49)

M �+1t =

∑m

(M �+1

omm1+M �+1

gmm1+M �+1

wmm1

)+

∑m

(M �+1

omm2+M �+1

gmm2+M �+1

wmm2

)+∑m

(q�+1om + q�+1

gm + q�+1wm

)×Δn (4.50)

The residual R�+1 is used to determine the best model in case the nonlinear iterations do not

converge.

R�+1 =M �+1

t −Mnt

Mnt

(4.51)

6. In psim convergence update primary ell, print any desired primary variables and any desired

secondary variables at �+1 for all grid cells. If desired, print information on the convergence

process, including the residual and the grid cells with maximum changes in P , S, Xm, Ym,

and WCO2 .

4.7 Update Primary Variables at Each Nonlinear Iteration: Flash

In psim new primary from flash ell and the subroutines it calls, calculate the primary variables

at �+ 1 which depend on flash calculations. These include:

• The primary variables calculated by psim new primary from flash ell: Som1 , Som2 ,Xm′=1...NC−2,m1,

Ym′=1...NC−2,m1 , Xm′=1...NC−2,m2 , Ym′=1...NC−2,m2

87


< εSw or S�wm1



< εSw or S�wm2


• The secondary variables calculated by psim new primary from flash ell: WCO2m1 , ξgm1 , ξom1 ,

ξwm1 , WCO2m2 , ξgm2 , ξom2 , and ξwm2 .

1. In psim new primary from flash ell call psim calc after TRANS ell one. Calculate primary vari-

ables at �+ 1.

1.1. Calculate U �+1mm1

using (4.38) and (4.52) as a function of P �+1om1

, P �+1om2

, and properties at

n.

U �+1mm1

=

0.006328 VR ∇ ·(Xn

mm1ξnom1

knrom1

μnom1

k#m1(∇P [�+1]

om1− γn

om1∇D#)

)+

0.006328 VR ∇ ·(Y n

mm1ξngm1

knrgm1

μngm1

k#m1(∇P [�+1]

om1+∇Pn

cgom1− γn

gm1∇D#)

)+

0.006328 VR ∇ ·(Wn

mm1ξnwm1

knrwm1

μnwm1

k#m1(∇P [�+1]

om1−∇Pn

cowm1− γn

wm1∇D#)

)+(

Xnmm1

ξnom1q[�+1]om1

+ Y nmm1

ξngm1q[�+1]gm1

+Wnmm1

ξnwm1q[�+1]wm1

)−

0.006328 VR σ#m1/m2

k#m1/m2

(P

[�+1]om1

− P[�+1]om2

)×(Xup,n

mm1/m2ξup,nom1/m2

kup,nrom1/m2

μup,nom1/m2

+Y up,nmm1/m2


μup,ngm1/m2

+W up,n

mm1/m2ξup,nwm1/m2

kup,nrwm1/m2

μup,nwm1/m2

)

(4.52)

1.2. Calculate U �+1mm2


, P �+1om2

, and properties at

n.

U �+1mm2

=

0.006328 VR σ#m1/m2

k#m1/m2

(P

[�+1]om1

− P[�+1]om2

)×(Xup,n

mm1/m2ξup,nom1/m2

kup,nrom1/m2

μup,nom1/m2

+Y up,nmm1/m2


μup,ngm1/m2

+W up,n

mm1/m2ξup,nwm1/m2

kup,nrwm1/m2

μup,nwm1/m2

)

(4.53)

1.3. Calculate U �+1WATm1


, P �+1om2

, and properties

at n.

88

U �+1WATm1

=

0.006328 VR ∇ ·(ξnwm1

knrwm1

μnwm1

k#m1(∇P [�+1]

om1−∇Pn

cowm1− γn

wm1∇D#)

)+(

ξnwm1q[�+1]wm1

)−

0.006328 VR σ#m1/m2

k#m1/m2

(P

[�+1]om1

− P[�+1]om2

)×(ξup,nwm1/m2

kup,nrwm1/m2

μup,nwm1/m2

) (4.54)

1.4. Calculate U �+1WATm2


, P �+1om2

, and properties

at n.

U �+1WATm2

= 0.006328 VR σ#m1/m2

k#m1/m2

(P

[�+1]om1

− P[�+1]om2

)×(ξup,nwm1/m2

kup,nrwm1/m2

μup,nwm1/m2

)(4.55)

2. In psim new primary from flash ell, load the previously calculated values of αnm1, αn

m2, βn

m1,

βnm2. These depend only on variables at n, which means they were already calculated in in

psim calc after TRANS n.

αnm1

= VRΔtφ

nm1(Sn

om1ξnom1

+ Sngm1

ξngm1) (4.56)

βnm1

= VRΔtφ

nm1Snwm1

ξnwm1(4.57)

αnm2

= VRΔtφ

nm2(Sn

om2ξnom2

+ Sngm2

ξngm2) (4.58)

βnm2

= VRΔtφ

nm2Snwm2

ξnwm2(4.59)

3. In psim new primary from flash ell, calculate additional properties at �+1 needed to calculate

Z�+1m,2ph,m1

and Z�+1m,2ph,m2

3.1. Calculate U �+1tm1


, P �+1om2

, and properties at n.

U �+1tm1

=∑m

U �+1mm1

(4.60)

3.2. Calculate U �+1tm2


, P �+1om2

, and properties at n.

U �+1tm2

=∑m

U �+1mm2

(4.61)

3.3. Solve (4.42) for β�+1m1

= VRΔtφ

�+1m1

S�+1wm1

ξ�+1wm1

= U �+1WAT,m1

+ βnm1

3.4. Solve (4.43) for β�+1m2

= VRΔtφ

�+1m2

S�+1wm2

ξ�+1wm2

= U �+1WAT,m2

+ βnm2

89

3.5. Solve (4.40) for α�+1m1

= VRΔtφ

�+1m1

(S�+1om1

ξ�+1om1

+ S�+1gm1

ξ�+1gm1

) = U �+1t,m1

+ αnm1

+ βnm1− β�+1

m1

3.6. Solve (4.41) for α�+1m2

= VRΔtφ

�+1m2

(S�+1om2

ξ�+1om2

+ S�+1gm2

ξ�+1gm2

) = U �+1t,m2

+ αnm2

+ βnm2− β�+1

m2

4. In psim new primary from flash ell, calculate Z�+1m,2ph,m1

and Z�+1m,2ph,m2

4.1. If α�+1m1

> εα and α�+1m2

> εα (under normal conditions), calculate Z�+1m,2ph,m1

and Z�+1m,2ph,m2

.

4.1.1. For m = 1 . . . NC − 2, solve (4.38) for Z�+1m,2ph,m1

=(U �+1m,m1

+ αnm1Znm,2ph,m1

)/α�+1

m1

4.1.2. Calculate Z�+1CO2,2ph,m1

= 1−NC−2∑m′=1

Z�+1m′,2ph,m1

.

4.1.3. Ensure that 0 ≤ Z�+1m,2ph,m1

≤ 1 and that∑

m Z�+1m,2ph,m1

= 1.


=(U �+1m,m2

+ αnm2Znm,2ph,m2

)/α�+1

m2


= 1−NC−2∑m′=1

Z�+1m′,2ph,m2

.


≤ 1 and that∑

m Z�+1m,2ph,m2

= 1.

4.2. If α�+1m1

< εα and α�+1m2

< εα,

4.2.1. Set Z�+1m�=CO2,2ph,m1

= 0 and Z�+1m=CO2,2ph,m1

= 1.

4.2.2. Set Z�+1m�=CO2,2ph,m2

= 0 and Z�+1m=CO2,2ph,m2

= 1.

4.3. If α�+1m1

> εα and α�+1m2

< εα, calculate Z�+1m,2ph,m1

.


=(U �+1m,m1

+ αnm1Znm,2ph,m1

)/α�+1

m1


= 1−NC−2∑m′=1

Z�+1m′,2ph,m1

.


≤ 1 and that∑

m Z�+1m,2ph,m1

= 1.

4.3.4. Set Z�+1m,2ph,m2

= Z�+1m,2ph,m1

4.4. If α�+1m1

< εα and α�+1m2

> εα, calculate Z�+1m,2ph,m2

.


=(U �+1m,m2

+ αnm2Znm,2ph,m2

)/α�+1

m2


= 1−NC−2∑m′=1

Z�+1m′,2ph,m2

.


≤ 1 and that∑

m Z�+1m,2ph,m2

= 1.

4.4.4. Set Z�+1m,2ph,m1

= Z�+1m,2ph,m2

5. In psim new primary from flash ell, flash Z�+1m,2ph,m1

and calculate X�+1m,m1

, Y �+1m,m1

, and S�+1wm1

.

90

5.1. Flash Z�+1m,2ph,m1

at P �+1om1

to calculate X�+1mm1

, Y �+1mm1

, ξ�+1om1

, ξ�+1gm1

, L�+1m1

, V �+1m1

.

5.2. Ensure that 0 ≤ X�+1m,m1

≤ 1 and that∑

mX�+1m,m1

= 1.

5.3. Ensure that 0 ≤ Y �+1m,m1

≤ 1 and that∑

m Y �+1m,m1

= 1.

5.4. If S�wm1

< εS or S�wm1

> 1− εS, calculate S�+1wm1

here.

5.4.1. Calculate an approximate value for ξ�+ 1

2w,m1

. Accumulate composition derivative at

this point.

ξ�+ 1

2w,m1

= ξ�wm1+

∂ξwm1

∂PδP +

∂ξwm1

∂Ym′δYm′ +

∂ξwm1

∂Xm′δXm′ (4.62)

5.4.2. Use ξ�+ 1

2w,m1

calculated above; if ξ�+ 1

2w,m1

< ε then set ξ�+ 1

2w,m1

= ξw [Patm, Tres,WCO2 = 0,WNaCl].

5.4.3. Use φ�+ 1

2m1

calculated above; if φ�+ 1

2m1

< ε then S�+1wm1

= 0.

5.4.4. If φ�+ 1

2m1

> ε then

S�+1wm1

=Δt

VR

β�+1m1

φ�+ 1

2m1

ξ�+ 1

2w,m1

(4.63)

5.4.5. Ensure that 0 ≤ S�+1wm1≤ 1

6. In psim new primary from flash ell, call psim primary iterate WCO2 ell. Use an iterative method

to calculate W �+1CO2,m1

and ξ�+1w,m1

. See Section 4.8 for the details.

7. In psim new primary from flash ell, update So, Sg, Xm′ , and Ym′ .


≤ 1 and that∑

mX�+1m,m1

= 1.


≤ 1 and that∑

m Y �+1m,m1

= 1.

7.3. Calculate S�+1om1

=(1− S�+1

w,m1)L�+1

m1ξ�+1g,m1

L�+1m1

ξ�+1g,m1

+ V �+1m1

ξ�+1o,m1

7.4. Calculate S�+1gm1

=(1− S�+1

w,m1)V �+1

m1ξ�+1o,m1

L�+1m1

ξ�+1g,m1

+ V �+1m1

ξ�+1o,m1

8. In psim new primary from flash ell, flash Z�+1m,2ph,m2

and calculate X�+1m,m2

, Y �+1m,m2

, and S�+1wm2

.

8.1. Flash Z�+1m,2ph,m2

at P �+1om2

to calculate X�+1mm2

, Y �+1mm2

, ξ�+1om2

, ξ�+1gm2

, L�+1m2

, V �+1m2

.


≤ 1 and that∑

mX�+1m,m2

= 1.


≤ 1 and that∑

m Y �+1m,m2

= 1.

91

8.4. If S�wm2

< εS or S�wm2

> 1− εS, calculate S�+1wm2

here.

8.4.1. Calculate an approximate value for ξ�+ 1

2w,m2

. Accumulate composition derivative at

this point.

ξ�+ 1

2w,m2

= ξ�wm2+

∂ξwm2

∂PδP +

∂ξwm2

∂Ym′δYm′ +

∂ξwm2

∂Xm′δXm′ (4.64)

8.4.2. Use ξ�+ 1

2w,m2

calculated above; if ξ�+ 1

2w,m2

< ε then set ξ�+ 1

2w,m2

= ξw [Patm, Tres,WCO2 = 0,WNaCl].

8.4.3. Use φ�+ 1

2m2

calculated above; if φ�+ 1

2m2

< ε then S�+1wm2

= 0.

8.4.4. If φ�+ 1

2m2

> ε then

S�+1wm2

=Δt

VR

β�+1m2

φ�+ 1

2m2

ξ�+ 1

2w,m2

(4.65)

8.4.5. Ensure that 0 ≤ S�+1wm2≤ 1

9. In psim new primary from flash ell, call psim primary iterate WCO2 ell. Use an iterative method

to calculate W �+1CO2,m2

and ξ�+1w,m2

. See Section 4.8 for the details.

10. In psim new primary from flash ell, update So, Sg, Xm′ , and Ym′ .


≤ 1 and that∑

mX�+1m,m2

= 1.


≤ 1 and that∑

m Y �+1m,m2

= 1.

10.3. Calculate S�+1om2

=(1− S�+1

w,m2)L�+1

m2ξ�+1g,m2

L�+1m2

ξ�+1g,m2

+ V �+1m2

ξ�+1o,m2

10.4. Calculate S�+1gm2

=(1− S�+1

w,m2)V �+1

m2ξ�+1o,m2

L�+1m2

ξ�+1g,m2

+ V �+1m2

ξ�+1o,m2

11. In psim new primary from flash ell, update φm1 , φm2 , φt.

11.1. Calculate φ�+1t as a nonlinear function of P �+1

om1

φ�+1t = φ#

ref exp[Cφ ×

(P �+1om1− P#

ref

)](4.66)

11.2. Calculate φ�+1m1

and φ�+1m2

:

92

φ�+1m1

= φ�m1× φ�+1

t

φ�t

(4.67)

φ�+1m2

= φ�m2× φ�+1

t

φ�t

(4.68)

4.8 Update WCO2

In psim primary iterate WCO2 ell, use an iterative method to calculate W �+1CO2,m1

and ξ�+1w,m1

. Up-

date S�+1w if necessary. The calculation of W �+1

CO2,m2and ξ�+1

w,m2follows the same procedure in

psim primary iterate trap WCO2 ell.

1. Calculate P �+1, αn, α�+1, βn, β�+1, Unm, U �+1

m , U �+1t , Un

WAT, U�+1WAT, Z

nm,2ph, Z

�+1m,2ph, Y

�+1CO2

, and

V �+1.

2. Iterative calculation of WCO2 . Each loop, Z�+1,em,2ph, Y

�+1,eCO2,2ph

, and V �+1,e are updated.

2.1. If β�+1 < εS, set WCO2 = 0, exit iterative loop.

2.2. Calculate W �+1,eCO2,mbal from the mass balance of the CO2 component equation:

W �+1,eCO2,mbal =

U �+1m − Z�+1,e

m,2phα�+1 + Zn

m,2phαn +W n

CO2βn

β�+1(4.69)

2.3. Calculate the CO2 solubility, W �+1,eCO2,sol

:

W �+1,eCO2,sol

=

⎧⎪⎨⎪⎩

WCO2 [P�+1o , Y �+1,e

CO2= 1] S�+1

w > 1− εS =⇒ Sw ≈ 1

WCO2 [P�+1b , Y �+1,e

CO2[Pb]] V �+1,e < εV =⇒ Sg ≈ 0

WCO2 [P�+1o , Y �+1,e

CO2] V �+1,e ≥ εV =⇒ Sg > 0

(4.70)

2.4. If∣∣∣W �+1,e

CO2,mbal −W �+1,eCO2,sol

∣∣∣ < εWCO2, set WCO2 = W �+1,e

CO2,mbal, exit iterative loop.

2.5. For m = 1 . . . NC − 1, compute M �+1,em,2ph,m1

:

M �+1,em,2ph = Z�+1,e−1

m,2ph α�+1 (4.71)

2.6. Calculate ΔM �+1,eCO2

:

ΔM �+1,eCO2

= M �+1,eCO2,mbal −M �+1,e

CO2,sol= β�+1

(W �+1,e

CO2,mbal −W �+1,eCO2,sol

)(4.72)

93

2.7. If W �+1,eCO2,mbal < W �+1,e

CO2,soland |ΔM �+1,e

CO2| > M �+1,e

CO2,2ph

ΔM �+1,eCO2

= −M �+1,eCO2,2ph

(4.73)

2.8. Update M �+1,eCO2,2ph

M �+1,eCO2,2ph

= M �+1,eCO2,2ph

+ΔM �+1,eCO2

(4.74)

2.9. If∑m′

M �+1,em′,2ph < ε, all mass is now in the water phase

• S�+1w = 1

• Z�+1,eCO2,2ph

= 1

Otherwise,

Z�+1,em,2ph =

M �+1,em,2ph∑

m′M �+1,e

m′,2ph

(4.75)

Ensure 0 ≤ Z�+1,em,2ph ≤ 1 and

∑m′

Z�+1,em′,2ph.

2.10. Flash Z�+1,em,2ph at P �+1

o to calculate X�+1,em , Y �+1,e

m , ξ�+1,eo , ξ�+1,e

g , L�+1,e, V �+1,e.

2.11. If S�w > 1− εS and ΔM �+1,e

CO2> 0

S�+1w =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

1− ΔM �+1,eCO2

ξ�+1,eo

, V �+1,e < εV

1− ΔM �+1,eCO2

ξ�+1,eg

, V �+1,e ≥ εV

(4.76)

3. Calculate ξ�+1w

94

CHAPTER 5

TRAPPING FORMULATION

This chapter describes different mathematical formulation options for including trapping in a

compositional reservoir simulation model.

5.1 Trapping Variables

• m1 mobile oil

• m2 trapped oil

• m matrix

• f fracture

• Wm[m1

m2],Xm[m1

m2], Ym[m1

m2], Zm[m1

m2] mole fractions

• ξo[m1m2], ξg,[m1

m2], ξw,[m1

m2] molar densities

• Som, Sgm, Swm matrix saturations

• Som1 , Sgm1 , Swm1 mobile saturations

• Som2 , Sgm2 , Swm2 immobile, trapped, or bypassed saturations

• Mom,[m1

m2], Mgm,[m1

m2], Mwm,[m1

m2] molar mass of each component in each phase

• τomm1/m2, τgmm1/m2

, τwmm1/m2transfer from mobile phase to trapped phase

• Vo,[m1

m2], Vg,[m1

m2], Vw,[m1

m2] volumes of each phase

• φm matrix porosity

• φm1 mobile matrix porosity

• φm2 trapped matrix porosity

• VR rock volume

95

5.2 Initialize Trapping

Initialize the properties for the total system:

1. Define the initial pressure at a specific depth Pom,ijk.

2. Define the initial water saturation for all grid cells Swm,ijk. The initial water saturation will

vary by rock type and depth.

3. Define the initial total composition Zmm as a constant in all grid cells.

4. Flash Zmm,ijk at Pom,ijk to calculate Xmm,ijk, Ymm,ijk, Wmm,ijk, ξom,ijk, ξgm,ijk, ξwm,ijk.

5. Compute the initial oil and gas saturation Som,ijk and Sgm,ijk based on lm,ijk, vm,ijk, ξom,ijk,

ξgm,ijk, and Swm,ijk.

Set the properties of the mobile phase m1 to the total properties.

Pom1,ijk = Pom,ijk Pom2,ijk = Pom,ijk (5.1)

φm1,ijk = φm,ijk φm2,ijk = 0 (5.2)

Xmm1,ijk = Xmm,ijk Xmm2,ijk = Xmm,ijk (5.3)

Ymm1,ijk = Ymm,ijk Ymm2,ijk = Ymm,ijk (5.4)

Wmm1,ijk = Wmm,ijk Wmm2,ijk = Wmm,ijk (5.5)

Som1,ijk = Som,ijk Som2,ijk = 0 (5.6)

Sgm1,ijk = Sgm,ijk Sgm2,ijk = 0 (5.7)

Swm1,ijk = Swm,ijk Swm2,ijk = 0 (5.8)

ξom1,ijk = ξom,ijk ξom2,ijk = ξom,ijk (5.9)

ξgm1,ijk = ξgm,ijk ξgm2,ijk = ξgm,ijk (5.10)

ξwm1,ijk = ξwm,ijk ξwm2,ijk = ξwm,ijk (5.11)

Specify the amount of trapping at initial conditions: Snew trapom , Snew trap

gm , Snew trapwm .

5.3 Update Trapping

This section describes the procedure for transferring mass between the mobile and trapped

phases. Portions of this procedure were also used to initialize the trapped and mobile saturations.

5.3.1 Input

At the time step level, update the amount of trapping. This may happen at initialization,

at specific transitions like the end of the waterflood or each WAG cycle. Trapping may also be

96

updated when a saturation switches from increasing to decreasing or increasing to decreasing for

any specific grid cell at time n. Trapping could also be updated at each timestep n.

When an incremental amount of trapping occurs, this transfers mass from the mobile m1 phase

to the immobile m2 phase. The amount of mass transfer is based on the newly trapped saturation

Snew trapgm , Snew trap

om , or Snew trapwm . The densities and mole fractions are based on the upstream

properties from m1.

5.3.2 Mass at Time n

The molar mass for each component and each phase is defined as follows:

Mnom[m1

m2]

= Xnm[m1

m2]ξno[m1m2]Sno[m1m2]φn

[m1m2]VR (5.12)

Mngm[m1

m2]

= Y nm[m1

m2]ξng[m1m2]Sng[m1m2]φn

[m1m2]VR (5.13)

Mnwm[m1

m2]

= W nm[m1

m2]ξnw[m1

m2]Snw[m1

m2]φn

[m1m2]VR (5.14)

The transfer from the mobile phases to the trapped phases happens when a saturation switches

from increasing to decreasing.

τnomm1/m2= Xn

mm1ξnom1

Snew trapom φn

mVR (5.15)

τngmm1/m2= Y n

mm1ξngm1

Snew trapgm φn

mVR (5.16)

τnwmm1/m2= W n

mm1ξnwm1

Snew trapwm φn

mVR (5.17)

5.3.3 Transfer Mass

If oil is trapped, adjust the oil phase masses in the following way:

Mnewomm1

= Mnomm1

− τnomm1/m2Mnew

omm2= Mn

omm2+ τnomm1/m2

(5.18)

If gas is trapped, adjust the gas phase masses in the following way:

Mnewgmm1

= Mngmm1

− τngmm1/m2Mnew

gmm2= Mn

gmm2+ τngmm1/m2

(5.19)

If water is trapped, adjust the gas phase masses in the following way:

Mnewwmm1

= Mnwmm1

− τnwmm1/m2Mnew

wmm2= Mn

wmm2+ τnwmm1/m2

(5.20)

97

5.3.4 Update Mole Fractions

If oil or gas is trapped, define Znewm[m1

m2],hc

as follows:

Znewm=1...NC−2,[m1

m2],hc

=

Mnewom[m1

m2]+Mnew

gm[m1m2]

NC−1∑m=1

(Mnew

om[m1m2]+Mnew

gm[m1m2]

) Znewm=NC−1,[m1

m2],hc

= 1−NC−2∑m=1

Znewm[m1

m2],hc

(5.21)

Flash Znewm[m1

m2],hc

at Pn to calculate Xnewm[m1

m2], Y new

m[m1m2], ξnew

o[m1m2], and ξnew

g[m1m2].

If water is trapped, define W newm[m1

m2]as follows:

W newm[m1

m2]=

Mnewwm[m1

m2]

NC∑m=NC−1

Mnewwm[m1

m2]

(5.22)

Compute the aqueous density using W newm[m1

m2]−→ ξnew

w[m1m2].

5.3.5 Compute the Volumes

The mobile and immobile volumes in each phase are calculated as follows:

V newo[m1m2]=

NC∑m=1

Mnewom[m1

m2]

ξnewo[m1m2]

V newg[m1m2]=

NC∑m=1

Mnewgm[m1

m2]

ξnewg[m1m2]

V neww[m1

m2]=

NC∑m=1

Mnewwm[m1

m2]

ξneww[m1

m2]

(5.23)

5.3.6 Compute the Saturations

Compute the saturations based on the volume fractions:

Snewo[m1m2]

=

V newo[m1m2]

V newo[m1m2]+ V new

g[m1m2]+ V new

w[m1m2]

(5.24)

Snewg[m1m2]

=

V newg[m1m2]

V newo[m1m2]+ V new

g[m1m2]+ V new

w[m1m2]

(5.25)

Sneww[m1

m2]

=

V neww[m1

m2]

V newo[m1m2]+ V new

g[m1m2]+ V new

w[m1m2]

(5.26)

Note that these saturation definitions have the following implications:

98

Snewo[m1m2]+ Snew

g[m1m2]+ Snew

w[m1m2]= 1 (5.27)

The total matrix saturations are also computed based on the volume fractions:

Snewom =

V newom1

+ V newom2

V newom1

+ V newgm1

+ V newwm1

+ V newom2

+ V newgm2

+ V newwm2

(5.28)

Snewgm =

V newgm1

+ V newgm2

V newom1

+ V newgm1

+ V newwm1

+ V newom2

+ V newgm2

+ V newwm2

(5.29)

Snewwm =

V newwm1

+ V newwm2

V newom1

+ V newgm1

+ V newwm1

+ V newom2

+ V newgm2

+ V newwm2

(5.30)

Note that these saturation definitions have the following implications:

Snewom + Snew

gm + Snewwm = 1 (5.31)

The mobile and immobile porosities are calculated based on volume fractions.

φnewm1

=V newom1

+ V newgm1

+ V newwm1

V newom1

+ V newgm1

+ V newwm1

+ V newom2

+ V newgm2

+ V newwm2

φm (5.32)

φnewm2

=V newom2

+ V newgm2

+ V newwm2

V newom1

+ V newgm1

+ V newwm1

+ V newom2

+ V newgm2

+ V newwm2

φm (5.33)

Note that these porosity definitions have the following implications:

φm1 + φm2 = φm (5.34)

Som1φm1 + Som2φm2 = Somφm (5.35)

Sgm1φm1 + Sgm2φm2 = Sgmφm (5.36)

Swm1φm1 + Swm2φm2 = Swmφm (5.37)

5.4 Single Porosity Irreversible Trapping

A system with irreversible trapping can be handled as a dual porosity system with a mobile

m1 pore system and an immobile m2 pore system. Hysteresis and trapping are handled in between

time steps as a separate calculation, so there is no transfer term in (5.38). Fluids become trapped

if their saturation changes from decreasing or constant to increasing.

99

0.006328 VR ∇ ·(Xn

mm1ξnom1

knrom1

μnom1

k#m1(∇P �+1

om1− γn

om1∇D#)

)+

0.006328 VR ∇ ·(Y n

mm1ξngm1

knrgm1

μngm1

k#m1(∇P �+1

om1+∇Pn

cgom1− γn

gm1∇D#)

)+

0.006328 VR ∇ ·(Wn

mm1ξnwm1

knrwm1

μnwm1

k#m1(∇P �+1

om1−∇Pn

cowm1− γn

wm1∇D#)

)+(

Xnmm1

ξnom1q�+1om1

+ Y nmm1

ξngm1q�+1gm1

+Wnmm1

ξnwm1q�+1wm1

)=

VR

Δt

(φ�+1m1

X�+1mm1

S�+1om1

ξ�+1om1

+ φ�+1m1

Y �+1mm1

S�+1gm1

ξ�+1gm1

+ φ�+1m1

W �+1mm1

S�+1wm1

ξ�+1wm1

)−

VR

Δt

(φnm1Xn

mm1Snom1

ξnom1+ φn

m1Y nmm1

Sngm1

ξngm1+ φn

m1Wn

mm1Snwm1

ξnwm1

)(5.38)

• m1 mobile oil: unless otherwise specified, all m1 properties are updated using (5.38) and the

trapping update procedure described in Section 5.3.

• m2 trapped oil: unless otherwise specified, all m2 properties are updated only using the

trapping update procedure described in Section 5.3.

• Som, Sgm, Swm: matrix saturations can be calculated using (5.35)–(5.37).

• λom1 = krom1/μom1 ; λgm1 = krgm1/μgm1 ; λwm1 = krwm1/μwm1

• krom1 , krgm1 , krwm1 , Pcowm1 , Pcgom1 are all calculated using the total matrix saturations Som,

Sgm, and Swm. This assumes that the trapping is representative of effects that are smaller than

the core-scale so that core measurements yield krom rather than krom1 . This could work with

commercial simulators only if the endpoints are adjusted based on the trapped saturations.

• φm, φm1 , φm2 : this formulation ignores the compressibility of the m2 portion of the porosity.

If we add the φm2Xmm2Som2ξom2 and similar terms to the right hand side of (5.38), then we

have too many unknowns for the number of equations. If we change the right hand side terms

to φmXmmSomξom then the formulation has an inconsistent mass balance.

5.5 Dual Porosity as Reversible Trapping

A system with reversible trapping can be handled as a dual porosity system with a mobile m1

pore system and an immobile m2 pore system. Hysteresis and trapping are handled in between

100

time steps as a separate calculation and also as a transfer function. Fluids become trapped if their

saturation changes from decreasing or constant to increasing. Fluids can also move from m1 to m2

if the potential Ψm1 > Ψm2 . Fluids can move from m2 to m1 if the potential Ψm2 > Ψm1 . (5.39)

represents the m1 pore system.

0.006328 VR ∇ ·(Xn

mm1ξnom1

knrom1

μnom1

k#m1(∇P �+1

om1− γn

om1∇D#)

)+

0.006328 VR ∇ ·(Y n

mm1ξngm1

knrgm1

μngm1

k#m1(∇P �+1

om1+∇Pn

cgom1− γn

gm1∇D#)

)+

0.006328 VR ∇ ·(Wn

mm1ξnwm1

knrwm1

μnwm1

k#m1(∇P �+1

om1−∇Pn

cowm1− γn

wm1∇D#)

)+(

Xnmm1

ξnom1q�+1om1

+ Y nmm1

ξngm1q�+1gm1

+Wnmm1

ξnwm1q�+1wm1

)− τ �+1mm1/m2

=

VR

Δt

(φ�+1m1

X�+1mm1

S�+1om1

ξ�+1om1

+ φ�+1m1

Y �+1mm1

S�+1gm1

ξ�+1gm1

+ φ�+1m1

W �+1mm1

S�+1wm1

ξ�+1wm1

)−

VR

Δt

(φnm1Xn

mm1Snom1

ξnom1+ φn

m1Y nmm1

Sngm1

ξngm1+ φn

m1Wn

mm1Snwm1

ξnwm1

)(5.39)

(5.40) represents the m2 pore system.

τ �+1mm1/m2

=

VR

Δt

(φ�+1m2

X�+1mm2

S�+1om2

ξ�+1om2

+ φ�+1m2

Y �+1mm2

S�+1gm2

ξ�+1gm2

+ φ�+1m2

W �+1mm2

S�+1wm2

ξ�+1wm2

)−

VR

Δt

(φnm2Xn

mm2Snom2

ξnom2+ φn

m2Y nmm2

Sngm2

ξngm2+ φn

m2Wn

mm2Snwm2

ξnwm2

)(5.40)

Evaluate Pcgom1 , Pcgom2 , Pcowm1 , and Pcowm2 using the total saturations. This means there

is no capillary pressure difference between the trapped phase and the mobile phase. Given this

assumption, the transfer function is defined by:

τ �+1mm1/m2

= 0.006328 VR σ#m1/m2

k#m1/m2

(P �+1om1− P �+1

om2

)×(Xup,n

mm1/m2ξup,nom1/m2

kup,nrom1/m2

μup,nom1/m2

+Y up,nmm1/m2


μup,ngm1/m2

+W up,n

mm1/m2ξup,nwm1/m2

kup,nrwm1/m2

μup,nwm1/m2

)(5.41)

• m1 mobile oil: unless otherwise specified, all m1 properties are updated using (5.39)–(5.41)

and the trapping update procedure described in Section 5.3.

101

• m2 trapped oil: unless otherwise specified, all m2 properties are updated using (5.39)–(5.41)

and the trapping update procedure described in Section 5.3.

• Som, Sgm, Swm: matrix saturations can be calculated using (5.35)–(5.37).

• λom1 = krom1/μom1 ; λgm1 = krgm1/μgm1 ; λwm1 = krwm1/μwm1

• kro[m1

m2], krg[m1

m2], krw[m1

m2], Pcow[m1

m2], Pcgo[m1

m2]: there are four options:

– Option 1, use total matrix saturations Som, Sgm, and Swm. This assumes that the

trapping is representative of effects that are smaller than the core-scale so that core

measurements yield krom, krgm, and krwm. If this option is used, note that Pcowm1 =

Pcowm2 , Pcgom1 = Pcgom2 , krom1 = krom2 , krgm1 = krgm2 , and krwm1 = krwm2 .

– Option 2, use Som1 , Sgm1 , and Swm1 to calculate krom1 , krgm1 , krwm1 , Pcowm1 , and Pcgom1 .

Use Som2 , Sgm2 , and Swm2 to calculate krom2 , krgm2 , krwm2 , Pcowm2 , and Pcgom2 . This

assumes that the trapping, or bypassing, is representative of effects that are between the

core-scale and the reservoir grid scale. This means that core measurements represent

krom1 , krgm1 , and krwm1 and there is no direct measurement of m2.

– Option 3, reset all endpoints and then use Som1 , Sgm1 , and Swm1 to calculate krom1 , krgm1 ,

krwm1 , Pcowm1 , and Pcgom1 . Use Som2 , Sgm2 , and Swm2 to calculate krom2 , krgm2 , krwm2 ,

Pcowm2 , and Pcgom2 . The difficulty with this method is determining how to adjust the

endpoints. For the m2 system, one approach is to assume all the endpoints are 0.

– Option 4, assume Pcgom2 = 0 and Pcowm2 = 0.

– Option 2, 3, and 4 are possible in commercial simulators, although they ignore the effects

of Section 5.3.

• φm, φm1 , φm2 : this formulation considers the compressibility of the m2 portion of the porosity.

• km1/m2; calculate equivalent k to a diffusion system.

k [md] =D[10−3 cm2/s

]μo

[cp · 10−3 kg/(ms)

cp

]k�roP

[psia6.894·106 kg/(ms2)

psia

] [md

9.869 · 10−12 cm2

]≈ 5 · 10−5 md (5.42)

102

• Define σm1/m2as follows:

σm1/m2= 4

(1

min(DX/2, 5)2+

1

min(DY/2, 5)2+

1

min(DZ/2, 1)2

)= 4

(1

52+

1

52+

1

12

)(5.43)

• Upstream weighting for all properties in the transfer function (5.41).

– Pom2 > Pom1 then m2 else m1

– Pom2 − Pcgom2 > Pom1 − Pcgom1 then m2 else m1

– Pom2 + Pcowm2 > Pom1 − Pcowm1 then m2 else m1

5.6 Dual Porosity Computation Options

This section describes the computation of the Um and Jacobian matrix for the dual porosity

option. Several definitions will help simplify the notation.

T nm1/m2,mo = 0.006328 VR σ#

m1/m2k#m1/m2

×Xup,n

mm1/m2ξup,nom1/m2

kup,nrom1/m2

μup,nom1/m2

(5.44)

T nm1/m2,mg = 0.006328 VR σ#

m1/m2k#m1/m2

×Y up,nmm1/m2


μup,ngm1/m2

(5.45)

T nm1/m2,mw = 0.006328 VR σ#

m1/m2k#m1/m2

×W up,n

mm1/m2ξup,nwm1/m2

kup,nrwm1/m2

μup,nwm1/m2

(5.46)

Accm1 =VR

Δt

(φm1Xmm1Som1ξom1 + φm1Ymm1Sgm1ξgm1 + φm1Wmm1Swm1ξwm1

)(5.47)

Accm2 =VR

Δt

(φm2Xmm2Som2ξom2 + φm2Ymm2Sgm2ξgm2 + φm2Wmm2Swm2ξwm2

)(5.48)

5.6.1 Implicit Pm2

For the implicit calculation of Pm2 , both Pm1 and Pm2 are evaluated at � + 1. The m1 and m2

equations are fully coupled; Pm1 and Pm2 appear in both the m1 and the m2 equations.

Component equations for m1 system.

103

0.006328 VR ∇ ·(Xn

mm1ξnom1

knrom1

μnom1

k#m1(∇P �+1

om1− γn

om1∇D#)

)+

0.006328 VR ∇ ·(Y n

mm1ξngm1

knrgm1

μngm1

k#m1(∇P �+1

om1+∇Pn

cgom1− γn

gm1∇D#)

)+

0.006328 VR ∇ ·(Wn

mm1ξnwm1

knrwm1

μnwm1

k#m1(∇P �+1

om1−∇Pn

cowm1− γn

wm1∇D#)

)+(

Xnmm1

ξnom1q�+1om1

+ Y nmm1

ξngm1q�+1gm1

+Wnmm1

ξnwm1q�+1wm1

)+

−τ�+1mm1/m2︷︸︸︷(

T nm1/m2,mo + T n

m1/m2,mg + T nm1/m2,mw

)(P �m1

+ δPm1− P �

m2− δPm2

)=

Acc�+1m1− Accnm1

(5.49)

Component equations for the m2 system.

τ�+1mm1/m2︷︸︸︷(

T nm1/m2,mo + T n


)(P �m1

+ δPm1 − P �m2− δPm2

)= Acc�+1

m2− Accnm2

(5.50)

5.6.2 Explicit Pm2

For the explicit calculation of Pm2 , Pm1 is evaluated at � + 1 and Pm2 is evaluated at n. This

decouples the m1 and m2 equations; the m1 equation still requires a global matrix solve, but the m2

equations are now a local matrix solve.


0.006328 VR ∇ ·(Xn

mm1ξnom1

knrom1

μnom1

k#m1(∇P �+1

om1− γn

om1∇D#)

)+

0.006328 VR ∇ ·(Y n

mm1ξngm1

knrgm1

μngm1

k#m1(∇P �+1

om1+∇Pn

cgom1− γn

gm1∇D#)

)+

0.006328 VR ∇ ·(Wn

mm1ξnwm1

knrwm1

μnwm1

k#m1(∇P �+1

om1−∇Pn

cowm1− γn

wm1∇D#)

)+(

Xnmm1

ξnom1q�+1om1

+ Y nmm1

ξngm1q�+1gm1

+Wnmm1

ξnwm1q�+1wm1

)+

−τ�+1mm1/m2︷︸︸︷(

T nm1/m2,mo + T n


)(P �m1

+ δPm1− Pn

m2

)=


(5.51)

104

Component equations for the m2 system.

τ�+1mm1/m2︷︸︸︷(

T nm1/m2,mo + T n


)(P �m1

+ δPm1− Pn

m2

)= Acc�+1

m2− Accnm2

(5.52)

5.6.3 Implicit τ = 0

For the implicit calculation of τ = 0, Pm1 is evaluated at �+1 and Pm2 is not used as a primary

variable. This means that Pm1 = Pm2 . The accumulation term evaluated at � is used in the m1

equations to account for the changes in compressibility of the system. The accumulation term is

evaluated at � instead of � + 1 because it is difficult to calculate the derivatives∂Accm2∂Pm1

,∂Accm2∂Swm1

,

∂Accm2∂Som1

,∂Accm2∂Xm′m1

, and∂Accm2∂Ym′m1

.


0.006328 VR ∇ ·(Xn

mm1ξnom1

knrom1

μnom1

k#m1(∇P �+1

om1− γn

om1∇D#)

)+

0.006328 VR ∇ ·(Y n

mm1ξngm1

knrgm1

μngm1

k#m1(∇P �+1

om1+∇Pn

cgom1− γn

gm1∇D#)

)+

0.006328 VR ∇ ·(Wn

mm1ξnwm1

knrwm1

μnwm1

k#m1(∇P �+1

om1−∇Pn

cowm1− γn

wm1∇D#)

)+(

Xnmm1

ξnom1q�+1om1

+ Y nmm1

ξngm1q�+1gm1

+Wnmm1

ξnwm1q�+1wm1

)=


+ Acc�m2− Accnm2

(5.53)

5.6.4 Explicit τ = 0

For the explicit calculation of τ = 0, Pm1 is evaluated at �+1 and Pm2 is not used as a primary

variable. This means that Pm1 = Pm2 . The m2 accumulation term is ignored for the m1 equations,

which means the compressibility of the m2 system is ignored.


105

0.006328 VR ∇ ·(Xn

mm1ξnom1

knrom1

μnom1

k#m1(∇P �+1

om1− γn

om1∇D#)

)+

0.006328 VR ∇ ·(Y n

mm1ξngm1

knrgm1

μngm1

k#m1(∇P �+1

om1+∇Pn

cgom1− γn

gm1∇D#)

)+

0.006328 VR ∇ ·(Wn

mm1ξnwm1

knrwm1

μnwm1

k#m1(∇P �+1

om1−∇Pn

cowm1− γn

wm1∇D#)

)+(

Xnmm1

ξnom1q�+1om1

+ Y nmm1

ξngm1q�+1gm1

+Wnmm1

ξnwm1q�+1wm1

)=


(5.54)

5.7 Computation of the Solution of a Dual Porosity System

The way of solving a dual porosity compositional system is similar to the way of solving a single

porosity system described in Chapter 4, with the following differences. Use (5.39)–(5.41) instead

of (3.1). This leads to twice the number of equations and primary variables as the single porosity

system. The order of the equations in the LU decomposition is slightly different. At each step,

both the m1 and the m2 properties are calculated, rather than only the m1 properties.

The process involves the following steps:

1. Assemble the Jacobian. The ordering of the primary variables and equations listed here is

for the gas/oil/water system with WCO2 as a primary variable. The simplified systems for

gas/oil, gas/water, water only, both with and without WCO2 as a primary variable can be

created by deleting the rows which aren’t applicable.

• The primary variables are ordered as follows: Pom1 , Pom2 , Swm1 , Swm2 , WCO2m1 , WCO2m2 ,

Som1 , Xm=1,m1 , Xm=2,m1 , Xm=3,m1 (up to Xm=NC−2,m1), Ym=1,m1 , Ym=2,m1 , Ym=3,m1

(up to Ym=NC−2,m1), Som2 , Xm=1,m2 , Xm=2,m2 , Xm=3,m2 (up to Xm=NC−2,m2), Ym=1,m2 ,

Ym=2,m2 , Ym=3,m2 (up to Ym=NC−2,m2)

• The equations are ordered as follows: CH2O,m1 , CH2O,m2 , CCO2,m1 , CCO2,m2 , Gg/w,CO2,m1,

Gg/w,CO2,m2, Cm=1,m1 , Cm=2,m1 , Cm=3,m1 (up to CNC−2,m1), Gm=1,m1 , Gm=2,m1 , Gm=3,m1 ,

Gm=4,m1 (up to GNC−1,m1), Cm=1,m2 , Cm=2,m2 , Cm=3,m2 (up to CNC−2,m2), Gm=1,m2 ,

Gm=2,m2 , Gm=3,m2 , Gm=4,m2 (up to GNC−1,m2)

2. Calculate Unmm1

, Unmm2

, αnm1, αn

m2, βn

m1, βn

m2.

106

3. Solve the matrix equation for P[�+1]om1

.

4. Back substitute to calculate P[�+1]om2

, S[�+1]wm1

, S[�+1]wm2

, W[�+1]CO2m1

, W[�+1]CO2m2

5. Compute U[�+1]mm1

, U[�+1]mm2

, U[�+1]tm1

, and U[�+1]tm2

as a function of P[�+1]om1

and P[�+1]om2

.

6. Compute φ[�+1]m1

and φ[�+1]m2


and P[�+1]om2

7. If WCO2 is a secondary variable, compute W[�+1]CO2m1

and W[�+1]CO2m2


and

P[�+1]om2

. Compute ξ[�+1]wm1

and ξ[�+1]wm2


, P[�+1]om2

, W[�+1]CO2m1

, and W[�+1]CO2m2

.

8. Compute β[�+1]m1

and β[�+1]m2

as a function of φ[�+1]m1

, S[�+1]wm1

, ξ[�+1]wm1

, φ[�+1]m2

, S[�+1]wm2

, and ξ[�+1]wm2

.

9. Compute α[�+1]m1

and α[�+1]m2

as a function of U[�+1]tm1

, β[�+1]m1

, U[�+1]tm2

, and β[�+1]m2

.

10. Compute Z[�+1]2ph,m,m1

and Z[�+1]2ph,m,m2

as a function of U[�+1]mm1

, α[�+1]m1

, U[�+1]mm2

, and α[�+1]m2

.

11. Flash Z[�+1]2ph,m,m1

at P[�+1]om1

to calculate X[�+1]mm1

, Y[�+1]mm1

, ξ[�+1]om1

, and ξ[�+1]gm1

. Flash Z[�+1]2ph,m,m2

at

P[�+1]om2

to calculate X[�+1]mm2

, Y[�+1]mm2

, ξ[�+1]om2

, and ξ[�+1]gm2

.

12. Calculate S[�+1]om1

, S[�+1]gm1

, S[�+1]om2

, S[�+1]gm2

.

107

CHAPTER 6

TIME DERIVATIVES FORMULATION

All the accumulation terms are local to a specific cell. The notation in this section uses i to

represent this cell. This applies equally well to a 1D, 2D, or 3D cell.

The accumulation term

∂Accmi

∂t=

∂

∂t

(φiξoiSoiXmi + φiξgiSgiYmi + φiξwiSwiWmi

)(6.1)

Evaluate the Taylor series expansion of (6.1).

∂Accmi

∂t=

Accn+1mi − Accnmi

tn+1 − tn≈ 1

Δt

(Acc�+1

mi − Accnmi

)(6.2)

Expand Acc�+1mi for NC components; all terms are evaluated for cell i and component m.

Acc�+1mi = Acc�mi+

∂Acc�mi

∂PiδPi+

∂Acc�mi

∂SoiδSoi+

∂Acc�mi

∂SgiδSgi+

NC−2∑m′=1

(∂Acc�mi

∂Xm′iδXm′i +

∂Acc�mi

∂Ym′iδYm′i

)

(6.3)

6.1 Pressure Derivatives



∂Acc�mi

∂P= ξ�oiS

�oiX

�mi

∂φ�i

∂P+ ξ�giS

�giY

�mi

∂φ�i

∂P+ φ�

iS�oiX

�mi

∂ξ�oi∂P

+ φ�iS

�giY

�mi

∂ξ�gi∂P

(6.4)



∂Acc�mi

∂P= ξ�oiS

�oiX

�mi

∂φ�i

∂P+ ξ�giS

�giY

�mi

∂φ�i

∂P+ ξ�wiS

�wiW

�mi

∂φ�i

∂P+

φ�iS

�oiX

�mi

∂ξ�oi∂P

+ φ�iS

�giY

�mi

∂ξ�gi∂P

+ φ�iS

�wiW

�mi

∂ξ�wi

∂P+ φ�

iξ�wiS

�wi

∂W �CO2,i

∂P(6.5)



108

∂Acc�mi

∂P= ξ�wiS

�wiW

�mi

∂φ�i

∂P+ φ�

iS�wiW

�mi

∂ξ�wi

∂P− φ�

iξ�wiS

�wi

∂W �CO2,i

∂P(6.6)

The porosity increases with depth at constant overburden stress.

φ[P ] = φref · exp [Cφ · (P − Pref)] ≈ φref · (1 + Cφ · (P − Pref)) (6.7)

Use the definition of Cφ.

Cφ =1

φ

∂φ

∂P

∂φ

∂P= φCφ (6.8)

6.2 Saturation Derivatives


.

∂Acc�mi

∂So= φ�

iξ�oiX

�mi − φ�

iξ�giY

�mi (6.9)

Evaluate∂Acc�mi∂Sw

.

∂Acc�mi

∂Sw= φ�

iξ�wiW

�mi − φ�

iξ�giY

�mi (6.10)


= 0.


= 0

6.3 Composition Derivatives



m.

∂Acc�mi

∂Xm′= φ�

iS�oiX

�mi

∂ξ�oi∂Xm′

+ φ�iξ

�oiS

�oiδm,m′ (6.11)



m.

∂Acc�mi

∂Ym′= φ�

iS�giY

�mi

∂ξ�gi∂Ym′

+ φ�iξ

�giS

�giδm,m′ (6.12)

109


m.

∂Acc�CO2,i

∂Xm′= φ�

iS�oiX

�CO2,i

∂ξ�oi∂Xm′

− φ�iS

�oiξ

�oi + φ�

iS�wiWCO2

∂ξ�wi

∂Xm′+ φ�

iS�wiξ

�wi

∂WCO2

∂Xm′(6.13)


m.

∂Acc�mi

∂Ym′= φ�

iS�giY

�CO2

∂ξ�gi∂Ym′

− φ�iS

�giξ

�gi + φ�

iS�wiWCO2

∂ξ�wi

∂Ym′+ φ�

iS�wiξ

�wi

∂WCO2

∂Ym′(6.14)


∂Acc�mi∂X′

m.

∂Acc�H2O,i

∂Xm′= φ�

iS�wiWH2O

∂ξ�wi

∂Xm′− φ�

iS�wiξ

�wi

∂WCO2

∂Xm′(6.15)


∂Acc�mi∂Y ′

m.

∂Acc�H2O,i

∂Ym′= φ�

iS�wiWH2O

∂ξ�wi

∂Ym′− φ�

iS�wiξ

�wi

∂WCO2

∂Ym′(6.16)

110

CHAPTER 7

SPACE DERIVATIVES FORMULATION

This chapter describes the mathematical expansion of the spatial derivatives in the finite dif-

ference expansion of the partial differential equations.

7.1 Initial Expansion

This section assumes implicit pressure, explicit saturation, and explicit composition. Start with

(3.1); multiply through by the rock volume VRi = ΔxiΔyiΔzi.

0.006328 · VRi∇ ·(Xmξoλok(∇Po − γo∇D)

)+

0.006328 · VRi∇ ·(Ymξgλgk(∇Po +∇Pcgo − γg∇D)

)+

0.006328 · VRi∇ ·(Wmξwλwk(∇Po −∇Pcow − γw∇D)

)+(Xmξoqo + Ymξgqg +Wmξwqw

)=

VRi∂

∂t


)(7.1)

Write the finite difference expansion of Um. Each of the terms are labeled.

Unm(Pn+1) =

LHS1︷︸︸︷(Xmξoqo + Ymξgqg +Wmξwqw)

ni

+

LHS2︷︸︸︷ΔyiΔzi

((Xmξoλokxx)

ni+ 1

2

Δxi+ 12

· (Pn+1i+1 − Pn+1

i

))−


((Xmξoλokxx)

ni− 1

2

Δxi− 12

· (Pn+1i − Pn+1

i−1

))

+


((Ymξgλgkxx)

ni+ 1

2

Δxi+ 12

· (Pn+1i+1 − Pn+1

i

))−


((Ymξgλgkxx)

ni− 1

2

Δxi− 12

· (Pn+1i − Pn+1

i−1

))

+


((Wmξwλwkxx)

ni+ 1

2

Δxi+ 12

· (Pn+1i+1 − Pn+1

i

))−


((Wmξwλwkxx)

ni− 1

2

Δxi− 12

· (Pn+1i − Pn+1

i−1

))

−


((Xmξoλokxx)

ni+ 1

2

Δxi+ 12

γno,i+ 1

2· (Di+1 −Di)

)−


((Xmξoλokxx)

ni− 1

2

Δxi− 12

γno,i− 1

2· (Di −Di−1)

)

+ · · ·Continued in next equation · · · (7.2)

111

Unm(Pn+1) = · · ·Continued from previous equation · · ·+

−


((Ymξgλgkxx)

ni+ 1

2

Δxi+ 12

γng,i+ 1

2· (Di+1 −Di)

)−


((Ymξgλgkxx)

ni− 1

2

Δxi− 12

γng,i− 1

2· (Di −Di−1)

)

−


((Wmξwλwkxx)

ni+ 1

2

Δxi+ 12

γnw,i+ 1

2· (Di+1 −Di)

)−


((Wmξwλwkxx)

ni− 1

2

Δxi− 12

γnw,i− 1

2· (Di −Di−1)

)

+


((Ymξgλgkxx)

ni+ 1

2

Δxi+ 12

· (Pncgo,i+1 − Pn

cgo,i

))−


((Ymξgλgkxx)

ni− 1

2

Δxi− 12

· (Pncgo,i − Pn

cgo,i−1

))

−


((Wmξwλwkxx)

ni+ 1

2

Δxi+ 12

· (Pncow,i+1 − Pn

cow,i

))+


((Wmξwλwkxx)

ni− 1

2

Δxi− 12

· (Pncow,i − Pn

cow,i−1

))(7.3)

7.2 Transmissibility

There are several transmissibilities used in this formulation. The following is an example in the

x-direction, where X = X|Y |W , ξo = ξo|ξg|ξw, μo = μo|μg|μw, and ± is either positive or negative

everywhere.

Tmnxo,i± 1

2

= 0.006328 ·ΔyiΔzi

Xnm,i± 1

2

ξno,i± 1

2

knro,i± 1

2

k#xx,i± 1

2

μno,i± 1

2

Δx#i± 1

2

(7.4)

This is an example in the y-direction:

Tmnyo,j± 1

2= 0.006328 ·ΔxjΔzj

Xnm,j± 1

2

ξno,j± 1

2

knro,j± 1

2

k#yy,j± 1

2

μno,j± 1

2

Δy#j± 1

2

(7.5)

This is an example in the z-direction:

Tmnzo,k± 1

2= 0.006328 ·ΔxkΔyk

Xnm,k± 1

2

ξno,k± 1

2

knro,k± 1

2

k#zz,k± 1

2

μno,k± 1

2

Δz#k± 1

2

(7.6)

7.3 Expand Deltas

Approximate Pn+1.

112

Pn+1 ≈ P �+1

δP = P �+1 − P �

Pn+1 ≈ P � + δP

(7.7)

7.4 Expand Terms on Left-Hand-Side

Substitute (7.7) into (7.2), LHS2.

LHS2 =

(Tmnxo,i+ 1

2

·(P �i+1 + δPi+1 − P �

i − δPi

))(7.8)


LHS3 =

(Tmnxo,i− 1

2·(P �i + δPi − P �

i−1 − δPi−1

))(7.9)


LHS4 =

(Tmnxg,i+ 1

2·(P �i+1 + δPi+1 − P �

i − δPi

))(7.10)


LHS5 =

(Tmnxg,i− 1

2

·(P �i + δPi − P �

i−1 − δPi−1

))(7.11)


LHS6 =

(Tmnxw,i+ 1

2·(P �i+1 + δPi+1 − P �

i − δPi

))(7.12)


LHS7 =

(Tmnxw,i− 1

2

·(P �i + δPi − P �

i−1 − δPi−1

))(7.13)


LHS8 =

(Tmnxo,i+ 1

2γno,i+ 1

2·(Di+1 −Di

))(7.14)

113


LHS9 =

(Tmnxo,i− 1

2γno,i− 1

2·(Di −Di−1

))(7.15)


LHS10 =

(Tmnxg,i+ 1

2γng,i+ 1

2·(Di+1 −Di

))(7.16)


LHS11 =

(Tmnxg,i− 1

2γng,i− 1

2·(Di −Di−1

))(7.17)


LHS12 =

(Tmnxw,i+ 1

2γnw,i+ 1

2·(Di+1 −Di

))(7.18)


LHS13 =

(Tmnxw,i− 1

2γnw,i− 1

2·(Di −Di−1

))(7.19)


LHS14 =

(Tmnxg,i+ 1

2·(Pncgo,i+1 − Pn

cgo,i

))(7.20)


LHS15 =

(Tmnxg,i− 1

2·(Pncgo,i − Pn

cgo,i−1

))(7.21)


114

LHS16 =

(Tmnxw,i+ 1

2·(Pncow,i+1 − Pn

cow,i

))(7.22)


LHS17 =

(Tmnxw,i− 1

2·(Pncow,i − Pn

cow,i−1

))(7.23)

7.5 Rearrange Terms

There are 24 + 20 terms in equations (7.8)–(7.23). These terms need to be rearranged in the

following way:

• Multiples of δP at i and i± 1; these will end up in A in the matrix equation.

• Terms which do not multiply a δ; these will end up in R in the matrix equation.


DPmnxt,i±1 =

(Tmnxo,i± 1

2+ Tmn

xg,i± 12+ Tmn

xw,i± 12

)(7.24)


DPmnxt,i = −

(DPmn


)=

−(Tmnxo,i+ 1

2

+ Tmnxo,i− 1

2

+ Tmnxg,i+ 1

2

+ Tmnxg,i− 1

2

+ Tmnxw,i+ 1

2

+ Tmnxw,i− 1

2

)(7.25)


DCmn�xt,i±1 = −Tmn

xo,i± 12·(P �i±1 − γn

o,i± 12Di±1

)− Tmn

xg,i± 12·(P �i±1 − γn

g,i± 12Di±1 + Pn

cgo,i±1

)+

− Tmnxw,i± 1

2·(P �i±1 − γn

w,i± 12Di±1 − Pn

cow,i±1

)(7.26)


115

DCmn�xt,i = Tmn

xo,i+ 12

·(P �i − γn

o,i+ 12

Di

)+ Tmn

xo,i− 12

·(P �i − γn

o,i− 12

Di

)+

Tmnxg,i+ 1

2·(P �i − γn

g,i+ 12Di + Pn

cgo,i

)+ Tmn

xg,i− 12·(P �i − γn

g,i− 12Di + Pn

cgo,i

)+

Tmnxw,i+ 1

2·(P �i − γn

w,i+ 12Di − Pn

cow,i

)+ Tmn

xw,i− 12·(P �i − γn

w,i− 12Di − Pn

cow,i

)(7.27)

7.6 Combine Terms

The final form of Umx is:

Umn�xi =

(DPmn

xt,i+1,jkδPi+1,jk +DPmnxt,ijkδPijk + DPmn

xt,i−1,jkδPi−1,jk+

−DCmn�xt,i+1,jk − DCmn�

xt,ijk − DCmn�xt,i−1,jk

)(7.28)

The final form of Umy is:

Umn�yi =

(DPmn

yt,ij+1,kδPij+1,k + DPmnyt,ijkδPijk + DPmn

yt,ij−1,kδPij−1,k+

−DCmn�yt,ij+1,k − DCmn�

yt,ijk − DCmn�yt,ij−1,k

)(7.29)

The final form of Umz is:

Umn�zi =

(DPmn

zt,ijk+1δPijk+1 + DPmnzt,ijkδPijk + DPmn

zt,ijk−1δPijk−1+

−DCmn�zt,ijk+1 −DCmn�

zt,ijk − DCmn�zt,ijk−1

)(7.30)

Umn�ti =

(Un�mxi + Un�

myi + Un�mzi

)+ (Xmξoqo + Ymξgqg +Wmξwqw)

ni (7.31)

7.7 Upstream Weighting

At time n, it is necessary to evaluate which cells are upstream of other cells in order to calculate

the appropriate i± 12 terms. This computation involves the following equations

Ψnoi± 1

2

=((Pn

i±1 − γnoi±1D#i±1)− (Pn

i − γnoiD#i ))

(7.32)

Ψngi± 1

2

=((Pn

i±1 − γngi±1D#i±1 + Pn

cgo,i±1)− (Pni − γngiD

#i + Pn

cgo,i))

(7.33)

Ψnwi± 1

2=((Pn

i±1 − γnwi±1D#i±1 − Pn

cow,i±1)− (Pni − γnwiD

#i − Pn

cow,i))

(7.34)

For normal evaluation, evaluate all the fluid properties at the upstream node. This applies to

the following properties.

• γnϕ,i± 1

2

, μnϕ,i± 1

2

, ξnϕ,i± 1

2

, knrϕ,i± 1

2

116

• Xnm,i± 1

2

, Y nm,i± 1

2

, W nm,i± 1

2

The upstream weighting for all of these properties is defined by (7.35), using a generic variable

χ.

χnϕ,i± 1

2=⟨χnϕ,i, χ

nϕ,i±1

⟩U=

{χnϕ,i±1 Ψn

ϕ,i± 12

> 0

χnϕ,i Ψn

ϕ,i± 12

≤ 0(7.35)

Evaluate the permeability as a weighted harmonic average:

kxx,i± 12

Δxi± 12

=

⟨kxx,iΔxi

,kxx,i±1

Δxi±1

⟩H

=2

Δxikxx,i

+ Δxi±1

kxx,i±1

(7.36)

Using a combination of (7.35) and (7.36),

Tmnxo,i± 1

2= 0.006328·ΔyiΔzi ·

⟨(Xn

mξno knro

μno

)i

,

(Xn

mξno knro

μno

)i±1

⟩U

·⟨(

kxxΔx

)i

,

(kxxΔx

)i±1

⟩H

(7.37)

7.8 Time Steps

The maximum time step size is determined by the “CFL” constraint, based on the original

paper, Courant et al. (1967). The basic CFL constraint is defined for IMPES models by:

uΔt

Δx≤ 1 (7.38)

For practical reasons, it is often better to use:

uΔt

Δx≤ 0.1 (7.39)

The CFL constraint is defined for each phase across each boundary.

Δtn+1[xyz

][gow

]i± 1

2

≤ 0.1φiVRi (ξoiSoi + ξgiSgi + ξwiSwi)∣∣∣∣∑m

Tmn[xyz

][gow

]i± 1

2

Ψn[xyz

][gow

]i± 1

2

∣∣∣∣(7.40)

Another constraint is that the volumes moving through a grid cell at any time should also be

less than 10% cell pore volumes in a time step. For a fixed rate injector:

117

Δtn+1well,i ≤

0.1φiVRi

|qt,w| (7.41)

For a fixed pressure injector:

Δtn+1well,i ≤

0.1φiVRi

| −WIiλnt,i(P

ni − Pn

w,i)|(7.42)

For a fixed rate producer:

Δtn+1well,i ≤

0.1φiVRi

|qnt,i|(7.43)

For a fixed pressure producer:

Δtn+1well,i ≤

0.1φiVRi

|(−WIiλnt,i(P

ni − Pn

w,i))|(7.44)

The eventual time step sized used is based on the minimum across all the wells and all the

interfaces:

Δtn+1 = MIN

[MIN∀w

[Δtn+1well,w],

MIN∀ijk

[Δtn+1xg,i+ 1

2,j,k

,Δtn+1xg,i− 1

2,j,k

,Δtn+1xo,i+ 1

2,j,k

,Δtn+1xo,i− 1

2,j,k

,Δtn+1xw,i+ 1

2,j,k

,Δtn+1xw,i− 1

2,j,k

],

MIN∀ijk

[Δtn+1yg,i,j+ 1

2,k,Δtn+1

yg,i,j− 12,k,Δtn+1

yo,i,j+ 12,k,Δtn+1

yo,i,j− 12,k,Δtn+1

yw,i,j+ 12,k,Δtn+1

yw,i,j− 12,k],

MIN∀ijk

[Δtn+1zg,i,j,k+ 1

2

,Δtn+1zg,i,j,k− 1

2

,Δtn+1zo,i,j,k+ 1

2

,Δtn+1zo,i,j,k− 1

2

,Δtn+1zw,i,j,k+ 1

2

,Δtn+1zw,i,j,k− 1

2

]

](7.45)

118

CHAPTER 8

EQUATION OF STATE FORMULATION

All the fugacity terms are local to a specific cell. The notation in this section uses i to represent

this cell. This applies equally well to a 1D, 2D, or 3D cell.

The Gm = 1 . . . NC − 1 fugacity equations are:

fn+1omi − fn+1

gmi = 0 (8.1)

Evaluate the Taylor series expansion for fn+1omi :

fn+1omi ≈ f�+1

omi = f�omi +∂f�omi

∂PiδPi +

NC−2∑m′=1

(∂f�omi

∂Xm′iδXm′i

)(8.2)

Evaluate the Taylor series expansion for fn+1gmi :

fn+1gmi ≈ f�+1

gmi = f�gmi +∂f�gmi

∂PiδPi +

NC−2∑m′=1

(∂f�gmi

∂Ym′iδYm′i

)(8.3)

8.0.1 Expand Fugacities

The fugacities are defined by (8.4).

flm = ΦlmXmP fvm = Φv

mYmP (8.4)

Evaluate∂fl�mi∂P , m = 1 . . . NC − 1:

∂fl�mi

∂P= X�

miP�i

∂Φl�mi

∂P+Φl�

miXm = fl�mi

(1

Φl�mi

∂Φl�mi

∂P

)+Φl�

miXm (8.5)

Evaluate∂fv�mi∂P , m = 1 . . . NC − 1:

∂fv�mi

∂P= Y �

miP�i

∂Φv�mi

∂P+Φv�

miY�mi = fv�mi

(1

Φv�mi

∂Φv�mi

∂P

)+Φv�

miY�mi (8.6)

For the normal hydrocarbon equations m = 1 . . . NC − 2, evaluate∂fl�mi∂Xm′ for m′ = 1 . . . NC − 2:

119

∂fl�mi

∂Xm′= X�

miP∂Φl�

mi

∂Xm′+Φl�

miP�i δm,m′ = fl�mi

(1

Φl�mi

∂Φl�mi

∂Xm′

)+Φl�

miPδm,m′ (8.7)

For the normal hydrocarbon equations m = 1 . . . NC − 2, evaluate∂fv�mi∂P for m′ = 1 . . . NC − 2:

∂fv�mi

∂Ym′= Y �

miP�i

∂Φv�mi

∂Ym′+Φv�

miP�i δm,m′ = fv�mi

(1

Φv�mi

∂Φv�mi

∂Ym′

)+Φv�

miPδm,m′ (8.8)


∂fl�mi

∂Xm′= X�

miP�i

∂Φl�mi

∂Xm′−Φl�

miP�i = fl�mi

(1

Φl�mi

∂Φl�mi

∂Xm′

)− Φl�

miP (8.9)

For the CO2 equations m = 1 . . . NC − 1, evaluate∂fv�mi∂P for m′ = 1 . . . NC − 2:

∂fv�mi

∂Ym′= Y �

miP�i

∂Φv�mi

∂Ym′− Φv�

miP�i = fv�mi

(1

Φv�mi

∂Φv�mi

∂Ym′

)− Φv�

miP (8.10)

8.1 Fugacity Equations - Above Bubble Point

Above the bubble point, Sg = 0. Sg is replaced by a new variable, the bubble point pressure

Pb. One of the fugacity equations (8.1) is replaced by (8.11).

GNC−1 : Pn+1b −

NC−1∑m=1

fn+1om [Pb, �X]

Φn+1gm [Pb, �Y ]

= Pn+1b −

NC−1∑m=1

fn+1om [Pb, �X ]Y n+1

m Pn+1b

fn+1gm [Pb, �Y ]

= 0 (8.11)

The other fugacity equations are evaluated at Pb for m from 1 to NC − 2.

G1...NC−2 : fn+1om [Pb, �X ]− fn+1

gm [Pb, �Y ] = 0 (8.12)

Evaluate the Taylor series expansion for GNC−1:

Gn+1NC−1,i ≈ G�+1

NC−1,i = G�NC−1,i +

∂G�NC−1,i

∂PbiδPbi+

NC−2∑m′=1

(∂G�

NC−1,i

∂Xm′iδXm′i

)+

NC−2∑m′=1

(∂G�

NC−1,i

∂Ym′iδYm′i

)(8.13)

120

The derivatives, ∂fom∂Pb

,∂fgm∂Pb

, ∂fom∂Xm′ , and

∂fgm∂Ym′ are evaluated using (8.5)–(8.8) with P → Pb. In

order to calculate both GNC−1 and the derivatives of GNC−1, define:

G�m =

f�om[Pb, �X ]Y �mP �

b

f�gm[Pb, �Y ](8.14)

Evaluate the derivative∂G�

NC−1

∂Pb:

∂GNC−1

∂Pb= 1−

NC−1∑m=1

Gm

(1

Pb+

1

fom

∂fom∂Pb

− 1

fgm

∂fgm∂Pb

)(8.15)

Evaluate the derivative for m′ = 1 . . . NC − 2, evaluate∂G�

NC−1

∂Xm′ :

∂GNC−1

∂Xm′= −

NC−1∑m=1

Gm

fom

∂fom∂Xm′

(8.16)


NC−1

∂Ym′ :

∂GNC−1

∂Ym′= −

NC−1∑m=1

(fomPbδm,m′ −Gm

∂fgm∂Ym′

)/fgm (8.17)

8.2 Fugacity Equations - Below Dew Point

Below the dew point, So = 0. So is replaced by a new variable, the dew point pressure Pd. One

of the fugacity equations (8.1) is replaced by (8.18).

GNC−1 : Pn+1d −

NC−1∑m=1

fn+1gm [Pd, �Y ]

Φn+1om [Pd, �X ]

= Pn+1d −

NC−1∑m=1

fn+1gm [Pd, �Y ]Xn+1

m Pn+1d

fn+1om [Pd, �X ]

= 0 (8.18)

The other fugacity equations are evaluated at Pd for m from 1 to NC − 2.

G1...NC−2 : fn+1om [Pd, �X ]− fn+1

gm [Pd, �Y ] = 0 (8.19)

Evaluate the Taylor series expansion for GNC−1:

121

Gn+1NC−1,i ≈ G�+1

NC−1,i = G�NC−1,i +

∂G�NC−1,i

∂PdiδPdi+

NC−2∑m′=1

(∂G�

NC−1,i

∂Xm′iδXm′i

)+

NC−2∑m′=1

(∂G�

NC−1,i

∂Ym′iδYm′i

)(8.20)

The derivatives, ∂fom∂Pd

,∂fgm∂Pd

, ∂fom∂Xm′ , and

∂fgm∂Ym′ are evaluated using (8.5)–(8.8) with P → Pd. In

order to calculate both GNC−1 and the derivatives of GNC−1, define:

G�m =

f�gm[Pd, �Y ]X�mP �

d

f�om[Pd, �X ](8.21)

Evaluate the derivative∂G�

NC−1

∂Pd:

∂GNC−1

∂Pd= 1−

NC−1∑m=1

Gm

(1

Pd+

1

fgm

∂fgm∂Pd

− 1

fom

∂fom∂Pd

)(8.22)


NC−1

∂Xm′ :

∂GNC−1

∂Xm′= −

NC−1∑m=1

(fgmPdδm,m′ −Gm

∂fom∂Xm′

)/fom (8.23)


NC−1

∂Ym′ :

∂GNC−1

∂Ym′= −

NC−1∑m=1

Gm

fgm

∂fgm∂Ym′

(8.24)

8.3 Method for Peng-Robinson Flash Calculation

The equation of state is a mechanism to calculate the Xm, Ym, ξo, and ξg.

The non-ideal gas law is defined by (8.25).

P =zRT

v(8.25)

The single component Peng-Robinson equation of state is defined by (8.26).

P =RT

v − b− a

v(v + b) + b(v − b)(8.26)

122

8.3.1 Peneloux Volume Adjustment

This section is based on Pedersen and Christensen (2007) and Ahmed (2007a). The Peng-

Robinson Equation of State with the Peneloux volume correction (Peneloux and Rauzy, 1982) is

defined by (8.27).

P =RT

v − b− a

(v + c)(v + 2c+ b) + (b+ c)(v − b)(8.27)

This results in an adjustment to the specific volumes and the densities, but does not adjust the

phase splitting.

vnew = vEOS −∑

Xmcm (8.28)

In some cases (for instance the Eclipse SSHIFT parameter sm) , the volume shift is defined as a

multiplier to the bm:

cm = bmsm (8.29)

The fugacities are adjusted as follows:

fm,new = fm,EOS exp[−cm P

RT] (8.30)

In practice, this is accomplished by adjusting the fugacity coefficient:

lnΦm,new = lnΦm,EOS − cmP

RT(8.31)

There are several correlations in the literature for initial values of the volume shift parameter.

In practice, they are typically used as fitting parameters for tuning the equation of state.

8.3.2 Constants for This Formulation

For this formulation, the temperature is constant it the initial value T#. It may be necessary

to vary the temperature with depth in the future.

