The Combined Influence of Gravity, Flow Rate and Small Scale … · 2012. 12. 18. · Chia-Wei Kuo...

THE COMBINED INFLUENCE OF

GRAVITY, FLOW RATE AND SMALL SCALE HETEROGENEITY

ON CORE-SCALE MULTIPHASE FLOW OF CO2 AND BRINE

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF ENERGY

RESOURCES ENGINEERING

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Chia-Wei Kuo

Dec 2012

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Abstract

The purpose of the research is to understand and predict the combined influences of

viscous, gravity and capillary forces in heterogeneous rocks over the range of conditions

relevant to storage of CO2 in deep underground geological formations. The study begins

by quantifying the separate and combined influences of flow rate, gravity, and sub-core

capillary heterogeneity on brine displacement using the 3-D simulator TOUGH2 (Kuo et

al. 2010). These studies demonstrate that the average saturation depends on the capillary

and gravity numbers in a predictable way. Based on the insight gained from numerical

simulation, this work develops an approximate semi-analytical solution for predicting the

average steady-state saturation during multiphase core flood experiments over a wide

range of capillary and gravity numbers as well as a wide range of heterogeneity.

Although computational technology has improved greatly, running high-resolution 3D

models including capillarity, gravity, and capillary heterogeneity still takes a significant

amount of computational effort. The new solution provided here is a quick and easy way

to estimate the flow regimes for horizontal core floods.

A two dimensional analysis of the governing equations for the multiphase flow system

at steady-state is used to develop the approximate semi-analytical solution. We have

developed a new criterion to identify the viscous-dominated regime at the core scale

where the average saturation is independent of flow rate. Variations of interfacial tension,

core permeability, length of the core, and the effects of buoyancy, capillary and viscous

forces are all accounted for in the semi-analytical solutions. We have also shown that

three dimensionless numbers (NB, Ngv, Rl) and two critical gravity numbers (Ngv,c1, Ngv,c2)

are required to properly capture the balance of viscous, gravity, and capillary forces.

There is good agreement between the average saturations calculated from the 3-D

simulations and the analytical model. This new model can be used to design and interpret

multiphase flow core-flood experiments, gain better understanding of multiphase flow

displacement efficiency over a wide range of conditions and for different fluid pairs, and

perhaps even provide a tool for studying the influence of sub-grid scale multiphase flow

phenomena on reservoir-scale simulations.

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One practical application for the new semi-analytical solution is to help design and

interpret core flood experiments, including assuring that relative permeability

measurements are made in the viscous dominated regime, evaluating potential flow rate

dependence, influence of core-dimension on a multiphase flow experiments, and

influence of fluid properties on the experiments.

New guidelines and suggestions for making relative permeability measurements are

presented. Results are based on a combination of high resolution of 3D simulations and

core-flooding experiments with X-ray CT scanning of saturation distributions. Effects of

flow rate, permeability, surface tension, core length, boundary conditions, sub-core scale

heterogeneity, and gravity over a range of fractional flows of CO2 are systematically

investigated. Synthetic “data sets” are generated using TOUGH2 and subsequently used

to calculate relative permeability curves. A comparison between the input relative

permeability curves and “calculated” relative permeability is used to assess the accuracy

of the “measured” values. Results show that for a modified capillary number

(Ncv=kLpc*A/H

2μCO2qt) smaller than 15, flows are viscous dominated. Under these

conditions, saturation depends only on the fractional flow and is independent of flow rate,

gravity, permeability, core length and interfacial tension. For modified capillary numbers

less than 15, accurate whole-core relative permeability measurements can be obtained

regardless of the orientation of the core and for a high degree of heterogeneity under a

range of relevant and practical conditions. Importantly, the transition from the viscous to

gravity/capillary dominated flow regimes occurs at much higher flow rates for

heterogeneous rocks. For modified capillary numbers larger than 15, saturation gradients

develop along the length of the core and accurate relative permeability measurements are

not obtained using traditional steady state methods. However, if capillary pressure

measurements at the end of the core are available, or can be estimated from

independently measured capillary pressure curves and the measured saturation at the inlet

and outlet of the core, accurate relative permeability measurements can be obtained even

when there is a small saturation gradient across the core.

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Acknowledgments

First and foremost, I owe my deepest gratitude to my advisor, Professor Sally M. Benson,

who not only served as my supervisor but also encouraged and challenged me throughout

my graduate program. Sally has been providing her wide knowledge and advice during

my Ph.D. study and gave me a lot of freedom to explore the research area what I am

interested in. I am truly thankful for her patience, inspiring guidance and the unlimited

time she has spent to correct my proposal, paper, and this dissertation. This dissertation

could not have been done without her encouragement and help and I could not express

my sincere appreciation to her.

I wish to thank all the members of my reading and examination committees:

Professor Tony Kovscek and Margot Gerritsen for their careful reading of the entire

dissertation and many constructive comments on this work as well as supervising my

qualification and Ph.D. dissertation. I am greatly indebted for their guidance and advice.

My special thanks to Professor Keith Loague for agreeing to chair my oral defense and

the examiner Professor Jef Caers who gave me insightful suggestions during my

qualification.

I am blessed with having kind and generous friends and colleagues during my Ph.D.

study. I like to thank to the former and current group members in the Benson’s group.

Special thanks to Ljuba Miljkovic who taught me at the very beginning for how to run the

TOUGH2 simulation step by step; Mike Krause who helped me a lot either for the

problems I accounted for and Boxiao Li who discuss the detail of the simulation; Jean-

Christophe Perrin, Sam Krevor, Ronny Pini, and Lin Zuo who gave me the general idea

of the experimental setting; Ariel Esposito, Christin Strandli, and Whitney Sargent who

shared the room together; Ethan Chabora, Karim Farhat, and Da Huo for their friendship

and useful suggestions. In summary, working with them was a valuable experience for

me.

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In addition, thank to Ni (Panithita Rochana), Prare (Monrawee Pancharoen),

Yangyang Liu, Xiaochen Wang, Yang Liu, Zhe Wang and all the other students of the

Department of Energy Resources Engineering for making my years at the department so

much easier and enjoyable. I also like to gratefully acknowledge the financial support of

the Global Climate and Energy Project (GCEP) at Stanford University.

Finally, I owe my deepest gratitude and respect to my family for their tremendous,

unconditional support during these years. I am deeply grateful to my dearest fiancé

Hsuan-Tung Chu, waiting for me all these years with his understanding, patience, and

love enabled me to complete this work. This dissertation is dedicated with love to my

family.

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Contents

ABSTRACT ................................................................................................................................................... V

ACKNOWLEDGMENTS ........................................................................................................................... VII

CONTENTS ................................................................................................................................................. IX

LIST OF TABLES ...................................................................................................................................... XII

LIST OF FIGURES .................................................................................................................................... XIV

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

1.1 BACKGROUND/STATEMENT OF THE PROBLEM ................................................................................................. 1 1.2 OUTLINE OF THE APPROACH......................................................................................................................... 5 1.3 DISSERTATION OUTLINE/ORGANIZATION ........................................................................................................ 6

CHAPTER 2 LITERATURE REVIEW .......................................................................................................... 8

2.1 MULTIPHASE FLOW IN THE VISCOUS AND CAPILLARY LIMITS ............................................................................... 8 2.2 MULTIPHASE FLOW WITH VISCOUS, CAPILLARY AND GRAVITY FORCES ................................................................. 9 2.3 CAPILLARY HETEROGENEITY ....................................................................................................................... 11 2.4 RELATIVE PERMEABILITY MEASUREMENTS .................................................................................................... 13 2.5 DISCUSSION ............................................................................................................................................ 16

CHAPTER 3 SIMULATION METHODOLOGY ........................................................................................ 18

3.1 TOUGH2 MP/ECO2N ........................................................................................................................... 20 3.2 CORE DESCRIPTIONS AND GRID SIZE ............................................................................................................ 24 3.3 INPUT PARAMETERS ................................................................................................................................. 25

3.3.1 Boundary Conditions ................................................................................................................... 25 3.3.2 Initial Conditions ......................................................................................................................... 27 3.3.3 Input Capillary Pressure .............................................................................................................. 27 3.3.4 Input Relative Permeability ......................................................................................................... 30 3.3.5 Input Injection Flow Rate ............................................................................................................ 31

3.4 OUTPUT PARAMETERS .............................................................................................................................. 33 3.4.1 Slice-Average Quantities (SCO2, PCO2, Pc) ...................................................................................... 33 3.4.2 Simulated CO2 Saturation Distributions ...................................................................................... 34

CHAPTER 4 3D NUMERICAL AND 2D ANALYTICAL STUDIES FOR THE HOMOGENEOUS

CORES .......................................................................................................................................................... 36

4.1 METHODOLOGY ....................................................................................................................................... 37 4.1.1 Simulation Study ......................................................................................................................... 37

4.2 THEORETICAL ANALYSIS ............................................................................................................................. 44 4.2.1 2D General Solution .................................................................................................................... 44 4.2.2 2D General Solution with Simulation Constraints ....................................................................... 46

4.3 RESULTS ................................................................................................................................................. 48

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4.3.1 Comparison between Simulations and the Approximate Semi-Analytical Solution ................... 48 4.3.2 Model Validation ........................................................................................................................ 50

4.4 DISCUSSION ........................................................................................................................................... 53 4.5 CONCLUSIONS ........................................................................................................................................ 56

CHAPTER 5 3D NUMERICAL AND 2D ANALYTICAL STUDIES FOR THE HETEROGENEOUS

CORES ......................................................................................................................................................... 58

5.1 HETEROGENEOUS REPRESENTATIONS .......................................................................................................... 59 5.1.1 Random Permeability Distribution (3D) ..................................................................................... 59 5.1.2 Porosity-based Permeability Distribution (3D) ........................................................................... 60 5.1.3 Permeability Distribution Summary ........................................................................................... 62

5.2 SIMULATION STUDIES............................................................................................................................... 64 5.2.1 Heterogeneity Effects ................................................................................................................. 64 5.2.2 Interfacial Tension and Permeability Effects .............................................................................. 65

5.3 THEORETICAL ANALYSIS OF MULTIPHASE DISPLACEMENT EFFICIENCY IN HETEROGENEOUS CORES ........................... 68 5.3.1 2D General Solution.................................................................................................................... 69 5.3.2 2D General Solution for Heterogeneous Rocks Using Simulation Constraints ........................... 71 5.3.3 Approximate Semi-Analytical Solution ....................................................................................... 74

5.4 VERIFICATION OF THE ANALYTICAL MODEL ................................................................................................... 77 5.4.1 Different Interfacial Tension and Permeability .......................................................................... 77 5.4.2 Different Fractional Flows of CO2 (HC Model) ............................................................................ 78 5.4.3 Different Core Dimensions .......................................................................................................... 79 5.4.4 Different Heterogeneity ............................................................................................................. 80

5.5 PROCEDURES FOR USING THE ANALYTICAL SOLUTIONS ................................................................................... 82

CHAPTER 6 CO2 AND BRINE RELATIVE PERMEABILITY IN HETEROGENEOUS ROCKS .......... 85

6.1 SIMULATION OUTPUTS ............................................................................................................................. 86 6.2 CALCULATION OF RELATIVE PERMEABILITY ................................................................................................... 89

6.2.1 Relative Permeability Calculated when ΔPw=ΔPCO2 .................................................................... 89 6.2.2 Relative Permeability Calculated by True Pressure Drops (ΔPw=ΔPCO2–ΔPc) .............................. 94

6.3 SENSITIVITY STUDIES FOR DIFFERENT CORE PROPERTIES ................................................................................. 97 6.3.1 Effects of Heterogeneity ............................................................................................................. 97 6.3.2 Effects of Core Length (15.24-45.72 cm) .................................................................................... 99 6.3.3 Effects of Absolute Permeability (31.8-3180 md) ....................................................................... 99 6.3.4 Effects of Interfacial Tension (7.49-67.41 mN/m) .................................................................... 100 6.3.5 Effects of Gravity ...................................................................................................................... 102

6.4 DISCUSSION OF RELATIVE PERMEABILITY MEASUREMENTS ............................................................................ 103 6.4.1 Observations from the Numerical and Semi-Analytical Models ............................................... 103 6.4.2 General Rule of Thumb for Reliable Relative Permeability Measurements .............................. 104

CHAPTER 7 CONCLUSIONS AND FUTURE WORK ........................................................................... 108

7.1 SUMMARY AND CONCLUSIONS OF THE PRESENT WORK ................................................................................ 108 7.2 DIRECTIONS FOR FUTURE RESEARCH ......................................................................................................... 110

APPENDIX A 2D ANALYTICAL DERIVATION FOR THE HOMOGENEOUS MODEL .................. 113

A.1 DETAIL OF 2D GENERAL SOLUTION .......................................................................................................... 113 A.2 DERIVATION OF PARAMETERS A AND B ...................................................................................................... 116

APPENDIX B 2D ANALYTICAL DERIVATION FOR THE HETEROGENEOUS MODEL ............... 118

B.1 GENERAL SOLUTION .............................................................................................................................. 118

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B.2 ANALYTICAL SOLUTION WITH SIMULATION CONSTRAINTS .............................................................................. 122 B.3 HETEROGENEOUS FACTOR, ΣLNK/LN(KMEAN)................................................................................................... 125

APPENDIX C FLOW REGIMES ............................................................................................................... 126

C.1 CRITICAL GRAVITY NUMBERS ................................................................................................................... 126

APPENDIX D PARAMETER TABLES FOR HETEROGENEOUS CORES.......................................... 128

APPENDIX E VALIDATION OF TOUGH2 SIMULATIONS ................................................................ 129

E.1 RELATIVE CONVERGENCE TOLERANCE Ε1=10-5

............................................................................................. 129 E.2 GPRS .................................................................................................................................................. 130

NOMENCLATURE .................................................................................................................................... 131

GREEKS ................................................................................................................................................... 132 SUBSCRIPTS ............................................................................................................................................ 133

BIBLIOGRAPHY ....................................................................................................................................... 134

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List of Tables

TABLE 1. 1 COMPARISON OF THE DIFFERENCES AND SIMILARITIES BETWEEN THE THREE-

DIMENSIONAL NUMERICAL SIMULATIONS AND TWO-DIMENSIONAL THEORETICAL

DERIVATION. .............................................................................................................................................. 6

TABLE 2. 1 SUMMARY OF THE LIMITING CASES (ZHOU ET AL. 1994). ........................................ 10

TABLE 2. 2 SUMMARY OF THE LIMITING CASES (YORTSOS 1995). .............................................. 11

TABLE 3. 1 SIMULATION SUMMARY. .................................................................................................. 19

TABLE 3. 2 SUMMARY OF ECO2N. ........................................................................................................ 20

TABLE 3. 3 BOUNDARY CONDITION SUMMARY. ............................................................................. 27

TABLE 3. 4 INPUT PARAMETER VALUES FOR RELATIVE PERMEABILITY AND CAPILLARY

PRESSURE CURVES FIT. .......................................................................................................................... 29

TABLE 3. 5 THE EFFECT OF POROSITY, PERMEABILITY, AND CAPILLARY PRESSURE ON CO2

SATURATION DISTRIBUTION. ............................................................................................................... 30

TABLE 4. 1: SUMMARY OF SENSITIVITY STUDIES ........................................................................... 39

TABLE 4. 2: VALUES OF UNKNOWN VARIABLES USED TO MATCH THE BASE CASE. ............ 48

TABLE 4. 3: BEREA CORE PROPERTIES AND FLUID PROPERTIES USED IN THE

HOMOGENEOUS CORES FOR THE BASE CASE .................................................................................. 49

TABLE 4. 4: SUMMARY OF FLOW REGIONS FOR GENERAL CASES .............................................. 54

TABLE 5. 1 SUMMARY OF SIMULATION CASES WITH DIFFERENT DEGREES OF

HETEROGENEITY. .................................................................................................................................... 63

TABLE 5. 2: SUMMARY OF SENSITIVITY STUDIES FOR HIGH CONTRAST AND RANDOM 3

MODELS. ..................................................................................................................................................... 66

TABLE 5. 3: SUMMARY OF CONSTANT COEFFICIENTS IN EQ. 5.12. ............................................. 74

TABLE 6. 1 SYNTHETIC INPUT PARAMETERS FOR EVERY GRID IN THE SIMULATIONS FOR

THREE DIFFERENT MODELS. ................................................................................................................ 87

TABLE 6. 2 SUMMARY OF DIFFERENT FLOW RATES FOR HOMOGENEOUS CORES WITH 95%

FRACTIONAL FLOW OF CO2. CAPILLARY NUMBERS ARE CALCULATED BASED ON EQ. 4.1

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AND EQ. 4.2, RESPECTIVELY. ................................................................................................................. 91

TABLE 6. 3 SUMMARY OF DIFFERENT FLOW RATES FOR HETEROGENEOUS CORES (HIGH

CONTRAST MODEL: ΣLNK/LNKMEAN=0.168) WITH 95% FRACTIONAL FLOW OF CO2. ................... 94

TABLE 6. 4 SUMMARY OF DIFFERENT FLOW RATES FOR DIFFERENT HETEROGENEOUS

CORES: HOMOGENEOUS, RANDOM 2, KOZENY-CARMAN, HIGH CONTRAST AND RANDOM 3

MODELS. ..................................................................................................................................................... 98

TABLE 6. 5 SUMMARY OF DIFFERENT LENGTHS OF HETEROGENEOUS CORES (HIGH

CONTRAST MODEL: ΣLNK/LNKMEAN=0.168). ........................................................................................... 99

TABLE 6. 6 SUMMARY OF THE FIRST CRITICAL CAPILLARY NUMBER NCV,C1 FOR DIFFERENT

HETEROGENEITY CORES. ..................................................................................................................... 104

TABLE D.1: SUMMARY OF SENSITIVITY STUDIES FOR HIGH CONTRAST MODELS (HC). ..... 128

TABLE D.2: SUMMARY OF SENSITIVITY STUDIES FOR RANDOM 3 MODELS. ......................... 128

TABLE D.3: SUMMARY OF SENSITIVITY STUDIES FOR DIFFERENT HETEROGENEOUS

MODELS. ................................................................................................................................................... 128

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List of Figures

FIGURE 1. 1 DIFFERENT FLOW REGIMES DOMINATED BY DIFFERENT FORCES IN THE

STORAGE RESERVOIR. .............................................................................................................................. 3

FIGURE 1. 2 VARIOUS RELATIVE PERMEABILITY CURVES AT INSITU CONDITIONS FOR CO2

/BRINE SYSTEM (A) BASAL CAMBRIAN, (B) COOKING LAKE, (C) NISKU, (D) WABAMUN

(LOW PERMEABILITY), (E) ELLERSLIE, (F) VIKING, (G) WABAMUN (HIGH PERMEABILITY),

AND (H) FRIO SANDSTONE ...................................................................................................................... 4

FIGURE 3. 1 THE EXPERIMENTAL STEADY STATE THREE-DIMENSIONAL VIEWS OF CO2

SATURATION IN THE CORE FOR A GIVEN FRACTIONAL FLOW OF CO2 AT A GIVEN FLOW

RATE. THE FLUIDS WERE INJECTED FROM RIGHT TO LEFT (PERRIN ET AL, 2009). ............... 18

FIGURE 3. 2 SIMULATION CORE OF 25 25 31 GRID BLOCKS OF UNIFORM SIZE. ................. 19

FIGURE 3. 3 SPACE DISCRETIZATION IN THE INTEGRAL FINITE DIFFERENCE METHOD. ..... 21

FIGURE 3. 4 CO2 SATURATION ALONG THE BEREA SANDSTONE CORE FOR DIFFERENT

FRACTIONAL FLOWS OF CO2 AT A TOTAL INJECTION FLOW RATE 2.6 ML/MIN: (A)

EXPERIMENTAL RESULTS; (B) HIGH CONTRAST MODEL WITH BOUNDARY CONDITION

PC=0; (C) HIGH CONTRAST MODEL WITH BOUNDARY CONDITION DPC/DX=0. ......................... 26

FIGURE 3. 5 LABORATORY CAPILLARY PRESSURE DATA WITH A CURVE FIT USED IN

SIMULATIONS. .......................................................................................................................................... 28

FIGURE 3. 6 THE RANGE OF CAPILLARY PRESSURE CURVES IN THE SIMULATIONS. THE

VALUES OF INPUT PARAMETERS ARE A1=0.007734, B1=0.307601, Λ1=2.881, Λ2=2.255, SP=0 AND

Σ =22.47 DYNES/CM. ................................................................................................................................. 29

FIGURE 3.7 INPUT RELATIVE PERMEABILITY CURVES FOR CO2 AND BRINE WITH SWR =0.15,

NW =7, AND NCO2=3 AND THE EXPERIMENTAL DATA. ..................................................................... 31

FIGURE 3.8 CONCEPTUAL MODEL OF THE RESERVOIR USED TO ESTIMATE THE RANGE OF

RELEVANT FLOW VELOCITIES. ............................................................................................................ 32

FIGURE 3.9 THE VOLUMETRIC FLOW RATE AND ITS CORRESPONDING DISTANCE OF CO2

PLUME AT THE RESERVOIR (100 METER THICKNESS AND 1 MT CO2/YR INJECTION RATE). 32

FIGURE 3.10 CO2 SATURATION DISTRIBUTION AT STEADY-STATE FOR 95% FRACTIONAL

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FLOW OF CO2 AT A TOTAL INJECTION FLOW RATE 1.2 ML/MIN. .................................................. 33

FIGURE 3.11 SATURATION DISTRIBUTION FOR HOMOGENEOUS AND HETEROGENEOUS

CORES WITH GRAVITY, USING GRID WITH PIXELS (150, 150, 31) TO GRID BLOCKS (25, 25, 31).