Compute κm, am, and bm, constant for a specific T and P . The Ωa (Eclipse OMEGAA) and Ωb

(Eclipse OMEGAB) are defined by (8.32).

123

Ωa = 0.4572355289 Ωb = 0.0777960739 (8.32)

The κ is defined by (8.33), where ω (Eclipse ACF) is the acentric factor.

κm = 0.37464 + 1.54226ωm − 0.26992ω2m (8.33)

In 1978, Peng and Robinson defined a new κ as follows:

κm =

{0.37464 + 1.54226ωm − 0.26992ω2

m ωm ≤ 0.4910.379642 + 1.48503ωm − 0.164423ω2

m + 0.016666ω3m, ωm > 0.491

(8.34)

The a is defined by (8.35). Tcm is the critical temperature (Eclipse TCRIT) and Pcm is the critical

pressure (Eclipse PCRIT).

am =

(Ωa

R2T 2cm

Pcm

)(1 + κm

(1−

√T

Tcm

))2

(8.35)

The b is defined by (8.36).

bm = ΩbRTcm

Pcm(8.36)

The amn is defined by (8.37). The binary interaction coefficient is δmn (Eclipse BIC). In PR78, δmn

is a function of temperature.

amn = (1− δmn)a1/2m a1/2n (8.37)

8.3.3 Initial Values, Compute Km, (Full Flash Only)

If there is a pre-existing value of Ke1m ,

Ke1+1m = Ke1

m

flmfvm

(8.38)

If there is no pre-existing value of Km, compute the initial estimate of Km based on (8.39),

Wilson’s equation.

Ke1m =

Pcm

P �exp

(5.3727(1 + ωm)

(1− Tcm

T

))(8.39)

124

8.3.4 Flash to Calculate the Vapor Fraction, (Full Flash Only)

The flash computation is defined by (8.40).

f(V e2) =∑m

((Ke1

m − 1)Z�m

(Ke1m − 1)V e2 + 1

)(8.40)

The flash derivative is

∂f

∂V= −

∑m

(Ke1m − 1)2Z�

m((Ke1

m − 1)V e2 + 1)2 (8.41)

If f(0) ≤ 0 and f(1) ≤ 0, then V e2 = 0 and the components are all liquid. Define Xe1m and Y e1

m

using (8.42).

V e2 = 0 Xe1m = Z�

m Y e1m = Ke1

mZ�m (8.42)

If f(1) ≥ 0 and f(0) ≥ 0, then V e2 = 1 and the components are all vapor. Define Xe1m and Y e1

m

using (8.43).

V e2 = 1 Y e1m = Z�

m Xe1m =

Z�m

Ke1m

(8.43)

Calculate V e2 using Newton Raphson iteration with a starting value of either V e2=0 = 0.5 or the

previous estimate of V e2 . Solve for V e2 when f(V e2) = 0.

V e2+1 = V e2 − f(V e2)(∂f(V e2)

∂V

) (8.44)

Convergence is defined by

∣∣∣∣V e2+1 − V e2

V e2

∣∣∣∣ ≤ εV (8.45)

Calculate Xe1m and Y e1

m based on (8.46).

Xe1m =

Z�m

V (Ke1m − 1) + 1

Y e1m = Ke1

m ·Xe1m (8.46)

125

If a temporary step for (8.46) evaluates V e2 ≤ 0, compute the next iteration based on f(0) and

f ′(0). If a temporary step for (8.46) evaluates V e2 ≥ 1, compute the next iteration based on f(1)

and f ′(1).

Normalize∑

mXe1m = 1 and

∑m Y e1

m = 1 to avoid any numerical errors. Calculate L.

Le2 = 1− V e2 (8.47)

8.3.5 Calculate the Mixing Parameters

The multi-component Peng-Robinson equation of state is defined by a set of mixing rules.

Compute the mixed a using (8.48).

ale1 =∑m,n

amnXe1mXe1

n ave1 =∑m,n

amnYe1m Y e1

n (8.48)

The mixing rule for b is defined by (8.49).

ble1 =∑m

bmXe1m bve1 =

∑m

bmY e1m (8.49)

Define A by (8.50).

Ale1 =ale1P �

R2T 2Ave1 =

ave1P �

R2T 2(8.50)

Define B by (8.51).

Ble1 =ble1P �

RTBve1 =

bve1P �

RT(8.51)

8.3.6 Calculate the z-factor

Calculate zle3 , the smallest positive real root in (8.52), using Ale1 and Ble1 . Calculate zve3 ,

the largest real root in (8.52), using Ave1 and Bve1 .

f(z) = z3 + (B − 1)z2 + (A− 3B2 − 2B)z + (B3 −AB +B2) = 0 (8.52)

Define the following coefficients of the terms of f(z).

a0 = (B3 −AB +B2) a1 = (A− 3B2 − 2B) a2 = (B − 1) (8.53)

126

Solve the cubic equation using the methods of Section 8.8.

When A and B are constants, ∂f∂z evaluates to (8.54). This is used for the Newton-Raphson

flash calculation.

∂f

∂z

∣∣∣∣A,B

= 3z2 + 2(B − 1)z + (A− 3B2 − 2B) (8.54)

8.3.7 Calculate the Fugacities f (Not if Only Computing z)

Compute the liquid fugacity coefficients, Φle1m using (8.55).

lnΦle1m =

bmble1

(zle3 − 1)− ln (zle3 −Ble1)+

− Ale1

2√2Ble1

(2

ale1

(∑n

Xe1n amn

)− bm

ble1

)· ln

(zle3 +

(√2 + 1

)Ble1

zle3 − (√2− 1)Ble1

)− cmP

RT(8.55)

Compute the vapor fugacity coefficients, Φve1m using (8.56).

lnΦve1m =

bmbve1

(zve3 − 1)− ln (zve3 −Bve1)+

− Ave1

2√2Bve1

(2

ave1

(∑n

Y e1n amn

)− bm

bve1

)· ln

(zve3 +

(√2 + 1

)Bve1

zve3 − (√2− 1)Bve1

)− cmP

RT(8.56)

The fugacities are defined by (8.57).

fle1m = Φle1m Xe1

mP � fve1m = Φve1m Y e1

m P � (8.57)

8.3.8 Calculate the Tolerance (Full Flash Only)

Compute Re1m using (8.58).

Re1m =

fle1m

fve1m(8.58)


|Re1m − 1| ≤ εf ∀m (8.59)

If |Re1m − 1| ≥ εf for any m, compute Ke1+1

m = Re1mKe1

m and return to Section 8.3.4.

127

Eclipse uses (8.60), but it has several problems. First, it requires values at lots of nodes, whereas

the convergence in (8.59) is local to one grid cell. Second, it does not identify which grid cells are

having problems with the flash computation.

∑m

(fle1m

fve1m− 1

)2

< εf (8.60)

8.3.9 Calculate the Densities

Once the fugacities converge in Step 8, calculate the ξ using (8.61).

ξo =P

zlRTξg =

P

zvRT(8.61)

Calculate vt.

vlt =1

ξovvt =

1

ξg(8.62)

If there is a Peneloux volume adjustment factor cm, then first calculate the specific volume vt

using (8.63) and then calculate ξ using (8.64).

vlt =zlRT

P−∑m

Xmcm vvt =zvRT

P−∑m

Ymcm (8.63)

ξo =1

vltξg =

1

vvt(8.64)

Compute the molecular weight for the liquid and gas from (8.65).

MWo =∑m

MWmXm MWg =∑m

MWmYm (8.65)

Compute the densities using (8.66).

ρo = ξoMWo ρg = ξgMWg (8.66)

Compute the specific gravities, in[psift

]based on ρ

[lbmft3

].

γo =

(0.433 · ρo

ρw,sc

)=

(0.433

62.4ρo

)=

(1

144ρo

)γg =

(1

144ρg

)(8.67)

128

8.3.10 Calculate the saturations (Full Flash Only)

Calculate the saturations for two phases only:

L =ξoSo

ξoSo + ξgSgV =

ξgSg

ξoSo + ξgSg(8.68)

Snoi =

ξngiLni

ξnoiVni + ξngiL

ni

Sngi =

ξnoiVni

ξnoiVni + ξngiL

ni

(8.69)

8.4 Evaluate Fugacity Derivatives

Compute the liquid fugacity coefficients, Φle1m using (8.70).

lnΦle1m =

bmble1

(zle3 − 1)− ln (zle3 −Ble1)+

− Ale1

2√2Ble1

(2

ale1

(∑n

Xe1n amn

)− bm

ble1

)· ln

(zle3 +

(√2 + 1

)Ble1

zle3 − (√2− 1)Ble1

)− cmP

RT(8.70)

Compute the vapor fugacity coefficients, Φve1m using (8.71).

lnΦve1m =

bmbve1

(zve3 − 1)− ln (zve3 −Bve1)+

− Ave1

2√2Bve1

(2

ave1

(∑n

Y e1n amn

)− bm

bve1

)· ln

(zve3 +

(√2 + 1

)Bve1

zve3 − (√2− 1)Bve1

)− cmP

RT(8.71)

Evaluate ∂Φl�m

∂P . Note that ∂A/B∂P = 0.

1

Φl�m

∂Φl�m

∂P=

bmbl�

∂zl�

∂P− 1

(zl� −Bl�)

(∂zl�

∂P− ∂Bl�

∂P

)+

−(

Al�

2√2Bl�

)·(

2

al�

(NC−1∑n=1

X�namn

)− bm

bl�

)·(

∂zl�

∂P +(√

2 + 1)

∂Bl�

∂P

zl� +(√

2 + 1)Bl�

−∂zl�

∂P −(√

2− 1)

∂Bl�

∂P

zl� − (√2− 1)Bl�

)− cmRT

(8.72)

Evaluate ∂Φv�m

∂P .

1

Φv�m

∂Φv�m

∂P=

bmbv�

∂zv�

∂P− 1

(zv� −Bv�)

(∂zv�

∂P− ∂Bv�

∂P

)+

−(

Av�

2√2Bv�

)·(

2

av�

(NC−1∑n=1

Y �namn

)− bm

bv�

)·(

∂zv�

∂P +(√

2 + 1)

∂Bv�

∂P

zv� +(√

2 + 1)Bv�

−∂zv�

∂P −(√

2− 1)

∂Bv�

∂P

zv� − (√2− 1)Bv�

)− cmRT

(8.73)

129

For the normal hydrocarbon equations m = 1 . . . NC − 2, evaluate ∂Φl�m

∂Xm′ for m′ = 1 . . . NC − 2.

Since amm = 0, the derivatives are the same for m = NC − 1.

1

Φl�m

∂Φl�m

∂Xm′=

bmbl�

(−(zl� − 1)

(1

bl�∂bl�

∂Xm′

)+

∂zl�

∂Xm′

)− 1

(zl� −Bl�)

(∂zl�

∂Xm′− ∂Bl�

∂Xm′

)+

−(

1

Al�

∂Al�

∂Xm′− 1

Bl�

∂Bl�

∂Xm′

)·(

Av�

2√2Bv�

)·(

2

al�

(NC−1∑n=1

X�namn

)− bm

bl�

)· ln(zl� +

(√2 + 1

)Bl�

zl� − (√2− 1)Bl�

)+

−(

Av�

2√2Bv�

)·(

2

al�

(NC−1∑n=1

X�namn

)(− 1

al�∂al�

∂Xm′

)+

2(am,m′ − am,M )

al�+

bmbl�

(1

bl�∂bl�

∂Xm′

))·

ln

(zl� +

(√2 + 1

)Bl�

zl� − (√2− 1)Bl�

)+

−(

Al�

2√2Bl�

)·(

2

al�

(NC−1∑n=1

X�namn

)− bm

bl�

)·⎛⎝ ∂zl�

∂Xm′ +(√

2 + 1)

∂Bl�

∂Xm′

zl� +(√

2 + 1)Bl�

−∂zl�

∂Xm′ −(√

2− 1)

∂Bl�

∂Xm′

zl� − (√2− 1)Bl�

⎞⎠

(8.74)

For the normal hydrocarbon equations m = 1 . . . NC − 2, evaluate ∂Φv�m

∂Ym′ for m′ = 1 . . . NC − 2.

Since amm = 0, the derivatives are the same for m = NC − 1.

1

Φv�m

∂Φv�m

∂Ym′=

bmbv�

(−(zv� − 1)

(1

bv�∂bv�

∂Ym′

)+

∂zv�

∂Ym′

)− 1

(zv� −Bv�)

(∂zv�

∂Ym′− ∂Bv�

∂Ym′

)+

−(

1

Av�

∂Av�

∂Ym′− 1

Bv�

∂Bv�

∂Ym′

)·(

Av�

2√2Bv�

)·(

2

av�

(NC−1∑n=1

Y �namn

)− bm

bv�

)· ln(zv� +

(√2 + 1

)Bv�

zv� − (√2− 1)Bv�

)+

−(

Av�

2√2Bv�

)·(

2

av�

(NC−1∑n=1

Y �namn

)(− 1

av�∂av�

∂Ym′

)+

2(am,m′ − am,M )

al�+

bmbv�

(1

bv�∂bv�

∂Ym′

))·

ln

(zv� +

(√2 + 1

)Bv�

zv� − (√2− 1)Bv�

)+

−(

Av�

2√2Bv�

)·(

2

av�

(NC−1∑n=1

Y �namn

)− bm

bv�

)·⎛⎝ ∂zv�

∂Ym′ +(√

2 + 1)

∂Bv�

∂Ym′

zv� +(√

2 + 1)Bv�

−∂zv�

∂Ym′ −(√

2− 1)

∂Bv�

∂Ym′

zv� − (√2− 1)Bv�

⎞⎠

(8.75)

8.5 Evaluate Peng-Robinson Pressure Derivatives

Required derivatives:

∂z

∂P

∂ξ

∂P

∂A

∂P

∂B

∂P(8.76)

130

All of these derivatives are required for both the oil phase and the gas phase.

8.5.1 Evaluate ∂ξ∂P

Use the definition of ξ = 1v in the equation of state; solve for ξ.

vPR =zRT

Pξ =

P

zRT=

1

vPR(8.77)

Derivative of ξ:

∂ξ

∂P= ξ

(1

P− 1

z

∂z

∂P

)(8.78)

Where there is a Peneloux volume adjustment, evaluate ∂ξ∂P as follows:

ξ =1

vPR −∑

mXmcm(8.79)

∂ξ

∂P= ξ2 · vPR ·

(1

P− 1

z

∂z

∂P

)(8.80)

8.5.2 Evaluate ∂z∂P

Evaluate cubic equation, solve for ∂z∂P . Evaluate the derivative of both sides of (8.52).

∂f(z, A,B)

∂P=

∂f

∂z

∂z

∂P+

∂f

∂A

∂A

∂P+

∂f

∂B

∂B

∂P= 0 (8.81)

Solve (8.81) for ∂z∂P

∂z

∂P= −

(∂f∂A

∂A∂P + ∂f

∂B∂B∂P

)(∂f∂z

) (8.82)

8.5.3 Evaluate Derivatives of f(z)

Evaluate ∂f∂z

∂f

∂z= 3z2 + 2(B − 1)z + (A− 3B2 − 2B) (8.83)

Evaluate ∂f∂A

131

∂f

∂A= z −B (8.84)

Evaluate ∂f∂B

∂f

∂B= z2 − 6Bz − 2z + 3B2 −A+ 2B (8.85)

8.5.4 Evaluate Derivatives of A and B

A is defined by (8.86):

Al =alP �

R2T 2(8.86)

Thus ∂A∂P is:

∂Al

∂P=

al

R2T 2

1

A

∂Al

∂P=

1

P(8.87)

B is defined by (8.88):

Bl =blP �

RT(8.88)

Thus ∂B∂P is:

∂Bl

∂P=

bl

RT

1

B

∂Bl

∂P=

1

P(8.89)

8.6 Evaluate Peng-Robinson Composition Derivatives

Required derivatives:

∂z

∂Xm′

∂ξ

∂Xm′

∂A

∂Xm′

∂B

∂Xm′

∂a

∂Xm′

∂b

∂Xm′(8.90)

All of these derivatives are required for both the oil phase and the gas phase.

132

8.6.1 Evaluate ∂ξ∂Xm′

Where there is no Peneloux volume adjustment, evaluate ∂ξ∂P using the definition of ξ = 1

v in

the equation of state; solve for ξ.

vPR =zRT

Pξ =

P

zRT=

1

vPR(8.91)

Derivative of ξ:

∂ξ

∂Xm′= −ξ

(1

z

∂z

∂Xm′

)(8.92)

Where there is a Peneloux volume adjustment, evaluate ∂ξ∂Xm′ as follows:

ξ =1

vPR −∑

mXmcm(8.93)

∂ξ

∂Xm′= −ξ2

(vPR

1

z

∂z

∂Xm′− (cm′ − cM )

)(8.94)

8.6.2 Evaluate ∂z∂Xm′

Evaluate the cubic equation; solve for ∂z∂Xm′ . Evaluate the derivative of both sides of (8.52).

∂f(z, A,B)

∂Xm′=

∂f

∂z

∂z

∂Xm′+

∂f

∂A

∂A

∂Xm′+

∂f

∂B

∂B

∂Xm′= 0 (8.95)

Solve (8.95) for ∂z∂Xm′ :

∂z

∂Xm′= −

(∂f∂A

∂A∂Xm′ +

∂f∂B

∂B∂Xm′

)(∂f∂z

) (8.96)

8.6.3 Evaluate Derivatives of A and B

A is defined by:

Al =alP �

R2T 2(8.97)

Thus ∂A∂Xm′ is

133

∂Al

∂Xm′=

P �

R2T 2

∂al

∂Xm′

1

A

∂Al

∂Xm′=

1

a

∂al

∂Xm′(8.98)

B is defined by:

Bl =blP �

RT(8.99)

Thus ∂B∂P is

∂Bl

∂Xm′=

P �

RT

∂bl

∂Xm′

1

B

∂Bl

∂Xm′=

1

b

∂bl

∂Xm′(8.100)

8.6.4 Evaluate ∂a∂Xm′

For this section, M = NC−1. Figure 8.1 illustrates how the amn is split into pieces. Figure 8.1(a)

shows the pieces which contain Xk, using k = m′ = 1 for this illustration. Figure 8.1(b) shows

the pieces for m = M or n = M (ie contains XM ). Figure 8.1(c) shows the overlap between

Figure 8.1(a) and Figure 8.1(b).

m:1..Mm=k m=M

n=k

n:1..M

n=M

(a) Values with Xk.

m:1..Mm=k m=M

n=k

n:1..M

n=M

(b) Values with XM .

m:1..Mm=k m=M

n=k

n:1..M

n=M

(c) Values with both Xk and XM .

Figure 8.1: Illustration of amn.

Recall the definition of a:

al =M∑

m=1

M∑n=1

(1− δmn)a1/2m a1/2n XmXn (8.101)

Expand (8.101) into the terms that contain Xk. Use Figure 8.1 for reference. Although akk = 0

and aMM = 0, the derivation is actually simpler if we ignore this.

134

a =

no Xk, no XM︷︸︸︷M−1∑m�=k

M−1∑n �=k

amnXmXn +

one Xk, no XM︷︸︸︷2 ·

M−1∑m�=k

amkXmXk +

no Xk, one XM︷︸︸︷2 ·

M−1∑m�=k

amMXmXM +

two Xk, no XM︷︸︸︷akkXkXk +

no Xk, two XM︷︸︸︷aMMXMXM +

one Xk , one XM︷︸︸︷2 · akMXkXM (8.102)

The derivative ∂XM∂Xk

is defined by

∂XM

∂Xk=

∂

∂Xk(1−X1 −X2 − · · · −Xk − · · · −XM−2 −XM−1) = −1 (8.103)

Calculate the derivatives ∂a∂Xk

, for k = m′ = 1 . . . NC − 2, using (8.102).

∂a

∂Xk= 2·

M−1∑m�=k

amkXm+−2·M−1∑m�=k

amMXm+2·akkXk+−2·aMMXM+2·akMXM−2·akMXk (8.104)

(8.104) can be simplified by using

M∑m=1

amkXm =

M−1∑m�=k

amkXm + akkXk + akMXM (8.105)

The final result is the derivative ∂a∂Xk

:

∂a

∂Xk= 2 ·

M∑m=1

Xm(amk − amM ) (8.106)

8.6.5 Evaluate ∂b∂Xm′

For this section, M = NC − 1. Recall the definition of b:

b =

M∑m=1

bmXm (8.107)

Expand (8.107) into (8.108).

b = b1X1 + · · ·+ bkXk + · · ·+ bM (1−X1 − · · · −XM−1) (8.108)

Evaluate ∂b∂Xk

, for k = m′ = 1 . . . NC − 2.

135

∂b

∂Xk= bk − bM = bm′ − bCO2 (8.109)

8.7 Check Fugacity Derivatives

This section provides some internal consistency checks for fugacity derivatives based on Michelsen

and Mollerup (2007) and Mollerup and Michelsen (1992). Mollerup and Michelsen (1992) provides

derivatives based on the Redlich-Kwong equation of state; this section provides the derivations for

the Peng-Robinson equation of state.

8.7.1 Introduction

The mixing rules for both equations of state are:

b =∑i

Xibi B = nb =∑i

nibi (8.110)

a =∑i

∑j

XiXjaij D = n2a =∑i

∑j

ninjaij (8.111)

The general form of the equation of state is

P =RT

v − b− a

(v + δ1b)(v + δ2b)(8.112)

For Redlich Kwong, δ1 = 1 and δ2 = 0. For Peng-Robinson, δ1 = 1 +√2 and δ2 = 1−√2.

The derivations are based on the reduced residual Helmholtz energy.

F = n ln

[V

V −B

]− D

B(δ1 + δ2)RTln

[V + δ1B

V + δ2B

](8.113)

8.7.2 Fugacity Coefficient

The fugacity coefficient is defined as

lnΦi =

(∂F

∂ni

)T,V

− ln z (8.114)

This is based on ∂F∂ni

136

(∂F

∂ni

)T,V

= Fn + FB∂B

∂ni+ FD

∂D

∂ni(8.115)

Evaluate ∂F∂n

Fn =∂F

∂n= ln

[V

V −B

](8.116)

Evaluate ∂F∂B

FB =∂F

∂B=

n

V −B− D

B(δ1 + δ2)RT

(− 1

Bln

[V + δ1B

V + δ2B

]+

δ1V + δ1B

− δ2V + δ2B

)(8.117)

Evaluate ∂F∂D

FD =∂F

∂D= − 1

B(δ1 + δ2)RTln

[V + δ1B

V + δ2B

](8.118)

Evaluate ∂D∂ni

∂D

∂ni= 2 ·

∑j

njaij (8.119)

Evaluate ∂B∂ni

∂B

∂ni= bi (8.120)

8.7.3 Pressure Derivative

Evaluate ∂ lnΦi∂P as

∂ ln Φi

∂P=

viRT− 1

P(8.121)

Evaluate vi

vi =

(∂V

∂ni

)T,P

= −

(∂P

∂ni

)T,V(

∂P

∂V

)n,T

(8.122)

137

Evaluate ∂P∂V

(∂P

∂V

)T,n

= −RT

(∂2F

∂ V 2

)T,n

− nRT

V 2(8.123)

Evaluate ∂P∂ni

(∂P

∂ni

)T,V,nj �=i

= −RT

(∂2F

∂V ∂ni

)T,nj

+RT

V(8.124)

Evaluate FV V

FV V =

(∂2F

∂ V 2

)T,n

= − 4Dδ1V

RT (δ1 + δ2)(V 2 −B2δ21

)2 +D

RT (V −Bδ1)2(Bδ2 + V )+

D

RT (V −Bδ1)(Bδ2 + V )2− Bn(B − 2V )

V 2(B − V )2(8.125)

Evaluate FV,ni

FV,ni =

(∂2F

∂V ∂ni

)T,nj

= FnV + FBV∂B

∂ni+ FDV

∂D

∂ni(8.126)

Evaluate FnV

FnV =∂2F

∂n∂V=

B

BV − V 2(8.127)

Evaluate FBV

FBV =∂2F

∂B∂V=

4DBδ31

RT (δ1 + δ2)(V 2 −B2δ21

)2 +DV − 2DBδ1

BRT (Bδ1 − V )2(Bδ2 + V )+

DV

BRT (Bδ1 − V )(Bδ2 + V )2− n

(B − V )2(8.128)

Evaluate FDV

FDV =∂2F

∂D∂V=

δ1 − δ2RT (δ1 + δ2)(Bδ1 + V )(Bδ2 + V )

(8.129)

8.7.4 Composition Derivative

Evaluate ∂ lnΦi∂nj

as

138

∂ ln Φi

∂nj=

(∂ ln Φi

∂nj

)T,P

=

(∂2F

∂ni∂nj

)T,V

+RT

(∂P

∂ni

)T,V

(∂P

∂nj

)T,V(

∂P

∂V

)T,n

+1

n(8.130)

Evaluate Fni,nj

Fni,nj =

(∂2F

∂ni∂nj

)T,V

= FnB

(∂B

∂ni+

∂B

∂nj

)+ FBB

∂B

∂ni

∂B

∂nj+

FB∂2B

∂ni∂nj+ FBD

(∂B

∂ni

∂D

∂nj+

∂B

∂nj

∂D

∂ni

)+ FD

∂2D

∂ni∂nj(8.131)

Evaluate FnB

FnB =∂2F

∂n∂B=

1

V −B(8.132)

Evaluate FBB

FBB =∂2F

∂ B2= −

2D log(Bδ1+VBδ2+V

)B3RT (δ1 + δ2)

+D(δ1V − δ2V )

B2RT (δ1 + δ2)(Bδ1 + V )(Bδ2 + V )+

DV (δ1 − δ2)

B2RT (δ1 + δ2)(Bδ1 + V )(Bδ2 + V )+

Dδ1V (δ1 − δ2)

BRT (δ1 + δ2)(Bδ1 + V )2(Bδ2 + V )+

Dδ2V (δ1 − δ2)

BRT (δ1 + δ2)(Bδ1 + V )(Bδ2 + V )2+

n

(B − V )2(8.133)

Evaluate FBD

FBD =∂2F

∂B∂D=

log(Bδ1+VBδ2+V

)B2RT (δ1 + δ2)

− V (δ1 − δ2)

BRT (δ1 + δ2)(Bδ1 + V )(Bδ2 + V )(8.134)

Evaluate Dni,nj

Dni,nj =∂2D

∂ni∂nj= 2 · aij (8.135)

Evaluate Bni,nj

Bni,nj =∂2B

∂ni∂nj= 0 (8.136)

139

8.7.5 Consistency Check

Internal consistency of the implementation of ∂ lnΦi∂P can be evaluated as follows:

∑i

ni∂ ln Φi

∂P=

(z − 1)n

P=⇒

∑i

Xi∂ ln Φi

∂P=

z − 1

P(8.137)

Mollerup and Michelsen (1992) is based on total number of moles. Two additional derivatives

are needed to create the derivatives with respect to mole fractions. Use∂nj

∂ni= 0, and also:

∂Xj

∂nj=

∂

∂nj

(nj

n1 + n2 + · · ·+ nj + · · ·+ nM

)=

1

N(1−Xj) (8.138)

Internal consistency of the implementation of ∂ lnΦi∂nj

can be evaluated as follows:

∂ ln Φi

∂nj=

∂ ln Φj

∂ni=⇒ (1−Xj)

∂ ln Φi

∂Xj= (1−Xi)

∂ ln Φj

∂Xi(8.139)

∑i

ni∂ ln Φi

∂nj= 0 =⇒

∑i

Xi(1−Xj)∂ ln Φi

∂Xj= 0 (8.140)

8.8 Solving Cubic Equations Numerically

There is a closed form solution for a cubic equation that involves complex variables. This section

describes how to computationally implement this closed form solution.

8.8.1 Initialize

For this section, the following variables are used with arbitrary units: x, a0, a1, a2, Q, R, θ, A,

B.

The following solution procedure is based on Press, Flannery, Tukolsky, and Vetterling (1992)

and Wang (2006) with a derivation in Weisstein (2006). A general cubic equation has the form:

x3 + a2x2 + a1x+ a0 = 0 (8.141)

Define Q and R as follows. Note that if a2, a1, and a0 are real then Q and R are real.

Q =a22 − 3a1

9R =

2a32 − 9a2a1 + 27a054

(8.142)

140

8.8.2 Three Distinct Real Roots

If R2 < Q3, then there are three distinct real roots. Note that since R2 ≥ 0, Q3 > 0 and

therefore Q > 0 and thus√

Q3 and√Q exist and

√Q3 > 0. Define the angle θ. Arccos represents

the principle arc-cosine.

θ = Arccos

[R

Q3/2

](8.143)

The real roots are

x1 = −2√

Q cos

[θ

3

]− a2

3(8.144)

x2 = −2√

Q cos

[θ + 2π

3

]− a2

3(8.145)

x3 = −2√

Q cos

[θ − 2π

3

]− a2

3(8.146)

8.8.3 One Real Root

If R2 > Q3, then there is one real root and two complex roots. For real Q and R, compute A

as follows:

A = −sgn[R](|R|+

√R2 −Q3

)1/3(8.147)

If A = 0, then there are three identical roots

x1 = x2 = x3 = −a23

(8.148)

If A �= 0, then the real root is x1 and the complex conjugate roots are x2 and x3:

x1 = A+Q

A− a2

3(8.149)

x2 = −1

2

(A+

Q

A

)− a2

3+ i

√3

2

(A− Q

A

)(8.150)

x3 = −1

2

(A+

Q

A

)− a2

3− i

√3

2

(A− Q

A

)(8.151)

141

8.8.4 Three Real Roots, Two or More Coincide

If R2 = Q3 then there are three real roots and two or more coincide. The approach for R2 > Q3

simplifies to the following. For this case, A = B.

A = B = −sgn[R] (|R|)1/3 (8.152)

If A = 0 then there are three identical real roots,

x1 = x2 = x3 = −a23

(8.153)

If A �= 0,

x1 = 2A− a23

(8.154)

x2 = x3 = −A− a23

(8.155)

8.8.5 Newton Raphson

The roots of a polynomial equation are subject to significant roundoff errors. This can easily

be illustrated with a polynomial such as

(x− 109)(x− 2)(x− 1) = x3 − (109 + 3)x2 + (3 · 109 + 2)x− (2 · 109) (8.156)

As a result of these numerical inaccuracies, it is common practice to “polish” numerical roots using

Newton Raphson. Newton Raphson is an excellent tool for this since it has quadratic convergence

close to a root. The down side of Newton Raphson is that it is sensitive to the initial estimate.

This is overcome by using an algebraic solution as a starting value.

xe3+1 = xe3 − f(x)(∂f(x)

∂x

) (8.157)


∣∣∣∣xe3+1 − xe3

xe3

∣∣∣∣ ≤ εx (8.158)

142

8.9 Fugacity Computations

The fugacity, fugacity coefficients, and fugacity derivatives are computed in the same subroutine

by first defining some temporary variables. These variables are highlighted in (8.159), (8.160), and

(8.161): variables a1, . . . , a12 do not depend on m; variables b1m, b2m, b3m, c1m, c2m, c4m depend

on m; variable c3mm′ depends on both m and m′. Compute the liquid fugacity coefficients, Φle1m

using (8.159).

c1m︷︸︸︷ln

c4m︷︸︸︷Φl�

m =

b1m︷︸︸︷bm

a1︷︸︸︷1

bl�

a2︷︸︸︷(zl� − 1

)−a4︷︸︸︷

ln

a3︷︸︸︷(zl� −Bl�

)−

a5︷︸︸︷Al�

2√2Bl�

b3m︷︸︸︷⎛⎜⎜⎜⎜⎜⎜⎜⎝

b2m︷︸︸︷a6︷︸︸︷2

al�

(NC−1∑n=1

X�namn

)−

b1m︷︸︸︷bmbl�

⎞⎟⎟⎟⎟⎟⎟⎟⎠×

a9︷︸︸︷

ln

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1/a11︷︸︸︷zl� +

a7︷︸︸︷(√2 + 1

)Bl�

1/a12︷︸︸︷zl� −

a8︷︸︸︷(√2− 1

)Bl�

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

+

b4m︷︸︸︷a14︷︸︸︷− 1

RTcm P (8.159)

Evaluate ∂Φl�m

∂P using (8.160).

c2m︷︸︸︷1

Φl�m

∂Φl�m

∂P=


∂zl�

∂P−

a10︷︸︸︷1a3︷︸︸︷(

zl� −Bl�)(∂zl�

∂P− ∂Bl�

∂P

)+

−

a5︷︸︸︷(Al�

2√2Bl�

)×

b3m︷︸︸︷⎛⎜⎜⎝2

a13︷︸︸︷1

al�

(NC−1∑n=1

X�namn

)− bm

bl�

⎞⎟⎟⎠×

⎛⎜⎜⎜⎜⎜⎜⎝

∂zl�

∂P +

a7︷︸︸︷(√2 + 1

)∂Bl�

∂P

1/a11︷︸︸︷zl� +

(√2 + 1

)Bl�

−∂zl�

∂P −a8︷︸︸︷(√2− 1

)∂Bl�

∂P

1/a12︷︸︸︷zl� −

(√2− 1

)Bl�

⎞⎟⎟⎟⎟⎟⎟⎠+

b4m︷︸︸︷a14︷︸︸︷− 1

RTcm (8.160)

For the normal hydrocarbon equations m = 1 . . . NC − 2, evaluate ∂Φl�m

∂Xm′ for m′ = 1 . . . NC − 2

using (8.161). Since amm = 0, the derivatives are the same for m = NC − 1.

143

c3mm′︷︸︸︷1

Φl�m

∂Φl�m

∂Xm′=


⎛⎝−

a2︷︸︸︷(zl� − 1)

(1

bl�∂bl�

∂Xm′

)+

∂zl�

∂Xm′

⎞⎠−

a10︷︸︸︷1

(zl� −Bl�)

(∂zl�

∂Xm′− ∂Bl�

∂Xm′

)+

−(

1

Al�

∂Al�

∂Xm′− 1

Bl�

∂Bl�

∂Xm′

)·

a5︷︸︸︷(Av�

2√2Bv�

)·

b3m︷︸︸︷(2

al�

(NC−1∑n=1

X�namn

)− bm

bl�

)·

a9︷︸︸︷ln

(zl� +

(√2 + 1

)Bl�

zl� − (√2− 1)Bl�

)+

−

a5︷︸︸︷(Av�

2√2Bv�

)·

⎛⎜⎜⎜⎜⎝

b2m︷︸︸︷2

al�

(NC−1∑n=1

X�namn

)(− 1

al�∂al�

∂Xm′

)+

a6︷︸︸︷2

al�(am,m′ − am,M ) +


(1

bl�∂bl�

∂Xm′

)⎞⎟⎟⎟⎟⎠ ·

a9︷︸︸︷ln

(zl� +

(√2 + 1

)Bl�

zl� − (√2− 1)Bl�

)+

−

a5︷︸︸︷(Al�

2√2Bl�

)·

b3m︷︸︸︷(2

al�

(NC−1∑n=1

X�namn

)− bm

bl�

)·

⎛⎜⎜⎜⎜⎜⎜⎝

∂zl�

∂Xm′ +

a7︷︸︸︷(√2 + 1

)∂Bl�

∂Xm′

1/a11︷︸︸︷zl� +

(√2 + 1

)Bl�

−∂zl�

∂Xm′ −a8︷︸︸︷(√2− 1

)∂Bl�

∂Xm′

1/a12︷︸︸︷zl� −

(√2− 1

)Bl�

⎞⎟⎟⎟⎟⎟⎟⎠

(8.161)

8.10 Flash Calculations

The flash calculations described here are based on Rachford and Rice (1952). Michelsen and

Mollerup (2007, pg 252) has a similar “successive substitution” algorithm. This algorithm has more

detailed checks for degenerate cases and division by zero.

1. Calculate initial values of Km values; use Wilson. (Commentary: at one point, I tried using

the previous Km values; most of the time this works; some of the time it caused convergence

failures or convergence to the wrong solution. If the previous solution had some problem, this

compounds that problem. Now I always use Wilson. Also note, storing all the Km for all the

grid cells in a big grid requires a lot of memory. )

Ke1m =

Pcm

P �exp

(5.3727(1 + ωm)

(1− Tcm

T

))(8.162)

2. Calculate a weighted critical temperature.

144

Tc,mix =

∑mMWmZmTc,m∑

mMWmZm(8.163)

3. itK = 0; convergedK = TRUE

4. Loop K: WHILE((itK < itK,max)OR(convergedK))

5. itK = itK + 1

6. Calculate f [V = 0], f [V = 1]

7. Initial estimate of V . Make single phase computation consistent. (Commentary: before doing

this, some single phase cases were giving inconsistent results. Single component cases were

not consistent with the rest of the phase diagram.)

IF((f [V = 0] >= 0)OR(f [V = 1] <= 0))THENIF(T >= Tc,mix) THEN V = 1 ELSE V = 0 ENDIF

ELSEV = 0.5

ENDIF

(8.164)

8. Loop f(V )

9. IF (V >= 1− εV ); current estimate is 100% gas.

10. Fill in later

11. ELSEIF (V <= εV ); current estimate is 100% liquid.

12. Fill in later

13. ELSE (εV < V < 1− εV ); current estimate is two-phase gas-liquid.

14. Initialize Vmin, Vmax, and V , current version. (Commentary: current code uses this version,

but version below is fine for negative flash; I don’t remember if there was a reason for changing

this.)

145

Vmin = 0 (8.165)

Vmax = 1 (8.166)

V = 0.5 (8.167)

15. Initialize Vmin, Vmax, and V , negative flash

Vmin =−1

Km,max − 1(8.168)

Vmax =−1

Km,min − 1(8.169)

V =Vmin + Vmax

2(8.170)

16. Loop V :

17. Calculate f(V )

18. IF (f(V ) > 0), update Vmin = max(V, Vmin)

19. IF (f(V ) <= 0), update Vmax = min(V, Vmax)

20. Vold = V

21. IF |f ′(V )| < εsmall, avoid dividing by zero; set f ′(V ) = εsmall.

22. Update V using Newton Raphson

V = Vold − f(V )

f ′(V )(8.171)

23. Check if V out of range; update using binary search if necessary.

IF ((V <= Vmin) OR (V >= Vmax))THEN V =Vmin + Vmax

2(8.172)

24. Calculate convergence criteria and avoid dividing by zero:

146

IF (|Vold| < εV ) THENconvV = |V − Vold|

ELSE

convV =

∣∣∣∣V − Vold

Vold

∣∣∣∣(8.173)

25. End of loop V

26. Loop m #1: (update Xm and Ym and avoid dividing by zero)

27. Calculate temporary variable a1 to avoid dividing by zero.

a1 = (Km − 1) ∗ V + 1 (8.174)

28. Update Xm and Ym

IF (|a1| < εsmall) THENXm = 1Ym = 0

ELSE

Xm =Zm

a1

Ym = Km ∗ Zm

a1

(8.175)

29. End of loop m #1

30. Loop m #2: (make sure all Xm and Ym are in range)

31. IF (Xm < εZm) THEN Xm = 0

32. IF (Xm > 1) THEN Xm = 1

33. IF (Ym < εZm) THEN Ym = 0

34. IF (Ym > 1) THEN Ym = 1

35. End of loop m #2

36. Renormalize Xm and Ym

147

IF (∑m′

Xm′ < εsmall) THEN

Xm = Zm

ELSE

Xm =Xm∑

m′Xm′

(8.176)

IF (∑m′

Ym′ < εsmall) THEN

Ym = Zm

ELSE

Ym =Ym∑

m′Ym′

(8.177)

37. End of loop f(V )

38. End of loop K

8.11 Flowchart

This section presents flow charts for the flash calculations, Figure 8.2 and Figure 8.3.

148

Flash Calculation Flowchart • INPUT

• OUTPUT

• Initial Values – Previous Km or Wilson

– Previous V or 0.5;

– Previous Pb or P

– Previous Pd or P

– Be careful of dividing by 0

, , mP T Z

Start

, , , , ,m m m b dX Y K P P V

1

( 1)Calculate : ( )( 1) 1

Mm m

m m

K Zf VK V

two phases : (0) 0 and (1) 0f f

liquid : (0) 0f vapor : (1) 0f

1 ( )'( )

ee e

ef VV Vf V

1e eV V

( 1) 1m

m m m mm

ZX Y K XK V

Normalize : 1 1m mm m

X YNormalize : 1mm

Y Normalize : 1mm

X

/ / /

Calculate :, , , ,L V L V L V

m m b m mz P X Yf / / /


m m d m mz P X Yf / / /


m m m mz P X Yf

/L Vm m mR f f

61 10mR 1e e em m mK R K

1 1Wilson : exp 5.3727(1 ) 1m mrm rm

KP T

Calculate :L Vm m

b dV Lm mm m

P Pf f

0

m m

m m m

VX ZY K Z

1

/m m

m m m

VY ZX Z K

End

1 610e eV V

,max ,min

1 1: ,( 1) ( 1)m m

VK K

1

2

3

3.1

3.2

3.3

4

4.1

4.2

4.3

5

5.1

5.2

5.3

5.4

5.5

5.6

6

7

8 9

Figure 8.2: Regular flash calculation flow chart.

149

Extended Flash Calculation Flowchart

• Used for thermodynamic MMP

• INPUT

• OUTPUT

• Initial Values – Previous Km or Wilson

– Previous V or 0.5

– Be careful of dividing by 0

• vaporizing gas drive – Solve

– MMP where

• condensing gas drive – Solve

– MMP where

Start

1

( 1)Calculate : ( )( 1) 1

Mm m

m m

K Zf VK V

1 ( )'( )

ee e

ef VV Vf V

1e eV V

( 1) 1m

m m m mm

ZX Y K XK V

Normalize : 1 1m mm m

X Y

/ / /


m m m mz P X Yf

/L Vm m mR f f

61 10mR 1e e em m mK R K

1 1Wilson : exp 5.3727(1 ) 1m mrm rm

KP T

, , mP T Z

, , ,m m mX Y K V

End

1 610e eV V

,max ,min

1 1: ,( 1) ( 1)m m

VK K

10

initial oil ( ( 1) 1)m m mZ K X

INJ ( ( 1) 1)m m mZ K X

10

1

2

5.1

5.2

5.3

5.4

5.5

5.6

7

8 9

Figure 8.3: Flash calculation flow chart for thermodynamic minimum miscibility pressure.

150

CHAPTER 9

FORMULATION OF WELLS

This chapter is based on Kazemi et al. (1978) and course notes from Dr. Kazemi’s classes

(Kazemi, 2008a, 2009, 2010). All of the equations were re-derived as part of this dissertation.

9.1 Well Notation

The following notations are specific to wells.

Table 9.1: Superscripts

variable units name

# script represents a constant term, one that does not vary with time

n index temporal index representing full time step; represents variablesevaluated explicitly at n

n+ 1 index temporal index representing full time step; represents variablesevaluated implicitly at n+ 1

� index temporal index representing nonlinear iteration level betweenn = (� = 0) and n+ 1 = (�+ 1).

e index represents iterative solution for well connections; this iteration is donefor each � step

emax index represents converged solution for well connections

w script indicates that a variable is within the wellbore, not the reservoir

w′ script represents the terms of the well equation that do not depend on theprimary variables at time �

� script represents total properties for well w

Table 9.2: Subscripts

variable units name

w index indicates this property is for well number 1, 2, . . .

α index index for completions in a well; starts at the toe of the well andincreases towards the heel of the well. For a fully penetrating verticalwell, α = kmax − k + 1.

α′ index index for completions in a well, used in summations

Table 22.2 identifies the variables used in this document. The units given are typical units. The

units for empirical correlations are listed in a particular section are listed within each section that

contains correlations.

151

Table 9.3: Well variables

variable units name

Cwα day/ft3 well bore storage coefficient

Dα ft total vertical depth of completion α in a well

dwα ft well inside diameter

γw psi/ft specific gravity of aqueous phase

γo psi/ft specific gravity of oil phase

γg psi/ft specific gravity of gas phase

hf,α+ 12

ft friction adjustment based on the length of the well segment

k md permeability

λw 1/cp mobility of water phase

λo 1/cp mobility of oil phase

λg 1/cp mobility of gas phase

μw cp viscosity of water phase

μo cp viscosity of oil phase

μg cp viscosity of gas phase

NRe unitless Reynold’s number

Q lbmol/day molar flux rate

q ft3/day volumetric rate

rw,α ft effective wellbore radius for flow between the reservoir andthe well

ρ lbmol/ft3 density

sα unitless skin factor for well

V wα ft3 volume within wellbore

vwϕα ft3/day velocity of phase ϕ in wellbore

WI#α (ft3/day)(cp/psi) well index

9.2 Flow from Node to Well

Since fluid flow in the reservoir is based on average pressures represented at the center of the

grid cell, Pw,α = Pw,α.

The flow rate from each perforated cell for producing wells is defined by:

qo,w,α = −WI#w,αλo,w,α(Pw,α − Pww,α)

qg,w,α = −WI#w,αλg,w,α(Pw,α − Pww,α)

qw,w,α = −WI#w,αλw,w,α(Pw,α − Pww,α)

(9.1)

The flow rate from each perforated cell for injection wells is defined by:

qt,w,α = −WI#w,αλt,w,α(Pw,α − Pww,α) (9.2)

152

9.3 Well Index

The well index for a vertical well. If it’s a fractured well, k → kf,eff = kfφf + (1− φf)km.

WI#w,α =0.006328(

√kx,w,α)(

√ky,w,α)(2π)(Δzw,α)

ln

[(√

Δxw,α)(√

Δyw,α)

(√π)rw,w,α

]− 1

2 + sw,α

(9.3)

For a horizontal well in the x-direction.

WI#w,α =0.006328(

√ky,w,α)(

√kz,w,α)(2π)(Δxw,α)

ln

[(√

Δyw,α)(√

Δzw,α)

(√π)rw,w,α

]− 1

2 + sw,α

(9.4)

9.4 Properties for Flow in Wellbore

Define the mass flow rates for producers as:

Qo,w,α = qo,w,αξo,w,α Qg,w,α = qg,w,αξg,w,α Qw,w,α = qw,w,αξw,w,α (9.5)

Define the mass flow rates for injectors as:

Qo,w,α = qo,w,αξwo,w,α Qg,w,α = qg,w,αξ

wg,w,α Qw,w,α = qw,w,αξ

ww,w,α (9.6)

For both injectors and producers:

Qww,w,α =

α∑α′=1

Qw,w,α′ Q�w,w = Qw

w,w,αmax(9.7)

For both injectors and producers:

Qwhc,w,α =

α∑α′=1

(Qo,w,α′ +Qg,w,α′

)(9.8)

Define the total mole fraction for each component for producers as follows. If the denominator

is approximately zero, then set Zwm,w,α = Zw

m,w,α−1.

Zwm,w,α =

∑αα′=1

(Xmw,α′Qo,w,α′ + Ymw,α′Qg,w,α′

)∑αα′=1


) (9.9)

153

Define the total mole fraction for each component for injectors as follows. If the denominator

is approximately zero, then set Zwm,w,α = Zw

m,w,α+1.

Zwm,w,α =

∑αmaxα′=α

(Xw

mw,α′Qo,w,α′ + Y wmw,α′Qg,w,α′

)∑αmax

α′=α


) (9.10)

Compute the well mole fractions and densities using flash:

{Pww,α, T

ww,α, Z

wmw,α} flash−−−→ {lww,α,v

ww,α,X

wm,w,α, Y

wm,w,α,W

wm,w,α, ξ

wo,w,α, ξ

wg,w,α, ξ

ww,w,α,

γwo,w,α, γwg,w,α, γ

ww,w,α, ρ

wo,w,α, ρ

wg,w,α, ρ

ww,w,α, μ

wo,w,α, μ

wg,w,α, μ

ww,w,α, C

wo,w,α, C

wg,w,α, C

ww,w,α} (9.11)

Qwo,w,α = lww,αQ

whc,w,α Q�

o,w = Qwo,w,αmax

Qwg,w,α = vw

w,αQwhc,w,α Q�

g,w = Qwg,w,αmax

(9.12)

Qwt,w,α = Qw

o,w,α +Qwg,w,α +Qw

w,w,α Q�t,w = Qw

t,w,αmax(9.13)

Define the flow rates

qwo,w,α =Qw

o,w,α

ξwo,w,α

qwg,w,α =Qw

g,w,α

ξwg,w,α

qww,w,α =Qw

w,w,α

ξww,w,α

(9.14)

qwt,w,α = qwo,w,α + qwg,w,α + qww,w,α q�t,w = qwt,w,αmax(9.15)

vwo,w,α =

(qwo,w,α

)π4 (d

ww,α)

2vwg,w,α =

(qwg,w,α

)π4 (d

ww,α)

2vww,w,α =

(qww,w,α

)π4 (d

ww,α)

2(9.16)

vwt,w,α = vwo,w,α + vwg,w,α + vww,w,α (9.17)

9.5 Pressure in Wellbore

The well pressures are defined relative to the reference well pressure, P �w, which is defined at

the heel of the well, or α = αmax for well w.

Pww,α = P �

w +

WBS→0︷︸︸︷α∑

α′=1

V ww,α′+ 1

2

· Cww,α′+ 1

2

Δt·(Pw,n+1w,α′ − Pw,n

w,α′

)+

gravity︷︸︸︷αmax−1∑α′=α

(γwt,w,α′

)·(Dw,α′ −Dw,α′+1 − hf,w,α′+ 1

2

)(9.18)

154

Cww,α+ 1

2

=Co,w,α+ 1

2Qw

o,w,α+ 12

+ Cg,w,α+ 12Qw

g,w,α+ 12

+ Cw,w,α+ 12Qw

w,w,α+ 12

Qwt,w,α+ 1

2

(9.19)

γwt,w,α+ 1

2=

γwo,w,α+ 1

2

Qwo,w,α+ 1

2

+ γwg,w,α+ 1

2

Qwg,w,α+ 1

2

+ γww,w,α+ 1

2

Qww,w,α+ 1

2

Qwt,w,α+ 1

2

(9.20)

V ww,α′+ 1

2=

π

4(Lw,α′ − Lw,α′+1) · 1

2

((dww,α′)2 + (dww,α′+1)

2)

(9.21)

hf,w,α+ 12= sign[q] · f[NRe,w,α+ 1

2] ·

(vwt,w,α+1

286,400

)2

(dww,α+ 1

2

) · (32.2) · (Lw,α − Lw,α+1) (9.22)

NRe,w,α+ 12= 0.017224·dw

w,α+ 12

·⎛⎝ρw

o,w,α+ 12

|vwo,w,α+ 1

2

|μwo,w,α+ 1

2

+ρwg,w,α+ 1

2

|vwg,w,α+ 1

2

|μwg,w,α+ 1

2

+ρww,w,α+ 1

2

|vww,w,α+ 1

2

|μww,w,α+ 1

2

⎞⎠

(9.23)

Compute the following α+ 12 terms using the following types of weighting: upstream weighting

for an injection well is α+ 1, upstream weighting for an production well is α.

• C[ogw

],w,α+ 1

2

: upstream weighted

• γ[ogw

],w,α+ 1

2

: upstream weighted

• Qw[ogw

],w,α+ 1

2

: upstream weighted

• Qwt,w,α+ 1

2

: upstream weighted

• vw[ogw

],w,α+ 1

2

: upstream weighted

• vwt,w,α+ 1

2

: upstream weighted

• ρ[ogw

],w,α+ 1

2

: upstream weighted

155

• μ[ogw

],w,α+ 1

2

: upstream weighted

• dww,α+ 1

2

: arithmetic average

9.6 Compute the Moody Friction Factor

Calculate the Moody friction factor as follows:

f[NRe < 2000] : f1 =64

NRe(9.24)

For large Reynold’s numbers, compute f using Newton Raphson iteration of the following:

f[NRe < 4000] :1

(√f2)

= 1.7384 − 2 · log10[2εRe

d+

18.574

(NRe)(√f2)

](9.25)

f[2000 < NRe < 4000] : f = f1[2000] + (f2[4000] − f1[2000]) ·(NRe − 2000

4000 − 2000

)(9.26)

9.7 Computation for Fixed Rate

When the rate q�w is fixed, calculate the flow rates, pressures, and other properties in the

following order. This requires an iterative approach because the friction term in the pressure

calculation depends on the flow rate in a nonlinear way.

1. Initialize qew,α using (9.27) for producers and (9.28) for injectors.

qe[ogw

],w,α

=1∑αmax


WI#w,αλn[ogw

],w,α

q�w (9.27)

qet,w,α =WI#w,αλ

nt,w,α∑αmax


q�w (9.28)

2. Calculate Pw,ew,α using (9.2).

Pw,ew,α =

qet,w,α +WI#w,αλnt,w,αP

n+1w,α

WI#w,αλnt,w,α

(9.29)

3. Calculate Zw,emw,α using (9.9); flash to calculate γw,e

t,w,α using (9.11).

156

4. Calculate Pw,e+1w,α from bottom to top, (9.18).

Pw,e+1w,α =

P �w︷︸︸︷

Pw,ew,αmax

+

Pw′,ew,α︷︸︸︷

αmax−1∑α′=α

(γw,et,w,α′

)·(Dw,α′ −Dw,α′+1 − hw,e

f,w,α′+ 12

)(9.30)

5. Calculate qw,α from bottom to top using (9.2).

qe+1t,w,α = −WI#w,αλ

nt,w,α(P

n+1w,α − Pw,e+1

w,α ) (9.31)

6. Calculate q�′

w .

q�′e+1

t,w =

αmax∑α′=1

qe+1t,w,α′ (9.32)

7. Repeat steps 2–6 until∣∣∣q�t,w − q�

′,e+1t,w

∣∣∣ < εq�w, using

qe+2t,w,α =

(q�t,w

q�′e+1

t,w

)· qe+1

t,w,α (9.33)

Define Pw′,ew,α .

Pw′,ew,α =


(γw,et,w,α′

)·(Dw,α′ −Dw,α′+1 − hw,e

f,w,α′+ 12

)(9.34)


fixed rate well:

−WI#w,α·(Xn

m,w,αξno,w,αλ

no,w,α + Y n

m,w,αξng,w,αλ

ng,w,α +W n

m,w,αξnw,w,αλ

nw,w,α

)·(Pn+1w,α − Pw,emax

w,α

)(9.35)

In terms of δP and δP �w:

−WI#w,α ·(Xn

m,w,αξno,w,αλ

no,w,α + Y n

m,w,αξng,w,αλ

ng,w,α +W n

m,w,αξnw,w,αλ

nw,w,α

) ·(( RHS︷︸︸︷

P �w,α +

diagonal︷︸︸︷δPw,α

)−( RHS︷︸︸︷P �,�w +

well︷︸︸︷δP �

w +

RHS︷︸︸︷Pw′,emaxw,α

))(9.36)

157



(Xn

m,w,αξno,w,αλ

no,w,α + Y n

m,w,αξng,w,αλ

ng,w,α +W n

m,w,αξnw,w,αλ

nw,w,α

)(9.37)



(Xn

m,w,αξno,w,αλ

no,w,α + Y n

m,w,αξng,w,αλ

ng,w,α +W n

m,w,αξnw,w,αλ

nw,w,α

)(9.38)



(Xn

m,w,αξno,w,αλ

no,w,α + Y n

m,w,αξng,w,αλ

ng,w,α +W n

m,w,αξnw,w,αλ

nw,w,α

) ·((P �w,α

)−(P �,�w + Pw′,emax

w,α

))(9.39)

Each well has a total rate equation. This equation has the following form for a fixed rate well:

q�t,w = −αmax∑α′=1

(WI#w,α′

)·(Pn+1w,α′ − Pw,emax

w,α′

)·


o,w,α′

ξw,emaxo,w,αmax


g,w,α′

ξw,emaxg,w,αmax


w,w,α′

ξw,emaxw,w,αmax

)(9.40)

In terms of δP and δP �w:

RHS︷︸︸︷q�t,w = −

αmax∑α′=1

(WI#w,α′

)·(( RHS︷︸︸︷

P �w,α′ +

all α︷︸︸︷δPw,α′

)−( RHS︷︸︸︷P �,�w +

well︷︸︸︷δP �

w +

RHS︷︸︸︷Pw′,emax

w,α′

))·


o,w,α′

ξw,emaxo,w,αmax


g,w,α′

ξw,emaxg,w,αmax


w,w,α′

ξw,emaxw,w,αmax

)(9.41)


QDPnw,α = −

(WI#w,α

)·(qemaxo,w,αξ

no,w,αλ

no,w,α

ξw,emaxo,w,αmax

+qemaxg,w,αξ

ng,w,αλ

ng,w,α

ξw,emaxg,w,αmax

+qemaxw,w,αξ

nw,w,αλ

nw,w,α

ξw,emaxw,w,αmax

)(9.42)


158

QDWnw,α =

αmax∑α′=1

(WI#w,α′

)×


o,w,α′

ξw,emaxo,w,αmax


g,w,α′

ξw,emaxg,w,αmax


w,w,α′

ξw,emaxw,w,αmax

)(9.43)


QCn�w,α = q�t,w +

αmax∑α′=1

(WI#w,α′

)×((

P �w,α′

)−(P �,�w + Pw′,emax

w,α′

))×


o,w,α′

ξw,emaxo,w,αmax


g,w,α′

ξw,emaxg,w,αmax


w,w,α′

ξw,emaxw,w,αmax

)(9.44)

9.8 Computation for Fixed Pressure

When the pressure P �w is fixed, calculate the flow rates, pressures, and other properties in the

following order for the first timestep only.

1. Initialize properties for grid cell w, αmax.

1.1. Pw,ew,αmax

= P �w.

1.2. Initialize qw,αmax using (9.2).

qet,w,αmax= −WI#w,αmax

λnt,w,αmax

(Pn+1w,αmax

− Pw,ew,αmax

) (9.45)

1.3. Calculate Zw,emw,αmax

using (9.9); flash to calculate γw,et,w,αmax

using (9.11).

2. Initialize from top to bottom:

2.1. Pw,ew,α assuming hf = 0 using (9.18).

Pw,ew,α = P �

w +


(γw,et,w,αmax

)·(Dw,α′ −Dw,α′+1 −

→0︷︸︸︷hw,e

f,w,α′+ 12

)(9.46)

2.2. Calculate qet,w,α using (9.2).

qet,w,α = −WI#w,αλnt,w,α(P

n+1w,α − Pw,e

w,α) (9.47)

159

3. Calculate Zw,e+1mw,α using (9.9); flash to calculate γw,e+1

t,w,α using (9.11).

4. Calculate Pww,α from bottom to top using (9.18).

Pw,e+1w,α = P �

w +


(γw,e+1t,w,α

)·(Dw,α′ −Dw,α′+1 − hw,e

f,w,α′+ 12

)(9.48)

5. Calculate qw,α from bottom to top using (9.2).


nt,w,α(P

n+1w,α − Pw,e+1

w,α ) (9.49)

6. Repeat 3–5 until max∣∣∣Pw,e+1

w,α − Pw,ew,α

∣∣∣ < εPww,α

When the pressure P �w is fixed, calculate the flow rates, pressures, and other properties in the

following order for all timesteps after the first.

1. Initialize properties from the previous timestep.

1.1. Initialize γw,etw,α = γw,n−1

tw,α

1.2. Initialize hw,ef,w,α = hw,n−1

f,w,α

2. Initialize from top to bottom

2.1. Pw,ew,α using (9.18).

Pw,ew,α = P �

w +


(γw,et,w,α

) ·(Dw,α′ −Dw,α′+1 − hw,e

f,w,α′+ 12

)(9.50)

2.2. Calculate qet,w,α using (9.2).

qet,w,α = −WI#w,αλnt,w,α(P

n+1w,α − Pw,e

w,α) (9.51)

3. Calculate Zw,e+1mw,α using (9.9); flash to calculate γw,e+1

tw,α using (9.11).

4. Calculate Pww,α from bottom to top using (9.18).

Pw,e+1w,α = P �

w +


(γw,e+1t,w,α

)·(Dw,α′ −Dw,α′+1 − hw,e

f,w,α′+ 12

)(9.52)

160

5. Calculate qe+1t,w,α from bottom to top using (9.2).


nt,w,α(P

n+1w,α − Pw,e+1

w,α ) (9.53)

6. Repeat 3–5 until max∣∣∣Pw,e+1

w,α − Pw,ew,α

∣∣∣ < εPww,α


fixed pressure well:

(WI#w,α∑αmax


)×(Xn

m,w,αξno,w,αλ

no,w,α + Y n

m,w,αξng,w,αλ

ng,w,α +W n

m,w,αξnw,w,αλ

nw,w,α

)×(q�t,w)(9.54)

In terms of δq�w:

(WI#w,α∑αmax


)×

(Xn

m,w,αξno,w,αλ

no,w,α + Y n

m,w,αξng,w,αλ

ng,w,α +W n

m,w,αξnw,w,αλ

nw,w,α

)× (RHS︷︸︸︷q�,�t,w +

well︷︸︸︷δq�t,w

)(9.55)

The coefficient of δP is 0.

WDPmnw,α = 0 (9.56)


WDWmnw,α =

(WI#w,α∑αmax


)×

(Xn

m,w,αξno,w,αλ

no,w,α + Y n

m,w,αξng,w,αλ

ng,w,α +W n

m,w,αξnw,w,αλ

nw,w,α

)(9.57)


WCmn�w,α = −

(WI#w,α∑αmax


)×

(Xn

m,w,αξno,w,αλ

no,w,α + Y n

m,w,αξng,w,αλ

ng,w,α +W n

m,w,αξnw,w,αλ

nw,w,α

) × q�,�t,w (9.58)

161


well:

q�t,w = −αmax∑α′=1

(WI#w,α′

)·(Pn+1w,α′ − Pw,n+1

w,α′

)×


o,w,α′

ξw,emaxo,w,αmax


g,w,α′

ξw,emaxg,w,αmax


w,w,α′

ξw,emaxw,w,αmax

)(9.59)

In terms of δP and δq�w:

RHS︷︸︸︷q�,�t,w +

well︷︸︸︷δq�t,w = −

αmax∑α′=1

(WI#w,α′

)×(( RHS︷︸︸︷

P �w,α′ +

all α︷︸︸︷δPw,α′

)−( RHS︷︸︸︷Pw,emax

w,α′

))×


o,w,α′

ξw,emaxo,w,αmax


g,w,α′

ξw,emaxg,w,αmax


w,w,α′

ξw,emaxw,w,αmax

)(9.60)


QDPnw,α =

(WI#w,α

)×(qemaxo,w,αξ

no,w,αλ

no,w,α

ξw,emaxo,w,αmax

+qemaxg,w,αξ

ng,w,αλ

ng,w,α

ξw,emaxg,w,αmax

+qemaxw,w,αξ

nw,w,αλ

nw,w,α

ξw,emaxw,w,αmax

)(9.61)


QDWnw,α = 1 (9.62)



αmax∑α′=1

(WI#w,α′

)×(P �w,α′ − Pw,emax

w,α′

)×


o,w,α′

ξw,emaxo,w,αmax


g,w,α′

ξw,emaxg,w,αmax


w,w,α′

ξw,emaxw,w,αmax

)(9.63)

9.9 Wells with Single Completions

For single well completions, the approach described in (9.27)–(9.63) can be simplified. The

source term for each component can be solved for directly, so the total well equations are not

necessary. The values of WDP and WC can be solved without iterations. There is no need for a well

variable, so the coefficient WDW = 0 for all wells. Because there are no total-well equations and

162

no well-specific primary variables, the Jacobian matrix will have a strictly block-banded structure.

9.9.1 Fixed Pressure Producer

For a fixed pressure producer with a single completion, the bottom hole producing pressure P �w

is specified at the elevation of the completion. There is no total well equation and no well variables,

so QDP = 0, QDW = 0, QC = 0, and WDW = 0.

WDPmnijk = −WI#w,ijk

(Xn

m,ijkξno,ijkk

nro,ijk

μno,ijk

+Y nm,ijkξ

ng,ijkk

nrg,ijk

μng,ijk

+W n

m,ijkξnw,ijkk

nrw,ijk

μnw,ijk

)(9.64)

WCm�ijk = WI#w,ijk

(Xn

m,ijkξno,ijkk

nro,ijk

μno,ijk

+Y nm,ijkξ

ng,ijkk

nrg,ijk

μng,ijk

+W n

m,ijkξnw,ijkk

nrw,ijk

μnw,ijk

)(P �ijk − P �

w

)(9.65)

9.9.2 Fixed Rate Producer

For a fixed rate producer with a single completion, the total bottom hole production rate q�w < 0

is specified. There is no total well equation and no well variables, so QDP = 0, QDW = 0, QC = 0,

and WDW = 0. The total rate is multiplied by various variables at n, so there is no dependence on

Pn+1ijk , and WDP = 0.

λnt,ijk = λn

o,ijk + λng,ijk + λn

w,ijk =knro,ijkμno,ijk

+knrg,ijkμng,ijk

+knrw,ijk

μnw,ijk

(9.66)

WDPm = 0 (9.67)

WCmnijk = − q�w

λnt,ijk

×(Xn

m,ijkξno,ijkk

nro,ijk

μno,ijk

+Y nm,ijkξ

ng,ijkk

nrg,ijk

μng,ijk

+W n

m,ijkξnw,ijkk

nrw,ijk

μnw,ijk

)(9.68)

9.9.3 Fixed Mole Rate Producer

For a fixed total molar rate producer with a single completion, the total bottom hole production

rate Q�w,lbmol[lbmol/day] < 0 is specified. There is no total well equation and no well variables, so

QDP = 0, QDW = 0, QC = 0, and WDW = 0. The total rate is multiplied by various variables at

n, so there is no dependence on Pn+1ijk , and WDP = 0.

163

λnt,ijk = λn



+knrg,ijkμng,ijk

+knrw,ijk

μnw,ijk

(9.69)

WDPm = 0 (9.70)

WCmnijk = −Q�

w,lbmol (9.71)

The following relates the molar rate to the volumetric rate.

q�w =Q�

w,lbmol × λnt,ijk

ξno,ijkλno,ijk + ξng,ijkλ

ng,ijk + ξnw,ijkλ

nw,ijk

(9.72)

9.9.4 Fixed Pressure Injector

For a fixed pressure injector with a single completion, the bottom hole injection pressure P �w

is specified at the elevation of the completion. As is typical for injectors, it is based on the total

mobility of the grid cell injected into. The total specified composition Zw,nm is flashed at Pn

ijk to

determine Xw,nm , Y w,n

m , Ww,nm , ξw,n

o , ξw,ng , and ξw,n

w . There is no total well equation and no well

variables, so QDP = 0, QDW = 0, QC = 0, and WDW = 0.

λnt,ijk = λn



+knrg,ijkμng,ijk

+knrw,ijk

μnw,ijk

(9.73)

WDPmnijk = −WI#w,ijkλ

nt,ijk

(Xw,n

m ξw,no + Y w,n

m ξw,ng +Ww,n

m ξw,nw

)(9.74)

WCm�ijk = WI#w,ijkλ

nt,ijk

(Xw,n

m ξw,no + Y w,n

m ξw,ng +Ww,n

m ξw,nw

) (P �ijk − P �

w

)(9.75)

9.9.5 Fixed Rate Injector

For a fixed rate producer with a single completion, the total bottom hole production rate q�w > 0

is specified. There is no total well equation and no well variables, so QDP = 0, QDW = 0, QC = 0,

and WDW = 0. The total rate is multiplied by various variables at n, so there is no dependence on

Pn+1ijk , and WDP = 0.

164

WDPm = 0 (9.76)

For a water injector, Wwm = Zw

m. The aqueous density ξw,nw,ijk is determined based on Ww

CO2,

WwNaCl, the reservoir temperature T#, and the reservoir pressure Pn

ijk.

WCmnijk = −q�wWw

mξw,nw (9.77)

For a gas injector, the total specified composition Zw,nm is flashed at Pn

ijk to determine Xw,nm ,

Y w,nm , ξw,n

o , ξw,ng , Lw,n

o , and V w,ng .

WCmnijk = −q�w ×

(Xw,n

m ξw,no Lw,n

o + Y w,nm ξw,n

g V w,ng

)(9.78)

9.9.6 Fixed Pressure Producer with Switch to Rate Control

Calculate the total mobility

λt =kroμo

+krgμg

+krwμw

(9.79)

Calculate the total flow rate, which should be less than zero.

qcheck = −WI · λt(P�+1 − P �) (9.80)

If the calculated total flow rate is greater than a maximum flow rate, |qcheck| > |qprodmax|, thenset the flow rate to qprodmax and calculate the well properties using rate control.

WDPm = 0 (9.81)

WCm = −qprodmax

λt

(Xn

m,ijkξno,ijkk

nro,ijk

μno,ijk

+Y nm,ijkξ

ng,ijkk

nrg,ijk

μng,ijk

+W n

m,ijkξnw,ijkk

nrw,ijk

μnw,ijk

)(9.82)

If the calculated total flow rate qcheck ≥ 0, then set the flow rate to 0 and calculate the well

properties using rate control.

WDPm = 0 (9.83)

165

WCm = 0 (9.84)

If the calculated total flow rate is between 0 and the maximum flow rate qprodmax, −|qprodmax| ≤qcheck ≤ 0, calculate the well properties using pressure control.


(Xn

m,ijkξno,ijkk

nro,ijk

μno,ijk

+Y nm,ijkξ

ng,ijkk

nrg,ijk

μng,ijk

+W n

m,ijkξnw,ijkk

nrw,ijk

μnw,ijk

)(9.85)

WCm = −WDPm

(P �ijk − P �

w

)(9.86)

9.9.7 Fixed Rate Producer with Switch to Pressure Control


λt =kroμo

+krgμg

+krwμw

(9.87)

Calculate the bottom hole producing pressure

Pcheck =q�

WI · λt+ P �+1 (9.88)

If the calculated bottom hole producing pressure Pcheck < PBHPmin, then set the producing

pressure to PBHPmin and calculate the well properties using pressure control.


(Xn

m,ijkξno,ijkk

nro,ijk

μno,ijk

+Y nm,ijkξ

ng,ijkk

nrg,ijk

μng,ijk

+W n

m,ijkξnw,ijkk

nrw,ijk

μnw,ijk

)(9.89)

WCm = −WDPm

(P �ijk − PBHPmin

)(9.90)

If the calculated bottom hole producing pressure Pcheck > PBHPmin, then calculate the well


WDPm = 0 (9.91)

166

WCmnijk = − q�w

λnt,ijk

×(Xn

m,ijkξno,ijkk

nro,ijk

μno,ijk

+Y nm,ijkξ

ng,ijkk

nrg,ijk

μng,ijk

+W n

m,ijkξnw,ijkk

nrw,ijk

μnw,ijk

)(9.92)

9.9.8 Fixed Pressure Injector with Switch to Rate Control


λt =kroμo

+krgμg

+krwμw

(9.93)

Calculate the total flow rate, which should be greater than zero.

qcheck = −WI · λt(P�+1 − P �) (9.94)

If the calculated total flow rate is greater than a maximum flow rate, |qcheck| > qinjmax, then set

the flow rate to qinjmax and calculate the well properties using rate control.

WDPm = 0 (9.95)



CO2,


ijk.

WCmnijk = −qinjmaxW

wmξw,n

w (9.96)



Y w,nm , ξw,n

o , ξw,ng , Lw,n

o , and V w,ng .

WCmnijk = −qinjmax ·

(Xw,n

m ξw,no Lw,n

o + Y w,nm ξw,n

g V w,ng

)(9.97)

If the calculated total flow rate qcheck ≤ 0, then set the flow rate to 0 and calculate the well


WDPm = 0 (9.98)

WCm = 0 (9.99)

167

If the calculated total flow rate is between 0 and the maximum flow rate qinjmax, 0 ≤ qcheck <

qinjmax, calculate the well properties using pressure control.


nt,ijk

(Xw,n

m ξw,no + Y w,n

m ξw,ng +Ww,n

m ξw,nw

)(9.100)

WCm = −WDPm

(P �ijk − P �

w

)(9.101)

9.9.9 Fixed Rate Injector with Switch to Pressure Control


λt =kroμo

+krgμg

+krwμw

(9.102)

Calculate the bottom hole injection pressure

Pcheck =q�

WI · λt+ P �+1 (9.103)

If the calculated bottom hole injection pressure Pcheck > Pinjmin, then set the injection pressure

to Pinjmin and calculate the well properties using pressure control.


nt,ijk

(Xw,n

m ξw,no + Y w,n

m ξw,ng +Ww,n

m ξw,nw

)(9.104)

WCm = −WDPm

(P �ijk − Pinjmin

)(9.105)

If the calculated bottom hole injection pressure Pcheck < Pinjmin, then calculate the well prop-

erties using rate control.

WDPm = 0 (9.106)



CO2,


ijk.

WCmnijk = −q�wWw

mξw,nw (9.107)

168



Y w,nm , ξw,n

o , ξw,ng , Lw,n

o , and V w,ng .

WCmnijk = −q�w ·

(Xw,n

m ξw,no Lw,n

o + Y w,nm ξw,n

g V w,ng

)(9.108)

169

CHAPTER 10

MASS BALANCE CALCULATIONS

This chapter describes methods to calculate the properties of well fluids at surface conditions

using separators, the calculation of original oil in place at surface conditions, and the mass balance

calculations used to evaluate the computational success of each nonlinear iteration and each time

step.

10.1 Calculate Surface Conditions of Well Fluids Using Separators

For this formulation, the WCO2 at surface conditions is assumed to be zero. All CO2 dissolved

in water at reservoir conditions is assumed to go into the gas line at surface conditions.

Define the mass flux at each well in lbmol/day.

Qwom = Xw

mξwo qwo (10.1)

Qwgm = Y w

m ξwg qwg (10.2)

Qwwm = Ww

mξww qww (10.3)

Qwhc,m = Qw

om +Qwgm (10.4)

Qwo = ξwo q

wo (10.5)

Qwg = ξwg q

wg (10.6)

Qww = ξww q

ww (10.7)

Qwhc = Qw

o +Qwg (10.8)

Define the hydrocarbon mole fractions:

Zwhc,m =

Qwhc,m

Qwhc

(10.9)

Flash Zwhc,m at the conditions of separator 1.

Flash Zwhc,m

P sep1,T sep1

−−−−−−→ Xsep1m , Y sep1

m , Lsep1, V sep1, ξsep1o , ξsep1g (10.10)

Define the new mass flux rates after separator 1.

170

Qsep1o = Lsep1Qw

hc (10.11)

Qsep1g = V sep1Qw

hc (10.12)

Qsep1om = Xsep1

m Qsep1o (10.13)

Qsep1gm = Y sep1

m Qsep1g (10.14)

The liquid output of separator 1 goes to separator 2. The gas output of separator 1 goes into

the gas line. Flash Xsep1m at the conditions of separator 2.

Flash Xsep1m

P sep2,T sep2

−−−−−−→ Xsep2m , Y sep2



Qsep2o = Lsep2Qsep1

o (10.16)

Qsep2g = V sep2Qsep1

o (10.17)

Qsep2om = Xsep2

m Qsep2o (10.18)

Qsep2gm = Y sep2

m Qsep2g (10.19)



Flash Xsep2m

P sep3,T sep3

−−−−−−→ Xsep3m , Y sep3



Qsep3o = Lsep3Qsep2

o (10.21)

Qsep3g = V sep3Qsep2

o (10.22)

Qsep3om = Xsep3

m Qsep3o (10.23)

Qsep3gm = Y sep3

m Qsep3g (10.24)

Compute the oil rate and molar oil rate at standard conditions.

qsco [SCF/day] = Qsep3o /ξsep3o = Lsep3Lsep2Lsep1Qw

hc/ξsep3o (10.25)

Qscom[lbmol/day] = Qsep3

om = Xsep3m Lsep3Lsep2Lsep1Qw

hc (10.26)

171

Compute the gas line molar rates:

Qgas lineg = Qsep1

g +Qsep2g +Qsep3

g +Qww,CO2

=

V sep1Qwhc + V sep2Lsep1Qw

hc + V sep3Lsep2Lsep1Qwhc +Qw

w,CO2(10.27)

Compute the gas line molar rates for each component (m = 1 . . . NC − 1):

Qgas linegm = Qsep1

gm +Qsep2gm +Qsep3

gm +Qww,m =

Y sep1m V sep1Qw

hc + Y sep2m V sep2Lsep1Qw

hc + Y sep3m V sep3Lsep2Lsep1Qw

hc +Qww,m (10.28)

Calculate the mole fractions in the gas line:

Zgas linem =

Qgas linegm

Qgas lineg

(10.29)

The non-ideal gas law is

PV = znRT (10.30)

To calculate the volume at standard conditions as a function of the number of moles, z = 1. In

imperial units (P [psia], V [ft3], n[lbmol], and T [R]), use the following:

V [ft3] = nzRT/P =

n[lbmol]×z︷︸︸︷(1) ×

R︷︸︸︷(10.731592

[ft3psia

R lbmol

])×

T︷︸︸︷(60◦F + 459.67◦F) /

P︷︸︸︷(14.7 psia) =

n[lbmol] × 379.38

[ft3

lbmol

](10.31)

Compute the gas rate at standard conditions.

qscg [SCF/day] = Qgas lineg × 379.38

[SCFlbmol

](10.32)

Qscgm[lbmol/day] = Qgas line

gm (10.33)

The water density at standard conditions is calculated from

ξscw = ξw[Psep3, T sep3,WCO2 = 0,WNaCl] (10.34)

172

Compute the water rate at standard conditions

qscw [SCF/day] = Qww,H2O

/ξscw (10.35)

Qscw,CO2

[lbmol/day] = 0 (10.36)

Qscw,H2O[lbmol/day] = Qw

w,H2O(10.37)

10.2 Calculate Surface Conditions of Original Oil in Place

For this formulation, the WCO2 at surface conditions is assumed to be zero. All CO2 dissolved

in water at reservoir conditions is assumed to go into the gas line at surface conditions.

Define the initial fluids in the reservoir in lbmol.

M initom = VRφinitSinit

o X initm ξinito (10.38)

M initgm = VRφinitSinit

g Y initm ξinitg (10.39)

M initwm = VRφinitSinit

w W initm ξinitw (10.40)

M inithc,m = M init

om +M initgm (10.41)

M inito = VRφinitSinit

o ξinito (10.42)

M initg = VRφinitSinit

g ξinitg (10.43)

M initw = VRφinitSinit

w ξinitw (10.44)

M inithc = M init

o +M initg (10.45)

Define the hydrocarbon mole fractions:

Z inithc,m =

M inithc,m

M inithc

(10.46)

Flash Z inithc,m at the conditions of separator 1.

Flash Z inithc,m

P sep1,T sep1

−−−−−−→ Xsep1m , Y sep1



M sep1o = Lsep1M init

hc (10.48)

M sep1g = V sep1M init

hc (10.49)

M sep1om = Xsep1

m M sep1o (10.50)

M sep1gm = Y sep1

m M sep1g (10.51)

173



Flash Xsep1m

P sep2,T sep2

−−−−−−→ Xsep2m , Y sep2



M sep2o = Lsep2M sep1

o (10.53)

M sep2g = V sep2M sep1

o (10.54)

M sep2om = Xsep2

m M sep2o (10.55)

M sep2gm = Y sep2

m M sep2g (10.56)



Flash Xsep2m

P sep3,T sep3

−−−−−−→ Xsep3m , Y sep3



M sep3o = Lsep3M sep2

o (10.58)

M sep3g = V sep3M sep2

o (10.59)

M sep3om = Xsep3

m M sep3o (10.60)

M sep3gm = Y sep3

m M sep3g (10.61)

Compute the initial volume of oil at standard conditions.

OOIP[SCF] = M sep3o /ξsep3o = Lsep3Lsep2Lsep1M init

hc /ξsep3o (10.62)

M scom[lbmol] = M sep3

om = Xsep3m Lsep3Lsep2Lsep1M init

hc (10.63)

Compute the gas line molar volumes:

Mgas lineg = M sep1

g +M sep2g +M sep3

g +M initw,CO2

=

V sep1M inithc + V sep2Lsep1M init

hc + V sep3Lsep2Lsep1M inithc +M init

w,CO2(10.64)

174

Mgas linegm = M sep1

gm +M sep2gm +M sep3

gm +M initw,m =

Y sep1m V sep1M init

hc + Y sep2m V sep2Lsep1M init

hc + Y sep3m V sep3Lsep2Lsep1M init

hc +M initw,m (10.65)

Calculate the mole fractions in the gas line:

Zgas linem =

Mgas linegm

Mgas lineg

(10.66)

Compute the initial volume of free and associated gas at standard conditions.

OGIP[SCF] = Mgas lineg × 379.38

[SCFlbmol

](10.67)

M scgm[lbmol] = Mgas line

gm (10.68)

The water density at standard conditions is calculated from

ξscw = ξw[Psep3, T sep3,WCO2 = 0,WNaCl] (10.69)

Compute the water rate at standard conditions

M scw [SCF] = Mw

w,H2O/ξscw (10.70)

M scw,CO2

[lbmol] = 0 (10.71)

M scw,H2O[lbmol] = Mw

w,H2O(10.72)

Calculate the recovery factor:

RF =cumulative qsco

OOIP(10.73)

10.3 Mass Balance Calculations

Define the mass in lbmol of each phase:

Motm1,ijk = Som1,ijkξom1,ijkφm1,ijkVR (10.74)

Mgtm1,ijk = Sgm1,ijkξgm1,ijkφm1,ijkVR (10.75)

Mwtm1,ijk = Swm1,ijkξwm1,ijkφm1,ijkVR (10.76)

Motm2,ijk = Som2,ijkξom2,ijkφm2,ijkVR (10.77)

175

Mgtm2,ijk = Sgm2,ijkξgm2,ijkφm2,ijkVR (10.78)

Mwtm2,ijk = Swm2,ijkξwm2,ijkφm2,ijkVR (10.79)

Define the total system mass as

M �ijk = M �

otm1,ijk +M �gtm1,ijk +M �

wtm1,ijk +M �otm2,ijk +M �

gtm2,ijk +M �wtm2,ijk (10.80)

M � =∑ijk

M �ijk (10.81)

Define the injection and production rates in lbmol of each phase:

Qotm1,ijk = qo,ijkξom1,ijkΔt (10.82)

Qgtm1,ijk = qg,ijkξgm1,ijkΔt (10.83)

Qwtm1,ijk = qw,ijkξwm1,ijkΔt (10.84)

Define the total flux in lbmol.

Q�ijk = Q�

otm1,ijk +Q�gtm1,ijk +Q�

wtm1,ijk (10.85)

Q� =∑ijk

Q�ijk (10.86)

Use (10.87) to determine the best solution, especially if there was no convergence.

massbal = M �+1 −Mn −Q�+1inj + |Q�+1

prod| (10.87)

Use (10.88) for the incremental mass balance.

massbalincr = 1− M �+1 −Mn

Q�+1inj − |Q�+1

prod|(10.88)

Use (10.89) for the cumulative mass balance.

massbalcum = 1−∑n

n′=0 Mn′ −M0∑n

n′=0Qn′inj − |Qn′

prod|(10.89)

176

CHAPTER 11

RELATIVE PERMEABILITY AND CAPILLARY PRESSURE

There are two principal ways to specify the capillary pressure; capillary pressure as a function of

saturation or J-function as a function of saturation. Relative permeability is specified as a function

of saturation. For a more generalized formulation, capillary pressure and relative permeability are

functions of the following:

• Saturation of oil, water, and gas

• Hysteresis: direction of change of oil, water, and gas

• Trapping: trapped oil, water, and gas

• Interfacial tension and miscibility: explicit functional dependence on interfacial tension for

J-function specification of capillary pressure; need to specify relationship for relative perme-

ability

• Rock Type, including porosity, permeability (explicit functional dependence for J-function

specification of capillary pressure), single porosity / dual porosity, and anisotropy. Rock type

may change with fluid-rock chemical interactions.

• Wettability: explicit functional dependence on contact angle for J-function specification of

capillary pressure; need to specify relationship for relative permeability

• Temperature

• Composition of fluids

• Fluid flow rate dependence

Traditionally, gas relative permeability is assumed to be a function of gas saturation only and

water relative permeability is assumed to be a function of water saturation only. Oil relative

permeability is usually assumed to be a function of all three phase saturations. For some oil wet

reservoirs, oil may be approximately a function of the oil saturation only. For mixed wet reservoirs

177

and for CO2 WAG scenarios in all kinds of reservoirs, the gas, oil, and water relative permeabilities

may be a function of all three phase saturations.

11.1 Three Phase Relative Permeability

Three-phase relative permeability models describe a way to use sets of two-phase relative per-

meabilities to calculate the three-phase relative permeability. These models almost always apply

to calculation of kro; some may also be applied to krg and krw. The following articles discuss

three-phase relative permeability models.

• Land (1968), SPE #1942

• Stone (1970), SPE #2116: “Stone 1”

• Stone (1973): “Stone 2”

• Fayers and Matthews (1984), SPE #11277: Analysis of 3-phase relative permeability experi-

ments. Modification of Stone 1.

• Baker (1988), SPE #17369: Analysis of 3-phase relative permeability experiments. Uses

saturation weighted relative permeabilities.

• Delshad and Pope (1989): Comparison of 7 different 3-phase relative permeability models2.

Presents a different model for calculating kro.

• Fayers (1989), SPE #16965: Describes alternate ways to calculate Sorm for Stone 1 algorithm.

• Larsen and Skauge (1998), SPE #38456

• Pope et al. (1998), SPE #49266: Gas condensate relative permeability model

• Paterson, Painter, Zhang, and Pinczewski (1998), SPE #50938: Gas condensate relative

permeability model

• van Dijke et al. (2000), SPE #59310


178

There are some limitations of these algorithms which are discussed in later literature. Many

articles mention difficulties with Stone 1 (Stone, 1970) and Stone 2 (Stone, 1973), including: Kleppe

et al. (1997), Larsen and Skauge (1998), Blunt (2000), Element et al. (2003), and Spiteri and Juanes

(2004). Drawbacks of Land (1968) are mentioned in Jerauld (1997), Blunt (2000), and Element

et al. (2003).

11.2 Hysteresis

Hysteresis means that a function depends not only on the current state but on some of the

past history. The term was originally coined in approximately 1800 to describe the lag in response

to magnetic forces. It was based on the Greek word “hysteresis”, which means “deficiency”, or

“the state of being behind or late” (Gove, 1986; Wikipedia, 2010a). The relative permeability

and capillary pressure functions may be different depending on the increasing (I), decreasing (D),

or constant (C) state of each of the phases. This leads to the following twelve legal states for a

two-phase or three-phase system, listed as a 3-tuple in the order water-oil-gas:

• IID, IDI, DII

• IDC, ICD, DIC, DCI, CID, CDI

• IDD, DID, DDI

The initial state of the system is determined by the geologic history of the formation. The state

of individual grid cells are dynamic effects which change with the reservoir simulation.

11.2.1 Hysteresis Applications

Hysteresis is important when coning of water or gas is present, during immiscible gas injection,

during miscible gas injection, and during water-alternating-gas (WAG) injection. Hysteresis effects

change the relative permeability and capillary pressure based on changes in the increasing or de-

creasing state of water, oil, and gas phases. Hysteresis causes changes in recovery and different

timing of breakthrough or coning.

11.2.2 Hysteresis Literature

The following articles discuss hysteresis algorithms.

179

• Killough (1976), SPE #5106: Hysteresis algorithm

• Carlson (1981), SPE #10157: Non-wetting phase hysteresis

• Delshad et al. (2003), SPE #86916: Mixed wet model with hysteresis

• Spiteri et al. (2005), SPE #96448: New hysteresis model

There are some limitations of these algorithms which are discussed in later literature. Articles

which discuss limitations of the other algorithms include: Kleppe et al. (1997), Element et al.

(2003), and Spiteri and Juanes (2004).

11.2.3 Combined Three-Phase Relative Permeability and Hysteresis

Many more recent articles discuss both both three-phase relative permeability and hysteresis,

including:

• Jerauld (1997), SPE #36178: Correlations for 3-phase relative permeability and hysteresis to

fit an extensive mixed wet data set in Alaska.

• Blunt (2000), SPE #67950: 3-phase relative permeability and hysteresis model; comparison

of different models

• Egermann et al. (2000), SPE #65127: 3-phase relative permeability model with hysteresis

• Hustad et al. (2002), SPE #75138: Presentation of what Eclipse calls the “IKU” method;

comparison of different models

• Hustad (2002), SPE #74705: Presentation of what Eclipse calls the “ODD3P” method;

comparison of different models.

• Element et al. (2003), SPE #84903: Evaluation of different relative permeability and hys-

teresis models

• Spiteri and Juanes (2004), SPE #89921: Evaluation of different relative permeability and

hysteresis models

180

11.2.4 Combined Analysis of Algorithms

The following articles include a discussion of many of the previous algorithms.

• Blunt (2000)

– Review of previous algorithms, plus presentation of new model

– multiple problems indicated with Stone 1 (Stone, 1970), Stone 2 (Stone, 1973), and

LandLand (1968)

– discussion of the other models: Vizika and Lombard (1996), Larsen and Skauge (1998),

Jerauld (1997)

– Element et al. (2003)

– Review of previous algorithms, using new data set

– Larsen and Skauge (1998): Problems because uses Stone 1; no trapping of water; no

variation of Land constant with cycle

– Blunt (2000): No wetting phase hysteresis; hysteresis is in a closed loop and shouldnt

be; no variation of Land constant with cycle

– Egermann et al. (2000): No variation of Land constant with cycle

• Spiteri and Juanes (2004)

– Review of previous algorithms, using Oaks data (Oak, 1990)

– Stone 1, Stone 2, Baker (1988): All are bad fits for kro

– Larsen and Skauge (1998): Not suitable for krg

– Other models: Killough (1976), Carlson (1981), Lenhard and Oostrom (1998), Jerauld

(1997), Blunt (2000)

11.3 Trapping

Trapping refers to the process of making a portion of one of the phases immobile. In two

dimensions trapping can be illustrated using capillary tubes of different sizes, Figure 11.1. In

three dimensions, variations in the pore size distribution, variation in the possible paths in three

181

dimensions, and the differing wettability along different mineral grains lead to many additional

ways for trapping to occur. Different histories of increasing and decreasing of each of the phase

saturations can lead to each of the following six possibilities, several of which may be present in

any grid cell.

• Gas trapped by oil

• Gas trapped by water

• Oil trapped by gas

• Oil trapped by water

• Water trapped by gas

• Water trapped by oil

Trapping typically occurs when one phase switches from increasing to decreasing saturation,

Figure 11.1(a) and Figure 11.1(b). Additional fluids may be trapped by later cycles. Trapped vol-

umes will only decrease by diffusive processes or compositional effects. Trapped fluid compositions

change only by diffusive processes. Trapping changes are dynamic effects which change with the

reservoir simulation.

(a) Early displacement of oil by water; watermoves faster through the smaller pore throats.

(b) Later displacement of oil by water, showingtrapped oil.

Figure 11.1: Illustration of pore doublet effect in a water-wet rock; green is oil and blue is water.

Algorithms to calculate trapping are typically combined with three-phase relative permeability

and hysteresis in the literature.

182

11.3.1 Composition of Trapped Phase

One of the fundamental assumptions of reservoir simulation is that the phases and components

are in equilibrium with each other within each grid cell for each time step. This assumption is not

very good when it comes to the interaction of components in a trapped phase with components in

the mobile phases.

When a phase is trapped, its starting composition is the same as the mobile phase. Later on, the

composition of the trapped phase may change through diffusive processes or may remain constant.

None of the published articles on compositional simulation include the different compositions of

the trapped phase from the mobile phase.

If the compositions of the trapped phases are not tracked separately (base case), then the

following are required for every grid cell:

• Gas, Sg, Ym

• Oil, So, Xm

• Water, Sw, Wm

11.3.2 Simple Trapping Composition

The easiest composition trapping option to implement is for the trapped compositions to be

locked in place. The memory requirements are smaller than the option including diffusion, and it

is computationally much closer to the base case. If additional oil is trapped in a cell that already

has trapped oil, then the trapped composition Xmt is a weighted average of the existing Xmt and

the added mobile oil Xm. This option only adds additional explicit computations when a phase

changes from increasing to decreasing, triggering trapping. This option requires the following in

addition to the base case:

• Trapped gas, Sgt, Ymt; only interaction Ym → Ymt

• Trapped oil, Sot, Xmt; only interaction Xm → Xmt

• Trapped water, Swt, Wmt; only interaction Wm →Wmt

183

To track hysteresis effects, the trapped saturations may also be required. These can be stored

individually for every grid cell, or a flag can track the presence of hysteresis in every grid cell and

then the trapped saturations are only stored for those cells where they are needed.

11.3.3 Complex Trapping Composition

An early reference for diffusion constrained trapping is Coats and Smith (1964). A more com-

plicated composition tracking option involves the diffusive mixing of the trapped compositions.

Here it is necessary to track both the trapped phases and also which phase is doing the trapping,

because the kinetics and relevant equations are different. In addition, it also adds another explicit

primary equation every time step for each diffusion option that is used. This requires the following

in addition to the base case:

• Gas trapped by oil, Sgto, Ymto; initial Ym → Ymto; later interaction Ymto ↔ Xm

• Gas trapped by water, Sgtw, Ymtw; initial Ym → Ymtw; later interaction Ymtw ↔ Wm; here

applies only to CO2 component

• Oil trapped by gas, Sotg, Xmtg; initial Xm → Xmtg ; later interaction Xmtg ↔ Ym

• Oil trapped by water, Sotw, Xmtw; initial Xm → Xmtw; later interaction Xmtw ↔ Wm; here


• Water trapped by gas, Swtg, Wmtg; initial Wm → Wmtg; later interaction Wmtg ↔ Ym; here


• Water trapped by oil, Swto, Wmto; initial Wm → Wmto; later interaction Wmto ↔ Xm; here


11.3.4 Composition Trapping Formulation

When trapping is added this yields (11.1).

0.006328∇ ·(Xmξoλok(∇Po − γo∇D)

)+ 0.006328∇ ·

(Ymξgλgk(∇Po +∇Pcgo − γg∇D)

)+


)+(Xmξoqo + Ymξg qg +Wmξw qw

)− τmtt =

∂

∂t


)(11.1)

184

The total transfer function for the trapped phase is (11.2).

τmtt = τmotw + τmotg + τmgtw + τmgto + τmwto + τmwtg (11.2)

(11.3)–(11.8) represents the transfer functions for each phase trapped by every other.

τmgto = Kmgtoσ((XmSoξo)− (YmtoSgtoξgto)

)=

∂

∂t

(φYmtoSgtoξgto)

)(11.3)

τmgtw = Kmgtwσ((WmSwξw)− (YmtwSgtwξgtw)

)=

∂

∂t

(φYmtwSgtwξgtw)

)(11.4)

τmotg = Kmotgσ((YmSgξg)− (XmtgSotgξotg)

)=

∂

∂t

(φXmtgSotgξotg)

)(11.5)

τmotw = Kmotwσ((WmSwξw)− (XmtwSotwξotw)

)=

∂

∂t

(φXmtwSotwξotw)

)(11.6)

τmwtg = Kmwtgσ((YmSgξg)− (WmtgSwtgξwtg)

)=

∂

∂t

(φWmtgSwtgξwtg)

)(11.7)

τmwto = Kmwtoσ((XmSoξo)− (WmtoSwtoξwto)

)=

∂

∂t

(φWmtoSwtoξwto)

)(11.8)

11.4 Interfacial Tension

As oil and gas phases transition from immiscible to miscible or from miscible to immiscible, the

relative permeability and capillary pressure can change significantly. Computationally, the miscible

hydrocarbon phase may be labeled as either “oil” or “gas”. The hydrocarbon and water relative

permeabilities need to be the same regardless of how this phase is labeled. Capillary pressure using

the J-function formulation explicitly accounts for the contact angle between the phases. Miscibility

changes are dynamic effects which change with the reservoir simulation.

11.4.1 Interfacial Tension Literature

For any simulation model that includes miscibility, it is necessary to consider the changes of

relative permeability with interfacial tension. Several authors have used the capillary number

185

(Nc = uμσ ) to scale the relative permeability, (Gibson, 2006; Stegemeier, 1977). Several authors

have used the ratio of the interfacial tension to a reference interfacial tension, σ/σ0, (Coats, 1980;

Hustad, 2002; Karimaie and Torsæter, 2008). Several authors have used a density weighting function

fh =ξh−ξgξo−ξg

in addition to the capillary number, (Blunt, 2000; Jerauld, 1997). Chase and Todd

(1984) use a “miscibility weighting function” α, which ranges between 0 for immiscible and 1 for

miscible. Schlumberger (2007a,b) use a ratio of the temperature to the critical temperature, T/Tc.

There does not seem to be any paper in the literature which compares these different techniques.

11.5 Rock Type and Wettability

This section describes the effects of rock type and wettability on relative permeability and

capillary pressure.

11.5.1 Rock Type

Different rock types are often used to initialize a simulation with different static relative per-

meability and capillary pressure curves. Different curves are also specified for the fracture system

and the matrix system in dual porosity or dual permeability regions. Different end points may

also be associated with individual grid cells to add additional variability to the properties. Rock

type changes are typically used to lump the effects of different mineralogy, permeability, porosity,

and pore size distribution. If the simulation grid is the result of upscaling a finer scaled geologic

distribution of properties, then it may be necessary to have the relative permeability and capillary

pressure functions differ in different directions.

11.5.2 Wettability Definitions

A good definition of wettability is “the tendency of one fluid to spread on or adhere to a solid

surface in the presence of other immiscible fluids” (Anderson, 1986a). There are also several other

definitions in 13933, including definitions based on contact angle, Amott index, and USBM index,

as well as some rules of thumb for wettability based on relative permeability, capillary pressure,

and residual saturations. The following six articles are a series of articles published in 1986-1987.

They review the wettability literature of the time and focus on different effects of wettability.

• Anderson (1986a), SPE #13932: General overview of wettability. Reference on wettability

definitions.

186

• Anderson (1986b), SPE #13933: Discussion of wettability measurement techniques. Good

discussion of different wettability indices.

• Anderson (1986c), SPE #13934: Discussion of how wettability affects the electrical properties.

Related to other work of the CSM/PI team.

• Anderson (1987a), SPE #15271: Discussion of how wettability affects capillary pressure.

• Anderson (1987b), SPE #16323: Discussion of how wettability effects relative permeability,

but does not define a functional relationship. As the wettability changes, the endpoints and

curvature both change.

• Anderson (1987c), SPE #16471: Discussion of how wettability affects waterflooding. This

has an impact on the total recovery from reservoirs as well as the remaining oil available for

enhanced oil recovery.

There are several illustrations of the variations of relative permeability with wettability, includ-

ing several figures in Anderson (1987b), for instance, Figure 11.2. None of these articles give a

formula for this change as a function of contact angle.

11.5.3 Static Wettability Changes

Rocks with different wettabilities have different relative permeability and capillary pressure

functions. If the wettability is constant within a given rock type then the different relative per-

meability and capillary pressure functions may be specified with static properties based on the

rock types. In some reservoirs, there are variations in wettability within the same rock type; this

may be a result of the deposition of different amounts of asphaltenes over geologic time based on

compositional gradients and variations in the oil water contact elevation. Capillary pressure using

the J-function methodology incorporates the changes in wettability explicitly through variations in

the contact angle. It is possible to specify the variation between relative permeability in a variety

of ways, but a comparison of these methods is not discussed in the literature.

11.5.4 Dynamic Wettability Changes

Wettability may also change dynamically with the deposition of asphaltenes, with temperature

changes, and with fluid-rock chemical interactions. If any of these dynamic effects are simulated,

187

Figure 11.2: Variation of relative permeability with wettability changes (Anderson, 1987b).

188

then it is also necessary to specify how the relative permeability curves change dynamically with

the changes in wettability. The same approach used for wettability gradients should also work for

dynamic changes in wettability.

The permeability and porosity will also change with the deposition of asphaltenes. Because

asphaltenes will not be deposited uniformly in all pores and pore throats, this could also cause a

dynamic variation in the relative permeability and capillary pressure functions. There is insufficient

experimental data to understand exactly how the functional form of the capillary pressure and

relative permeability changes with this deposition. If asphaltene deposition is simulated, then the

dynamic impacts of this deposition will be accounted for using dynamic wettability changes and

an additional scaling of the end points.

11.6 Temperature

The relative permeability and capillary pressure functions change with temperature variations.

There is insufficient data in the literature to document whether these changes are solely a result

of the changes in the interfacial tension and wettability with changes in temperature. If there are

additional variations, then a method to change capillary pressure and relative permeability functions

would need to be created. Under static conditions, most reservoirs experience a temperature

gradient. For many simulation models, these temperature changes may be ignored, but they are

significant for some reservoirs. There also are some large reservoirs which have lateral variations in

initial reservoir temperature. Temperature may also change dynamically by the injection of cold

water into a initially hot reservoir, or by the injection of steam or other hot fluid into a reservoir. For

this dissertation, initial temperature variations may be simulated but time dependent temperature

changes will not be.

11.7 Composition

Relative permeability and capillary pressure are a function of different fluid compositions. This

is why most experimental relative permeability and capillary pressure functions are determined

using oil, gas, and brine typical of the producing formation. In addition to these effects, there also

seem to be different relative permeabilities for a gaseous CO2:water system and a gaseous H2S:water

system, (Bennion and Bachu, 2008b). This indicates that there are probably dynamic variations

189

in the relative permeability and capillary pressure functions with the changes in the composition

of the reservoir fluids, but insufficient experimental data exists to create a general compositional

model of these variations.

11.8 Flow Rates

Vary rapid flow rates will generate non-Darcy effects. It is assumed that these non-Darcy effects

are simulated using a different technique than variations in the relative permeability. Rapid flow

rates or geomechanical effects may also change the porosity, permeability, and structure of the

rock. These changes would also cause changes in the capillary pressure and relative permeability,

but these effects are beyond the scope of this work.

11.9 Brooks-Corey Properties for Mixed Wet Rock

The definitions of the saturations, porosities, and other terms used to describe the single poros-

ity, dual porosity, and dual permeability systems with various amounts of trapping are described

in Chapter 5.

11.9.1 Simplified Three-Phase Relative Permeability

Relative permeability varies with the rock type; the wettability; the interfacial tension; the

previous maxima and minima of each saturation; and the increasing, decreasing, or constant status

of each phase. If all of these changes can be expressed as adjustments to the saturation endpoints

or the relative permeability value at the maximum saturation, then the following equations can be

used for all relative permeability calculations.

The total water relative permeability is defined by (11.9). For many systems, the water relative

permeability curve does not have significant hysteresis, so even when other forms are used to

calculate the three-phase relative permeability for the oil system, (11.9) may still be used for the

water system.

krw = k�rw

(Sw − Sw,min

Sw,max − Sw,min

)nw

= k�rw

(Sw − Swrm1 − Swm2

1− Sorm1 − Som2 − Sgrm1 − Sgm2 − Swrm1 − Swm2

)nw

(11.9)

krw[Sw ≤ Sw,min] = 0 krw[Sw ≥ Sw,max] = k�rw (11.10)

190

The total oil relative permeability may be defined by (11.11). For many systems, it is defined

instead as a mixture of krow and krog.

kro = k�ro

(So − So,min

So,max − So,min

)no

= k�ro

(So − Sorm1 − Som2

1− Sorm1 − Som2 − Sgrm1 − Sgm2 − Swrm1 − Swm2

)no

(11.11)

kro[So ≤ So,min] = 0 kro[So ≥ So,max] = k�ro (11.12)

The total gas relative permeability may be defined by (11.13). The form of this equation is

different from (11.9) and (11.11) in order to ensure that the derivative ∂kr∂S as S → Smin is non-zero

for at least one of the phases. Having one non-zero derivative and two zero derivatives stabilizes

the mathematics of the three-phase relative permeability calculations. Although the trapped and

residual properties change with the system, Sg,min stays the same for some systems. In this case,

krg is often specified using (11.13) rather than as a mixture of krgo and krgw.

krg = k�rg1

(Sg − Sg,min

Sg,max − Sg,min

)+ (k�rg − k�rg1)

(Sg − Sg,min

Sg,max − Sg,min

)ng

(11.13)

krg[Sg ≤ Sg,min] = 0 krg[Sg ≥ Sg,max] = k�rg (11.14)

For a spreading oil, it is logical to rewrite kro using a non-zero derivative and write krg with a

zero derivative. In this case, (11.11) is rewritten as (11.15) and (11.13) is rewritten as (11.16).

kro = k�ro1

(So − So,min

So,max − So,min

)+ (k�ro − k�ro1)

(So − So,min

So,max − So,min

)no

(11.15)

krg = k�rg

(Sg − Sg,min

Sg,max − Sg,min

)ng

(11.16)

Some cases with hysteresis require S-shaped scanning curves, for instance an increasing scanning

curve where the Smin endpoint was not reached, (11.17). This format is flexible; if krL[Smax1] =

krR[Smax1],∂krL∂S [Smax1] =

∂krR∂S [Smax1], krR[Smax2] is specified, and n2 is negative, then this gener-

ates an S-shaped curve.

⎧⎨⎩ krL = k�r

(S−Smin

Smax1−Smin

)n1

, S < Smax1

krR = k�r2 + k�r3

(S−Smin

Smax2−Smin

)n2

, S ≥ Smax1

(11.17)

191

Another approach used to define three-phase relative permeabilities is to specify individual

two-phase properties and then mix them in some way to obtain three-phase relative permeabilities.

11.9.2 Derivatives of Simplified Three-Phase Relative Permeability

For the IMPSEC formulation, it is necessary to calculate the derivatives of the relative per-

meability with respect to saturation. The following are the derivatives of the equations in Sec-

tion 11.9.1. All adjustments to the Smin, Smax, and k�r are made at time n, so these terms are

constants for purposes of the derivative calculations.

The derivatives of krw with respect to saturation from (11.9) are defined by:

krw = k�rw

(Sw − Sw,min

Sw,max − Sw,min

)nw

(11.18)

∂krw∂Sw

=k�rwnw

(Sw,max − Sw,min)

(Sw − Sw,min

Sw,max − Sw,min

)nw−1

=nw

(Sw − Sw,min)krw (11.19)

∂krw∂Sw

[Sw → Sw,min] = 0 (11.20)

∂krw∂So

= −∂krw∂Sw

(11.21)

∂krw∂Sg

= −∂krw∂Sw

(11.22)

The derivatives of kro with respect to saturation from (11.11) for a non-spreading oil are defined

by:

kro = k�ro

(So − So,min

So,max − So,min

)no

(11.23)

∂kro∂So

=k�rono

(So,max − So,min)

(So − So,min

So,max − So,min

)no−1

=no

(So − So,min)kro (11.24)

∂kro∂So

[So → So,min] = 0 (11.25)

∂kro∂Sw

= −∂krw∂So

(11.26)

∂kro∂Sg

= −∂krw∂So

(11.27)

The derivatives of krg with respect to saturation from (11.13) for the gas associated with a

non-spreading oil are defined by:

192

krg = krg1 + krg2 (11.28)

krg1 = k�rg1

(Sg − Sg,min

Sg,max − Sg,min

)(11.29)

krg2 = (k�rg − k�rg1)

(Sg − Sg,min

Sg,max − Sg,min

)ng

(11.30)

∂krg1∂Sg

=k�rg1

(Sg,max − Sg,min)(11.31)

∂krg2∂Sg

=(k�rg − k�rg1)ng

(Sg,max − Sg,min)

(Sg − Sg,min

Sg,max − Sg,min

)ng−1

=ng

(Sg − Sg,min)· krg2 (11.32)

∂krg∂Sg

=k�rg1

(Sg,max − Sg,min)+

ng

(Sg − Sg,min)· krg2 (11.33)

∂krg∂Sg

[Sg → Sg,min] =k�rg1

(Sg,max − Sg,min)(11.34)

∂krg∂So

= −∂krw∂Sg

(11.35)

∂krg∂Sw

= −∂krw∂Sg

(11.36)

The derivative of the S-shaped relative permeabilities are:

⎧⎨⎩ krL = k�r

(S−Smin

Smax1−Smin

)n1

, S < Smax1

krR = k�r2 + krR3 = k�r2 + k�r3

(S−Smin

Smax2−Smin

)n2

, S ≥ Smax1

(11.37)

∂krL∂S

=k�rLn

(Smax − Smin)

(S − Smin

Smax − Smin

)n−1

=n

(S − Smin)krL (11.38)

∂krL∂S

[S → Smin] = 0 (11.39)

∂krR∂S

=k�r3n2

(Smax2 − Smin)

(S − Smin

Smax2 − Smin

)n2−1

=n2

(S − Smin)krR3 (11.40)

11.9.3 Two-Phase Relative Permeabilities

The two-phase oil-water relative permeability is defined by (11.41). Note that this is a subset

of (11.11), but (11.11) explicitly accounts for the effects of trapped phases and the presence of gas.

krow = k∗row

(So − Sowr

1− Swr − Sowr

)now

(11.41)

The two-phase water-oil relative permeability is defined by (11.42). Note that this is a subset

of (11.9), but (11.9) explicitly accounts for the effects of trapped phases and the presence of gas.

193

krw = k∗rw

(Sw − Swr

1− Swr − Sowr

)nw

(11.42)

The two-phase oil-gas relative permeability is defined by (11.43). Note that this is a subset of

(11.11), but (11.11) explicitly accounts for the effects of trapped phases and the presence of residual

phases of each type. (11.11) is also written in terms of So as a primary variable.

krog = k∗rog

(1− Sg − Swr − Sogr

1− Swr − Sogr

)nog

(11.43)

The two-phase gas-liquid relative permeability is defined by (11.44). Note that this is a subset

of (11.13), but (11.13) explicitly accounts for the effects of trapped phases and the presence of

residual phases of each type.

krg = k∗rg

(Sg

1− Swr − Sogr

)ng

(11.44)

There are several ways of combining separate two-phase relative permeabilities into three phase

relative permeabilities. One of these approaches is based on Stone’s method.

kro,mix =Sgkrog + (Sw − Swr)krow

Sg + Sw − Swr(11.45)

11.9.4 Water-Oil Capillary Pressure for Mixed-Wet Systems

As usual, the following definitions are used for Sw,min and Sw,max:

Sw,min = Sw,m1,r + Sw,m2 (11.46)

Sw,max = 1− So,m1,r − So,m2 − Sg,m1,r − Sg,m2 (11.47)

The oil-water capillary pressure for a mixed wet system is defined by both of these equations:

Pcow1 = Pcow,thr − αow (Swx − Sw,min) ln

[Sw − Sw,min

Swx − Sw,min

](11.48)

Pcow2 = Pcow,thr + αow (Sw,max − Swx) ln

[Sw,max − Sw

Sw,max − Swx

](11.49)

194

Pcow =

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

Pcow,max, Sw ≤ Sw,min

Pcow,max, Sw,min < Sw ≤ Sw,min,clip

Pcow1, Sw,min,clip < Sw ≤ Swx

Pcow2, Swx < Sw ≤ Sw,max,clip

Pcow,min, Sw,max,clip < Sw ≤ Sw,max

Pcow,min, Sw ≥ Sw,max

(11.50)

An alternate definition uses εow ≈ 10−4 to avoid calculating ln[0].

Pcow1 = Pc,thr − αow (Swx − Sw,min + εow) ln

[Sw − Sw,min + εowSwx − Sw,min + εow

](11.51)

Pcow2 = Pc,thr + αow (Sw,max − Swx + εow) ln

[Sw,max − Sw + εowSw,max − Swx + εow

](11.52)

Pcow =

⎧⎪⎪⎨⎪⎪⎩Pcow,max, Sw ≤ Sw,min

Pcow1, Sw,min < Sw ≤ Swx

Pcow2, Swx < Sw ≤ Sw,max

Pcow,min, Sw ≥ Sw,max

(11.53)

Note that for both of these equations, Pcow1[Swx] = Pcow2[Swx] = 0.

11.9.5 Gas-Oil Capillary Pressure

As usual, the following definitions are used for Sg,min and Sg,max:

Sg,min = Sg,m1,r + Sg,m2 (11.54)

Sg,max = 1− So,m1,r − So,m2 − Sw,m1,r − Sw,m2 (11.55)

The gas-oil capillary pressure for the primary drainage cycle is defined using Pcgo1 with an extra

threshhold term.

Pcgo1 = −αgo (Sg,max − Sg,min) ln

[Sg − Sg,min

Sgmax − Sg,min

](11.56)

Pcgo =

⎧⎪⎪⎨⎪⎪⎩

Pcgo,max, Sg ≤ Sg,min

Pcgo,max, Sg,min < Sg ≤ Sg,min,clip

Pcgoth + Pcgo1, Sg,min,clip ≤ Sg ≤ Sg,max

0, Sg ≥ Sg,max

(11.57)

195

11.9.6 Derivatives of Capillary Pressure

The derivatives of the mixed wet oil-water capillary pressure are

∂Pcow1

∂Sw= −αow

(Swx − Sw,min

Sw − Sw,min

)(11.58)

∂Pcow1

∂So= −∂Pcow1

∂Sw(11.59)

∂Pcow1

∂Sg= −∂Pcow1

∂Sw(11.60)

∂Pcow2

∂Sw= −αow

(Sw,max − Swx

Sw,max − Sw

)(11.61)

∂Pcow2

∂So= −∂Pcow2

∂Sw(11.62)

∂Pcow2

∂Sg= −∂Pcow2

∂Sw(11.63)

Note that the derivatives∂Pcow1

∂Sw[Swx] =

∂Pcow2

∂Sw[Swx] = −αow.

The derivatives of the gas-oil capillary pressure are

∂Pcgo1

∂Sg= −αgo

(Sg,max − Sg,min

Sg − Sg,min

)(11.64)

∂Pcgo1

∂So= −∂Pcgo1

∂Sg(11.65)

∂Pcgo1

∂Sw= −∂Pcgo1

∂Sg(11.66)

11.10 Three Phase Relative Permeability References

The following articles discuss three-phase relative permeability models.

• Land (1968), SPE #1942: 3-phase relative permeability model

• Stone (1970), SPE #2116: “Stone 1”, 3-phase relative permeability model

• Stone (1973): “Stone 2”, 3-phase relative permeability model

• Fayers and Matthews (1984), SPE #11277: Analysis of 3-phase relative permeability experi-

ments. Modification of Stone 1.

• Baker (1988), SPE #17369: Analysis of 3-phase relative permeability experiments. Uses

saturation weighted relative permeabilities.

196

• Delshad and Pope (1989): Comparison of 7 different 3-phase relative permeability models.

Presents a different model for calculating kro.

• Fayers (1989), SPE #16965: Describes alternate ways to calculate Sorm for Stone 1 algorithm.

• Pope et al. (1998), SPE #49266: Gas condensate relative permeability model

• Paterson et al. (1998), SPE #50938: Gas condensate relative permeability model

• van Dijke et al. (2000), SPE #59310: 3-phase relative permeability model


The following articles discuss problems with some of the previous algorithms: Selected literature

indicating problems with Stone 1 (Stone, 1970), Stone 2 (Stone, 1973)

• Kleppe et al. (1997)

• Larsen and Skauge (1998)

• Blunt (2000)

• Element et al. (2003)


11.11 Hysteresis References

Hysteresis models:

• Killough (1976), SPE #5106: Hysteresis calculation

• Carlson (1981), SPE #10157: Non-wetting phase hysteresis


of different models

• Delshad et al. (2003), SPE #86916: Mixed wet model with hysteresis

• Spiteri et al. (2005), SPE #96448: New hysteresis model

197

11.12 Combined Three-Phase Relative Permeability and Hysteresis References

Models for both three-phase relative permeability and hysteresis:

• Jerauld (1997), SPE #36178: Correlations for 3-phase relative permeability and hysteresis to

fit an extensive mixed wet data set.

• Larsen and Skauge (1998), SPE #38456: 3-phase relative permeability model


of different models


• Hustad et al. (2002), SPE #75138: Presentation of the “IKU” method (named by Eclipse);

comparison of different models

• Hustad (2002), SPE #74705: 3-phase relative permeability and capillary pressure model with

hysteresis; called the “ODD3P” model by Eclipse.

• Element et al. (2003), SPE #84903: Evaluation of different relative permeability and hys-

teresis models

• Spiteri and Juanes (2004), SPE #89921: Evaluation of different relative permeability and

hysteresis models

Many of the algorithms presented in the literature have some limitations or drawbacks. The

following articles discuss problems with some of the previous algorithms:

• Selected literature indicating problems with Stone 1 (Stone, 1970), Stone 2 (Stone, 1973)

– Kleppe et al. (1997)

– Larsen and Skauge (1998)

– Blunt (2000)

– Element et al. (2003)

– Spiteri and Juanes (2004)

198

• Jerauld (1997): Problems with Land (1968)

• Kleppe et al. (1997): Problems with Killough (1976)

• Blunt (2000): Review of previous algorithms, plus presentation of new model; multiple prob-

lems indicated with Stone 1 (Stone, 1970), Stone 2 (Stone, 1973), LandLand (1968); discussion

of the other models: Vizika and Lombard (1996), Larsen and Skauge (1998), Jerauld (1997)

• Element et al. (2003)

– Review of previous algorithms, using new data set

– Larsen and Skauge (1998): Problems because uses Stone 1; no trapping of water; no

variation of Land constant with cycle

– Blunt (2000): No wetting phase hysteresis; hysteresis is in a closed loop and shouldnt

be; no variation of Land constant with cycle

– Egermann et al. (2000): No variation of Land constant with cycle


– Review of previous algorithms, using Oaks data (Oak, 1990)

– Stone 1, Stone 2, Baker (1988): All are bad fits for kro

– Larsen and Skauge (1998): Not suitable for krg

– Other models: Killough (1976), Carlson (1981), Lenhard and Oostrom (1998), Jerauld

(1997), Blunt (2000)

199

CHAPTER 12

VISCOSITY FORMULATION

This chapter describes several different mathematical formulations to calculate the oil and gas

viscosity.

12.1 Treatment of Viscosity by Commercial Applications

There are two primary models for calculating the viscosity for a compositional model. The LBC

model, Lohrenz, Bray, and Clark (1964), has five regular tuning parameters plus the critical volumes

for each parameter. It is a reasonably good model if tuned, but is frequently off by 50% if it is not

tuned. The Pedersen model, Pedersen and Fredenslund (1987), is based on a corresponding states

model which maps the viscosity of a mixture to the viscosity of methane. It is tuned in different

ways by different programs. The Aasberg model, Aasberg-Petersen, Knudsen, and Fredenslund

(1991), is based on a corresponding states model which maps the viscosity of a mixture to the

viscosity of methane and the viscosity of n-decane. None of the commercial software packages

implement the Aasberg model.

Schlumberger Eclipse implements the Pedersen and the LBC models. It is possible to tune the

parameters for the LBC model by adjusting the 5 values in the polynomial correlation based on

the reduced density or by adjusting the vc values. No tuning is allowed for the Pedersen model.

Haliburton Landmark VIP implements the Pedersen and the LBC models. It is possible to tune

the parameters for the LBC model by adjusting the 5 values in the polynomial correlation based

on the reduced density or by adjusting the vc values. It is possible to tune the parameters for the

Pedersen model by adjusting the parameter values in μCH4,1 and μCH4,2. VIP also allows binary

interaction parameters for the viscosity equation.

Computer Modeling Group GEM and Winprop implement the Pedersen model but does not

implement the LBCmodel. It is possible to tune the parameters for the Pedersen model by adjusting

the parameter values in the MWmix and the parameters in calculating α.

Calsep PVTsim implements the Pedersen and the LBC models. It is possible to tune the

parameters for the LBC model by adjusting the 5 values in the polynomial correlation based on

200

the reduced density or by adjusting the vc values. The Pedersen model can be tuned by adjusting

the parameters for calculating the MWmix and the parameters in μCH4,2.

Powers implements the LBC model, but does not implement the Pedersen model. It is possible

to tune the parameters for the LBC model by adjusting the 5 values in the polynomial correlation

based on the reduced density or by adjusting the vc values.

12.2 Other Viscosity Models

Several other viscosity models that may have predictive capabilities were discovered in the

literature search.

The f -theory model, Quinones-Cisneros, Zeberg-Mikkelsen, and Stenby (2001a), defines vis-

cosity correlations based on friction theory using a dilute gas viscosity correlation and correla-

tions based on the attractive and repulsive forces of the Peng-Robinson model. There are sixteen

parameters which have been determined based on fitting viscosity data for various hydrocarbon

components. There are another eleven parameters which have been determined for the dilute gas

viscosity correlation. There is also the option to tune the data using a single linear parameter, plus

the possibility of adjusting the critical volume and/or critical viscosity of various components. In

its simplest form, the model requires the Peng-Robinson correlation (including the critical temper-

ature, critical pressure, acentric factor, molecular weight), plus the critical volume and the critical

viscosity. If the critical volume and critical viscosity are not known for a component, then they can

be calculated from the provided self-consistent correlations.

12.3 Lohrenz-Brae-Clark Model

This section discusses how viscosity changes with composition, based on correlations of Lohrenz,

Brae, and Clark (Lohrenz et al., 1964). The correlations in this section use T [R], P [psi], MW[lb/lbmol],

and μ[cp], ξ[lbmol/ft3].

12.3.1 Constants

The Lohrenz-Brae-Clark model requires the molecular weight, MWm, the critical pressure Pcm,

the critical temperature Tcm, and the critical volume vcm for each component. Given a phase density

ξϕ, the mole fractions Xm, and a temperature T , the method will then calculate the viscosity of

201

the phase μϕ. Typically, ξϕ[P, T,X] is obtained from the equation of state, requiring ωm, δmn, Ωa,

Ωb, and cm in addition to Pcm and Tcm.

Define λm, with T in R and P in psia.

1

λ#m

=

(MW#

m

)1/2 (P#cm

14.7

)2/3(T#cm1.8

)1/6 (12.1)

Define the relative temperature

T#rm =

T#

T#cm

(12.2)

Calculate the component viscosity at low pressure.

μ#m[cp] =

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩34 · 10−5 ·

(T#rm

)0.94λ#m

T#rm ≤ 1.5

17.78 · 10−5 ·(4.58 · T#

r − 1.67)5/8

λ#m

T#rm > 1.5

(12.3)

12.3.2 Time-Dependent

For the oil phase, the pseudo-reduced properties of mixtures are defined as follows:

pPnc =

∑Xn

mP#cm pT n

c =∑

XnmT#

cm pξnc =1∑

Xnmv#cm

(12.4)

For the gas phase, the pseudo-reduced properties of mixtures are defined as follows:

pPnc =

∑Y nmP#

cm pT nc =

∑Y nmT#

cm pξnc =1∑Y nmv#cm

(12.5)

For the oil phase, the total properties are defined as follows

MWnt =

∑m

XnmMW#

m (12.6)

202

μnt,low =

∑Xn

mμ#m

(MW#

m

)1/2∑

Xnm

(MW#

m

)1/2 (12.7)

For the gas phase, the total properties are defined as follows

MWnt =

∑m

Y nmMW#

m (12.8)

μnt,low =

∑Y nmμ#

m

(MW#

m

)1/2∑

Y nm

(MW#

m

)1/2 (12.9)

The λt uses the same equation for both the oil and gas phases:

λnt =

(pT n

c

1.8

)1/6

(MWnt)

−1/2

(pPn

c

14.7

)−2/3

(12.10)

Calculate the relative density, using ϕ = o, g:

ξnϕr =ξnϕpξnc

(12.11)

Calculate the viscosity. The Lohrenz, Brae, and Clark model uses the correlation presented by

Jossi, Stiel, and Thodos (1962).

((μn

ϕ[cp]− μnt,low) · λn

t + 10−4)1/4

=

0.10230 + 0.023364 · ξϕr + 0.058533 · ξ2ϕr +−0.040758 · ξ3ϕr + 0.0093324 · ξ4ϕr (12.12)

12.4 Jossi plus Lee

CMG does not provide the Lohrenz, Brae, and Clark (Lohrenz et al., 1964) model. The Jossi

model (Jossi et al., 1962) with low pressure viscosity calculated based on Lee and Eakin (1964) is

a model that can be implemented relatively easily to compare with CMG models. The correlations

in this section use T [R], P [psi], MW[lb/lbmol], and μ[cp], ξ[lbmol/ft3]; CMG uses T [K], P [atm],

MW[g/gmol], μ[cp], and ξ[kmol/m3]

For the oil phase, the pseudo-reduced properties of mixtures are defined as follows:

203

pPnc =

∑Xn

mP#cm pT n

c =∑

XnmT#

cm pξnc =1∑

Xnmv#cm

(12.13)

For the gas phase, the pseudo-reduced properties of mixtures are defined as follows:

pPnc =

∑Y nmP#

cm pT nc =

∑Y nmT#

cm pξnc =1∑Y nmv#cm

(12.14)

For the oil phase, the total properties are defined as follows

MWnt =

∑m

XnmMW#

m (12.15)

For the gas phase, the total properties are defined as follows

MWnt =

∑m

Y nmMW#

m (12.16)

Based on Lee and Eakin (1964), the low pressure viscosity is calculated based on

μt,low =10−4 (7.43 + 0.0133MWt) (T [R])

32

T [R] + 75.4 + 13.9MWt(12.17)

The λt uses the same equation for both the oil and gas phases:

λnt =

(pT n

c

1.8

)1/6

(MWnt)

−1/2

(pPn

c

14.7

)−2/3

(12.18)

Calculate the relative density, using ϕ = o, g:

ξnϕr =ξnϕpξnc

(12.19)

Calculate the viscosity.

((μn

ϕ[cp]− μnt,low) · λn

t + 10−4)1/4

=

0.10230 + 0.023364 · ξϕr + 0.058533 · ξ2ϕr +−0.040758 · ξ3ϕr + 0.0093324 · ξ4ϕr (12.20)

204

12.5 Corresponding States

The corresponding states model, as presented by Pedersen and Christensen (2007), relates the

viscosity of an oil or gas mixture to the viscosity of methane. The viscosity of methane is calculated

from a very detailed correlation of methane density and viscosity as a function of pressure and

temperature, with some additional adjustments below the freezing point of methane. The units in

this section use T [K], P [atm], ρ[g/cm3], ξ[gmol/L], and μ[cp].

Table 12.1: Units for (12.21).

variable units name

P atm pressure P [atm] = P [psi]/14.6959488

P bar pressure P [bar] = P [atm] ∗ 1.01325ξ gmol/L molar density ξ[gmol/L] = 16.0184634 · ξ[lbmol/ft3]

gmol = lbmol/453.59237

liter = ft3/28.31684659

T K temperature T [K] = 273.15 + (T [F ]− 32) · 59μ cP viscosity

The corresponding states model requires the critical pressure Pcm, the critical temperature

Tcm, the molecular weight MWm, and a detailed model of the viscosity and density as a function

of temperature and pressure for methane. Given the pressure P , the temperature T , and the mole

fractions of each component Xm the method will then calculate the viscosity of the phase μϕ.

12.5.1 Methane Density

The following correlation for pressure as a function of methane density is based on Pedersen

and Christensen (2007) and McCarty (1974). This correlation is in the form of P [ξ, T ] but what we

need is ξ[P, T ]; this is calculated using Newton-Raphson iterations. The solution seems to converge

to the correct value for a wide range of pressure and temperatures if ξ = 25gmol/L is used as an

initial estimate. It is important not to use the value estimated from the Peng-Robinson equation

of state, since this correlation is more accurate for methane than the more general Peng-Robinson

EOS module.

205

FξCH4[ξ, P, T ] :

0 = −P + 0.08205616 · ξ · T+ξ2 · (−0.018439486666 · T + 1.0510162064 · T 1/2+

− 16.057820303+ 848.44027562 · T−1 +−4.2738409106 · 104 · T−2)+

ξ3 · (7.6565285254 · 10−4 · T +−0.48360724197+ 85.195473835 · T−1 +−1.6607434721 · 104 · T−2)+

ξ4 · (−3.7521074532 · 10−5 · T + 2.8616309259 · 10−2 +−2.868528973 · T−1)+

ξ5(1.1906973942 · 10−4)+

ξ6 · (−8.5315715699 · 10−3 · T−1 + 3.8365063841 · T−2)+

ξ7 · (2.4986828379 · 10−5 · T−1)+

ξ8 · (5.7974531455 · 10−6 · T−1 +−7.1648329297 · 10−3 · T−2)+

ξ9 · (1.2577853784 · 10−4 · T−2)+

exp[−0.0096 · ξ2] ·(ξ3 · (2.2240102466 · 104 · T−2 +−1.4800512328 · 106 · T−3)+

ξ5 · (50.498054887 · T−2 + 1.6428375992 · 106 · T−4)+

ξ7 · (0.21325387196 · T−2 + 37.791273422 · T−3)+

ξ9 · (−1.1857016815 · 10−5 · T−2 +−31.630780767 · T−4)+

ξ11 · (−4.1006782941 · 10−6 · T−2 + 1.4870043284 · 10−3 · T−3)+

ξ13 · (3.1512261532 · 10−9 · T−2 +−2.1670774745 · 10−6 · T−3 + 2.4000551079 · 10−5 · T−4)

)(12.21)

The Newton-Raphson evaluation of ξ is as follows

ξ�+1 = ξ� −(FξCH4 [ξ

�, P, T ]

)/

(∂FξCH4

∂ξ[ξ�, P, T ]

)(12.22)

With convergence criteria

∣∣∣∣ξ�+1 − ξ�

ξ�

∣∣∣∣ < εξ (12.23)

Figure 12.1 shows that this correlation for density has been properly coded using the correct

units.

12.5.2 Methane Viscosity

The critical properties for methane, from Hanley, McCarty, and Haynes (1975) are the following.

It is important to use these values exactly to reproduce the correlation accurately.

• molecular weight MW = 16.043g/gmol.

206

24 26 28 30 32 340

200

400

600

800

1000

CH4 density �mol�L�

P�b

ar�

Compare to Pedersen Figure 10.3

Figure 12.1: Compare the density correlation for methane to Pedersen Figure 10.3. Each set ofdots represents steps of 30bar for a specific temperature from the code.

207

• freezing temperature TF = 91K

• critical temperature Tc = 190.55K

• critical pressure Pc = 45.387atm

• critical molar density ξc = 10.15gmol/L

• critical density ρc = 0.162836g/cm3

There is extensive data on the viscosity of methane. This correlation is based on Pedersen

and Christensen (2007) and Hanley et al. (1975), with a low temperature adjustment based on

Pedersen and Fredenslund (1987). The tanh terms are adjusted to fit Pedersen and Christensen

(2007), Figure 10.4. The coefficient of 1.0ΔT does not reproduce Figure 10.4.

μCH4 [ρ[g/cm3], T [K]] = 10−4 ·

(μCH4,0[T ] + μCH4,1[T ]+

1 + tanh[0.1(T − TF,CH4)]

2μCH4,2[ρ, T ] +

1− tanh[0.1(T − TF,CH4)]

2μCH4,3[ρ, T ]

)(12.24)

μCH4,0[T [K]] =

− 2.090975 · 105 · T−1 + 2.647269 · 105 · T−2/3 +−1.472818 · 105 · T−1/3+

4.716740 · 104 +−9.491872 · 103 · T 1/3 + 1.219979 · 103 · T 2/3+

− 9.627993 · 101 · T + 4.274152 · T 4/3 + −8.141531 · 10−2 · T 5/3 (12.25)

μCH4,1[T [K]] = 1.696985927 +−0.133372346 ·(1.4 − ln

[T

168.0

])2

(12.26)

μCH4,2[ρ[g/cm3], T [K]] = exp

[−10.35060586 + 188.73011594 · T−1]×(

exp[ρ0.1 · (17.571599671 +−3019.3918656 · T−3/2)+

ρ− ρcρc

· √ρ · (0.042903609488 + 145.29023444 · T−1 + 6127.6818706 · T−2)]− 1.0

)(12.27)

208

μCH4,3[ρ[g/cm3], T [K]] = exp

[−9.74602 + 44.6055 · T−1]×(

exp[ρ0.1 · (18.0834 +−4126.66 · T−3/2)+

ρ− ρcρc

· √ρ · (0.976544 + 81.8134 · T−1 + 15649.9 · T−2)]− 1.0

)(12.28)

Figure 12.2 shows that this correlation for viscosity has been properly coded using the correct

units. The Pedersen and Fredenslund (1987) model is very good for 100bar and 800bar. The

coefficient of the tanh term was adjusted to get as good a fit as possible at 2000bar. All fits were

very good for the Hanley et al. (1975) model, provided that the identical numerical values for the

critical pressure, temperature, and density values are used.

70 80 90 100 110 1200.

0.4

0.8

1.2

1.670 80 90 100 110 120

0.

0.4

0.8

1.2

1.6

T�K�

Μ�cp�

Compare to Pedersen Figure 10.4

Figure 12.2: Compare the viscosity correlation for methane to Pedersen Figure 10.4. The dots aretemperature steps of 1K. The red dots are for the Pedersen 1987 model. The green dots are forthe Hanley 1975 model.

Figure 12.3 shows that the Hanley et al. (1975) correlation is a also good predictor of the

experimental data at higher temperatures and pressures specified in Gonzalez, Bukacek, and Lee

(1967). Note that the Hanley et al. (1975) correlation falls within the range of the experimental

209

data, although at higher pressures it is further from the Gonzalez et al. (1967) correlation.

40 80 120 160 200 240 280 320 360 400 440150

170

190

210

230

250

270

290

310

330

350

370

39040 80 120 160 200 240 280 320 360 400 440

150

170

190

210

230

250

270

290

310

330

350

370

390

T�°F�

Μ�10�4cp�

Compare to Gonzalez Figure 2

Figure 12.3: Compare the Hanley viscosity correlation for methane to Gonzalez Figure 2. The dotsare temperature steps of 5◦F.

12.5.3 Corresponding States Calculations

The mixed critical temperature is defined as

T nc,mix[X

n�ı ] =

∑i

∑j

Xni X

nj

⎛⎝(T#

ci

P#ci

)1/3

+

(T#cj

P#cj

)1/3⎞⎠3√

T#ci · T#

cj

∑i

∑j

Xni X

nj

⎛⎝(T#

ci

P#ci

)1/3

+

(T#cj

P#cj

)1/3⎞⎠3 (12.29)

The mixed critical pressure is defined as

210

Pc,mix[Xn�ı ] = 8 ·

∑i

∑j

Xni X

nj

⎛⎝(T#

ci

P#ci

)1/3

+

(T#cj

P#cj

)1/3⎞⎠3√

T#ci · T#

cj

⎛⎜⎝∑

i

∑j

Xni X

nj

⎛⎝(T#

ci

P#ci

)1/3

+

(T#cj

P#cj

)1/3⎞⎠3⎞⎟⎠

2 (12.30)

The following properties are not time-dependent and can be pre-calculated for Tc,mix and Pc,mix

TcPc#ij =

⎛⎝(T#

ci

P#ci

)1/3

+

(T#cj

P#cj

)1/3⎞⎠3

TcTc#ij = TcPc#ij ·√T#ci · T#

cj (12.31)

T nc,mix[X

n�ı ] =

∑i

∑j

Xni X

nj TcTc

#ij∑

i

∑j

Xni X

nj TcPc

#ij

Pnc,mix[X

n�ı ] = 8 ·

∑i

∑j

Xni X

nj TcTc

#ij

⎛⎝∑

i

∑j

Xni X

nj TcPc

#ij

⎞⎠2 (12.32)

The reduced density is defined by

ξnr [Pn, T#,Xn

�ı ] =

ξCH4

[Pn · P#

c,CH4

Pnc,mix

,T# · T#

c,CH4

T nc,mix

]

ξ#c,CH4

(12.33)

The molecular weight of the mixture is defined by

MWnmix[X

n�ı ] = 1.304 · 10−4 ·

((〈MW〉nw)2.303 − (〈MW〉nn)2.303

)+ 〈MW〉nn (12.34)

Using the weight averaged and number averaged molecular weights:

〈MW〉nw [Xn�ı ] =

∑i

Xni

(MW#

i

)2∑i

Xni MW#

i

〈MW〉nn [Xn�ı ] =

∑i

Xni MW#

i (12.35)

The adjustment factor αmix is defined by:

αnmix[P

n, T#,Xn�ı ] = 1.00 + 7.378 · 10−3 · (ξnr )1.847 (MWn

mix)0.5173 (12.36)

211

The methane adjustment factor αCH4 is defined by:

αnCH4

[Pn, T#] = 1.00 + 7.378 · 10−3 ·(ξCH4

[Pn, T#

]ξ#c,CH4

)1.847 (MW#

CH4

)0.5173(12.37)

The adjusted pressure and temperature are:

Pn0 [P

n, T#,Xn�ı ] =

Pn · P#c,CH4

· αnCH4

Pnc,mix · αn

mix

T n0 [P

n, T#,Xn�ı ] =

T# · T#c,CH4

· αnCH4

T nc,mix · αn

mix

(12.38)

The viscosity of the mix is defined by:

μnmix,L[P

n, T#,Xn�ı ] =

(T nc,mix

T#c,CH4

)−1/6

·(

Pnc,mix

P#c,CH4

)2/3

·(

MWnmix

MW#CH4

)1/2

·(

αnmix

αnCH4

)·μCH4 [P

n0 , T

n0 ] (12.39)

12.5.4 Heavy oil adjustment

For heavy oils, the corresponding states model based on methane is not accurate. The following

adjusted viscosity for heavy oils is based on Pedersen and Christensen (2007) and Rønningesen

(1993).

μmix,H [Pn, T#,Xn�ı ] = 10

(−0.07995−0.1101Mn−371.8(T#)

−1+6.215Mn(T#)

−1)exp[0.008 · (Pn − 1.000)]

(12.40)

Here, M is defined by the following

Mn[Xn�ı ] =

⎧⎪⎪⎪⎨⎪⎪⎪⎩〈MW〉nw〈MW〉nn

≤ 1.5, 〈MW〉nn〈MW〉nw〈MW〉nn

≥ 1.5, 〈MW〉nn ·√

1

1.5· 〈MW〉nw〈MW〉nn

(12.41)

Define the viscosity of the mixture in the following way, using (12.38) to define the mix tem-

perature.

• If T0 ≥ 75K, μmix = μmix,L based on (12.39).

• If T0 ≤ 65K, μmix = μmix,H based on (12.40).

• If 65K < T0 < 75K, μmix = (μmix,H + μmix,L)/2.

212

Since the Rønningesen (1993) model is an adjustment to the corresponding states model, it

requires the same things as the corresponding states model: the critical pressure Pcm, the critical

temperature Tcm, the molecular weight MWm, and a detailed model of the viscosity and density

as a function of temperature and pressure for methane. Given the pressure P , the temperature T ,

and the mole fractions of each component Xm the method will then calculate the viscosity of the

phase μϕ.

12.6 Extended Corresponding States

The extended corresponding states model, as presented by Aasberg-Petersen et al. (1991),

relates the viscosity of an oil or gas mixture to the viscosity of methane and n-decane. It is based

on Pedersen and Fredenslund (1987). As presented here, the methane correlations from Pedersen

and Fredenslund (1987) are used directly. For n-decane, the viscosity correlation from Aasberg-

Petersen et al. (1991) is used, and compared graphically to Lee and Ellington (1965). The density

correlation in Aasberg-Petersen et al. (1991) does not provide the full details of the method, and

references other works that are not available or not cited in their bibliography. As a result, n-

decane density correlations from Audonnet and Padua (2004), Cibulka and Hnedkovsky (1996),

and Assael, Dymond, and Exadaktilou (1994). These are compared graphically to density values

from Sage and Lacey (1950).

The corresponding states model requires the critical pressure Pcm, the critical temperature

Tcm, the molecular weight MWm, and a detailed model of the viscosity and density as a function

of temperature and pressure for methane. Given the pressure P , the temperature T , and the mole

fractions of each component Xm the method will then calculate the viscosity of the phase μϕ.

12.6.1 n-Decane Density

Aasberg-Petersen et al. (1991) defines a correlation for n-decane, but it references a dissertation

by Jensen at the Technical University of Denmark which was not available and an article by Chueh

and Prausnitz which is not listed in their bibliography. As a result, there is insufficient information

to be able to implement the Aasberg-Petersen et al. (1991) n-decane density correlation.

Audonnet and Padua (2004) uses T [K], P [MPa], ρ[kg/m3]. Audonnet and Padua (2004) uses

the Tait equation, Dymond and Malhotra (1988), to define the pressure and temperature variation

213

of density.

ρ[T, P ]− ρ0[T, P0]

ρ[T, P ]= C log10

[B + P

B + P0

](12.42)

For the n-decane correlation by Audonnet and Padua (2004), the parameter values are:

P0 = 25MPa (12.43)

C = 0.2252 (12.44)

B = 436.929 − 2.17957T + 0.00272365T 2 (12.45)

ρ0 = 1133.36 − 2.39023T + 0.00225011T 2 (12.46)

Cibulka and Hnedkovsky (1996) collected a lot of density data for various n-paraffins, including

n-decane; in addition, the article provides a correlation for n-decane density based on the Tait

equation. Cibulka and Hnedkovsky (1996) uses T [K], P [MPa], ρ[kg/m3].

ρ[T, P ] =ρ0[T, P0]

1− C ln[B+PB+P0

] (12.47)

For n-decane correlation by Cibulka and Hnedkovsky (1996), the parameter values are:

P0 = 0.101325MPa (12.48)

T0 = 294.35K (12.49)

C = 0.087992 − 0.000816

(T − T0

100

)(12.50)

B = 83.5746 − 61.9418

(T − T0

100

)+ 21.8935

(T − T0

100

)2

+ (12.51)

− 6.4316

(T − T0

100

)3

+ 1.0545

(T − T0

100

)4

(12.52)

Cibulka and Hnedkovsky (1996) uses Assael et al. (1994) as a correlation for ρ0. Cibulka and

Hnedkovsky (1996) uses T [K], P [MPa], ρ[kg/m3].

ρ0[T, P0] = 239

(1 + 0.329139 + 7.364340

(1− T

617.65

) 13

+

− 9.985096

(1− T

617.65

) 23

5.283608

(1− T

617.65

))(12.53)

214

12.6.2 n-Decane Viscosity

The viscosity model of Aasberg-Petersen et al. (1991) uses T [K], P [atm], ρ[g/cm3], and μ[cP].

• molecular weight MW = 142.284g/gmol.

• freezing temperature TF = 243.5K

• critical temperature Tc = 617.40K

• critical pressure Pc = 20.18atm

• critical density ρc = 0.2269g/cm3

μnC10 [ρ[g/cm3], T [K]] = 10−4 ·

(μnC10,0[T ] + ρ · μnC10,1[T ] + μnC10,2[ρ, T ]

)(12.54)

μnC10,0[T [K]] = 0.2640 · T−1 + 0.9487 · T−2/3 + 71.0 · T−1/3 (12.55)

μnC10,1[T [K]] = 2.48 · 10−4 + 81.35 ·(5.9583 − ln

[T

490

])2

(12.56)

μnC10,2[ρ[g/cm3], T [K]] = exp

[−11.739 − 811.3 · T−1] ·(exp[ρ0.1 · (16.092+−18464 ·T−3/2)+

ρ− ρcρc

· √ρ · (1.9745 + 898.45 · T−1 + 119620 · T−2)]− 1.0

)(12.57)

12.6.3 Calculations

The extended corresponding states model of Aasberg-Petersen et al. (1991) is a specific example

of the generalized corresponding states model described in Teja and Rice (1981).