....................................................................................................................................................................... 35

FIGURE 4. 1: AVERAGE CO2 SATURATION AS A FUNCTION OF (A) TRADITIONAL CAPILLARY

NUMBER CA, (B) ALTERNATIVE CAPILLARY NUMBER NCV, AND (C) ALTERNATIVE

GRAVITY NUMBER NGV. THE INTERFACIAL TENSION Σ AND THE PERMEABILITY OF THE

BASE CASE ARE 22.47 MN/M AND 430 MD, RESPECTIVELY. THE SENSITIVITY STUDIES OF

INTERFACIAL TENSION ARE ILLUSTRATED IN THE LEFT HAND SIDE OF THE FIGURES AND

PERMEABILITY ON THE RIGHT. ............................................................................................................ 41

FIGURE 4. 2: PRESSURE GRADIENTS FOR THREE DIFFERENT FORCES AS A FUNCTION OF

ALTERNATIVE GRAVITY NUMBER NGV. THE INTERFACIAL TENSION AND THE

PERMEABILITY OF THE 3Σ CASE ARE 67.41 MN/M AND 430 MD, RESPECTIVELY. .................... 42

FIGURE 4. 3: THE FUNCTION ΔJ HAS A LINEAR DEPENDENCY ON THE SQUARE ROOT OF

ASPECT RATIO, RL. .................................................................................................................................... 44

FIGURE 4. 4 (LHS) COMPARISON OF AVERAGE CO2 SATURATION AS A FUNCTION OF

CAPILLARY NUMBER NCV BETWEEN THEORETICAL VALUES AND SIMULATION RESULTS

FOR THE HOMOGENEOUS OR BASE CASE (Σ, K); (RHS) FRACTIONAL FLOW CURVE BASED

ON OUR INPUT RELATIVE PERMEABILITY CURVES (EQ.3.6). ........................................................ 48

FIGURE 4. 5 COMPARISON OF AVERAGE CO2 SATURATION AS A FUNCTION OF CAPILLARY

NUMBER NCV BETWEEN THEORETICAL VALUES AND SIMULATION RESULTS FOR THE

SENSITIVITY CASES OF INTERFACIAL TENSION. LHS: (3Σ, K), RHS: (Σ/3, K) .............................. 50



SENSITIVITY CASES OF DIFFERENT DIMENSIONS OF THE CORE. ................................................ 51



SENSITIVITY CASES OF DIFFERENT FRACTIONAL FLOWS OF CO2. ............................................. 52


NUMBER NCV BETWEEN THEORETICAL VALUES AND SIMULATION RESULTS FOR SIX

DIFFERENT INPUT RELATIVE PERMEABILITY CURVES. THOSE SIMULATIONS SHARE THE

SAME CONSTANT PARAMETERS IN TABLE 4.3. ................................................................................ 53

FIGURE 4. 9 (LHS) THE FINER GRIDS HAVE GRID DIMENSIONS 0.884MM X 0.884MM X 5.08MM

(GRID NUMBERS: 53X53X31) WHILE THE COARSER GRIDS HAVE DIMENSIONS 1.874MM X

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1.874MM X 5.08MM (GRID NUMBERS: 25X25X31); (RHS) CFL NUMBER (UXΔT/ΔX<1) FOR

DIFFERENT FLOW RATE. ........................................................................................................................ 56

FIGURE 5. 1: FOUR DIFFERENT RANDOM PERMEABILITY PROFILES AND THE

CORRESPONDING AVERAGE CO2 SATURATIONS IN THE VISCOUS-DOMINATED REGIME. .. 60

FIGURE 5. 2: AVERAGE PERMEABILITY AND AVERAGE CO2 SATURATION ALONG THE

FLOW DIRECTION FOR THREE DIFFERENT HETEROGENEOUS CORES (HOMOGENEOUS

MODEL, KOZENY-CARMAN MODEL, AND HIGH CONTRAST MODEL). ....................................... 62

FIGURE 5. 3: AVERAGE CO2 SATURATION AS A FUNCTION OF ALTERNATIVE CAPILLARY

NUMBER NCV. THE INTERFACIAL TENSIONS Σ FOR ALL CASES IS 22.47 MN/M AND THE

EFFECTIVE PERMEABILITY VARIES FROM 254 TO 570 MD. ........................................................... 65

FIGURE 5. 4: AVERAGE CO2 SATURATION AS A FUNCTION OF TRADITIONAL CAPILLARY

NUMBER CA, ALTERNATIVE CAPILLARY NUMBER NCV, AND ALTERNATIVE GRAVITY

NUMBER NGV FOR TWO HETEROGENEOUS CORES: HIGH CONTRAST MODEL (LEFT) AND

RANDOM 3 MODEL (RIGHT). THE INTERFACIAL TENSION Σ IS KEPT AS A CONSTANT, 22.47

MN/M. .......................................................................................................................................................... 67

FIGURE 5. 5: PRESSURE GRADIENTS AS A FUNCTION OF ALTERNATIVE CAPILLARY

NUMBER NCV FOR THREE PHYSICAL FORCES. .................................................................................. 68

FIGURE 5. 6: COMPARISON OF AVERAGE CO2 SATURATION AS A FUNCTION OF CAPILLARY


SENSITIVITY CASES OF DIFFERENT DEGREES OF HETEROGENEOUS CORES. ......................... 74


NUMBER NCV BETWEEN THEORETICAL VALUES AND SIMULATION RESULTS FOR HIGH

CONTRAST MODEL AT A WIDE RANGE OF PERMEABILITY AND INTERFACIAL TENSION. .. 75

FIGURE 5. 8: THE DEPENDENCE OF Ε AND Ω IN TERMS OF THE MODIFIED BUCKLEY-

LEVERETT SATURATION (SBLHETE

) AND THE BOND NUMBER (NB) FOR HIGH CONTRAST AND

RANDOM 3 MODELS. ............................................................................................................................... 76


NUMBER NCV BETWEEN THEORETICAL VALUES AND SIMULATION RESULTS FOR (A) HIGH

CONTRAST MODEL AND (B) RANDOM PERMEABILITY MODEL IN A WIDE RANGE OF

PERMEABILITY AND INTERFACIAL TENSION. ................................................................................. 78

FIGURE 5. 10: COMPARISON OF AVERAGE CO2 SATURATION AS A FUNCTION OF


FOR HIGH CONTRAST MODEL AT DIFFERENT FRACTIONAL FLOWS OF CO2. .......................... 79


xvii


FOR THE SENSITIVITY CASES OF DIFFERENT ASPECT RATIO. ..................................................... 80


CAPILLARY NUMBER NCV BETWEEN HOMOGENEOUS ANALYTICAL PREDICTIONS AND

SIMULATION RESULTS FOR THE SENSITIVITY CASES OF SMALL DEGREES OF

HETEROGENEOUS CORES. ...................................................................................................................... 81



FOR THE SENSITIVITY CASES OF DIFFERENT DEGREES OF HETEROGENEOUS CORES. ........ 82

FIGURE 6. 1 OVERVIEW OF SCIENTIFIC APPROACH. ........................................................................ 86

FIGURE 6. 2 3D POROSITY DISTRIBUTION OF BEREA SANDSTONE. ............................................. 87

FIGURE 6. 3 CO2 SATURATION DISTRIBUTION AT STEADY-STATE FOR 95% FRACTIONAL

FLOW OF CO2 AT A TOTAL INJECTION FLOW RATE 1.2 ML/MIN. .................................................. 88

FIGURE 6. 4 FLOW RATE EFFECT ON CO2 SATURATION ALONG THE HOMOGENEOUS CORE

AT A 95% FRACTIONAL FLOW OF CO2 WITH FLOW RATES RANGING FROM 0.1 ML/MIN TO 6

ML/MIN. ....................................................................................................................................................... 90

FIGURE 6. 5 RELATIVE PERMEABILITY CALCULATED BY THE SAME PRESSURE DROP

(ΔPW=ΔPCO2) FOR HOMOGENEOUS CORE WITH 430 MD PERMEABILITY AT DIFFERENT FLOW

RATES. ......................................................................................................................................................... 92

FIGURE 6. 6 THE EFFECT OF HETEROGENEITY ON CO2 SATURATION ALONG THE CORE AT A

FRACTIONAL FLOW OF 95% OVER A WIDE RANGE OF FLOW RATES. ........................................ 93

FIGURE 6. 7 RELATIVE PERMEABILITY CALCULATED BY THE SAME PRESSURE DROP

(ΔPW=ΔPCO2) FOR HETEROGENEOUS CORE WITH 318 MD PERMEABILITY AT DIFFERENT

FLOW RATES. SMALL PICTURE SHOWS THE SAME RELATIVE PERMEABILITY CURVES AT

LOG SCALE. ................................................................................................................................................ 94

FIGURE 6. 8 FLOW RATE EFFECT ON RELATIVE PERMEABILITY CALCULATED BY THE TRUE

PRESSURE DROPS (ΔPW ΔPCO2) FOR HOMOGENEOUS AND HETEROGENEOUS CORE AT

VARIOUS FLOW RATES. .......................................................................................................................... 95

FIGURE 6. 9 (LHS) RELATIVE PERMEABILITY CALCULATED USING THE SAME PRESSURE

DROP FOR BOTH FLUIDS (FIGURE 6.7); (RHS) RELATIVE PERMEABILITY CALCULATED BY

THE CORRECTED PRESSURE DROPS FOR HIGH CONTRAST MODELS AT VARIOUS FLOW

RATES. ......................................................................................................................................................... 96

FIGURE 6. 10: RELATIVE PERMEABILITY CALCULATED BY THE TRUE PRESSURE DROPS

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FOR FIVE DIFFERENT HETEROGENEOUS CORES IN THE VISCOUS-DOMINATED REGIMES:

HOMOGENEOUS AND THE RANDOM 2 CORES (Q=0.5 ML/MIN), KOZENY-CARMAN MODELS

(Q=1.2 ML/MIN), HIGH CONTRAST MODELS (Q=2.6 ML/MIN), THE RANDOM 3 CORES (Q=6

ML/MIN) AND THE INPUT RELATIVE PERMEABILITY CURVES. ................................................... 98

FIGURE 6. 11 (LHS) BRINE DISPLACEMENT EFFICIENCIES FOR THREE DIFFERENT LENGTHS

OF HOMOGENEOUS CORE WITH CAPILLARY NUMBER RANGING FROM 10 TO 107; (RHS)

RELATIVE PERMEABILITY CALCULATED BY THE TRUE PRESSURE DROPS FOR

HOMOGENEOUS CORES AT 0.1 ML/MIN FLOW RATES. ................................................................... 99

FIGURE 6. 12 (LHS) BRINE DISPLACEMENT EFFICIENCIES FOR THREE DIFFERENT

PERMEABILITY VALUES WITH CAPILLARY NUMBER NCV RANGING FROM 10 TO 105; (RHS)

PERMEABILITY EFFECTS ON RELATIVE PERMEABILITY CALCULATED IN THE TRUE

PRESSURE DROPS FOR HETEROGENEOUS CORE (HIGH CONTRAST MODEL) AT 6 ML/MIN

FLOW RATES. .......................................................................................................................................... 100

FIGURE 6. 13 (LHS) AVERAGE CO2 SATURATION AS A FUNCTION OF CAPILLARY NUMBER

NCV FOR HOMOGENEOUS AND HIGH CONTRAST MODELS WITH THREE DIFFERENT VALUES

OF INTERFACIAL TENSIONS; (RHS) INTERFACIAL TENSION EFFECTS ON RELATIVE

PERMEABILITY CALCULATED IN THE TRUE PRESSURE DROPS FOR HETEROGENEOUS

CORE (HIGH CONTRAST MODEL) AT 6 ML/MIN FLOW RATES. ................................................... 101

FIGURE 6. 14 AVERAGE CO2 SATURATION AS A FUNCTION OF CAPILLARY NUMBER NCV FOR

HOMOGENEOUS, KOZENY-CARMAN (SMALL HETEROGENEITY) AND HIGH CONTRAST

(LARGE HETEROGENEITY) MODELS WITH AND WITHOUT GRAVITY (1G/0G). ...................... 102

FIGURE 6. 15 AVERAGE CO2 SATURATION AS A FUNCTION OF ALTERNATIVE CAPILLARY

NUMBER NCV, AND ALTERNATIVE GRAVITY NUMBER NGV FOR HOMOGENEOUS AND HIGH

CONTRAST MODELS. ............................................................................................................................. 105

FIGURE 6. 16 LABORATORY CAPILLARY PRESSURE DATA. ........................................................ 105

FIGURE 6. 17 HETEROGENEOUS PARAMETER Τ IN TERMS OF NORMALIZED STANDARD

DEVIATION. ............................................................................................................................................. 107

FIGURE C.1: PRESSURE GRADIENTS FOR VISCOUS, CAPILLARY AND GRAVITY FORCES FOR

DIFFERENT ASPECT RATIOS. .............................................................................................................. 126


DIFFERENT INTERFACIAL TENSION AND PERMEABILITY VALUES. ........................................ 127


DIFFERENT PERMEABILITY VALUES FOR HIGH CONTRAST MODEL. ...................................... 127

FIGURE E. 1 CO2 SATURATION AS A FUNCTION OF PORE VOLUME INJECTED (PVI) FOR

xix

DIFFERENT FLOW RATES. .................................................................................................................... 130

FIGURE E. 2 GPRS VS TOUGH2 RESULTS FOR SLICE X=16 (LHS) AND X=21 (RHS) .................. 130

1

Chapter 1

Introduction

CO2 sequestration in saline aquifers requires knowledge of multiphase flow of CO2 and

brine, specifically, the interaction between viscous, gravity, and capillary forces in

heterogeneous rocks at the reservoir scale. This is an important and challenging task, to

which insights from core-scale experiments and simulations can be applied. The

purposes of this work are to first, gain insight about these processes by examining core

scale experiments, second, quantify the influence and interplay of these forces using

numerical simulation, and finally, to develop a theoretical model to predict the average

saturation during multiphase displacements and identify different flow regimes based on

dimensional variables. As an application of these ideas, the results are used to design

core laboratory experiments for obtaining accurate relative permeability data for

heterogeneous and homogeneous cores.

1.1 Background/Statement of the Problem

Carbon dioxide capture and sequestration in deep geological formations is one of the

most important technologies for climate change mitigation (IPCC, 2005). Although

depleting or abandoned oil and gas reservoirs are available in some regions like Texas in

the US, the Middle East, Russia, and Alberta in Canada, they have lower capacity than is

needed to store worldwide CO2 emissions from large stationary sources. Furthermore,

these depleted reservoirs are not common all over the world. Moreover, their capacities

are available only when the reservoirs are depleted or if CO2 sequestration is combined

with CO2–EOR. Thus, storage of CO2 in depleted or abandoned oil and gas fields is

2

limited. Therefore, the large storage capacity of worldwide distributed saline aquifers

makes them good options to store CO2 captured from power generation stations and

industrial sources, which are the largest stationary sources emitting CO2 (IPCC, 2005).

Moreover, sequestration of CO2 in deep saline formations is immediately accessible

compared to the other options.

Carbon dioxide and water are immiscible under the conditions anticipated in

geological sequestration reservoirs (1,000 to 3,000 m depth). Therefore, the migration of

injected CO2 and brine displacement will be governed by the physics of multiphase flow.

There are many studies of multiphase flow of oil and water in petroleum reservoirs, but

because the viscosity, density, interfacial tension of CO2 is different that oil, conclusions

drawn from oil and water systems may not be applicable in CO2 and brine systems. For

example, since the density difference between CO2 and brine is typically much larger

than between oil and water, buoyancy forces will play a greater role. Similarly, the low

viscosity CO2 relative to oil, results in smaller pressure gradients and higher mobility to

CO2. Moreover, the flow velocities in a typical CO2 storage project will be much lower

than during oilfield recovery, especially far from the injection well and during the post-

injection period.

To date, actual experience with multiphase flow of CO2 in saline aquifers is limited to

the Sleipner Project (Kaarstad, 2004) and a number of smaller pilot tests (ex: Frio Project,

2006). Key issues related to the feasibility of deep saline aquifer storage include:

1. What will be the average saturation of CO2 in the plume?

2. How far will the injected CO2 migrate?

3. How much CO2 will be dissolved in the native brine and how quickly?

4. How much CO2 will be immobilized by capillary trapping?

5. What fraction, if any, of the CO2 will leak up to the subsurface and how fast?

Answering these questions relies on proper understanding of the multiphase flow systems

and the complex interplay of viscous, gravitational, and capillary forces as well as large

and small scale heterogeneity. Figure 1.1 illustrates the conceptual CO2 migration in

deep saline aquifers. Three physical forces dominate CO2 flow behavior in different flow

3

regimes.

Figure 1. 1 Different flow regimes dominated by different forces in the storage reservoir.

The relative permeability of CO2/brine systems is an essential element to determine CO2

injectivity and migration, and to assess the safety of potential CO2 sequestration sites. To

date, very few relative permeability measurements have been made for CO2/brine

systems (Bennion and Bachu, 2005, 2006; Benson et al, 2006; Krevor et al., 2012). This

work was initially motivated by the need to increase the number of relative permeability

measurements for CO2/brine systems under realistic reservoir conditions. However, in

the course of these investigations the important role of sub-core scale heterogeneity

became evident (Perrin and Benson, 2009). This observation motivated a systematic

evaluation of sub-core scale heterogeneity and flow rates on brine displacement

efficiency during horizontal core floods. Figure 1.2 shows some relative permeability

data measured on a range of rock types. The wide range of rock types results in a high

degree of variability. Also, unusually high residual brine saturations appear in most of

samples, the reason for which needs to be understood.