The mixed critical temperature is defined as

T ncx[X

n�ı ] =

∑i

∑j

Xni X

nj

⎛⎝(T#

ci

P#ci

)1/3

+

(T#cj

P#cj

)1/3⎞⎠3√

T#ci · T#

cj

∑i

∑j

Xni X

nj

⎛⎝(T#

ci

P#ci

)1/3

+

(T#cj

P#cj

)1/3⎞⎠3 (12.58)

215

The mixed critical pressure is defined as

Pcx[Xn�ı ] = 8 ·

∑i

∑j

Xni X

nj

⎛⎝(T#

ci

P#ci

)1/3

+

(T#cj

P#cj

)1/3⎞⎠3√

T#ci · T#

cj

⎛⎜⎝∑

i

∑j

Xni X

nj

⎛⎝(T#

ci

P#ci

)1/3

+

(T#cj

P#cj

)1/3⎞⎠3⎞⎟⎠

2 (12.59)

The molecular weight of the mixture is defined by

MWnx[X

n�ı ] = 0.00867358 ·

((〈MW〉nw)1.56079 − (〈MW〉nn)1.56079

)+ 〈MW〉nw (12.60)

Using the weight averaged and number averaged molecular weights:

〈MW〉nw [Xn�ı ] =

∑i

Xni

(MW#

i

)2∑i

Xni MW#

i

〈MW〉nn [Xn�ı ] =

∑i

Xni MW#

i (12.61)

Define the mixture viscosity using the reference viscosity of methane and n-decane using

μmix =μcx · μCH4 [T1, P1]

μc1·(μnC10 [T2, P2] · μc1

μCH4 [T1, P1] · μc2

)(MWx−MWCH4

MWnC10−MWCH4

)(12.62)

Define the reference temperatures and pressures as follows:

T1 =T · Tc,CH4

TcxT2 =

T · Tc,nC10

TcxP1 =

P · Pc,CH4

PcxP2 =

P · Pc,nC10

Pcx(12.63)

Define the reference mixture properties

μcx = (MWx)12 (Pcx)

23 (Tcx)

− 16 (12.64)

μc1 = (MWCH4)12 (Pc,CH4)

23 (Tc,CH4)

− 16 (12.65)

μc2 = (MWnC10)12 (Pc,nC10)

23 (Tc,nC10)

− 16 (12.66)

12.7 f -Theory Model

The f -theory model is described by Quinones-Cisneros et al. (2001a). There are additional

derivations and comparisons to experimental data in Quinones-Cisneros, Zeberg-Mikkelsen, and

216

Stenby (2000) and Quinones-Cisneros, Zeberg-Mikkelsen, and Stenby (2001b). For this model, the

viscosity is in μP, the specific volume is in cm3/mol, the temperature is in K, and the pressure is

in bar.

The viscosity is split into components depending on the dilute gas viscosity μ0 and the friction-

based viscosity μf . The friction-based viscosity is split into a portion that multiplies the Peng-

Robinson repulsive pressure, the Peng-Robinson repulsive pressure squared, and the Peng-Robinson

attractive pressure.

μ = μ0 + μf = μ0 + κaPPRa + κrPPRr + κrr (PPRr)2 (12.67)

12.7.1 Dilute Gas Viscosity and General Properties

The dilute gas viscosity is defined based on the work of Chung.

μ0 = 40.785(MW · T )1/2Fc

(vc)2/3 Ω�

(12.68)

Where Fc and Ω� are defined using

Fc = 1− 0.2756ω T � = 1.2593T

Tc(12.69)

Ω� = 1.16145 (T �)−0.14874 + 0.52487 exp [−0.7732 · T �] + 2.16178 exp [−2.43787 · T �]

− 6.435 · 10−4 (T �)0.14874 · sin[18.0323 · (T �)−0.76830 − 7.27371

](12.70)

The critical density may be calculated from the following correlation if it has not been defined from

another correlation:

vc

[cm3

mol

]=

1

0.000235751 + 3.42770(

PcRTc

) (12.71)

12.7.2 f -Theory Friction Properties

The critical viscosity is defined using the following correlation based on Uyehara or the tabulated

values.

217

μc = 7.9483 (MW)1/2 (Pc)2/3 (Tc)

−1/6 (12.72)

• N2, μc[μP] = 174.179

• CO2, μc[μP] = 376.872

• CH4, μc[μP] = 152.930

• C2, μc[μP] = 217.562

• C3, μc[μP] = 249.734

• iC4, μc[μP] = 271.155

• nC4, μc[μP] = 257.682

• iC5, μc[μP] = 275.073

• nC5, μc[μP] = 258.651

• C6, μc[μP] = 257.841

The Peneloux adjusted Peng-Robinson terms are defined by:

PPRa =−a

(v + c)(v + 2c+ b) + (b+ c)(v − b)PPRr =

RT

v − b(12.73)

The following definitions use the reduced variables Γ and Ψ:

Γ =Tc

TΨ =

RTc

Pc(12.74)

The attractive term is defined by the following equation, using seven pre-fit coefficients.

κa · Pc

μc= −0.140464 +−4.89197 · 10−2 (Γ− 1)+(

0.270572 +−1.10473 · 10−4Ψ) · (exp [Γ− 1]− 1)+(−4.48111 · 10−2 + 4.08972 · 10−5Ψ+−5.79765 · 10−9Ψ2

) · (exp [2Γ− 2]− 1) (12.75)

The repulsive term is defined by the following equation, using seven pre-fit coefficients.

218

κr · Pc

μc= 1.19902 · 10−2 +−0.357875 (Γ− 1)+(

0.637572 +−6.02128 · 10−5Ψ) · (exp [Γ− 1]− 1)+(−7.9024 · 10−2 + 3.72408 · 10−5Ψ+−5.561 · 10−9Ψ2

) · (exp [2Γ− 2]− 1) (12.76)

The squared repulsive term is defined by the following equation, using two pre-fit coefficients.

κrr · (Pc)2

μc= 8.55115 · 10−4 + 1.37290 · 10−8 ·Ψ · (exp [2Γ]− 1) · (Γ− 1)2 (12.77)

12.7.3 Mixing Rules

The following mixing rules apply to the f -theory model.

μ0,mx = exp

[∑i

Xi ln[μ0,i]

](12.78)

Define X� by weighting the mole fractions with the molecular weight raised to a power, here

−0.30.

X�i =

(Xi(MWi)

−0.3)/

⎛⎝∑

j

Xj(MWj)−0.3

⎞⎠ (12.79)

For κa, κr, and κrr, the mixture properties are defined as:

κmx =∑i

κiX�i (12.80)

219

CHAPTER 13

FORMULATION FOR PROPERTIES OF WATER CONTAINING CO2

There are several ways CO2 differs from the hydrocarbon components of oil and natural gas.

CO2 is much more soluble in water than hydrocarbon components, so for simulation of CO2 injection

it is necessary to include this solubility effect. There are some adjustments to the Peng-Robinson

equation of state which make the EOS more accurate in the presence of CO2. Asphaltene deposition

may be significant when the CO2 composition of the oil phase is between certain thresholds. When

mixed with certain oils at temperatures below 150◦F, CO2 can cause the formation of two liquid

hydrocarbon phases, plus a gas phase, plus an aqueous phase. The critical point for CO2 is within

the normal operating conditions; as a result CO2 injection is normally as a supercritical fluid that

has some properties of a liquid and some properties of a gas.

13.1 CO2 Solubility in Water

Figure 13.1 shows the solubility of methane in water. Figure 13.2 shows the solubility of CO2

in water. Note that the solubility of CO2 in water is about ten times the solubility of methane in

water. For a CO2 flood, it is necessary to consider the CO2 solubility in water, but is not necessary

to consider the solubility of methane. There are additional properties defined in Klins (1984) which

define the variation in the water viscosity based on dissolved CO2, water compressibility, water

formation volume factor, water density, and adjustments to solution gas-water ratio with salinity.

13.2 Adjustments to Equation of State

There are some improvements to the Peng-Robinson equation of state discussed in the literature.

Two of these references with extra details include Ahmed (1989) and Ahmed (2007b). There are

also some versions of Peng-Robinson that handle the water phase (Whitson and Brule, 2000).

13.3 Other Special Properties of CO2

CO2 has some unusual properties. Lake (1989) discusses some of these effects. Rogers and

Grigg (2001) provides a literature survey of the variation in injection of CO2. Because the criti-

220

Figure 13.1: Solubility of methane in water (Klins, 1984).

Figure 13.2: Solubility of CO2 in water (Klins, 1984).

221

cal temperature for CO2 is 87.91◦F, the z-factor for CO2 dips very steeply at low temperatures,

Figure 13.3.

Figure 7-4 Compressibility chart for carbon dioxide (CO2) (from Gibbs, 1971)

Figure 13.3: Change in z-factor as a function of pressure for CO2 (Lake, 1989).

13.4 Properties of Water Containing CO2, Overview

There are several approaches to calculate the solubility of non-water components in the aqueous

phase. The approaches described here include Henry’s Law correlations and adjustments to the

Peng-Robinson or other equation of state for systems containing H2O. Solubility may be based on

three different approaches: a pure H2O aqueous phase; only CO2 is soluble in the aqueous phase;

and multiple components are soluble in the aqueous phase. For this work, only CO2 is soluble in

the aqueous phase. If H2S were present, it would also need to be soluble in the aqueous phase.

There are also two options for the vapor phase: H2O may be present or absent in the vapor phase.

For this work, H2O is assumed to be absent from the vapor phase.

The equation of state based models are more general than the Henry’s Law correlations, but

they are also more time consuming and harder to validate for the case where only CO2 is soluble

in the aqueous phase and H2O is not present in the oleic or vapor phases. In the case where the

water content in the vapor phase is neglected and only CO2 and H2O is present in the aqueous

phase, Henry’s Law correlations seem to yield sufficiently accurate predictions.

222

13.5 Commercial Simulators

Schlumberger (2007b) describes several methods for calculating the properties of CO2 in the

aqueous phase. The CO2SOL option uses the Henry’s law based model of Chang et al. (1998). It

is described by the Eclipse manual as the most applicable method for enhanced oil recovery. This

method is also used by VIP (Landmark, 2000). The CO2STORE option is designed for two phases,

a CO2-rich phase and a H2O-rich phase. It uses the equation of state procedure by Spycher and

Pruess (2005) to calculate the mole fraction in the aqueous phase. It uses the method by Kell and

Whalley (1975) to calculate the pure water density and then the method of Ezrokhi described in

Zaytsev and Aseyev (1992) to adjust for the salt content. The viscosity is calculated using Vesovic,

Wakeham, Olchowy, Sengers, Watson, and Millat (1990) and Fenghour, Wakeham, and Vesovic

(1998). The GASWAT option is most applicable to CO2 storage in an aquifer or a depleted gas

reservoir. It accounts for the presence of H2O in the gas phase. It uses the Søreide and Whitson

(1992) modifications to the Peng-Robinson EOS.

13.6 Properties of Water Containing CO2, CMG GEM

In CMG GEM, CMG (2010), the aqueous viscosity may be specified as a simple function

of pressure or calculated by Kestin, Khalifa, and Correia (1981). The mole fractions of CO2

are calculated from Henry’s Law, using Li and Nghiem (1986) or Harvey (1996). The Harvey

(1996) calculations also require several additional correlations: the saturation pressure for water

is calculated using Saul and Wagner (1987); the partial molar volume for CO2 is calculated using

Garcıa (2001); the salinity adjustment is calculated using Bakker (2003). The fugacity of saturated

water is calculated using Canjar and Manning (1967). The molar volume of water is calculated

using Rowe and Chou (1970).

13.7 Units of concentration

The following concentration units are used, with appropriate conversions:

• mi represents the molality, moli/masssolvent. For this chapter, the solvent is H2O. Note that

the denominator is the mass of the solvent, not the total mass of the solution. Typically

measured in m = mol/kg or mol/lbm. For the following conversions, mi is in units of mol/kg.

223

• ci represents the molarity, moli/volt. Typically measured in mol/m3, mol/L, mol/cm3, or

mol/ft3. For the following conversions, ci is in units of mol/L.

• Xi, or WCO2 represents the mole fraction, moli/molt. For the following conversions, Wi is in

units of mol/mol.

• wi represents the mass fraction, massi/masst. Typically measured in m3/m3, ft3/ft3, or

ppmw. For the following conversions, wi is in units of kg/kg.

• Rsw represents the gas solubility in scf/stb.

For the following conversions, MWi is in units of g/mol, and ρ is in units of kg/m3 = g/L. Most

of the correlations use the equivalent concentration of NaCl rather than the specific composition of

the salts. Based on Duan and Sun (2003), this is a good assumption for most cations and anions

except for SO−−4 . The molecular weight of H2O is 18.0153, the molecular weight of CO2 is 44.0096,

and the molecular weight of NaCl is 58.4430. The molality of an aqueous phase containing only

H2O is 55.5084 mol/kg.

To convert from weight fraction wi into mole fraction Xi

αi =wi

MWi

∏j

MWj Xi =αi∑j αj

(13.1)

To convert from weight fraction wi into molal units mi

mi = 1000

(wi

MWi

)/wsolvent (13.2)

To convert from weight fraction wi into molar units ci

ci =wiρtMWi

(13.3)

To convert from mole fraction Xi into weight fraction wi

wi =XiMWi∑j XjMWj

(13.4)

To convert from mole fraction Xi into molal units mi, first convert Xi into wi, then convert wi

into mi.

224

wi =XiMWi∑j XjMWj

mi = 1000

(wi

MWi

)/wsolvent (13.5)

To convert from mole fraction Xi into molar units ci

ci =Xi∑

j XjMWjρt (13.6)

To convert from molal units mi into weight fraction wi

wi =miMWi∑j mjMWj

(13.7)

To convert from molal units mi into mole fraction Xi

Xi =mi∑j mj

(13.8)

To convert from molal units mi into molar units ci

ci =

(mi∑

j mjMWj

)ρt (13.9)

To convert from molar units ci into weight fraction wi

wi =ciMWi

ρt(13.10)

To convert from molar units ci into weight fraction xi

Xi =ci∑j cj

(13.11)

To convert from molar units ci into molal units mi, first convert from molar units ci into mole

fraction Xi, then convert from mole fraction Xi into weight fraction wi, then convert from weight

fraction wi into molal units mi.

Xi =ci∑j cj

wi =XiMWi∑j XjMWj

mi = 1000

(wi

MWi

)/wsolvent (13.12)

225

To convert from weight fraction wi in kg/kg into the gas solubility in Rsw,i in scf/stb uses the

following equation. The conversion constant 2130.3 is based on 379.423 scf/mol from Klins (1984).

Not all of the correlations are clear about which “standard conditions” are used for temperature

and pressure.

Rsw,i =2130.3ρtwi

MWi(13.13)

To convert from gas solubility in Rsw,i in scf/stb into weight fraction wi in kg/kg

wi =MWiRsw,i

2130.3ρt(13.14)

13.8 Selection Process

This section describes how the algorithms for calculating brine density, brine viscosity, and CO2

solubility in water were selected.

13.8.1 Rowe, Brine Density

Rowe and Chou (1970) is used by all of the commercial simulators to calculate brine density as

a function of H2O and NaCl content. It is also the preferred method by many other authors which

need a method for calculating brine density, including Kestin, Khalifa, Abe, Grimes, Sookiazian,

and Wakeham (1978), Kestin et al. (1981), Kestin and Shankland (1984), Chang et al. (1998), Enick

and Klara (1992), and Li and Nghiem (1986). Rowe and Chou (1970) reports that their correlation

is within 3% of the experimental data for both density and the derivatives of density with respect

to pressure and temperature for a range of temperatures from 0◦C to 150◦C, 0 to 25% weight

percent NaCl, and pressures from 1 to 350 kg/cm2 = 4978 psia. To check the implementation, the

correlations were validated using all of the figures and tables in Rowe and Chou (1970). It compares

favorably to figures 3.42 and 3.43A in Klins (1984). The values of the density and compressibility

were also compared favorably with several web sources, including:

• http://en.wikipedia.org/wiki/Properties_of_water

• http://www.engineeringtoolbox.com/fluid-density-temperature-pressure-d_309.html

• http://www.searchanddiscovery.com/documents/2006/06015powley/images/a03.htm

It also compares favorably to the data in ASME (1935).

226

13.8.2 Garcıa, CO2 Brine Density and Partial Molar Volume

Garcıa (2001) provides a way to calculate the density of a brine containing NaCl, H2O, and

CO2. It uses Rowe and Chou (1970) to calculate the NaCl plus H2O brine density. This method

is referred to in many more recent articles as a way to calculate the partial molar volume of CO2.

Garcıa (2001) reports that their correlation is valid for temperatures between 0◦C and 300◦C,

and from 0 to 0.05 mole fraction CO2. The authors compare their correlations to four previous

correlations. To check the implementation, the correlations were validated using all of the figures

and tables in Garcıa (2001).

13.8.3 Kestin, Brine Viscosity

Kestin et al. (1978) describes a correlation for the viscosity of NaCl brine solutions for 20–150◦C

and pressures of 0.1–35 MPa, and 0–5.4 molal. Kestin et al. (1978) report that their correlation has

a maximum deviation of 1.4% with a standard deviation of 0.5%. Sayegh and Najman (1987) shows

that CO2 has a negligible impact on the viscosity of the H2O+NaCl system. The Kestin correlations

are used by Eclipse and VIP. To check the implementation, the correlations were validated using all

of the figures and tables in Kestin et al. (1978), Kestin et al. (1981), Kestin and Shankland (1984),

and figures 3.44 and 3.45 from Klins (1984).

13.8.4 Duan, Henry’s Law

There are a lot of different methods for solubility calculations. The methods of the commercial

simulators Eclipse (Schlumberger, 2007b), VIP (Landmark, 2000), and CMG CMG (2010) were

reviewed, plus all the articles they cite related to CO2 solubility. An independent literature search

was also conducted. Methods were selected for further study which have CO2 solubility as a function

of temperature, pressure, and NaCl salinity.

Several methods are based on adjustments of an equation of state. These include Søreide and

Whitson (1992), Delshad et al. (2011), Yan and Stenby (2009), Melham and Little (1989), Spycher

and Pruess (2005), and Li and Nghiem (1986). Because it is more difficult to validate these methods

and because the presence of H2O in the vapor phase is neglected, these methods were described

but not implemented.

227

Four methods based on Henry’s Law calculations were also selected for evaluation. These

includes the methods of Duan and Sun (2003); Chang et al. (1998); Enick and Klara (1990); and

CMG; including the methods of Harvey (1996), Saul and Wagner (1987), Garcıa (2001), and Bakker

(2003). All four of these methods were implemented, and Duan, Møller, and Weare (1992) was

used to calculate fugacity (accurate to within 5%) for comparison purposes. Each of the methods

were validated using the figures and tables presented in the articles that describe the correlations.

The method of Duan and Sun (2003) was selected based on the best fit with data from a variety of

sources. First, Duan and Sun (2003) was validated using plots from Duan and Sun (2003). Duan

and Sun (2003), Spycher, Pruess, and Ennis-King (2003), and Spycher and Pruess (2005) have

detailed figures including solubility data from a large number of sources. The correlations from

Duan and Sun (2003) were compared to figures and tables from those sources as well as Harvey

(1996), Klins (1984), Obeida, Kalam, Al-Sahn, Gibson, Masaleeh, and Zhang (2009), Rumpf,

Nicolaisen, Ocal, and Maurer (1994), Li and Nghiem (1986), Søreide and Whitson (1992), Yan and

Stenby (2009), and Zeebe and Wolf-Gladrow (2001). Duan and Sun (2003) is valid for 273 K to

533 K, 0 to 2000 bar = 29007 psia, and 0 to 4 mol/kg = 0.189 kg/kg NaCl, and with a correlation

and experimental accuracy of 7% CO2 solubility.

13.9 Correlations for this Project

The following is a short summary of the procedure for calculating the viscosity, solubility, and

density of the aqueous phase containing CO2, H2O, and NaCl. These correlations involve conversion

between different units for pressure, temperature, and concentration. The mole fraction of NaCl,

W nNaCl, in a NaCl-H2O system is used as the concentration input. W n

NaCl is calculated explicitly at

each time step n after all other properties have been calculated at n. The weight fractions for a

NaCl-H2O system are defined with a � superscript:

w�,nNaCl =

W �,nNaClMWNaCl

W �,nNaClMWNaCl + (1−W �,n

NaCl)MWH2O(13.15)

w�,nH2O

=(1−W �,n

NaCl)MWH2O

W �,nNaClMWNaCl + (1−W �,n

NaCl)MWH2O(13.16)

The molalities for a NaCl-H2O system are defined with a � superscript:

228

m�,nNaCl =

1000

w�,nH2O

w�,nNaCl

MWNaCl(13.17)

m�,nH2O

=1000

w�,nH2O

w�,nH2O

MWH2O(13.18)

The viscosity is calculated using Kestin et al. (1981) and Kestin et al. (1978). For the IMPES

formualtion, it is only computed at the time step level n and no derivatives are required.

μnw = μn

w[Pn, T#,m�,n

NaCl[WnNaCl]] (13.19)

The solubility WCO2 in a CO2+NaCl+H2O system is calculated using the following procedure

from Duan and Sun (2003).

WCO2 = WCO2 [P, T#,m�

NaCl[W�,nNaCl],m

�H2O[W

�,nNaCl], fCO2 ] =

mCO2 [P, T#, fCO2 ,m

�NaCl]

mCO2 [P, T#, fCO2 ,m

�NaCl] +m�,n

NaCl +m�,nH2O

(13.20)

The mnCO2

is calculated as follows:

mnCO2

= mCO2 [P, T, fCO2 ,m�NaCl[WNaCl]] =

fCO2

HCO2

=Y nCO2

PnΦnCO2

[Pn, T#, Y nm′ ]

HnCO2

[Pn, T#,m�,nNaCl]

(13.21)

The molar density of the aqueous phase ξw in a CO2 +NaCl +H2O system is calculated based

on correlations by Rowe and Chou (1970) and Garcıa (2001), using the WNaCl and WCO2 . ρbrine

represents the density of a NaCl + H2O system.

ξnw = ξw[P, T,WCO2 , w�NaCl[WNaCl],MWw,t[WCO2 ,WNaCl]] =

ρnbrine[Pn, T#, w�,n

NaCl]/MWnw,t

1 +Wn

CO2MWn

w,t

(−MWCO2 + ρnbrine[Pn, T#, w�,n

NaCl]vCO2 [T#]10−3

) (13.22)

The aqueous density ξn+1w is evaluated as follows:

ξn+1w =

ρn+1brine[P

n+1, T#, w�,nNaCl]/MWn+1

w,t

1 +Wn+1

CO2

MWn+1w,t

(−MWCO2 + ρn+1brine[P

n+1, T#, w�,nNaCl]vCO2 [T

#]10−3) (13.23)

229

The total molecular weight is defined in the following way because the experiments were first

conducted to measure the properties of the NaCl+H2O system, with an adjustment added for the

NaCl + H2O+CO2 system.

MWw,t = MWw,t[WCO2 ,WNaCl] = WCO2MWCO2+(1−WCO2)(W�NaClMWNaCl+(1−W �

NaCl)MWH2O)

(13.24)

13.10 Computational Forms of WCO2

Recall that Duan and Sun (2003) uses the following version of Henry’s Law for mCO2 :

mCO2 =fCO2

HCO2

(13.25)

To calculate the derivative of WCO2 with respect to pressure,∂WCO2

∂P , use the conversion from

molality to mole fraction.

Wi =mi∑j mj

(13.26)

After solving∂WCO2

∂P and∂mH2O

∂P simultaneously, this yields

∂WCO2

∂P=

1∑j mj

∂mCO2

∂P(13.27)

The derivative of WCO2 with respect to Ym′ ,∂WCO2∂Ym′ is

∂WCO2

∂Ym′=

1∑j mj

∂mCO2

∂Ym′(13.28)

The derivative of mCO2 with respect to pressure,∂mCO2

∂P is

∂mCO2

∂P=

1

HCO2

(∂fCO2

∂P−mCO2

∂HCO2

∂P

)(13.29)

The derivative of mCO2 with respect to Ym′ is

∂mCO2

∂Ym′=

1

HCO2

∂fCO2

∂Ym′(13.30)

230

13.10.1 Option 0: WCO2 = 0

This option is the simplest to implement and compute. It completely neglects the solubility of

CO2 in the aqueous phase.

WCO2 = 0∂WCO2

∂P= 0

∂WCO2

∂Ym′= 0 (13.31)

This option has a cumulative mass balance error less than 10−4 for all components but does

not accurately represent the physics of CO2 solubility.

13.10.2 Option C: Constant WCO2

This option is the simplest to implement and compute. The CO2 solubility is not a function of

pressure or composition, but is non-zero.

WCO2 = constant (13.32)

∂WCO2

∂P= 0 (13.33)

∂WCO2

∂Xm′= 0 (13.34)

∂WCO2

∂Ym′= 0 (13.35)

For the flash calculation, use

β�+1w =

VR

Δtφ�+1S�+1

w ξ�+1w [P �+1,W#

CO2] (13.36)

This option has a low cumulative mass balance error and the appropriate behavior for the water

saturation, but does not accurately represent the physics of CO2 solubility.

13.10.3 Option ZW0: Compute ξw using WCO2 = 0

This computation uses Rowe and Chou (1970) rather than Rowe and Chou (1970) plus Garcıa

(2001). The following pressure derivatives are the following; all other derivatives are zero.

∂ρbrine∂P

= ρbrineCw (13.37)

∂ξw∂P

=1

MWbrine

∂ρbrine∂P

= ξwCw (13.38)

231

13.10.4 Option ZW1: Compute ξw using WCO2

ρw,t =ρbrine

1 +WCO2

MWw,t· (−MWCO2 + ρbrine · vCO2 · 10−3

) (13.39)

MWw,t = MWCO2WCO2 +MWH2O(1−WCO2) (13.40)

The molar density ξw is defined by

ξw =ρw,t

MWw,t(13.41)

The derivative of ξw with respect to pressure ∂ξw∂P is

∂ξw∂P

=1

MWw,t

(∂ρw,t

∂P− ξw

∂MWw,t

∂P

)(13.42)

The derivative of ξw with respect to mole fraction ∂ξw∂Ym′ is

∂ξw∂Ym′

=1

MWw,t

(∂ρw,t

∂Ym′− ξw

∂MWw,t

∂Ym′

)(13.43)

The derivative of ξw with respect to mole fraction ∂ξw∂Xm′ is

∂ξw∂Xm′

=1

MWw,t

(∂ρw,t

∂Xm′− ξw

∂MWw,t

∂Xm′

)(13.44)

13.10.5 Option KP1: Use a simplified model for WCO2 using YCO2 [Pb]

If there is gas in the system, use YCO2 . If there is no gas in the system, use YCO2 [Pb]. Use the

brine density rather than the total aqueous density. Define WCO2 as follows:

Rsw = 200379 (1− exp[−0.001386P ]) (13.45)

α = Rsw + 5.6146ξbrine (13.46)

WCO2 = YCO2

Rsw

Rsw + 5.6146ξbrine(13.47)

The derivatives are calculated as follows:

232

∂Rsw

∂P= 200

379 · 0.001386 · exp[−0.001386P ] (13.48)

∂WCO2

∂P=

YCO2α

(∂Rsw∂P − Rsw

α∂Rsw∂P − 5.6146Rsw

α∂ξbrine∂P

)(13.49)

∂WCO2

∂Ym′= − Rsw

Rsw + 5.6146ξbrine(13.50)

(13.51)


saturation. Option KP1 is computationally stable with both option ZW0 and ZW1. This approach

is good if the salinity and temperature are constant, but the correlation needs to be updated for a

specific salinity and temperature.

13.10.6 Option KP2: Use a simplified model for WCO2 using YCO2 below the bubblepoint and YCO2 = 0 above the bubble point

If there is gas in the system, use YCO2 . If there is no gas in the system, use YCO2 = 0 =⇒WCO2 = 0. Use the brine density rather than the total aqueous density. Define WCO2 as follows:

Rsw = 200379 (1− exp[−0.001386P ]) (13.52)

α = Rsw + 5.6146ξbrine (13.53)

WCO2 = YCO2

Rsw

Rsw + 5.6146ξbrine(13.54)


∂Rsw

∂P= 200

379 · 0.001386 · exp[−0.001386P ] (13.55)

∂WCO2

∂P=

YCO2α

(∂Rsw∂P − Rsw


α∂ξbrine∂P

)(13.56)

∂WCO2

∂Ym′= − Rsw

Rsw + 5.6146ξbrine(13.57)

(13.58)



is good if the salinity and temperature are constant, but the correlation needs to be updated for a

specific salinity and temperature.

233

13.10.7 Option KP3: Use a simplified model for WCO2 using YCO2 [Pb]

If there is gas in the system, use YCO2 . If there is no gas in the system, use YCO2 [Pb]. Use the

brine density rather than the total aqueous density. This model of WCO2 is based on a fit of Duan

and Sun (2003) using T = 200◦F and ws = 0.225, for pressures from P = 100 psia to P = 5000 psia

in steps of 100 psia, and for YCO2 ranging from 0 to 1 in steps of 0.05.

Rsw = 0.195028 (1− exp[−0.000571152P ]) (13.59)

α = Rsw + 5.6146ξbrine (13.60)

WCO2 = YCO2

Rsw

Rsw + 5.6146ξbrine(13.61)


∂Rsw

∂P= 0.195028 · 0.000571152 · exp[−0.000571152P ] (13.62)

∂WCO2

∂P=

YCO2α

(∂Rsw∂P − Rsw


α∂ξbrine∂P

)(13.63)

∂WCO2

∂Ym′= − Rsw

Rsw + 5.6146ξbrine(13.64)

(13.65)



is better than KP1 because it is based on a specific temperature and salinity relevant for offshore

Abu Dhabi.

13.10.8 Option 1: W n+1CO2

fully implicit

The mn+1CO2

is calculated as follows, using a implicit approach:

mn+1CO2

=fn+1CO2

Hn+1CO2

=Y n+1CO2

Pn+1Φn+1CO2

[Pn+1, T#, Y n+1m′ ]

Hn+1CO2

[Pn+1, T#,m�,nNaCl]

(13.66)

∂Wn+1CO2

∂P , ∂ξw∂P , are computed using

∂mn+1CO2

∂P .∂Wn+1

CO2∂Ym′ , ∂ξw

∂Ym′ , are computed using∂mn+1

CO2∂Ym′ .

234

∂mn+1CO2

∂P= 1

Hn+1CO2

(∂fn+1

CO2∂P −mn+1

CO2

∂Hn+1CO2∂P

)(13.67)

∂mn+1CO2

∂Ym′= 1

Hn+1CO2

∂fn+1CO2

∂Ym′ (13.68)

This option causes a steady decrease in the water saturation, even when no water is injected

or produced. This inconsistency arises because the CO2 solubility experiments used only a pure

supercritical or vapor CO2 phase and a water phase to measure the water solubility.

13.10.9 Option 2: W n+1CO2

implicit pressure, explicit fugacity coefficient

The mn+1CO2

is calculated as follows, using an implicit pressure explicit fugacity coefficient ap-

proach:

mn+1CO2

=Y nCO2

Pn+1ΦnCO2

[Pn, T#, Y nm′ ]

Hn+1CO2


(13.69)

Using this approach,∂Wn+1

CO2∂Ym′ = 0 and ∂ξn+1

w∂Ym′ = 0, and

∂Wn+1CO2

∂P and ∂ξw∂P are computed using

∂mn+1CO2

∂P .

∂mn+1CO2

∂P=

same︷︸︸︷1

Hn+1CO2

⎛⎜⎜⎜⎝

different︷︸︸︷fnCO2

Pn−

same︷︸︸︷mn+1

CO2

∂Hn+1CO2

∂P

⎞⎟⎟⎟⎠ (13.70)

∂mn+1CO2

∂Ym′= 0 (13.71)




13.10.10 Option 3: W n+1CO2

implicit pressure, explicit fugacity

The mn+1CO2

is calculated as follows, using an implicit pressure explicit fugacity approach:

mn+1CO2

=fnCO2

Hn+1CO2

=Y nCO2

PnΦnCO2

[Pn, T#, Y nm′ ]

Hn+1CO2


(13.72)



w∂Ym′ = 0, and

∂Wn+1CO2


∂mn+1CO2

∂P .

235

∂mn+1CO2

∂P=

same︷︸︸︷1

Hn+1CO2

⎛⎜⎜⎜⎝

different︷︸︸︷0 −


CO2

∂Hn+1CO2

∂P

⎞⎟⎟⎟⎠ (13.73)

∂mn+1CO2

∂Ym′= 0 (13.74)




13.10.11 Option 4: W n+1CO2

implicit pressure, fugacity at �

The mn+1CO2


mn+1CO2

=f�CO2

Hn+1CO2

=Y �CO2

P �Φ�CO2

[P �, T#, Y �m′ ]

Hn+1CO2


(13.75)



w∂Ym′ = 0, and

∂Wn+1CO2


∂mn+1CO2

∂P .

∂mn+1CO2

∂P=

same︷︸︸︷1

Hn+1CO2

⎛⎜⎜⎜⎝



CO2

∂Hn+1CO2

∂P

⎞⎟⎟⎟⎠ (13.76)

∂mn+1CO2

∂Ym′= 0 (13.77)





implicit pressure, explicit fugacity coefficient

The mn+1Z,CO2

is calculated as follows, using an implicit pressure explicit fugacity coefficient

approach:

mn+1Z,CO2

=ZnCO2

Pn+1ΦnZ,CO2

[Pn, T#, Znm′ ]

Hn+1CO2


(13.78)

236


Z,CO2∂Ym′ = 0 and ∂ξn+1

w∂Ym′ = 0, and

∂Wn+1Z,CO2∂P and ∂ξw

∂P are computed using∂mn+1

Z,CO2∂P .

∂mn+1Z,CO2

∂P=

same︷︸︸︷1

Hn+1CO2

⎛⎜⎜⎜⎝

different︷︸︸︷fnZ,CO2

Pn−


Z,CO2

∂Hn+1CO2

∂P

⎞⎟⎟⎟⎠ (13.79)

∂mn+1Z,CO2

∂Ym′= 0 (13.80)

This option has a mass balance error of 10−2 in the CO2 component.


implicit pressure, explicit fugacity

The mn+1Z,CO2


mn+1Z,CO2

=fnZ,CO2

Hn+1CO2

=ZnCO2

PnΦnZ,CO2

[Pn, T#, Znm′ ]

Hn+1CO2


(13.81)



w∂Ym′ = 0, and



Z,CO2∂P .

∂mn+1Z,CO2

∂P=

same︷︸︸︷1

Hn+1CO2

⎛⎜⎜⎜⎝



Z,CO2

∂Hn+1CO2

∂P

⎞⎟⎟⎟⎠ (13.82)

∂mn+1Z,CO2

∂Ym′= 0 (13.83)



implicit pressure, fugacity at �

The mn+1CO2


mn+1Z,CO2

=f�Z,CO2

Hn+1CO2

=Z�CO2

P �Φ�CO2

[P �, T#, Z�m′ ]

Hn+1CO2


(13.84)



w∂Ym′ = 0, and



Z,CO2∂P .

237

∂mn+1Z,CO2

∂P=

same︷︸︸︷1

Hn+1CO2

⎛⎜⎜⎜⎝



Z,CO2

∂Hn+1CO2

∂P

⎞⎟⎟⎟⎠ (13.85)

∂mn+1Z,CO2

∂Ym′= 0 (13.86)


13.10.15 Option 1XY: W n+1CO2

partially implicit, function of both Xm′ and Ym′

Here, we assume that WCO2 is a linear combination of the WCO2 [P, T,Xm′ ] and WCO2 [P, T, Ym′ ].

This means the derivatives ofWCO2 are also linear combinations of the derivatives ofWCO2 [P, T,Xm′ ]

and WCO2 [P, T, Ym′ ].

WCO2 = V ·WCO2,v︷︸︸︷

WCO2 [P, T, Ym′ ] +L ·WCO2,l︷︸︸︷

WCO2 [P, T,Xm′ ] (13.87)

Expand this in the following way, ignoring the derivatives of V and L.

W n+1CO2

= V �

(W �

CO2[P, T, Ym′ ] +

∂W �CO2

[P, T, Ym′ ]

∂PδP +

∂W �CO2

[P, T, Ym′ ]

∂Ym′δYm′

)+

L�

(W �

CO2[P, T,Xm′ ] +

∂W �CO2

[P, T,Xm′ ]

∂PδP +

∂W �CO2

[P, T,Xm′ ]

∂Xm′δXm′

)(13.88)

The W �CO2

is calculated as

W �CO2

= V � ·W �CO2,v + L� ·W �

CO2,l (13.89)

W �CO2,v =

m�CO2,v∑j m

�j,v

W �CO2,l =

m�CO2,l∑j m

�j,l

(13.90)

m�CO2,v


m�CO2,v[P, T, Ym′ ] =

f�CO2,v

H�CO2

=Y �CO2

P �Φ�CO2,v

[P �, T#, Y �m′ ]

H�CO2

[P �, T#,m�,nNaCl]

(13.91)

238

m�CO2,l


m�CO2,l[P, T,Xm′ ] =

f�CO2,l

H�CO2

=X�

CO2P �Φ�

CO2,l[P �, T#,X�

m′ ]

H�CO2

[P �, T#,m�,nNaCl]

(13.92)

The∂W �

CO2∂P is calculated as:

∂W �CO2

∂P= V � · ∂W

�CO2,v

∂P+ L� · ∂W

�CO2,l

∂P(13.93)

The∂W �

CO2∂P are calculated using

∂m�CO2

∂P :

∂WCO2

∂P=

1∑j m

�j

∂m�CO2

∂P(13.94)

The∂m�

CO2∂P are calculated as:

∂m�CO2,v

∂P= 1

H�CO2

(∂f�CO2,v

∂P −m�CO2,v

∂H�CO2∂P

)(13.95)

∂m�CO2,l

∂P= 1

H�CO2

(∂f�CO2,l

∂P −m�CO2,l

∂H�CO2∂P

)(13.96)

The∂W �

CO2∂Ym′ is calculated as:

∂W �CO2

∂Ym′= V � · ∂W

�CO2,v

∂Ym′(13.97)

The∂W �

CO2,v

∂Ym′ are calculated using∂m�

CO2,v

∂Ym′ :

∂WCO2,v

∂Ym′=

1∑j m

�j,v

∂m�CO2,v

∂Ym′(13.98)

∂m�CO2,v

∂Ym′ is calculated as:

∂m�CO2,v

∂Ym′=

1

H�CO2

∂f�CO2,v

∂Ym′(13.99)

The∂W �

CO2∂Xm′ is calculated as:

239

∂W �CO2

∂Xm′= L� · ∂W

�CO2,l

∂Xm′(13.100)

The∂W �

CO2,l

∂Xm′ are calculated using∂m�

CO2,l

∂Xm′ :

∂WCO2,l

∂Xm′=

1∑j m

�j,l

∂m�CO2,l

∂Xm′(13.101)

∂m�CO2,l

∂Xm′ is calculated as:

∂m�CO2,l

∂Xm′=

1

H�CO2

∂f�CO2,l

∂Xm′(13.102)

13.10.16 Option Y1: W n+1CO2

fully implicit

When there is no gas in the system, WCO2 = 0. When there is gas in the system, use a fully

implicit calculation of WCO2 [Yn+1m′ , Pn+1]. For the flash calculation, use β�+1

w = VRΔtφ

�+1S�+1w ζ�w.

The mn+1CO2


mn+1CO2

=fn+1CO2

Hn+1CO2

=Y n+1CO2

Pn+1Φn+1CO2

[Pn+1, T#, Y n+1m′ ]

Hn+1CO2


(13.103)

∂Wn+1CO2

∂P , ∂ξw∂P , are computed using

∂mn+1CO2

∂P .∂Wn+1

CO2∂Ym′ , ∂ξw

∂Ym′ , are computed using∂mn+1

CO2∂Ym′ .

∂mn+1CO2

∂P= 1

Hn+1CO2

(∂fn+1

CO2∂P −mn+1

CO2

∂Hn+1CO2∂P

)(13.104)

∂mn+1CO2

∂Ym′= 1

Hn+1CO2

∂fn+1CO2

∂Ym′ (13.105)

This option causes a non-physical change in the water saturation, even when no water is injected

or produced.

13.10.17 Option Y5: W nCO2

explicit

When there is no gas in the system, WCO2 = 0. When there is gas in the system, use an explicit

calculation of WCO2 [Ynm′ , Pn].

240

W �CO2

= W nCO2

= WCO2 [Pn, T#, Y n

m] (13.106)

∂W nCO2

∂P= 0 (13.107)

∂W nCO2

∂Xm′= 0 (13.108)

∂W nCO2

∂Ym′= 0 (13.109)

Because the composition derivatives of WCO2 are zero,

∂ξw∂Xm′

= 0 (13.110)

∂ξw∂Ym′

= 0 (13.111)

Everywhere ξ�w occurs, use

ξ�w = ξw[P�,W n

CO2] (13.112)


β�+1w =

VR

Δtφ�+1S�+1

w ξ�+1w [P �+1,W n

CO2] (13.113)


or produced. There is also a large mass balance error in the CO2.

13.10.18 Option P1: W n+1CO2

[P only] fully implicit

This option defines WCO2 [Pn+1, T#, Y #]. Typically use the constant Y #

CO2= 1.

WCO2 = WCO2 [P, Y#CO2

] (13.114)

∂WCO2

∂P= 1∑

j mj

∂mCO2∂P (13.115)

∂WCO2

∂Xm′= 0 (13.116)

∂WCO2

∂Ym′= 0 (13.117)

The mn+1CO2


241

mn+1CO2

=fn+1CO2

Hn+1CO2

=Y #CO2

Pn+1Φn+1CO2

[Pn+1, T#, Y #m′ ]

Hn+1CO2


(13.118)

The pressure derivative of mCO2 is defined as:

∂mn+1CO2

∂P=

1

Hn+1CO2

(∂fn+1

CO2

∂P−mn+1

CO2

∂Hn+1CO2

∂P

)(13.119)


β�+1w =

VR

Δtφ�+1S�+1

w ξ�+1w [P �+1,W �+1

CO2[P �+1, Y #

CO2]] (13.120)


or produced. There is also a large mass balance error in the CO2.

13.10.19 Option K1: W n+1CO2

[YCO2 only], evaluate Y at �

This option defines WCO2 [YCO2 only]. Typically use the constant K#CO2

= 0.005.

WCO2 = KCO2YCO2 (13.121)

∂WCO2

∂P= 0 (13.122)

∂WCO2

∂Xm′= 0 (13.123)

∂WCO2

∂Ym′= −KCO2 (13.124)

If there is gas in the system, use YCO2 . If there is no gas in the system, use YCO2 [Pb].

For the flash calculation, use the following:

β�+1w =

VR

Δtφ�+1S�+1

w ξ�+1w [P �+1,W �+1

CO2[Y �

CO2]] (13.125)


saturation, but does not represent the pressure dependence of CO2 solubility.

At a temperature of 200◦F a salinity of 0.225 and an average reservoir pressure of 2000 psia,

K = 0.007.

242

13.10.20 Option K2: W n+1CO2

[YCO2 only], evaluate Y at �

This option defines WCO2 [YCO2 only]. Typically use the constant K#CO2

= 0.005.

WCO2 = KCO2YCO2 (13.126)

∂WCO2

∂P= 0 (13.127)

∂WCO2

∂Xm′= 0 (13.128)

∂WCO2

∂Ym′= −KCO2 (13.129)

If there is gas in the system, use YCO2 . If there is no gas in the system, use YCO2 = 0.

For the flash calculation, use the following:

β�+1w =

VR

Δtφ�+1S�+1

w ξ�+1w [P �+1,W �+1

CO2[Y �

CO2]] (13.130)

The water saturation behavior for this model is good. There are significant mass balance errors

introduced when a two phase oil-water system transitions to a three-phase oil-water-gas system.

13.10.21 Using WCO2 as a Transfer Term

(13.131) represents the m1 pore system for the gas and oil phases:

0.006328 VR ∇ ·(Xn

mm1ξnom1

knrom1

μnom1

k#m1(∇P �+1

om1− γn

om1∇D#)

)+

0.006328 VR ∇ ·(Y n

mm1ξngm1

knrgm1

μngm1

k#m1(∇P �+1

om1+∇Pn

cgom1− γn

gm1∇D#)

)+(

Xnmm1

ξnom1q�+1om1

+ Y nmm1

ξngm1q�+1gm1

)− τ �+1mm1/m2

− τmm1,hc/w =

VR

Δt

(φ�+1m1

X�+1mm1

S�+1om1

ξ�+1om1

+ φ�+1m1

Y �+1mm1

S�+1gm1

ξ�+1gm1

)−VR

Δt

(φnm1Xn

mm1Snom1

ξnom1+ φn

m1Y nmm1

Sngm1

ξngm1

)(13.131)

The water equation uses a water component that contains NaCl, H2O, and CO2.

0.006328 VR ∇ ·(ξnwm1

knrwm1

μnwm1

k#m1(∇P �+1

om1−∇Pn

cowm1− γn

wm1∇D#)

)+

(ξnwm1

q�+1wm1

)− τ �+1mm1/m2

+ τmm1,hc/w =VR

Δt

(φ�+1m1

S�+1wm1

ξ�+1wm1

)− VR

Δt

(φnm1Snwm1

ξnwm1

)(13.132)

243

The WCO2 equation

0.006328 VR ∇ ·(Wn

mm1ξnwm1

knrwm1

μnwm1

k#m1(∇P �+1

om1−∇Pn

cowm1− γn

wm1∇D#)

)+(

Wnmm1

ξnwm1q�+1wm1

)− τ �+1W,mm1/m2

+ τW,mm1,hc/w =

VR

Δt

(φ�+1m1

W �+1mm1

S�+1wm1

ξ�+1wm1

)− VR

Δt

(φnm1Wn

mm1Snwm1

ξnwm1

)(13.133)

(13.134) represents the m2 pore system for the hydrocarbon components

τ �+1mm1/m2

− τmm2,hc/w =VR

Δt

(φ�+1m2

X�+1mm2

S�+1om2

ξ�+1om2

+ φ�+1m2

Y �+1mm2

S�+1gm2

ξ�+1gm2

)−

VR

Δt

(φnm2Xn

mm2Snom2

ξnom2+ φn

m2Y nmm2

Sngm2

ξngm2

)(13.134)

(13.135) represents the m2 pore system for the aqueous component.

τ �+1mm1/m2

+ τmm2,hc/w =VR

Δt

(φ�+1m2

S�+1wm2

ξ�+1wm2

)− VR

Δt

(φnm2Snwm2

ξnwm2

)(13.135)

(13.136) represents the m2 pore system for WCO2 .

τ �+1W,mm1/m2

+ τW,mm2,hc/w =VR

Δt

(φ�+1m2

W �+1mm2

S�+1wm2

ξ�+1wm2

)− VR

Δt

(φnm2Wn

mm2Snwm2

ξnwm2

)(13.136)

(13.137) represents the m1/m2 transfer for the hydrocarbon components.

τ �+1mm1/m2

= 0.006328 VR σ#m1/m2

k#m1/m2

(P �+1om1− P �+1

om2

)×(Xup,n

mm1/m2ξup,nom1/m2

kup,nrom1/m2

μup,nom1/m2

+Y up,nmm1/m2


μup,ngm1/m2

)(13.137)

(13.138) represents the m1/m2 transfer for the aqueous component.

τ �+1mm1/m2

= 0.006328 VR σ#m1/m2

k#m1/m2

(P �+1om1− P �+1

om2

)×(ξup,nwm1/m2

kup,nrwm1/m2

μup,nwm1/m2

)(13.138)

244

(13.139) represents the m1/m2 transfer for the aqueous component.

τ �+1W,mm1/m2

= 0.006328 VR σ#m1/m2

k#m1/m2

(P �+1om1− P �+1

om2

)×(W up,n

mm1/m2ξup,nwm1/m2

kup,nrwm1/m2

μup,nwm1/m2

)(13.139)

13.10.22 Rowe, Brine Density, Eclipse + VIP+CMG, H2O+NaCl, ρw + Cw

Rowe and Chou (1970) defines a correlation for water density using specific volume v[cm3/g],

P [kg/cm2], T [◦K], ws[wt fraction], and Cw[cm2/kg] . This correlation is used by Eclipse, VIP, and

CMG. It is based on experimental data from T = 0◦C to T = 180◦C, pressures up to 400 kg/cm2,

and salt concentrations up to 25 wt%. The ws used here is based on a system with only H2O and

NaCl. Since this is also the basis of the measurement of salinity, use wH2O = 1− wNaCl.

v[P, T,ws] = A[T ]−P ·B[T ]−P 2·C[T ]+ws·D[T ]+w2s ·E[T ]−ws·P ·F [T ]−w2

s ·P ·G[T ]−1

2ws·P 2·H[T ]

(13.140)

A[T ] =(5.916365 − 0.01035794T + 0.9270048 · 10−5T 2 − 1127.522T−1 + 100674.1T−2

)(13.141)

B[T ] =

(0.5204914 · 10−2 − 0.10482101 · 10−4T+

0.8328532 · 10−8T 2 − 1.1702939T−1 + 102.2783T−2

)(13.142)

C[T ] =(0.118547 · 10−7 − 0.6599143 · 10−10T

)(13.143)

D[T ] =(−2.5166 + 0.0111766T − 0.170552 · 10−4T 2

)(13.144)

E[T ] =(2.84851 − 0.0154305T + 0.223982 · 10−4T 2

)(13.145)

F [T ] =(−0.0014814 + 0.82969 · 10−5T − 0.12469 · 10−7T 2

)(13.146)

245

G[T ] =(0.0027141 − 0.15391 · 10−4T + 0.22655 · 10−7T 2

)(13.147)

H[T ] =(0.62158 · 10−6 − 0.40075 · 10−8T + 0.65972 · 10−11T 2

)(13.148)

The compressibility is defined as

Cw[P, T,ws, v] = −1

v

(∂v

∂P

)T,ws

=

− 1

v

(−B[T ]− 2 · P · C[T ]− ws · F [T ]− w2s ·G[T ]− ws · P ·H[T ]

)(13.149)

The density is defined as

ρbrine = v−1 (13.150)

The derivative of the density with respect to pressure, ∂ρbrine∂P is defined as

∂ρbrine∂P

=∂1/v

∂P= −v−2 ∂v

∂P=

Cw

v= ρbrineCw (13.151)

The derivative of the density with respect to mole fraction, ∂ρbrine∂Ym′ is defined as

∂ρbrine∂Ym′

= 0 (13.152)

13.10.23 Garcıa, CMG, Brine Density, H2O+CO2 +NaCl, ρw + vCO2

Garcıa (2001) defines the density and partial molar volume of H2O plus CO2. The units are

T [◦C], ρ[kg/m3], vCO2 [cm3/mol], c[mol/L], MW[g/mol]. The partial molar volume is calculated

based on temperatures from 0◦C to 300◦C and from 0 to 5% molarity.

vCO2 [T ] = 37.51 − 9.585 · 10−2T + 8.740 · 10−4T 2 − 5.044 · 10−7T 3 (13.153)

Garcıa (2001) also presents the following equation for calculating the aqueous density.

ρaq = ρH2O +MWCO2 · cCO2 − cCO2 · ρH2O · vCO2 · 10−3 (13.154)

246

If we can assume that the H2O-CO2 system can be decoupled from the H2O-NaCl, then we can

use the following, where ρbrine comes from Rowe and Chou (1970). Since the molar concentration

cCO2 is a function of the total density ρw,t, this is an iterative calculation.

ρw,t = ρbrine +MWCO2 · cCO2 − cCO2 · ρbrine · vCO2 · 10−3 (13.155)

Rewrite cCO2 in terms of WCO2 :

cCO2 =WCO2∑j WjMWj

ρw,t =WCO2ρw,t

MWw,t(13.156)

Substitute for cCO2 in (13.155).

ρw,t =ρbrine

1 +WCO2

MWw,t· (−MWCO2 + ρbrine · vCO2 · 10−3

) (13.157)

The total molecular weight for the aqueous phase, MWw,t is defined as follows:

MWw,t = MWCO2WCO2 +MWbrine(1−WCO2) (13.158)

Where the brine molecular weight is defined by:

MWbrine = MWNaClWNaCl +MWH2O(1−WNaCl) (13.159)

The derivative of the total molecular weight with respect to WCO2 is

∂MWw,t

∂WCO2

= MWCO2 −MWbrine (13.160)

Garcia uses units of T [◦C], MW[ gmol ], ρ[

kgm3 ], and cCO2 [

molL ]. (13.157) converts the units to field

units, MW[ lbmlbmol ], ρ[

lbmft3

], and WCO2 [molmol ]. A � represents converting from kg

m3 to lbmft3

, so ρ� is in

units of lbmft3

. Two terms in (13.157) are defined locally as α and β to simplify the derivatives.

ρ�w,t =ρ�brine

α︷︸︸︷1 +

WCO2

MWw,t·

β︷︸︸︷(−MWCO2 + ρ�brine · v1/�Garcia

)(13.161)

247

The derivative of ρw,t with respect to WCO2 is:

∂ρ�w,t

∂WCO2

= −ρ�w,t

α

β

MWwt

(1− WCO2

MWwt

∂MWwt

∂WCO2

)(13.162)

The derivative of ρw,t with respect to ρbrine is:

∂ρ�w,t

∂ρ�brine= ρ�wt

(1

ρ�brine− WCO2

MWwt

v1/�Garcia

α

)(13.163)

The derivative of ρw,t with respect to pressure∂ρw,t

∂P is defined by

∂ρ�w,t

∂P=

∂ρ�w,t

∂ρ�brine

∂ρ�brine∂P

+∂ρ�w,t

∂WCO2

∂WCO2

∂P(13.164)

The derivative of ρw,t with respect to mole fraction∂ρw,t

∂Ym′ is defined by

∂ρ�w,t

∂Ym′=

∂ρ�w,t

∂WCO2

∂WCO2

∂Ym′(13.165)

The derivative of ρw,t with respect to mole fraction∂ρw,t

∂Xm′ is defined by

∂ρ�w,t

∂Xm′=

∂ρ�w,t

∂WCO2

∂WCO2

∂Xm′(13.166)

The molar density ξw is defined by

ξw =ρ�w,t

MWw,t(13.167)

The derivative of ξw with respect to WCO2 is

∂ξw∂WCO2

=1

MWw,t

(∂ρ�w,t

∂WCO2

− ξw∂MWw,t

∂WCO2

)(13.168)

The derivative of ξw with respect to pressure ∂ξw∂P is

∂ξw∂P

=1

MWw,t

(∂ρ�w,t

∂P− ξw

∂MWw,t

∂WCO2

∂WCO2

∂P

)(13.169)

The derivative of ξw with respect to mole fraction ∂ξw∂Ym′ is

248

∂ξw∂Ym′

=1

MWw,t

(∂ρ�w,t

∂Ym′− ξw

∂MWw,t

∂WCO2

∂WCO2

∂Ym′

)=

1

MWw,t

(∂ρ�w,t

∂WCO2

− ξw∂MWw,t

∂WCO2

)∂WCO2

∂Ym′

(13.170)

The derivative of ξw with respect to mole fraction ∂ξw∂Xm′ is

∂ξw∂Xm′

=1

MWw,t

(∂ρ�w,t

∂Xm′− ξw

∂MWw,t

∂WCO2

∂WCO2

∂Xm′

)=

1

MWw,t

(∂ρ�w,t

∂WCO2

− ξw∂MWw,t

∂WCO2

)∂WCO2

∂Xm′

(13.171)

When WCO2 is a primary variable, the derivative of ρw,t with respect to pressure∂ρw,t

∂P is defined

by

∂ρ�w,t

∂P=

∂ρ�w,t

∂ρ�brine

∂ρ�brine∂P

(13.172)

When WCO2 is a primary variable, the derivative of ρw,t with respect to mole fraction∂ρw,t

∂Ym′ is

defined by

∂ρ�w,t

∂Ym′= 0 (13.173)

When WCO2 is a primary variable, the derivative of ρw,t with respect to mole fraction∂ρw,t

∂Xm′ is

defined by

∂ρ�w,t

∂Xm′= 0 (13.174)

When WCO2 is a primary variable, the derivative of ξw with respect to pressure ∂ξw∂P is

∂ξw∂P

=1

MWw,t

(∂ρ�w,t

∂P

)(13.175)

When WCO2 is a primary variable, the derivative of ξw with respect to mole fraction ∂ξw∂Ym′ is

∂ξw∂Ym′

= 0 (13.176)

When WCO2 is a primary variable, the derivative of ξw with respect to mole fraction ∂ξw∂Xm′ is

249

∂ξw∂Xm′

= 0 (13.177)

13.10.24 Kestin, Brine Viscosity, Eclipse+VIP, H2O+NaCl, μw

Kestin et al. (1981) and Kestin et al. (1978) describe the viscosity of NaCl brine solutions for 20–

150◦C and pressures of 0.1–35 MPa, and 0–5.4 molal. These correlations are based on Kestin et al.

(1978) and Rowe and Chou (1970). Sayegh and Najman (1987) shows that CO2 has a negligible

impact on the viscosity of the H2O+NaCl system. These correlations are defined using the following

units: ms[molNaCl/kgH2O], μ[μPa · s], P [MPa], T [◦C], β[1/GPa] The Kestin correlations are based

on a system with H2O and NaCl only. As a result, the mNaCl and mH2O are calculated using WNaCl

and WH2O = 1−WNaCl.

μ[P, T,ms] = μ0[T,ms] ·(1 + β[T,ms] · 10−3 · P ) (13.178)

μ0[T,ms] = μ0w[T ] · μ0

r[T,ms] (13.179)

μ0w[20

◦C] = 1002.0 (13.180)

log10

[μ0w[T ]

μ0w[20

◦C]

]= log10

[μ0w[T ]

1002.0

]=(

1.2378(20 − T )− 1.303 · 10−3(20 − T )2 + 3.06 · 10−6(20− T )3 + 2.55 · 10−8(20 − T )4)

96 + T(13.181)

log10[μ0r[T,ms]

]=(3.324 · 10−2ms + 3.624 · 10−3m2

s − 1.879 · 10−4m3s

)+(−3.96 · 10−2ms + 1.02 · 10−2m2

s − 7.02 · 10−4m3s

) ∗ log10[

μ0w[T ]

μ0w[20

◦C]

](13.182)

β[T,ms] = βEs [T ]β

�[T,ms] + βw[T ] (13.183)

250

βw[T ] =(−1.297 + 5.74 · 10−2T − 6.97 · 10−4T 2 + 4.47 · 10−6T 3 − 1.05 · 10−8T 4

)(13.184)

βEs [T ] = 0.545 + 2.8 · 10−3T − βw[T ] (13.185)

m�s[T ] = 6.044 + 2.8 · 10−3T + 3.6 · 10−5T 2 (13.186)

β�[T,ms] = 2.5

(ms

m�s[T ]

)− 2.0

(ms

m�s[T ]

)2

+ 0.5

(ms

m�s[T ]

)3

(13.187)

13.10.25 Duan, Henry’s Law, H2O+CO2 +NaCl, WCO2 +HCO2

The Duan and Sun (2003) model presents correlations for calculating the Henry’s Law constants

for aqueous solutions containing CO2, NaCl, plus additional salts. The following units are used in

this section: P [bar], T [K], m[mol/kg], v[L/mol], fCO2 [barmolmol ], and R[ bar·Lmol·K ] = 0.08314467.

The Duan correlation is based on experimental data. These data were collected by first creating

a H2O and NaCl mixture and then measuring the solubility after equilibrium was reached with a

CO2 vapor system. Because of this experimental process and for consistency with the Kestin and

Rowe models, mNaCl and mH2O are calculated based on WNaCl and WH2O = 1−WNaCl. Next, mCO2

is calculated using the Duan and Sun (2003) correlation, and then WCO2 is calculated using mNaCl,

mCO2 , and mH2O. The value of WNaCl is not changed. The value of WH2O = 1−WNaCl −WCO2 is

used for further calculations.

ln

[YCO2PΦCO2

mCO2

]= ln

[fCO2

mCO2

]= lnHCO2 =

μl(0)CO2

RT+ 2mNaλCO2,Na +mNamClζCO2,Na,Cl

= A[P, T ] + 2mNaClB[P, T ] +m2NaClC[P, T ] (13.188)

251

μl(0)CO2

RT= A[P, T ] =(

28.9447706 +−0.0354581768T +−4770.67077T−1 + 1.02782768 · 10−5T 2+

33.8126098

630− T+ 9.04037140 · 10−3P +−1.14934031 · 10−3P ln[T ]+

−0.307405726PT

+−0.0907301486P

630 − T+

9.32713393 · 10−4P 2

(630 − T )2

)(13.189)

∂A

∂P= 9.04037140 · 10−3 +−1.14934031 · 10−3 ln[T ]+

−0.307405726T

+−0.0907301486

630 − T+ 2 · 9.32713393 · 10

−4P

(630 − T )2(13.190)

λCO2,Na = B[P, T ] =

(−0.411370585 + 6.07632013 · 10−4T + 97.5347708T−1+

−0.0237622469PT

+0.0170656236P

630 − T+ 1.41335834 · 10−5T ln[P ]

)(13.191)

∂B

∂P=−0.0237622469

T+

0.0170656236

630 − T+ 1.41335834 · 10−5 T

P(13.192)

ζCO2,Na,Cl = C[P, T ] =

(3.36389723 · 10−4 +−1.98298980 · 10−5T+

2.12220830 · 10−3P

T+−5.24873303 · 10−3P

630− T

)(13.193)

∂C

∂P=

2.12220830 · 10−3

T+−5.24873303 · 10−3

630− T(13.194)

Use the following definition of mCO2 :

mCO2 =fCO2

HCO2

(13.195)

The derivative of mCO2 with respect to pressure,∂mCO2

∂P is

∂mCO2

∂P=

1

HCO2

(∂fCO2

∂P−mCO2

∂HCO2

∂P

)(13.196)

252

The derivative of HCO2 with respect to pressure,∂HCO2∂P is

1

HCO2

∂HCO2

∂P=

∂A

∂P+ 2mNaCl

∂B

∂P+m2

NaCl

∂C

∂P(13.197)

To calculate the derivative of WCO2 with respect to pressure,∂WCO2

∂P , use the conversion from

molality to mole fraction.

Wi =mi∑j mj

(13.198)

After solving∂WCO2

∂P and∂mH2O

∂P simultaneously, this yields

∂WCO2

∂P=

1∑j mj

∂mCO2

∂P(13.199)

The derivative of mCO2 with respect to Ym′ ,∂mCO2∂Ym′ is

∂mCO2

∂Ym′=

1

HCO2

∂fCO2

∂Ym′(13.200)

The derivative of WCO2 with respect to Ym′ ,∂WCO2∂Ym′ is

∂WCO2

∂Ym′=

1∑j mj

∂mCO2

∂Ym′(13.201)

Duan and Sun (2003) also tested extending the model to solutions containing Ca2+, K+, Mg2+,

SO2−4 , CO2−

3 , and HCO−3 . They were able to approximate all monovalent cations with mNa+ and

all divalent cations with 2 ∗mNa+ . Only SO2−4 required an adjustment factor. (13.188) becomes:

ln

[YCO2PΦCO2

mCO2

]= ln

[fCO2

mCO2

]= lnH =

μl(0)CO2

RT+ 2 (mNa +mK + 2 ∗mMg + 2 ∗mCa)λCO2,Na+

(mNa +mK +mMg +mCa)mClζCO2,Na,Cl − 0.07mSO4 (13.202)

13.11 Correlations Used to Evaluate Other Correlations

The correlations described in this section are used to evaluate the other correlations.

253

13.11.1 Zeebe, Henry’s Law for Seawater, H2O+CO2 +NaCl, H

Zeebe and Wolf-Gladrow (2001) describes properties of seawater, as well as a Henry’s Law

correlation for CO2 in seawater. The units of the following correlation are m[molCO2/kgH2O],

fCO2 [atm], H−1[(molCO2/kgH2O)/atm], ws[wt fraction] (assumes salinity S[pptw]), T [◦C]. This

correlation is the one recommended by Zeebe and Wolf-Gladrow (2001) based on Weiss (1974).

fCO2 = H ·mCO2 (13.203)

lnH−1 = −60.2409 + 9345.17/T + 23.3585 ln

[T

100

]+

ws · 1000 ·(0.023517 − 0.00023656 · T + 0.0047036

(T

100

)2)

(13.204)

13.11.2 Duan, Fugacity, H2O+CO2

To compare Henry’s Law computations with direct computations of CO2 solubility, Duan and

Sun (2003) presents correlations for the fugacity of CO2 based on Duan et al. (1992). These

fugacities are only used to check the other correlations. Their accuracy for the CO2-H2O system

is reported as within 5%. For simulation with the three-phase system, the fugacity of CO2 is

calculated using the Peng-Robinson equation of state for the gas-oil system. The correlation is

valid in the range 36 to 1000◦C and 0 to 8000 bar.

fCO2 = YCO2PΦCO2 (13.205)

YCO2 =P − P s

H2O

P(13.206)

The water saturation pressure is calculated using Pc,H2O = 220.85 bar and Tc,H2O = 647.29 K.

P sH2O =

PcT

Tc

(1 +−38.640844

(−T − Tc

Tc

)1.9

+ 5.8948420

(T − Tc

Tc

)+

59.876516

(T − Tc

Tc

)2

+ 26.654627

(T − Tc

Tc

)3

+ 10.637097

(T − Tc

Tc

)4)(13.207)

254

The fugacity coefficient for the CO2-H2O system is calculated using Tr = T/Tc, Pr = P/Pc,

vr =vPcRTc

, Tc,CO2 = 304.2 K, and Pc,CO2 = 73.825 bar.

lnΦCO2 = zCO2 − 1− ln zCO2 +A1v−1r +

1

2A2v

−2r +

1

4A3v

−4r +

1

5A4v

−5r +

1

2

a13T 3r a15

(a14 + 1−

(a14 + 1 +

a15v2r

)exp

[−a15

v2r

])(13.208)

zCO2 =PrvrTr

= 1+A1v−1r +A2v

−2r +A3v

−4r +A4v

−5r +

a13T 3r v

2r

(a14 +

a15v2r

)exp

[−a15

v2r

](13.209)

A1 = 8.99288497 · 10−2 +−4.94783127 · 10−1T−2r + 4.77922245 · 10−2T−3

r (13.210)

A2 = 1.0380883 · 10−2 +−2.8516861 · 10−2T−2r + 9.49887563 · 10−2T−3

r (13.211)

A3 = 5.20600880 · 10−4 +−2.93540971 · 10−4T−2r +−1.77265112 · 10−3T−3

r (13.212)

A4 = −2.51101973 · 10−5 + 8.93353441 · 10−5T−2r + 7.88998563 · 10−5T−3

r (13.213)

a13 = −1.66727022 · 10−2 a14 = 1.398 a15 = 2.96 · 10−2 (13.214)

13.12 Henry’s Law Correlations

The three methods in this section use Henry’s Law to calculate the solubility of CO2 in the

aqueous phase.

13.12.1 Chang, Mole Fraction, Eclipse + VIP, H2O+CO2 +NaCl, WCO2 +Rsw +Bw

Chang et al. (1998) defines a correlation for the solubility of CO2 in H2O. This method is used

by VIP and Eclipse. The units of these correlations use Rsw[SCFCO2/STBH2O], T [◦F], P [psia],

Bw[RB/STB], ρ[lbm/ft3], ws[wt fraction], and Cw[psi−1]. This model uses local constants a, b, c,

d, and P 0.

255

Rsw[P, T ] =

{P < P 0 : a · P ·

(1− b · sin

[π2 · c·P

c·P+1

])P ≥ P 0 : a · P 0(1− b3) + d(P − P 0)

(13.215)

The brine solubility is defined by

log10

[Rsb[P, T,ws]

Rsw

]= −2.8037 · ws · T−0.12039 (13.216)

The parameters in (13.215) are defined by the following, using

a[T ] = 1.16306+−16.6304·10−3T+111.07305·10−6T 2+−376.85925·10−9T 3+524.88916·10−12T 4

(13.217)

b[T ] = 0.96509+−0.27255·10−3T+0.09234·10−6T 2+−0.10083·10−9T 3+0.09979·10−12T 4 (13.218)

c[T ] = 1.28030 · 10−3 +−10.75660 · 10−6T + 52.69622 · 10−9T 2+

− 222.39488 · 10−12T 3 + 462.67255 · 10−15T 4 (13.219)

P 0[T ] =2

π· sin−1[b2]

c(1− 2

π · sin−1[b2]) (13.220)

d[T ] = a− ab

(sin

[π

2· cP 0

1 + cP 0

]+

π

2· cP 0

(1 + cP 0)2cos

[π

2· cP 0

1 + cP 0

])(13.221)

The formation volume factor Bw is defined as follows, with ρw,sc and ρw,atm defined by Rowe

and Chou (1970).

Bw[P, T,ws] =ρw,sc[P = 14.7 psia, T = 60◦F, ws] + 0.02066 · Rsb

ρw,atm[P = 14.7 psia, T, ws] + 0.0058 ·Rsb(13.222)

The water compressibility Cw is defined based on Rowe and Chou (1970) and the following:

P > 5000 :1

Cw[P, T,ws]=

1

Cw,5000[P = 5000 psia, T, ws]+ 7.033(P − 5000) (13.223)

Chang et al. (1998) suggests using Kestin et al. (1978) for the viscosity correlation.

256

13.12.2 CMG, Henry’s Law, H2O+CO2 +NaCl, WCO2 +HCO2

The GEM User’s Manual, CMG (2010), specifies several correlations for calculating the mole

fraction of CO2 in the aqueous phase using Henry’s Law correlations. There are also correlations

listed for N2, H2S, and CH4. These correlations use Hsi [MPa], P [MPa], T [K], v[cm3/mol], and

Cs[mol/kgH2O]. The critical properties of water are Tc,H2O = 647.14 K and Pc,H2O = 22.064 MPa

The following correlation for the Henry’s Law is based on Harvey (1996), based on the H2O satu-

ration pressure.

lnHsCO2

[T ] = lnP sH2O +−9.4234

(T

Tc,H2O

)−1

+ 4.0087

(1−

(T

Tc,H2O

))0.355( T

Tc,H2O

)−1

+

10.3199 exp

[1−

(T

Tc,H2O

)](T

Tc,H2O

)−0.41

(13.224)

The saturation pressure is defined by Saul and Wagner (1987). Saul and Wagner (1987) also defines

the density, specific enthalpy, and specific entropy of water at the saturation pressure, but these

properties are not needed here.

ln

[P sH2O

[T ]

Pc,H2O

]=

(Tc,H2O

T

)(−7.85823

(1− T

Tc,H2O

)+ 1.83991

(1− T

Tc,H2O

)1.5

+

− 11.7811

(1− T

Tc,H2O

)3

+ 22.6705

(1− T

Tc,H2O

)3.5

+

− 15.9393

(1− T

Tc,H2O

)4

+ 1.77516

(1− T

Tc,H2O

)7.5)(13.225)

The Henry’s Law constant is defined by

lnHCO2 [P, T ] = lnHsCO2

[T ] +1

RT

∫ P

P sH2O

vCO2 [T ]dP

= lnHsCO2

[T ] +1

RTvCO2 [T ]

(P − P s

H2O[T ]) (13.226)

The partial molar volume is calculated using Garcıa (2001). The salinity effects are calculated

using a “salting out” coefficient ksalt,CO2 .

ln

[Hsalt,CO2

HCO2

]= ksalt,CO2Cs (13.227)

257

The salting out coefficient is defined based on Bakker (2003), for temperatures from 0◦C to 300◦C.

ksalt,CO2 [T ] = 0.11572−6.0293·10−4(T−273.15)+3.5817·10−6(T−273.15)2−3.7772·10−9(T−273.15)3(13.228)

The fugacity of the water phase is defined by

faq,CO2 = WCO2PΦCO2 = WCO2HCO2 (13.229)

13.12.3 Enick, Henry’s Law, H2O+CO2 +NaCl, H +Rsw +WCO2 + μw

Enick and Klara (1992) and Enick and Klara (1990) describe ways to calculate the solubility of

CO2.