4

Figure 1. 2 Various relative permeability curves at insitu conditions for CO2 /brine

system (a) Basal Cambrian, (b) Cooking Lake, (c) Nisku, (d) Wabamun (low

permeability), (e) Ellerslie, (f) Viking, (g) Wabamun (high permeability), and (h) Frio

Sandstone

Several potential methods can be applied to study CO2/brine systems:

Core-Scale Laboratory Experiments: Experimental studies for both drainage and

imbibition displacements at representative reservoir pressure and temperatures are

needed to improve our ability to answer these questions

Petrophysical Studies: Pore-scale features such as size and shapes of the pore

spaces in the rock can be used to study multiphase flow properties using pore-

network models.

Numerical Simulation: Numerical simulations intended to replicate multiphase

flow experiments, in which CO2 is injected into a brine saturated rock, is one way

to test our understanding of the underlying physics. Numerical simulation also

can be used to develop scaling relationships and interpret field-scale behavior of

injected CO2.

Multiphase Flow Theory: Improving fundamental understanding of multi-phase

flow in CO2/brine systems to better predict both the experimental and numerical

results, and developing approaches for up-scaling laboratory measured relative

5

permeability curves for use in reservoir simulations will also be needed.

Predicting experimental or numerical results quantitatively can save a significant

amount of computational time and provide systematic insights into the physical

controls on multiphase flow under a wide range of conditions and for different

fluid pairs.

Field Scale Studies: it is important to study the CO2 distribution, migration, and

flow movement at the reservoir scale to test our knowledge gained from

numerical and theoretical investigation.

1.2 Outline of the Approach

In this work, numerical simulations and theoretical analysis are performed to study the

issues described in the previous section. First, we perform 3D high resolution

simulations to match the experimental data qualitatively and quantitatively. Second, we

do sensitivity studies including a wide range of rock types with different types of

heterogeneity to investigate the potential effects on the CO2/brine multiphase flow

system. Based on a series of numerical experiments, a two-dimensional theoretical

model incorporating gravity, capillary pressure, and sub-core heterogeneity is developed

to predict the three-dimensional results. Finally, the volume-averaged relative

permeability values calculated from the simulation outputs are investigated. Table 1.1

compares the similarity and the difference between these two different approaches. In

summary, this comprehensive study presents numerical and theoretical research efforts

that provide new understanding of CO2 migration in heterogeneous reservoirs.

6

Table 1. 1 Comparison of the differences and similarities between the three-dimensional

numerical simulations and two-dimensional theoretical derivation.

Miscible Diffusion Compressible Isotropic Gravity Capillarity Heterogeneity

3D

Numerical

Simulation

NO NO YES YES YES YES YES

2D

Theoretical

Derivation

NO NO NO YES YES YES YES

1.3 Dissertation Outline/Organization

The dissertation is organized as follows. Chapter 2 summarizes the relevant literature on

multiphase flow and the existing theoretical analysis pertinent to this work. In the first

part of this chapter, research focused on flow systems in which viscous or capillary forces

are dominant, the so-called “viscous and capillary limits” are reviewed. The second

section reviews studies considering the combined effects of capillarity and gravity. The

last section reviews a series of relative permeability studies focusing on several issues

including flow injection rate, gravity segregation, and sub-core scale heterogeneity.

Chapter 3 presents the methodology for numerical simulations performed in this

study. The short term goal of core-scale numerical simulations is to model the behavior

of CO2 at steady-state while the long term goal is be able to generalize small scale results

to large scale or at least gain insight from core-scale studies. Through these studies we

will ultimately develop a better understanding of interactions between physical forces as

well as the role of sub-core scale heterogeneity on multi-phase flow. A description of

the simulator, initial and boundary conditions, and required input parameters are

presented.

A numerical and theoretical investigation of the influence of flow rate, gravity, and

capillarity on brine displacement efficiency in homogeneous porous medium is presented

in Chapter 4. This chapter starts with the results of three-dimensional high resolution

numerical simulations. Sensitivity studies on core absolute permeability and interfacial

tension are presented to illustrate three regimes dominates by the viscous force, buoyancy

force, and capillary force, respectively. A two-dimensional semi-analytical solution that

7

includes gravity and capillarity is developed based on relevant dimensionless numbers.

Model validation is applied to a wide range of core dimensions, different fractional flows

of CO2 as well as different input relative permeability functions.

Based on the knowledge gained from the homogeneous studies, a further

investigation in heterogeneous porous media both numerically and analytically is

presented in Chapter 5. The first part of Chapter 5 shows results of 3D numerical

simulations with a wide range of degree of capillary heterogeneity simulated at core-scale

to understand the role of sub-core scale heterogeneities. Two types of heterogeneity are

studied: one uses a random log-normal distribution to generate 3-D permeability maps.

The other type studied is based on a real rock where the heterogeneous porosity

distribution measured using a CT scanner is used to generate the permeability map. The

second part of this chapter presents a 2D analytical solution incorporating the influence

of heterogeneity. A general form of the solution is first derived and the constraints from

3D high-resolution simulations are applied to obtain the final semi-analytical solution.

The validation of analytical model for heterogeneous cores is presented next. Finally, a

summary of how to use these homogeneous and heterogeneous analytical results are

provided.

Chapter 6 presents one of applications for the theoretical model developed in

Chapters 4 and 5, specifically, sensitivity studies on volume-averaged (up-scaled) relative

permeability that accounts for the role of sub-core scale heterogeneity on multi-phase

flow in CO2/brine systems. Correct procedures for calculating relative permeability are

discussed. After that, a general rule for calculating reliable relative permeability

measurements, even in heterogeneous cores, is provided.

Finally, conclusions and recommendations for future work are discussed in Chapter 7.

8

Chapter 2

Literature Review

Chapter 2 summarizes the relevant literature on multiphase flow and the existing

theoretical analyses pertinent to this work. In the first part of this chapter, research

focused on flow systems in which viscous or capillary forces are dominant, the so-called

“viscous and capillary limits” are reviewed. The second section reviews studies

considering the combined effects of capillarity and gravity. The third section reviews the

small-scale heterogeneity. The fourth section reviews a series of relative permeability

studies focusing on several issues including flow injection rate, gravity segregation, and

sub-core scale heterogeneity. Finally, a discussion based on the literature is provided that

identifies gaps in current understanding.

2.1 Multiphase Flow in the Viscous and Capillary

Limits

Recently, the influence of flowrate, gravity, and capillarity on immiscible CO2/brine two-

phase flow system at the core scale has been studied in laboratory experiments (Perrin et

al. 2009; Perrin and Benson 2010; Krevor et al. 2011) as well as using 3-D numerical

simulation (Krause et al. 2011; Kuo et al. 2010; Shi et al. 2010). Although theoretical

analysis of multiphase flow systems has identified flow regions controlled by different

types of flow behavior, most of them have focused on oil/water systems where the effect

of gravity has been neglected (Lenormand et al. 1988; Pickup and Stephen 2000; Jonoud

and Jackson 2008; Hussain et al. 2011). Without considering gravity, the flow regime is

often studied for two extreme limits: one is when the viscous force is sufficiently large

9

such that the capillary pressure is negligible. In this viscous limit, fractional flow is

uniform. The other limit is when the flow rate is very low and capillary forces dominate

the fractional flow distribution. In the capillary limit, capillary equilibrium is assumed.

Characterizing different flow regimes is important and very useful for upscaling.

Dimensionless numbers are often used to identify different regimes since they combine

the effects of flow rate, interfacial tension, the permeability of the core, and gravity

(Fulcher et al. 1985; Avraam and Payatakes 1995; Skauge et al. 1997; Skauge et al. 2000;

Virnovsky et al. 2004; Cinar et al. 2006).

However, the boundaries between viscous and capillary limits are ambiguous in the

literature. Jonoud and Jackson (2008) develop a new dimensionless group using three

dimensionless numbers (transverse and longitudinal Peclet numbers and end-point

mobility ratio: PeT, Pe

L, M

e) to characterize the balance of viscous and capillary forces

and use numerical simulations to determine empirically the threshold values of these two

limits. Their results show that the balance between viscous forces along the direction of

flow and capillary forces perpendicular to the flow control the boundaries.

2.2 Multiphase Flow with Viscous, Capillary and

Gravity Forces

For gas/oil or gas/water systems, the density difference between two the fluids is large,

and the effect of gravity becomes important (Rossen and Duijn 2004; Nordbotten et al.

2005; Kopp et al. 2009). Strong gravitational forces could lead to distinct gravity

segregation between the injected nonwetting phase and wetting phase. In addition, when

flow rates are low, gravity-capillary equilibrium is established.

Comprehensive 2D analyses of the combined effects of viscous, gravity and capillary

forces on the fluid flow in the transverse direction have been performed (Zhou et al.

1994; Yortsos 1995). Both of these works study incompressible flow in an anisotropic

system, and identify flow regions dominated by different forces in the vertical direction.

The corresponding simplified equations and conditions based on dimensionless groups.

Zhou et al. (1994) use modified gravity number NgvM/(1+M), modified capillary number

10

NcvM/(1+M), and the shape factor RL2 while Yortsos (1995) use gravity number NG,

transverse capillary number NCT and aspect ratios H/L, kav/kah as well as the RL2 as the

average scaling parameters. Definitions of these dimensionless parameters are provided

below.

*

2av c

cv CT L2

o t

k LpN = =N R

H μ u (2.1)

2av

gv G L

o t

Δρgk LN = =N R

Hμ u (2.2)

av

L

ah

kLR =

H k (2.3)

w

o

λM=

λ (2.4)

Zhou et al. (1994) establish approximate bounds for the transitions between regions by

examining a wide range of experimental data in the literature. However, most of the data

sources are focused on imbibition and miscible flow types. Table 2.1 lists five flow

regimes and their conditions.

Table 2. 1 Summary of the limiting cases (Zhou et al. 1994).

Flow Region Conditions

Capillary-dominated Crossflow Ncv>>Ngv and NcvM/(1+M)>>1

Gravity-dominated Crossflow Ngv>>Ncv and NgvM/(1+M)>>1

Capillary-gravity Equilibrium Ncv≈Ngv and NgvM/(1+M)>>1

Viscous-dominated Crossflow (Ncv+Ngv)M/(1+M)<<1

Vertical Crossflow Equilibrium (VCE) (Ncv+Ngv)M/(1+M)<<1 and RL2>>1

No-communication

(Viscous crossflow can be neglected) (Ncv+Ngv)M/(1+M)<<1 and RL

2<<1

Yortsos (1995) takes another approach to identify these flow regimes by using asymptotic

analysis. Table 2.2 summarizes five flow regimes and the corresponding conditions as

well as the qualitative comparison between three forces. Both the Zhou et al (1994)

11

model and the Yortsos (1995) model only characterize these different flow regions

without solving the simplified equations and they consider only transport in the

transverse direction.

Table 2. 2 Summary of the limiting cases (Yortsos 1995).

Flow Region Conditions Viscous Gravity Capillary

Viscous Fingering Ngv<<1, NCT<<H/L Strong Negligible Negligible

Viscous Fingering

with Dispersion Ngv~1, NCT~ H/L Strong Moderate Moderate

Gravity Tonguing Ngv~1/ RL2, NCT<<(kav/kah)L/H Moderate Strong Negligible

Capillary-gravity

Equil. Ngv~1/ RL

2, NCT~(kav/kah)L/H Moderate Strong Moderate

Capillary Equil.

Ngv~1/ RL

2, NCT>>(kav/kah)L/H Moderate Strong Strong

2.3 Capillary Heterogeneity

Capillary heterogeneity is known to be an important parameter affecting multiphase flow

of CO2 and brine, affecting properties such as saturation profile, capillary pressure and

relative permeability (Perrin and Benson 2010; Krevor et al. 2011; Pini et al. 2012;

Krause et al. 2011; Kuo et al. 2010; Shi et al. 2010). Moreover, capillary barriers within

the rock can affect flow behavior significantly. If the orientation of a capillary barrier is

perpendicular to the flow direction, non-wetting phase fluid in a water-wet system may

be trapped inside high permeability zones surrounded by low permeability ones

(Honarpour et al. 1995; Hamon and Roy 2000; Pickup and Stephen 2000). It is also

suggested that capillary heterogeneity provides a new trapping mechanism in carbon

sequestration (Saadatpoor et al 2010) in addition to structural trapping, residual trapping,

dissolution trapping and mineral trapping.

Rock heterogeneity is very common in geological formation and exists at every scale:

12

pore-scale (~m), grain scale (~mm), Core-scale (~ 0.1 to 10 cm), and field-scale (~ 10

cm and larger). These different scales of heterogeneity result in complexity when solving

multiphase flow problems. In this work, we are focused on understanding the influence

of spatial heterogeneity at core-scale on multiphase flow of CO2 and brine.

The analytical study of the capillary heterogeneity has been very limited due to the

complexity it introduces to the multiphase flow problem. Most studies of capillary

heterogeneity are limited to 1D or 2D analyses (e.g. Yortsos and Chang 1989, 1990;

Chang and Yortsos 1992; Chaouche et al. 1994; Hussain et al. 2011; Chen 2012). The

major conclusion from the 1D studies is that capillary heterogeneity leads to variable

saturation distributions. In general, for very low flow rates, the saturation distribution

follows the heterogeneity variation. For higher flow rates the situation is more complex:

viscous forces compete with capillary forces to control the saturation distribution at the

interfaces between different regions in the rock.

Yortsos and Chang (1990) studied the one-dimensional steady-state saturation

response to different heterogeneity both analytically and numerically. Permeability

variations are used to describe the heterogeneity of the core, the Leverett scaling

relationship is used to characterize capillary heterogeneity, and relative permeability is

assumed to be uniform. The 1D analytical model is accurate for large capillary numbers

Ncv (low enough flow rates). In addition, they also study the effect of permeability

heterogeneity on viscous dominated Buckley-Leverett type displacements (Chang and

Yortsos 1992) based on a similar 1D analysis. Three permeability distribution models are

studied: random step, sinusoidal, and correlated models. They conclude that the effect of

capillary heterogeneity on the saturation profile of a Buckley-Leverett displacement

(relatively high flow rates, small capillary number Ncv) is non-negligible once the

amplitude of permeability variation is high enough and the spatial correlation is small.

They also conclude that “further study of this problem and development of some practical

approaches would be desirable under-takings.”

Studying the influence of capillary heterogeneity effect in two-dimensions analytically

is even more difficult and complicated. Simple layered heterogeneous porous media

considering anisotropic permeability has been investigated in 2D (Zhou et al. 1994;

Yortsos 1995). However, its application is limited to miscible displacements and

13

immiscible displacements without gravity effects. In a recent study (Kuo and Benson,

2012), the combined effect of viscous, gravitational, and capillary forces at the core-scale

has been studied numerically and analytically. However, this study only considered

homogeneous porous media.

2.4 Relative Permeability Measurements

Saline aquifers have the largest potential capacity to store CO2 (IPCC 2005). However,

as compared to oil and gas reservoirs, where a century of experience exists regarding

multiphase displacement processes, our understanding of the fate and transport of CO2

and brine in saline aquifers is still limited. When CO2 migrates through a saline aquifer,

the interplay between viscous, capillary, buoyancy forces as well as structural

heterogeneities will determine how far and how fast the plume will move, how much CO2

will dissolve, and how much will be immobilized by residual trapping (Ide et al. 2007;

Kopp et al. 2009). An increasing number of studies based on numerical simulations both

at the reservoir (Bryant et al. 2006; Juanes et al. 2006; Flett et al. 2007; Han et al. 2011)

and core scale (Krause et al. 2011; Kuo et al. 2010; Shi 2010; Kuo and Benson, 2012) are

being widely applied to describe and quantify these processes: critical input parameters to

any simulation are the multiphase flow properties for the CO2/brine/rock system, such as

the capillary pressure and the relative permeability curves. The latter, specifically the

drainage relative permeability, is the main subject this literature review.

There are two main categories of laboratory techniques to measure relative

permeability curves: steady-state methods (Muskat 1937; Morse et al. 1947; Osoba et al.

1951; Abaci et al. 1992; Perrin and Benson 2010; Krevor et al. 2011) and unsteady-state

methods (Welge 1952; Jonhson et al. 1959; Bennion and Bachu 2006; Chalbaud et al.

2007). In the unsteady-state method, only a single phase is injected into the core to

displace in-situ fluids. Saturation equilibrium is not attained and thus it can significantly

reduce the time needed to measure the relative permeability curves. In the steady-state

method, two fluid phases are injected simultaneously at a fixed volumetric ratio and

constant rate until saturation and differential pressure along the core become constant.

14

Although the attainment of equilibrium for steady-state method might be time

consuming, the data can be interpreted directly with the multiphase flow extension of

Darcy’s law using the measured saturation and pressure drop (Abaci et al. 1992; Avraam

and Payatakes 1995). In this review, we focus on the steady-state relative permeability

measurement technique.

Currently there are limited laboratory data on CO2-brine relative permeability

(Bennion and Bachu 2005, 2006, 2007, 2008, 2010; Bachu and Bennion 2007; Perrin and

Benson 2010; Akbarabadi and Piri 2011, Krevor et al., 2012). However, the reliability of

such published relative permeability is directly affected by the quality of the measured

relative permeability curves, as recently highlighted in a review of published relative

permeability measurements (Müller, 2010). In particular, factors that could affect these

measurements are (a) the core heterogeneity that may be responsible for flow rate

dependency and incomplete fluid displacement; (b) capillary end effects that are not

properly accounted for; and (c) gravity segregation that may occur when relatively long

cores are used in a horizontal core-flooding system. In the following, the above

mentioned issues are discussed in more detail.

One of the biggest concerns about relative permeability measurements is the capillary

end effect (Leverett 1941). Whenever a capillary pressure gradient exists along the

porous medium, traditional approaches to calculate the relative permeability are

insufficient. One of the standard techniques used to minimize the capillary end effect is

injecting high flow rate since the capillary forces are small compared to viscous forces at

high flow rates. The influence of end effect on relative permeability becomes significant

at low rates or for low pressure gradients as saturation gradients increase with decreasing

flow rate (Leverett, 1941; Morse et al, 1947; Osoba et al., 1951; Caudle et al., 1951; Kyte

and Rapoport, 1958; Henderson et al., 1997). Based on that, early studies have argued

that relative permeability should be independent of flow rates and that if flow rate

dependency is observed, this should be attributed to the boundary effect (Osoba et al.

1951; Sandberg et al. 1958; Fulcher et al. 1985). However, there are a number of

research papers in the literature suggesting that relative permeability does depend on the

flow rate even when the end effect is carefully avoided (Henderson and Yuster 1948;

Caudle et al. 1951; Avraam and Payatakes 1995; Skauge et al. 2000; Virnovsky et al.

15

2004). Flow rate dependent relative permeability curves are attributed to inadequacy of

the multiphase extension of Darcy’s law for transient flow also upscaling (volume

averaging) of heterogeneous rocks with capillary heterogeneity (Ringrose et al., 1993;

Saad et al., 1995; Dale et al., 1997; Pickup and Stephen, 2000; Pickup et al., 2000;

Virnovsky et al., 2004; Lohne et al., 2006; Jonoud and Jackson, 2008).