The following correlations are from Enick and Klara (1990). This uses T [K], P [MPa], v[cm3/mol],

ws[wt fraction], wCO2 [wt fraction], ws[wt fraction], WCO2 [mol fraction]. Cs is the total dissolved

solids in weight percent excluding dissolved gases. Several correlations are defined for H∗i and v∞i .

These were evaluated by Enick and Klara (1990) for temperatures between 298 K–523 K and a

pressure range from 3.4 MPa–85 MPa.

ln

[fCO2

WCO2

]= lnH = lnH∗

CO2+

A

RT(W 2

H2O − 1) +v∞CO2

P

RT(13.230)

H∗CO2

= −5076.29 + 31.9877T − 0.057691T 2 + 3.18012 · 10−5T 3 (13.231)

A = −2.08184 · 106 + 2.13034 · 104T − 79.8190T 2 + 0.129991T 3 − 7.76471 · 10−5T 4 (13.232)

v∞CO2[T ] = 1799.36− 17.8218T +0.0659297T 2 − 1.05786 · 10−4T 3 +6.200275 · 10−8T 4 (13.233)

The following equation is for weight fractions wCO2 .

wCO2,b = wCO2,w

(1− 4.893414 · 10−2(100ws)+

0.1302838 · 10−2(100ws)2 − 0.1871199 · 10−4(100ws)

3

)(13.234)

258

Enick and Klara (1992) uses the model by Enick and Klara (1990).

WCO2,b =

wCO2,b

44wCO2,b

44 +(1−wCO2,b

)

MWb

(13.235)

MWb =1051.2

58.4 − 0.404CTDS(13.236)

ρb = ξbMWb (13.237)

μb = μw

(1 + 1.892 · 10−2Cs + 1.215 · 10−4(Cs)

2 + 1.941 · 10−5(Cs)3)

(13.238)

Cs[wt%] =58.4WNaCl

18WH2O + 58.4WNaCl(13.239)

13.13 Adjustments to Peng-Robinson Equation of State

The methods in this section use modifications to the equation of state to calculate the CO2

solubility in the aqueous phase.

13.13.1 Peng-Robinson Equation of State Paramters

The Peng-Robinson Equation of State Peng and Robinson (1976) with the Peneloux volume

correction (Peneloux and Rauzy, 1982) is defined by:

P =RT

v − b− a

(v + c)(v + 2c+ b) + (b+ c)(v − b)(13.240)

The am are defined by:

am =

(Ωa

R2T 2cm

Pcm

)(1 + κm[ωm]

(1−

√T

Tcm

))2

(13.241)

The bm are defined by:

bm = ΩbRTcm

Pcm(13.242)

259

This results in an adjustment to the specific volumes and the densities, but does not adjust the

phase splitting.

vnew = vEOS −∑

Xmcm (13.243)

In some cases (for instance the Eclipse SSHIFT parameter sm) , the volume shift is defined as a

multiplier to the bm:

cm = bmsm (13.244)

The amn is defined by (13.245). The binary interaction coefficient is δmn (Eclipse BIC). In the

Peng-Robinson 1978 version, δmn is a function of temperature. δmn is often labeled kmn, but the

symbol δmn is used here to avoid confusion with permeability.

amn = (1− δmn)a1/2m a1/2n (13.245)

Compute the mixed a using:

ale1 =∑m,n

amnXe1mXe1

n ave1 =∑m,n

amnYe1m Y e1

n (13.246)

Compute the mixed b using:

ble1 =∑m

bmXe1m bve1 =

∑m

bmY e1m (13.247)

Compute the mixed c using:

cle1 =∑m

cmXe1m cve1 =

∑m

cmY e1m (13.248)

13.13.2 Soreide, EOS, Eclipse, H2O+CO2 +NaCl, WCO2 + ρaq

Eclipse describes modifications of the Peng-Robinson equation of state based on Søreide and

Whitson (1992). Redefine the am for the water component, using the following units: P [bar],

Cs[molCO2/kgH2O], T [◦C] for temperatures from 0◦C to 325◦C and salt concentrations from 0 to

5 mol/kg.

260

am =

(Ωa

R2T 2cm

Pcm

)(1 + 0.4530

(1− (1− 0.0103(Cs)

1.1)( T

Tcm

))+

0.0034(( T

Tcm

)−3

− 1))2

(13.249)

The unit conversions for molality are:

molality = 1000 ∗ molsaltmasswater

Cmolals =

Cppmws

58440.0 − 0.05844 ∗ Cppmws

(13.250)

The binary interaction coefficients in the aqueous phase for water are defined with a temperature

and salinity dependence, based on Søreide and Whitson (1992).

δaqj,H2O=(1.112 − 1.7369ω−0.1

j

)(1 + 0.017407Cs)+

(1.1001 + 0.836ωj) (1 + 0.033516Cs)

(T

Tc,H2O

)+

(−0.15742 − 1.0988ωj) (1 + 0.011478Cs)

(T

Tc,H2O

)2

(13.251)

The binary interaction coefficients in the aqueous phase for CO2 are defined with a temperature

and salinity dependence, based on Søreide and Whitson (1992).

δaqCO2,H2O= −0.31092

(1 + 0.15587 (Cs)

0.7505)+

0.23580(1 + 0.17837 (Cs)

0.979)( T

Tc,CO2

)+

− 21.2566 exp

[−6.7222

(T

Tc,CO2

)− Cs

](13.252)

13.13.3 Delshad, EOS and IFT, H2O+CO2 +NaCl, WCO2 + ρaq + σgw

Delshad et al. (2011) describes adjustments to the EOS and calculation of calculation of the

interfacial tension. These equations use T [◦F] and total dissolved solids Cs[ppm].

δH2O,CO2 = −0.093625 + 4.861 · 10−4(T − 113) + 2.29 · 10−7Cs (13.253)

It uses a volume shift defined by

261

cH2O = 0.179 + 2.2222 · 10−4(T − 113) + 4.9867 · 10−7Cs (13.254)

The gas-water interfacial tension is calculated using the following correlations. This correlation

uses T [◦C], P [MPa], salinity Cs[wt%], and σ[mN/m]. It reproduces the trend in reduced interfacial

tension, but the absolute magnitudes do not fit the experimental data of Bennion and Bachu (2008b)

very well (see Delshad et al. (2011), figure 1–2).

σwg = 71.69243P 0.432629 + 0.210558T 0.900261 + 0.075859C1.457937s (13.255)

13.13.4 Yan, EOS, H2O+CO2 +NaCl, WCO2 + ρaq

Yan and Stenby (2009) uses the Søreide and Whitson (1992) model with some adjustments. For

a system where the CO2 is soluble in water but water is not present in the hydrocarbon phase, the

binary interaction term is adjusted using the following equation. Yan and Stenby (2009) makes the

comment that assuming no H2O in the vapor phase leads to large inaccuracies in the CO2 fugacity

at temperatures above 150◦C and/or low pressures. The units are Cs[molal] and T [◦C].

δH2O,CO2 = −0.00739470Cs − 0.443752 +(4.55173 · 10−5Cs + 0.00111209

)T (13.256)

13.13.5 Melhem, EOS, H2O+CO2, WCO2 + ρaq

Melham and Little (1989) defines some modifications to the Peng-Robinson equation of state.

Redefine the am for water and CO2, using T [K], P [atm]. Tc,CO2 [K] = 304.2, Pc,CO2 [atm] = 72.8,

Tc,H2O[K] = 647.3, and Pc,H2O[atm] = 217.6.

aCO2 =

(Ωa

R2T 2c,CO2

Pc,CO2

)exp

[0.6877

(1− T

Tc,CO2

)+ 0.3813

(1−

√T

Tc,CO2

)2](13.257)

aH2O =

(Ωa

R2T 2c,H2O

Pc,H2O

)exp

[0.8893

(1− T

Tc,H2O

)+ 0.0151

(1−

√T

Tc,H2O

)2](13.258)

262

13.13.6 Spycher, EOS, Eclipse, H2O+CO2 +NaCl, WCO2 + ρaq

The mole fraction of H2O in the gas phase and CO2 in the aqueous phase in the presence of NaCl

is defined by Spycher and Pruess (2005). The correlations require values of the thermodynamic

equilibrium constant at temperature T and reference pressure P 0 = 1 bar (K0H2O

,K0CO2

) , the

fugacity coefficient of each species in the gas phase (ΦH2O, ΦCO2), the average partial molar volume

over the pressure range P − P 0 (VH2O, VCO2) from Spycher et al. (2003).

The activity coefficient of CO2 in a mixture containing various salts can be calculated using

several different techniques from the literature. The following two methods give the best results

according to Spycher and Pruess (2005). Duan and Sun (2003) defines the activity coefficient in

terms of pressure, temperature, molality of various salts, and the molality in a pure CO2-H2O

mixture. Rumpf et al. (1994) defines the activity coefficient in terms of temperature and the

molality of various salts, and the molality in a pure CO2-H2O mixture.

Together, Spycher and Pruess (2005) and Spycher et al. (2003) describe a Redlich-Kwong equa-

tion of state model for the H2O+CO2+NaCl system. Because the model is based on Redlich-Kwong

rather than Peng-Robinson, another option is preferred if possible.

13.14 Models Considered But Not Used

Li and Nghiem (1986) defines correlations for Henry’s Law constants, for H2O +CO2 + NaCl.

CMG uses this model to calculate three-phase equilibria. The model is more complicated than the

other models in Section 13.12. It requires parameters from scaled particle theory which may not

be commonly available. Because of the additional data requirements, added complexity, and focus

on three-phase equilibrium calculations, this technique was not selected for this project.

The methods of Kell and Whalley (1975) and Zaytsev and Aseyev (1992) define methods for

calculating the density as a function of the detailed salt composition. The method is used by the

CO2STORE option of Eclipse, but is overly complicated for this work. Kell and Whalley (1975)

defines a correlation for the pure water density as a function of temperature and pressure for 0–

1000 bar and 0–150◦C. Zaytsev and Aseyev (1992) describes a method originally based on Erzokhi

to adjust the density of water based on the concentrations of various salts. These specify different

correlation for each salt to adjust the density.

263

The methods of Vesovic et al. (1990) and Fenghour et al. (1998) define correlations for the

viscosity of pure CO2. These methods are used by the CO2STORE option of Eclipse, but is too

specialized for this work. Vesovic et al. (1990) describes a correlation for the viscosity of CO2 in

terms of μ0 (which is defined as a correlation in terms of temperature), a complex correlation for

near-critical behavior in terms of density and temperature, and a correlation in terms of density

and temperature. Fenghour et al. (1998) provides a simpler correlation for μ0 and Δμ, but uses

the same correlation near the critical region.

Duan, Hu, Li, and Mao (2008) presents a detailed review of the experimental data for the

H2O+CO2 +NaCl system. The data review is good, but the correlations are not well presented.

Majer, Sedlbauer, and Bergin (2008) presents a detailed model for calculating the Henry’s Law

constant for aqueous H2O + CO2, plus various other components. It is a complicated model that

does not include the effect of salinity, so it will not be used here.

Fernandez-Prini, Alvarez, and Harvey (2003) provides an updated correlation for Henry’s Law

constants for aqueous H2O+CO2, plus various other components. Since the model does not account

for salinity and requires an additional saturation pressure correlation, it is not used here.

Rumpf et al. (1994) conducted experiments of the solubility in the H2O+CO2 +NaCl system.

They present a complex model to calculate this solubility. Their data is used by several of the more

recent articles. Other articles provide a simpler approach which is more applicable to this project.

264

CHAPTER 14

COMPUTATION: ASSEMBLY OF JACOBIAN

0.006328∇ ·(Xn

mξnoμno

knrok#(∇Pn+1

o − γno∇D#)

)+

0.006328∇ ·(Y n

mξngμng

knrgk#(∇Pn+1

o +∇Pncgo − γn

g∇D#))+

0.006328∇ ·(Wn

mξnwμnw

knrwk#(∇Pn+1

o −∇Pncow − γn

w∇D#))+(Xn

mξno qno + Y n

mξng qng +Wn

mξnw qnw

)=

1

Δt

(φn+1(Xn+1

m Sn+1o ξn+1

o + Y n+1m Sn+1

g ξn+1g +Wn+1

m Sn+1w ξn+1

w ))−

1

Δt

(φn(Xn

mSno ξ

no + Y n

mSng ξ

ng +Wn

mSnwξ

nw))

(14.1)

14.1 Diagonal Terms

Figure 14.1: Block 1: block geometry for the main block diagonal of a NC = 5 problem. Blackrepresents non-zero values; gray represents zero values.

Diagonal terms have the following form, Figure 14.1.


Cm X X X X X

Gm X 0 0 X X

(14.2)

Diagonal terms have the following form. If there are no well connections to cell ijk, then

WDP = 0.

265


Cm −VRΔt

∂Accm�ijk

∂P + −VRΔt

∂Accm�ijk

∂So−VR

Δt

∂Accm�ijk

∂Sg−VR

Δt

∂Accm�ijk

∂Xm′ −VRΔt

∂Accm�ijk

∂Ym′

DPmnxt,ijk + DPmn

yt,ijk + DPmnzt,ijk+

WDPmnijk

Gm∂fm�

o,ijk

∂P − ∂fm�g,ijk

∂P 0 0∂fm�

o,ijk

∂Xm′ −∂fm�

g,ijk

∂Xm′∂fm�

o,ijk

∂Ym′ −∂fm�

g,ijk

∂Ym′

(14.3)

14.2 Diagonal Terms Above the Bubble Point

Figure 14.2: Block 7: block geometry above the bubble point for the main block diagonal of aNC = 5 problem. Black represents non-zero values; gray represents zero values.


Po So Pb Xm′ Ym′

Cm X X 0 X X

Gm 0 0 X X X

(14.4)

Diagonal terms above the bubble point (Sg = 0) have the following form. If there are no well

connections to cell ijk, then WDP = 0.

Po So Pb Xm′ Ym′

Cm −VRΔt

∂Accm�ijk

∂P + −VRΔt

∂Accm�ijk

∂So0 −VR

Δt

∂Accm�ijk

∂Xm′ −VRΔt

∂Accm�ijk

∂Ym′

DPmnxt,ijk + DPmn


WDPmnijk

Gm 0 0∂fm�

o,ijk

∂Pb− ∂fm�

g,ijk

∂Pb

∂fm�o,ijk


g,ijk

∂Xm′∂fm�

o,ijk


g,ijk

∂Ym′

G′NC−1 0 0

∂G�NC−1

∂Pb

∂G�NC−1,ijk

∂Xm′∂G�

NC−1,ijk

∂Ym′

(14.5)

266

14.3 Diagonal Terms Below the Dew Point

Figure 14.3: Block 7: block geometry below the dew point for the main block diagonal of a NC = 5problem. Black represents non-zero values; gray represents zero values.


Po Pd Sg Xm′ Ym′

Cm X 0 X X X

Gm 0 X 0 X X

(14.6)

Diagonal terms below the dew point (So = 0) have the following form. If there are no well

connections to cell ijk, then WDP = 0.

Po Pd Sg Xm′ Ym′

Cm −VRΔt

∂Accm�ijk

∂P + 0 −VRΔt

∂Accm�ijk

∂Sg−VR

Δt

∂Accm�ijk

∂Xm′ −VRΔt

∂Accm�ijk

∂Ym′

DPmnxt,ijk + DPmn


WDPmnijk

Gm 0∂fm�

o,ijk

∂Pd− ∂fm�

g,ijk

∂Pd0

∂fm�o,ijk


g,ijk

∂Xm′∂fm�

o,ijk


g,ijk

∂Ym′

G′NC−1 0

∂G�NC−1,ijk

∂Pd0

∂G�NC−1,ijk

∂Xm′∂G�

NC−1,ijk

∂Ym′

(14.7)

14.4 Off-Diagonal Terms

Off-diagonal terms have the following form, Figure 14.4.


Cm X 0 0 0 0

Gm 0 0 0 0 0

(14.8)

267


Off-diagonal bands have the following form, here illustrated for i+ 1, j, k.


Cm DPmn[xyz

]t,i+1,jk

0 0 0 0

Gm 0 0 0 0 0

(14.9)

14.5 Well Terms

Figure 14.5: Block 4: well terms for the component equations for a NC = 5 problem. Blackrepresents non-zero values; gray represents zero values.

Well unknowns have the following form, Figure 14.5.

P �t,w|q�t,w

Cm X

Gm 0

(14.10)

Well unknowns have the following form.

P �t,w|q�t,w

Cm WDWmnijk

Gm 0

(14.11)

268

14.6 Right Hand Side

Figure 14.6: Block 6: right-hand-side terms for the component equations for a NC = 5 problem.Black represents non-zero values; gray represents zero values.

Right-hand-side, constant terms have the following form, Figure 14.6.

R

Cm X

Gm X

(14.12)

Right-hand-side, constant terms without well connections have the following form.

R

CmVRΔtAcc

m�ijk − VR

ΔtAccmnijk + DCmn�

xt,ijk +DCmn�yt,ijk +DCmn�

zt,ijk


g,ijk

(14.13)

Right-hand-side, constant terms with well connections have the following form.

R

CmVRΔtAcc

m�ijk − VR


xt,ijk +DCmn�yt,ijk +DCmn�

zt,ijk +WCmn�ijk


g,ijk

(14.14)

Right-hand-side, constant terms above the bubble point or below the dew point without well


R

CmVRΔtAcc

m�ijk − VR


xt,ijk + DCmn�yt,ijk + DCmn�

zt,ijk


g,ijk

G′NC−1 −G�

NC−1,ijk

(14.15)

Right-hand-side, constant terms above the bubble point or below the dew point with well


269

R

CmVRΔtAcc

m�ijk − VR


xt,ijk + DCmn�yt,ijk + DCmn�

zt,ijk +WCmn�ijk


g,ijk

G′NC−1 −G�

NC−1,ijk

(14.16)

14.7 Total Rate Equations

Figure 14.7: Block 5: blocks for the well equations for a NC = 5 problem. Black represents non-zerovalues; gray represents zero values.

Total rate equations for each well have the following form, Figure 14.7.


Qw X 0 0 0 0(14.17)

Total rate equations for each well have the following form.


Qw QDPnijk 0 0 0 0

(14.18)


P �t,w|q�t,w

Qw X(14.19)


P �t,w|q�t,w

Qw QDWnijk

(14.20)


R

Qw X(14.21)


R

Qw QCn�ijk

(14.22)

270

14.8 Accumulation

Define the accumulation term

Accmi =(φiξoiSoiXmi + φiξgiSgiYmi + φiξwiSwiWmi

)(14.23)

14.9 Accumulation Derivatives: Pressure



∂Acc�mi

∂P= ξ�oiS

�oiX

�mi

∂φ�i

∂P+ ξ�giS

�giY

�mi

∂φ�i

∂P+ φ�

iS�oiX

�mi

∂ξ�oi∂P

+ φ�iS

�giY

�mi

∂ξ�gi∂P

(14.24)



∂Acc�mi

∂P= ξ�oiS

�oiX

�mi

∂φ�i

∂P+ ξ�giS

�giY

�mi

∂φ�i

∂P+ ξ�wiS

�wiW

�mi

∂φ�i

∂P+

φ�iS

�oiX

�mi

∂ξ�oi∂P

+ φ�iS

�giY

�mi

∂ξ�gi∂P

+ φ�iS

�wiW

�mi

∂ξ�wi

∂P+ φ�

iξ�wiS

�wi

∂W �CO2,i

∂P(14.25)



∂Acc�mi

∂P= ξ�wiS

�wiW

�mi

∂φ�i

∂P+ φ�

iS�wiW

�mi

∂ξ�wi

∂P− φ�

iξ�wiS

�wi

∂W �CO2,i

∂P(14.26)

14.10 Accumulation Derivatives: Saturation


.

∂Acc�mi

∂So= φ�

iξ�oiX

�mi − φ�

iξ�wiW

�mi (14.27)

Evaluate∂Acc�mi∂Sg

.

∂Acc�mi

∂Sg= φ�

iξ�giY

�mi − φ�

iξ�wiW

�mi (14.28)


= 0.


= 0

271

14.11 Accumulation Derivatives: Composition



m.

∂Acc�mi

∂Xm′= φ�

iS�oiX

�mi

∂ξ�oi∂Xm′

+ φ�iξ

�oiS

�oiδm,m′ (14.29)



m.

∂Acc�mi

∂Ym′= φ�

iS�giY

�mi

∂ξ�gi∂Ym′

+ φ�iξ

�giS

�giδm,m′ (14.30)


m.

∂Acc�CO2,i

∂Xm′= φ�

iS�oiX

�CO2,i

∂ξ�oi∂Xm′

− φ�iS

�oiξ

�oi (14.31)


m.

∂Acc�mi

∂Ym′= φ�

iS�giY

�CO2

∂ξ�gi∂Ym′

− φ�iS

�giξ

�gi + φ�

iS�wiWCO2

∂ξ�wi

∂Ym′+ φ�

iS�wiξ

�wi∂WCO2

∂Ym′(14.32)


∂Acc�mi∂X′

m.

∂Acc�H2O,i

∂Xm′= 0 (14.33)


∂Acc�mi∂Y ′

m.

∂Acc�H2O,i

∂Ym′= φ�

iS�wiWH2O

∂ξ�wi

∂Ym′− φ�

iS�wiξ

�wi

∂WCO2

∂Ym′(14.34)

14.12 Spatial Derivatives: Pressure

The following derivatives are written in terms of x and i± 1. The same approach applies to y

and j ± 1 and z and k ± 1.


272

DPmnxt,i±1 =

(Tmnxo,i± 1

2+ Tmn

xg,i± 12+ Tmn

xw,i± 12

)(14.35)


DPmnxt,i = −

(DPmn


)=

−(Tmnxo,i+ 1

2

+ Tmnxo,i− 1

2

+ Tmnxg,i+ 1

2

+ Tmnxg,i− 1

2

+ Tmnxw,i+ 1

2

+ Tmnxw,i− 1

2

)(14.36)


DCmn�xt,i±1 = Tmn

xo,i± 12·(P �i±1 − γn

o,i± 12Di±1

)+ Tmn

xg,i± 12·(P �i±1 − γn

g,i± 12Di±1 + Pn

cgo,i±1

)+

Tmnxw,i± 1

2·(P �i±1 − γn

w,i± 12Di±1 − Pn

cow,i±1

)(14.37)


DCmn�xt,i = −Tmn

xo,i+ 12·(P �i − γn

o,i+ 12Di

)− Tmn

xo,i− 12·(P �i − γn

o,i− 12Di

)+

− Tmnxg,i+ 1

2·(P �i − γn

g,i+ 12Di + Pn

cgo,i

)− Tmn

xg,i− 12·(P �i − γn

g,i− 12Di + Pn

cgo,i

)+

− Tmnxw,i+ 1

2·(P �i − γn

w,i+ 12Di − Pn

cow,i

)− Tmn

xw,i− 12·(P �i − γn

w,i− 12Di − Pn

cow,i

)(14.38)

14.13 Fugacity Equations

The fugacities are defined by

fmo = Φmo XmP fmg = Φm

g YmP (14.39)

Evaluate∂fm�

oi∂P , m = 1 . . . NC − 1:

∂fm�oi

∂P= fm�

oi

(1

Φm�oi

∂Φm�oi

∂P

)+Φm�

oi Xm (14.40)

Evaluate∂fm�

gi

∂P , m = 1 . . . NC − 1:

∂fm�gi

∂P= fm�

gi

(1

Φm�gi

∂Φm�gi

∂P

)+Φm�

gi Y�mi (14.41)

273

For the normal hydrocarbon equations m = 1 . . . NC − 2, evaluate∂fo�gi∂Xm′ for m′ = 1 . . . NC − 2:

∂fm�oi

∂Xm′= fm�

oi

(1

Φm�oi

∂Φm�oi

∂Xm′

)+Φm�

oi Pδm,m′ (14.42)

For the normal hydrocarbon equations m = 1 . . . NC − 2, evaluate∂fm�

gi

∂P for m′ = 1 . . . NC − 2:

∂fm�gi

∂Ym′= fm�

gi

(1

Φm�gi

∂Φm�gi

∂Ym′

)+Φm�

gi Pδm,m′ (14.43)


∂fm�oi

∂Xm′= fm�

oi

(1

Φm�oi

∂Φm�oi

∂Xm′

)− Φm�

oi P (14.44)

For the CO2 equations m = 1 . . . NC − 1, evaluate∂fm�

gi

∂P for m′ = 1 . . . NC − 2:

∂fm�gi

∂Ym′= fm�

gi

(1

Φm�gi

∂Φm�gi

∂Ym′

)− Φm�

gi P (14.45)

14.14 Fugacity Equations - Above Bubble Point

Above the bubble point, Sg = 0. Sg is replaced by a new variable, the bubble point pressure

Pb.

GNC−1 = Pb −NC−1∑m=1

fom[Pb, �X ]

Φgm[Pb, �Y ](14.46)

Evaluate the derivative∂GNC−1

∂Pb:

∂GNC−1

∂Pb= 1−

NC−1∑m=1

1

Φvm[Pb, �Y ]

(∂flm[Pb, �X ]

∂Pb− flm[Pb, �X ]

Φvm

∂Φvm[Pb, �Y ]

∂Pb

)(14.47)

Evaluate the derivative for m′ = 1 . . . NC − 2, evaluate∂GNC−1

∂Xm′ :

∂GNC−1

∂Xm′= −

NC−1∑m=1

1

Φvm[Pb, �Y ]

(∂flm[Pb, �X ]

∂Xm′

)(14.48)


∂Ym′ :

274

∂GNC−1

∂Ym′=

NC−1∑m=1

(flm[Pb, �X ]

(Φvm)2

∂Φvm[Pb, �Y ]

∂Ym′

)(14.49)

14.15 Fugacity Equations - Below Dew Point

Above the bubble point, So = 0. So is replaced by a new variable, the dew point pressure Pd.

GNC−1 = Pd −NC−1∑m=1

fgm[Pd, �Y ]

Φom[Pd, �X ](14.50)

Evaluate the derivative∂GNC−1

∂Pb:

∂GNC−1

∂Pd= 1−

NC−1∑m=1

1

Φlm[Pd, �X ]

(∂fvm[Pd, �Y ]

∂Pd− fvm[Pd, �Y ]

Φlm

∂Φlm[Pd, �X ]

∂Pd

)(14.51)


∂Xm′ :

∂GNC−1

∂Xm′=

NC−1∑m=1

(fvm[Pd, �Y ]

(Φlm)2

∂Φlm[Pd, �X]

∂Xm′

)(14.52)


∂Ym′ :

∂GNC−1

∂Ym′= −

NC−1∑m=1

1

Φlm[Pd, �X ]

(∂fvm[Pd, �Y ]

∂Ym′

)(14.53)

14.16 Computation for Fixed Rate Wells

Each component equation Cw,α,m has a source term. The coefficient of δP is


(Xn

m,w,αξno,w,αλ

no,w,α + Y n

m,w,αξng,w,αλ

ng,w,α +W n

m,w,αξnw,w,αλ

nw,w,α

)(14.54)



(Xn

m,w,αξno,w,αλ

no,w,α + Y n

m,w,αξng,w,αλ

ng,w,α +W n

m,w,αξnw,w,αλ

nw,w,α

)(14.55)


275


(Xn

m,w,αξno,w,αλ

no,w,α + Y n

m,w,αξng,w,αλ

ng,w,α +W n

m,w,αξnw,w,αλ

nw,w,α

) ·((P �w,α

)−(P �,�w + Pw′,n

w,α

))(14.56)

Each well has a total rate equation. This equation has the following form for a fixed rate well.


QDPnw,α = −

(WI#w,α

)·(qemaxo,w,αξ

no,w,αλ

no,w,α

ξw,emaxo,w,αmax

+qemaxg,w,αξ

ng,w,αλ

ng,w,α

ξw,emaxg,w,αmax

+qemaxw,w,αξ

nw,w,αλ

nw,w,α

ξw,emaxw,w,αmax

)(14.57)


QDWnw,α =

αmax∑α′=1

(WI#w,α′

)·(qemaxo,w,α′ξno,w,α′λn

o,w,α′

ξw,emaxo,w,αmax


g,w,α′

ξw,emaxg,w,αmax


w,w,α′

ξw,emaxw,w,αmax

)

(14.58)


QCn�w,α =

RHS︷︸︸︷q�t,w +

αmax∑α′=1

(WI#w,α′

)·((

P �w,α′

)−(P �,�w + Pw,const @ n

w,α′

))·


o,w,α′

ξw,emaxo,w,αmax


g,w,α′

ξw,emaxg,w,αmax


w,w,α′

ξw,emaxw,w,αmax

)(14.59)

14.17 Computation for Fixed Pressure Wells


fixed pressure well. The coefficient of δP is 0.

WDPmnw,α = 0 (14.60)


WDWmnw,α =

(WI#w,α∑αmax


)×

(Xn

m,w,αξno,w,αλ

no,w,α + Y n

m,w,αξng,w,αλ

ng,w,α +W n

m,w,αξnw,w,αλ

nw,w,α

)(14.61)

276


WCmn�w,α = −

(WI#w,α∑αmax


)×

(Xn

m,w,αξno,w,αλ

no,w,α + Y n

m,w,αξng,w,αλ

ng,w,α +W n

m,w,αξnw,w,αλ

nw,w,α

)(14.62)


well. The coefficient of δP is

QDPnw,α =

(WI#w,α

)×(qemaxo,w,αξ

no,w,αλ

no,w,α

ξw,emaxo,w,αmax

+qemaxg,w,αξ

ng,w,αλ

ng,w,α

ξw,emaxg,w,αmax

+qemaxw,w,αξ

nw,w,αλ

nw,w,α

ξw,emaxw,w,αmax

)(14.63)


QDWnw,α = 1 (14.64)



αmax∑α′=1

(WI#w,α′

)×(P �w,α′ − Pw,n

w,α′

)×


o,w,α′

ξw,emaxo,w,αmax


g,w,α′

ξw,emaxg,w,αmax


w,w,α′

ξw,emaxw,w,αmax

)(14.65)

14.18 Additional Comments on Computation

This chapter contains additional information and illustrations written since the April 6, 2011

report. The first section describes various efforts we have already made to make the program

computationally efficient. There is more work to be done in several of these categories. The

second section shows graphical illustrations of the process of solving the linear system of equations

←→A �δ = �b. This linear system of equations is solved at each nonlinear iteration �, where �δ represents

the differences between nonlinear iteration �+1 and nonlinear iteration � for the primary variables.

Hopefully the new figures will provide a better understanding of the solution process.

14.19 Computational Efficiency

Several steps were taken to ensure efficient computations.

277

1. The appropriate physics were selected: the model assumes a constant temperature; the com-

positional formulation assumes that the mole fraction of the components other than CO2 and

H2O are not present in the aqueous phase; it is also assumed that H2O is not present in the

oleic phase or the vapor phase.

2. Each calculation was written in a computationally efficient way.

3. Nonlinear iterations are used in order to linearize the partial differential equations.

4. The matrix equations are written in terms of nonlinear differences rather than the variables

directly, δP = P �+1−P � rather than P �+1. This normalizes the units of the different primary

variables and acts as a pre-conditioner for the linear solver.

5. The well terms are eliminated to reduce the bandwidth of the matrix and to regularize the

sparsity structure of the matrix.

6. Local LU decomposition is conducted on the equations for each grid cell. This serves to

extract the largest eigenvalues of the sparse matrix and greatly reduce the size of the linear

system, from (2NC−1)∗Nxyz to between 1∗Nxyz and 3∗Nxyz depending on the formulation.

It also acts as another pre-conditioner.

7. For small models, where small is determined by memory requirements and computation time,

use a direct sparse parallel solver.

8. For larger models, use an iterative solver with a pre-conditioner. For example, GMRES with an

ILU(0) pre-conditioner or BICGSTAB with an ILU(0) pre-conditioner. Either of these solution

approaches may be faster depending on the model size and the number of iterations required.

14.20 Illustration of Solution Procedure

Two models are used for illustrations. A small model with Nx = 5, Ny = 5, and Nz = 3 is

used for most of the illustrations. A larger, but still very small model with Nx = 16, Ny = 16, and

Nz = 3 is used for some illustrations.

There are several other typical problem sizes, but they are too large to illustrate the full structure

of the problem at the resolution of the images. One typical problem has Nx = 80, Ny = 80, and

278

Nz = 15. For this system, Nxyz = 96000 and half bandwidth β = Ny ∗Nz +Nz + 1 = 1216. This

problem can easily be scaled up or down by keeping Nz fixed and varying Nx = Ny. Nz can also

be increased to 30, 45, or 60 layers. The typical number of components ranges from NC = 5 to

NC = 15, with NC = 8 components as the most common. This leads to block sizes of Nb = 2·NC−1from 9 to 29 with 15 most common. Some formulations for natural fracture systems can double

the block size, from 18 to 58 with 30 most common. If the model uses horizontal wells, there are

typically Nw = 3 horizontal wells aligned with the y-axis, with Nwc = Ny = 80 well completions

for each well. If the model uses vertical wells, there are typically Nw = 5 vertical wells arranged

in a 5-spot pattern aligned with the z-axis, with Nwc = Nz = 15 well completions for each well.

Another problem that has the same aspect ratio as this one has Nx = 16, Ny = 16, and Nz = 3.

This is used for some of the illustrations.

Another typical problem has Nx = 320, Ny = 320, and Nz = 15. For this system, Nxyz =

1563000 and half bandwidth β = Ny ∗Nz +Nz +1 = 4816. This problem can easily be scaled up or

down by keeping Nz fixed and varying Nx = Ny. Nz can also be increased to 30 or 45 or 60 layers.

The number of equations per grid cell vary in the same way as described above for the 80× 80× 15

model. If the model uses horizontal wells, there are typically Nw = 36 horizontal wells aligned

with the y-axis, with Nwc = Ny = 80 well completions for each well. If the model uses vertical

wells, there are typically Nw = 81 vertical wells aligned in 16 5-spot patterns with the z-axis, with

Nwc = Nz = 15 well completions for each well.

14.20.1 Illustration of a 5× 5× 3 Model, Well Geometry

This section illustrates the solution procedure for a small problem. The problem dimensions

were selected so that it is still possible to see the structure at the resolution of the images. The

system illustrated here has Nx = 5, Ny = 5, and Nz = 3. For this system, Nxyz = 75 and half

bandwidth β = Ny∗Nz+Nz+1 = 19. There are Nw = 3 horizontal wells aligned with the y-axis, see

Figure 14.8, with Nwc = 5 well completions for each well. There are NC = 5 components, leading to

a block size of Nb = 2 ·NC − 1 = 9. Unless otherwise specified, the formulation is IMPES (implicit

pressure, explicit saturation and composition), based on an 11-point finite difference scheme (9

points in the xy plane and 2 points in ±z).

279

Figure 14.8: Geometry of three horizontal wells for a 5× 5× 3 problem.

14.20.2 Illustration of a 5× 5× 3 Model, Block Values

Figure 14.9 shows the block banded matrix structure which is sent to the solver. The rest of this

section describes how this matrix is created. Each block on the main diagonal (red in Figure 14.9)

has the structure illustrated in Figure 14.10 for a NC = 5, Nb = 2NC − 1 = 9 problem. Each block

on the off-diagonal (blue in Figure 14.9) has the structure illustrated in Figure 14.11 for a NC = 5

problem. For the IMPSEC formulation, the off-diagonal blocks have the structure illustrated in

Figure 14.12 for a NC = 5 problem. The well terms for the component equations are represented

by Figure 14.13. The well equations are represented by Figure 14.14.

Figure 14.9: Matrix 0: block banded matrix for a 5 × 5 × 3 problem. Red cells are on the mainblock diagonal; blue cells are non-zero values off of the main block diagonal.

280

Figure 14.10: Block 1: block geometry for the main block diagonal of a NC = 5 problem. Blackrepresents non-zero values; gray represents zero values.


Figure 14.12: Block 3: block geometry for the off-block diagonal values with the IMPSEC for-mulation for a NC = 5 problem. Black represents non-zero values; gray represents zero values.

Figure 14.13: Block 4: well terms for the component equations for a NC = 5 problem. Blackrepresents non-zero values; gray represents zero values.

281

Figure 14.14: Block 5: blocks for the well equations for a NC = 5 problem. Black representsnon-zero values; gray represents zero values.

14.20.3 Illustration of a 5× 5× 3 Model, Matrix Assembly

The spatial derivatives are illustrated in Matrix 1, Figure 14.15. The time derivatives of the

accumulation term are illustrated in Matrix 2, Figure 14.16. When Matrix 1 and Matrix 2 are added

together, they yield Matrix 3, Figure 14.17. Matrix 4 shows the well coefficients, Figure 14.18.

Matrix 5 shows Matrix 3 combined with Matrix 4, Figure 14.19. Matrix 6 shows the results of

eliminating the q�w or P �well well terms from the component equations, Figure 14.20. This generates

some terms which are not on the block-banded structure of Matrix 3. Matrix 7 shows the results

of eliminating the off-band well terms from the component equations, Figure 14.21. The off-band

terms are eliminated using δP �−1.

14.20.4 Illustration of a 5× 5× 3 Model, Local LU Decomposition

One way to simplify the matrix solve for the system described in Figure 14.20 is to perform a

local LU decomposition on the equations for each grid cell and then extract the upper left corner of

each block. If LU with partial pivoting is used, this corresponds to extracting the largest eigenvalue

for each grid cell. This local LU decomposition operates on Matrix 7 one block-row at a time.

Figure 14.22 shows one row extracted from Matrix 7, Figure 14.21, called Row 1 for this discus-

sion. Figure 14.23 shows the results of removing the zero blocks from Figure 14.22. Figure 14.24

shows the results of removing the zero columns from Figure 14.23. Figure 14.25 shows the way Row

1 is stored for the local LU decomposition, with the main block diagonal values on the left followed

by the off-diagonal terms. Figure 14.26 shows the result of applying local LU decomposition to Row

1 as they are stored in memory. Figure 14.27 shows the result of applying local LU decomposition

to Row 1, showing only the non-zero blocks in the same format as Figure 14.23. The red and blue

values in Figure 14.26 and Figure 14.27 are used for the global matrix solution.

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Figure 14.15: Matrix 1: Spatial derivatives for a 5× 5× 3× 9 problem. Black represents non-zerovalues; gray represents zero values within each block; white represents zero values.

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Figure 14.16: Matrix 2: Time derivatives for a 5 × 5 × 3 × 9 problem. Black represents non-zerovalues; gray represents zero values within each block; white represents zero values.

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Figure 14.17: Matrix 3: Combined matrix for a 5 × 5 × 3 × 9 problem. Black represents non-zerovalues; gray represents zero values within each block; white represents zero values.

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Figure 14.18: Matrix 4: Well matrix for a 5× 5× 3× 9 problem with three horizontal wells. Blackrepresents non-zero values; gray represents zero values within each block; white represents zerovalues.

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Figure 14.19: Matrix 5: Combined matrix with wells for a 5 × 5 × 3 × 9 problem with threehorizontal wells. Black represents non-zero values; gray represents zero values within each block;white represents zero values.

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Figure 14.20: Matrix 6: Eliminate the q�w or P �well well terms from the component equations for a

5× 5× 3× 9 problem with three horizontal wells. Black represents non-zero values; gray representszero values within each block; white represents zero values.

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Figure 14.21: Matrix 7: Eliminate the off-band well terms from the component equations for a5× 5× 3× 9 problem with three horizontal wells. Black represents non-zero values; gray representszero values within each block; white represents zero values.

Figure 14.22: Row 1: An example of a row without well terms for a 5 × 5× 3× 9 problem. Blackrepresents non-zero values; gray represents zero values within each block; white represents zerovalues.

Figure 14.23: The non-zero blocks of Row 1. Black represents non-zero values; gray represents zerovalues within each block.

Figure 14.24: The non-zero columns of Row 1. Black represents non-zero values; gray representszero values within each block.

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Figure 14.25: The non-zero columns from Row 1 are stored with the main block diagonal first.Black represents non-zero values; gray represents zero values within each block.

Figure 14.26: The result of local LU decomposition on Row 1 in the order they are stored. Blackrepresents non-zero values; gray represents zero values within each block; red values are extractedfrom the main block diagonal; blue values are extracted from the off-diagonal.

14.20.5 Illustration of a 5× 5× 3 Reduced Model

After the local LU decomposition, Matrix 7, Figure 14.21 becomes Matrix 8, Figure 14.28. The

direct solvers do not require any additional steps, but for the iterative solvers it may be a good idea

to eliminate the well values, green in Figure 14.28. For vertical wells, this will often be worthwhile.

For horizontal wells, it may be worthwhile or it may be better to evaluate wells at � rather than at

n+ 1. After elimination, this results in Matrix 9, Figure 14.29.

14.20.6 Illustration of a 16× 16× 3 Model

This illustration has Nx = 16, Ny = 16, and Nz = 3. For this system, Nxyz = 768 and half

bandwidth β = Ny ∗Nz +Nz + 1 = 52. This problem was selected because it has the same aspect

ratio as a typical problem with Nx = 80, Ny = 80, and Nz = 15. This model has Nw = 3 horizontal

wells aligned with the y-axis, with Nwc = Ny = 16 well completions for each well, Figure 14.30.

Matrix 10, Figure 14.31 shows the banded matrix for this system with the well terms. Figure 14.32

zooms in on the upper left corner of Matrix 10, since it is hard to see the well terms in Figure 14.31.

Figure 14.27: The result of local LU decomposition on Row 1. Black represents non-zero values;gray represents zero values within each block; red values are extracted from the main block diagonal;blue values are extracted from the off-diagonal.

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Figure 14.28: Matrix 8: banded matrix for a 5×5×3 problem without eliminating wells. Red cellsare on the main block diagonal; blue cells are non-zero values off of the main block diagonal; greencells are the result of wells.

Figure 14.29: Matrix 9: banded matrix for a 5 × 5 × 3 problem. Red cells are on the main blockdiagonal; blue cells are non-zero values off of the main block diagonal.

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Matrix 11, Figure 14.33 shows the banded matrix for this system without the well terms. This is

the form that is typically solved.

Figure 14.30: Geometry of three horizontal wells for a 16× 16× 3 problem.

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Figure 14.31: Matrix 10: banded matrix for a 16× 16× 3 problem without eliminating wells. Redcells are on the main block diagonal; blue cells are non-zero values off of the main block diagonal;green cells are the result of wells.

Figure 14.32: Upper left corner of Matrix 10. Red cells are on the main block diagonal; blue cellsare non-zero values off of the main block diagonal; green cells are the result of wells.

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Figure 14.33: Matrix 11: banded matrix for a 16× 16× 3 problem. Red cells are on the main blockdiagonal; blue cells are non-zero values off of the main block diagonal.

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CHAPTER 15

COMPUTATION: DESCRIPTION OF LINEAR SOLVERS

Because in our simulator the linear solver lies within the inner most loop, the quality of the

simulator depends heavily on the quality of the linear solvers. This report analyzes various linear

solvers and evaluates their computation complexity, accuracy, robustness, memory requirement

and scalability in parallel environment. An extremely small test case with NX = 3, NY = 1, and

NZ = 1 is used to illustrate these solvers.

Figure 15.1: Jacobian matrix for a 3× 1× 1 system with NC = 5 and Nblock = 9. Black representsnon-zero values; gray represents zero values within the block diagonals; white represents other zerovalues.

15.1 Serial Solvers

Three serial solvers are currently proposed and implemented: dense Gaussian elimination, band

Gauss elimination and a special LU solver.

15.1.1 Dense Gaussian Elimination

This is the simple LU with partial pivoting based linear solver, with the matrix in a traditional

2-D array dense form. It corresponds to the DGESV subroutine of the high quality LAPACK

package. This solver is stable and has been used for decades in scientific computing. The error

source in this case is only roundoff error, which is nicely bounded most of the time in practice.

However, since the matrix we are dealing with is highly structured and sparse, it’s a huge waste of

computation and storage to store a such sparse matrix in a dense form.

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The computation complexity of dense Gaussian elimination solver is around 486 · n3xyz. The

memory requirement is 81 · n2xyz.

The solution procedure starts with the Jacobian matrix in Figure 15.1. The first stage of

Gaussian elimination creates zeroes above the diagonal, Figure 15.2. The second stage of Gaussian

elimination creates zeroes below the diagonal, Figure 15.3, resulting in a solution for each unknown.

Figure 15.2: The test case after the first stage of Gaussian elimination. The above diagonal valuesare eliminated first by column and then by row, moving from lower left to upper right. Blackrepresents non-zero values; gray represents zero values within the block diagonals; white representsother zero values; cyan represents zero values that have been created.

Figure 15.3: The test case after the second stage of Gaussian elimination. The below diagonalvalues are eliminated by back substitution, first by column and then by row, moving from upperleft to lower right. Black represents non-zero values; gray represents zero values within the blockdiagonals; white represents other zero values; cyan represents zero values that have been created.

15.1.2 Band Gaussian Elimination

A slightly better format to store the matrix band storage format, which then can be solved by

LAPACK DGBSV subroutine. Since in our matrix the furthest subdiagonal or superdiagonal is

9·nz ·ny away from main diagonal, it’s meaningful to arrange the matrix into a band format(despite

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that it’s still quite sparse in band format) which ignores all left-bottom and right-top zeros. The

algorithm and accuracy behavior is similar to dense Gaussian elimination but the computation and

storage requirements are greatly reduced.

The computation complexity of band Gaussian elimination solver is around 486 · n2y · n2

z · nxyz.

The memory requirement is 27 · ny · nz · nxyz.

The solution procedure starts with the Jacobian matrix in Figure 15.1. The matrix is stored

based on diagonals as shown in Figure 15.4. The first stage of Banded Gaussian elimination creates

zeroes above the diagonal, Figure 15.5. The second stage of Banded Gaussian elimination creates

zeroes below the diagonal, Figure 15.6, resulting in a solution for each unknown.

Figure 15.4: The banded structure for the test case. Everything not in purple is stored in memoryby diagonal and manipulated by the band solver. Black represents non-zero values; gray representszero values within the block diagonals; white represents other zero values; purple represents zerovalues outside of the bandwidth.

Figure 15.5: The test case after the first stage of Banded Gaussian elimination. The above diagonalvalues are eliminated first by column and then by row, moving from lower left to upper right. Blackrepresents non-zero values; gray represents zero values within the block diagonals; white representsother zero values; purple represents zero values outside of the bandwidth; cyan represents zerovalues that have been created.

297

Figure 15.6: The test case after the second stage of Banded Gaussian elimination. The belowdiagonal values are eliminated by back substitution, first by column and then by row, moving fromupper left to lower right. Black represents non-zero values; gray represents zero values within theblock diagonals; white represents other zero values; purple represents zero values outside of thebandwidth; cyan represents zero values that have been created.

15.1.3 Special Gaussian Elimination

For our particular matrix structure there’s a trick which essentially LU decomposes each block

and generate a substantially more compact linear system. After solving the compact system a

cheap back substitution will solve the whole system.

Suppose we are using the band solver for the compact system and standard back substitution,

the computation complexity is (1440 + 2ny · nz) · nxyz, and the memory storage requirement is

(153 + 2ny · nz) · nxyz.

Another nice property is that the compact system has a diagonal dominant matrix which means

to solve it the LU decomposition process does not need any pivoting - a great potential for perfor-

mance improvement since pivoting involves communication which degrades performance consider-

ably, especially for parallel computations.

The solution procedure starts with the Jacobian matrix in Figure 15.1, but because this solution

procedure utilizes the block structure of the matrix, it is stored as Figure 15.7.

First, conduct a local LU decomposition on each grid cell, reducing the system from Nblock = 9

to Nblock = 1; Figure 15.8. Next, extract the upper left corner of each block and assemble a

new, condensed matrix, Figure 15.9. For a condensed banded solve, store the matrix diagonals,

Figure 15.10. The first stage of Banded Gaussian elimination creates zeroes above the diagonal,

Figure 15.11. The second stage of Banded Gaussian elimination creates zeroes below the diagonal,

Figure 15.12, resulting in a solution for the pressures. Using the results of the condensed matrix

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Figure 15.7: The sparse storage structure of the local LU solvers. Everything in purple are zerovalues that are not stored explicitly. Black represents non-zero values; gray represents zero valueswithin the block diagonals; white represents other zero values.

solution for pressures, locally back substitute in each grid cell to obtain solutions for the other

primary variables, Figure 15.13.

Figure 15.8: The first step of the local LU solvers. Everything in purple are zero values that arenot stored explicitly. Black represents non-zero values; gray represents zero values within the blockdiagonals; white represents other zero values; cyan represents zero values that have been created;red represents values on the main diagonal to be extracted; blue represents values off of the maindiagonal to be extracted.

15.1.4 Summary

From the above discussion we can see that the special Gaussian elimination has much less

computation complexity and memory requirements, as shown in Table 15.1. There is a large piece

of memory associated with the Jacobian calculations for each grid cell.

Table 15.1: Computation and memory requirement for 3 different solvers

Jacobian Dense Gauss Band Gauss Special Gauss

Computation 486 · n3xyz 486 · n2

y · n2z · nxyz (1440 + 2ny · nz) · nxyz

Memory 608 · nxyz 81 · n2xyz 27 · ny · nz · nxyz (153 + 2ny · nz) · nxyz

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Figure 15.9: The condensed matrix after the local LU decomposition. Red represents values onthe main diagonal that were extracted from the full matrix; blue represents values off of the maindiagonal that were extracted from the full matrix; white represents other zero values.

Figure 15.10: The banded structure of the condensed matrix. Everything not in purple is storedin memory by diagonal and manipulated by the band solver. Red represents values on the maindiagonal that were extracted from the full matrix; blue represents values off of the main diagonalthat were extracted from the full matrix; purple represents zero values outside of the bandwidth.

Figure 15.11: The condensed matrix after the first step of the band solve. The above diagonalvalues are eliminated first by column and then by row, moving from lower left to upper right. Redrepresents values on the main diagonal that were extracted from the full matrix; blue representsvalues off of the main diagonal that were extracted from the full matrix; purple represents zerovalues outside of the bandwidth; cyan represents zero values that have been created.

Figure 15.12: The condensed matrix after the second step of the band solve. The below diagonalvalues are eliminated by back substitution, first by column and then by row, moving from upperleft to lower right. Red represents values on the main diagonal that were extracted from the fullmatrix; blue represents values off of the main diagonal that were extracted from the full matrix;purple represents zero values outside of the bandwidth; cyan represents zero values that have beencreated.

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Figure 15.13: Use the values from the condensed matrix solve to perform a back substitution oneach grid cell. Everything in purple are zero values that are not stored explicitly. Black representsnon-zero values; gray represents zero values within the block diagonals; white represents otherzero values; cyan represents zero values that have been created; red represents values on the maindiagonal that were extracted and solved in the condensed version; blue represents values off of themain diagonal that were extracted and solved in the condensed version.

When the problem scales(nx, ny, nz gets bigger) we can see that the special Gaussian elimination

approach is far more favorable and thus should be used in practice. It’s also worth noting that

these 3 solvers generally solves the linear system with almost the same accuracy subject to roundoff

errors.

15.2 Parallel Solvers

When going into parallel the choices of solvers become much more subtle, mainly because the

computation and memory requirements can not be as easily determined as in serial case. When we

are in the parallel realm we must be solving a much bigger problem which means a dense or band

matrix format are simply too expensive; we need a sparse matrix format. This not only changes the

algorithms and storage scheme, it also induces some uncertainty as to computation and memory

requirements. Even a direct solver on sparse format matrix(like LU solver) will have different

computation and memory behavior depending on the input matrix and the specific implementation

of the algorithm. More oftentimes, an iterative solver is desired because a direct solver may not scale

well. Iterative solvers are especially hard to predict its execution time and memory consumption

because the number of steps it requires to converge is not known before actually executing it.

From the discussion of the serial solver section we are convinced that the special Gaussian

elimination should be used. We thus only focus on solving the compact system(which can yet be

very large in bigger problems) since this part is the dominant computation task in that approach.

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15.2.1 Direct LU Solver

Since the compact matrix is highly structured and sparse and diagonal dominant, there’s a good

chance a direct sparse LU solver(like SuperLU) can perform quite well in this case. However, it

still remains to be seen whether direct solvers are appropriate for our problem.

A direct solver is generally very robust and stable like the serial solvers discussed earlier; but its

performance depends heavily on the matrix and the implementation. We propose to use SuperLU

and UMFPACK to test the feasibility to use a direct solver.

15.2.2 Iterative Solvers

Many iterative methods exist for solving large sparse systems. Typically iterative methods enjoy

a relatively low memory footprint and are more scalable than direct sparse solvers. The trick is

how to find the most efficient method for the problem at hand. Unfortunately no universal method

works well for all problems, thus insights into the problem and iterative methods are required.

Another issue is that iterative methods often depends on preconditioner to be effective. The choice

of preconditioners, again, is subtle and no universal scheme proves to work for all problems. All

this requires deeper understanding of the problem and algorithms.

For a starting point we propose to use BICGSTAB or GMRES with ILU or block Jacobi

preconditioner.

15.2.3 Parallel Framework

PETSc is the planed framework that we are going to work with developing the parallel simu-

lator. PETSc includes an expanding suite of parallel linear, nonlinear equation solvers and time

integrators that may be used in application codes written in Fortran, C, C++, Python, and MAT-

LAB (sequential). PETSc provides many of the mechanisms needed within parallel application

codes, such as parallel matrix and vector assembly routines. The library is organized hierarchically,

enabling users to employ the level of abstraction that is most appropriate for a particular prob-

lem. By using techniques of object-oriented programming, PETSc provides enormous flexibility for

users.

The PETSc package provides a infrastructure for our parallel programming, and it provides all

our proposed solvers built-in which makes it an ideal platform for developing and testing various

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approaches.

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CHAPTER 16

COMPUTATATION: PARALLEL COMPUTING

The following is an outline of the overall parallel solution approach. For each of the steps,

the amount of expected parallelism is described, including a reference size and an estimate of

the memory, computation time, and communication time if applicable. These sizes have various

constant multipliers, some of which can be quite large. There are also lower order polynomial terms

not represented by the O notation. The combination of all of all these steps is memory limited.

The problem has been demonstrated by Saudi Aramco (Dogru et al., 2008) to be scalable for sizes

up to Nxyz = 109. At Saudi Aramco they have spent approximately 150 man-years developing

Powers, so I would not expect my results in a few months to be as good as theirs3.

The total number of grid cells is Nxyz = Nx ×Ny ×Nz. The bandwidth β is used the banded

solver algorithms; for a system with Nx = Ny > Nz, β = Nx×Nz. The total number of components,

NC , includes the hydrocarbon components, CO2 component, and H2O component. The number of

processing nodes, Nn, represents the number of different machines involved in the computations.

The number of processing cores on each node is Np. The total number of cores on all nodes is Nnp.

Message Passing Interface (Gabriel, Fagg, Bosilca, Angskun, Dongarra, Squyres, Sahay, Kam-

badur, Barrett, Lumsdaine, Castain, Daniel, Graham, and Woodall, 2004; Wikipedia, 2010b,c,

MPI, ), is a language independent communications protocol for parallel computers, including both

shared and distributed memory computers. Open Multi Processing (OpenMP, 2008; Wikipedia,

2010d, OpenMP, ), is a different language independent communications protocol applicable only

to shared memory computers. The two can be combined in a hybrid MPI/OpenMP framework,

using the MPI interface to communicate between nodes and the OpenMP interface to communi-

cate between processors on the same node. For an MPI parallel implementation, all computations,

communication, and memory are divided among Nnp nodes. For a hybrid MPI/OpenMP imple-

mentation, memory and communication are among Nn nodes. Computations are divided among

Nn nodes and then further divided among Np processing cores.


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16.1 Computation Grid

The matrix coefficients are defined using a grid of parallel processors. The following figures use

Nx = Ny = 100, Nn = 9, Np = 8, and Nnp = 72. Figure 16.1 shows a grouping of 9 nodes with

8 cores. Figure 16.2 shows the hybrid approach, with a 3 × 3 × 8 grid of processors. Figure 16.3

shows the pure MPI approach, with a 9× 8 grid of processors. The matrix solver requires a linear

processor array of all 72 processors, Figure 16.4.

Figure 16.1: Illustration of a group of 9 nodes with 8 processor cores each.

Figure 16.2: Illustration of computations with a hybrid MPI/openMP 3× 3× 8 processor grid.

The normal boundary computations are illustrated in Figure 16.5. Figure 16.6 shows how this

applies to a 3× 3 processor grid, and Figure 16.7 shows how this applies to a 9× 8 processor grid.

Because there are normally a lot of computations for each grid cell for compositional simulation, it

is normally better to compute them on one processor and then send them to the adjacent processor

rather than to make the computations twice.

When the number of grid cells on each processor are not identical, it may be beneficial to

subcontract the grid cells to another processor. Figure 16.8 shows how these computations may be

305

Figure 16.3: Illustration of computations with an MPI 9× 8 processor grid.

Figure 16.4: Illustration of computations with a linear array of 72 processors.

central area central area

inner border inner borderouter border outer border

Figure 16.5: Parallel boundary computations.

306

Figure 16.6: Parallel boundary computations for a 3× 3 processor grid.

Figure 16.7: Parallel boundary computations for a 9× 8 processor grid.

307

subcontracted. Using Nx = Ny = 100 and Nn = 9, the processor grid is a 3× 3 array, Figure 16.2.

If the load on each processor were completely balanced, then there would be 1111.11 2-D grid cells

in on each processor. Using the most efficient grid of processors, there will be one 34× 34 = 1156,

four 34× 33 = 1122, and four 33× 33 = 1089. Using Nx = Ny = 100 and Nnp = 72, the processor

grid is a 9×8 array, Figure 16.3. If the load on each processor were completely balanced, then there

would be 138.89 grid cells on each processor. Using the most efficient grid of processors, there will

be four 13× 12 = 156, four 12× 12 = 144, thirty-two 13× 11 = 143, and thirty-two 12× 11 = 132.

Subcontracting for load balancing means that the computations for some of the grid cells on the

nodes with more cells are sent to the nodes with fewer processors.

central area central area

inner border inner borderouter border outer border

subcon

tract

Figure 16.8: Parallel computations for load balancing.

16.2 Solution Steps

The following steps are involved in the solution procedure.

1. Initialization - calculate the grid geometry and distribute the data among the processors

• Reference size α = Nxyz

• Computation: O [α]; computation time: some parts O [α/Nnp], other parts O [α] de-

pending on details of implementation

• Memory: O [α]; local memory: O [α/Nn]

• Communication (one-to-many): O [α]; communication time: O [α log2 Nnp] for the MPI

approach and O [α log2 Nn] for the hybrid approach

2. Start of time step n or nonlinear iteration �

308

3. Calculating the coefficients of the matrix equation that depend only on the local grid cell

(block-diagonal terms)

• Reference size α = 7 · (2NC − 1)2 ·Nxyz, using a 7-point finite difference stencil

• Computation: (big constant) ∗ O [α]; computation time: (big constant) ∗ O [α/Nnp]


• Communication is only required if subcontracting is required for load balancing. Assume

5% of the cells require subcontracting for balancing.

• Communication (one-to-one): O [0.05 · α]; communication time: O [0.05 · α/Nnp] for the

MPI approach and O [0.05 · α/Nn] for the hybrid approach

4. Perform local LU decomposition to transform the fully implicit matrix into an IMPES or

IMPSEC matrix. (Press, Teukolsky, Vetterling, and Flannery, 2007)

• Reference size α = 7 · (2NC − 1)2 ·Nxyz, using a 7-point finite difference stencil

• Computation: O [(2NC − 1)α]; computation time: O [(2NC − 1)α/Nnp]


5. Calculating the off block-diagonal coefficients of the matrix equation

• Reference size: α = 4β√

Nnp; hybrid reference size: α = 4β√Nn

• Computation: O [α]; computation time: O [α/Nnp]


• Communication (one-to-one): O [α]; communication time: O [α/Nnp] for the MPI ap-

proach and O [α/Nn] for the hybrid approach

6. Set up the matrix solver

• Reference size: IMPES α = 7Nxyz ; IMPSEC: α = 7 · 32 ·Nxyz

• Memory: O [2α]; local memory: O [2α/Nn]

• Communication (many-to-many): O [α]; communication time: O [α log2Nnp] for the

MPI approach and O [α log2 Nn] for the hybrid approach

309

7. Perform matrix solve; (Gauss: Press et al. (2007), Banded: Arbenz, Cleary, Dongarra, and

Hegland (2001); Cleary and Dongarra (1997))

• Reference size, IMPES α = Nxyz; IMPSEC α = 3 ·Nxyz; banded IMPES β = Nx ·Nz;

banded IMPSEC β = 2 · 3 ·Nx ·Nz

• Gauss computation: O [α3]; computation time: O [α3/Nnp

]; IMPES = 27 · IMPSEC

• Gauss memory: O [α2]; local memory: O [α2/Nnp

]; IMPES = 9 · IMPSEC

• Gauss communication: O [α2]; communication time: O [α2 log2 Nnp

]; IMPES = 9 ·

IMPSEC

• Banded computation: O [β2α]; computation time: O [β2α/Nnp

]; IMPES = 108·IMPSEC

• Banded memory: O [βα]; local memory: O [βα/Nnp]; IMPES = 18 · IMPSEC

• Banded communication: O [βα]; communication time: O [βα log2 Nnp]; IMPES = 18 ·IMPSEC

8. Transfer results of matrix solve back to grid cells

• Reference size: IMPES α = Nxyz; IMPSEC: α = 3 ·Nxyz

• Memory: O [2α]; local memory: O [2α/Nn]

• Communication (many-to-many): O [α]; communication time: O [α log2Nnp] for the

MPI approach and O [α log2 Nn] for the hybrid approach

9. local LU back substitution

• Timing already accounted for in Item 4

10. For each new time step or nonlinear iteration, go back to Item 2

16.3 Initialize

Initialization - calculate the grid geometry and distribute the data among the processors

• Reference size α = Nxyz

• Computation: O [α]; computation time: some parts O [α/Nnp], other parts O [α] depending

on details of implementation

310


• Communication (one-to-many): O [α]; communication time: O [α log2 Nnp] for the MPI ap-

proach and O [α log2 Nn] for the hybrid approach

Some of the properties need to be distributed to all nodes. Others will be stored only on the

appropriate node. There are some simple initialization parameters that have to be distributed

to all the processors, such as Nx, Ny, Nz, and NC . If grid-based properties, such as porosity φ

and permeability k, are constants or vary only with z, then these need to be distributed to all

processors as well. This type of distribution is best handled by MPI_BROADCAST, which has total

communication Nnp log2Nnp or communication timing log2Nnp. Fortunately, this initialization

only needs to occur once at the beginning of the run.

If grid-based properties vary in 3-D, then only the portion of the grid that lives on each processor

needs to be distributed or it needs to be read locally from a file on that node. Wells only need to

live the specific processor that contains the well. If all the data of a particular kind is loaded on

one processor, then this communication is best handled by MPI_SPLIT.

Information relating to variable time step size needs to be distributed to all processors or

calculated locally on each processor. This needs to happen with every time step. Determination of

convergence also needs to happen across all processors.

16.4 Scalability

Scalability evaluations start with the evaluation of the computation time, communication time,

and memory demands for an algorithm on various numbers of processors for various sizes of prob-

lems. The following variables are used in this description:

• O [x]: computational order of x.

• Coff [x]: off-node communication time for Ra for data size x.

• Nx: number of grid cells in x-direction.

• Ny: number of grid cells in y-direction.

• Nz: number of grid cells in z-direction.

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• Nxyz: total number of grid cells.

• β: bandwidth for banded solver

• P : number of processing cores

• n: number of time steps

• �: average number of nonlinear iterations for each time step

• COMPT1: the computation time for a single processor

• COMPT1/P :T1P ; used because it is part of the efficiency calculations and the total TP .

• COMPTP : the computation time for multiple processors

• COMMTP : the communication time for multiple processors

• MP : the memory requirement for each processor for a model

16.4.1 Computation Magnitude

Two clusters at Colorado School of Mines were used as part of this dissertation: RA and MIO.

The following problems are illustrated using Nx = Ny, and β = Nx×Nz. It’s based on properties of

RA thin nodes: 16GB per 8-core node. Values of the coefficients are estimated based on theoretical

calculations and some timing estimates on MIO. Additional calculations are necessary on MIO and

RA. The computation order for a single processor, written as T1/P = T1P :

COMPT1/P = O [Nxyz] +O[Nxyz

P

]+O

[nNxyz

P

]+O

[n�

Nxyz

P

]+O

[n�

β2Nxyz

P

]

≈ 104·O [Nxyz]+103·O[Nxyz

P

]+15∗103·O

[nNxyz

P

]+5∗103·O

[n�

Nxyz

P

]+8·O

[n�

β2Nxyz

P

](16.1)

Rewriting (16.1) using some estimated coefficients, n = 200, and � = 5, for NC = 8 components,

(16.1) becomes (16.2).

COMPT1/P = 104 · O [Nxyz] + 8 ∗ 106 · O[Nxyz

P

]+ 8 ∗ 103 · O

[β2Nxyz

P

](16.2)

The computation order for P processors is

312

COMPTP = COMPT1/P +O[n

β√P

]+O

[n�

β√P

]+O [n�β3 log2[P − 1]

]≈ COMPT1/P + 2000 · O

[n

β√P

]+ 100 · O

[n�

β√P

]+ 10 · O [n�β3 log2[P − 1]

](16.3)

Using estimated constants in (16.3):

COMPTP = COMPT1/P + 5 ∗ 105 · O[

β√P

]+ 104 · O [β3 log2[P − 1]

](16.4)

The communication order for P processors, using the communication time for each transmission

of N double precision numbers as COMM = ts +Ntp.

COMMTP = O [log2[P − 1]] +O[Nxyz

P

]∗ P +O

[β√P

]∗√P ∗ n+

O[

β√P

]∗√P ∗ n�+O

[Nxyz

P

]∗ P ∗ n�+O

[β2

P

]∗ P log2[P − 1] ∗ n�

≈ Coff[103] ∗ log2[P − 1] + Coff

[102

Nxyz

P

]∗ P + 4 ∗ Coff

[15

β√P

]∗√P ∗ n+

4∗Coff[

β√P

]∗√P∗n�+Coff

[8Nxyz

P

]∗P∗n�+Coff

[Nxyz

P

]∗P∗n�+4∗Coff

[β2

P

]∗P log2[P−1]∗n�

(16.5)

Using estimated constants in (16.5), using the bandwidth computations from Figure 16.9.

COMMTP = Coff[103] ∗ log2[P − 1] + Coff

[102

Nxyz

P

]∗ P+

800 · Coff[15

β√P

]∗√P + 4000 · Coff

[β√P

]∗√P + 1000 · Coff

[8Nxyz

P

]∗ P+

1000 · Coff[Nxyz

P

]∗ P + 4000 · Coff

[β2

P

]∗ P log2[P − 1] (16.6)

The total memory required for each processor in a system using P processors is defined by:

313

Figure 16.9: Ra bandwidth.

MP = O[Nxyz

P

]+O

[βNxyz

P

]+O

[β√P

]+O

[Nxyz

P

]+O

[β2

P

]

≈ 1500 · O[Nxyz

P

]+O

[βNxyz

P

]+ 120 · O

[β√P

]+ 4 · O

[β2

P

](16.7)

Using estimated constants in (16.7) yields the following in gigabytes.

MP = 8 · 10−9 ·(104 + 1500 · O

[Nxyz

P

]+O

[βNxyz

P

]+ 120 · O

[β√P

]+ 4 · O

[β2

P

])(16.8)

16.4.2 Analysis

The efficiency EP is defined by (16.9). Figure 16.10 shows the efficiency versus the number of

processors for a model with 80×80×15 grid cells, using (16.9) with the constants in (16.2), (16.4),

and (16.6).

EP =T1

P · TP=

T1/P

TP=

COMPT1/P

COMPT1/P + COMPTP + COMMTP(16.9)

314

2 5 10 20

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

P

Ep

Efficiency for 80x80x15 Model

Figure 16.10: Efficiency plot for Nx = 80, Ny = 80, and Nz = 15.

The speedup SP is defined by (16.10). Figure 16.11 shows the speedup versus the number of

processors for a model with 80×80×15 grid cells, using (16.9) with the constants in (16.2), (16.4),

and (16.6). A typical rule of thumb is to use the number of processors at the inflection point. For

this case, that would be between 100 and 400 cores, or 10 to 50 nodes on Ra.

SP =T1

TP=

T1/P · PTP

= EP · P (16.10)

2 5 10 202

3

4

5

6

7

8

9

P

Sp

Speedup for 80x80x15 Model

Figure 16.11: Speedup plot for Nx = 80, Ny = 80, and Nz = 15.

The number of processors for efficiency EP = 0.1 as a function of the number of grid cells Nxyz

is shown in Figure 16.12. This uses a model with Nx × Nx × 15 grid cells, with the constants in

(16.2), (16.4), and (16.6). Since there exists a number of processors P [Nxyz] for all numbers of cells

315

Nxyz, this illustrates that the algorithms are scalable. The plot has a similar shape for all values

of EP .

104 105 106 107 108 109 101010

100

1000

104

105

Nxyz

P

Number of Processors for Ep�0.1

Figure 16.12: Scalability plot for Nx = Ny, Nz = 15, and EP = 0.1.

Figure 16.13 shows the memory constrained scalability for a model with Nx×Nx×15 grid cells

with the constants in (16.2), (16.4), and (16.6).

316

104 105 106 107 1081

5

10

50

100

500

1000

Nxyz

Nod

es

Memory Constrained Scaling for Ep�0.1

Figure 16.13: Memory constrained scalability plot. The red line shows the upper limit of applica-bility of the banded solver. The purple line shows the minimum number of processors required forthe memory needs. The green line shows the maximum number of processors for EP ≥ 0.1. Thedashed black line shows the maximum number of processors for EP ≥ 0.01. The solid black lineshows the maximum number of processors for EP ≥ 0.50. The blue shading shows the valid regionfor EP ≥ 0.1.

317

CHAPTER 17

VALIDATION CASES

This chapter describes a series of comparison cases with GEM, a commercial compositional

simulator by Computer Modeling Group. Roughly a hundred models comparing CMG and my

code were run, including 1-D homogeneous, 2-D homogeneous, 2-D heterogeneous 5-spot models

based on a Middle East field, and 2-D heterogeneous 5-spot models of a fluvial system. Nine models

were selected to present here.

Before running the cases described here, the initialization of the simulations and the status after

a one-day time step were compared. The following items matched exactly between the two models

without any adjustments:

• The initial cell volumes, pore volumes, porosities, pressures, mole fractions, and saturations

were the same.

• The base relative permeability curves without trapping or hysteresis were the same.

• The base capillary pressure curves without trapping or hysteresis were the same.

After evaluating the initialization and the simulations after a one-day time step, several modi-

fications were made to make the simulations more comparable.

• EOS was modified so my model had identical properties for CO2, CH4, nC4, and nC10 as

the default pure-component properties in GEM. Turn off volume shift since the two codes

calculate it differently.

• Oil and gas viscosity model: implemented GEM’s viscosity model in my code since GEM does

not support LBC.

• Water viscosity and density model: implemented GEM’s water viscosity and density model

in my code.

• Well index: calculated well index in my code and then assigned GEM’s well index to this

calculated value.

318

• After adjusting the EOS, the initial moles in the system are off by less than 0.05%.

• After adjusting the EOS, the fugacities of each component are off by less than 0.1%.

Even after making the above adjustments to make the two simulators as comparable as possible,

there were still the following differences.