Another method to minimize the capillary end effect is to increase the length of the

core. However, experiments carried out with long cores in a horizontal core-flooding set-

up may encounter the issue of gravity segregation (Skauge et al. 1997). Gravity

segregation needs to be considered when working with a fluid pair characterized by a

significant density difference, such as the oil/gas or supercritical CO2/brine system

(Chang et al. 1994; Rossen and Duijn 2004; Ide et al. 2007; Hesse et al. 2008) since the

large density difference can lead to gravity override based on the high Bond number and

hence causes both horizontal and vertical saturation gradients. Although using vertical

experiments to measure relative permeability can avoid gravity segregation (Cinar et al.

2006), the use of a vertical arrangement would make the use of X-ray CT scanning to

observe fluid saturation is quite challenging without purpose-designed equipment.

In addition, spatial variation of rock properties affects both the capillary pressure and

relative permeability-saturation relations (Honarpour et al. 1995; Hamon and Roy 2000;

Ataie-Ashtiani et al. 2002), but it also influences the spatial distribution of saturation

(Chaouche 1994; Perrin and Benson 2010; Krause et al. 2010; Shi et al. 2010). Various

degrees of heterogeneity affect CO2 trapping capacity (Oloruntobi and LaForce 2009)

and may cause flow rate dependency, high residual water saturation and low end-point

relative permeabilities observed from the CO2/brine core flood experiment (Bennion and

Bachu 2005; Perrin and Benson 2010; Akbarabadi and Piri 2011, Krevor et al., 2012). It

has been shown that including heterogeneity characteristics in numerical simulator grid

blocks can improve the accuracy of simulation prediction and enable reliable relative

permeability measurements (Honarpour et al. 1995; Virnovsky et al. 2004; Krause et al.

2011; Kuo et al. 2010).

16

2.5 Discussion

Based on this literature review and the introductory discussion, the following conclusions

can be drawn:

A comprehensive analysis including gravity, capillarity and local heterogeneity is

needed to understand the fundamental physics of multiphase flow of CO2 and

brine in reservoirs and is needed to potentially reduce the computational effort

when performing high-resolution large-scale simulations;

Although there is an increasing number of measurements of multi-phase flow of

CO2 and brine in reservoir rocks (Bennion and Bachu, 2005,2006; Perrin and

Benson 2010; Krevor et al. 2011; Pini et al. 2012), given the emerging importance

of this topic, studies are needed to develop a strong scientific foundation to

support sequestration in saline aquifers;

The large body of multiphase flow studies, particularly relative permeability in

oil/water and gas/liquid systems provides a good starting point for understanding

CO2/brine systems;

Since the fluid properties of CO2 are very different than oil, and because of the

fundamentally empirical nature of the relative permeability concept, studies are

needed to establish similarities and differences between multiphase flow

oil/water and CO2/brine systems;

Potential and unresolved influences of flow rate, capillary number and small scale

heterogeneity on relative permeability in CO2/brine systems need to be

investigated;

The end effect is an important factor that could lead experimental error. If we

want to investigate the flow rate dependence on relative permeability curves, the

end effect must be carefully understood and compensated for;

Based on recently studies of heterogeneity, the effect of heterogeneity may be the

reason for observed dependence of relative permeability on flow rate; and

Since the experiments to measure the influence of heterogeneity on relative

permeability are time consuming, numerical simulations can be used to simulate,

understand and interpret laboratory experiments of multiphase flow in typical

reservoir rocks.

17

Based on above conclusions, and the motivation to investigate the importance of

heterogeneity in CO2 flooding at multiple scales, the goal of this work is to provide a

comprehensive study of the fundamental physics both numerically and analytically, of the

combined influence of heterogeneity, viscous forces, gravity, and capillarity on

multiphase flow of CO2 and brine. Specifically, steady-state displacements in both

homogeneous and heterogeneous cores over a range of relevant conditions are studied.

For example, 3D coreflood simulations have studied the impact of sub-core heterogeneity

on CO2/brine displacements over a wide range of flowrates. Various degrees of

heterogeneity are generated based on the normal random distribution as well as for real

models of cores based on 3-D X-Ray tomography. A 2D analytical model considering

gravity and permeability heterogeneity is developed for predicting brine displacement

efficiency over a wide range of capillary numbers that a good agreement with the 3D

results. The analytical derivation is general and suitable for all flow rates, and is not

limited to high or low capillary and gravity number regimes.

18

Chapter 3

Simulation Methodology

Chapter 3 presents methodology for the numerical simulations performed in this study.

The goal of the simulations is to replicate CO2/brine core flood experiments that are

representative of typical reservoir condition occurring at sites of CO2 sequestration in a

saline aquifer. Once experimental data are simulated qualitatively, a series of sensitivity

studies are performed to investigate the effects of flow rate, gravity, permeability,

interfacial tension, and core length as well as sub-core heterogeneity on multiphase flow

of CO2 and brine. Through these studies we will ultimately develop a better

understanding of interactions between physical forces as well as the role of sub-core scale

heterogeneity on multi-phase flow of CO2 and brine. A typical CO2 saturation

distribution inside a Berea Sandstone rock sample is shown in Figure 3. 1. This

saturation distribution was measured while a mixture of CO2 (95%) and brine (5%) were

injected into the core and after the saturation distribution was no longer changing (Perrin

et al., 2009). One interesting and important feature of the experiment is that there is a

large portion of the core near the outlet end that is almost completely bypassed by the

CO2.

Figure 3. 1 The experimental steady state three-dimensional views of CO2 saturation in

the core for a given fractional flow of CO2 at a given flow rate. The fluids were injected

from right to left (Perrin et al, 2009).

19

The laboratory conditions, core properties as well as grid information listed in Table 3.1

are selected to replicate the laboratory experiments. The Berea sandstone core used in

the experiment is modeled by a three dimensional roughly cylindrical core (Figure 3. 2).

A total of 31 slices is used in the flow direction, including 29 rock slices, an “inlet” slice

at the upstream end of the core, and an “outlet” slice at the downstream end. All of the

simulations are carried out by co-injecting known quantities of CO2 and brine at a

constant flow rate into the inlet end of the core (Figure 3. 1). The downstream end of the

core is maintained at a constant pressure by a back-pressure pump.

TOUGH2-MP with the ECO2N module is used to conduct core-scale multiphase flow

simulations and will be discussed more detail in the next section. The goal of these

simulations is 3-fold: first, the experiments are simulated with the goal of qualitatively

replicating the major features of the experiments; second, simulations are used to conduct

sensitivity studies regarding the influence of viscous, gravity and capillary forces over a

wide parameter space of interest for CO2 sequestration; and finally, the simulations are

used to help develop and then to validate semi-analytical solutions for predicting the

average saturation in a core as a function of the gravity and capillary numbers.

Figure 3. 2 Simulation core of 25 25 31 grid blocks of uniform size.

Table 3. 1 Simulation summary.

Temperature and

Pressure

Fluid properties Berea Core Properties Simulation Grid Data

Tres=50˚C

Pres=12.4 MPa

μCO2=0.046 cp

μw=0.558 cp

ρCO2=608 kg/m3

ρw=1005 kg/m3

σ=22.4 mN/m

L=14.73 cm

H=4.69 cm

Φmean=0.202

kmean=430 mD

25x25x31 grids

Total 19375 grids

5.08 mm grid length

1.874 mm grid width

20

3.1 TOUGH2 MP/ECO2N

TOUGH2 is a general purpose numerical simulator for multi-component, multi-

dimensional, multi-phase fluids flowing in porous and fractured media (Pruess et al.,

1999). TOUGH2-MP, a massively parallel (MP) version of TOUGH2 (Zhang et al.,

2003), is used due to the intensive computational requirement in areas such as reservoir

engineering and CO2 geological sequestration. ECO2N, one of the fluid property

modules of the current version of TOUGH2-MP V2.0, can model thermodynamic and

thermophysical properties of CO2, H2O and NaCl accurately under the range of

conditions where CO2 storage in saline aquifers is likely (Pruess, 2005; Zhang et al.,

2007). Table 3. 2 lists the summary of ECO2N parameters implemented in all the

simulations performed in this work. There are three mass components (water, NaCl and

CO2) in the simulation. Flow systems are assumed to be isothermal, hence three mass

conservation equations are needed to be solved per grid block. Molecular diffusion is not

included in the simulations.

Table 3. 2 Summary of ECO2N.

Components

#1: Water #2: NaCl #3: CO2

Parameter Choices

NK=3 # of mass component

NEQ=3 # of balance equations per grid block (isothermal)

NPH=3 # of phases that can be present

NB=6 no diffusion

Primary Variables

two fluid phases (aqueous and gas)

P Pressure

T Temperature

SG Gas Phase Saturation

XNACL Salt Mass Fraction in Brine

A general description of the differential equations solved in TOUGH2 is provided in the

TOUGH2 User’s Guide (Pruess et al. 1999). The mass conservation equations in

TOUGH2 are discretized in space and time and solved by using the fully implicit finite-

21

difference method.

The mass conservation equations are discretized in space using the integral finite-

difference (IFD) method (Pruess, 1991) and discretized in time using first-order

backward finite difference:

κ

κ κn

nm nm n

mn

dM 1= A F +q

dt V (3.1)

κ,k+1 κ,k

κ,k+1 κ,k+1n n

nm nm n

mn

M -M 1= A F +q

Δt V (3.2)

The superscripts, κ, represent the mass component κ. The subscripts, nm, indicate a

suitable averaging at the interface between grid blocks n and m, (Anm), illustrated in

Figure 3. 3. In this work, all grid blocks have the same size of rectangular shape,

therefore the volume of each grid block is the same (Vn=Vm).

Figure 3. 3 Space discretization in the integral finite difference method.

Mn is the average value of M over Vn. The surface integrals are approximated as a sum

of the average flux over Anm. Fnm is the average value of the inward, normal component

of the flow term, F, over the surface Anm. qn denotes sinks and sources in grid block n.

Mass flux Fnm is a sum over individual phase fluxes Fβ,nm, given by Darcy’s law, times

mass fraction of component κ present in phase β,β

κx :

β

κ κ

nm β,nm

β

F = x F (3.3)

The discretized Darcy flux term is as follows:

22

rβ β β,n β,m

β,nm nm β,nm nm

β nmnm

k ρ P -PF =-k -ρ g

μ D

(3.4)

The discretized flux is expressed in terms of averages over the parameters for element Vn

and Vm. knm and krβ,nm are average absolute permeability and average relative

permeability to phase β over Anm, respectively. ρβ,nm is the interface density of phase β,

defined as the arithmetic average of densities between grid blocks n and m, (ρβ,n+ρβ,m)/2.

Dnm is the distance between the nodal point n and m, and gnm is the component of

gravitational acceleration in the direction from m to n. In this work, evaluation of

mobilities and permeability at interfaces are fully upstream weighted.

In addition, Eq. 3.2 shows that the flux and the sink/source terms are evaluated at the

next time step, tk+1

. This so-called fully implicit method can provide numerical stability

(unconditionally stable). Based on Eq. 3.2, the discrete mass conservation equations can

be written in a residual form (Pruess, 1991; Pruess et al., 1999):

κ k+1 κ k+1 κ k κ k+1 κ,k+1

n n n nm nm n n

mn

ΔtR x =M x -M x - A F x +V q =0

V

(3.5)

where κ

nR are the residuals of component κ for grid block n, the vector xk consists of

primary variables at the time tk, and Δt denotes the current time step size. These strongly

coupled nonlinear equations are solved simultaneously using the Newton-Raphson

iterative scheme:

k k+1 k kJ x x -x =-R x (3.6)

where ij i jJ = R / x is the elements of the Jacobian matrix. Equation (3.6) is solved

leading to:

κ,k+1

κ,k+1n

p i,p+1 i,p n i,p

i i

R- | x -x =R x

x

(3.7)

where xi,p represents the value of ith

primary variable at the pth

iteration step. All the

Jacobian matrix elements are evaluated by numerical differentiation. The Jacobian

matrix as well as κ,t+1

n i,pR x need to be recalculated at each iteration, and this may lead

23

to massive computational work for a large simulation. In the parallel code, equation (3.7)

is computed by all processors. Consequently, computational performance is significantly

improved compared to the TOUGH2 code.

Iteration is continued until the residuals κ,k+1

nR are smaller than the absolute

convergence value

κ,k+1

n 1 2|R | ε ε (3.8)

where ε1 is relative convergence tolerance while ε2 is the convergence criterion for the

accumulation terms κ

nM . These two parameters are TOUGH2 input parameters, and

defaults are ε2=1 and ε1=10-5

, respectively. However, it is hard to converge using such

small tolerance once the small-scale capillary heterogeneity is included. Therefore, after

comparing simulation with high and low tolerances, we use ε1=10-2

to increase

computational speed while preserving adequate computational rigor (validation of these

simulations can be found in Appendix E.1). A comparison between results from

TOUGH2 and another simulator, Stanford’s Genreal Purpose Reservoir Simulator

(GPRS), has been performed and the results are almost identical (Li, 2011) which

provides confidence in the accuracy of TOUGH2 (Appendix E.2). Note that even with

such high tolerance, it still requires significant computational effort to simulate highly

heterogeneous cores. Although convergence usually takes 3-4 iterations, the time step

size Δt is reduced if the convergence cannot be achieved after 8 iterations (default). The

time step size Δt we use here is such that the CFL number (uxΔt/Δx) is much smaller than

1, for example, 0.04-0.3. More details about the CFL numbers are provided in the

discussion section of Chapter 4. Having a low CFL avoids numercal errors caused by the

large time step size although it also requires significant computational power. To achieve

steady-state saturation distributions for typical core flooding experiments in

heterogeneous rocks requires about 24-48 hours on a cluster with 4-8 processors.

There are a number of reasons why TOUGH2 was selected for these calculations.

First, although we are ultimately interested in steady-state conditions, we are also

interested in the transient flow behavior that occurs prior to the establishment of the

steady state. Second, TOUGH2 is the most widely used simulator for studying

multiphase flow of CO2 and brine, and as such, has well documented performance over a

24

wide range of conditions relevant to this study. Finally, TOUGH2 has a robust equation

of state for CO2/Water/Salt mixtures, again a capability needed for simulating the core

flooding experiments which are conducted at reservoir pressures and temperatures.

3.2 Core Descriptions and Grid Size

In order to simulate the laboratory experiments, the Berea core must be replicated in the

simulation grid. The core is divided into NYNZNX grid elements in Y, Z, and X

direction respectively. X-ray CT scans of the core prior to injection provide a three

dimensional spatial porosity map. The simulation grid blocks are assigned unique

porosity values based on the experimental data. Correlations between permeability and

porosity are applied to every grid block to generate its corresponding permeability values.

Therefore, each grid now has been assigned its unique rock properties. From the

TOUGH2-MP simulation, given a fractional flow of CO2 and specific flow rate, the

corresponding pressure drop and CO2 saturation after the system reaches steady state can

be obtained.

The dimensions of simulated core are 4.69 cm in diameter and 14.73 cm long, and the

core is modeled by 25 25 31 grid blocks with dimensions 1.884mm 1.884mm

5.08mm. The grid size chosen for these experiments balances three factors. First the grid

size must be large enough for the continuum representation of the flow and transport

equations to be valid (i.e. Darcy’s law can be used). Microtomography of the Berea

sandstone indicates that a 1mm cube of the grid is large enough to create a representative

elementary volume for the continuum assumption to be valid (Silin et al. 2011; Pini et al.

2012). Second, the grid cells must be sufficiently small to avoid numerical dispersion

and preserve accurate spatial gradients in saturation and capillary pressure. Finally, since

these simulations are computationally intensive, the number of grid cells needs to be

limited to keep the simulations tractable. In light of these considerations, the dimensions

of the grid cells used in this paper are 1.8mm x 1.8mm x 5mm. A comparison showing

that the solutions obtained with this grid are the same as for a grid that is twice as fine is

provided in Chapter 4.

25

3.3 Input Parameters

Before performing numerical simulations, each grid block is assigned values for porosity,

permeability, capillary pressure and relative permeability. The permeability of each grid

element is assumed to be isotropic for the rock slices. In this section, detail of boundary

conditions, initial conditions as well as the input capillary pressure and relative

permeability curves will be discussed.

3.3.1 Boundary Conditions

The boundary conditions used in this study are selected to replicate the core flooding

experiments (Perrin and Benson 2010; Krevor et al. 2012; Pini et al. 2011). In the

experiments, CO2 and brine are mixed and co-injected through a tube and enter into a

diffuser plate to distribute evenly before entering into the upstream end of the core. To

avoid dry-out, carbon dioxide and water are pre-equilibrated at high pressure and

temperature (in this case, 50˚C and 12.4 MPa) prior to starting the experiment. The

amounts of CO2 and brine that enter each pixel are controlled by the relative mobility of

CO2 and brine (Eq. 3.1) such that the total is equal to the injection rate of each phase (Eq.

3.2):

r,β β r,β β,inlet β,1

β,i

β βi i

kk Δp kk p -pq =- =

μ Δx μ Δx

where β=w, CO2 (3.1)

i T

i

q =q

where qi=qw,i+qCO2,i (3.2)

To replicate the inlet boundary condition with the simulator, anisotropic permeability is

implemented in the inlet slice such that the total injected fluids is free to spread out over

the cross section area evenly and enter each cell in accordance with its mobility.

In the experiment, the downstream end of the core is maintained at a constant

pressure by a back-pressure pump. Under this situation, it’s not apparent which boundary

conditions will most closely replicate the experimental measurements. Therefore we test

two numerical boundary conditions to determine which one would most closely replicate

26

the saturation distributions observed at the outlet of the core (Figure 3. 4a). Both test

cases have imposed a time-independent Dirichlet boundary condition: the primary

thermodynamic variables (for example, P and T) remain unchanged in the outlet. One

boundary condition sets the capillary pressure to zero in the outlet slice of the core

(Figure 3. 4b):

c outletP | =0 (3.3)

The other boundary condition imposed the condition that there is no capillary pressure

gradient between the last rock slice and the outlet slice of the model (Figure 3. 4c):

c

outlet

dP| =0

dx

(3.4)

An example of the measured saturation distribution along the core for different fractional

flows of CO2 at a total injection rate of 2.6 ml/min flow rate is shown in Figure 3. 4a.

Similar saturation distributions have been measured for other rocks as described by

Krevor et al. (2012). A relatively uniform saturation profile is observed over the whole

core; in particular, there is no large saturation gradient at the outlet. If the boundary

condition with Pc=0 at the downstream end is imposed, a large saturation gradient occurs,

which is not observed in the experiments (Figure 3. 4b). The Dirichlet boundary

condition with the added constraint that dPc/dx=0 between the last slice in the core and

the outlet provides a much better match to the data (Figure 3. 4c). Consequently we use

this boundary condition in the rest of the simulations. Specifications for the boundary

conditions are listed in Table 3. 3.

(a) (b) (c)

Figure 3. 4 CO2 saturation along the Berea Sandstone core for different fractional flows

of CO2 at a total injection flow rate 2.6 ml/min: (a) Experimental results; (b) High

contrast model with boundary condition Pc=0; (c) High contrast model with boundary

condition dPc/dx=0.

outlet

27

Table 3. 3 Boundary condition summary.

Inlet slice Rock slices (29 slices) Outlet slice

Φmean,

kmean: Anisotropic

(kz=ky=100kx)

Pc=Pc,mean

Φi,

ki: Isotropic

Pc i ik

Φmean,

kmean: Isotropic

Dirichlet boundary condition

dPc/dx=0

3.3.2 Initial Conditions

All the TOUGH2 simulations are conducted at 50˚ C temperature and 12.4 MPa initial

pore pressure. Before injecting CO2 and brine into the core, all the pore-space in the

simulated core is saturated with brine as the initial condition. Water is pre-equilibrated

with dissolved CO2 and 10,000 ppm NaCl. Pre-equilibrating CO2 and water before

injection is important for avoiding mass transfer between phases. Specifically, if dry CO2

is injected into the core, water is vaporized into dry CO2 and results in lower water

saturation near the inlet. Lower saturation of water results in the higher capillary

pressure at the inlet. This “dry-out” phenomenon could lead to imbibition of water from

the downstream portions of the core to the inlet. To avoid these complexities, we pre-

equilibrate the fluids, as is done in the laboratory experiments. At the experimental

condition of 50˚C and 12.4 MPa, the mass fraction of dissolved CO2 in brine is about

4.8%.

When CO2 and brine are injected into the core, the brine is partially displaced by

CO2. A small amount of the CO2 dissolved in the brine may be transferred to the gas

phase, but this is only significant at very small fractional flows of gas (1%). Preliminary

simulations were performed iteratively to determine the maximum CO2 mass fraction

before exsolution occurs. At the experimental condition of 50˚C and 12.4 MPa, the mass

fraction of dissolved CO2 in brine is about 4.8%.

3.3.3 Input Capillary Pressure

Eq. 3.5 show the capillary pressure equation for each grid element i, Pc,i, and the

corresponding modified J-function (Silin et al. 2009):

28

i

c,i w w

i

φP (S )=σcosθ J(S )

k

(3.5a)

2 2

1

λ 1/λ w P

w 1 1 * *λ

P*

S -S1J(S )=A ( -1)+B (1-S ) , S =

1-SS

(3.5b)

where Sw is the average brine saturation; σ is the CO2-brine interfacial tension; θ is the

contact angle, a function of properties of the solid and two fluids. φi and ki are the

porosity and permeability values of grid element i. A1, B1, λ1, λ2 and Sp are five fitting

parameters to match the experimental capillary pressure curves.

For the rock used in the core flood experiment, the capillary pressure was measured

using mercury injection porosimetry (Figure 3. 5). Eq. 3.5 is used to curve fit the

laboratory-measured capillary pressure curve obtained from a sub-sample of the rock

core based on the measured interfacial tension (σmean), measured porosity and

permeability values (φmean and kmean). θ=0 is used in the simulation based on the

assumption of strongly water wet core. Once the five fitting parameters of the J-function

are adjusted to match the measured capillary pressure data, they remain constant for all of

the simulations. The capillary pressure curve for each element in the simulation is then

scaled by the relationship (Eq. 3.5). The fitting parameters in J-function are shown in

Table 3. 4. Figure 3. 5 shows a plot of Pc,i versus Sw. The match between the measured

and fitted curve is very good over the majority of the saturation range, and in particular

the saturation range observed in the experiments. Note that the entry capillary pressure is

zero in this work based on Eq. 3.5.

Figure 3. 5 Laboratory capillary pressure data with a curve fit used in simulations.

29

To determine the effect of heterogeneity on brine displacement efficiency, homogeneous

core and heterogeneous core are simulated at the same reservoir condition, with the same

initial and boundary conditions, same grid sizes and same fitting parameters for relative

permeabilities and capillary pressure. For the homogeneous core, φi = φmean and ki

=kmean, and therefore result in a uniform capillary pressure assigned to each grid cell. On

the other hand, the capillary pressure curve of each grid element is scaled by its porosity

and permeability values for the heterogeneous core. Hence, each grid element has its

own capillary pressure curve. The unique pair of porosity and permeability values results

in a range of capillary pressure curves. Figure 3. 6 illustrates the mean capillary pressure

curve (Pc,mean) and the bounds for the capillary pressure curves in the heterogeneous core.

These capillary pressure curves are a function of core heterogeneity. A higher degree of

heterogeneity will result in a wider range of saturations for a given capillary pressure.

Figure 3. 6 The range of capillary pressure curves in the simulations. The values of input parameters are

A1=0.007734, B1=0.307601, λ1=2.881, λ2=2.255, Sp=0 and σ =22.47 dynes/cm.

Table 3. 4 Input parameter values for relative permeability and capillary pressure curves

fit. Parameters of Relative Permeability

SCO2,r Swr nw nCO2

0 0.15 7 3

Parameters of Capillary Pressure Curve

A1 B1 λ1 λ2 Sp

0.007734 0.307601 2.881 2.255 0

30

The Influence of Capillary Pressure on Saturations

A unique capillary pressure curve assigned to each grid is a key parameter to enable

simulating experimental core flood saturations qualitatively. Table 3. 5 shows the results

of four simulations performed at the 50% fractional flow of CO2 with different input

parameters of the rock such as porosity, permeability and capillary pressure curves. The

results show that even when the permeability and porosity vary within the core, a uniform

capillary pressure curve across the core leads in all cases to a uniform saturation

distribution. The last column demonstrates that the heterogeneity observed in CO2

distribution is introduced only when assigning a unique capillary pressure curve to each

grid cell. Therefore, to replicate the kind of spatial variations in CO2 saturation observed

in the experiments, the capillary pressure characteristic curve must be different in each

grid element.

Table 3. 5 The effect of porosity, permeability, and capillary pressure on CO2 saturation

distribution.

Porosity Φ

Uniform Measured Measured Measured

Permeability k

Uniform Uniform

K-C Model

K-C Model

Capillary

Pressure Pc

Uniform

Uniform

Uniform

Various

Saturation

Distribution

Average

Saturation SCO2 24.38% 24.38% 24.26% 21.24%

3.3.4 Input Relative Permeability

The relative permeability curves used in the simulations for both homogeneous and

31

heterogeneous cores are power-law functions:

CO w2

2

2

n n

w CO ,r w wr

r,CO r,w

wr wr

1-S -S S -Sk = , k =

1-S 1-S

(3.6)

where Swr is the residual brine saturation, and nw

and nCO2 are the functional exponents

for the brine and CO2 curves respectively (Figure 3.7). These three parameters in the

relative permeability functions were chosen to fit the relative permeability data calculated

from experimental measurements conducted by Perrin and Benson (2010).

Figure 3.7 Input relative permeability curves for CO2 and brine with Swr =0.15, nw =7,

and nCO2=3 and the experimental data.

3.3.5 Input Injection Flow Rate

The range of injection flow rates is chosen from 0.001 ml/min to up to 24 ml/min (four

orders of magnitudes). In the laboratory, flow velocity (v) can be calculated based on

volumetric flow rates (qlab) and core cross section (Acore). Consider a homogeneous and

isotropic reservoir with a thickness of 100 meters and a fully penetrated injection well

where CO2 is injected uniformly at a constant rate of 1 Mt CO2/yr (Figure 3.8); We can

calculate the flow velocity as a function of distance from the well using a 1-D radial

geometry:

well lab

core

Q qv= =

2πrH A (3.7)

32

Figure 3.8 Conceptual model of the reservoir used to estimate the range of relevant flow

velocities.

For example, Figure 3.9 shows that laboratory flow rate 1.2 ml/min corresponds to CO2

plume at 4.2 meters away from the injection well; 0.1 ml/min corresponds to 100 meters

away and 0.01 ml/min corresponds to 1 kilometer away.

Therefore these flow rates span the range of conditions expected in the near-will

region (1.2 ml/min) to the leading edge of the plume (0.001 ml/min), which may be up to

5 km or more from the injection well. Based on this range of values, the interplay of

viscous, capillary and gravity forces in core flood experiments is investigated at different

gravity and capillary numbers representative of those expected for a typical sequestration

project.

Figure 3.9 The volumetric flow rate and its corresponding distance of CO2 plume at the

reservoir (100 meter thickness and 1 Mt CO2/yr injection rate).

33

3.4 Output Parameters

For a given flow rate, simulations of co-injection of CO2 and brine are run until the

pressure drop and core-averaged saturation stabilize. All of the simulations have been

confirmed to run long enough (more than at least 10 pore volumes injected) to reach

steady-state. Important output parameters include grid cell CO2 saturation, CO2 pressures

and capillary pressures.

3.4.1 Slice-Average Quantities (SCO2, PCO2, Pc)

Here we also evaluate slice average quantities along the length of the core such as the

slice average CO2 saturation (SCO2), slice-average pressure in the CO2 phase (PCO2), and

slice-average capillary pressure (Pc). Figure 3.10 shows a typical simulation result: with

the CO2 saturation distribution, pressure drop across the core, and the core-averaged CO2

saturation.

Figure 3.10 CO2 saturation distribution at steady-state for 95% fractional flow of CO2 at a

total injection flow rate 1.2 ml/min.

The pressure drops across the core are defined as the difference between the inlet and the

outlet values:

2 2 2CO CO ,inlet CO ,outletΔP =P -P (6.1a)

w w,inlet w,outletΔP =P -P (6.1b)

Since Pc= PCO2–Pw, the water pressure drop can be rewritten in terms of the two output

parameters ΔPCO2 and ΔPc:

34

2 2 2w CO ,inlet CO ,outlet c,inlet c,outlet CO cΔP = P -P - P -P =ΔP -ΔP (6.2)

3.4.2 Simulated CO2 Saturation Distributions

In addition to measuring the average saturation of the core, we can also measure the

saturation distribution. For the homogeneous cores with no buoyancy force, the CO2

saturation distribution is uniform and equal after steady state for every flow rate.

Saturation gradients only occur along the horizontal direction during the transient period

while the front is advancing through the core, and shortly thereafter when the fractional is

approaching the fractional flow at the inlet.

However, once gravity is included, the saturation distribution changes significantly.

Figure 3.11 compares the saturation distribution for the homogeneous and the

heterogeneous cores. It shows that not only does the average brine displacement

efficiency decrease when flow rates are lower, but the distribution also changes. For high

flow rates, the saturation gradient is very small and only in the vertical direction due to

the density difference between CO2 and brine. However, saturation gradients occur along

both vertical and horizontal directions when gravity effect is relevant.

When comparing the saturation distribution between the homogenous and

heterogeneous cores, it is evident that the combination of gravity and heterogeneities

influences the CO2 distribution.

35

Figure 3.11 Saturation distribution for homogeneous and heterogeneous cores with

gravity, using grid with pixels (150, 150, 31) to grid blocks (25, 25, 31).

In summary, simulations are performed to replicate CO2/brine core flood experiments

that are representative of the typical reservoir condition occurring at sites of CO2

sequestration in saline aquifers. The conditions for the laboratory core flood experiments

are chosen to replicate temperatures and pressures typical of reservoir conditions (Perrin

et al 2009; Perrin and Benson 2010). For example, for the simulations described here, the

pressure is 124 bars (1800 psi) and the temperature is 50˚C. For the experiments,

injection rates are chosen to correspond to the flow rates in the near well region in the

field. Pore velocities range from 10-5

m/s to 10-3

m/s, as would be expected in the near-

well region for a typical storage project.

36

Chapter 4

3D Numerical and 2D Analytical Studies

for the Homogeneous Cores

In this chapter, we are interested in the drainage flow behavior in the CO2/brine two-

phase flow in the horizontal core. The main objective of this paper is to develop a greater

understanding of steady-state displacement mechanisms relevant to CO2 sequestration at

the core scale under reservoir conditions and to understand the transitions between

different flow regimes. Potential applications of the results include: 1) establishing the

bounds over which relative permeability can be accurately measured in horizontal core-

flood experiments; and 2) developing model experimental systems for studying

multiphase flows for a wide range of fluid pairs, geometric configurations and rock

properties. Additionally, the results have relevance for understanding reservoir-scale

processes, particularly at the sub-grid scale, where intra-gridblock processes may have an

influence on flow and transport parameterizations. Here we provide a new analytical

solution to identify different flow regimes in homogeneous and isotropic systems based

on the similar dimensionless groups used in Zhou et al. (1994). A two dimensional

analysis of the governing equations accounting for viscous, gravity, and capillary forces

at steady-state is used to develop an approximate semi-analytical solution for predicting

non-wetting phase saturations during core-flood experiments. A systematic parametric

study of the flow regimes is performed numerically to help identify flow regions. For

example, a wide range of flow rates, permeability, interfacial tension, and different

lengths of the core are investigated. Finally, the semi-analytical solution is validated by

comparing the results under a wide range of parameters, including different fractional

37

flows and relative permeability curves.

4.1 Methodology

The overall methodology for this study contains two parts. First, we conduct 3D high

resolution simulations of core scale CO2/brine flow over a wide range of injection

flowrates. We simulate the injection of a constant ratio of CO2/brine at a given flowrate

into a horizontal and initially brine saturated core. The corresponding brine displacement

efficiency is assessed when the system reaches steady-state (defined as the time when the

saturation is no longer changing and the fractional flows of CO2 and brine are equal at the

inlet and outlet of the core). Sensitivity studies incorporating a wide range of

permeability, core lengths and interfacial tension values have also been studied to

generalize these results. Second, a theoretical analysis of the multiphase flow equations

is used to develop an approximate semi-analytical solution for predicting the average

saturation in the core as a function of gravity number, Bond number and several

dimensionless parameters. Finally, we compare the results of simulations and

approximate semi-analytical solutions to test the validity of the solution over a wide

range of parameters.

4.1.1 Simulation Study

Initially, we focus on 95% fractional flow of CO2 and 5% brine, which are injected

simultaneously into a simulated core at a wide range of flowrates. Later we consider a

wide range of fractional flows. In this chapter, we assume a homogeneous core with the

average petrophysical properties of a Berea Sandstone (mean porosity of 0.202 and mean

permeability of 430 md). The capillary pressure gradient between the last slice of the

core and the outlet slice is set to zero to minimize end effects. This boundary condition

has been found to most accurately replicate saturation distributions measured during core

flooding experiments. The input curves for the capillary pressure and relative

permeability used in this section have been illustrated in Figure 3. 4 and Figure 3. 6.

The details regarding the simulation have already explained in Chapter 3. The

influences of flow rate, interfacial tension, and core permeability on the CO2/brine flow

38

systems are studied: flow rate is changed by several orders, from 0.001 ml/min up to 60

ml/min which corresponds the CO2 velocity from near well region (~m) to the leading

front of the plume (~10 km) based on the analysis shown in Section 3.3.5; interfacial

tension and core permeability are varied by two orders of magnitudes to study a range of

values. Additionally, a range of core lengths and heights is also included.

Figure 4. 1 a through c show the core average CO2 saturations for the homogeneous

cores as a function of traditional capillary number Ca, alternative capillary number Ncv,

and gravity number Ngv for two sensitivity studies using the definitions shown in Eqs.

4.1-4.3 (Fulcher et al. 1985; Lake 1989; Zhou et al. 1994).

2t coCa= u μ σ (4.1)

2

*

c

cv 2

co t

kLpN =

H μ u (4.2)

2

gv

co t

ΔρgkLN =

Hμ u (4.3)

where ut is the total average Darcy flow velocity, μCO2 is CO2 viscosity, pc* is a

characteristic capillary pressure and is defined as the displacement capillary pressure

(more detail in Chapter 6), L the length of the core, H the height of the core, Δρ=ρw-ρg is

the density difference between CO2 and brine, and g is the acceleration of gravity. The

values of Ca, Ncv and Ngv are controlled by varying injection rates, interfacial tension,

and permeability where the other parameters are kept as constants. The wide range of

IFT values is purely hypothetical and used solely to explore the sensitivity to IFT values.

Higher flowrates, lower permeability, smaller density differences and lower capillary

pressure have higher Ca, or lower Ncv and Ngv. The high flowrate regime is

representative of the near well region. Lower flowrates, hence the lower Ca or higher Ncv

and Ngv, are representative of the leading edge of the plume and during fluid

redistribution in the post-injection period. We can also define Bond number NB as a ratio

of gravity to capillary numbers:

gv

B *

cv c

N ΔρgHN =

N p (4.4)

39

Since capillary pressure depends on the interfacial tension and permeability (Eq. 3.5),

different pairs of σ and k result in different pc*, and hence different Bond number NB.

Table 4.1 shows the summary of four sensitivity studies and their corresponding Bond

number.

Table 4. 1: Summary of Sensitivity Studies

σ, mN/m k, md pc*, Pa NB

Base Case (σ, k) 22.47 430 3000 0.061

Sensitivity 1 (3σ, k) 67.41 430 9000 0.022

Sensitivity 2 (σ/3, k) 7.49 430 1000 0.198

Sensitivity 3 (σ, 0.1k) 22.47 43 9487 0.019

Sensitivity 4 (σ, 10k) 22.47 4300 947 0.192

All the simulation results shown in Figure 4. 1 are for a fractional flow of 95% CO2. The

sensitivity studies for interfacial tension are illustrated in the left hand side of the figures

and permeability on the right. When comparing the average saturation in terms of

traditional capillary number Ca, alternative capillary number Ncv and gravity number Ngv,

the efficiency of brine displacement clearly falls into three separate regimes: a viscous-

dominated regime where the saturation is independent of Ca, Ngv and Ncv; a gravity-

dominated regime where the average saturation is strongly dependent on the Ngv and Ncv;

and a capillary-dominated regime characterized by lower average saturations and smaller

sensitivity to variations in Ngv and Ncv.

The transitions from the viscous- to gravity-dominated regimes and from the gravity-

to capillary-dominated regimes are dependent on the interfacial tension, permeability and

flow rate (Figure 4. 1a and Figure 4. 1b). Results show that the transition points occur

earlier when gravity has a stronger influence (large Bond number), for example, for the

lower interfacial tension and higher permeability cases. Plotting the same data in terms

of gravity number (Ngv) results in the same transitions for different interfacial tensions or

core permeabilities, or simply, different Bond numbers (Figure 4. 1c). Here we define

the first transition, from the viscous- to gravity dominated regime, as the critical number

1 and the second transition, from gravity- to capillary-dominated regime, as the critical

number 2. The dashed lines shown in Figure 4. 1c indicate the two critical numbers.

40

In the viscous-dominated regime, defined as Ngv ≤ Ngv,c1, the brine displacement

efficiency is independent of Bond number. Average saturations are also accurately

predicted based on the Buckley-Leverett theory which neglects capillarity and gravity.

Macroscopically, the capillary pressure and CO2 saturation are uniform along the vertical

and horizontal axes of the core. The saturation gradient or the capillary pressure gradient

are very small (<5%) and only in the vertical direction.

As the flow rate decreases, gravity starts to have a significant effect on the multiphase

flow behavior and the viscous and gravity forces become similar in magnitude. In the

gravity-dominated regime (Ngv,c1 < Ngv < Ngv,c2), the brine displacement efficiency is

highly flowrate dependent and insensitive to the Bond number (Figure 4. 1c). Gravity

override leads to saturation gradients in the vertical direction, with higher saturation at

the top of the core, as would be expected for buoyancy driven flows. As a result of this, a

saturation gradient is also established along the horizontal axis of the core. In this

regime, the saturation gradient is large and exists along both vertical and horizontal

directions. This decreasing trend of brine displacement efficiency changes gradually over

two orders of magnitude of flow rate.

Capillary forces dominate when flowrates are reduced further. In this capillary-

dominated regime (Ngv,c2 ≤ Ngv), the effect of gravity is small and the brine displacement

efficiency is strongly dependent on capillary forces, and hence the Bond number.

Capillary pressure gradients control the displacement of the fluids, tending towards

achieving capillary-gravitational equilibrium based on the capillary pressure curve of the

core. In this regime the saturation gradient is small and mainly in the horizontal

direction.

41

(a)

(b)

(c)

Figure 4. 1: Average CO2 saturation as a function of (a) traditional capillary number Ca,

(b) alternative capillary number Ncv, and (c) alternative gravity number Ngv. The

interfacial tension σ and the permeability of the base case are 22.47 mN/m and 430 md,

respectively. The sensitivity studies of interfacial tension are illustrated in the left hand

side of the figures and permeability on the right.