• Well constraints are the same but GEM’s algorithm to enforce mixed pressure and rate

boundary conditions are different than mine.

• GEM requires well grid cells to be fully implicit even if the rest of the model is IMPES. Over

time, GEM will add additional fully implicit grid cells. There also seem to be differences in

how the pressure is calculated for the production grid cells.

• GEM’s time stepping algorithm is different from mine.

• If my model fails to converge, it takes the value of the best nonlinear iteration and then

continues. If GEM fails to converge, it tries to reduce the time step. If after several reductions

in time step size it still hasn’t converged then the model stops completely.

• GEM calculates hysteresis differently than my code.

17.1 Validation Cases

All of the validation cases described here, Table 17.1, are 1-D homogeneous models with 101 grid

cells in the x-direction. Each model was 1000 ft×100 ft×44 ft, with each grid cell 10 ft×100 ft×44 ft.Initial reservoir pressure was 3850 psia with a reservoir temperature of 210◦F. The system has four

components, CO2, CH4, nC4, and nC10. System permeability is 200 md, system porosity is 17.2%.

CO2 solubility in water was set to zero to simplify the comparisons.

17.2 Description of model 760E

Model 760E is a 1-D model with primary production and no trapping or hysteresis. Since this

is a primary production case, the injection rate is 0. The production well is constrained initially

by a maximum rate of 100 RB/day, Figure 17.1. At about 1100 days the well switches from rate

control to bottom hole producing pressure control, Figure 17.2. The system is above the bubble

319

Table 17.1: Validation cases

Name Production Scenario Hysteresis and Trapping

760E primary production no hysteresis, no trapping761E primary production gas hysteresis, no trapping762E primary production gas hysteresis, compositional trapping

760F waterflood no hysteresis, no trapping761F waterflood gas hysteresis, no trapping762F waterflood gas hysteresis, compositional trapping

760G primary production then waterflood no hysteresis, no trapping761G primary production then waterflood gas hysteresis, no trapping762G primary production then waterflood gas hysteresis, compositional trapping

point until about 100 RB/day; there is a large bend in the pressure curve (Figure 17.2) as gas

production starts (Figure 17.1).

0 200 400 600 800 1000 1200 1400

0

20

40

60

80

100

t�day�

�q��R

BPD�

Production Rate �RBPD�

�qTOT�

�qo�

�qg�

�qw�

Figure 17.1: Production rates at reservoir conditions for model 760E. Black is total production inRB/day; green is oil production; red is gas production; blue is water production.

The average saturations in the reservoir are shown in Figure 17.3. For this model, the water

saturation stays approximately constant. The gas saturation increases as the pressure drops below

the bubble point and stabilizes when the well switches to pressure control.

As shown in Figure 17.4, the average mole fraction of methane decreases with time, the CO2

stays approximately constant, the mole fraction of nC10 increases, and the mole fraction of nC4

increases slightly. Figure 17.5 shows the recovery factor for each of the hydrocarbon components.

Methane has the highest recovery, followed by nC4 and nC10.

320

0 200 400 600 800 1000 1200 14000

1000

2000

3000

4000

t�day�

PB

HP,P

ores

Production Pressure �psia�

PBHP

Pcell

Figure 17.2: Production pressure for model 760E. Blue is bottom hole injection pressure from mymodel; orange is grid cell injection pressure for my model. At this scale, the blue and orange curvevisually overlay each other.

0 200 400 600 800 1000 1200 14000.0

0.2

0.4

0.6

0.8

1.0

t�day�

Tot

alS

Saturation for Equivalent One�Cell Model

SwM2

SwM1

SoM2

SoM1

SgM2

SgM1

Figure 17.3: Saturation for equivalent one-cell model for model 760E. For each time step, green isthe volume of oil in all the grid cells divided by the total volume of fluids in all the grid cells; blueis water and red is gas.

0 200 400 600 800 1000 1200 14000.0

0.2

0.4

0.6

0.8

1.0

t�day�

CH

4,n

C4,n

C10

,CO

2

Mole Fraction in Reservoir

CH4�total moles

nC4�total moles

nC10�total moles

CO2�total moles

Figure 17.4: Mole fraction for equivalent one-cell model for model 760E. For each time step, red isthe ratio of the moles of methane to the total number of moles; nC4 is in green; nC10 is in cyan;CO2 is in orange.

321

0 200 400 600 800 1000 1200 14000.0

0.2

0.4

0.6

0.8

1.0

t�day�

CH

4,n

C4,n

C10

Produced Fraction by Component

RF CH4

RF nC4

RF nC10

Figure 17.5: Molar recovery factor for model 760E. For each time step, red is the ratio of thecumulative produced moles of methane to the original number of moles of methane in the reservoir;nC4 is in green; nC10 is in cyan.


Model 761E is a 1-D model with primary production and gas hysteresis. Model 761E gives

almost identical results to model 760E; because the gas saturation is always increasing there is no

liquid phase to induce the trapping of gas. Different values of the critical gas saturation would

affect these cases.


Model 762E is a 1-D model with primary production with compositional trapping and gas

hysteresis.

The injection rates and pressures for model 762E are visually the same as model 760E. The

production rates and production pressures for model 762E are visually the same as model 760E as

shown in Figure 17.1 and Figure 17.2.

For model 762E, the total model water saturation stays approximately constant. The gas

saturation increases as the pressure drops below the bubble point and stabilizes when the well

switches to pressure control, Figure 17.6. Figure 17.6 is very similar to Figure 17.3

17.5 Description of model 760F

Model 760F is a 1-D waterflood model with no trapping or hysteresis.

The injector has both a maximum bottom hole injection pressure of 3850 psia and a rate

constraint of 100 RB/day, Figure 17.7 and Figure 17.8.

322

0 200 400 600 800 1000 1200 14000.0

0.2

0.4

0.6

0.8

1.0

t�day�

Tot

alS


SwM2

SwM1

SoM2

SoM1

SgM2

SgM1

Figure 17.6: Saturation for equivalent one-cell model for model 762E. For each time step, greenis the volume of oil in all the grid cells divided by the total volume of fluids in all the grid cells;blue is water and red is gas. Purple is the trapped water, cyan is the trapped oil, and yellow is thetrapped gas.

0 500 1000 1500 2000

0

20

40

60

80

100

t�day�

q wor

q g�b

bl�

Injection Rate �RBPD�

qTOT,inj

qo,inj

qg,inj

qw,inj

Figure 17.7: Injection rates at reservoir conditions for model 760F. Black is total injection inRB/day; red is gas injection; blue is water injection.

0 500 1000 1500 20000

1000

2000

3000

4000

t�day�

PB

HP,P

ores

Injection Pressure �psia�

PBHP

Pcell

Figure 17.8: Injection pressures for model 760F. Blue is bottom hole injection pressure from mymodel; orange is grid cell injection pressure for my model.

323

The production well has a maximum total production rate of 100 RB/day, Figure 17.9. There

is also a minimum bottom hole producing pressure of 500 psia, but for this case it does not control

the produciton well, Figure 17.10. The system is above the bubble point for the entire simulation.

0 500 1000 1500 2000

0

20

40

60

80

100

t�day�

�q��R

BPD�


�qTOT�

�qo�

�qg�

�qw�

Figure 17.9: Production rates at reservoir conditions for model 760F. Black is total production inRB/day; green is oil production; red is gas production; blue is water production.

0 500 1000 1500 20000

1000

2000

3000

4000

t�day�

PB

HP,P

ores


PBHP

Pcell

Figure 17.10: Production pressure for model 760F. Blue is bottom hole injection pressure from mymodel; orange is grid cell injection pressure for my model.

The average saturations in the reservoir are shown in Figure 17.3. For this model, the water

saturation progressively increases as the oil saturation decreases. After water breakthrough there

is only a little additional recovery of oil.

As shown in Figure 17.12, the mole fractions of each component remain nearly constant through

the simulation. The compositional recovery factors of each component are also nearly the same,

Figure 17.13.

324

0 500 1000 1500 20000.0

0.2

0.4

0.6

0.8

1.0

t�day�

Tot

alS


SwM2

SwM1

SoM2

SoM1

SgM2

SgM1

Figure 17.11: Saturation for equivalent one-cell model for model 760F. For each time step, green isthe volume of oil in all the grid cells divided by the total volume of fluids in all the grid cells; blueis water and red is gas.

0 500 1000 1500 20000.0

0.2

0.4

0.6

0.8

1.0

t�day�

CH

4,n

C4,n

C10

,CO

2


CH4�total moles

nC4�total moles

nC10�total moles

CO2�total moles

Figure 17.12: Mole fraction for equivalent one-cell model for model 760F. For each time step, redis the ratio of the moles of methane to the total number of moles; nC4 is in green; nC10 is in cyan;CO2 is in orange.

0 500 1000 1500 20000.0

0.2

0.4

0.6

0.8

1.0

t�day�

CH

4,n

C4,n

C10


RF CH4

RF nC4

RF nC10

Figure 17.13: Molar recovery factor for model 760F. For each time step, red is the ratio of thecumulative produced moles of methane to the original number of moles of methane in the reservoir;nC4 is in green; nC10 is in cyan. In this figure, all three compositional recovery factors visuallyoverlay each other.

325


Model 761F is a 1-D waterflood model with gas hysteresis. Model 761F gives almost identical

results to model 760F; because the system is always above the bubble point the gas saturation is

always 0, so the option for gas hysteresis is not relevant.


Model 762F is a 1-D waterflood model with compositional trapping and gas hysteresis. Model

762F gives almost identical results to model 760F; because the system is always above the bubble

point the gas saturation is always 0, so compositional trapping is not relevant.

For model 762F, the water saturation progressively increases as the oil saturation decreases,

Figure 17.14. Because the pressure remains above the bubble point, splitting the water and oil into

trapped and mobile fractions has no visual impact on the results (Figure 17.11).

0 500 1000 1500 20000.0

0.2

0.4

0.6

0.8

1.0

t�day�

Tot

alS


SwM2

SwM1

SoM2

SoM1

SgM2

SgM1

Figure 17.14: Saturation for equivalent one-cell model for model 762F. For each time step, greenis the volume of oil in all the grid cells divided by the total volume of fluids in all the grid cells;blue is water and red is gas. Purple is the trapped water, cyan is the trapped oil, and yellow is thetrapped gas.

The results as a function of time and a function of space at a fixed time are visually the same

between model 762F and model 760F.

17.8 Description of model 760G

Model 760G is a 1-D model with primary production followed by a waterflood with no trapping

or hysteresis.

326


constraint of 100 RB/day, Figure 17.15 and Figure 17.16. For this case only the rate constraint is

needed.

0 500 1000 1500 2000 2500 3000

0

20

40

60

80

100

t�day�

q wor

q g�b

bl�


qTOT,inj

qo,inj

qg,inj

qw,inj

Figure 17.15: Injection rates at reservoir conditions for model 760G. Black is total injection inRB/day; red is gas injection; blue is water injection.

0 500 1000 1500 2000 2500 30000

1000

2000

3000

4000

t�day�

PB

HP,P

ores


PBHP

Pcell

Figure 17.16: Injection pressures for model 760G. Blue is bottom hole injection pressure from mymodel; orange is grid cell injection pressure for my model.

The production well has a maximum total production rate of 100 RB/day, Figure 17.17. There

is also a minimum bottom hole producing pressure of 500 psia, but for this case it does not control

the produciton well, Figure 17.18.

The average saturations in the reservoir are shown in Figure 17.3. For this model, the oil

saturation decreases through the entire simulation. The gas saturation increases initially and then

decreases, going to zero at about the time of water breakthrough. The water saturation is initially

constant and then increases during the waterflood.

327

0 500 1000 1500 2000 2500 3000

0

20

40

60

80

100

t�day�

�q��R

BPD�


�qTOT�

�qo�

�qg�

�qw�

Figure 17.17: Production rates at reservoir conditions for model 760G. Black is total production inRB/day; green is oil production; red is gas production; blue is water production.

0 500 1000 1500 2000 2500 30000

1000

2000

3000

4000

t�day�

PB

HP,P

ores


PBHP

Pcell

Figure 17.18: Production pressure for model 760G. Blue is bottom hole injection pressure from mymodel; orange is grid cell injection pressure for my model.

328

As shown in Figure 17.20, the mole fraction of CH4 decreases with time. The nC4 and nC10

mole fractions increase with time, with the increase in nC10 bigger than for nC4. The CO2 is

approximately constant. The compositional recovery factors of CH4 is greater than the recovery

factor for nC4 which is greater than the recovery factor for nC10, Figure 17.21.

0 500 1000 1500 2000 2500 30000.0

0.2

0.4

0.6

0.8

1.0

t�day�

Tot

alS


SwM2

SwM1

SoM2

SoM1

SgM2

SgM1

Figure 17.19: Saturation for equivalent one-cell model for model 760G. For each time step, greenis the volume of oil in all the grid cells divided by the total volume of fluids in all the grid cells;blue is water and red is gas.

0 500 1000 1500 2000 2500 30000.0

0.2

0.4

0.6

0.8

1.0

t�day�

CH

4,n

C4,n

C10

,CO

2


CH4�total moles

nC4�total moles

nC10�total moles

CO2�total moles

Figure 17.20: Mole fraction for equivalent one-cell model for model 760G. For each time step, redis the ratio of the moles of methane to the total number of moles; nC4 is in green; nC10 is in cyan;CO2 is in orange.


Model 761G is a 1-D model with primary production followed by a waterflood with gas hys-

teresis.


constraint of 100 RB/day, Figure 17.22 and Figure 17.23. For this case only the rate constraint

329

0 500 1000 1500 2000 2500 30000.0

0.2

0.4

0.6

0.8

1.0

t�day�

CH

4,n

C4,n

C10


RF CH4

RF nC4

RF nC10

Figure 17.21: Molar recovery factor for model 760G. For each time step, red is the ratio of thecumulative produced moles of methane to the original number of moles of methane in the reservoir;nC4 is in green; nC10 is in cyan.

is needed. The injection profile is the same as the injection profile for model 760G, Figure 17.15.

The injection pressure profiles are different right after the start of water injection, Figure 17.16

0 500 1000 1500 2000 2500 3000

0

20

40

60

80

100

t�day�

q wor

q g�b

bl�


qTOT,inj

qo,inj

qg,inj

qw,inj


The production well has a maximum total production rate of 100 RB/day, Figure 17.24. The

oil response to the waterflood is earlier for model 761G (Figure 17.24) than for model 760G (Fig-

ure 17.17). There is also a minimum bottom hole producing pressure of 500 psia; this controls

production shortly after water breakthrough, Figure 17.25. This is different from the producer in

model 760G, Figure 17.18.


saturation decreases through the entire simulation. The gas saturation increases initially and then

330

0 500 1000 1500 2000 2500 30000

1000

2000

3000

4000

t�day�

PB

HP,P

ores


PBHP

Pcell


0 500 1000 1500 2000 2500 3000

0

20

40

60

80

100

t�day�

�q��R

BPD�


�qTOT�

�qo�

�qg�

�qw�


0 500 1000 1500 2000 2500 30000

1000

2000

3000

4000

t�day�

PB

HP,P

ores


PBHP

Pcell


331

decreases, going to zero at about the time of water breakthrough. The water saturation is initially

constant and then increases during the waterflood. The gas saturation profile is different between

500 days and 1000 days between model 761G (Figure 17.26) and 760G (Figure 17.19).

As shown in Figure 17.27, the mole fraction of CH4 decreases and then increases again. This

is different than model 760G (Figure 17.20) where it just decreases. This shows the importance of

gas hysteresis. The nC10 mole fraction increases and then decreases with time, also different from

model 760G. The nC4 and CO2 mole fractions are both approximately constant.

The compositional recovery factors of CH4 is greater than the recovery factor for nC4 which is

greater than the recovery factor for nC10, Figure 17.28. The difference between the CH4 recovery

and the nC10 recovery is much less for model 761G than for model 760G (Figure 17.21). This shows

that a moderate amount of methane is trapped based on the gas hysteresis effects.

0 500 1000 1500 2000 2500 30000.0

0.2

0.4

0.6

0.8

1.0

t�day�

Tot

alS


SwM2

SwM1

SoM2

SoM1

SgM2

SgM1

Figure 17.26: Saturation for equivalent one-cell model for model 761G. For each time step, greenis the volume of oil in all the grid cells divided by the total volume of fluids in all the grid cells;blue is water and red is gas.


Model 762G is a 1-D model with primary production followed by a waterflood with composi-

tional trapping and gas hysteresis.


constraint of 100 RB/day, Figure 17.29 and Figure 17.30. For this case only the rate constraint

is needed. The injection profile is the same as the injection profile for model 761G, Figure 17.22.

The injection pressure profiles are quite different during water injection, Figure 17.23.

332

0 500 1000 1500 2000 2500 30000.0

0.2

0.4

0.6

0.8

1.0

t�day�

CH

4,n

C4,n

C10

,CO

2


CH4�total moles

nC4�total moles

nC10�total moles

CO2�total moles


0 500 1000 1500 2000 2500 30000.0

0.2

0.4

0.6

0.8

1.0

t�day�

CH

4,n

C4,n

C10


RF CH4

RF nC4

RF nC10


0 500 1000 1500 2000 2500 3000

0

20

40

60

80

100

t�day�

q wor

q g�b

bl�


qTOT,inj

qo,inj

qg,inj

qw,inj


333

0 500 1000 1500 2000 2500 3000

1000

1500

2000

2500

3000

3500

t�day�

PB

HP,P

ores


PBHP

Pcell


The production well has a maximum total production rate of 100 RB/day, Figure 17.31. The oil

response to the waterflood is earlier and higher for model 762G (Figure 17.31) than for model 761G

(Figure 17.24). There is also a minimum bottom hole producing pressure of 500psia; this controls

production shortly after water breakthrough, Figure 17.32. This is similar to the producer in model

761G, Figure 17.25 but the duration is much shorter. Production pressures are much higher in the

model with compositional trapping than in model 761G without compositional trapping.

0 500 1000 1500 2000 2500 3000

0

20

40

60

80

100

t�day�

�q��R

BPD�


�qTOT�

�qo�

�qg�

�qw�



saturation decreases through the entire simulation. The portion of the oil that is trapped steadily

increases after the start of water injection. The gas saturation increases initially and then decreases.

After the start of water injection most of the gas is trapped. The water saturation is initially

334

0 500 1000 1500 2000 2500 3000

500

1000

1500

2000

2500

3000

3500

t�day�

PB

HP,P

ores


PBHP

Pcell


constant and then increases during the waterflood. The saturation profile is similar to model 761G

(Figure 17.26), although the shape of the water saturation profile is smoother before and after

water breakthrough.

As shown in Figure 17.34, the mole fraction of CH4 decreases and then increases again. In

model 762G it actually increases above the initial mole fraction. This is because the gas which

becomes trapped gas has a high CH4 content; after it is trapped it can only get produced through

a slow transfer back to the mobile system. The nC10 and nC4 mole fraction increase and then

decrease with time to a value lower than the initial mole fraction; this is different than model

761G, Figure 17.27. The CO2 mole fraction increases slightly with time.

The compositional recovery factors of nC4 and nC10 (Figure 17.35) is visually similar to model

761G (Figure 17.28). The CH4 recovery is much lower for model 762G than for model 761G; the

CH4 recovery in model 761G is lower than model 760G. Both gas hysteresis and compositional

trapping increase the amount of methane that remains in the reservoir.

17.11 Compare CMG Model with my Model 760E and 761E

The production wells have a mixed pressure and rate constraint; they start out controlled by

the production rate, at around 1100 days they switch to bottom hole producing pressure control.

Figure 17.36 shows the bottom hole production rate for my model and the GEMmodel. Figure 17.37

shows the grid cell pressure for the production cell for my model and the GEM model. There is

a big change in the shape of the pressure profile as the system drops below the bubble point at

335

0 500 1000 1500 2000 2500 30000.0

0.2

0.4

0.6

0.8

1.0

t�day�

Tot

alS


SwM2

SwM1

SoM2

SoM1

SgM2

SgM1

Figure 17.33: Saturation for equivalent one-cell model for model 762G. For each time step, greenis the volume of mobile oil in all the grid cells divided by the total volume of fluids in all the gridcells; cyan is the volume of trapped oil; red is the mobile gas; yellow is the trapped gas; blue is themobile water; purple is the trapped water.

0 500 1000 1500 2000 2500 30000.0

0.2

0.4

0.6

0.8

1.0

t�day�

CH

4,n

C4,n

C10

,CO

2


CH4�total moles

nC4�total moles

nC10�total moles

CO2�total moles


0 500 1000 1500 2000 2500 30000.0

0.2

0.4

0.6

0.8

1.0

t�day�

CH

4,n

C4,n

C10


RF CH4

RF nC4

RF nC10


336

around 100 days.

The pressures in Figure 17.37 are very similar; Figure 17.38 illustrates the difference between the

two models is consistently less than 15 psia with a moderately constant offset after the well drops

below the bubble point pressure. These slight differences in pressure lead to different times when the

well transitions from rate control to pressure control (Figure 17.36). The different pressures leads

to different flash conditions, which leads to the variations in molar rate shown in Figure 17.39. The

difference in flash conditions also leads to a different amount of produced oil compared to produced

gas, which leads to a different oil recovery factor, Figure 17.40.

0 200 400 600 800 1000 1200 14000

20

40

60

80

100

t�day�

q

Producer Rate �RBD�

JSB

CMG

Figure 17.36: Comparison of production rates for model 760E. Green is from my model; purple isfrom the GEM model.

0 200 400 600 800 1000 1200 14000

1000

2000

3000

4000

t�day�

P

Producer Cell Pressure �psia�

JSB

CMG

Figure 17.37: Comparison of producer grid cell pressures for model 760E. Green is from my model;purple is from the GEM model.

Model 761E gives almost identical results to model 760E; because the gas saturation is always

increasing there is no liquid phase to induce the trapping of gas. Different values of the critical gas

337

0 200 400 600 800 1000 1200 1400

�10

�5

0

5

10

t�day�

P


Figure 17.38: Difference of producer grid cell pressures for model 760E.

0 200 400 600 800 1000 1200 14000

50

100

150

200

250

t�day�

HC

lbm

ol�d

ay

Molar Rate

JSB

CMG

Figure 17.39: Comparison of total molar rates for model 760E. Green is from my model; purple isfrom the GEM model.

0 200 400 600 800 1000 1200 14000.0

0.2

0.4

0.6

0.8

1.0

t�day�

RF

prod

uced

oil�

RC

F�

PV�R

CF�

Recovery Factor, CMG vs JSB

JSB

CMG

Figure 17.40: Comparison of recovery factors for model 760E. Green is from my model; purple isfrom the GEM model.

338

saturation would affect these cases. The difference in flash conditions leads to a different amount of

produced oil compared to produced gas, which leads to a different oil recovery factor, Figure 17.41.

Figure 17.41: Comparison of recovery factors for model 761E. Green is from my model; purple isfrom the GEM model.

17.12 Compare CMG Model with my Model 762E

Figure 17.42 shows the bottom hole production rate for my model and the GEM model. There

is more pressure difference between 762E and the GEM model, Figure 17.43, than there is between

760E and GEM, Figure 17.43. This difference is even more obvious in Figure 17.44. The recovery

factors are shown in Figure 17.45.

0 200 400 600 800 1000 1200 14000

20

40

60

80

100

t�day�

q


JSB

CMG

Figure 17.42: Comparison of production rates for model 762E. Green is from my model; purple isfrom the GEM model.

339

0 200 400 600 800 1000 1200 14000

1000

2000

3000

4000

t�day�

P


JSB

CMG

Figure 17.43: Comparison of producer grid cell pressures for model 762E. Green is from my model;purple is from the GEM model.

0 200 400 600 800 1000 1200 1400�40

�20

0

20

40

t�day�

P


Figure 17.44: Difference of producer grid cell pressures for model 762E.

0 200 400 600 800 1000 1200 14000.0

0.2

0.4

0.6

0.8

1.0

t�day�

RF

prod

uced

oil�

RC

F�

PV�R

CF�


JSB

CMG

Figure 17.45: Comparison of recovery factors for model 762E. Green is from my model; purple isfrom the GEM model 761E.

340

17.13 Compare CMG Model with my Model 760F, 761F, and 762F

Figure 17.46 shows the bottom hole production rate for my model and the GEM model; note

that GEM handles mixed pressure and rate constraints differently than my model; it has trouble

finding a solution at the time of water breakthrough. Figure 17.47 shows the grid cell pressure for

the production cell for my model and the GEM model. There is a difference of around 100 psia

between the two models after the initial time steps. This pressure difference may be a result of the

convergence failure in the first few time steps. The recovery factor (Figure 17.48) are very similar

between the two models because the system always stays above the bubble point.

0 500 1000 1500 2000

99.95

100.00

100.05

t�day�

q


JSB

CMG

Figure 17.46: Comparison of production rates for model 760F. Green is from my model; purple isfrom the GEM model.

0 500 1000 1500 20000

1000

2000

3000

4000

t�day�

P


JSB

CMG

Figure 17.47: Comparison of producer grid cell pressures for model 760F. Green is from my model;purple is from the GEM model.

The water saturation profile is very similar, with small differences in the shape of the front;

these differences may be a result of GEM using a mixed fully implicit and IMPES scheme while

341

0 500 1000 1500 20000.0

0.2

0.4

0.6

0.8

1.0

t�day�

RF

prod

uced

oil�

RC

F�

PV�R

CF�


JSB

CMG

Figure 17.48: Comparison of recovery factors for model 760F. Green is from my model; purple isfrom the GEM model.

my code uses an IMPES scheme, Figure 17.49.

0 200 400 600 800 10000.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x�ft�

Sw

Sw � t�500

JSB

CMG

Figure 17.49: Comparison of water saturation for model 760F at 500 days. Green is from my model;purple is from the GEM model.

Model 761F gives almost identical results to model 760F; because the system is always above the

bubble point the gas saturation is always 0, so gas hysteresis is not relevant. The recovery factors

(Figure 17.50) are very similar between the two models because the system always stays above

the bubble point. Model 762F gives almost identical results to model 760F; because the system

is always above the bubble point the gas saturation is always 0, so compositional trapping is not

relevant. The recovery factors (Figure 17.51) are very similar between the two models because the

system always stays above the bubble point.

342


0 500 1000 1500 20000.0

0.2

0.4

0.6

0.8

1.0

t�day�

RF

prod

uced

oil�

RC

F�

PV�R

CF�


JSB

CMG


343

17.14 Compare CMG Model with my Model 760G


that GEM handles mixed pressure and rate contraints differently than my model. Figure 17.53

shows the grid cell pressure for the production cell for my model and the GEM model. The

recovery factors (Figure 17.54) are similar between the two models. The differences are likely tied

to the pressure differences in Figure 17.53, which lead to different flash conditions.

0 500 1000 1500 2000 2500 3000 3500

200

400

600

800

1000

t�day�

q


JSB

CMG

Figure 17.52: Comparison of production rates for model 760G. Green is from my model; purple isfrom the GEM model.

0 500 1000 1500 2000 2500 3000 35000

1000

2000

3000

4000

t�day�

P


JSB

CMG

Figure 17.53: Comparison of producer grid cell pressures for model 760G. Green is from my model;purple is from the GEM model.

The waterflood starts at 500 days. The gas saturation profile before the start of water injection

is different, probably because of the difference in flash pressures, Figure 17.55.

The waterflood starts at 500 days. At 1000 days is after some water injection but before

water breakthrough. The water saturation profiles are very similar between GEM and my model,

344

0 500 1000 1500 2000 2500 3000 35000.0

0.2

0.4

0.6

0.8

1.0

t�day�

RF

prod

uced

oil�

RC

F�

PV�R

CF�


JSB

CMG

Figure 17.54: Comparison of recovery factors for model 760G. Green is from my model; purple isfrom the GEM model.

0 200 400 600 800 10000.00

0.05

0.10

0.15

0.20

x�ft�

Sg

Sg � t�500

JSB

CMG

Figure 17.55: Comparison of gas saturation for model 760G at 500 days. Green is from my model;purple is from the GEM model.

345

Figure 17.56. The gas saturations are different due to differences in flash pressures, Figure 17.57.

0 200 400 600 800 10000.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x�ft�

Sw

Sw � t�1000

JSB

CMG

Figure 17.56: Comparison of water saturation for model 760G at 1000 days. Green is from mymodel; purple is from the GEM model.

0 200 400 600 800 10000.00

0.02

0.04

0.06

0.08

0.10

x�ft�

Sg

Sg � t�1000

JSB

CMG




that GEM handles mixed pressure and rate constraints differently than my model; this leads to

large differences in the production rates between 1200 days and 1400 days when water breakthrough

occurs. Figure 17.59 shows the grid cell pressure for the production cell for my model and the GEM

model. There are some differences during the waterflood and larger differences during and after

water breakthrough. The recovery factor (Figure 17.60) are similar between the two models. The

differences are likely tied to the pressure differences in Figure 17.59, which lead to different flash

346

conditions.

0 500 1000 1500 2000 2500 3000

100

200

300

400

500

600

700

800

t�day�

q


JSB

CMG

Figure 17.58: Comparison of production rates for model 761G. Green is from my model; purple isfrom the GEM model.

0 500 1000 1500 2000 2500 30000

1000

2000

3000

4000

t�day�

P


JSB

CMG

Figure 17.59: Comparison of producer grid cell pressures for model 761G. Green is from my model;purple is from the GEM model.

The waterflood starts at 500 days. At 1000 days is after some water injection but before wa-

ter breakthrough. During waterflood, the pressure differences are bigger after the water front has

passed, Figure 17.61. The water saturation profiles are similar between GEM and my model, Fig-

ure 17.62. My model is smoother than the GEM model, probably a result of the difference between

IMPES and the mixture of IMPES and fully implicit that GEM uses. The gas saturations are

different due to differences in flash pressures, Figure 17.63. There may also be other computational

differences; again my model has a much more smooth distribution than the GEM model. The GEM

model has spikes which do not make physical sense in a homogenous model.

347

0 500 1000 1500 2000 2500 30000.0

0.2

0.4

0.6

0.8

1.0

t�day�

RF

prod

uced

oil�

RC

F�

PV�R

CF�


JSB

CMG

Figure 17.60: Comparison of recovery factors for model 761G. Green is from my model; purple isfrom the GEM model.

0 200 400 600 800 10000

1000

2000

3000

4000

x�ft�

P�ps

ia�

Grid Cell Pressure � t�1000

JSB

CMG

Figure 17.61: Comparison of pressure profiles for model 761G at 1000 days. Green is from mymodel; purple is from the GEM model.

0 200 400 600 800 10000.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x�ft�

Sw

Sw � t�1000

JSB

CMG


348

0 200 400 600 800 10000.00

0.05

0.10

0.15

0.20

x�ft�

Sg

Sg � t�1000

JSB

CMG


1500 days is after water breakthrough. There are still pressure differences between GEM and

my model, Figure 17.64, but they are smaller than at 1000 days. The water saturation profiles

are similar between GEM and my model, Figure 17.65. After water breakthrough my model shows

some weird variations in water saturations. This may be related to the gas saturations still present

in my model but absent from the GEM model, Figure 17.66.

0 200 400 600 800 10000

1000

2000

3000

4000

x�ft�

P�ps

ia�

Grid Cell Pressure � t�1500

JSB

CMG

Figure 17.64: Comparison of pressure profiles for model 761G at 1500 days. Green is from mymodel; purple is from the GEM model.


The recovery factor (Figure 17.67) are different between the two models as a result of the

compositional trapping changing the mobile saturations at different times; compare Figure 17.33

to Figure 17.26.

349

0 200 400 600 800 10000.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x�ft�

Sw

Sw � t�1500

JSB

CMG


0 200 400 600 800 1000

0.000

0.005

0.010

0.015

0.020

0.025

x�ft�

Sg

Sg � t�1500

JSB

CMG


0 500 1000 1500 2000 2500 30000.0

0.2

0.4

0.6

0.8

1.0

t�day�

RF

prod

uced

oil�

RC

F�

PV�R

CF�


JSB

CMG

Figure 17.67: Comparison of recovery factors for model 762G. Green is from my model; purple isfrom the GEM model 761G.

350

CHAPTER 18

CASE STUDIES

The test cases used in this thesis are based on a low permeability carbonate reservoir in Abu

Dhabi. The work for this study was conducted in collaboration with the CSM/PI Integrated Car-

bonate Reservoir Research Group, and some of the information comes from discussion with members

of this research group. Portions of proprietary reservoir studies conducted by past operators of the

field were used for some properties of the reservoir simulation. Several publications were especially

valuable for data used here, including Alameri (2010), Jobe (2013), and Shibasaki, Edwards, Qotb,

and Akatsuka (2006).

18.1 Initial Conditions

The following initial conditions are specified.

• Reservoir depth 7550 ft, based on reservoir study information and Alameri (2010).

• Based on the reservoir studies, Jobe (2013), and Shibasaki et al. (2006) the expected dip is

less than 1◦; 0◦ is used here.

• Initial reservoir pressure P initom1

= 3842 psia, based on reservoir study information.

• Reservoir temperature T = 210◦F, based on reservoir study information.

• WNaCl = 0.082142, from 200, 000 ppm, based on reservoir study information.

• Z0m = {CH4,nC4,nC10,CO2} = {0.25, 0.25, 0.45, 0.05}, based on reservoir study information

and converted from a 8-hydrocarbon component EOS to a 4-hydrocarbon component EOS.

• Initial water saturation Sinitw = Swr = 0.059.

The following well constraints are specified. Wells in 1-D simulations and wells in the corner of

a quarter five-spot or a five-spot are 1/4 of these rates.

• Fracture pressure Pfrac = 5662 psia, corresponding to a fracture gradient of 0.75 psia/ft based

on Alameri (2010).

351

• Maximum injection pressure PBHIP = 5000 psia, based on Alameri (2010).

• Injection rate qinj = 400 RB/day = 2245.84 RCF/day, based on Alameri (2010).

• Bottom hole producing pressure PBHPP = 500 psia, based on Alameri (2010).

• Production rate qprod = 400 RB/day = 2245.84 RCF/day, based on Alameri (2010).

• Well radius rw = 0.5 ft.

• Wellbore skin s = 0.

The following are the grid properties for a 2-D 5-spot pattern:

• Δx = 100 m = 328 ft, NX = 21, based on the current 2 km × 2 km development pattern in

Shibasaki et al. (2006).

• Δy = 100 m = 328 ft, NY = 21, based on the current 2 km × 2 km development pattern in


• Δz = 44 ft, NZ = 1 based on Jobe (2013) and Shibasaki et al. (2006).

The following are the grid properties for a 2-D 1/4 5-spot pattern:

• Δx = 100 m = 328 ft, NX = 11, based on the current 2 km × 2 km development pattern in


• Δy = 100 m = 328 ft, NY = 11, based on the current 2 km × 2 km development pattern in


• Δz = 44 ft, NZ = 1 based on Jobe (2013) and Shibasaki et al. (2006).

The following are the grid properties for a 1-D pattern:

• Δx = 141.4 m = 464 ft, NX = 11, based on the 1414 m diagonal of the 1/4 5-spot.

• Δy = 141.4 m = 464 ft, NY = 1, based on the 1414 m diagonal of the 1/4 5-spot.

• Δz = 44 ft, NZ = 1 based on the total thickness of the reservoir in Jobe (2013) and Shibasaki

et al. (2006).

352

Summary of rock properties; variations in permeability and porosity are described in Sec-

tion 18.2.

• kxx = kyy = kzz = 5.6 md, based on average values for facies “L5A” from Shibasaki et al.

(2006) which corresponds to facies “F5” Jobe (2013). Shibasaki et al. (2006) indicates that

effective permeability from well tests may be up to 5×kcore. The maximum listed permeability

in “L5A” from Shibasaki et al. (2006) is 63.6 md.

• φ = 0.19 based on average values for facies “L5A” from Shibasaki et al. (2006).

• Cφ = 4 · 10−6 psi−1

Trapping properties are defined as follows

• km2 = 5×10−4 md; this is computed to match a liquid diffusion coefficient of D = 10−5 cm2/s.

k[md] =(

D︷︸︸︷10−5 cm2/s)× (

μo︷︸︸︷1cp )

(

kro︷︸︸︷0.3 )× (

ΔP︷︸︸︷1000 psia)

× 1 md

9.9869 × 10−12 cm2×10−3 kg/(ms)

1 cp× 1 psia

6894 kg/(ms2)(18.1)

• σm1/m2= 4.32 ft−2 is the shape factor. This is calculated in the same way as a fracture-matrix

shape factor:

σm1/m2= 4

(1

(5 ft)2+

1

(5 ft)2+

1

(1 ft)2

)(18.2)

Summary of relative permeability; refer to Section 18.4 for a full description.

• Swr = 0.059

• Sorw = 0.231

• Sorg = 0.15

• Sgr = 0.00

• nw = 4.49

• now = 3.76

353

• nog = 4.18

• ng = 2.3147

• k�rw = 0.093

• k�ro = 0.3

• k�rg = 0.3

The gas-oil capillary pressure is assumed to be 0, based on the assumptions of the reservoir

studies and lack of data. See Section 18.5 for a full description of the water-oil capillary pressures;

the parameters are as follows.

• Swr = 0.059

• Sorw = 0.231

• SIwx = 0.28

• SDwx = 0.33

• αIow = 5

• αDow = 6.5

• P Ic,offset = −3.7

• PDc,offset = −1.7

• Pcow,min = −20

• Pcow,max = 20

The following Peng-Robinson Equation of State properties are used:

• MWm = {16.043, 58.124, 142.285, 44.010}

• Pcm = {667.2, 551.1, 305.68, 1069.87}

• Tcm = {343.08, 765.36, 1111.68, 547.56}

354

• ωm = {0.008, 0.193, 0.49, 0.225}

• Pm = {77.3, 191.7, 431.0, 78.0}

• sm = {−0.19404,−0.08625, 0.08563,−0.06155} (cm = bmsm)

• vm = {1.59, 4.08, 9.66, 1.51}

• δmn =

⎛⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0.0422 0.12

0 0 0.0078 0.12

0.0422 0.0078 0 0.1141

0.12 0.12 0.1141 0

⎞⎟⎟⎟⎟⎟⎟⎟⎠

Initial conditions based on flash of Z0m.

• So = 0.941

• Sg = 0

• V = 0

• Estimated bubble point Pb = 1268.

• W 0m = {0, 0, 0, 0.000525, 0.999475}

• X0m = {0.25, 0.25, 0.45, 0.05, 0.0}

• ξw = 3.311

• ξo = 0.461

• ρw = 70.4281 lbmol/ft3

• ρo = 39.09 lbmol/ft3

• γw = 0. lbmol/ft3

• γo = 0.271 lbmol/ft3

• μw = 0.517

• μo = 0.233

355

• kro = 0.3

• krg = 0

• krw = 0

• Pcow = 20

• Pcgo = 0

Initial injection conditions based on flash of {0, 0, 0, 1, 0}.

• ξg = 0.175

18.2 Variations in Porosity and Permeability

Jobe (2013) described 12 cores from different parts of the field of interest. Routine porosity and

permeability analysis was previously conducted on these cores, but it was discovered later that the

CMS300 used for the analysis had not been calibrated for 20 years.

One of the facies from Jobe (2013) was selected for the 2-D studies in this dissertation. Facies 5

of Jobe (2013) corresponds to facies “L5A” and “L5B” of Shibasaki et al. (2006) and lithotype “23”

of the previous reservoir studies. Based on Jobe (2013), Facies 5 is a Lithocodium-Bacinella Wacke-

stone with abundant oncoidal Lithocodium-Bacinella, common echinoderm, coral, bivalve skeletal

debris, and benthic forams including Miliolida, Textularia, and Orbitolina. It is a heterogenous

bioclastic boundstone with both micro and macro porosity.

The porosity distribution for Facies 5 is shown in Figure 18.1. After the outliers beyond three

standard deviations were excluded from the analysis, the mean porosity was 22.3% with a standard

deviation of 3.957. The distribution is symmetric and approximately normal. Based on the known

calibration errors in the porosity measurements, the porosity distribution was shifted based on the

mean values of Shibasaki et al. (2006). The new distribution had a mean of 19% and a standard

deviation of 3.957.

φF5[%] = Normal[μ = 19.0%, σ = 3.957%] (18.3)

356

0 10 20 30 400.00

0.02

0.04

0.06

0.08

0.10

porosity ��

freq

uenc

y

Porosity Distribution, Facies 5

Figure 18.1: Porosity distribution for Facies 5 of Jobe (2013).

Within Facies 5, the permeability is weakly correlated with the porosity, Figure 18.2. The out-

liers beyond three standard deviations in the original fit were eliminated (cyan dots in Figure 18.2).

A new log-normal fit was created, shown in blue in Figure 18.2. This fit is not exact, so the ad-

ditional variability is represented by a normal distribution with mean 0 and standard deviation

0.166.

kF5[md] = 10(−0.498+0.0427×φ%F5+Normal[0,0.166]) (18.4)

Heterogeneity is expected to have a significant effect on reservoir performance. To understand

the effect of heterogeneity, geostatistical analysis was conducted and geostatistical realizations were

created for a typical 5-spot pattern.

The spatial variability of the porosity and permeability was simulated using geostatistics. We

did not have access to enough data to conduct variogram analysis, so a semivariogram was created

that yielded distributions of porosity and permeability that looked reasonable. This semivariagrom

is based on a spherical variogram with a longer range in the NW-SE direction and a lower range

in the NE-SW direction. The distances are all represented in units of m here.

• h = lag; distance between two points.

357

0 5 10 15 20 25 30

0.5

1.0

2.0

5.0

10.0

porosity ��

perm

eabi

lity�m

d�

Porosity�Permeability Distribution, Facies 5

Figure 18.2: Porosity-permeability correlation for Facies 5, Jobe (2013). The blue line is a log-normal fit to the permeability-porosity trend. The red lines represent one and two standard devia-tions away from this primary trend. The blue dots are the core plug measurements. The cyan dotswere more than three standard deviations away from the original trend.

• a = range; beyond this distance points are not correlated.

– aNW = 2000 m

– aNE = 1000 m

• c = 0.7 = sill; constant value beyond range.

γ[h] =

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

0.3 + 0.7×(1.5 h

aNW− 0.5

(h

aNW

)3), h ≤ aNW,primary direction, NW-SE

1.0, h > aNW,primary direction, NW-SE

0.3 + 0.7×(1.5 h

aNE− 0.5

(h

aNE

)3), h ≤ aNE, secondary direction, NE-SW

1.0, h > aNE, secondary direction, NE-SW

(18.5)

Based on the variograms, a series of 100 Sequential Gaussian Simulations were conducted using

GSLIB (Deutsch and Journel, 1992). These simulations were based on a normal distribution with

mean 0 and standard deviation 1. The control data assumed that all the wells of the 5-spot pattern

have average properties (mean 0). Although the realizations were simulated using Normal[0, 1], the

sample mean for a particular realization will not be equal to 0. The realizations were first shifted to

have a mean of 0, and then transformed into the Normal[19, 3.957] distribution of the porosity for

F5. Six of the one hundred realizations were selected for further use. For each of the six realizations,

358

10 sequences of uncorrelated random numbers were used to generate the permeability using (18.4).

One of the 10 permeability distributions was selected for further use. Figure 18.3–Figure 18.8 show

the selected porosity and permeability spatial distributions as well as the porosity and permeability

histograms.

0 10 20 30 400.00

0.02

0.04

0.06

0.08

0.10

0.12

porosity ��

freq

Porosity Distribution; Facies 5; ID � 14

(a) Porosity distribution

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.00

0.05

0.10

0.15

1 1.5 2 2.5 3 4 5 7 10 15 20 25 30

log10�permeability� �md�

coun

t

Permeability �md�; Facies 5; ID � 14

(b) Permeability distributionPorosity; Facies 5; ID � 14

Φ ��

5

10

15

20

25

30

(c) Porosity map

log10�Permeability�; Facies 5; ID � 14

log10�k�

0

0.25

0.50

0.75

1.00

1.25

1.50

(d) Permeability map

Figure 18.3: Porosity and permeability for Geostatistical Realization # 1.

18.3 Relative Permeability Test Case Literature Review

Previously, a literature search was conducted for “three-phase relative permeability”, “relative

permeability hysteresis”, “relative permeability in carbonates”, “mixed wettability”, and “Abu

Dhabi fields”. These papers were reviewed again looking for data that might add additional con-

straints on the relative permeability curves for this test case.

359

0 10 20 30 400.00

0.02

0.04

0.06

0.08

0.10

0.12

porosity ��

freq



0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.00

0.05

0.10

0.15

1 1.5 2 2.5 3 4 5 7 10 15 20 25 30


coun

t



Φ ��

5

10

15

20

25

30

(c) Porosity map


log10�k�

0

0.25

0.50

0.75

1.00

1.25

1.50



360

0 10 20 30 400.00

0.02

0.04

0.06

0.08

0.10

porosity ��

freq



0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.00

0.05

0.10

0.15

1 1.5 2 2.5 3 4 5 7 10 15 20 25 30


coun

t



Φ ��

5

10

15

20

25

30

(c) Porosity map


log10�k�

0

0.25

0.50

0.75

1.00

1.25

1.50



361

0 10 20 30 400.00

0.02

0.04

0.06

0.08

0.10

0.12

porosity ��

freq



0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.00

0.05

0.10

0.15

1 1.5 2 2.5 3 4 5 7 10 15 20 25 30


coun

t



Φ ��

5

10

15

20

25

30

(c) Porosity map


log10�k�

0

0.25

0.50

0.75

1.00

1.25

1.50



362

0 10 20 30 400.00

0.02

0.04

0.06

0.08

0.10

0.12

porosity ��

freq



0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.00

0.05

0.10

0.15

0.20

1 1.5 2 2.5 3 4 5 7 10 15 20 25 30


coun

t



Φ ��

5

10

15

20

25

30

(c) Porosity map


log10�k�

0

0.25

0.50

0.75

1.00

1.25

1.50



363

0 10 20 30 400.00

0.02

0.04

0.06

0.08

0.10

0.12

porosity ��

freq



0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.00

0.05

0.10

0.15

1 1.5 2 2.5 3 4 5 7 10 15 20 25 30


coun

t



Φ ��

5

10

15

20

25

30

(c) Porosity map


log10�k�

0

0.25

0.50

0.75

1.00

1.25

1.50



364

18.3.1 Water-Oil Data

Spiteri et al. (2005) present simulation models applied to CO2 injection with relative perme-

ability hysteresis. Masalmeh (2002) presents capillary pressure and relative permeability data for

mixed wet and oil wet Middle East carbonates. Masalmeh (2003) presents capillary pressure and

relative permeability data and their variations with wettability. Lamy et al. (2010) presents capil-

lary pressure data for carbonate cores. Honarpour et al. (1996) presents an experimental apparatus

to simultaneously measure relative permeability, capillary pressure, and electrical resistivity during

a core flood. Hysteresis data is presented for Berea sandstone. Jerauld (1997) presents three-phase

relative permeability data and curve fits for mixed wet Prudhoe Bay sandstone. Kralik et al. (2000)

presents the results of three-phase relative permeability experiments on an oil-wet sandstone. Der-

naika et al. (2012) presents relative permeability data with hysteresis for various carbonate rocks.

18.3.2 Gas-Oil Data

Fatemi et al. (2012a) presents a history match of experimental three-phase relative permeability

data and a good literature review. Fatemi et al. (2012b) presents three-phase relative permeability

data for water wet and mixed wet cores.

18.3.3 Gas-Water Data

Levine (2011) presents CO2/brine relative permeability in sandstone and constructed cores.

Bennion and Bachu (2005) and Bennion and Bachu (2008b) present CO2/brine relative permeability

data for carbonate and sandstone cores in Canada.

18.3.4 Two-Phase Experiments with Different Phases

Aljarwan, Belhaj, Haroun, and Ghedan (2012) present oil/water and gas/oil data for an Abu

Dhabi reservoir. Ehrlich et al. (1984) presents laboratory data for a dolomite reservoir subjected

to a lab-based CO2 WAG flood. Bhatti et al. (2012) presents relative permeability and capillary

pressure data for Abu Dhabi carbonates.

18.3.5 Three-Phase Experiments

Spiteri and Juanes (2004) and Spiteri and Juanes (2006) present simulation of WAG injection

with different three-phase relative permeability models. Al-Dhahli, Geiger, and van Dijke (2012),

365

and Fatemi and Sohrabi (2012), and Element et al. (2003) present experimental results which

show cycle dependent residual oil saturations. Oak (1990) presents the results of very thorough

experiments of three-phase relative permeability on water-wet Berea sandstone.

18.3.6 Relative Permeability Formulations

For the test cases used here, krow is very close to krog for all values of So. For this specific

application oil relative permeability was calculated as a function of the oil saturation only. The

following articles are a selected group of three-phase relative permeability and kro calculation

methods. One of these methods would most likely be selected to calculate oil relative permeability

if the SCAL data did not support the specialized simplification.

Spiteri and Juanes (2004), presents an evaluation of different relative permeability and hys-

teresis models, including the presentation of a new method for three-phase relative permeability

hysteresis. Hustad et al. (2002) presents the results of 2D cross-section simulation models of WAG

with hysteresis. Hustad (2002) presents a three-phase capillary pressure and relative permeability

model with hysteresis. Larsen and Skauge (1998) presents a three-phase relative permeability for-

mulation. Dietrich and Bondor (1976) presents a three-phase relative permeability model. Coats

and Smith (1964) describes dead-end space using a diffusion model. Hustad and Browning (2009)

presents a relative permeability and capillary pressure formulation with hysteresis. Blunt (2000)

presents an analysis of three-phase relative permeability and capillary pressure experiments, in-

cluding a discussion of trapped oil, spreading oil, and mobile oil. Baker (1988) presents an analysis

of different three-phase relative permeability formulations. Fayers and Matthews (1984) analyzes

three-phase relative permeability data from various literature sources. Kossack (2000) presents a

comparison three-phase relative permeability models with hysteresis as implemented in Eclipse.

Kokal and Maini (1990) presents analysis of several three-phase relative permeability experiments

and a modified Stone’s method. Delshad and Pope (1989) presents an analysis of seven different

three-phase relative permeability formulations. Killough (1976), present a hysteresis algorithm.

18.3.7 Relative Permeability Observations

The following articles are related to mixed wet and/or carbonate reservoir relative permeability

and capillary pressure: Masalmeh (2001), Byrnes and Bhattacharya (2006), Syed, Ghedan, Al-

366

Hage, and Tariq (2012), Dabbouk, Liaqat, Williams, and Beattie (2002), Keelan and Pugh (1975),

Wegener and Harpole (1996) After evaluating these articles and the articles listed above, trends

from these articles were used but the data was not directly used.

Ghomian et al. (2008) presents simulations of CO2 WAG for EOR and sequestration using

different three-phase relative permeability and capillary pressure models. Iglauer et al. (2009)

presents a review of capillary trapping in sandstones along with some new data. Shahverdi and

Sohrabi (2012) presents an analysis of three-phase relative permeability data. Masalmeh and Wei

(2010) presents a study of WAG options using three-phase relative permeability and capillary

pressure hysteresis. Ahmed Elfeel, Al-Dhahli, Geiger, and van Dijke (2013) uses tables of three-

phase relative permeability from pore-network models to simulate WAG.

18.4 Relative Permeability

This section describes the relative permeability curves used in the test cases.

18.4.1 Experimental Data

All the SCAL data we had access to from previous reservoir studies was reviewed. Lithotype

23, 6.49 md corresponds to facies F5 of Jobe (2013) and facies ”L5a” and ”L5b” of Shibasaki et al.

(2006) . Based on data from Tadesse Teklu and Waleed Al-Ameri, k�row = k�rog = 0.3. Both the

oil-water and gas-oil kr values are multiplied by 0.3 before curve fitting.

We don’t have measurements of the interfacial tensions to allow for the calculation of the

spreading coefficient. The values of the relative permeability to oil are very small (kro[So = 0.265] =

1.09×10−5 for the water oil F5 experiment). This very small kro makes it difficult to justify a linear

layer flow model for the oil. As a result, kro is fit using a Corey model without an additional linear

flow component. Based on recommendations of Dr. Kazemi, the presence of capillary pressure

makes it unnecessary to have

limS→Smin

∂kr∂S

= 0 (18.6)

for any of the saturations. As a result, Corey curves were also used for krg and krw.

van Dijke et al. (2000) and van Dijke et al. (2001) present a formulation for three-phase relative

permeability. Juanes and Patzek (2004b) and Juanes and Patzek (2004a) present a theoretical

367

discussion under what conditions three-phase relative permeability models transition between hy-

perbolic and elliptic regions.

18.4.2 Oil/Water Experiment

The kDrow fit based on data for the water-oil system SCAL (WOG=IDC), scaled to k�row = 0.3.

The data and the fits are shown in Figure 18.9. A standard Corey curve fits the krow.

krow =

⎧⎨⎩

0, So ≤ Sorw

k�row

(So − Sorw

1− Swr − Sorw

)now

, So > Sorw(18.7)

• Swr = 0.059

• Sorw = 0.231

• k�row = 0.3

• now = 3.76

The kIrw fit based on data for the water-oil system SCAL (WOG=IDC), scaled to k�row = 0.3 :

The data and the fits are shown in Figure 18.9. A standard Corey curve fits the krw.

krw =

⎧⎨⎩

0, Sw ≤ Swr

k�rw

(Sw − Swr

1− Swr − Sorw

)nw

, Sw > Swr(18.8)

• Swr = 0.059

• Sorw = 0.231

• k�rw = 0.093

• nw = 4.49

18.4.3 Gas/Oil Experiment

The kDrog fit based on data for the gas-oil system SCAL (WOG=IDC), scaled to k�rog = 0.3. The

data and the fits are shown in Figure 18.10. A standard Corey curve fits the krog.

368

Swr Sorw

�Swr,krow� �

�1�Sowr,krwo� �

krow krw

0.0 0.2 0.4 0.6 0.8 1.00.0

0.1

0.2

0.3

0.4

0.5

Sw

kr

Oil�Water Relative Permeability

Figure 18.9: Oil and water relative permeability curves including the data points. The green curveand data points are krow. The blue curve and data points are krw.

krog =

⎧⎨⎩

0, So ≤ Sorg

k�rog

(So − Sorg

1− Swr − Sorg

)nog

, So > Sorg(18.9)

• Swr = 0.059

• Sorg = 0.15

• k�rog = 0.3

• nog = 4.18

The kIrg fit based on data for the gas-oil system SCAL (WOG=IDC), scaled to k�rog = 0.3. The

data and the fits are shown in Figure 18.10. A standard Corey curve fits the kIrg.

kIrg =

⎧⎨⎩

0, Sg ≤ 0

k�rg

(Sg

1− Swr − Sorg

)ng

, Sg > 0(18.10)

• Swr = 0.059

• Sorg = 0.15

• k�rg = 0.3

• ng = 2.3147

369

SwrSorg

�Sgr,krog� � �1�Sorg�Swr,krgo

� �

krog krg

0.0 0.2 0.4 0.6 0.8 1.00.0

0.1

0.2

0.3

0.4

0.5

0.6

Sg

kr

Gas�Oil Relative Permeability

Figure 18.10: Gas and oil relative permeability curves including the data points. The green curveand data points are krog. The red curve and data points are krg.

18.4.4 Trapped Gas

We did not have access to any trapped gas or gas hysteresis measurements for the field test cased

used in this thesis. Based on a literature review of mixed wet sandstones and carbonates, typical

maximum trapped gas saturation is between 0.2 and 0.3. The shape of this curve for carbonates

and mixed wet sandstones is better fit by Jerauld (1997) than by Land (1968). Specify the trapping

function based on Jerauld (1997), illustrated in Figure 18.11.

Sg,trap[Sg] =Sg

1 +

(1

Sgt,max− 1

)× S

(1+b

Sgt,max1−Sgt,max

)g

(18.11)

• b = 1, indicates 0 slope at Sg = 1.

• Sgt,max = 0.25; the maximum amount of trapped gas.

18.4.5 Trapped Oil

The values of trapped oil saturations vary significantly in the literature. Mixed wet sandstones

and carbonates have relatively low trapped oil saturations, with approximately 0.10− 0.15 typical.

The literature often does not report the trapped oil or kro hysteresis for mixed wet reservoirs. The

trapping function based on Jerauld (1997) is illustrated in Figure 18.12.

370

0.0 0.2 0.4 0.6 0.8 1.00.00

0.05

0.10

0.15

0.20

0.25

0.30

Sg

S gtrap

Trapped Gas

Figure 18.11: Trapped gas saturation as a function of maximum achieved gas saturation.

So,trap[So] =So

1 +

(1

Sot,max− 1

)× S

(1+b

Sot,max1−Sot,max

)o

(18.12)

• b = 1, indicates 0 slope at So = 1.

• Sot,max = 0.10; the maximum amount of trapped oil.

0.0 0.2 0.4 0.6 0.8 1.00.00

0.05

0.10

0.15

0.20

So

S otrap

Trapped Oil

Figure 18.12: Trapped oil saturation as a function of maximum oil saturation achieved after theinitial oil saturation.

371

18.4.6 Cycle Dependent Residual Oil Saturations

Al-Dhahli et al. (2012), and Fatemi and Sohrabi (2012), and Element et al. (2003) present

experimental results which show cycle dependent residual oil saturations. The following values

were selected based on the trends in these articles.

• Saturation path 1, water injection: Sorw = S0orw = S1

orw = S2orw = 0.231 based on a fit to the

oil/water SCAL data for rock type F5.

• Saturation path 2, gas injection: Sorg = S0org = S1

org = S2org = 0.15 based on Dr. Kazemi’s

experience that the gas/oil Sorg from the SCAL data of 0.05 was too low. Individual cores

often have a very low Sorg, but this is not representative of a reservoir scale simulation.

• Saturation path 3, water injection: S3orw = 0.13

• Saturation path 4, gas injection: S4org = 0.12

• Saturation path 5, water injection: S5orw = 0.11

• Saturation path 6, gas injection: S6org = 0.10

18.4.7 Water Relative Permeability

The following assumptions were selected for the water relative permeability.

• krw is a function of Sw only. This is valid for all water wet reservoirs and seems valid for

mixed wet reservoirs also. It is not a good assumption for strongly oil wet reservoirs.

• There is no water trapping by oil or gas.

• No physical/chemical process considered here reduces Swr.

• There is no water hysteresis.

• The krwg = krwo = krw and is based on krw calculated from the WOG=IDC waterflood

process for a water-oil system.

• When Sor changes from Sorw to Sorg, the krw follows the krwo curve to a higher endpoint

saturation if necessary. In this case, krw[1− Sorg] > k�rw.

372

• The krw is illustrated in Figure 18.13.

Specify the reference function k0rw for the water relative permeability based on the fit to the

water/oil data, Figure 18.13. It is only necessary to specify one reference function because the

water relative permeability curve does not change even if the residual saturations change.

• S0w,min = Swr = 0.059

• S0w,max = 1− Sorw = 1− 0.231 = 0.791

• k0rw,max = k�rw = 0.093

• n0w = nw = 4.49

k0rw[Sw] =

⎧⎪⎨⎪⎩

0, Sw < S0w,min

k0rw,max ×(

Sw − S0w,min

S0w,max − S0

w,min

)nw

, Sw ≥ S0w,min

(18.13)

Swr Sorw

�1�Sorw,krwo� �

�1�Sorg,krw�1�Sorg��

�Swr,0�

0.0 0.2 0.4 0.6 0.8 1.00.00

0.05

0.10

0.15

0.20

0.25

0.30

Sw

krw

Water Relative Permeability

Figure 18.13: Water relative permeability based on a fit to the oil-water data in Figure 18.9.

18.4.8 Gas Relative Permeability

The following assumptions were selected for the gas relative permeability.

• krg is a function of Sg only. This is valid for all the reviewed experiments in the literature.

• Gas is trapped using a formulation by Jerauld (1997); this fits the observed data better for

mixed wet carbonate reservoirs than the formulation by Land (1968).

373

• The trapped gas specified by Land (1968) and Jerauld (1997) refer to the total saturations.

To convert between total saturations and the saturations in the m2 systems, use the following

equation.

Sgt = Sg,m2

φm2

φt(18.14)

• The hysteresis in the gas relative permeability is related to the trapping of gas. Additional

gas is trapped when switching from an increasing scanning curve to a decreasing scanning

curve.

• The krg is based on data for a WOG=CDI gas-oil experiment.

• When Sor changes from Sorw to Sorg, the krg curve extends to a higher endpoint saturation

(1− Sor − Swr) if necessary. In this case, krg[1− Sor − Swr] > k�rg.

The gas relative permeability bounding curves are shown in Figure 18.14. The bounding curves

are the increasing relative permeability at zero initial gas saturation and the decreasing relative

permeability curve at maximum trapped gas saturation.

SwrSorgSgtmax

�SgG0,krg

G0�

�SgG1,krg

G1�

0.0 0.2 0.4 0.6 0.8 1.00.0

0.1

0.2

0.3

0.4

0.5

0.6

Sg

krg

Gas Relative Permeability

Figure 18.14: Bounding scanning curves for gas relative permeability based on Figure 18.10.

The following steps are associated with gas relative permeability hysteresis. The case described

first has an initial water flood with the initial Sg = 0, followed by alternating WAG cycles of gas

374

and water. It is assumed for this description that no gas comes out of solution during the initial

water flood (WOG=IDC).

First, specify the reference function k0rg for the gas relative permeability based on the fit to

the gas/oil data, Figure 18.14. It is only necessary to specify one reference function because the

drainage and imbibition bounding curves and all the scanning curves have the same curvature

ng. The parameters S0g,min, S

0g,max, k

0rg,max, and ng do not change even if the residual saturations

change. If the initial gas saturation is 0, then use the reference curve during the initial waterflood.

If the gas saturation is still 0 at the end of the initial waterflood, also use the reference curve for

the first gas injection cycle.

• S0g,min = Sgr = 0

• S0g,max = 1− Swr − S0

ogr = 1− 0.059 − 0.15 = 0.791

• k0rg,max = k�rg = 0.3

• n0g = ng = 2.3147

k0rg[Sg] =

⎧⎪⎨⎪⎩

0, Sg < S0g,min

k0rg,max ×(

Sg − S0g,min

S0g,max − S0

g,min

)ng

, Sg ≥ S0g,min

(18.15)

During water injection, the gas saturation decreases (WOG=IDD). Figure 18.15 illustrates a

decreasing scanning curve in green. Start by calculating the trapped gas based on Jerauld (1997),

Figure 18.11. Specify the minimum gas saturation for the scanning curve based on the trapped gas.

The maximum gas saturation and relative permeability for the scanning curve are the values at

the end of the previous increasing cycle. The two known points on the scanning curve are (Sgm2 , 0)

and (SAg , k

Arg), points 3© and 2© in Figure 18.15.

• SAg is the gas saturation at the end of the previous cycle, point 2© in Figure 18.15.

• kArg is the gas relative permeability at the end of the previous cycle, point 2© in Figure 18.15.

• SAgt = Sg,trap

[SAg

].

• Calculate the trapped gas

375

Sngt = max

[Sng,m2

φnm2

φnt

, SAgt

](18.16)

Note that Sngt may have decreased from Sprev max

gt based on flash changes and transfer between

the trapped and mobile systems.

• SDg,min = Sgm2 , point 3© in Figure 18.15.

• SDg,max = SA

g , point 2© in Figure 18.15.

• kDrg,max = kArg, point 2© in Figure 18.15.

• nDg = ng

kDrg[Sg] =

⎧⎪⎨⎪⎩

0, Sg < SDg,min

kDrg,max ×(

Sg − SDg,min

SDg,max − SD

g,min

)ng

, Sg ≥ SDg,min

(18.17)

SwrSorgSgtmax

1

2: �SgA,krg

A �

3: �SgM2,0�

4: �Sgmax,krgmax�

14: �Sgtmax,0�

0.0 0.2 0.4 0.6 0.8 1.00.0

0.1

0.2

0.3

0.4

Sg

krg


Figure 18.15: A decreasing gas relative permeability scanning curve shown in green, 2© decr−−→ 3©.This assumes that the previous values increased to 2© with values of (SA

g , kArg). Bounding curves

from Figure 18.14 are shown in red, 1© incr−−→ 4© and 4© decr−−→ 14©.

During gas injection, the gas saturation increases (WOG=DDI). Figure 18.16 illustrates an

increasing scanning curve in cyan. Start by specifying the SIg,max and kIrg,max for this scanning

376

curve. If Sorg is constant, this is (S0g,max, k

0rg,max). Calculate SI

gmin so that the new scanning curve

passes through the values at the end of the previous decreasing cycle (SAg , k

Arg), or (0, 0) if this is

the first gas injection cycle and there was no initial free gas. The two known points on the scanning

curve are (SAg , k

Arg) and (SI

g,max, kIrg,max), points 3© and 4© in Figure 18.16.

• SAg is the gas saturation at the end of the previous cycle, point 3© in Figure 18.16.

• kArg is the gas relative permeability at the end of the previous cycle, point 4© in Figure 18.16.

• SIg,max = 1−Swr−Sn

org, point 4© in Figure 18.16. Note that Snorg may change with the WAG

cycle.

• kIrg,max = k0rg[SIgmax], point 4© in Figure 18.16.

• Use the constraint that kIrg[SAg ] = kArg to obtain SI

gmin, point 13© in Figure 18.16.

α =

(kArg

kIrg,max

) 1ng

(18.18)

SIgmin =

α · SIgmax − SA

g

α− 1(18.19)

• nIg = ng

kIrg[Sg] =

⎧⎪⎨⎪⎩

0, Sg < SIg,min

kIrg,max ×(

Sg − SIg,min

SIg,max − SI

g,min

)ng

, Sg ≥ SIg,min

(18.20)

18.4.9 Oil Relative Permeability

The following assumptions were selected for the oil relative permeability.

• krow and krog are very close to each other (Figure 18.17,Figure 18.18,Figure 18.19). This

means that kro is a function of So if the Sor is adjusted appropriately.

• Oil is trapped using a formulation by Jerauld (1997); this fits the observed data better for

mixed wet carbonate reservoirs than the formulation by Land (1968).

377

SwrSorgSgtmax

1

2

3: �SgA,krg

A �

4: �Sgmax,krgmax�

14: �Sgtmax,0�12:�SgM2

A ,0�13:�Smin

I ,0�

0.0 0.2 0.4 0.6 0.8 1.00.0

0.1

0.2

0.3

0.4

Sg

krg


Figure 18.16: An increasing gas relative permeability scanning curve shown in cyan, 3© decr−−→ 4©.This assumes that the previous values decreased to 3© with values of (SA

g , kArg). The previous

decreasing scanning curve was 2© decr−−→ 3© decr−−→ 12©, but the saturation did not drop below (SAg , k

Arg).

Bounding curves from Figure 18.14 are shown in red, 1© incr−−→ 4© and 4© decr−−→ 14©.

• The trapped oil specified by Land (1968) and Jerauld (1997) refer to the total saturations.

To convert between total saturations and the saturations in the m2 systems, use the following

equation.

Sot = So,m2

φm2

φt(18.21)

• The maximum trapped oil Sot,max is less than the residual oil saturation Sorg or Sorw, even

when Sor is cycle-dependent. This means there is no hysteresis in the kro.

• The krog is based on data for a WOG=CDI gas-oil experiment.

• The krow is based on data for a WOG=IDC water-oil experiment.

• When Sor changes from Sorw to Sorg, and may also continue to decrease with the WAG cycle.

The oil relative permeability krow from the oil-water SCAL data is illustrated in Figure 18.17.

The oil relative permeability krog from the gas-oil SCAL data is illustrated in Figure 18.18. The

krow and krog curves are very similar for this data, as illustrated in Figure 18.19. The properties

of the krow and krog reference curves are as follows:

378

• S0orw = 0.231; if Sorw is cycle-dependent, then SW2

orw = 0.13 and SW3orw = 0.11.

• S0org = 0.15; if Sorg is cycle-dependent, then SG2

org = 0.12 and SG3org = 0.10.

• S0o,max = 1− Swr = 1− 0.059 = 0.941

• k0ro,max = k�rog = k�row = 0.3

• n0ow = now = 3.76

• n0og = now = 4.17

k0row[So] =

⎧⎨⎩

0, So < S0orw

k0ro,max ×(

So − S0orw

S0o,max − S0

orw

)now

, So ≥ S0orw

(18.22)

k0rog[So] =

⎧⎪⎨⎪⎩

0, So < S0org

k0ro,max ×(

So − S0org

S0o,max − S0

org

)nog

, So ≥ S0org

(18.23)

Sorw

Sorg

Swr

�Sorw,0�

�1�Swr,krow� �

0.0 0.2 0.4 0.6 0.8 1.00.0

0.1

0.2

0.3

0.4

So

krow

Oil Relative Permeability

Figure 18.17: Oil relative permeability based on from the oil/water SCAL, Figure 18.9.

The maximum amount of trapped oil, Smaxot < Sorg < Sorw, Figure 18.20 and Figure 18.21. Using

the approach for scanning curves described in Section 18.4.8 would mean that the krog and krow

would follow the same scanning curves, illustrated in Figure 18.20 and Figure 18.21. Because the

residual oil saturation Sorw or Sorg changes it is still necessary to calculate increasing and decreasing

scanning curves. The trapped oil is still calculated at the end of oil-increasing saturation paths;

379

Sorw

Sorg

Swr

�Sorg,0�


0.0 0.2 0.4 0.6 0.8 1.00.0

0.1

0.2

0.3

0.4

So

krog


Figure 18.18: Oil relative permeability based on from the gas/oil SCAL, Figure 18.10.

Sorw

Sorg

Swr

0.0 0.2 0.4 0.6 0.8 1.00.0

0.1

0.2

0.3

0.4

So

krow

,kro

g


Figure 18.19: Compare the krow in cyan, Figure 18.17, and the krog in purple, Figure 18.18. Forthis data set the curves are very similar.

380

this trapped oil saturation effects the composition of Som1 and Som2 even though it does not effect

the kro. The trapped oil, Sot renormalized as Som2 , only interacts with the mobile system through

a transfer function. Mobile oil Sot < S < Sor is in thermodynamic equilibrium with the additional

mobile oil, gas, and water; the compositions flash between Som1 , Sgm1 , and Swm1 at each nonlinear

iteration.

Sorw

Sorg

Swr

Sotmax

�Sorw,0�


�Sotmax,0�

0.0 0.2 0.4 0.6 0.8 1.00.0

0.1

0.2

0.3

0.4

So

krow


Figure 18.20: The krow scanning curves have no hysteresis because Smaxot ≤ Sorg < Sorw.

Sorw

Sorg

Swr

Sotmax

�Sorg,0�


�Sotmax,0�

0.0 0.2 0.4 0.6 0.8 1.00.0

0.1

0.2

0.3

0.4

So

krog


Figure 18.21: The krog scanning curves have no hysteresis because Smaxot ≤ Sorg < Sorw.

Although the trapped oil does not effect the oil relative permeability hysteresis, the oil relative

permeability curves still change based on changes in the residual oil saturation.

Figure 18.22 illustrates a decreasing scanning curve in black. The two known points on the

scanning curve are (Sor, 0) and (SAo , k

Aro), points 5© and 3© in Figure 18.15.

• SAo is the oil saturation at the end of the previous cycle, point 3© in Figure 18.22.

• kAro is the oil relative permeability at the end of the previous cycle, point 3© in Figure 18.22.

381

• SAot = So,trap

[SAo

].

• Calculate the trapped oil

Snot = max

[Sno,m2

φnm2

φnt

, SAot

](18.24)

Note that Snot may have decreased from Sprev max

ot based on flash changes and transfer between

the trapped and mobile systems.

• SDo,min = Scycle

or , point 5© in Figure 18.22. For the first water injection, SDo,min = SW1

orw. For the

first gas injection, SDo,min = SG1

org.

• SDo,max = 1− Swr, point 2© in Figure 18.22.