Figure 4. 1 illustrates the pressure gradients associated with the viscous, gravity and

capillary forces as a function of gravity numbers for the high interfacial tension case (3σ).

The values in Figure 4. 1 are obtained directly from the output of the high resolution 3-D

simulations using TOUGH2 described in Chapter 3. The viscous and capillary pressure

drops in the flow direction (ΔPv,x and ΔPc,x) are calculated based on the slice-average

pressure difference between inlet and outlet grid cells while the pressure drop due to

gravity (ΔPgrav) is defined as ΔρgH. Note the pressure obtained from TOUGH2 is the gas

42

phase pressure. It is clear that the viscous force is dominant before the first transition.

The transition between the viscous and gravity dominated flow regime occurs when the

viscous pressure gradient in the flow direction (ΔPv,x/L) and the pressure gradient of

gravity in the vertical direction (ΔPgrav/H) have the same magnitude. In the second

regime, gravity force eventually exceeds the viscous force while capillary force also

increases to have the similar magnitude. The second transition, from the gravity to the

capillary dominated regime, occurs when the viscous pressure drop and the capillary

pressure drop in the flow direction have the same magnitude (ΔPv,x≈ΔPc,x), which implies

the capillary-dominated regime occurs when the water pressure drop across the core has

almost vanished. The pressure gradients for more sensitivity cases can be referred to

Appendix C.

Figure 4. 2: Pressure gradients for three different forces as a function of alternative

gravity number Ngv. The interfacial tension and the permeability of the 3σ case are 67.41

mN/m and 430 md, respectively.

Based on Figure 4. 1c and Figure 4. 2, the transition between the viscous and gravity-

dominated regimes occurs when Ngv ≤ 54 while the capillary-dominated regime begins

when Ngv ≥ 220. The first critical gravity number can be derived from first principles,

that is, when the buoyancy pressure gradient equals the viscous pressure gradient.

Assuming uniform saturation, the viscous pressure drop across the core can be calculated

based on the multiphase extension of Darcy’s law:

43

2 2 2

2

2 2

r,CO CO t COv,x

CO v,x

CO r,CO

kk f u μΔpu =- Δp =

μ L kk

Ｌ (4.5)

uCO2 is Darcy flow velocity of CO2 and fCO2 is CO2 fractional flow. The pressure gradient

due to buoyancy forces is ΔPgrav= ΔρgH. Therefore, the first critical gravity number

(Ngv,c1) can be calculated by:

2 2

2 2 2 2 2 2

r,CO r,COgrav

gv

CO t CO r,CO CO t CO CO lv,x

Δρgkk kΔp H Δρg= = N 1

f u μ kk f u μ f RΔp L (4.6)

2

2

CO l

gv,c1

r,CO

f RN

k (4.7)

Rl =L/H is the shape factor or the so-called the aspect ratio. Using Eq. 4.7, the first

critical number is 53.9, which agrees well with the results from the simulations shown in

Figure 4. 2. Similar analysis can be used to derive the second critical number, Ngv,c2.

The capillary pressure of multiphase flow system is given by pc=pc*J. The ratio of the

viscous pressure drop to the capillary pressure drop is

2 2 2 2 2 2

22

2v,x CO t CO r,CO CO t CO CO

* * 2

c r,CO cvc r,CO c

Δp f u μ kk f u μ fL L= =

|Δp | ΔJ k ΔJNp ΔJ kk p H

Ｌ

(4.8)

2 2 2

2 2 2

2 2 2

v,x CO l CO l CO l

cv,c2 gv,c2 B

c r,CO cv r,CO r,CO

Δp f R f R f R= =1 N = or N =N

|Δp | k ΔJN k ΔJ k ΔJ

(4.9)

However, it is more difficult to predict Ngv,c2 since ΔJ depends on the average saturation

at the inlet and outlet ends of the core, which depends on a number of variables.

Therefore, an empirical method, based on an evaluation of the sensitivity studies, is used

to calculate the second critical number:

3/2

l

gv,c2

BL

RN =α where α=12.79737

S (4.10)

The second critical number depends on the aspect ratio and the saturation predicted by

the Buckley-Leverett solution (SBL). Comparing Eq. 4.9 and Eq. 4.10 implies that ΔJ is

proportional to Rl0.5

. If we treat ΔJ as a fitting parameter and analyze a series of

sensitivity studies on the aspect ratio, it indeed shows a linear dependency on Rl0.5

(Figure

4.3). Therefore, it provides confidence in Eq. 4.10. The concept of three different

44

regimes and the associated critical numbers will be used in the analytical solution in the

following sections.

Figure 4. 3: The function ΔJ has a linear dependency on the square root of aspect ratio,

Rl.

4.2 Theoretical Analysis

4.2.1 2D General Solution

We first simplify our 3-dimensional problem into 2 dimensions (x-z direction). The

properties of this 2D porous medium are homogeneous and isotropic. Porosity and

permeability are constant. Mass conservation equations and pressure equation for

incompressible flow are:

j j j,x j,z

j

S S u uφ u =φ =0

t t x z

(4.11)

tu 0 (4.12)

with the condition Sw+2COS =1. uj,x and uj,z are Darcy velocity of phase j in the x and z

direction, respectively. Darcy flow velocities for both phases are given by

j j rj

j j j j,x j,z j

j

p p ku =-λ k , +ρ g u , u where λ =

x z μ

(4.13)

where uj, λj, pj, ρj and μj are Darcy flow velocity, relative mobility, pressure, density, and

45

viscosity of phase j, respectively. Boundary conditions used in the simulation are given

as followings. At the top and the bottom boundaries,

uw,z=ug,z=0 (4.14)

In the flow direction x, total volumetric flow rate is sum of water and gas flow:

uw,x+ug,x=ut (4.15)

Using Darcy’s equation and the boundary conditions, the mass conservation of gas phase

now becomes:

g g gt c c

w

S Mkλ Mkλu p p1 1- + Δρg- =0

t φ kλ x 1+M φ z z 1+Mx

(4.16)

M= λw/λg is mobility ratio. To non-dimensionlise the equation, we define xD=x/L,

zD=z/H, tD= t/tchar= tut/φL, and pc(Sw) =pc*J (Sw). The characteristic time tchar is chosen as

the time it would take for one pore volume to flow through the core at a velocity of ut.

Substituting all the defined terms into Eq. 4.16 yields the dimensionless mass

conservation equation:

2

g rg rg rg

gv cv

D D D l D D D D

S Mk Mk Mk1 1 J J+ +N -N + =0

t x 1+M z 1+M R x x 1+M z z 1+M

(4.17)

where Ncv and Ngv are shown in Eq. 4.2 and 4.3. Assuming steady-state (Eq. 4.18) and

using separation of variables, we can obtain 2D time independent CO2 saturation SG (Eq.

4.19). The derivation is detailed in Appendix A for reference.

g

D

S=0

t

at steady state (4.18)

2

lD

cvB D B D

R- ax

N-N bz -N bz

1 2 3 4SG C e +C e +C e +C (4.19)

Unknown variables a and b are functions of saturation and mobility ratio M (see

Appendix A.2). C1, C2, C3 and C4 are functions of xD and zD. Based on this solution, the

time independent CO2 saturation SG depends only on its position xD and zD, M, as well as

46

the dimensionless numbers Rl, Ncv and NB.

4.2.2 2D General Solution with Simulation Constraints

Since it is difficult to integrate Eq. 4.19 to obtain the average core saturation, we assume

that the saturation at a particular point (x0,D, z0,D) is representative of the core average

saturation2COS :

2CO 0,D 0,DS SG x ,z (4.20)

Now C1, C2, C3 and C4 become constants and the two terms in the exponent ax0,D and

bz0,D are unknown variables which will be determined later by fitting the simulation

results. To eliminate some unknown parameters, we use several observations from the

simulation results. First, Figure 4. 1c shows that average CO2 saturations are independent

of gravity number Ngv when Ngv ≤ Ngv,c1:

2l

B 0,D2 gv,c1

R- N ax

CO N

gv gv,c1

gv

S 0 when N N e 0

N

(4.21)

Therefore, in the viscous-dominated regime, Ngv ≤ Ngv,cl, the core average saturation

becomes the Buckley-Leverett solution SBL:

B 0,D

2

-N bz

CO 3 4 BLS C e +C S . (4.22)

The average saturation in this regime is independent of Bond number, which implies

C3=0 and C4= SBL. Second, when Bond number equals to zero (g=0), saturations are

observed to be a constant SBL (Kuo et al. 2010):

2l

0,Dcv

2

R- axN

CO 1 2 BL BLS C +C e +S S (4.23)

Eq. 4.23 results in C2= -C1. Based on above constraints, Eq. 4.19 can be rewritten as the

following form:

2

l0,D

B 0,D cv

2

R- ax

-N bz N

CO 1 BLS C e -1 e +S (4.24)

Letting the term ax0,D ≡d1Ngv,cl/NB, we need to satisfy2

l 1-R de 0 based on Eq. 4.21. Once

47

we choose an appropriate value of d1 to satisfy this condition, we can replace the

unknown term ax0,D in terms of d1, NB, and the critical gravity number Ngv,c1. Similarly,

Figure 4. 1c also illustrates that saturations are not sensitive to Bond number when Ngv ≤

Ngv,c2, which leads to

2 B 0,DCO -N bz

gv gv,c2

B

S0 when N N e 0

N

(4.25)

Defining the exponent term bz0,D≡d2/Ngv,c2, choosing an appropriate value of d2 to satisfy

2 B-d N Ngv,c2e 0 , and the saturations are now determined by Bond number and gravity

number, shown in Eq. 4.26.

2l gv,c1B

2 1gv,c2 gv

2

R NN-d -d

N N

CO 1 BLS =C e -1 e +S

(4.26)

C1, d1, and d2 are fitting parameters which can be determined by simulation results or

experimental data. Eq. 4.26 shows that the core average saturation depends on two terms.

First, in the high flowrate regime or for Ngv ≤ Ngv,c1, the average saturation (and hence,

SBL) is determined solely from the fractional flow curve based on Buckley-Leverett

theory (1942) which neglect gravity and capillary pressure:

SBL = SBL(2COf ) (4.27)

where CO2 fractional flow, fCO2, is a function of mobility ratio, M:

2COf =1 1+M (4.28)

If we know the input relative permeability curves, the viscosity of the two fluids and the

CO2 fractional flow, we can determine the corresponding SBL.

Second, the core average saturation also depends on a correctional term which

combines the interactions between all three forces through the dimensionless numbers

and the aspect ratio. Therefore, Eq. 4.26 is the modified Buckley-Leverett solution; it not

only depends on the viscous, capillary and gravity forces but also depends on the size of

multiphase flow system. In the literature, people often solve mass conservation equation

numerically by simplifying the conservation equations into different flow regimes (Zhou

48

et al. 1994; Yortsos 1995) or by generalizing the traditional mass conservation equations

without providing the explicit form of solutions (Hassanizadeh and Gray 1993b). Here

we provide a tool which can actually predict the solution of the CO2 saturation explicitly.

4.3 Results

4.3.1 Comparison between Simulations and the Approximate Semi-

Analytical Solution

Figure 4. 4 shows that the average CO2 saturation of the base case as a function of

capillary number Ncv (LHS) and the CO2 fractional flow curve (RHS). To compare our

theoretical predictions with the simulation results shown earlier, in addition to SBL and

two critical numbers Ngv,c1 and Ngv,c2 described above, we need to determine variables C1,

d1 and d2. Table 4. 2 provides the values of dimensionless parameters and the fitting

parameter (C1, d1 and d2) for the base case with interfacial tension σ and core

permeability k (see Table 4.3 for parameter values). As shown, the semi-analytical

solution compares well with the simulation results and captures the major features and

transitions of the curve.

Figure 4. 4 (LHS) Comparison of average CO2 saturation as a function of capillary

number Ncv between theoretical values and simulation results for the homogeneous or

base case (σ, k); (RHS) fractional flow curve based on our input relative permeability

curves (Eq.3.6).

Table 4. 2: Values of unknown variables used to match the base case.

Ngv,c1 Ngv,c2 C1 d1 d2

54 220 934.18 0.3747 1

49

Table 4. 3: Berea core properties and fluid properties used in the homogeneous cores for

the base case

mean φ mean k,

md σ, mN/m L, cm H, cm Rl μbrine, cp μCO2, cp

0.202 430 22.47 14.73 4.69 3.1445 0.558 0.046

By analyzing the fitting parameters C1, d1, d2 for a wide range of input parameters, we

have been able to establish that the fitting parameters d1 and d2 depend on the aspect ratio

of the core and that C1 depends on the Bond number:

11

1 11

l l

dd = where d =2.0894

R R (4.29a)

l l

2 22

22

R Rd = where d =5.57607

d (4.29b)

11

1 11

B

CC = where C 56.78

N (4.29c)

Ideally, one would derive the constant coefficients such as α, d11, d22 and C11 from first

principles. However, it is not clear how to develop a closed-form solution for

determining these coefficients. Consequently, we use an empirically based approach to

identify these coefficients based on curve fitting to the simulation cases. The general

form of core average CO2 saturation now can be predicted based on Eq. 4.26 and 4.29 as

well as the critical numbers in Eq. 4.7 and 4.10.

Figure 4. 5 show the sensitivity studies for permeability (43 md and 4300 md) and

interfacial tension (7.49 mN/m and 67.41 mN/m) respectively. This wide range of

sensitivity studies covers a realistic range of relevant parameters such as permeability in

the field. The range of interfacial tension is much larger than expected and selected

solely to explore how variations in interfacial tension would affect the results. Those

figures compare the simulation results and the predicted values based on Eq. 4.26 and Eq.

4.29. As shown, we can replicate the simulation results quite well, especially the

transition from the viscous- to gravity-dominated regime. However, a slight mismatch in

the capillary-dominated regime occurs for the cases when the capillary force is strong

(0.1k and 3σ). When the capillary force is relatively small such as 10k (4300 md) and

50

σ/3 (7.49 mN/m), the semi-analytical solution matches best in the transition from

viscous- to gravity-dominated regime but deviates slightly in the gravity- and capillary-

dominated regime.

Figure 4. 5 Comparison of average CO2 saturation as a function of capillary number Ncv

between theoretical values and simulation results for the sensitivity cases of interfacial

tension. LHS: (3σ, k), RHS: (σ/3, k)

4.3.2 Model Validation

Here we test the validity of Eq. 4.26 by using different core dimensions (Figure 4. 6),

different fractional flows of CO2 (Figure 4. 7), and different input relative permeability

curves (Figure 4. 8).

Different Core Dimensions

We have carried out simulations with the aspect ratio Rl ranging from 1.6 to 16. The

solution provides accurate values for the average core saturation within this range (Figure

4. 6). Since the input relative permeability curves and the fractional flow of CO2

(fCO2=0.95) for these cases are the same as for the base case, the Buckley-Leverett

solution SBL is still 0.324. As shown, the semi-analytical solution predicts the average

saturation quite well for different core geometries. In addition, Eq.4.7 shows that the

51

larger aspect ratio results in a bigger Ngv,c1, which implies that the gravity-dominated

regime occurs at larger values of Ngv if the system has a larger aspect ratio.


between theoretical values and simulation results for the sensitivity cases of different

dimensions of the core.

Different Fractional Flows of CO2

Figure 4. 7 shows the average CO2 saturations as a function of capillary numbers for CO2

fractional flows of 0.79, 0.51 and 0.34. Although the same input relative permeability

curves are used, different fractional flow of CO2 results in different Buckley-Leverett

solution SBL and hence different krg(SBL) values. Again, the semi-analytical model

predicts average saturations very well for the entire range of fractional flows.

52



fractional flows of CO2.

Different Relative Permeability Curves

The final cases to test our semi-analytical solution use different relative permeability

curves with 0.95 fractional flow of CO2. The other parameters such as those in Table 4.3

are the same. For these cases, the Buckley-Leverett solution SBL and krg(SBL) are

different due to various relative permeability curves. Figure 4. 8 shows the average CO2

saturations as a function of capillary numbers for six different input relative permeability

curves. The first four cases (the 1st and 2

nd rows of Figure 4. 8) use the same form of

relative permeability functions shown in Eq. 3.6. The base cases use parameters nw=7,

nCO2=3, and Swr=0.15 (IRP1) while the other three cases change one or two parameters at

a time. One has a lower residual brine saturation (IRP2), one has a higher exponent of

brine nw (IRP3), and the last one has a higher exponent of nCO2 as well as a lower residual

brine saturation (IRP5). The 2D model predicts the lower residual brine saturation

(IRP2) and the higher nw (IRP3) very well while the predictions for the higher nCO2 and

lower residual brine saturation (IRP5) have slight deviations in the gravity- and capillary-

dominated regime. However, the semi-analytical solution still captures the transition

from viscous- to gravity-dominated regime quite accurately. The last two cases (the 3rd

53

row of Figure 4. 8) use Corey’s equation with residual saturation 0.05 and 0.15,

respectively. The results show that even with different relative permeability functions,

the semi-analytical solutions still match the simulation results quite well.


between theoretical values and simulation results for six different input relative

permeability curves. Those simulations share the same constant parameters in Table 4.3.

4.4 Discussion

In previous sections, we have shown that simulation results and theoretical predictions

agree quite well. The average CO2 saturation is determined by three forces. The relative

magnitudes of gravity to capillary forces and viscous to gravity forces are characterized

by the Bond number NB and gravity number Ngv. The combined effect of these forces

results in Eq. 4.26. The semi-analytical model provides a simple tool to determine

54

approximate values for the critical numbers. Eq. 4.29 works well for all the cases tested

in this paper.

Table 4. 4 summarizes three different flow regimes and the corresponding core

average CO2 saturation expressions based on Eq. 4.26. At high flow rates or the so-called

viscous-dominated regime (Ngv ≤ Ngv,c1), the brine displacement efficiencies for

homogeneous cores are flowrate independent at the given fractional flow of CO2. The

value of SBL can be predicted from Buckley-Leverett theory neglecting capillarity and

gravity. When buoyancy forces begin to dominate multiphase flow (Ngv,c1 < Ngv < Ngv,c2),

average saturations are highly flow rate dependent. In this gravity-dominated regime,

the average saturation is mainly dependent on gravity number. When Ngv > Ngv,c2,

saturation becomes less sensitive to the gravity number. In the capillary-dominated

regime, the average saturation asymptotically approaches to a constant value.

Table 4. 4: Summary of flow regions for general cases

Flow region Conditions Steady State CO2 saturation

Viscous-dominated regime Ngv ≤ Ngv,c1

2CO BLS S

Gravity-dominated regime Ngv,c1 < Ngv < Ngv,c2

2l gv,cl

1gv

2

R N-d

N2 11

CO BL

gv,c2

d CS S - e

N

Capillary-dominated regime Ngv,c2 ≤ Ngv

2

2

1 l gv,cl2 11

CO BL

gv,c2 gv

d R Nd CS S - 1-

N N

Some degree of mismatch between the semi-analytical solution and TOUGH2

simulations are observed in both the gravity- and capillary-dominated regimes. The

differences between the 2D analytical solution and 3D simulation may be explained by a

number of reasons. First, we assume that all of the cases share the same fitting

parameters C11, d11, α and d22 once they are adjusted to match the simulation results of a

reference case. However, C1 is a function of (x0,D, z0,D) and hence a function of NB,

Ngv,cl, and Ngv,c2. Based on the derivation in Appendix A.2, the coefficients d11 and d22

are functions of several parameters such as mobility ratio M, the derivative of J-function,

and relative permeability coefficients aw and ag since the first critical number Ngv,cl

depends on fCO2/krg while the second critical number depends on SBL and hence fCO2 (see

55

Eqs. A-19). In addition, the fitting parameter α also changes slightly for different cases:

g w

11

a +ad =f

J'

(4.30a)

g w

22

a M-ad =f

J'

(4.30b)

2B rg CO Lα=f N ,k ,f ,R (4.30c)

Therefore, the assumption used to develop this solution requires that the combined effects

of other parameters are constant. We show that this assumption is reasonably good for

the cases studied here, but it is not strictly correct. The semi-analytical predictions for

the lower flowrate regimes can be improved by slightly adjusting these coefficients.

However, the differences are small and the correct functional form to adjust these

coefficients systematically is not evident. Further investigation could generalize

solutions by exploring the sensitivity to these parameters.

Second, the mismatch in the capillary-dominated regime (Figure 4. 4) could be

influenced by the outlet boundary condition we use for the simulation (no Pc gradient in

the flow direction at the outlet). Although this boundary is chosen to match the

experiments at moderate (~cm3/min) flow rates, we find that even for lower velocities (in

the gravity-dominated regime), the analytical predictions are close to the simulation

results. The potential artifact may occur only at very low flowrate, for example, when

Ngv > 105.

Finally, the gridding and the time step size could potentially contribute to the

mismatch observed. As discussed in Section 3.2.1, the size of the grid used in this paper

(1.8mm x 1.8mm x 5mm) is chosen to balance several considerations. The LHS of

Figure 4. 9, which compares results from the grid used for these studies to another with ½

the size in the y and z directions (0.88 mm) yields essentially identical results, with a

maximum deviation in the average saturation in the core of less than 2%. The good

agreement between the higher and lower grid resolution simulations suggests that grid

refinement is not the cause of the discrepancy between the analytical solution and

numerical results. In addition, to avoid numerical artifacts caused by the time-step size,

the initial time step is chosen to have a small CFL number (uxΔt/Δx<1) for every flow

56

rate, shown on the RHS of Figure 4. 9. For subsequent time steps, it is automatically

adjusted by TOUGH2 to higher or lower values during a simulation run dependent on the

convergence rate. After breakthrough, the time step size will increase up to the

maximum time step size Δtmax set up at the beginning.

For instance, for the case of the viscous-dominated regime such as 1.2 ml/min, the

initial time step is set as 8 seconds (CFL number=0.05) and it reduces to the smaller time

step size (4 seconds) or increases to the larger time step size (16, 32, or 64 seconds)

dependent on the convergence rate. The corresponding CFL numbers are 0.025, 0.1, 0.2,

and 0.4, which all smaller than 1.

Figure 4. 9 (LHS) The finer grids have grid dimensions 0.884mm x 0.884mm x 5.08mm

(grid numbers: 53x53x31) while the coarser grids have dimensions 1.874mm x 1.874mm

x 5.08mm (grid numbers: 25x25x31); (RHS) CFL number (uxΔt/Δx<1) for different flow

rate.

4.5 Conclusions

A new semi-analytical solution has been developed to predict the influence of gravity and

capillary numbers on the average saturation expected during multiphase flow

experiments. Although computational technology has improved greatly, running high

resolution 3D models including capillarity and gravity still takes a significant amount of

computational effort. The new solution provided here is a quick and easy way to

estimate the flow regimes for horizontal core floods. Practical applications include

helping to design core flood experiments, including assuring that relative permeability

measurements are made in the viscous dominated regime, evaluating potential flow rate

dependence, influence of core-dimension on a multiphase flow experiments, and

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

1 10 100 1000 10000 100000

Ave

rage

CO

2 Sa

tura

tio

n

Capillary Number, Ncv

Grid Effect (Homogeneous)

CoarseScale: 25x25x31

FineScale: 53x53x31 0

0.2

0.4

0.6

0.8

1

0.01 0.1 1 10 100

CFL

nu

mb

er

flow rate q (ml/min)

CFL number

57

influence of fluid properties on the experiments. In addition, having a semi-analytical

solution provides a useful tool for investigating multiphase fluid displacement efficiency

over a wide parameter space of practical interest. Other applications, not explored in this

paper, include investigation of upscaling strategies in the transition between the viscous,

gravity and capillary dominated regimes. The new semi-analytical solution can be used

to estimate the average saturation over a wide range of conditions in terms of several

important dimensionless numbers such as mobility ratio, relative permeability to gas

evaluated at SBL, aspect ratio Rl, Bond number NB and gravity number Ngv.

The solution has been compared to 3D high resolution simulations to study the effects

of flowrate, gravity, interfacial tension, core-length, and core permeability on two-phase

immiscible flow. The proposed 2D semi-analytical technique predicts the brine

displacement efficiency for 3D homogeneous CO2/brine two-phase flow simulations very

well when the Bond number ranges from 0.02-0.2 and aspect ratio ranges from 1.57-

15.72. As mentioned before, the sensitivity studies cover the wide range of permeability

and interfacial tension values and hence results in this range of Bond number, which is in

the parameter range of interest for CO2 sequestration in deep aquifers.

Theoretical predictions match the corresponding simulation results not only for the

base case but also for many sensitivity cases. It can also apply to various fractional flows

of CO2 as well as different input relative permeability curves. It is expected that the first

critical number can be defined accurately even for the larger aspect ratios outside the

range studied in this work, for example, Rl=100, which could be happened in the sub-grid

cell of field simulations with 100 meters in length and 1 meter in height. However,

further investigation is needed to confirm this conclusion. The limitations of the

analytical solution will be discussed in detail in Chapter 7.

58

Chapter 5

3D Numerical and 2D Analytical Studies

for the Heterogeneous Cores

In the Chapter 4, a 2D analytical solution was developed to predict the average saturation

in homogeneous cores during steady state core flood experiments over a wide range of

relevant conditions. The analytical predictions show excellent agreement with the

simulation results. Based on the knowledge gained from the homogeneous studies,

further investigations into the behavior of heterogeneous rocks are presented in this

chapter. Another important variable, capillary heterogeneity, is added into the 2D

analytical analysis to provide more realistic predictions for heterogeneous rocks.

In order to provide a direct comparison to the results for homogeneous rocks, the

same grid size, initial and boundary conditions, fitting parameters for capillary pressure

and relative permeability curves for the base case illustrated in Chapter 4 are used. For

example, the capillary pressure gradient between the last slice of the core and the outlet

slice is set to zero to minimize end effects. The input curves for the capillary pressure

and relative permeability implemented in all the simulations shown in Eq. 3.5 and Eq. 3.6

are used. Every simulation is performed at the reservoir condition 50˚C and 12.4 MPa.

We also focus on 95% fractional flow of CO2 and 5% brine injected simultaneously into

a simulated core at a wide range of flow rates. Additional details regarding the

simulations have been provided in Chapter 3. The only parameters changed are rock

properties of porous media such as the porosity, permeability and capillary pressure

curves of each grid cell in the simulation.

59

5.1 Heterogeneous Representations

The first part of Chapter 5 will present results of 3D numerical simulations with a wide

range of heterogeneity to understand the role of sub-core scale heterogeneities. To

generate different heterogeneity distributions, two methods are used. The first method

for assigning heterogeneity uses a constant porosity distribution (φ=0.202) and the

permeability distribution is generated based on a random log-normal distribution with a

standard deviation of σlnk. The other type of heterogeneity uses measured values of Berea

porosity distribution obtained from X-ray CT scanner to generate its corresponding

permeability distribution based on a porosity-permeability relation. Both methods are

implemented with isotropic permeability (kx=ky=kz) for every grid cell and the mean

permeability ranges from about 250 md to 570 md. In the heterogeneous cases, the

capillary pressure curve is calculated based on a modified form of the Leverett J-function

(Eq. 3.5) using the known porosity and permeability distributions. Each grid element

has a unique pair of porosity and permeability values; hence a unique capillary pressure

curve.

5.1.1 Random Permeability Distribution (3D)

Four different degrees of permeability heterogeneity are created based on a random log-

normal distribution. As mentioned above, porosity is uniform, which is a reasonable

assumption, since the variation of porosity is often small, unlike the variation in

permeability, which could be up to several orders of magnitude. Once the porosity and

permeability of a grid element is assigned, the corresponding capillary pressure curve is

generated based on Eq. 3.5. The relative permeability of each grid cell is identical (Eq.

3.6). With these inputs, the steady-state average CO2 saturation from a series of core-

scale simulations can be obtained. The standard deviations for the four different cases

and their permeability distributions are listed in Table 5.1.

Figure 5. 1 illustrates both the slice-average permeability and the corresponding

steady-state average saturation (at the 6 ml/min total injection rate) along the flow

direction for four random distributions. Case 1 (lnk=0.02) has the most uniform

permeability profile while the Case 4 (lnk=2.65) has the widest range of slice averaged

60

permeability values along the length of the core. It is observed that large variations over

small length scales result in a relatively large variation in the saturation profile, similar to

1D result in the literature (Yortsos and Chang 1989, 1990; Chang and Yortsos 1992). It

is also consistent with the experimental results showing that relatively homogeneous

cores result in the smoothly varying saturation profiles, while the saturation distributions

of very heterogeneous cores are variable (Krevor et al. 2012). Importantly, these graphs

also show that for randomly distributed heterogeneity, the larger the degree of

heterogeneity, the lower average saturation in the core.

Figure 5. 1: Four different random permeability profiles and the corresponding average

CO2 saturations in the viscous-dominated regime.

5.1.2 Porosity-based Permeability Distribution (3D)

The other type of heterogeneity distribution is generated using a porosity-based approach,

which is commonly used in the literature (Mavko and Nur 1997; Pape et al. 1999; Krause

et al. 2008). We have two permeability fields generated from porosity-permeability

61

models based on the measured porosity values of Berea Sandstone (Perrin et al. 2009).

One of three-dimensional permeability maps is generated from the Kozeny-Carman

equation (Kozeny, 1927; Carman, 1937):

3i

i 2i

φk =S

(1-φ )

(5.1)

S is a scaling factor that assures that the average of all permeability values is equal to the

average permeability of the core, 430 md. φi is a pixel value in the porosity map, and ki

is the corresponding value calculated using Eq. (5.1). To increase the contrast in

permeability and hence increase the degree of heterogeneity, an alternative empirical rock

property model is created as follows:

4

ii=exp 64φ -6k (5.2)

Figure 5. 2 illustrates the slice-averaged porosity profile and its two corresponding

permeability profiles based on Eq. 5.1 and 5.2. The porosity of Berea Sandstone varies

within a small range. Comparing Figure 5. 1 and Figure 5. 2, the permeability

distributions generated from the porosity values of a real rock change more smoothly

than the random distributions and have spatially correlated low permeability features

aligned sub-parallel to the axis of the core. The corresponding saturation profile is shown

in the bottom of Figure 5.2. For the Kozeny-Carman permeability model, the results are

nearly indistinguishable from the homogeneous case, which is consistent with the results

from the random distribution with a small value of lnk. For the model with a higher

degree of heterogeneity (High Contrast model), the slice-average saturation varies

significantly along the length of the core and the average saturation is lower. These

results are qualitatively consistent with the results from the random distribution.

62

Figure 5. 2: Average permeability and average CO2 saturation along the flow direction

for three different heterogeneous cores (Homogeneous model, Kozeny-Carman model,

and High Contrast model).

5.1.3 Permeability Distribution Summary

Table 5. 1 illustrates all the 3D permeability distributions used for this study: Random 1-

4 cases and the two porosity-based permeability fields (Eq. 5.1 and 5.2). Although the

isotropic permeability is implemented for all the grid element, there is some degree of

anisotropy observed for the Random 3 and Random 4 cores due to the shape of the grid

cells used in the simulation (rectangular shape). Therefore it results in some spatial

correlation of properties in the flow direction. However, the effect of anisotropy caused

by the shape of grids would not affect our results significantly; hence “Random”

distribution cores are used in this work to refer these quasi-random distribution models.

The standard deviation of permeability ranges from 0 to 2.65. Although the statistical

mean permeability is chosen to be 430 md for most of the cases, the absolute

permeability or the so-called effective permeability for different heterogeneous

63

representations are different. Effective permeability kmean is calculated based on the

single phase Darcy’s law: injecting solely brine into a brine-saturated core at a given flow

rate and calculating pressure drop across the core after steady-state. The effective

permeability is around 430 md for the cores with a small degree of heterogeneity. The

degree of heterogeneity is represented by the parameter σlnk/ln(kmean). A relatively

homogeneous medium has a small σlnk/ln(kmean) and its permeability variation approaches

to zero, while a highly heterogeneous porous medium can have large values of

σlnk/ln(kmean). In this study, σlnk/ln(kmean) ranges from about 0 to 0.5.

Table 5. 1 Summary of Simulation Cases with Different Degrees of Heterogeneity.

Permeability

Realizations

Porosity

φ

Effective

Permeability

kmean (md)

Standard

Deviation

σlnk

Heterogeneous

Factor

σlnk/ln(kmean)

Homogeneous

0.202 430 0 0

Random 1

0.202 432 0.0236 0.0039

Random 2

0.202 430 0.2540 0.0419

Berea

(Kozeny-

Carman)

various 430 0.2757 0.0455

Berea

(High

Contrast) various 318 0.9602 0.1680

Random 3

0.202 366 1.3679 0.2343

Random 4

0.202 254 2.6542 0.4747

64

5.2 Simulation Studies

Similar to the approach used to study homogeneous cores, we investigate the significance

of flow rate, interfacial tension, and core permeability on the CO2/brine flow systems for

moderately heterogeneous cores, or specifically, the High Contrast and Random 3

models. Flow rate is changed by several orders, from 0.005 ml/min up to 24 ml/min;

interfacial tension and core permeability are again varied by two orders of magnitudes to

study a range of values. Sensitivity studies for a wide range of permeability values are

provided later in this section. Different values of CO2 fractional flows are also studied.

5.2.1 Heterogeneity Effects

For the base case of the homogeneous study, we injected a fractional flow of 95% CO2

over a wide range of flow rates at reservoir conditions. To study the effects of

heterogeneity on brine displacement efficiency, we keep the conditions the same as for

the homogeneous study. Figure 5. 3 shows the core average CO2 saturations at 95%

fractional flow for the two cores over a wide range of flow rates, hence capillary

numbers. Although the two types of heterogeneity exhibit different behaviors, it is clear

that a higher degree of heterogeneity (higher σlnk/lnkmean) results in a lower average

saturation in the core. It also increases the flowrate dependency of brine displacement

efficiency, specifically the transition from the viscous dominated regime to gravity and

the capillary controlled regimes occurs at lower capillary numbers.

Note that the saturations for the Kozeny-Carman model are lower than for the

Random 2 model even though they have similar heterogeneity factor (σlnk/lnkmean~0.05).

This is mainly due to the presence of a capillary barrier close to the outlet end of core that

prevents CO2 from entering a portion of the core (not visible in Table 5.1 but shown later

in Figure 6.2).

65

Figure 5. 3: Average CO2 saturation as a function of alternative capillary number Ncv.

The interfacial tensions σ for all cases is 22.47 mN/m and the effective permeability

varies from 254 to 570 md.

5.2.2 Interfacial Tension and Permeability Effects

A series of sensitivity studies on interfacial tension and core permeability for the

heterogeneous cores are performed. Table 5.2 shows the summary of sensitivity studies

and their corresponding Bond numbers for the High Contrast model and Random 3 model

respectively. Based on capillary pressure function (Eq. 3.5), the effect of interfacial

tension changes from 3σ to σ/3 is similar to the effect of permeability changes from 0.1k

to 10k. Simulations have confirmed this conclusion and therefore only results for a wide

range of permeability values are shown in Figure 5. 4. However, the results of brine

displacement efficiency with three different interfacial tension values in terms of

alternative capillary number Ncv will be discussed later in Chapter 6.

66

Table 5. 2: Summary of Sensitivity Studies for High Contrast and Random 3 Models.

High Contrast Models σ, mN/m k, md pc*, Pa NB

Base Case (σ, k) 22.47 318 3489 0.0523

Sensitivity 1 (3σ, k) 67.41 318 10466 0.0174

Sensitivity 2 (σ/3, k) 7.49 318 1163 0.1568

Sensitivity 3 (σ, 0.1k) 22.47 31.8 11032 0.0165

Sensitivity 4 (σ, 10k) 22.47 3180 1103 0.1654

Random 3 Models σ, mN/m k, md pc*, Pa NB

Base Case (σ, k) 22.47 366 3252 0.056

Sensitivity 1 (σ/3, k) 7.49 366 1084 0.168

Sensitivity 2 (σ, 0.1k) 22.47 36.6 10283 0.018

Sensitivity 3 (σ, 10k) 22.47 3680 1028 0.177

The core-averaged CO2 saturations as a function of traditional capillary number Ca,

alternative capillary number Ncv, and gravity number Ngv using the definitions shown in

Eqs. 4.1-4.3 are plotted in Figure 5. 4. The left hand side provides the results for the

High Contrast model while the right hand side shows the results of the Random 3 model.

For ease of comparison, sensitivity studies on permeability for the homogeneous cores

are also illustrated in the same graph. Solid lines represent the homogeneous saturations

while dashed lines represent the heterogeneous ones.

As discussed extensively in Chapter 4, plotting the average saturations for

homogeneous cores in terms of gravity number can distinguish three flow regimes

clearly. For the heterogeneous cores, the efficiency of brine displacement also falls into

three separate regimes: a viscous-dominated regime where the saturation is independent

or nearly independent of Ca, Ngv and Ncv; a viscous-capillary transition regime where the

average saturation is strongly dependent on the dimensionless numbers; and a capillary-

dominated regime characterized by low saturations with a small dependence on the

dimensionless variables. However, unlike the case for homogeneous cores where the

gravity number can be used to normalize the results, for heterogeneous permeability

distributions, sensitivity studies are normalized better in terms of the transverse capillary

number Ncv (Eq. 4.2), shown in Figure 5. 4. This suggests that Ncv is a better

dimensionless number to distinguish the transitions when considering capillary

heterogeneity. The ratio of capillary to viscous forces is a critical measure for assessing

the influence of capillary trapping in heterogeneous systems (Rapoport 1955; Ringrose et

67

al 1996)

Results show the transitions from the viscous- to viscous-capillary transition regimes

and from the transition to the capillary-dominated regimes for both types of

heterogeneous cores occur earlier for higher degrees of heterogeneity. Capillary

heterogeneity not only reduces the average saturation in the viscous-dominated regime

but also increases the flowrate dependency, which implies that higher flow rates are

required to reach the viscous-dominated regime.

Figure 5. 4: Average CO2 saturation as a function of traditional capillary number Ca,

alternative capillary number Ncv, and alternative gravity number Ngv for two

heterogeneous cores: High Contrast model (left) and Random 3 model (right). The

interfacial tension σ is kept as a constant, 22.47 mN/m.

68

Figure 5.5 illustrates the pressure gradients for the three forces as a function of capillary

numbers for the High Contrast and the Random 3 models, respectively. These

demonstrate that the viscous force is always greater than the buoyancy force within the

flowrate range we are interested in. Viscous and capillary forces are much more relevant

than the gravity force once considering capillary heterogeneity. Unlike for the

homogeneous cores, where the first transition can be defined by the point at which

gravity and viscous forces are equal, for heterogeneous cores, viscous forces are always

greater than the gravity force. However, the second transition occurs when the viscous

pressure drop and the capillary pressure drop in the flow direction have the similar

magnitude, similar to the homogeneous results shown in Figure 4. 1.

Figure 5. 5: Pressure gradients as a function of alternative capillary number Ncv for three

physical forces.