• If the current cycle is gas injection, nDo = nog. If the current cycle is water injection, n

Do = now.

• Solve for kDro,max, point 2© in Figure 18.22, using the constraint that kDro[SAo ] = kAro.

kDro,max = kAro ×(

SAo − SD

o,min

SDo,max − SD

o,min

)−nDo

(18.25)

kDro[So] =

⎧⎪⎨⎪⎩

0, So < SDo,min

kDro,max ×(

So − SDo,min

SDo,max − SD

o,min

)nDo

, So ≥ SDo,min

(18.26)

Figure 18.23 illustrates an increasing scanning curve in black. Start by specifying the SIo,max

and kIro,max for this scanning curve. Calculate SIomin so that the new scanning curve passes through

the values at the end of the previous cycle (SAo , k

Aro). The two known points on the scanning curve

are (SAo , k

Aro) and (SI

o,max, kIro,max), points 3© and 2© in Figure 18.23.

• SAo is the oil saturation at the end of the previous cycle, point 3© in Figure 18.23.

• kAro is the oil relative permeability at the end of the previous cycle, point 3© in Figure 18.23.

• SIo,max = 1− Swr, point 2© in Figure 18.23.

382

Sorw

Sorg

Swr

4:�Sorw,0�

1:�1�Swr,krow� �

5:�SominD ,0�

3:�SoA,kro

A �

2:�SomaxD ,kromax

D �

0.0 0.2 0.4 0.6 0.8 1.00.0

0.1

0.2

0.3

0.4

So

krow


Figure 18.22: A decreasing oil relative permeability scanning curve is shown in black, 3© decr−−→ 5©.This assumes the values at the end of the previous cycle, 3©, achieved values of (SA

o , kAro). The

reference curve is shown in green, 1© decr−−→ 3© decr−−→ 4©.

• kIro,max = k0ro[SIomax] = k�ro, point 2© in Figure 18.23.

• If the current cycle is gas injection, nIo = nog. If the current cycle is water injection, n

Io = now.

• Use the constraint that kIro[SAg ] = kAro to obtain SI

omin, point 4© in Figure 18.23.

α =

(kAro

kIro,max

) 1

nIo

(18.27)

SIomin =

α · SIomax − SA

o

α− 1(18.28)

kIro[So] =

⎧⎪⎨⎪⎩

0, So < SIo,min

kIro,max ×(

So − SIo,min

SIo,max − SI

o,min

)nIo

, So ≥ SIo,min

(18.29)

18.5 Capillary Pressure

The capillary pressure curves are represented by equations of the following form.

Pc1 = Pc,offset − α(Sx − Smin)× log

[S − Smin

Sx − Smin

](18.30)

383

Sorw

Sorg

Swr

4:�Sorw,0�

1:�1�Swr,krow� �

5:�SominI ,0�

3:�SoA,kro

A �

2:�SomaxI ,kromax

I �

0.0 0.2 0.4 0.6 0.8 1.00.0

0.1

0.2

0.3

0.4

So

krow


Figure 18.23: An increasing oil relative permeability scanning curve is shown in black, 3© incr−−→ 2©.This assumes the values at the end of the previous cycle, 3©, achieved values of (SA

o , kAro). The

reference curve is shown in green, 1© decr−−→ 3© decr−−→ 4©.

Pc2 = Pc,offset + α(Smax − Sx)× log

[Smax − S

Smax − Sx

](18.31)

Pcow =

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

Pcow,max, S ≤ Smin

Pcow,max, Pc1 > Pcow,max

Pc1, Smin < S < Sx

Pc2, Sx < S < Smax

Pcow,min, Pc2 < Pcow,min

Pcow,min, S ≥ Smax

(18.32)

The increasing water (imbibition) oil-water capillary pressure data is shifted to correspond to

the Swr measured for the relative permeability data. After this shift, a capillary pressure curve

P I0cow of the following form is fit to the data. Figure 18.24 illustrates both the increasing capillary

pressure curve and the data points.

• Smin = Swr = 0.059

• Sorw = 0.231; Smax = 1− Sorw

• SIwx = 0.28

• αIow = 5

384

• P Ic,offset = −3.7


• Pcow,max = 20

A bounding decreasing capillary pressure curve PD0cow was then estimated so that the capillary

pressure hysteresis is consistent with the literature. Figure 18.24 illustrates the decreasing capillary

pressure curve.

• Smin = Swr = 0.059

• Sorw = 0.231; Smax = 1− Sorw

• SDwx = 0.33

• αDow = 6.5

• PDc,offset = −1.7


• Pcow,max = 20

Scanning curves are calculated based on interpolating between the bounding curves; see Fig-

ure 18.25. When switching from increasing to decreasing or decreasing to increasing scanning

curves, the last achieved saturation SA is used in the following interpolation.

PScow[S, SA] = P I0

cow[S] + (PD0cow[S]− P I0

cow[S])×(

SA − S0min

S0max − S0

min

)(18.33)

Decreasing capillary pressure scanning curves are calculated as illustrated in Figure 18.26. The

following procedure is used.

1. The previous achieved value of SA corresponds to the previous capillary pressure PAcow based

on the previous scanning curve, the black star in Figure 18.26.

2. Calculate the new scanning curve PScow[S, SA], the green curve in Figure 18.26.

385

0.0 0.2 0.4 0.6 0.8 1.0�20

�10

0

10

20

Sw

Pcow

Oil�Water Capillary Pressure

Figure 18.24: Oil-water capillary pressure curves including the data points. The blue curve anddata points represent increasing water saturation, decreasing oil saturation (imbibition). The redcurve represents decreasing water saturation, increasing oil saturation (drainage).

386

0.0 0.2 0.4 0.6 0.8 1.0

�20

�10

0

10

20

Sw

Pcow


Figure 18.25: Capillary pressure bounding curves and interpolated scanning curves. The blue curveis the bounding increasing water curve. The red curve is the bounding decreasing water curve. Thegreen curves are interpolated scanning curves.

387

3. Calculate the saturation SB where PScow[SB , SA] = PA

cow.

4. Shift the scanning curve to match the points (Smin, Pcow,max) and (SA, PAcow). Define the new

increasing scanning curve, the black curve in Figure 18.26, as follows.

PDcow = PS

cow

[S0min + (S − S0

min)×(SB − S0

min

SA − S0min

), SA

](18.34)

�SA,PcowA �

�SB,PcowB �

�Swr,Pcowmax�

0.0 0.2 0.4 0.6 0.8 1.0

�20

�10

0

10

20

Sw

Pcow


Figure 18.26: Decreasing capillary pressure scanning curve in black from the blue star to the blackstar and back towards the blue star. The blue curve is the bounding increasing water curve. Thered curve is the bounding decreasing water curve. The green curve is the interpolated decreasingscanning curve corresponding to SA. The green curve between S0

min and SB is mapped onto theblack curve between S0

min to SA.

Increasing capillary pressure scanning curves are calculated as illustrated in Figure 18.27. The

following procedure is used.

388

1. The previous achieved value of SA corresponds to the previous capillary pressure PAcow based

on the previous scanning curve, the black star in Figure 18.27.

2. Calculate the new scanning curve PScow[S, SA], the green curve in Figure 18.27.

3. Calculate the saturation SB where PScow[SB , SA] = PA

cow.

4. Shift the scanning curve to match the points (SA, PAcow) and (Smax, Pcow,min). Define the new

decreasing scanning curve, the black curve in Figure 18.27, as follows.

P Icow = PS

cow

[SB + (S − SA)×

(S0max − SB

S0max − SA

), SA

](18.35)

18.6 Future Test Case Scenarios

Various additional test cases could be run to understand the sensitivities.

1. Easiest to evaluate; no code changes required

1.1. Review additional cases with heterogeneity.

1.2. An alternate reservoir depth of for instance 5000 ft could be used to simulate a reservoir

with similar properties at a shallower depth. The reservoir pressure is based on the

same gradient of 0.508 psia/ft would remain the same, leading to a reservoir pressure

of 2550 psia. The fracture gradient of 0.75 psia/ft would remain the same, leading to a

fracture pressure of 3750 psia. The temperature gradient of 0.019◦F/ft would remain the

same, leading to a reservoir temperature of 162◦F. If there is an initial gas saturation,

may need to adjust the krg curves.

1.3. Alternate horizontal grid spacing; for instance 21 × 21 → 42 × 42. Very easy if either

homogeneous or integer multiple of previous model

1.4. Alternate injection rates, production rates, and production schemes: initial waterflood,

initial gas flood, initial WAG.

1.5. Vary WAG ratio and length.

1.6. Evaluate different criteria to switch from primary production to waterflood to gas flood

to WAG.

389

�SA,PcowA �

�SB,PcowB �

�1�Sorw,Pcowmin �

0.0 0.2 0.4 0.6 0.8 1.0

�20

�10

0

10

20

Sw

Pcow


Figure 18.27: Increasing capillary pressure scanning curve in black from the red star to the blackstar and back towards the red star. The blue curve is the bounding increasing water curve. Thered curve is the bounding decreasing water curve. The green curve is the interpolated decreasingscanning curve corresponding to SA. The green curve between SB and S0

max is mapped onto theblack curve between SA and S0

max

390

1.7. Add horizontal anisotropy to the permeability.

2. Moderately easy to evaluate; some small code changes

2.1. Default: change kro, krg, Pcow, Sorw and Sorg everywhere when switch between water

and gas injection. Another option is to switch when the gas/water starts increasing in

a cell.

2.2. For the base case, the krow and krog are very close, so kro[So only]. A different three-phase

relative permeability curve could be used, such as a renormalized Stone II Stone (1973),

a saturation weighted approach such as Baker (1988), or a new published approach.

2.3. An alternate reservoir dip of 1◦. Need to initialize P as a function of depth.

2.4. Evaluate case with 2 km × 2 km development; for initial tests it was difficult to get

realistic well performance without 10× or more increase in effective permeability

3. More difficult to evaluate; more code changes and testing

3.1. 2-D cross-section or 3-D model. Need to initialize P as a function of depth. May need a

Pcgo. Need a different set of kr and Pc curves for each layer. Adjust the thicknesses of

each layer to match Shibasaki et al. (2006). Need permeability and porosity distribution

for each layer. Vary the vertical permeability based on the presence of stylolites. Use

Zm′ as constant throughout and temperature as constant.

3.2. Add simulation of a natural fracture system.

3.3. Add horizontal wells.

3.4. Add hydraulically fractured horizontal wells

3.5. Use a tracer to identify when injected gas arrives in a cell (as opposed to solution gas).

Use this to switch relative permeability curves.

391

CHAPTER 19

DISCUSSION OF RESULTS

Different scenarios were created and simulated with varying formulation options expected to

have an impact on the trapped fluids:

• Trapping: model with no compositional trapping (single media system m1) versus a model

with compositional trapping (dual media system m1 and m2).

• Heterogeneity: homogeneous or heterogeneous.

• Geometry: 2-D 1/4 5-spot pattern or 2-D injector centered 5-spot pattern.

• Mass Transfer: Vary km2 to represent a slower or faster transfer rate between the m1 and m2

systems.

• WCO2 : No aqueous CO2 solubility (WCO2 = 0), or with the most stable and accurate formu-

lation for WCO2 > 0.

• Gas relative permeability hysteresis: No gas relative permeability hysteresis and no gas trap-

ping (krg = kI0rg); with gas relative permeability hysteresis but without the compositional

variations of the trapped gas; or with gas relative permeability hysteresis and compositional

trapped gas.

• Trapped oil: Trapped oil based only on Jerauld (1997), or trapped oil based on Jerauld (1997)

plus an additional 0.10 saturation units immediately after the waterflood.

• Vary Sor: With or without cycle-dependent residual oil saturations. With either option the

Sorw and Sorg are different, but with cycle-dependent residual saturations the Sorw and Sorg

decrease during the first three WAG cycles.

Table 19.1 provides a description of the specific test cases selected with comments on the

purpose of each one. Models with names starting with W are homogeneous 2-D models without

compositional trapping. Models with names starting with X are homogeneous 2-D models with

392

compositional trapping. Models with names starting with Y are heterogeneous 2-D models without

compositional trapping. Models with names starting with Z are heterogeneous 2-D models with

compositional trapping.

Each scenario is run under four different production schemes.

• Primary production (40 acre): Primary production to an economic limit of 10 RBOPD.

• Waterflood (20 acre): Primary production followed by initiation of water injection approx-

imately 180 days after the economic limit for primary production is reached. Simulate the

waterflood until the rate drops again to an economic limit of 10 RBOPD.

• CO2 injection (20 acre): Primary production followed by waterflood followed by initiation of

CO2 injection approximately 360 days after the economic limit for the waterflood is reached.

Simulate the CO2 injection until the rate drops again to an economic limit of 10 RBOPD,

which typically occurs at a minimum point in the CO2 utitlization curve. Production also

typically stops when gas represents 100% of the production. Although this may occur simply

due to the single production phase getting labeled as “gas” rather than “oil”, the producing

compositions confirm that the production is almost all CO2 at this point.

• CO2 WAG (20 acre): Primary production followed by waterflood followed by CO2 injection

followed by initiation of WAG when the oil rate increases again above 10 RBOPD. Simulate

the CO2 WAG injection until the rate drops again to an economic limit of 10 RBOPD.

The scenarios were evaluated based on several different criteria. The most important criterion

is the recovery factor at the economic limit for each production scheme, which was evaluated

at reservoir conditions, but may be flashed to surface conditions in a separate calculation. The

following criteria were evaluated at the economic limit of each production scheme.

• Recovery factor (RB/RB)

• Time to economic limit.

• CO2 storage (lbmol/lbmol)

CO2 storage =cumulative CO2 injected− cumulative CO2 produced

cumulative CO2 injected(19.1)

393

Table 19.1: Description of test case scenariosModel

type

Model

#

Pattern

Com

positional

trap

ping

Heterogen

eity

Transfer:km

2

WCO

2op

tion

Hysteresis

ofkrg

Extratrap

ped

oilafterW

F

Cycledep

endentSor

Purpose

W 551 1/4 no no NA 0 no trap + no hysteresis NA no Least base: single-media + least trappingW 561 1/4 no no NA > 0 no trap + no hysteresis NA no Least base + WCO2

W 562 1/4 no no NA 0 trap + hysteresis NA no Least base + gas hysteresisW 563 1/4 no no NA 0 no trap + no hysteresis NA yes Least base + cycleW 564 1/4 no no NA > 0 trap + hysteresis NA yes Single-media: most trapping

X 550 1/4 yes no 5 · 10−5 > 0 trap + hysteresis yes yes Most base: dual-media + with most trappingX 552 1/4 yes no 5 · 10−5 0 trap + hysteresis no no Dual-media + mostly gas trapX 553 1/4 yes no 5 · 10−5 0 no trap + no hysteresis yes yes Dual-media + mostly oil trapX 554 1/4 yes no 5 · 10−5 > 0 trap + hysteresis no no Dual-media + mostly CO2 trapX 571 1/4 yes no 5 · 10−7 > 0 trap + hysteresis yes yes Most base + lower km2

X 572 1/4 yes no 5 · 10−9 > 0 trap + hysteresis yes yes Most base + much lower km2

X 573 1/4 yes no 5 · 10−3 > 0 trap + hysteresis yes yes Most base + higher km2

X 574 1/4 yes no 5 · 10−5 0 trap + hysteresis yes yes Most base: WCO2 = 0X 575 1/4 yes no 5 · 10−5 > 0 hysteresis + no trap yes yes Most base: hysteresis + no trap gasX 576 1/4 yes no 5 · 10−5 > 0 no trap + no hysteresis yes yes Most base: no trap gas + no hysteresisX 577 1/4 yes no 5 · 10−5 > 0 trap + hysteresis no yes Most base: no extra oil trap after WFX 578 1/4 yes no 5 · 10−5 > 0 trap + hysteresis yes no Most base + no cycle Sor

X 579 1/4 yes no 5 · 10−5 0 no trap + no hysteresis no no Dual-media: least trapping

Y 560 1/4 no yes NA 0 no trap + no hysteresis NA no Least base + heterogeneityZ 570 1/4 yes yes 5 · 10−5 > 0 trap + hysteresis yes yes Most base + heterogeneityY 580 5-spot no yes NA 0 no trap + no hysteresis NA no Least base + heterogeneityZ 581 5-spot yes yes 5 · 10−5 > 0 trap + hysteresis yes yes Most base + heterogeneity

394

• CO2 utilization (MCF/RB)

CO2 storage =cumulative CO2 injected[

cumulative oilproduced with CO2

]− [cumulative oilproduced with waterflood

] (19.2)

• Compositional recovery factor (lbmol/lbmol) for CH4, nC4, and nC10

19.1 Evaluation of Primary Production Performance

Table 19.2 presents the recovery factor (RF) and time to economic limit (EL) for primary

production for all of the scenarios. The economic limit for primary production is the same for all

of the homogeneous models without trapping. The economic limit for primary production is the

same for all of the homogeneous models with compositional trapping. The models with trapping

include a 0.01 SU of water and gas during the primary production to stabilize the computations.

This small amount of trapping leads to 90 days of additional time before the economic limit is

reached. For heterogeneous cases, the time to the economic limit varies because of variations in the

porosity, permeability, transmissibility, and flow paths.

The primary recovery factor for the homogeneous models without trapping is the lowest. The

presence of a little bit of trapped water and oil causes more time to pass before the economic limit

is reached and this also corresponds to a larger recovery factor than without trapping.

19.2 Evaluation of Waterflood Performance

Table 19.3 presents the time to economic limit (EL) for primary and waterflood (WF) and the

recovery factor (RF) for the primary and waterflood, plus the incremental time and incremental

recovery. Table 19.3 is ordered based on the incremental time between the economic limit of

primary production and the economic limit of waterflood production. The timing for the end of

primary production was similar for all the models, so the ranking of the economic limit at the end

of the waterflood and the additional days of production between the end of primary production and

the end of the waterflood are the same. The economic limit for the waterflood is reached earliest

for the homogeneous cases without trapping. For the cases with trapping, the economic limit for

the waterflood is reached earliest for the cases with no trapped gas or gas relative permeability

hysteresis. Although the times to the waterflood economic limit varies with the value of km2 , the

changes in the times do not follow an obvious pattern.

395

Table 19.2: Primary production: recovery factor and time to economic limit

Model

Type

Model

#

Pattern

Com

positional

Trapping

Heterogen

eity

Transfer:km

2

WCO

2Option

Hysteresis

ofkrg

ELPrimary(10days)

RFPrimary(R

CF)

Y 560 1/4 no yes NA 0 no trap + no hysteresis 662 0.192441Z 570 1/4 yes yes 5 · 10−5 > 0 trap + hysteresis 671 0.203522Y 580 5-spot no yes NA 0 no trap + no hysteresis 680 0.179983Z 581 5-spot yes yes 5 · 10−5 > 0 trap + hysteresis 689 0.190667

W 551 1/4 no no NA 0 no trap + no hysteresis 720 0.187443W 561 1/4 no no NA > 0 no trap + no hysteresis 720 0.191192W 562 1/4 no no NA 0 trap + hysteresis 720 0.191192W 563 1/4 no no NA 0 no trap + no hysteresis 720 0.191192W 564 1/4 no no NA > 0 trap + hysteresis 720 0.191192

X 572 1/4 yes no 5 · 10−9 > 0 trap + hysteresis 722 0.199968X 550 1/4 yes no 5 · 10−5 > 0 trap + hysteresis 729 0.202195X 552 1/4 yes no 5 · 10−5 0 trap + hysteresis 729 0.202195X 553 1/4 yes no 5 · 10−5 0 no trap + no hysteresis 729 0.202195X 554 1/4 yes no 5 · 10−5 > 0 trap + hysteresis 729 0.202195X 571 1/4 yes no 5 · 10−7 > 0 trap + hysteresis 729 0.202042X 573 1/4 yes no 5 · 10−3 > 0 trap + hysteresis 729 0.202195X 574 1/4 yes no 5 · 10−5 0 trap + hysteresis 729 0.202195X 575 1/4 yes no 5 · 10−5 > 0 hysteresis + no trap 729 0.202195X 576 1/4 yes no 5 · 10−5 > 0 no trap + no hysteresis 729 0.202195X 577 1/4 yes no 5 · 10−5 > 0 trap + hysteresis 729 0.202195X 578 1/4 yes no 5 · 10−5 > 0 trap + hysteresis 729 0.202195X 579 1/4 yes no 5 · 10−5 0 no trap + no hysteresis 729 0.202195

396

Table 19.3: Waterflood time to economic limit

Model

Type

Model

#

Pattern

CompositionalTrapping

Heterogen

eity

Transfer:km

2

WCO

2Option

Hysteresis

ofkrg

ELPrimary

(10day

s)

ELW

F(10day

s)

Tim

eW

F−Primary

(10day

s)

RFPrimary

(RCF)

RFW

F(R

CF)

RFW

F−Primary

(RCF)

Y 580 5-spot no yes NA 0 no trap + no hysteresis 680 1489 789 0.179983 0.562401 0.382418W 561 1/4 no no NA > 0 no trap + no hysteresis 720 1539 789 0.191192 0.638989 0.447797W 551 1/4 no no NA 0 no trap + no hysteresis 720 1564 814 0.187443 0.603017 0.415574W 563 1/4 no no NA 0 no trap + no hysteresis 720 1564 814 0.191192 0.615077 0.423885W 562 1/4 no no NA 0 trap + hysteresis 720 1572 822 0.191192 0.620464 0.429272W 564 1/4 no no NA > 0 trap + hysteresis 720 1573 823 0.191192 0.643650 0.452458X 576 1/4 yes no 5 · 10−5 > 0 no trap + no hysteresis 729 1606 856 0.202195 0.588463 0.386268X 575 1/4 yes no 5 · 10−5 > 0 hysteresis + no trap 729 1646 896 0.202195 0.660872 0.458677X 553 1/4 yes no 5 · 10−5 0 no trap + no hysteresis 729 1668 918 0.202195 0.570173 0.367978X 579 1/4 yes no 5 · 10−5 0 no trap + no hysteresis 729 1668 918 0.202195 0.570173 0.367978X 552 1/4 yes no 5 · 10−5 0 trap + hysteresis 729 1896 1146 0.202195 0.678063 0.475868X 574 1/4 yes no 5 · 10−5 0 trap + hysteresis 729 1896 1146 0.202195 0.678063 0.475868Z 581 5-spot yes yes 5 · 10−5 > 0 trap + hysteresis 689 1915 1215 0.190667 0.646166 0.455499X 572 1/4 yes no 5 · 10−9 > 0 trap + hysteresis 722 1999 1249 0.199968 0.733150 0.533182X 573 1/4 yes no 5 · 10−3 > 0 trap + hysteresis 729 2002 1252 0.202195 0.650930 0.448735X 550 1/4 yes no 5 · 10−5 > 0 trap + hysteresis 729 2054 1304 0.202195 0.683429 0.481234X 554 1/4 yes no 5 · 10−5 > 0 trap + hysteresis 729 2054 1304 0.202195 0.683429 0.481234X 577 1/4 yes no 5 · 10−5 > 0 trap + hysteresis 729 2054 1304 0.202195 0.683429 0.481234X 578 1/4 yes no 5 · 10−5 > 0 trap + hysteresis 729 2054 1304 0.202195 0.683429 0.481234Y 560 1/4 no yes NA 0 no trap + no hysteresis 662 2073 1383 0.192441 0.625756 0.433315X 571 1/4 yes no 5 · 10−7 > 0 trap + hysteresis 729 2215 1465 0.202042 0.726529 0.524487Z 570 1/4 yes yes 5 · 10−5 > 0 trap + hysteresis 671 2194 1504 0.203522 0.673626 0.470104

397

For both the cases with and without compositional trapping, the cases are also ordered from

earliest time to economic limit to latest time to economic limit as follows:

1. WCO2 > 0, no gas relative permeability hysteresis

2. WCO2 > 0, no compositional gas trapping but gas relative permeability hysteresis

3. WCO2 = 0, no gas relative permeability hysteresis

4. WCO2 = 0, with gas trapping and hysteresis

5. WCO2 > 0, with gas trapping and hysteresis

If two cases with compositional trapping are compared, the change in producing time is bigger

than if two cases without compositional trapping are compared. The presence of CO2 solubility in

water makes the difference in the calculation of the gas relative permeability hysteresis much more

significant, especially when compositional trapping effects are considered.

Table 19.4 presents the time to economic limit (EL) for primary and waterflood (WF) and the

recovery factor (RF) for the primary and waterflood, plus the incremental time and incremental

recovery. Table 19.4 is ordered based on the incremental recovery between the economic limit of

primary production and the economic limit of waterflood production. The lowest recovery factor

at the economic limit of the waterflood and also the lowest incremental waterflood recovery over

primary production occurs for the cases with compositional trapping but with no gas hysteresis.

Next are the cases with no compositional trapping. The compositional trapping cases with different

km2 increase their waterflood recovery as the km2 decreases. The cases with WCO2 = 0 have lower

recovery than the cases with WCO2 > 0. The cases with gas relative permeability hysteresis have

higher recoveries at the end of waterflood than cases with no gas relative permeability hysteresis.

For the compositional trapping cases, the gas relative permeability hysteresis has a larger impact

than the WCO2 . For the system without compositional trapping, the WCO2 is more important than

the hysteresis in the gas relative permeability.

19.3 Evaluation of Continuous CO2 Injection

Table 19.5 presents the start time of the waterflood, the start time of the continuous CO2

injection, the time of the increased oil production corresponsding to CO2 response, and the time

398

Table 19.4: Waterflood recovery factor

Model

Type

Model

#

Pattern


Heterogen

eity

Transfer:km

2

WCO

2Option

Hysteresis

ofkrg

ELPrimary

(10day

s)

ELW

F(10day

s)

Tim

eW

F−Primary

(10day

s)

RFPrimary

(RCF)

RFW

F(R

CF)

RFW

F−Primary

(RCF)

X 553 1/4 yes no 5 · 10−5 0 no trap + no hysteresis 729 1668 918 0.202195 0.570173 0.367978X 579 1/4 yes no 5 · 10−5 0 no trap + no hysteresis 729 1668 918 0.202195 0.570173 0.367978Y 580 5-spot no yes NA 0 no trap + no hysteresis 680 1489 789 0.179983 0.562401 0.382418X 576 1/4 yes no 5 · 10−5 > 0 no trap + no hysteresis 729 1606 856 0.202195 0.588463 0.386268W 551 1/4 no no NA 0 no trap + no hysteresis 720 1564 814 0.187443 0.603017 0.415574W 563 1/4 no no NA 0 no trap + no hysteresis 720 1564 814 0.191192 0.615077 0.423885W 562 1/4 no no NA 0 trap + hysteresis 720 1572 822 0.191192 0.620464 0.429272Y 560 1/4 no yes NA 0 no trap + no hysteresis 662 2073 1383 0.192441 0.625756 0.433315W 561 1/4 no no NA > 0 no trap + no hysteresis 720 1539 789 0.191192 0.638989 0.447797X 573 1/4 yes no 5 · 10−3 > 0 trap + hysteresis 729 2002 1252 0.202195 0.650930 0.448735W 564 1/4 no no NA > 0 trap + hysteresis 720 1573 823 0.191192 0.643650 0.452458Z 581 5-spot yes yes 5 · 10−5 > 0 trap + hysteresis 689 1915 1215 0.190667 0.646166 0.455499X 575 1/4 yes no 5 · 10−5 > 0 hysteresis + no trap 729 1646 896 0.202195 0.660872 0.458677Z 570 1/4 yes yes 5 · 10−5 > 0 trap + hysteresis 671 2194 1504 0.203522 0.673626 0.470104X 552 1/4 yes no 5 · 10−5 0 trap + hysteresis 729 1896 1146 0.202195 0.678063 0.475868X 574 1/4 yes no 5 · 10−5 0 trap + hysteresis 729 1896 1146 0.202195 0.678063 0.475868X 550 1/4 yes no 5 · 10−5 > 0 trap + hysteresis 729 2054 1304 0.202195 0.683429 0.481234X 554 1/4 yes no 5 · 10−5 > 0 trap + hysteresis 729 2054 1304 0.202195 0.683429 0.481234X 577 1/4 yes no 5 · 10−5 > 0 trap + hysteresis 729 2054 1304 0.202195 0.683429 0.481234X 578 1/4 yes no 5 · 10−5 > 0 trap + hysteresis 729 2054 1304 0.202195 0.683429 0.481234X 571 1/4 yes no 5 · 10−7 > 0 trap + hysteresis 729 2215 1465 0.202042 0.726529 0.524487X 572 1/4 yes no 5 · 10−9 > 0 trap + hysteresis 722 1999 1249 0.199968 0.733150 0.533182

399

Table 19.5: Continuous CO2 recovery factorModel

Type

Model

#

Pattern


Heterogen

eity

Transfer:km2

WCO

2Option

Hysteresis

ofkrg

ExtraTrapped

OilAfter

WF

Start

WF

(10day

s)

Start

GF

(10day

s)

OilResponse

GF

(10day

s)

Economic

Lim

itGF

(10day

s)

Tim

eGF−

WF

(10day

s)

Tim

eGF−

GF

resp

onse

(10day

s)

RF

WF

(RCF)

RF

GF

(RCF)

RF

GF−

WF

(RCF)

X 552 1/4 yes no 5 · 10−5 0 trap + hysteresis no 750 1930 2774 2785 844 11 0.678063 0.691281 0.013218X 574 1/4 yes no 5 · 10−5 0 trap + hysteresis yes 750 1930 2314 2352 384 38 0.678063 0.698792 0.020729X 573 1/4 yes no 5 · 10−3 > 0 trap + hysteresis yes 750 2040 2468 2525 428 57 0.650930 0.680371 0.029441X 578 1/4 yes no 5 · 10−5 > 0 trap + hysteresis yes 750 2090 2524 2580 434 56 0.683429 0.713990 0.030561X 550 1/4 yes no 5 · 10−5 > 0 trap + hysteresis yes 750 2070 2497 2552 427 55 0.683429 0.714275 0.030846X 553 1/4 yes no 5 · 10−5 0 no trap + no hysteresis yes 750 1700 2272 2382 572 110 0.570173 0.619315 0.049142Z 570 1/4 yes yes 5 · 10−5 > 0 trap + hysteresis yes 690 2230 2672 2848 442 176 0.673626 0.724226 0.050600X 575 1/4 yes no 5 · 10−5 > 0 hysteresis + no trap yes 750 1680 2299 2464 619 165 0.660872 0.732055 0.071183X 576 1/4 yes no 5 · 10−5 > 0 no trap + no hysteresis yes 750 1640 2211 2381 571 170 0.588463 0.660004 0.071541X 577 1/4 yes no 5 · 10−5 > 0 trap + hysteresis no 750 2090 2505 2628 415 123 0.683429 0.757512 0.074083X 579 1/4 yes no 5 · 10−5 0 no trap + no hysteresis no 750 1700 2241 2545 541 304 0.570173 0.749044 0.178871

W 563 1/4 no no NA 0 no trap + no hysteresis NA 750 1600 2172 2574 572 402 0.615077 0.846343 0.231266W 551 1/4 no no NA 0 no trap + no hysteresis NA 750 1580 2145 2545 565 400 0.603017 0.847951 0.244934Y 560 1/4 no yes NA 0 no trap + no hysteresis NA 690 2110 2629 3182 519 553 0.625756 0.883803 0.258047W 564 1/4 no no NA > 0 trap + hysteresis NA 750 1610 2105 2609 495 504 0.643650 0.907230 0.263580W 561 1/4 no no NA > 0 no trap + no hysteresis NA 750 1570 2055 2555 485 500 0.638989 0.906359 0.267370

400

to the economic limit (EL) of the CO2 flood. Table 19.5 also presents the recovery factor (RF) for

the waterflood (WF), continuous CO2 injection gas flood (GF), and the incremental recovery due

to the CO2 flood. Table 19.5 is ordered based on the incremental recovery between the waterflood

and the gas flood.

The system without compositional trapping has significantly higher recoveries at the economic

limit of CO2 injection than the cases which account for trapping. This is true both for the incre-

mental recovery above the waterflood and for the total recovery from the start of the simulation.

For both the cases with and without compositional trapping, accounting for the CO2 solubility in

water increases the continuous CO2 recovery factor. For both the cases with and without composi-

tional trapping, gas relative permeability hysteresis decreases the continuous CO2 recovery factor.

Gas relative permeability hysteresis is more significant than the WCO2 for a system with composi-

tional trapping, but WCO2 is more significant than krg hysteresis for systems without compositional

trapping.

For the system with dual-media compositional trapping, the incremental recovery from CO2

injection over the waterflood case varies between 0.01 and 0.18. The presence or absence of addi-

tional trapped oil after the waterflood causes a large amount of variability in the recovery factor

but does not follow a trend.




the waterflood (WF), continuous CO2 injection gas flood (GF) and the incremental recovery due

to the CO2 flood. Table 19.6 is ordered based on the time between the start of CO2 injection and

the time of the CO2 response.

The time from the end of waterflood to the increase in oil production corresponding to CO2

response varies between 3840 days and 8440 days. The results for models with compositional

dual-media trapping and without trapping are intermingled in CO2 response time. Cases with

gas relative permeability hysteresis have faster response times than cases without. Cases with

no compositional gas trapping but with gas relative permeability hysteresis seem to have longer

response times than any of the other cases, but since only one case was run with this option it is

difficult to evaluate.

401

Table 19.6: Continuous CO2 response time

Model

Type

Model

#

Pattern


Heterogen

eity

Transfer:km2

WCO

2Option

Hysteresis

ofkrg

ExtraTrapped

OilAfter

WF

Start

WF

(10day

s)

Start

GF

(10day

s)

OilResponse

GF

(10day

s)

Economic

Lim

itGF

(10day

s)

Tim

eGF−

WF

(10day

s)

Tim

eGF−

GF

resp

onse

(10day

s)

RF

WF

(RCF)

RF

GF

(RCF)

RF

GF−

WF

(RCF)

X 574 1/4 yes no 5 · 10−5 0 trap + hysteresis yes 750 1930 2314 2352 384 38 0.678063 0.698792 0.020729X 577 1/4 yes no 5 · 10−5 > 0 trap + hysteresis no 750 2090 2505 2628 415 123 0.683429 0.757512 0.074083X 550 1/4 yes no 5 · 10−5 > 0 trap + hysteresis yes 750 2070 2497 2552 427 55 0.683429 0.714275 0.030846X 573 1/4 yes no 5 · 10−3 > 0 trap + hysteresis yes 750 2040 2468 2525 428 57 0.650930 0.680371 0.029441X 578 1/4 yes no 5 · 10−5 > 0 trap + hysteresis yes 750 2090 2524 2580 434 56 0.683429 0.713990 0.030561Z 570 1/4 yes yes 5 · 10−5 > 0 trap + hysteresis yes 690 2230 2672 2848 442 176 0.673626 0.724226 0.050600W 561 1/4 no no NA > 0 no trap + no hysteresis NA 750 1570 2055 2555 485 500 0.638989 0.906359 0.267370W 562 1/4 no no NA 0 trap + hysteresis NA 750 1610 2100 NA 490 NA 0.620464 0.626867 0.006403W 564 1/4 no no NA > 0 trap + hysteresis NA 750 1610 2105 2609 495 504 0.643650 0.907230 0.263580Y 560 1/4 no yes NA 0 no trap + no hysteresis NA 690 2110 2629 3182 519 553 0.625756 0.883803 0.258047X 579 1/4 yes no 5 · 10−5 0 no trap + no hysteresis no 750 1700 2241 2545 541 304 0.570173 0.749044 0.178871W 551 1/4 no no NA 0 no trap + no hysteresis NA 750 1580 2145 2545 565 400 0.603017 0.847951 0.244934X 576 1/4 yes no 5 · 10−5 > 0 no trap + no hysteresis yes 750 1640 2211 2381 571 170 0.588463 0.660004 0.071541W 563 1/4 no no NA 0 no trap + no hysteresis NA 750 1600 2172 2574 572 402 0.615077 0.846343 0.231266X 553 1/4 yes no 5 · 10−5 0 no trap + no hysteresis yes 750 1700 2272 2382 572 110 0.570173 0.619315 0.049142X 575 1/4 yes no 5 · 10−5 > 0 hysteresis + no trap yes 750 1680 2299 2464 619 165 0.660872 0.732055 0.071183X 552 1/4 yes no 5 · 10−5 0 trap + hysteresis no 750 1930 2774 2785 844 11 0.678063 0.691281 0.013218

402

Table 19.7: Continuous CO2 response durationModel

Type

Model

#

Pattern


Heterogen

eity

Transfer:km2

WCO

2Option

Hysteresis

ofkrg

ExtraTrapped

OilAfter

WF

Start

WF

(10day

s)

Start

GF

(10day

s)

OilResponse

GF

(10day

s)

Economic

Lim

itGF

(10day

s)

Tim

eGF−

WF

(10day

s)

Tim

eGF−

GF

resp

onse

(10day

s)

RF

WF

(RCF)

RF

GF

(RCF)

RF

GF−

WF

(RCF)

X 552 1/4 yes no 5 · 10−5 0 trap + hysteresis no 750 1930 2774 2785 844 11 0.678063 0.691281 0.013218X 574 1/4 yes no 5 · 10−5 0 trap + hysteresis yes 750 1930 2314 2352 384 38 0.678063 0.698792 0.020729X 550 1/4 yes no 5 · 10−5 > 0 trap + hysteresis yes 750 2070 2497 2552 427 55 0.683429 0.714275 0.030846X 578 1/4 yes no 5 · 10−5 > 0 trap + hysteresis yes 750 2090 2524 2580 434 56 0.683429 0.713990 0.030561X 573 1/4 yes no 5 · 10−3 > 0 trap + hysteresis yes 750 2040 2468 2525 428 57 0.650930 0.680371 0.029441X 553 1/4 yes no 5 · 10−5 0 no trap + no hysteresis yes 750 1700 2272 2382 572 110 0.570173 0.619315 0.049142X 577 1/4 yes no 5 · 10−5 > 0 trap + hysteresis no 750 2090 2505 2628 415 123 0.683429 0.757512 0.074083X 575 1/4 yes no 5 · 10−5 > 0 hysteresis + no trap yes 750 1680 2299 2464 619 165 0.660872 0.732055 0.071183X 576 1/4 yes no 5 · 10−5 > 0 no trap + no hysteresis yes 750 1640 2211 2381 571 170 0.588463 0.660004 0.071541Z 570 1/4 yes yes 5 · 10−5 > 0 trap + hysteresis yes 690 2230 2672 2848 442 176 0.673626 0.724226 0.050600X 579 1/4 yes no 5 · 10−5 0 no trap + no hysteresis no 750 1700 2241 2545 541 304 0.570173 0.749044 0.178871

W 551 1/4 no no NA 0 no trap + no hysteresis NA 750 1580 2145 2545 565 400 0.603017 0.847951 0.244934W 563 1/4 no no NA 0 no trap + no hysteresis NA 750 1600 2172 2574 572 402 0.615077 0.846343 0.231266W 561 1/4 no no NA > 0 no trap + no hysteresis NA 750 1570 2055 2555 485 500 0.638989 0.906359 0.267370W 564 1/4 no no NA > 0 trap + hysteresis NA 750 1610 2105 2609 495 504 0.643650 0.907230 0.263580Y 560 1/4 no yes NA 0 no trap + no hysteresis NA 690 2110 2629 3182 519 553 0.625756 0.883803 0.258047

403




the waterflood (WF), continuous CO2 injection gas flood (GF), and the incremental recovery due

to the CO2 flood. Table 19.7 is ordered based on the time between the time of the CO2 response

and the time when the economic limit is reached.

The time from the continuous CO2 response until the economic limit of 10 RBOPD varies

between 380 days and 5040 days. For the cases with no compositional trapping the response lasts

significantly longer than when dual-media compositional trapping is considered. For cases with

compositional trapping, accounting for WCO2 increases the response time; some of the injected

CO2 is stored in the water phase. For cases with no compositional trapping, accounting for WCO2

decreases response time. For cases with compositional trapping, gas relative permeability hysteresis

decreases the response time. For cases with no compositional trapping, gas relative permeability

hysteresis increases response time.

19.4 Evaluation of CO2 WAG

Water-alternating-gas injection is used for several different reasons in field development. The

cost of water is often cheaper than the cost of CO2; as a result if the recovery is similar between gas

injection and WAG injection it is often cheaper to operate a WAG flood. WAG also helps to lower

the amount of produced gas, which decreases the cost for processing the gas in order to extract the

CO2 for re-injection. WAG helps control the mobility of the fluids; this causes the CO2 to follow a

different path through the reservoir. Mobility control helps both areal and vertical sweep efficiency.

With more heterogeneity, mobility control becomes more important. Mobility control is also very

important in thicker reservoirs where gas override is a larger problem. If there is cycle-dependent

Sor, then WAG also performs better than continuous CO2 injection. Vertical 2D cross sections or

3D cases would emphasize the differences between WAG and continuous CO2 injection, but these

cases were not simulated here.

The CO2 WAG cases start with primary production to the economic limit, followed by water-

flood to the economic limit, followed by continuous CO2 injection until response is observed, fol-

lowed by water-alternating-gas injection. In the cases described here, water is injected for 200 days

404

followed by gas for 200 days, followed by repeating cycles of the same length. It is typical to start

industry WAG at a 1 : 1 ratio, either by time or volume. Changes may then be made based on

observations of the wells and varying CO2 or water supply.

Table 19.8 presents the recovery factors (RF) for the waterflood (WF), continuous CO2 gas

flood (GF), and CO2 WAG plus the incremental recovery factors of gas flood versus waterflood,

WAG versus waterflood, and WAG versus continuous CO2 injection. Table 19.8 is ordered based

on the incremental recovery of the WAG flood versus the waterflood.

The cases without compositional trapping have consistently higher recovery factors after the

waterflood as a result of combined CO2 injection and WAG. Gas relative permeability hysteresis

also significantly decreases the effectiveness of CO2 and WAG injection for the cases with dual-

media compositional trapping. More trapped oil after the waterflood decreases the effectiveness of

CO2 and WAG injection. Although CO2 solubility in water causes variations in the response to

CO2 and WAG injection, the models do not follow an observable trend.

Table 19.9 presents the recovery factors (RF) for the waterflood (WF), continuous CO2 gas

flood (GF), and CO2 WAG plus the incremental recovery factors of gas flood versus waterflood,

WAG versus waterflood, and WAG versus continuous CO2 injection. Table 19.9 is ordered based

on the difference between the WAG flood recovery and the continuous CO2 flood recovery.

Comparing the incremental recovery after waterflood, some of the models have more incremen-

tal recovery from continuous CO2 injection and some have more incremental recovery from CO2

injection followed by CO2 WAG. The effects likely to most significantly effect WAG versus contin-

uous CO2 injection include heterogeneity, 3D gravity effects, economics, and operational flexibility.

Although not many heterogeneous simulations were conducted, the largest observed incremental

oil recovery from WAG is for case Y560.

Table 19.10 presents the start times of the waterflood (WF), continuous CO2 gas flood (GF),

and WAG with the economic limits (EL) for each production phase. Table 19.10 is ordered based

on the difference between the WAG flood recovery and the continuous CO2 flood recovery.

The amount of time between the start of WAG and reaching the economic limit varies signifi-

cantly between 120 days and 5610 days. There are some models that have no incremental produc-

tion from WAG and some models that have more than 6000 days of incremental production from

WAG, but there were computational difficulties with the models at both extremes that may mask

405

Table 19.8: WAG recovery factorModel

Type

Model

#

Pattern


Heterogen

eity

Transfer:km2

WCO

2Option

Hysteresis

ofkrg

ExtraTrapped

OilAfter

WF

Cycledep

enden

tSor

RF

WF

(RCF)

RF

GF

(RCF)

RF

WAG

(RCF)

RF

GF−

WF

(RCF)

RF

WAG

−W

F(R

CF)

RF

WAG

−GF

(RCF)

X 554 1/4 yes no 5 · 10−5 > 0 trap + hysteresis no no 0.683429 0.684152 0.684133 0.000723 0.000704 -0.000019W 562 1/4 no no NA 0 trap + hysteresis NA no 0.620464 0.626867 0.630089 0.006403 0.009625 0.003222X 574 1/4 yes no 5 · 10−5 0 trap + hysteresis yes yes 0.678063 0.698792 0.690718 0.020729 0.012655 -0.008074X 573 1/4 yes no 5 · 10−3 > 0 trap + hysteresis yes yes 0.650930 0.680371 0.676915 0.029441 0.025985 -0.003456X 550 1/4 yes no 5 · 10−5 > 0 trap + hysteresis yes yes 0.683429 0.714275 0.709605 0.030846 0.026176 -0.004670X 578 1/4 yes no 5 · 10−5 > 0 trap + hysteresis yes no 0.683429 0.713990 0.713955 0.030561 0.030526 -0.000035Z 570 1/4 yes yes 5 · 10−5 > 0 trap + hysteresis yes yes 0.673626 0.724226 0.724035 0.050600 0.050409 -0.000191X 577 1/4 yes no 5 · 10−5 > 0 trap + hysteresis no yes 0.683429 0.757512 0.735339 0.074083 0.051910 -0.022173

X 553 1/4 yes no 5 · 10−5 0 no trap + no hysteresis yes yes 0.570173 0.619315 0.624546 0.049142 0.054373 0.005231X 575 1/4 yes no 5 · 10−5 > 0 hysteresis + no trap yes yes 0.660872 0.732055 0.736300 0.071183 0.075428 0.004245X 576 1/4 yes no 5 · 10−5 > 0 no trap + no hysteresis yes yes 0.588463 0.660004 0.664551 0.071541 0.076088 0.004547X 579 1/4 yes no 5 · 10−5 0 no trap + no hysteresis no no 0.570173 0.749044 0.750321 0.178871 0.180148 0.001277

W 564 1/4 no no NA > 0 trap + hysteresis NA yes 0.643650 0.907230 0.863659 0.263580 0.220009 -0.043571W 563 1/4 no no NA 0 no trap + no hysteresis NA yes 0.615077 0.846343 0.847772 0.231266 0.232695 0.001429W 551 1/4 no no NA 0 no trap + no hysteresis NA no 0.603017 0.847951 0.853358 0.244934 0.250341 0.005407W 561 1/4 no no NA > 0 no trap + no hysteresis NA no 0.638989 0.906359 0.914345 0.267370 0.275356 0.007986Y 560 1/4 no yes NA 0 no trap + no hysteresis NA no 0.625756 0.883803 0.911696 0.258047 0.285940 0.027893

406

Table 19.9: WAG recovery factor versus continuous CO2 recovery factorModel

Type

Model

#

Pattern


Heterogen

eity

Transfer:km2

WCO

2Option

Hysteresis

ofkrg

ExtraTrapped

OilAfter

WF

Cycledep

enden

tSor

RF

WF

(RCF)

RF

GF

(RCF)

RF

WAG

(RCF)

RF

GF−

WF

(RCF)

RF

WAG

−W

F(R

CF)

RF

WAG

−GF

(RCF)

W 564 1/4 no no NA > 0 trap + hysteresis NA yes 0.643650 0.907230 0.863659 0.263580 0.220009 -0.043571X 577 1/4 yes no 5 · 10−5 > 0 trap + hysteresis no yes 0.683429 0.757512 0.735339 0.074083 0.051910 -0.022173X 574 1/4 yes no 5 · 10−5 0 trap + hysteresis yes yes 0.678063 0.698792 0.690718 0.020729 0.012655 -0.008074X 550 1/4 yes no 5 · 10−5 > 0 trap + hysteresis yes yes 0.683429 0.714275 0.709605 0.030846 0.026176 -0.004670X 573 1/4 yes no 5 · 10−3 > 0 trap + hysteresis yes yes 0.650930 0.680371 0.676915 0.029441 0.025985 -0.003456Z 570 1/4 yes yes 5 · 10−5 > 0 trap + hysteresis yes yes 0.673626 0.724226 0.724035 0.050600 0.050409 -0.000191X 578 1/4 yes no 5 · 10−5 > 0 trap + hysteresis yes no 0.683429 0.713990 0.713955 0.030561 0.030526 -0.000035X 554 1/4 yes no 5 · 10−5 > 0 trap + hysteresis no no 0.683429 0.684152 0.684133 0.000723 0.000704 -0.000019X 579 1/4 yes no 5 · 10−5 0 no trap + no hysteresis no no 0.570173 0.749044 0.750321 0.178871 0.180148 0.001277W 563 1/4 no no NA 0 no trap + no hysteresis NA yes 0.615077 0.846343 0.847772 0.231266 0.232695 0.001429W 562 1/4 no no NA 0 trap + hysteresis NA no 0.620464 0.626867 0.630089 0.006403 0.009625 0.003222X 575 1/4 yes no 5 · 10−5 > 0 hysteresis + no trap yes yes 0.660872 0.732055 0.736300 0.071183 0.075428 0.004245X 576 1/4 yes no 5 · 10−5 > 0 no trap + no hysteresis yes yes 0.588463 0.660004 0.664551 0.071541 0.076088 0.004547X 553 1/4 yes no 5 · 10−5 0 no trap + no hysteresis yes yes 0.570173 0.619315 0.624546 0.049142 0.054373 0.005231W 551 1/4 no no NA 0 no trap + no hysteresis NA no 0.603017 0.847951 0.853358 0.244934 0.250341 0.005407W 561 1/4 no no NA > 0 no trap + no hysteresis NA no 0.638989 0.906359 0.914345 0.267370 0.275356 0.007986Y 560 1/4 no yes NA 0 no trap + no hysteresis NA no 0.625756 0.883803 0.911696 0.258047 0.285940 0.027893

407

Table 19.10: WAG response durationModel

Type

Model

#

Pattern

Com

positional

Trapping

Heterogen

eity

Transfer:km

2

WCO

2Option

Hysteresis

ofkrg

ExtraTrapped

OilAfter

WF

Cycledep

endentSor

Start

WF(10days)

Start

GF(10days)

OilRespon

seGF(10days)

Econom

icLim

itGF(10days)

Econom

icLim

itWAG

(10days)

Tim

eWAG−

GF(10days)

X 554 1/4 yes no 5 · 10−5 > 0 trap + hysteresis no no 750 2090 2058 2070 2070 12X 574 1/4 yes no 5 · 10−5 0 trap + hysteresis yes yes 750 1930 2314 2352 2355 41X 578 1/4 yes no 5 · 10−5 > 0 trap + hysteresis yes no 750 2090 2524 2580 2600 76X 573 1/4 yes no 5 · 10−3 > 0 trap + hysteresis yes yes 750 2040 2468 2525 2588 120X 550 1/4 yes no 5 · 10−5 > 0 trap + hysteresis yes yes 750 2070 2497 2552 2619 122X 553 1/4 yes no 5 · 10−5 0 no trap + no hysteresis yes yes 750 1700 2272 2382 2410 138X 577 1/4 yes no 5 · 10−5 > 0 trap + hysteresis no yes 750 2090 2505 2628 2705 200X 576 1/4 yes no 5 · 10−5 > 0 no trap + no hysteresis yes yes 750 1640 2211 2381 2418 207X 575 1/4 yes no 5 · 10−5 > 0 hysteresis + no trap yes yes 750 1680 2299 2464 2509 210Z 570 1/4 yes yes 5 · 10−5 > 0 trap + hysteresis yes yes 690 2230 2672 2848 2946 274X 579 1/4 yes no 5 · 10−5 0 no trap + no hysteresis no no 750 1700 2241 2545 2550 309

W 551 1/4 no no NA 0 no trap + no hysteresis NA no 750 1580 2145 2545 2530 385W 561 1/4 no no NA > 0 no trap + no hysteresis NA no 750 1570 2055 2555 2537 482W 563 1/4 no no NA 0 no trap + no hysteresis NA yes 750 1600 2172 2574 2667 495Y 560 1/4 no yes NA 0 no trap + no hysteresis NA no 690 2110 2629 3182 3190 561W 564 1/4 no no NA > 0 trap + hysteresis NA yes 750 1610 2105 2609 2705 600

408

the actual production performance. For the successful models, the models without compositional

trapping have consistently longer incremental production from WAG. Although the calculation of

WCO2 , the presence or absence of gas relative permeability hysteresis, the presence or absence of

additional trapped oil after the waterflood, and cycle-dependent residual oil saturations change the

amount of time of incremental production, there is no trend in how these change for the test cases

simulated here.

19.5 Evaluation of Compositional Recovery Factor

Table 19.11, Table 19.12, and Table 19.13 present the compositional recovery factors for CH4,

nC4, and nC10 and the difference between nC10 and CH4 for the waterflood (WF), continuous CO2

gas flood (GF), and WAG. The trends for waterflood, continuous CO2 injection, and WAG are all

the same; Table 19.11, Table 19.12, and Table 19.13 would be combined in one table if it could

fit on one page. Table 19.11 is ordered based on the difference between the nC10 and CH4 for

the waterflood. Table 19.12 is ordered based on the difference between the nC10 and CH4 for the

continuous CO2 gas flood. Table 19.13 is ordered based on the difference between the nC10 and

CH4 for WAG.

The compositional recovery factors represent the number of moles of methane, butane, or decane

produced as a fraction of the original number of moles in the reservoir. The compositional variation

follows the same trend for the waterflood, CO2 flood, and WAG flood. All of the models without

compositional trapping and none of the models with compositional trapping have approximately the

same recovery for methane, butane, and decane. All the models with compositional trapping that

include gas relative permeability hysteresis have more decane production than methane production.

All the models with compositional trapping but with no gas relative permeability hysteresis have

more methane production than decane production. Decreasing the km2 causes an increase in the

difference between decane production and methane production because it is more difficult for the

methane in the trapped gas to move back into the mobile fluid.

19.6 Evaluation of CO2 Storage

Table 19.14 and Table 19.15 present the CO2 storage and CO2 utilization for continuous CO2

injection and WAG. Table 19.14 is ordered based on the amount of CO2 storage at the economic

409

Table 19.11: Compositional recovery factor for waterflood

Model

Type

Model

#

Pattern


Heterogen

eity

Transfer:km2

WCO

2Option

Hysteresis

ofkrg

ExtraTrapped

OilAfter

WF

Cycledep

enden

tSor

RF

WF

(lbmolCH

4)

RF

WF

(lbmolnC

4)

RF

WF

(lbmolnC

10)

RF

nC

10−

CH

4W

F

RF

nC

10−

CH

4GF

RF

nC

10−

CH

4WAG

X 553 1/4 yes no 5 · 10−5 0 no trap + no hysteresis yes yes 0.674197 0.644331 0.637915 -0.036282 -0.037973 -0.037967X 579 1/4 yes no 5 · 10−5 0 no trap + no hysteresis no no 0.674197 0.644331 0.637915 -0.036282 -0.031835 -0.032329X 576 1/4 yes no 5 · 10−5 > 0 no trap + no hysteresis yes yes 0.668508 0.659455 0.658303 -0.010205 -0.012460 -0.012478

W 551 1/4 no no NA 0 no trap + no hysteresis NA no 0.648968 0.643662 0.642873 -0.006095 -0.003265 -0.003228W 563 1/4 no no NA 0 no trap + no hysteresis NA yes 0.648968 0.643662 0.642873 -0.006095 -0.003333 -0.003329W 561 1/4 no no NA > 0 no trap + no hysteresis NA no 0.657580 0.656146 0.655923 -0.001657 -0.000086 -0.000069Y 560 1/4 no yes NA 0 no trap + no hysteresis NA no 0.636077 0.634787 0.634527 -0.001550 -0.001264 -0.001180Y 580 5-spot no yes NA 0 no trap + no hysteresis NA no 0.641361 0.643041 0.643449 0.002088 -0.004327 NAW 562 1/4 no no NA 0 trap + hysteresis NA no 0.641383 0.644134 0.644470 0.003087 -0.004739 0.001640W 564 1/4 no no NA > 0 trap + hysteresis NA yes 0.650195 0.656060 0.656853 0.006658 0.003705 0.004633

X 573 1/4 yes no 5 · 10−3 > 0 trap + hysteresis yes yes 0.611407 0.674177 0.683346 0.071939 0.059675 0.068333X 575 1/4 yes no 5 · 10−5 > 0 hysteresis + no trap yes yes 0.596050 0.677093 0.690110 0.094060 0.072808 0.072551Z 570 1/4 yes yes 5 · 10−5 > 0 trap + hysteresis yes yes 0.540548 0.632271 0.635686 0.095138 0.110104 0.136710Z 581 5-spot yes yes 5 · 10−5 > 0 trap + hysteresis yes yes 0.630521 0.717750 0.731673 0.101152 0.062464 NAX 552 1/4 yes no 5 · 10−5 0 trap + hysteresis no no 0.546910 0.641899 0.655798 0.108888 0.109196 0.171066X 574 1/4 yes no 5 · 10−5 0 trap + hysteresis yes yes 0.546910 0.641899 0.655798 0.108888 0.113115 0.111579X 550 1/4 yes no 5 · 10−5 > 0 trap + hysteresis yes yes 0.564804 0.660843 0.674870 0.110066 0.115095 0.107757X 554 1/4 yes no 5 · 10−5 > 0 trap + hysteresis no no 0.564804 0.660843 0.674870 0.110066 0.109401 0.110748X 577 1/4 yes no 5 · 10−5 > 0 trap + hysteresis no yes 0.564804 0.660843 0.674870 0.110066 0.120713 0.097278X 578 1/4 yes no 5 · 10−5 > 0 trap + hysteresis yes no 0.564804 0.660843 0.674870 0.110066 0.115175 0.113153X 572 1/4 yes no 5 · 10−9 > 0 trap + hysteresis yes yes 0.464729 0.641281 0.667150 0.202421 0.230762 0.230764X 571 1/4 yes no 5 · 10−7 > 0 trap + hysteresis yes yes 0.481982 0.665055 0.691852 0.209870 0.229581 0.230495

410

Table 19.12: Compositional recovery factor for continuous CO2 injection

Model

Type

Model

#

Pattern


Heterogen

eity

Transfer:km2

WCO

2Option

Hysteresis

ofkrg

ExtraTrapped

OilAfter

WF

Cycledep

enden

tSor

RF

WF

(lbmolCH

4)

RF

WF

(lbmolnC

4)

RF

WF

(lbmolnC

10)

RF

nC

10−

CH

4W

F

RF

nC

10−

CH

4GF

RF

nC

10−

CH

4WAG


W 562 1/4 no no NA 0 trap + hysteresis NA no 0.656507 0.652391 0.651768 0.003087 -0.004739 0.001640Y 580 5-spot no yes NA 0 no trap + no hysteresis NA no 0.653966 0.650930 0.649639 0.002088 -0.004327 NAW 563 1/4 no no NA 0 no trap + no hysteresis NA yes 0.829678 0.826777 0.826345 -0.006095 -0.003333 -0.003329W 551 1/4 no no NA 0 no trap + no hysteresis NA no 0.832960 0.830118 0.829695 -0.006095 -0.003265 -0.003228Y 560 1/4 no yes NA 0 no trap + no hysteresis NA no 0.866301 0.865216 0.865037 -0.001550 -0.001264 -0.001180W 561 1/4 no no NA > 0 no trap + no hysteresis NA no 0.865994 0.865923 0.865908 -0.001657 -0.000086 -0.000069W 564 1/4 no no NA > 0 trap + hysteresis NA yes 0.864346 0.867607 0.868051 0.006658 0.003705 0.004633

X 573 1/4 yes no 5 · 10−3 > 0 trap + hysteresis yes yes 0.674242 0.726323 0.733917 0.071939 0.059675 0.068333Z 581 5-spot yes yes 5 · 10−5 > 0 trap + hysteresis yes yes 0.749390 0.804886 0.811854 0.101152 0.062464 NAX 575 1/4 yes no 5 · 10−5 > 0 hysteresis + no trap yes yes 0.656086 0.718585 0.728894 0.094060 0.072808 0.072551X 552 1/4 yes no 5 · 10−5 0 trap + hysteresis no no 0.558095 0.654891 0.667291 0.108888 0.109196 0.171066X 554 1/4 yes no 5 · 10−5 > 0 trap + hysteresis no no 0.567815 0.663246 0.677216 0.110066 0.109401 0.110748Z 570 1/4 yes yes 5 · 10−5 > 0 trap + hysteresis yes yes 0.564331 0.660318 0.674435 0.095138 0.110104 0.136710X 574 1/4 yes no 5 · 10−5 0 trap + hysteresis yes yes 0.552456 0.651136 0.665571 0.108888 0.113115 0.111579X 550 1/4 yes no 5 · 10−5 > 0 trap + hysteresis yes yes 0.568922 0.669351 0.684017 0.110066 0.115095 0.107757X 578 1/4 yes no 5 · 10−5 > 0 trap + hysteresis yes no 0.568981 0.669479 0.684156 0.110066 0.115175 0.113153X 577 1/4 yes no 5 · 10−5 > 0 trap + hysteresis no yes 0.598775 0.704106 0.719488 0.110066 0.120713 0.097278X 571 1/4 yes no 5 · 10−7 > 0 trap + hysteresis yes yes 0.441645 0.641932 0.671226 0.209870 0.229581 0.230495X 572 1/4 yes no 5 · 10−9 > 0 trap + hysteresis yes yes 0.409479 0.610758 0.640241 0.202421 0.230762 0.230764

411

Table 19.13: Compositional recovery factor for WAG

Model

Type

Model

#

Pattern


Heterogen

eity

Transfer:km2

WCO

2Option

Hysteresis

ofkrg

ExtraTrapped

OilAfter

WF

Cycledep

enden

tSor

RF

WF

(lbmolCH

4)

RF

WF

(lbmolnC

4)

RF

WF

(lbmolnC

10)

RF

nC

10−

CH

4W

F

RF

nC

10−

CH

4GF

RF

nC

10−

CH

4WAG


W 563 1/4 no no NA 0 no trap + no hysteresis NA yes 0.829989 0.827096 0.826660 -0.006095 -0.003333 -0.003329W 551 1/4 no no NA 0 no trap + no hysteresis NA no 0.834826 0.832016 0.831598 -0.006095 -0.003265 -0.003228Y 560 1/4 no yes NA 0 no trap + no hysteresis NA no 0.890282 0.889393 0.889102 -0.001550 -0.001264 -0.001180W 561 1/4 no no NA > 0 no trap + no hysteresis NA no 0.869894 0.869838 0.869825 -0.001657 -0.000086 -0.000069W 562 1/4 no no NA 0 trap + hysteresis NA no 0.654133 0.655605 0.655773 0.003087 -0.004739 0.001640W 564 1/4 no no NA > 0 trap + hysteresis NA yes 0.845257 0.849335 0.849890 0.006658 0.003705 0.004633

X 573 1/4 yes no 5 · 10−3 > 0 trap + hysteresis yes yes 0.675334 0.734113 0.743667 0.071939 0.059675 0.068333X 575 1/4 yes no 5 · 10−5 > 0 hysteresis + no trap yes yes 0.657157 0.719434 0.729708 0.094060 0.072808 0.072551X 577 1/4 yes no 5 · 10−5 > 0 trap + hysteresis no yes 0.618137 0.709026 0.715415 0.110066 0.120713 0.097278X 550 1/4 yes no 5 · 10−5 > 0 trap + hysteresis yes yes 0.610048 0.706101 0.717805 0.110066 0.115095 0.107757X 554 1/4 yes no 5 · 10−5 > 0 trap + hysteresis no no 0.563680 0.660314 0.674428 0.110066 0.109401 0.110748X 574 1/4 yes no 5 · 10−5 0 trap + hysteresis yes yes 0.554238 0.651575 0.665817 0.108888 0.113115 0.111579X 578 1/4 yes no 5 · 10−5 > 0 trap + hysteresis yes no 0.577063 0.675877 0.690216 0.110066 0.115175 0.113153Z 570 1/4 yes yes 5 · 10−5 > 0 trap + hysteresis yes yes 0.569071 0.683153 0.705781 0.095138 0.110104 0.136710X 552 1/4 yes no 5 · 10−5 0 trap + hysteresis no no 0.521725 0.669230 0.692791 0.108888 0.109196 0.171066X 571 1/4 yes no 5 · 10−7 > 0 trap + hysteresis yes yes 0.438057 0.639127 0.668552 0.209870 0.229581 0.230495X 572 1/4 yes no 5 · 10−9 > 0 trap + hysteresis yes yes 0.409475 0.610755 0.640239 0.202421 0.230762 0.230764

412

Table 19.14: CO2 storage for continuous CO2 injection

Model

Type

Model

#

Pattern

Com

positional

Trapping

Heterogen

eity

Transfer:km

2

WCO

2Option

Hysteresis

ofkrg

ExtraTrapped

OilAfter

WF

Cycledep

endentSor

Storage

GF(lbmol)

Storage

WAG

(lbmol)

Storage

WAG−GF(lbmol)

Utilization

GF(M

CF/R

B)

Utilization

WAG

(MCF/R

B)

Utilization

WAG−

GF(M

CF/R

B)

Y 560 1/4 no yes NA 0 no trap + no hysteresis NA no 0.479 0.398 -0.081 7.06 4.04 -3.02X 554 1/4 yes no 5 · 10−5 > 0 trap + hysteresis no no 0.492 NA NA 684.12 150.71 -533.41Z 570 1/4 yes yes 5 · 10−5 > 0 trap + hysteresis yes yes 0.702 0.804 0.102 16.93 11.27 -5.66X 552 1/4 yes no 5 · 10−5 0 trap + hysteresis no no 0.805 0.805 0.000 289.64 NA NAX 577 1/4 yes no 5 · 10−5 > 0 trap + hysteresis no yes 0.864 0.894 0.030 9.09 11.43 2.34

W 561 1/4 no no NA > 0 no trap + no hysteresis NA no 0.870 0.829 -0.040 4.38 3.25 -1.13X 579 1/4 yes no 5 · 10−5 0 no trap + no hysteresis no no 0.872 0.851 -0.021 5.68 4.60 -1.08W 551 1/4 no no NA 0 no trap + no hysteresis NA no 0.877 0.844 -0.033 4.89 3.75 -1.14W 564 1/4 no no NA > 0 trap + hysteresis NA yes 0.877 0.893 0.015 4.48 4.15 -0.33W 563 1/4 no no NA 0 no trap + no hysteresis NA yes 0.879 0.845 -0.033 5.07 3.93 -1.14

X 578 1/4 yes no 5 · 10−5 > 0 trap + hysteresis yes no 0.890 0.883 -0.006 24.22 22.48 -1.74X 550 1/4 yes no 5 · 10−5 > 0 trap + hysteresis yes yes 0.894 0.891 -0.003 24.51 30.63 6.12X 576 1/4 yes no 5 · 10−5 > 0 no trap + no hysteresis yes yes 0.899 0.884 -0.016 11.98 10.16 -1.83X 575 1/4 yes no 5 · 10−5 > 0 hysteresis + no trap yes yes 0.906 0.894 -0.012 13.57 11.58 -1.99X 573 1/4 yes no 5 · 10−3 > 0 trap + hysteresis yes yes 0.914 0.921 0.007 27.61 31.78 4.17X 553 1/4 yes no 5 · 10−5 0 no trap + no hysteresis yes yes 0.916 0.903 -0.014 17.79 14.63 -3.15X 574 1/4 yes no 5 · 10−5 0 trap + hysteresis yes yes 0.918 0.931 0.013 43.60 115.99 72.39

413

Table 19.15: CO2 storage for WAG injection

Model

Type

Model

#

Pattern

Com

positional

Trapping

Heterogen

eity

Transfer:km

2

WCO

2Option

Hysteresis

ofkrg

ExtraTrapped

OilAfter

WF

Cycledep

endentSor

Storage

GF(lbmol)

Storage

WAG

(lbmol)

Storage

WAG−

GF(lbmol)

Utilization

GF(M

CF/R

B)

Utilization

WAG

(MCF/R

B)

Utilization

WAG−

GF(M

CF/R

B)

Y 560 1/4 no yes NA 0 no trap + no hysteresis NA no 0.479 0.398 -0.081 7.06 4.04 -3.02Z 570 1/4 yes yes 5 · 10−5 > 0 trap + hysteresis yes yes 0.702 0.804 0.102 16.93 11.27 -5.66X 552 1/4 yes no 5 · 10−5 0 trap + hysteresis no no 0.805 0.805 0.000 289.64 NA NAW 561 1/4 no no NA > 0 no trap + no hysteresis NA no 0.870 0.829 -0.040 4.38 3.25 -1.13W 551 1/4 no no NA 0 no trap + no hysteresis NA no 0.877 0.844 -0.033 4.89 3.75 -1.14W 563 1/4 no no NA 0 no trap + no hysteresis NA yes 0.879 0.845 -0.033 5.07 3.93 -1.14X 579 1/4 yes no 5 · 10−5 0 no trap + no hysteresis no no 0.872 0.851 -0.021 5.68 4.60 -1.08X 578 1/4 yes no 5 · 10−5 > 0 trap + hysteresis yes no 0.890 0.883 -0.006 24.22 22.48 -1.74X 576 1/4 yes no 5 · 10−5 > 0 no trap + no hysteresis yes yes 0.899 0.884 -0.016 11.98 10.16 -1.83X 550 1/4 yes no 5 · 10−5 > 0 trap + hysteresis yes yes 0.894 0.891 -0.003 24.51 30.63 6.12W 564 1/4 no no NA > 0 trap + hysteresis NA yes 0.877 0.893 0.015 4.48 4.15 -0.33X 577 1/4 yes no 5 · 10−5 > 0 trap + hysteresis no yes 0.864 0.894 0.030 9.09 11.43 2.34X 575 1/4 yes no 5 · 10−5 > 0 hysteresis + no trap yes yes 0.906 0.894 -0.012 13.57 11.58 -1.99X 553 1/4 yes no 5 · 10−5 0 no trap + no hysteresis yes yes 0.916 0.903 -0.014 17.79 14.63 -3.15X 573 1/4 yes no 5 · 10−3 > 0 trap + hysteresis yes yes 0.914 0.921 0.007 27.61 31.78 4.17X 574 1/4 yes no 5 · 10−5 0 trap + hysteresis yes yes 0.918 0.931 0.013 43.60 115.99 72.39

414

limit of continuous CO2 injection. Table 19.15 is ordered based on the amount of CO2 storage at

the economic limit of WAG.

CO2 storage is a measure of how much injected CO2 remains in the reservoir when an economic

limit is reached. For the homogeneous cases, the amount of CO2 storage is typically 80%–93%.

Heterogeneity can cause a significantly reduced amount of CO2 storage.

For the CO2 storage after continuous CO2 injection or WAG injection, there are variations

based on the WCO2 , gas relative permeability hysteresis, and km2 , but no obvious trends. There is a

small increase in the CO2 storage when the trapped oil increases and for cycle-dependent residual

oil saturations.

Table 19.16 presents the CO2 storage and CO2 utilization for continuous CO2 injection and

WAG. Table 19.16 is ordered based on the difference between CO2 storage at the economic limit

of WAG and at the economic limit of continuous CO2 injection.

For the homogeneous cases, CO2 storage varies between slightly less and slightly more storage

with the WAG flood than with pure CO2 injection. For the heterogeneous cases, the CO2 is utilized

better and less is stored during WAG than with continuous CO2 injection. For case Y560, the CO2

storage is much lower than in the homogeneous cases.

Cases with gas relative permeability hysteresis have more storage during WAG relative to con-

tinuous CO2 injection. If CO2 is soluble in water it causes a slight increase in the WAG storage

relative to the continuous CO2 storage. Cycle dependent residual oil saturation causes a slight

increase in the WAG storage relative to the continuous CO2 storage.

19.7 Evaluation of CO2 Utilization

CO2 utilization is a measure of how much CO2 injection it takes to produce an incremental

amount of oil. The lower the CO2 utilization, the better the performance. CO2 utilization values

of 10 MCF/STB are typically economical in the USA.

CO2 utilization is lower (better) for the cases without compositional trapping than for the cases

with compositional trapping. This is the case for both continuous CO2 utilization and CO2 WAG

utilization. The priority of the other options are different for continuous CO2 utilization and CO2

WAG utilization.

415

Table 19.16: CO2 storage difference for continuous vs WAG CO2 injection

Model

Type

Model

#

Pattern

Com

positional

Trapping

Heterogen

eity

Transfer:km

2

WCO

2Option

Hysteresis

ofkrg

ExtraTrapped

OilAfter

WF

Cycledep

endentSor

Storage

GF(lbmol)

Storage

WAG

(lbmol)

Storage

WAG−GF(lbmol)

Utilization

GF(M

CF/R

B)

Utilization

WAG

(MCF/R

B)

Utilization

WAG−

GF(M

CF/R

B)

Y 560 1/4 no yes NA 0 no trap + no hysteresis NA no 0.479 0.398 -0.081 7.06 4.04 -3.02W 561 1/4 no no NA > 0 no trap + no hysteresis NA no 0.870 0.829 -0.040 4.38 3.25 -1.13W 563 1/4 no no NA 0 no trap + no hysteresis NA yes 0.879 0.845 -0.033 5.07 3.93 -1.14W 551 1/4 no no NA 0 no trap + no hysteresis NA no 0.877 0.844 -0.033 4.89 3.75 -1.14X 579 1/4 yes no 5 · 10−5 0 no trap + no hysteresis no no 0.872 0.851 -0.021 5.68 4.60 -1.08X 576 1/4 yes no 5 · 10−5 > 0 no trap + no hysteresis yes yes 0.899 0.884 -0.016 11.98 10.16 -1.83X 553 1/4 yes no 5 · 10−5 0 no trap + no hysteresis yes yes 0.916 0.903 -0.014 17.79 14.63 -3.15X 575 1/4 yes no 5 · 10−5 > 0 hysteresis + no trap yes yes 0.906 0.894 -0.012 13.57 11.58 -1.99X 578 1/4 yes no 5 · 10−5 > 0 trap + hysteresis yes no 0.890 0.883 -0.006 24.22 22.48 -1.74X 550 1/4 yes no 5 · 10−5 > 0 trap + hysteresis yes yes 0.894 0.891 -0.003 24.51 30.63 6.12X 552 1/4 yes no 5 · 10−5 0 trap + hysteresis no no 0.805 0.805 0.000 289.64 NA NAX 573 1/4 yes no 5 · 10−3 > 0 trap + hysteresis yes yes 0.914 0.921 0.007 27.61 31.78 4.17X 574 1/4 yes no 5 · 10−5 0 trap + hysteresis yes yes 0.918 0.931 0.013 43.60 115.99 72.39W 564 1/4 no no NA > 0 trap + hysteresis NA yes 0.877 0.893 0.015 4.48 4.15 -0.33X 577 1/4 yes no 5 · 10−5 > 0 trap + hysteresis no yes 0.864 0.894 0.030 9.09 11.43 2.34Z 570 1/4 yes yes 5 · 10−5 > 0 trap + hysteresis yes yes 0.702 0.804 0.102 16.93 11.27 -5.66X 554 1/4 yes no 5 · 10−5 > 0 trap + hysteresis no no 0.492 NA NA 684.12 150.71 -533.41

416

Table 19.17: CO2 utilization for continuous CO2 injection

Model

Type

Model

#

Pattern

Com

positional

Trapping

Heterogen

eity

Transfer:km

2

WCO

2Option

Hysteresis

ofkrg

ExtraTrapped

OilAfter

WF

Cycledep

endentSor

Storage

GF(lbmol)

Storage

WAG

(lbmol)

Storage

WAG−GF(lbmol)

Utilization

GF(M

CF/R

B)

Utilization

WAG

(MCF/R

B)

Utilization

WAG−

GF(M

CF/R

B)

W 561 1/4 no no NA > 0 no trap + no hysteresis NA no 0.870 0.829 -0.040 4.38 3.25 -1.13W 564 1/4 no no NA > 0 trap + hysteresis NA yes 0.877 0.893 0.015 4.48 4.15 -0.33W 551 1/4 no no NA 0 no trap + no hysteresis NA no 0.877 0.844 -0.033 4.89 3.75 -1.14W 563 1/4 no no NA 0 no trap + no hysteresis NA yes 0.879 0.845 -0.033 5.07 3.93 -1.14

X 579 1/4 yes no 5 · 10−5 0 no trap + no hysteresis no no 0.872 0.851 -0.021 5.68 4.60 -1.08Y 560 1/4 no yes NA 0 no trap + no hysteresis NA no 0.479 0.398 -0.081 7.06 4.04 -3.02

X 577 1/4 yes no 5 · 10−5 > 0 trap + hysteresis no yes 0.864 0.894 0.030 9.09 11.43 2.34X 576 1/4 yes no 5 · 10−5 > 0 no trap + no hysteresis yes yes 0.899 0.884 -0.016 11.98 10.16 -1.83X 575 1/4 yes no 5 · 10−5 > 0 hysteresis + no trap yes yes 0.906 0.894 -0.012 13.57 11.58 -1.99Z 570 1/4 yes yes 5 · 10−5 > 0 trap + hysteresis yes yes 0.702 0.804 0.102 16.93 11.27 -5.66X 553 1/4 yes no 5 · 10−5 0 no trap + no hysteresis yes yes 0.916 0.903 -0.014 17.79 14.63 -3.15X 578 1/4 yes no 5 · 10−5 > 0 trap + hysteresis yes no 0.890 0.883 -0.006 24.22 22.48 -1.74X 550 1/4 yes no 5 · 10−5 > 0 trap + hysteresis yes yes 0.894 0.891 -0.003 24.51 30.63 6.12X 573 1/4 yes no 5 · 10−3 > 0 trap + hysteresis yes yes 0.914 0.921 0.007 27.61 31.78 4.17X 574 1/4 yes no 5 · 10−5 0 trap + hysteresis yes yes 0.918 0.931 0.013 43.60 115.99 72.39X 552 1/4 yes no 5 · 10−5 0 trap + hysteresis no no 0.805 0.805 0.000 289.64 NA NAX 554 1/4 yes no 5 · 10−5 > 0 trap + hysteresis no no 0.492 NA NA 684.12 150.71 -533.41

417


WAG. Table 19.17 is ordered based on the CO2 utilization at the economic limit of continuous CO2

injection.

Continuous CO2 utilization is lower (better) with CO2 solubility in water than with WCO2 = 0.

This effect is much bigger for cases with compositional trapping than for cases without composi-

tional trapping. For the dual-media compositional trapping cases, additional trapped oil leads to

increased (worse) continuous CO2 utilization. Gas relative permeability hysteresis leads to much

worse CO2 utilization for the cases with dual-media compositional trapping.


WAG. Table 19.18 is ordered based on the CO2 utilization at the economic limit of WAG.

WAG CO2 utilization is lower (better) with CO2 solubility in water than with WCO2 = 0. This

effect is much bigger for cases with compositional trapping than for cases without compositional

trapping. WAG CO2 utilization is higher (worse) with gas relative permeability hysteresis. This

effect is much bigger for cases with compositional trapping than for cases without compositional

trapping. Changing the trapped oil after waterflood or adding cycle-dependent residual oil sat-

uration causes variations in the WAG CO2 utilization but no trend was observed in these test

cases.


WAG. Table 19.19 is ordered based on the difference between CO2 utilization at the economic limit

of WAG and at the economic limit of continuous CO2 injection.

WAG CO2 utilization is lower (better) than continuous CO2 utilization in some cases and higher

(worse) in others. For the cases without compositional trapping and the heterogeneous cases, the

WAG CO2 utilization is lower (better) than the continuous CO2 utilization. The other properties

cause variations in the CO2 utilization between WAG and continuous CO2 utilization but no trend

is observed.

418

Table 19.18: CO2 utilization for WAG injection

Model

Type

Model

#

Pattern

Com

positional

Trapping

Heterogen

eity

Transfer:km

2

WCO

2Option

Hysteresis

ofkrg

ExtraTrapped

OilAfter

WF

Cycledep

endentSor

Storage

GF(lbmol)

Storage

WAG

(lbmol)

Storage

WAG−GF(lbmol)

Utilization

GF(M

CF/R

B)

Utilization

WAG

(MCF/R

B)

Utilization

WAG−

GF(M

CF/R

B)

W 561 1/4 no no NA > 0 no trap + no hysteresis NA no 0.870 0.829 -0.040 4.38 3.25 -1.13W 551 1/4 no no NA 0 no trap + no hysteresis NA no 0.877 0.844 -0.033 4.89 3.75 -1.14W 563 1/4 no no NA 0 no trap + no hysteresis NA yes 0.879 0.845 -0.033 5.07 3.93 -1.14Y 560 1/4 no yes NA 0 no trap + no hysteresis NA no 0.479 0.398 -0.081 7.06 4.04 -3.02W 564 1/4 no no NA > 0 trap + hysteresis NA yes 0.877 0.893 0.015 4.48 4.15 -0.33

X 579 1/4 yes no 5 · 10−5 0 no trap + no hysteresis no no 0.872 0.851 -0.021 5.68 4.60 -1.08X 576 1/4 yes no 5 · 10−5 > 0 no trap + no hysteresis yes yes 0.899 0.884 -0.016 11.98 10.16 -1.83Z 570 1/4 yes yes 5 · 10−5 > 0 trap + hysteresis yes yes 0.702 0.804 0.102 16.93 11.27 -5.66X 577 1/4 yes no 5 · 10−5 > 0 trap + hysteresis no yes 0.864 0.894 0.030 9.09 11.43 2.34X 575 1/4 yes no 5 · 10−5 > 0 hysteresis + no trap yes yes 0.906 0.894 -0.012 13.57 11.58 -1.99X 553 1/4 yes no 5 · 10−5 0 no trap + no hysteresis yes yes 0.916 0.903 -0.014 17.79 14.63 -3.15X 578 1/4 yes no 5 · 10−5 > 0 trap + hysteresis yes no 0.890 0.883 -0.006 24.22 22.48 -1.74X 550 1/4 yes no 5 · 10−5 > 0 trap + hysteresis yes yes 0.894 0.891 -0.003 24.51 30.63 6.12X 573 1/4 yes no 5 · 10−3 > 0 trap + hysteresis yes yes 0.914 0.921 0.007 27.61 31.78 4.17X 574 1/4 yes no 5 · 10−5 0 trap + hysteresis yes yes 0.918 0.931 0.013 43.60 115.99 72.39X 554 1/4 yes no 5 · 10−5 > 0 trap + hysteresis no no 0.492 NA NA 684.12 150.71 -533.41

419

Table 19.19: CO2 utilization difference for continuous vs WAG CO2 injection

Model

Type

Model

#

Pattern

Com

positional

Trapping

Heterogen

eity

Transfer:km

2

WCO

2Option

Hysteresis

ofkrg

ExtraTrapped

OilAfter

WF

Cycledep

endentSor

Storage

GF(lbmol)

Storage

WAG

(lbmol)

Storage

WAG−GF(lbmol)

Utilization

GF(M

CF/R

B)

Utilization

WAG

(MCF/R

B)

Utilization

WAG−

GF(M

CF/R

B)

X 554 1/4 yes no 5 · 10−5 > 0 trap + hysteresis no no 0.492 NA NA 684.12 150.71 -533.41Z 570 1/4 yes yes 5 · 10−5 > 0 trap + hysteresis yes yes 0.702 0.804 0.102 16.93 11.27 -5.66X 553 1/4 yes no 5 · 10−5 0 no trap + no hysteresis yes yes 0.916 0.903 -0.014 17.79 14.63 -3.15Y 560 1/4 no yes NA 0 no trap + no hysteresis NA no 0.479 0.398 -0.081 7.06 4.04 -3.02X 575 1/4 yes no 5 · 10−5 > 0 hysteresis + no trap yes yes 0.906 0.894 -0.012 13.57 11.58 -1.99X 576 1/4 yes no 5 · 10−5 > 0 no trap + no hysteresis yes yes 0.899 0.884 -0.016 11.98 10.16 -1.83X 578 1/4 yes no 5 · 10−5 > 0 trap + hysteresis yes no 0.890 0.883 -0.006 24.22 22.48 -1.74W 551 1/4 no no NA 0 no trap + no hysteresis NA no 0.877 0.844 -0.033 4.89 3.75 -1.14W 563 1/4 no no NA 0 no trap + no hysteresis NA yes 0.879 0.845 -0.033 5.07 3.93 -1.14W 561 1/4 no no NA > 0 no trap + no hysteresis NA no 0.870 0.829 -0.040 4.38 3.25 -1.13X 579 1/4 yes no 5 · 10−5 0 no trap + no hysteresis no no 0.872 0.851 -0.021 5.68 4.60 -1.08W 564 1/4 no no NA > 0 trap + hysteresis NA yes 0.877 0.893 0.015 4.48 4.15 -0.33X 577 1/4 yes no 5 · 10−5 > 0 trap + hysteresis no yes 0.864 0.894 0.030 9.09 11.43 2.34X 573 1/4 yes no 5 · 10−3 > 0 trap + hysteresis yes yes 0.914 0.921 0.007 27.61 31.78 4.17X 550 1/4 yes no 5 · 10−5 > 0 trap + hysteresis yes yes 0.894 0.891 -0.003 24.51 30.63 6.12X 574 1/4 yes no 5 · 10−5 0 trap + hysteresis yes yes 0.918 0.931 0.013 43.60 115.99 72.39

420

CHAPTER 20

CONCLUSIONS

The three-phase compositional reservoir simulator developed here was used to evaluate the

effects of compositional trapping, gas relative permeability hysteresis, the solubility of CO2 in water,

and areal heterogeneity. Other options evaluated include cycle-dependent residual oil saturations,

mass transfer between the trapped and mobile systems, and additional mechanisms for trapped oil.

Compositional recovery factors are different if and only if compositional trapping is used. Com-

positional trapping is the most significant option for differences in waterflood duration (more trap-

ping is better), gas flood recovery factor (less trapping is better), CO2 response duration (less

trapping is better), WAG recovery factor (less trapping is better), WAG duration (more trapping

is better), and CO2 utilization for WAG and continuous CO2 injection (WAG is better with more

trapping). Compositional trapping has a secondary effect on waterflood recovery (more trapping

is better).

(1) My results indicate that compositional trapping, gas relative permeability hysteresis, and

the solubility of CO2 in water, have a significant impact on the volume of oil produced, the timing

of oil, water, and gas production, and the amount of CO2 stored and CO2 utilized. Primary pro-

duction, waterflood, continuous CO2 injection, and CO2 WAG production schemes were evaluated.

Permeability and porosity heterogeneity are important to the timing, recovery, CO2 storage, and

CO2 utilization; the effects of heterogeneity need to be evaluated more thoroughly in future work.

(2) Gas relative permeability hysteresis is the most significant parameter in waterflood recovery

(more trapped gas is better) and WAG recovery (with compositional trapping, more trapped gas

is better). Gas relative permeability hysteresis has a secondary effect on the waterflood timing

(more trapped gas is better), gas flood recovery (more trapped gas is worse), duration of gas flood

response (with compositional trapping, more trapped gas is better), compositional recovery factor,

and CO2 utilization for WAG and continuous CO2 injection (more trapped gas is worse). Gas

relative permeability hysteresis was more important than expected.

(3) Solubility of CO2 in water is not the most important option for any of the evaluation

criteria, but it is of secondary importance for waterflood duration (more WCO2 is better), gas flood

421

recovery (more WCO2 is better), gas flood response duration (with compositional trapping, more

WCO2 is better), CO2 storage, and CO2 utilization (more WCO2 is worse). Solubility of CO2 was

less important than expected.

(4) The cycle-dependent residual oil saturations, mass transfer between the trapped and mobile

systems, and additional mechanisms for trapped oil caused small variations in the observations but

were never as significant as compositional trapping, gas relative permeability hysteresis, solubility

of CO2 in water, or heterogeneity. This was less impact than expected.

422

CHAPTER 21

RECOMMENDED FUTURE WORK

The recommended future work includes the following categories.

• Use of this model

• Formulation enhancements

• Computation enhancements

• Laboratory experiments

21.1 Use of This Model

The three-phase parallel compositional simulator developed here can be used to evaluate addi-

tional fields or projects. For the test cases described here, more detailed evaluation of the hetero-

geneity and how it interacts with compositional trapping would be beneficial.

21.2 Formulation and Computation Enhancements

Adding the dual-porosity simulation of naturally or hydraulically fractured reservoirs would add

flexibility to the evaluation of CO2 WAG cases in carbonate reservoirs. Running additional simu-

lations at different scales between the pore-scale and field-scale would be valuable in characterizing

the importance of measurements at different scales.

The simulator developed here is built on a parallel framework. Additional work to improve the

performance of the simulator would benefit future users of the simulator.

21.3 Phase Labeling and Relative Permeability Experiments

When miscibility develops, a two-phase oil-gas system becomes a single hydrocarbon phase.

This can present a problem in calculating relative permeability. If the single phase is labeled as a

gas, then kr,hc = krg whereas if the single phase is labeled as oil then kr,hc = kro. Often these are

simulated by weighting the krg and kro curves using an interfacial tension.

423

If miscibility develops gradually and the transition occurs from two-phase to one-phase, then the

mixture relative permeability could be estimated using interfacial tension. Unfortunately interfacial

tension is not a reliable measure of the change in relative permeability. There are some difficulties

in the correlations for interfacial tension at low interfacial tension values. It is also difficult to

experimentally measure very low interfacial tensions.

Transitions can also occur from single phase “oil” to single phase “gas” (or vice versa) in

the supercritical region of the fluids. These are actually gradual changes with no phase change

present, but depending on how the phases are labeled it may lead to inconsistencies in the relative

permeabilities. Part of the problem is that relative permeability is usually measured for a oil-water

system and a gas-oil system or for a gas-water system, but is not measured for a “supercritical

fluid”-water system. Supercritical fluids are common, especially when dealing with CO2 injection.

Adjusting the relative permeability based on interfacial tension won’t detect this change because

it was a single phase before and after the “transition”.

Experiments by Bennion and Bachu (2005) and Bennion and Bachu (2008b) illustrate the

differences between CO2-water and H2S-water relative permeability. It is likely that other kinds of

gas-water and gas-oil systems will have differing relative permeability based on the composition of

the gas.

For the life cycle of a reservoir undergoing a CO2 flood, several different gas relative permeabil-

ities are needed. During primary production, the gas is a hydrocarbon gas in equilibrium with the

oil; this is either part of an initial gas cap or solution gas that forms as the pressure drops near the

producer. The gas is increasing during this stage. During water injection it is necessary to have a

decreasing relative permeability to gas. The gas is still a hydrocarbon gas in equilibrium with the

oil.

If CO2 is injected, then it would be nice to have measurements of the CO2-oil-water relative

permeability. As the CO2 mixes with the oil, it will vaporize some of the components of the oil.

It would also be nice to have a (CO2 + hydrocarbon)-oil-water relative permeability. Three phase

relative permeability measurements would be helpful. It would also be useful to have measurements

of the CO2-water relative permeability and CO2-water-residual oil relative permeability. The resid-

ual oil changes depending on the saturation history, so several different CO2-water-residual oil

experiments would be necessary.

424

During CO2 WAG operations, the gas increases and decreases in different parts of the reservoir.

It would be nice to have measurements of the hysteresis of the relative permeability and capillary

pressure as well as measurements of how the composition varies. As miscibility develops, it would

be nice to have measurements of the relative permeability for the hydrocarbon-water system.

There is limited three-phase or compositional two-phase relative permeability data available,

especially for mixed-wet carbonate rocks. Over the years there have been many proposed formu-

lations for three-phase relative permeability and hysteresis, but without experimental data it is

difficult to evaluate the different methods or propose a new high-quality physically-based method.

425

CHAPTER 22

NOMENCLATURE

This chapter identifies all variables used in this document. Table 22.1 identifies all subscripts

and superscripts.

Table 22.1: Subscripts and superscripts

variable units name

# script represents a constant term, one that does not varywith time

i index spatial index in x-direction

j index spatial index in y-direction

k index spatial index in z-direction

n index temporal index representing full time step

� index temporal index representing nonlinear iteration levelbetween n = (� = 0) and n+ 1 = (�+ 1).

m index component index, typically runs from 1..NC − 1

m′ index primary variable component index, typically runsfrom 1..NC − 2

α index index for completions in a well

w index indicates that a variable is within the wellbore, notthe reservoir

ϕ index generic phase; may be o, w, g, or t.

θ index orientation of directional permeability

x script represents properties specific to x-direction

y script represents properties specific to y-direction

z script represents properties specific to z-direction

w script water phase

o script oil phase

g script gas phase

a script asphaltene phase

t script trapped phase

gto script gas trapped by oil

gtw script gas trapped by water

otg script oil trapped by gas

otw script oil trapped by water

wtg script water trapped by gas

wto script water trapped by oil

Continued.

426


Table 22.1: Subscripts and superscripts (continued)

variable units name

t script total, refers to sum of phases or sum of m1 and m2

systems

m script matrix properties

m1 script interconnected matrix properties

m2 script trapped matrix properties

f script fracture properties

f/m script transfer from fracture into matrix

m/f script transfer from matrix into fracture

l script liquid phase

v script vapor phase

CH4 script methane component

C1 script methane component

CI1 script intermediate hydrocarbon pseudo-component 1

CI2 script intermediate hydrocarbon pseudo-component 2

CH1 script heavy hydrocarbon pseudo-component 1



CO2 script carbon dioxide component

H2O script water component

Table 22.2 identifies the variables used in this document. The units given are typical units. Theunits for empirical correlations are listed in a particular section are listed within each section thatcontains correlations.

Table 22.2: Variables used in this document

variable units name

Accnmi lbmol/ft3 accumulation term

A varies general matrix

Al ft3/lbmol Peng-Robinson parameter

Av ft3/lbmol Peng-Robinson parameter

amn psi · ft3/lbmol Peng-Robinson coefficient

al psi · ft3/lbmol Peng-Robinson parameter

av psi · ft3/lbmol Peng-Robinson parameter

α varies General parameter

αow unitless capillary pressure coefficient

Bl ft3/lbmol Peng-Robinson parameter

Bv ft3/lbmol Peng-Robinson parameter

bm ft3/lbmol Peng-Robinson coefficient

Continued.

427


Table 22.2: Variables used in this document (continued)

variable units name

bl ft3/lbmol Peng-Robinson parameter

bv ft3/lbmol Peng-Robinson parameter

β unitless bandwidth

Cw 1/psi water compressibility

Co 1/psi oil compressibility

Cg 1/psi gas compressibility

Cφ 1/psi formation compressibility

Cwi fraction concentration, used to track mixing between injectedbrine and reservoir brine

Cs fraction salt concentration; units include weight fraction,mole fraction, volume fraction, mass/volume, andmole/volume

Cwα day/ft3 well bore storage coefficient

Coff [f ] time communication time between cores on differentnodes for f

Con [f ] time communication time between cores on the samenode for f

Cm lbmol/day component equation

cm ft3/lbmol Peneloux volume adjustment

DPmnxt (lbmol/day)/psi coefficient of δP

DCmn�xt (lbmol/day) coefficient which does not multiply δ

DSOmn�xt (lbmol/day) coefficient which multiplies δSo

DSGmn�xt (lbmol/day) coefficient which multiplies δSg

Dmol ft2/day molecular diffusion coefficient

Ddisp ft2/day dispersivity coefficient

Dl/lowo,mm1/m2

ft2/day liquid-liquid molecular diffusion forom2 → wm1 → om1

Dl/vgwg,mm1/m2

ft2/day gas-liquid molecular diffusion for gm2 → wm1 → gm1

Dl/lwow,mm1/m2

ft2/day liquid-liquid molecular diffusion forwm2 → om1 → wm1

D ft depth

Dα ft total vertical depth of completion α in a well

δ varies general solution vector

δmn unitless binary interaction coefficient

δPo psi primary variable; change in oil pressure over anonlinear iteration

δSo unitless primary variable; change in oil saturation over anonlinear iteration

Continued.

428



variable units name

δSg unitless primary variable; change in gas saturation over anonlinear iteration

δXm′ unitless primary variable; change in liquid mole fraction ofcomponent m′ over a nonlinear iteration

δYm′ unitless primary variable; change in gas mole fraction ofcomponent m′ over a nonlinear iteration

δm,m′ unitless Kronecker delta function, evaluates to 1 if m = m′

and 0 otherwise

dwα ft well inside diameter

εRe unitless pipe roughness for Reynold’s number calculation

ε varies tolerance or threshhold

fom psi oil phase fugacity

fgm psi gas phase fugacity

f unitless Moody friction factor

Gm′ psi thermodynamic constraint equation

γw psi/ft specific gravity of aqueous phase

γo psi/ft specific gravity of oil phase

γg psi/ft specific gravity of gas phase

hf,α+ 12

ft friction adjustment based on the length of the wellsegment

hϕ,f ft fluid height for phase ϕ in the fracture

hϕ,m1 ft fluid height for phase ϕ in the matrix

K ft2/day diffusion coefficient for mass transfer betweentrapped and mixable phases

Ke1m multiplier in solution of Peng-Robinson equation of

state

k md permeability

kθ md permeability in direction θ

kxx md permeability in x-direction

kyy md permeability in y-direction

kzz md permeability in z-direction

kr unitless relative permeability

k�rφ md value of relative permeability at maximumsaturation for phase ϕ

krw unitless total relative permeability to water

kro unitless total relative permeability to oil

krg unitless total relative permeability to gas

krow unitless relative permeability to oil in presence of water

krwo unitless relative permeability to water in presence of oil

Continued.

429



variable units name

krog unitless relative permeability to oil in presence of gas

krgo unitless relative permeability to gas in presence of oil

krwg unitless relative permeability to water in presence of gas

krgw unitless relative permeability to gas in presence of water

κm unitless Peng-Robinson parameter

LHS varies term on left hand side of equation

Lα ft measured depth along wellbore

l fraction mole fraction of liquid phase after flash

lwα fraction mole fraction of liquid phase after flash ofcumulative fluid in wellbore

λw 1/cp mobility of water phase

λo 1/cp mobility of oil phase

λg 1/cp mobility of gas phase

MW lbm/mol molecular weight

μw cp viscosity of water phase

μo cp viscosity of oil phase

μg cp viscosity of gas phase

Nx unitless number of grid cells in x direction

Ny unitless number of grid cells in y direction

Nz unitless number of grid cells in z direction

Nxyz unitless total number of grid cells

NC unitless number of components, including H2O

Nc unitless capillary number

Nb unitless bond number

Nn unitless number of processing nodes

Np unitless number of processing cores per node

Nnp unitless total number of processing cores on all nodes

NRe unitless Reynold’s number

nϕ unitless relative permeability exponent

O [f ] time computational order of f

P psi pressure; if no phase subscript, measured in oil phase

Pb psi bubble point pressure

Pd psi dew point pressure

Pcm psi critical pressure

Pw psi pressure measured in water phase

Po psi pressure measured in oil phase

Pg psi pressure measured in gas phase

Pc psi capillary pressure

Pcow psi water-oil capillary pressure

Continued.

430



variable units name

Pcgo psi gas-oil capillary pressure

Pwα psi pressure in wellbore

P � psi reference producing pressure at the heel of the well

Φm unitless fugacity coefficient

φ fraction porosity

Ψϕ psi potential of phase ϕ

Q lbmol/day molar flux rate

Q�t lbmol/day molar flux rate at heel of well

Qo,α lbmol/day molar flux rate from the well into the reservoir atcompletion α

Qwo,α lbmol/day cumulative molar flux rate in the wellbore at

completion α

q ft3/day volumetric rate

qnwi 1/day water source volumetric rate per reservoir volume

qnoi 1/day oil source volumetric rate per reservoir volume

qngi 1/day gas source volumetric rate per reservoir volume

qw ft3/day water rate

qo ft3/day oil rate

qg ft3/day gas rate

qt ft3/day total rate

q� ft3/day total volumetric flow rate at heel of well

RHS varies term on right hand side of equation

R psi · ft3/lbmol · ◦F Ideal gas law coefficient

R varies general right-hand-side of matrix equation

Re1m unitless convergence criteria for flash

Rsw SCF/STB solubility of gas in water

rw,α ft effective wellbore radius for flow between thereservoir and the well

ρ lbmol/ft3 density�S fraction short notation for So, Sg, Sw

Sw fraction water saturation

So fraction oil saturation

Sg fraction gas saturation

S∗ϕ fraction normalized saturation for phase ϕ

Swt fraction total trapped water saturation

Sot fraction total trapped oil saturation

Sgt fraction total trapped gas saturation

Swot fraction water trapped by oil phase

Swgt fraction water trapped by gas phase

Continued.

431



variable units name

Sowt fraction oil trapped by water phase

Sogt fraction oil trapped by gas phase

Sgwt fraction gas trapped by water phase

Sgot fraction gas trapped by oil phase

sα unitless skin factor for well

sm unitless Peneloux volume shift

σ dyne/cm interfacial tension

σ 1/ft2 shape factor

T R temperature

Tcm R critical temperature

Tmnxw,i± 1

2

(lbmol/ft2)psi−1 transmissibility term for water phase in x direction

Tmnxo,i± 1

2

(lbmol/ft2)psi−1 transmissibility term for oil phase in x direction

Tmnxg,i± 1

2

(lbmol/ft2)psi−1 transmissibility term for gas phase in x direction

Tmnyw,i± 1

2

(lbmol/ft2)psi−1 transmissibility term for water phase in y direction

Tmnyo,i± 1

2

(lbmol/ft2)psi−1 transmissibility term for oil phase in y direction

Tmnyg,i± 1

2

(lbmol/ft2)psi−1 transmissibility term for gas phase in y direction

Tmnzw,i± 1

2

(lbmol/ft2)psi−1 transmissibility term for water phase in z direction

Tmnzo,i± 1

2

(lbmol/ft2)psi−1 transmissibility term for oil phase in z direction

Tmnzg,i± 1

2

(lbmol/ft2)psi−1 transmissibility term for gas phase in z direction

t days time

ts days time step size

Δt days time step size

τ lbmol/day transfer function

τt,m1/m2lbmol/day transfer function between trapped and mixable

matrix phases

τt,f/m1lbmol/day transfer function between fracture and mixable

matrix phases

τmgto lbmol/day transfer function for component m for gas trappedby oil

τmgtw lbmol/day transfer function for component m for gas trappedby water

τmotg lbmol/day transfer function for component m for oil trapped bygas

τmotw lbmol/day transfer function for component m for oil trapped bywater

τmwtg lbmol/day transfer function for component m for water trappedby gas

Continued.

432



variable units name

τmwto lbmol/day transfer function for component m for water trappedby oil

Un�mi (lbmol/ft3)/day total of spatial terms and source terms

Un�mx,i (lbmol/ft3)/day total of spatial terms in x direction

Un�my,i (lbmol/ft3)/day total of spatial terms in y direction

Un�mz,i (lbmol/ft3)/day total of spatial terms in z direction

VR ft3 rock volume

V wα ft3 volume within wellbore

vwα fraction mole fraction of vapor phase after flash

v ft3/lbmol specific volume

vEOS ft3/lbmol specific volume calculated by Peng-Robinsonequation of state before Peneloux volume adjustment

vwϕα ft3/day velocity of phase ϕ in wellbore

WI#α (ft3/day)(cp/psi) well index

Wm unitless mole fraction in aqueous phase

Ωa lbmol/ft3 Peng-Robinson constant

Ωb unitless Peng-Robinson constant

ωm unitless Peng-Robinson acentric factor�X fraction short notation for X1,X2, . . . ,XNC−1,XNC

Xm fraction mole fraction in oil phase

Δx ft grid cell size in x direction

χ varies general variable

ξw lbmol/ft3 molar density of aqueous phase

ξo lbmol/ft3 molar density of oil phase

ξg lbmol/ft3 molar density of gas phase�Y fraction short notation for Y1, Y2, . . . , YNC−1, YNC

Ym fraction mole fraction in gas phase

Δy ft grid cell size in y direction

Zm fraction total mole fraction

Δz ft grid cell size in z direction

zl unitless Peng-Robinson z-factor

zv unitless Peng-Robinson z-factor

433

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APPENDIX - RESULTS FOR SPECIFIC TEST CASES

A.1 Primary Production Results

Figure A.1 illustrates the primary production pressures for W551 and X550.

0 2000 4000 6000 8000 10 000

1000

1500

2000

2500

t�day�

PB

HP,P

ores


(a) Least trapping base, W551.

0 2000 4000 6000 8000 10 000

1000

1500

2000

2500

t�day�

PB

HP,P

ores


(b) Most trapping base, X550.

Figure A.1: Primary Production Pressures.

Figure A.2 illustrates the primary production rates for W551 and X550.

0 2000 4000 6000 8000 10 000

0

50

100

150

200

t�day�

�q T

OT,q

o,q

g,q

w



0 2000 4000 6000 8000 10 000

0

50

100

150

200

t�day�

�q T

OT,q

o,q

g,q

w



Figure A.2: Primary Production Rates.

Figure A.3 illustrates the primary production ratios for W551 and X550.

Figure A.4 illustrates the primary nonlinear iteration convergence for W551 and X550.

Figure A.5 illustrates the primary CFL criteria (Courant et al., 1967) on time step size for

W551 and X550.

480

0 2000 4000 6000 8000 10 0000.0

0.2

0.4

0.6

0.8

1.0

t�day�

q o�q

T,q

g�q

T,q

w�q

T

Production Ratio


0 2000 4000 6000 8000 10 0000.0

0.2

0.4

0.6

0.8

1.0

t�day�

q o�q

T,q

g�q

T,q

w�q

T

Production Ratio


Figure A.3: Primary Production Ratios.

0 2000 4000 6000 8000 10 0000.0

0.5

1.0

1.5

2.0

2.5

3.0

t�day�

nonl

inea

rit

Average it � 1.046


0 2000 4000 6000 8000 10 0000

1

2

3

4

5

6

t�day�

nonl

inea

rit



Figure A.4: Primary nonlinear iteration convergence.

0 2000 4000 6000 8000 10 0000

5

10

15

20

t�day�

max

tssi

ze�d

ay�

Maximum ts from CFL


0 2000 4000 6000 8000 10 0000

5

10

15

20

t�day�

max

tssi

ze�d

ay�


Figure A.5: Primary time step criteria.

481

Figure A.6 illustrates the primary pressure for cells along the diagonal between wells for W551

and X550.

0 2000 4000 6000 8000 10 000

1000

1500

2000

2500

3000

3500

t�day�

P�p

si�

Pressure Across All Cells


0 2000 4000 6000 8000 10 000

1000

1500

2000

2500

3000

3500

t�day�

P�p

si�



Figure A.6: Primary pressure for cells along diagonal between wells.

Figure A.7 illustrates the primary total mass of CO2 for cells along diagonal between wells for

W551 and X550.

0 2000 4000 6000 8000 10 0000

100

200

300

400

500

600

700

t�day�

lbm

olC

O2

Total CO2 Across All Cells


0 2000 4000 6000 8000 10 0000

100

200

300

400

500

600

700

t�day�

lbm

olC

O2



Figure A.7: Primary total mass of CO2 for cells along diagonal between wells.

Figure A.8 illustrates the primary total mass of hydrocarbons (no CO2) for cells along diagonal

between wells for W551 and X550.

Figure A.9 illustrates the primary saturation for equivalent one cell model for W551 and X550.

Figure A.10 illustrates the primary total mole fraction in the reservoir for W551 and X550.

Figure A.11 illustrates the primary recovery factor for W551 and X550.

Figure A.12 illustrates the primary compositional recovery factor for W551 and X550.

482

0 2000 4000 6000 8000 10 0000

2000

4000

6000

8000

10 000

12 000

14 000

t�day�

lbm

olH

C

Total HC �no CO2� Across Diagonal Cells


0 2000 4000 6000 8000 10 0000

2000

4000

6000

8000

10 000

12 000

14 000

t�day�

lbm

ol�H

Cno

CO

2��tra

pm

obile�



Figure A.8: Primary total mass of hydrocarbons (no CO2) for cells along diagonal between wells.

0 5000 10 000 15 000 20 0000.0

0.2

0.4

0.6

0.8

1.0

t�day�

Tot

alS



0 5000 10 000 15 000 20 000 25 0000.0

0.2

0.4

0.6

0.8

1.0

t�day�

Tot

alS



Figure A.9: Primary saturation for equivalent one cell model. Purple is trapped water, blue ismobile water, cyan is trapped oil, green is mobile oil, yellow is trapped gas, red is mobile gas.

0 2000 4000 6000 8000 10 0000.0

0.2

0.4

0.6

0.8

1.0

t�day�

CH

4,n

C4,n

C10

,CO

2



0 2000 4000 6000 8000 10 0000.0

0.2

0.4

0.6

0.8

1.0

t�day�

CH

4,n

C4,n

C10

,CO

2



Figure A.10: Primary total mole fraction in the reservoir.

483

0 2000 4000 6000 8000 10 0000.0

0.2

0.4

0.6

0.8

1.0

t�day�

RF

prod

uced

oil�

RC

F�

PV�R

CF�

Recovery Factor � Econ Limit � 0.191192


0 2000 4000 6000 8000 10 0000.0

0.2

0.4

0.6

0.8

1.0

t�day�

RF

prod

uced

oil�

RC

F�

PV�R

CF�



Figure A.11: Primary recovery factor.

0 2000 4000 6000 8000 10 0000.0

0.2

0.4

0.6

0.8

1.0

t�day�

CH

4,n

C4,n

C10



0 2000 4000 6000 8000 10 0000.0

0.2

0.4

0.6

0.8

1.0

t�day�

CH

4,n

C4,n

C10



Figure A.12: Primary compositional recovery factor.

484

Figure A.13 illustrates the distribution of pressures at the economic limit of primary for W551

and X550.

1000 2000 3000 4000 50000.000

0.002

0.004

0.006

0.008

0.010

P �psia�

freq

Presure Distribution � time � 7200.


1000 2000 3000 4000 50000.000

0.002

0.004

0.006

0.008

0.010

P �psia�

freq



Figure A.13: Distribution of pressures at primary economic limit.

Figure A.14 illustrates the 2-D pressure distribution at the economic limit of primary for W551

and X550.

Presure � time � 7200.

P �psia�

1000

2000

3000

4000

5000



P �psia�

1000

2000

3000

4000

5000


Figure A.14: 2-D pressure distribution at primary economic limit.

Figure A.15 illustrates the distribution of oil saturations at the economic limit of primary for

W551 and X550.

Figure A.16 illustrates the 2-D oil saturation distribution at the economic limit of primary for

W551 and X550.

485

0.0 0.2 0.4 0.6 0.8 1.00

20

40

60

80

100

So

freq

Oil Saturation Distribution � time � 7200.


0.0 0.2 0.4 0.6 0.8 1.00

10

20

30

40

50

SoT

freq

Total Oil Saturation Distribution � time � 7290.


Figure A.15: Distribution of oil saturation at primary economic limit.

Oil Saturation � time � 7200.

So

0

0.2

0.4

0.6

0.8

1.0


Total Oil Saturation � time � 7290.

SoT

0

0.2

0.4

0.6

0.8

1.0


Figure A.16: 2-D oil saturation distribution at primary economic limit.

486

Figure A.17 illustrates the distribution of gas saturations at the economic limit of primary for

W551 and X550.

0.0 0.2 0.4 0.6 0.8 1.00

20

40

60

80

100

Sg

freq

Gas Saturation Distribution � time � 7200.


0.0 0.2 0.4 0.6 0.8 1.00

10

20

30

40

50

SgT

freq

Total Gas Saturation Distribution � time � 7290.


Figure A.17: Distribution of gas saturation at primary economic limit.

Figure A.18 illustrates the 2-D gas saturation distribution at the economic limit of primary for

W551 and X550.

Gas Saturation � time � 7200.

Sg

0

0.2

0.4

0.6

0.8

1.0


Total Gas Saturation � time � 7290.

SgT

0

0.2

0.4

0.6

0.8

1.0


Figure A.18: 2-D gas saturation distribution at primary economic limit.

Figure A.19 illustrates the distribution of water saturations at the economic limit of primary

for W551 and X550.

Figure A.20 illustrates the 2-D water saturation distribution at the economic limit of primary

for W551 and X550.

487

0.0 0.2 0.4 0.6 0.8 1.00

20

40

60

80

100

Sw

freq

Water Saturation Distribution � time � 7200.


0.0 0.2 0.4 0.6 0.8 1.00

10

20

30

40

50

SwT

freq

Total Water Saturation Distribution � time � 7290.


Figure A.19: Distribution of water saturation at primary economic limit.

Water Saturation � time � 7200.

Sw

0

0.2

0.4

0.6

0.8

1.0


Total Water Saturation � time � 7290.

SwT

0

0.2

0.4

0.6

0.8

1.0


Figure A.20: 2-D water saturation distribution at primary economic limit.

488

A.2 Waterflood Results

Figure A.21 illustrates the waterflood injection pressures for W551 and X550.

0 5000 10 000 15 000 20 000 25 000 30 000 35 0001000

1500

2000

2500

3000

3500

t�day�

PB

HP,P

ores



0 5000 10 000 15 000 20 000 25 000 30 0001000

1500

2000

2500

3000

3500

t�day�

PB

HP,P

ores



Figure A.21: Waterflood Injection Pressure.

Figure A.22 illustrates the waterflood injection rates for W551 and X550.

0 5000 10 000 15 000 20 000 25 000 30 000 35 000

0

50

100

150

200

t�day�

q wor

q g�b

bl�



0 5000 10 000 15 000 20 000 25 000 30 000

0

50

100

150

200

t�day�

q wor

q g�b

bl�



Figure A.22: Waterflood Injection Rates.

Figure A.23 illustrates the waterflood production pressures for W551 and X550.

Figure A.24 illustrates the waterflood production rates for W551 and X550.

Figure A.25 illustrates the waterflood production ratios for W551 and X550.

Figure A.26 illustrates the waterflood oil production rate minus the primary production rate

for W551 and X550.

Figure A.27 illustrates the waterflood nonlinear iteration convergence for W551 and X550.

Figure A.28 illustrates the waterflood CFL criteria (Courant et al., 1967) on time step size for

W551 and X550.

489

0 5000 10 000 15 000 20 000 25 000 30 000 35 000

1000

1500

2000

2500

t�day�

PB

HP,P

ores



0 5000 10 000 15 000 20 000 25 000 30 000

1000

1500

2000

2500

t�day�

PB

HP,P

ores



Figure A.23: Waterflood Production Pressures.

0 5000 10 000 15 000 20 000 25 000 30 000 35 000

0

50

100

150

200

t�day�

�q T

OT,q

o,q

g,q

w



0 5000 10 000 15 000 20 000 25 000 30 000

0

50

100

150

200

t�day�

�q T

OT,q

o,q

g,q

w



Figure A.24: Waterflood Production Rates.

0 5000 10 000 15 000 20 000 25 000 30 000 35 0000.0

0.2

0.4

0.6

0.8

1.0

t�day�

q o�q

T,q

g�q

T,q

w�q

T

Production Ratio


0 5000 10 000 15 000 20 000 25 000 30 0000.0

0.2

0.4

0.6

0.8

1.0

t�day�

q o�q

T,q

g�q

T,q

w�q

T

Production Ratio


Figure A.25: Waterflood Production Ratios.

490

0 5000 10 000 15 000 20 000 25 000 30 000 35 000

0

10

20

30

40

t�day�

q o�R

B�

New�Old Production Rate �RBPD�


0 5000 10 000 15 000 20 000 25 000 30 000

0

5

10

15

20

25

30

t�day�

q o�R

B�



Figure A.26: WF− Primary Oil Rate .

0 5000 10 000 15 000 20 000 25 000 30 000 35 0000.0

0.5

1.0

1.5

2.0

2.5

3.0

t�day�

nonl

inea

rit



0 5000 10 000 15 000 20 000 25 000 30 0000

10

20

30

40

50

t�day�

nonl

inea

rit



Figure A.27: Waterflood nonlinear iteration convergence.

0 5000 10 000 15 000 20 000 25 000 30 000 35 0000

5

10

15

20

t�day�

max

tssi

ze�d

ay�

Maximum ts from CFL


0 5000 10 000 15 000 20 000 25 000 30 0000

5

10

15

20

t�day�

max

tssi

ze�d

ay�


Figure A.28: Waterflood time step criteria.

491

Figure A.29 illustrates the waterflood pressure for cells along the diagonal between wells for

W551 and X550.

0 5000 10 000 15 000 20 000 25 000 30 000 35 000

1000

1500

2000

2500

3000

3500

t�day�

P�p

si�



0 5000 10 000 15 000 20 000 25 000 30 000

1000

1500

2000

2500

3000

3500

t�day�

P�p

si�



Figure A.29: Waterflood pressure for cells along diagonal between wells.

Figure A.30 illustrates the waterflood total mass of CO2 for cells along diagonal between wells

for W551 and X550.

0 5000 10 000 15 000 20 000 25 000 30 000 35 0000

100

200

300

400

500

600

700

t�day�

lbm

olC

O2



0 5000 10 000 15 000 20 000 25 000 30 0000

500

1000

1500

2000

2500

3000

t�day�

lbm

olC

O2



Figure A.30: Waterflood total mass of CO2 for cells along diagonal between wells.

Figure A.31 illustrates the waterflood total mass of hydrocarbons (no CO2) for cells along

diagonal between wells for W551 and X550.

Figure A.32 illustrates the waterflood saturation for equivalent one cell model for W551 and

X550.

Figure A.33 illustrates the waterflood total mole fraction in the reservoir for W551 and X550.

Figure A.34 illustrates the waterflood recovery factor for W551 and X550.

Figure A.35 illustrates the waterflood compositional recovery factor for W551 and X550.

492

0 5000 10 000 15 000 20 000 25 000 30 000 35 0000

2000

4000

6000

8000

10 000

12 000

14 000

t�day�

lbm

olH

C



0 5000 10 000 15 000 20 000 25 000 30 0000

2000

4000

6000

8000

10 000

12 000

14 000

t�day�

lbm

ol�H

Cno

CO

2��tra

pm

obile�



Figure A.31: Waterflood total mass of hydrocarbons (no CO2) for cells along diagonal betweenwells.

0 5000 10 000 15 000 20 000 25 000 30 000 35 0000.0

0.2

0.4

0.6

0.8

1.0

t�day�

Tot

alS



0 5000 10 000 15 000 20 000 25 000 30 0000.0

0.2

0.4

0.6

0.8

1.0

t�day�

Tot

alS



Figure A.32: Waterflood saturation for equivalent one cell model. Purple is trapped water, blue ismobile water, cyan is trapped oil, green is mobile oil, yellow is trapped gas, red is mobile gas.

0 5000 10 000 15 000 20 000 25 000 30 000 35 0000.0

0.2

0.4

0.6

0.8

1.0

t�day�

CH

4,n

C4,n

C10

,CO

2



0 5000 10 000 15 000 20 000 25 000 30 0000.0

0.2

0.4

0.6

0.8

1.0

t�day�

CH

4,n

C4,n

C10

,CO

2



Figure A.33: Waterflood total mole fraction in the reservoir.

493

0 5000 10 000 15 000 20 000 25 000 30 000 35 0000.0

0.2

0.4

0.6

0.8

1.0

t�day�

RF

prod

uced

oil�

RC

F�

PV�R

CF�



0 5000 10 000 15 000 20 000 25 000 30 0000.0

0.2

0.4

0.6

0.8

1.0

t�day�

RF

prod

uced

oil�

RC

F�

PV�R

CF�



Figure A.34: Waterflood recovery factor.

0 5000 10 000 15 000 20 000 25 000 30 000 35 0000.0

0.2

0.4

0.6

0.8

1.0

t�day�

CH

4,n

C4,n

C10



0 5000 10 000 15 000 20 000 25 000 30 0000.0

0.2

0.4

0.6

0.8

1.0

t�day�

CH

4,n

C4,n

C10



Figure A.35: Waterflood compositional recovery factor.

494

Figure A.36 illustrates the distribution of pressures at the economic limit of waterflood for W551

and X550.

1000 2000 3000 4000 50000.0000

0.0005

0.0010

0.0015

0.0020

0.0025

P �psia�

freq



1000 2000 3000 4000 50000.0000

0.0005

0.0010

0.0015

0.0020

0.0025

P �psia�

freq



Figure A.36: Distribution of pressures at waterflood economic limit.

Figure A.37 illustrates the 2-D pressure distribution at the economic limit of waterflood for

W551 and X550.


P �psia�

1000

2000

3000

4000

5000



P �psia�

1000

2000

3000

4000

5000


Figure A.37: 2-D pressure distribution at waterflood economic limit.

Figure A.38 illustrates the distribution of oil saturations at the economic limit of waterflood for

W551 and X550.

Figure A.39 illustrates the 2-D oil saturation distribution at the economic limit of waterflood

for W551 and X550.

495

0.0 0.2 0.4 0.6 0.8 1.00

2

4

6

8

10

12

So

freq



0.0 0.2 0.4 0.6 0.8 1.00

2

4

6

8

10

SoT

freq



Figure A.38: Distribution of oil saturation at waterflood economic limit.


So

0

0.2

0.4

0.6

0.8

1.0



SoT

0

0.2

0.4

0.6

0.8

1.0


Figure A.39: 2-D oil saturation distribution at waterflood economic limit.

496

Figure A.40 illustrates the distribution of gas saturations at the economic limit of waterflood

for W551 and X550.

0.0 0.2 0.4 0.6 0.8 1.00

20

40

60

80

100

Sg

freq



0.0 0.2 0.4 0.6 0.8 1.00

10

20

30

40

SgT

freq



Figure A.40: Distribution of gas saturation at waterflood economic limit.

Figure A.41 illustrates the 2-D gas saturation distribution at the economic limit of waterflood

for W551 and X550.


Sg

0

0.2

0.4

0.6

0.8

1.0



SgT

0

0.2

0.4

0.6

0.8

1.0


Figure A.41: 2-D gas saturation distribution at waterflood economic limit.

Figure A.42 illustrates the distribution of water saturations at the economic limit of waterflood

for W551 and X550.

Figure A.43 illustrates the 2-D water saturation distribution at the economic limit of waterflood

for W551 and X550.

497

0.0 0.2 0.4 0.6 0.8 1.00

2

4

6

8

10

12

Sw

freq



0.0 0.2 0.4 0.6 0.8 1.00

2

4

6

8

SwT

freq



Figure A.42: Distribution of water saturation at waterflood economic limit.


Sw

0

0.2

0.4

0.6

0.8

1.0



SwT

0

0.2

0.4

0.6

0.8

1.0


Figure A.43: 2-D water saturation distribution at waterflood economic limit.

498

A.3 Continuous CO2 Injection Results

Figure A.44 illustrates the continuous CO2 injection pressures for W551 and X550.

0 5000 10 000 15 000 20 000 25 000 30 000 35 000

1000

2000

3000

4000

5000

t�day�

PB

HP,P

ores



0 5000 10 000 15 000 20 000 25 000 30 000 35 000

1000

2000

3000

4000

5000

t�day�

PB

HP,P

ores



Figure A.44: Continuous CO2 Injection Pressure.

Figure A.45 illustrates the continuous CO2 injection rates for W551 and X550.

0 5000 10 000 15 000 20 000 25 000 30 000 35 000

0

50

100

150

200

t�day�

q wor

q g�b

bl�



0 5000 10 000 15 000 20 000 25 000 30 000 35 000

0

50

100

150

200

t�day�

q wor

q g�b

bl�



Figure A.45: Continuous CO2 Injection Rates.

Figure A.46 illustrates the continuous CO2 production pressures for W551 and X550.

Figure A.47 illustrates the continuous CO2 production rates for W551 and X550.

Figure A.48 illustrates the continuous CO2 production ratios for W551 and X550.

Figure A.49 illustrates the continuous CO2 oil production rate minus the waterflood production

rate for W551 and X550.

Figure A.50 illustrates the continuous CO2 nonlinear iteration convergence for W551 and X550.

499

0 5000 10 000 15 000 20 000 25 000 30 000 35 000

1000

1500

2000

2500

3000

3500

4000

4500

t�day�

PB

HP,P

ores



0 5000 10 000 15 000 20 000 25 000 30 000 35 000

1000

1500

2000

2500

3000

t�day�

PB

HP,P

ores



Figure A.46: Continuous CO2 Production Pressures.

0 5000 10 000 15 000 20 000 25 000 30 000 35 000

0

50

100

150

200

t�day�

�q T

OT,q

o,q

g,q

w



0 5000 10 000 15 000 20 000 25 000 30 000 35 000

0

50

100

150

200

t�day�

�q T

OT,q

o,q

g,q

w



Figure A.47: Continuous CO2 Production Rates.

0 5000 10 000 15 000 20 000 25 000 30 000 35 0000.0

0.2

0.4

0.6

0.8

1.0

t�day�

q o�q

T,q

g�q

T,q

w�q

T

Production Ratio


0 5000 10 000 15 000 20 000 25 000 30 000 35 0000.0

0.2

0.4

0.6

0.8

1.0

t�day�

q o�q

T,q

g�q

T,q

w�q

T

Production Ratio


Figure A.48: Continuous CO2 Production Ratios.

500

0 5000 10 000 15 000 20 000 25 000 30 000 35 000

0

50

100

150

200

t�day�

q o�R

B�



0 5000 10 000 15 000 20 000 25 000 30 000 35 000

0

20

40

60

80

t�day�

q o�R

B�



Figure A.49: GF−WF Oil Rate .

0 5000 10 000 15 000 20 000 25 000 30 000 35 0000

2

4

6

8

t�day�

nonl

inea

rit



0 5000 10 000 15 000 20 000 25 000 30 000 35 0000

10

20

30

40

50

60

70

t�day�

nonl

inea

rit



Figure A.50: Continuous CO2 nonlinear iteration convergence.

501

Figure A.51 illustrates the continuous CO2 CFL criteria (Courant et al., 1967) on time step size

for W551 and X550.

0 5000 10 000 15 000 20 000 25 000 30 000 35 0000

5

10

15

20

t�day�

max

tssi

ze�d

ay�

Maximum ts from CFL


0 5000 10 000 15 000 20 000 25 000 30 000 35 0000

5

10

15

20

t�day�

max

tssi

ze�d

ay�


Figure A.51: Continuous CO2 time step criteria.

Figure A.52 illustrates the continuous CO2 pressure for cells along the diagonal between wells

for W551 and X550.

0 5000 10 000 15 000 20 000 25 000 30 000 35 000

1000

2000

3000

4000

5000

t�day�

P�p

si�



0 5000 10 000 15 000 20 000 25 000 30 000 35 000

1000

2000

3000

4000

5000

t�day�

P�p

si�



Figure A.52: Continuous CO2 pressure for cells along diagonal between wells.

Figure A.53 illustrates the continuous CO2 total mass of CO2 for cells along diagonal between

wells for W551 and X550.

Figure A.54 illustrates the continuous CO2 total mass of hydrocarbons (no CO2) for cells along

diagonal between wells for W551 and X550.

Figure A.55 illustrates the continuous CO2 saturation for equivalent one cell model for W551

and X550.

502

0 5000 10 000 15 000 20 000 25 000 30 000 35 0000

5000

10 000

15 000

20 000

t�day�

lbm

olC

O2



0 5000 10 000 15 000 20 000 25 000 30 000 35 0000

5000

10 000

15 000

t�day�

lbm

olC

O2



Figure A.53: Continuous CO2 total mass of CO2 for cells along diagonal between wells.

0 5000 10 000 15 000 20 000 25 000 30 000 35 0000

2000

4000

6000

8000

10 000

12 000

14 000

t�day�

lbm

olH

C



0 5000 10 000 15 000 20 000 25 000 30 000 35 0000

2000

4000

6000

8000

10 000

12 000

14 000

t�day�

lbm

ol�H

Cno

CO

2��tra

pm

obile�



Figure A.54: Continuous CO2 total mass of hydrocarbons (no CO2) for cells along diagonal betweenwells.

0 5000 10 000 15 000 20 000 25 000 30 000 35 0000.0

0.2

0.4

0.6

0.8

1.0

t�day�

Tot

alS



0 5000 10 000 15 000 20 000 25 000 30 000 35 0000.0

0.2

0.4

0.6

0.8

1.0

t�day�

Tot

alS



Figure A.55: Continuous CO2 saturation for equivalent one cell model. Purple is trapped water,blue is mobile water, cyan is trapped oil, green is mobile oil, yellow is trapped gas, red is mobilegas.

503

Figure A.56 illustrates the continuous CO2 total mole fraction in the reservoir for W551 and

X550.

0 5000 10 000 15 000 20 000 25 000 30 000 35 0000.0

0.2

0.4

0.6

0.8

1.0

t�day�

CH

4,n

C4,n

C10

,CO

2



0 5000 10 000 15 000 20 000 25 000 30 000 35 0000.0

0.2

0.4

0.6

0.8

1.0

t�day�

CH

4,n

C4,n

C10

,CO

2



Figure A.56: Continuous CO2 total mole fraction in the reservoir.

Figure A.57 illustrates the continuous CO2 recovery factor for W551 and X550.

0 5000 10 000 15 000 20 000 25 000 30 000 35 0000.0

0.2

0.4

0.6

0.8

1.0

t�day�

RF

prod

uced

oil�

RC

F�

PV�R

CF�



0 5000 10 000 15 000 20 000 25 000 30 000 35 0000.0

0.2

0.4

0.6

0.8

1.0

t�day�

RF

prod

uced

oil�

RC

F�

PV�R

CF�



Figure A.57: Continuous CO2 recovery factor.

Figure A.58 illustrates the continuous CO2 compositional recovery factor for W551 and X550.

Figure A.59 illustrates the continuous CO2 storage of CO2 for W551 and X550.

Figure A.60 illustrates the continuous CO2 utilization of CO2 for W551 and X550.

Figure A.61 illustrates the distribution of pressures at the economic limit of continuous CO2

for W551 and X550.

Figure A.62 illustrates the 2-D pressure distribution at the economic limit of continuous CO2

for W551 and X550.

504

0 5000 10 000 15 000 20 000 25 000 30 000 35 0000.0

0.2

0.4

0.6

0.8

1.0

t�day�

CH

4,n

C4,n

C10



0 5000 10 000 15 000 20 000 25 000 30 000 35 0000.0

0.2

0.4

0.6

0.8

1.0

t�day�

CH

4,n

C4,n

C10



Figure A.58: Continuous CO2 compositional recovery factor.

0 5000 10 000 15 000 20 000 25 000 30 000 35 0000.0

0.2

0.4

0.6

0.8

t�day�

CO

2St

orag

e�lbm

ol�lbm

ol�

CO2 Storage � 0.876518


0 5000 10 000 15 000 20 000 25 000 30 000 35 0000.0

0.2

0.4

0.6

0.8

t�day�

CO

2St

orag

e�lbm

ol�lbm

ol�



Figure A.59: Continuous CO2 storage of CO2.

0 5000 10 000 15 000 20 000 25 000 30 000 35 0000

10

20

30

40

50

t�day�

CO

2U

tiliz

atoi

n�M

CF�R

B�

CO2 Utilizaiotn � Econ Limit � 4.89089


0 5000 10 000 15 000 20 000 25 0000

10

20

30

40

50

t�day�

CO

2U

tiliz

atoi

n�M

CF�R

B�



Figure A.60: Continuous CO2 utilization of CO2.

505

1000 2000 3000 4000 50000.000

0.001

0.002

0.003

0.004

0.005

P �psia�

freq



1000 2000 3000 4000 50000.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

P �psia�

freq



Figure A.61: Distribution of pressures at Continuous CO2 economic limit.


P �psia�

1000

2000

3000

4000

5000



P �psia�

1000

2000

3000

4000

5000


Figure A.62: 2-D pressure distribution at Continuous CO2 economic limit.

506

Figure A.63 illustrates the distribution of oil saturations at the economic limit of continuous

CO2 for W551 and X550.

0.0 0.2 0.4 0.6 0.8 1.00

10

20

30

40

50

60

So

freq



0.0 0.2 0.4 0.6 0.8 1.00

5

10

15

20

SoT

freq



Figure A.63: Distribution of oil saturation at Continuous CO2 economic limit.

Figure A.64 illustrates the 2-D oil saturation distribution at the economic limit of continuous



So

0

0.2

0.4

0.6

0.8

1.0



SoT

0

0.2

0.4

0.6

0.8

1.0


Figure A.64: 2-D oil saturation distribution at Continuous CO2 economic limit.

Figure A.65 illustrates the distribution of gas saturations at the economic limit of continuous


Figure A.66 illustrates the 2-D gas saturation distribution at the economic limit of continuous


507

0.0 0.2 0.4 0.6 0.8 1.00

10

20

30

40

Sg

freq



0.0 0.2 0.4 0.6 0.8 1.00

2

4

6

8

10

12

14

SgT

freq



Figure A.65: Distribution of gas saturation at Continuous CO2 economic limit.


Sg

0

0.2

0.4

0.6

0.8

1.0



SgT

0

0.2

0.4

0.6

0.8

1.0


Figure A.66: 2-D gas saturation distribution at Continuous CO2 economic limit.

508

Figure A.67 illustrates the distribution of water saturations at the economic limit of continuous


0.0 0.2 0.4 0.6 0.8 1.00

5

10

15

20

25

30

Sw

freq



0.0 0.2 0.4 0.6 0.8 1.00

2

4

6

8

10

SwT

freq



Figure A.67: Distribution of water saturation at Continuous CO2 economic limit.

Figure A.68 illustrates the 2-D water saturation distribution at the economic limit of continuous



Sw

0

0.2

0.4

0.6

0.8

1.0



SwT

0

0.2

0.4

0.6

0.8

1.0


Figure A.68: 2-D water saturation distribution at Continuous CO2 economic limit.

A.4 WAG Results

Figure A.69 illustrates the WAG injection pressures for W551 and X550.

Figure A.70 illustrates the WAG injection rates for W551 and X550.

Figure A.71 illustrates the WAG production pressures for W551 and X550.

509

0 5000 10 000 15 000 20 000 25 000

1000

2000

3000

4000

5000

t�day�

PB

HP,P

ores



0 5000 10 000 15 000 20 000 25 000 30 000

2000

4000

6000

8000

10 000

t�day�

PB

HP,P

ores



Figure A.69: WAG Injection Pressure.

0 5000 10 000 15 000 20 000 25 000

0

50

100

150

200

t�day�

q wor

q g�b

bl�



0 5000 10 000 15 000 20 000 25 000 30 000

0

50

100

150

200

t�day�

q wor

q g�b

bl�



Figure A.70: WAG Injection Rates.

0 5000 10 000 15 000 20 000 25 000

1000

1500

2000

2500

3000

3500

t�day�

PB

HP,P

ores



0 5000 10 000 15 000 20 000 25 000 30 000

2000

4000

6000

8000

10 000

t�day�

PB

HP,P

ores



Figure A.71: WAG Production Pressures.

510

Figure A.72 illustrates the WAG production rates for W551 and X550.

0 5000 10 000 15 000 20 000 25 000

0

50

100

150

200

t�day�

�q T

OT,q

o,q

g,q

w



0 5000 10 000 15 000 20 000 25 000 30 000

0

50

100

150

200

t�day�

�q T

OT,q

o,q

g,q

w



Figure A.72: WAG Production Rates.

Figure A.73 illustrates the WAG production ratios for W551 and X550.

0 5000 10 000 15 000 20 000 25 0000.0

0.2

0.4

0.6

0.8

1.0

t�day�

q o�q

T,q

g�q

T,q

w�q

T

Production Ratio


0 5000 10 000 15 000 20 000 25 000 30 0000.0

0.2

0.4

0.6

0.8

1.0

t�day�

q o�q

T,q

g�q

T,q

w�q

T

Production Ratio


Figure A.73: WAG Production Ratios.

Figure A.74 illustrates the WAG oil production rate minus the waterflood production rate for

W551 and X550.

Figure A.75 illustrates the WAG nonlinear iteration convergence for W551 and X550.

Figure A.76 illustrates the WAG CFL criteria (Courant et al., 1967) on time step size for W551

and X550.

Figure A.77 illustrates the WAG pressure for cells along the diagonal between wells for W551

and X550.

Figure A.78 illustrates the WAG total mass of CO2 for cells along diagonal between wells for

W551 and X550.

511

0 5000 10 000 15 000 20 000 25 000

0

50

100

150

200

t�day�

q o�R

B�



0 5000 10 000 15 000 20 000 25 000 30 000

0

50

100

150

200

t�day�

q o�R

B�



Figure A.74: WAG−WF Oil Rate .

0 5000 10 000 15 000 20 000 25 0000

1

2

3

4

5

6

7

t�day�

nonl

inea

rit



0 5000 10 000 15 000 20 000 25 000 30 0000

10

20

30

40

50

60

70

t�day�

nonl

inea

rit



Figure A.75: WAG nonlinear iteration convergence.

0 5000 10 000 15 000 20 000 25 0000

5

10

15

20

t�day�

max

tssi

ze�d

ay�

Maximum ts from CFL


0 5000 10 000 15 000 20 000 25 000 30 0000

5

10

15

20

t�day�

max

tssi

ze�d

ay�


Figure A.76: WAG time step criteria.

512

0 5000 10 000 15 000 20 000 25 000

1000

2000

3000

4000

5000

t�day�

P�p

si�



0 5000 10 000 15 000 20 000 25 000 30 000

2000

4000

6000

8000

10 000

t�day�

P�p

si�



Figure A.77: WAG pressure for cells along diagonal between wells.

0 5000 10 000 15 000 20 000 25 0000

5000

10 000

15 000

20 000

t�day�

lbm

olC

O2



0 5000 10 000 15 000 20 000 25 000 30 0000

2000

4000

6000

8000

10 000

12 000

14 000

t�day�

lbm

olC

O2



Figure A.78: WAG total mass of CO2 for cells along diagonal between wells.

513

Figure A.79 illustrates the WAG total mass of hydrocarbons (no CO2) for cells along diagonal

between wells for W551 and X550.

0 5000 10 000 15 000 20 000 25 0000

2000

4000

6000

8000

10 000

12 000

14 000

t�day�

lbm

olH

C



0 5000 10 000 15 000 20 000 25 000 30 0000

2000

4000

6000

8000

10 000

12 000

14 000

t�day�

lbm

ol�H

Cno

CO

2��tra

pm

obile�



Figure A.79: WAG total mass of hydrocarbons (no CO2) for cells along diagonal between wells.

Figure A.80 illustrates the WAG saturation for equivalent one cell model for W551 and X550.

0 5000 10 000 15 000 20 000 25 0000.0

0.2

0.4

0.6

0.8

1.0

t�day�

Tot

alS



0 5000 10 000 15 000 20 000 25 000 30 0000.0

0.2

0.4

0.6

0.8

1.0

t�day�

Tot

alS



Figure A.80: WAG saturation for equivalent one cell model. Purple is trapped water, blue is mobilewater, cyan is trapped oil, green is mobile oil, yellow is trapped gas, red is mobile gas.

Figure A.81 illustrates the WAG total mole fraction in the reservoir for W551 and X550.

Figure A.82 illustrates the WAG recovery factor for W551 and X550.

Figure A.83 illustrates the WAG compositional recovery factor for W551 and X550.

Figure A.84 illustrates the WAG storage of CO2 for W551 and X550.

Figure A.85 illustrates the WAG utilization of CO2 for W551 and X550.

Figure A.86 illustrates the distribution of pressures at the economic limit of WAG for W551

and X550.

514

0 5000 10 000 15 000 20 000 25 0000.0

0.2

0.4

0.6

0.8

1.0

t�day�

CH

4,n

C4,n

C10

,CO

2



0 5000 10 000 15 000 20 000 25 000 30 0000.0

0.2

0.4

0.6

0.8

1.0

t�day�

CH

4,n

C4,n

C10

,CO

2



Figure A.81: WAG total mole fraction in the reservoir.

0 5000 10 000 15 000 20 000 25 0000.0

0.2

0.4

0.6

0.8

1.0

t�day�

RF

prod

uced

oil�

RC

F�

PV�R

CF�



0 5000 10 000 15 000 20 000 25 0000.0

0.2

0.4

0.6

0.8

1.0

t�day�

RF

prod

uced

oil�

RC

F�

PV�R

CF�



Figure A.82: WAG recovery factor.

0 5000 10 000 15 000 20 000 25 0000.0

0.2

0.4

0.6

0.8

1.0

t�day�

CH

4,n

C4,n

C10



0 5000 10 000 15 000 20 000 25 000 30 0000.0

0.2

0.4

0.6

0.8

1.0

t�day�

CH

4,n

C4,n

C10



Figure A.83: WAG compositional recovery factor.

515

0 5000 10 000 15 000 20 000 25 0000.0

0.2

0.4

0.6

0.8

t�day�

CO

2St

orag

e�lbm

ol�lbm

ol�



0 5000 10 000 15 000 20 000 25 0000.0

0.2

0.4

0.6

0.8

t�day�

CO

2St

orag

e�lbm

ol�lbm

ol�



Figure A.84: WAG storage of CO2.

0 5000 10 000 15 000 20 000 25 0000

10

20

30

40

50

t�day�

CO

2U

tiliz

atoi

n�M

CF�R

B�



0 5000 10 000 15 000 20 000 25 000 30 0000

10

20

30

40

50

t�day�

CO

2U

tiliz

atoi

n�M

CF�R

B�



Figure A.85: WAG utilization of CO2.

1000 2000 3000 4000 50000.000

0.001

0.002

0.003

0.004

P �psia�

freq



1000 2000 3000 4000 50000.0000

0.0005

0.0010

0.0015

0.0020

P �psia�

freq



Figure A.86: Distribution of pressures at WAG economic limit.

516

Figure A.87 illustrates the 2-D pressure distribution at the economic limit of WAG for W551

and X550.


P �psia�

1000

2000

3000

4000

5000



P �psia�

1000

2000

3000

4000

5000


Figure A.87: 2-D pressure distribution at WAG economic limit.

Figure A.88 illustrates the distribution of oil saturations at the economic limit of WAG for

W551 and X550.

0.0 0.2 0.4 0.6 0.8 1.00

10

20

30

40

50

60

So

freq



0.0 0.2 0.4 0.6 0.8 1.00

1

2

3

4

5

SoT

freq



Figure A.88: Distribution of oil saturation at WAG economic limit.

Figure A.89 illustrates the 2-D oil saturation distribution at the economic limit of WAG for

W551 and X550.

Figure A.90 illustrates the distribution of gas saturations at the economic limit of WAG for

W551 and X550.

517


So

0

0.2

0.4

0.6

0.8

1.0



SoT

0

0.2

0.4

0.6

0.8

1.0


Figure A.89: 2-D oil saturation distribution at WAG economic limit.

0.0 0.2 0.4 0.6 0.8 1.00

10

20

30

40

Sg

freq



0.0 0.2 0.4 0.6 0.8 1.00

10

20

30

40

SgT

freq



Figure A.90: Distribution of gas saturation at WAG economic limit.

518

Figure A.91 illustrates the 2-D gas saturation distribution at the economic limit of WAG for

W551 and X550.


Sg

0

0.2

0.4

0.6

0.8

1.0



SgT

0

0.2

0.4

0.6

0.8

1.0


Figure A.91: 2-D gas saturation distribution at WAG economic limit.

Figure A.92 illustrates the distribution of water saturations at the economic limit of WAG for

W551 and X550.

0.0 0.2 0.4 0.6 0.8 1.00

5

10

15

20

Sw

freq



0.0 0.2 0.4 0.6 0.8 1.00

1

2

3

4

5

SwT

freq



Figure A.92: Distribution of water saturation at WAG economic limit.

Figure A.93 illustrates the 2-D water saturation distribution at the economic limit of WAG for

W551 and X550.

519


Sw

0

0.2

0.4

0.6

0.8

1.0



SwT

0

0.2

0.4

0.6

0.8

1.0


Figure A.93: 2-D water saturation distribution at WAG economic limit.

520

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