5.3 Theoretical Analysis of Multiphase

Displacement Efficiency in Heterogeneous Cores

The second part of this chapter presents a 2D analytical solution incorporating the

influence of heterogeneity. The analysis is an extension of the methods developed in

Chapter 4 for homogeneous cores. A general form of the solution is first derived and the

constraints from 3D high-resolution simulations are applied to obtain the final semi-

analytical solution. The concept of derivation is explained generally here. The detailed

derivation is provided in Appendix B.

69

5.3.1 2D General Solution

To predict the simulation results observed in the previous section, we consider steady-

state, two-dimensional flow (x-z direction) of two immiscible phases. The properties of

this 2D porous medium are heterogeneous and each grid cell has isotropic properties.

Using standard terminology and notation, the mass conservation equation of gas phase is

described by the following:

g g gt c c

w



(5.3)

The approach used to develop the analytical solution is similar to the derivation in

Chapter 4, except now the permeability and hence the capillary pressure values vary

spatially:

c w w

φp S , x, z σcosθ J S

k x, z (5.4)

For the analysis in this chapter, we consider only permeability heterogeneity (and

consequent capillary heterogeneity) while porosity, interfacial tension, and contact angle

remain constant. However, the same approach could also be applied to these different

types of heterogeneities. To non-dimensionlize the equation, we define xD=x/L, zD=z/H,

and tD=tut/φL as well as introducing a dimensionless variable τ(xD, zD) to represent

heterogeneity:

D D

D D

mean

k x ,zτ x ,z

k (5.5)

Based on the definition of Eq. 5.4, we can obtain pc= [σcosθ(φ/kmean)0.5

]J/τ. In addition,

for simplification, we introduce variables defined by Chang and Yortsos (1992) as H=-

fwkrgJ’ and G=fwkrgJ. Substituting all the variables into Eq. 5.3, the dimensionless mass

conservation equation at steady-state becomes:

2

2cv cv

CO cv B rg w cv2 2

D D D D D D D Dl l

N Nτ τ SG SGf G +N N k f τ +G = Hτ +N Hτ

x x z z x x z zR R

(5.6)

70

where the capillary and gravity numbers have the same definitions in Chapter 4 except

kmean is the permeability:

22

*

mean c mean

cv gv2

co tco t

k Lp Δρgk LN = and N =

Hμ uH μ u (5.6b)

fCO2=1/(1+M) and fw=M/(1+M) are fractional flows of CO2 and water respectively.

Important variables are the alternative capillary number or the so-called inverse

macroscopic capillary number Ncv (Eq. 5.6b), gravity number Ngv (Eq. 5.6b), aspect ratio

Rl =L/H, and the heterogeneity function τ(xD, zD). Incorporating capillary heterogeneity

into the mass conservation equations results in a high degree of complexity. The right

hand side of Eq. 5.6 is known as the capillary dispersion term. Using a similar strategy to

that used for development of the semi-analytical solution for homogeneous cores and

assuming all the variables are continuous functions (for saturation this will only be true if

all grid cells have a zero capillary entry pressure); we can obtain the general form of two-

dimensional time-independent CO2 saturation SG for the heterogeneous cores in terms of

Bond number NB, capillary number Ncv, aspect ratio Rl, and the heterogeneity term τ (Eq.

5.7a). The unknown coefficients C1Hete

, C2Hete

, C3Hete

and C4 not only depend on location

but also on the heterogeneity of the rock (Eq. 5.7b).

2l

DcvB D B D

R1- axτ N-τN bz -τN bzHete Hete Hete

1 2 3 4SG= C e +C e +C e +C (5.7a)

D D D Dz x x z-ε lnτ z -ε lnτ x -ε lnτ x -ε lnτ zHete Hete Hete

1 D D 1 2 D D 2 3 D D 3C x ,z ,τ =C e e , C x ,z ,τ =C e , and C x ,z ,τ =C e (5.7b)

The definitions of two variables, a and b, in the exponent terms are the same as defined in

the homogeneous analysis (Eq. A-19). C1, C2, C3 and C4 are functions of coordinates xD

and zD. (lnτ)x and (lnτ)z are the heterogeneity gradients in the flow direction x and

vertical direction z, respectively. The variable ε is defined as Jb-1 where J is the

Leverett-J function. A detailed derivation of Eq. 5.7 is provided in Appendix B.1.

71

5.3.2 2D General Solution for Heterogeneous Rocks Using

Simulation Constraints

Since it is difficult to integrate Eq. 5.7 to calculate the average core saturation, we use the

same strategy in the Chapter 4 to assume that the saturation at a particular point (x0,D,

z0,D) is representative of the core average saturation2COS :

2CO 0,D 0,DS SG x ,z (5.8)

Now C1Hete

, C2Hete

, C3Hete

, C4 and τ become some single valued functions of x0,D and z0,D.

Similarly, to eliminate some unknown coefficients, we apply several observations from

the simulation results to the general solution. First, in the viscous-dominated regime

(Ncv<Ncv,c1), the average saturation is independent of capillary and Bond numbers:

2 2CO CO

cv B

S S 0 and 0

N N

(5.9)

Second, in the capillary-dominated regime (Ncv>Ncv,c2), the average saturation is

independent of the Bond number:

2CO

B

S0

N

(5.10)

Applying simulation constraints to the general equation and using the homogeneous

solution derived in the previous chapter as a model, we can eliminate several unknown

coefficients. A detailed derivation is provided in Appendix B.2.

As mentioned before, the parameter τ (Eq. 5.5) is related to the heterogeneity factor,

σlnk/ln(kmean). The mathematical representation of the parameter τ in terms of the

heterogeneity factor is provided in Eq. 5.11a. See Appendix B.3 for a detailed

description of the derivation.

1 0,D lnk lnk2 0,D 2 0,D

2 0,D mean mean

β x σ σ1+ β z ωβ z

β z ln k ln k

0,D 0,D 0,D o,D meanτ x ,z k x ,z k τ=e =e

(5.11a)

β1 and β2 are proportionality factors between σlnk/ln(kmean) and the spatial derivatives of

the permeability field, (lnτ)x and (lnτ)z, shown in Eq. 5.11b. ω is a fitting parameter

dependent on β1 and β2 as well as x0,D and z0,D (Eq. 5.11c).

72

lnk lnk

2 1z x

D mean D mean

σ σln τ ln τlnτ β and lnτ β

z ln k x ln k

(5.11b)

1 0,D

2 0,D

β xω 1+

β z (5.11c)

More investigation and discussion about τ and ω will be provided in the next section.

Applying all the simulation constraints to the general solution (Eq. 5.7), we can predict

the average CO2 saturation for heterogeneous cores in terms of several dimensionless

numbers:

2 Hete2 l cv,c1

Hete 1cv,c2 cv

2

d R N- dN NHete HeteBL

CO 1 BLHete

BL

SS C e - e +S

S

(5.12a)

lnk

2 0,D0,D meanz

σ-εβ z

-ε lnτ z ln kHete

BL 1 BL 1S h τ S where h τ e =e (5.12b)

2

3/2CO lHete Hete l

cv,c1 cv,c2 HeteHeteB B BLrg BL

f R R1 1N = and N = α

τN τN Sk S

(5.12c)

l lHete 11 11

1 1 2ε

B 22l l

R RC d1C = , d = and d =

N dτ R R (5.12d)

The average saturation of the heterogeneous core is controlled by the Bond number NB,

capillary number Ncv, the Buckley-Leverett solutions SBL and SBLHete

, and the aspect ratio

Rl once the two critical capillary numbers (Ncv,c1Hete

and Ncv,c2Hete

) are calculated (Eq.

5.12a). Moreover, the heterogeneous solution naturally reduces to the homogeneous

solution when τ=1.

Eq. 5.12b shows that the average CO2 saturation of the heterogeneous core in the

viscous-dominated regime is SBLHete

, which is a modified Buckley-Leverett solution that

takes into account heterogeneity. The capillary heterogeneity has a significant effect on

the average saturation of a BL displacement. The effect is important even for large flow

rates within the viscous-dominated regime if the degree of heterogeneity is large (Figure

5. 3). This is because the capillary heterogeneity terms GNcv(τ/xD)/Rl2 and

GNcv(τ/zD) in the mass conservation equation (Eq. 5.6) are not negligible even for

small values of Ncv once we have large variations of heterogeneity (large τ/xD and

τ/zD). h1 is a heterogeneous factor, dependent on the heterogeneity gradient in the

73

vertical direction (lnτ)z as well as the parameter ε, shown in Eq. 5.12b. The variable ε is

considered as a fitting parameter to match the simulation results. Since h1<1, the average

saturation for the heterogeneous cores is always lower than the Buckley-Leverett

solution, which is reasonable and consistent with the simulation results (Figure 5. 3).

The definitions of two critical capillary numbers, Ncv,c1Hete

and Ncv,c2Hete

, are shown in

Eq. 5.12c. They can be related to the critical gravity number for the homogeneous

studies based on the Eq. 4.4, Ncv=Ngv/NB. Eq. 5.13 relates the two critical numbers

between the homogeneous cores and the heterogeneous cores for easier comparison.

Homo Hete

gv,c1 BLHete Homo Hete

cv,c1 cv,c1 BL

B

N S1 1N = = N S

τ N τ (5.13a)

Homo HeteHetegv,c2 BLgv,c2Hete Homo Hete

cv,c2 cv,c2 BL

B B

N SN 1N = = = N S

N τN τ (5.13b)

For the homogeneous porous medium, τ=1, hence lnτ =0, we can obtain SBLHete

=SBL.

Therefore Ncv,c1/2Hete

=Ncv,c1/2Homo

. Note that the smaller critical numbers are obtained

with a larger degree of heterogeneity (large τ).

Coefficients C1Hete

, d1 and d2 are shown in Eq. 5.12d where d1 and d2 are the same

functions used in the homogeneous solution. C1Hete

can be related to the C1Homo

as

follows:

Hete Homo

1 1ε

1C = C

τ (5.14)

C11, α, d11, and d22 are constant parameters, which have already been determined

previously (Eq. 4.27) by curve matching the semi-analytical solution with the simulation

results for homogeneous cores. The values are provided in Table 5.3.

In summary, the average saturation for the heterogeneous cores can be predicted once

we can obtain the correct values of τ and ε. These parameters will be determined by

matching the semi-analytical solution with the numerical results.

74

Table 5. 3: Summary of constant coefficients in Eq. 5.12.

d11 d22 C11 α

2.0894 5.57607 56.78 12.79737

5.3.3 Approximate Semi-Analytical Solution

From the simulation results shown in Figure 5. 3, we can obtain the modified Buckley-

Leverett solutions SBLHete

from cores with different degrees of heterogeneity. Figure 5.6

plots SBLHete

as a function of normalized standard deviation factor σlnk/ln(kmean) for

random permeability cores (LHS) and all the permeability cores listed in Table 5.1

(RHS).

Figure 5. 6: Comparison of average CO2 saturation as a function of capillary number Ncv


degrees of heterogeneous cores.

A near-perfect linear correlation between SBLHete

and σlnk/ln(kmean) for cores with

randomly distributed heterogeneity is observed in Figure 5.6:

Hete lnk lnk

BL BL

mean mean

σ σS =0.324-0.1788 1-0.55185 S

ln k ln k

(5.15)

For cores with structured heterogeneities a reasonably good correlation is also observed.

Based on the linear relation shown earlier, the heterogeneous function h1 can be

approximated as follows:

75

lnk2 0,D

mean

σHete -εβ zln kBL lnk

1 2 0,D

BL mean

S σ=h =e 1-εβ z

S ln k (5.16)

Once we obtain the value of h1 based on Eq. 5.16, we can evaluate the heterogeneous

factor τ in terms of this known parameter:

lnk

2 0,D1

mean

σωω β z 1-hln k ετ=e =e

(5.17)

In conclusion, the only fitting parameters introduced in Eq. 5.12 for the heterogeneous

cores now become ω and ε while the other parameters have been already introduced in

the homogeneous studies. By adjusting these two coefficients, the semi-analytical

predictions agree with the simulation results quite well. Figure 5.7 shows good

agreement between the simulated results and the theoretical predictions for the average

CO2 saturations of base cases for the High Contrast model (LHS) and the Random3

model (RHS), respectively.


between theoretical values and simulation results for High Contrast model at a wide

range of permeability and interfacial tension.

However, it is difficult to find a general functional form for ω and ε since they depend on

several factors. For example, they depend on the spatial gradients in heterogeneity for

both the x and z directions. For the random fields, the proportional factor β1 is equal to

β2. We can expect to have different values of ω for spatially correlated permeability

distributions. In addition, Figure 5.8 plots the dependence of ε and ω in terms of

potential factors such as the modified Buckley-Leverett saturation (SBLHete

), and the Bond

76

number (NB) based on the sensitivity studies of CO2 fractional flow, permeability and

interfacial tension. It shows that different heterogeneous cores have different

dependency. Further investigation is needed in the future to generalize solutions by

exploring the sensitivity to these parameters.

Figure 5. 8: The dependence of ε and ω in terms of the modified Buckley-Leverett

saturation (SBLHete

) and the Bond number (NB) for High Contrast and Random 3 models.

77

5.4 Verification of the Analytical Model

The validation of analytical model for heterogeneous cores based on Eq. 5.12 is presented

in this section. As mentioned earlier, it is difficult to obtain the correct functional form of

ε and ω; hence these two parameters are treated as fitting parameters. These two values

are different for different sensitivity cases and are provided in Appendix D with other

relevant dimensionless numbers.

5.4.1 Different Interfacial Tension and Permeability

Figure 5.9a shows the sensitivity studies of High Contrast model for permeability (31.8

md and 3180 md) and interfacial tension (7.49 mN/m and 67.41 mN/m) respectively.

Those figures compare the simulation results and the predicted values based on Eq. 5.12.

As shown, we can replicate the simulation results in general quite well, especially for the

high permeability and low interfacial tension cores. When the capillary force is relatively

small, the semi-analytical solution matches best in the transition from viscous- to

viscous/capillary-dominated regimes. Although a slight mismatch to the simulation

results occurs in the cases of High Contrast model when capillary force is strong (0.1k

and 3σ), the overall predictions are captured the transitions between regimes. The

prediction is even better for the random distribution core (Random 3) shown in Figure

5.9b for both permeability values (36.6 md and 3660 md).

The differences at the higher capillary numbers of High Contrast model could be due

to the lack of information related to the correlated permeability distribution observed in

the cores. Further studies to include the correlation length in the semi-analytical solution

could generalize these results to a wider range of heterogeneity distributions.

78

(a) Porosity-based permeability: High Contrast model

(b) Random permeability distribution: Random 3


between theoretical values and simulation results for (a) High Contrast model and (b)

Random permeability model in a wide range of permeability and interfacial tension.

5.4.2 Different Fractional Flows of CO2 (HC Model)

Figure 5.10 shows the average CO2 saturations of the High Contrast model as a function

of capillary numbers for four different CO2 fractional flows, 0.95, 0.79, 0.51 and 0.34.

The heterogeneous factor τ stays constant for four cases since it is the same core.

Although the same input relative permeability curves are used, different fractional flow

of CO2 results in different Buckley-Leverett solution SBLHete

and hence different

krg(SBLHete

) values. Again, the semi-analytical model predicts average saturations very

well.

79


between theoretical values and simulation results for High Contrast model at different

fractional flows of CO2.

5.4.3 Different Core Dimensions

One sensitivity study on the aspect ratio Rl is performed to test the semi-analytical model.

The solution works well for the limited cases studied (Figure 5. 11). Since the input

relative permeability curves and the fractional flow of CO2 (fCO2=0. 95) for this case are

the same as for the base case of the High Contrast model, we can expect the modified

Buckley-Leverett solution SBLHete

to remain the same.

80



aspect ratio.

5.4.4 Different Heterogeneity

The final cases to test the semi-analytical solution use different degrees of heterogeneity,

listed in Table 5. 1, with 0.95 CO2 fractional flow. All of the cases have the same

simulation parameters, namely, relative permeability curves, capillary pressure curves,

same reservoir conditions, and same grid size, etc. The only thing changed is the

permeability distribution of porous medium which is characterized by the heterogeneity

factor, σlnk/ln(kmean). The other important input parameters such as those in Table 4. 3 are

the same.

For these cases, different heterogeneity results in different values of τ. Therefore the

modified Buckley-Leverett solution SBLHete

and krg(SBLHete

), the slope coefficient C1Hete

as

well as the two critical numbers Ncv,c1Hete

and Ncv,c2Hete

are expected to be different for

each case. Figure 5.12 and Figure 5.13 shows the average CO2 saturations as a function

of capillary numbers for six models. We use the semi-analytical solution for the

homogeneous cores (Eq. 4.26) to predict the small degree of heterogeneity cases

(σlnk/lnkmean < 0.0455) shown in Figure 5.12 while the heterogeneous semi-analytical

solution is used to predict the large degree of heterogeneity cases (σlnk/lnkmean > 0.0455)

shown in Figure 5.13. The values of ω and ε for each case are listed in Table D.3

(Appendix D).

Figures 5.12 and 5.13 show that the 2D semi-analytical model predicts the average

81

saturation for different heterogeneous cores very well. Specifically, the average

saturation in the viscous dominated region is accurately predicted and the transitions

between different flow regimes occur at the correct capillary number. The results show

that even for different types of permeability distributions, the semi-analytical solutions

still match the simulation results well.


between homogeneous analytical predictions and simulation results for the sensitivity

cases of small degrees of heterogeneous cores.

82



degrees of heterogeneous cores.

5.5 Procedures for Using the Analytical Solutions

In this section, instructions for how to use the semi-analytical solutions for predicting

average saturations and flow regimes during homogeneous and heterogeneous core floods

are provided.

The following input parameters are required:

Porosity

Permeability

Capillary pressure curve (Eq. 3.5)

Relative permeability function (Eq. 3.6)

If your core is relatively homogeneous (σlnk/lnkmean < 0.05), the average saturation can be

predicted quite well based on the homogeneous analytical solution:

83

First determine SBL based on fractional flow curves

Calculate dimensionless numbers (Rl, Ngv, NB), two critical numbers, and

coefficients based on

2

2

CO l

gv,c1

r,CO

f RN

k (4.7)

3/2

l

gv,c2

BL

RN =α where α=12.79737

S (4.10)

l l11 11

1 1 2

B 22l l

R RC dC = , d = and d =

N dR R (4.29)

Obtain the average saturation for the homogeneous cores based on Eq. 4.26:

2l gv,c1B

2 1gv,c2 gv

2

R NN-d -d

N N

CO 1 BLS =C e -1 e +S

(4.26)

To predict the average saturation for moderately to highly heterogeneous cores:

First determine the heterogeneous factor σlnk/ln(kmean)

Calculate SBLHete

using Eq. 5.15

Hete lnk lnk

BL BL

mean mean

σ σS =0.324-0.1788 1-0.55185 S

ln k ln k

(5.15)

Once we obtain SBLHete

, we can evaluate

Hete

BL

1

BL

Sh

S (5.16)

Calculate τ based on Eq. 5.17a. For Bond numbers in the range of 0.017 to 0.17,

the average values of parameter and based on all the sensitivity studies in this

work can be estimated as 3.3 and 89, respectively. The percentage errors of

range from 2.6%-42% while the percentage errors of range from 21%-74%.

Further investigation of and on relevant dimensionless numbers such as

SBLHete

and NB are required to approximately predict the values of and .

84

lnk

2 0,D1

mean

σωω β z 1-hln k ετ=e =e

(5.17a)

Calculate dimensionless numbers (Rl, Ngv, NB) and two critical numbers based on

Eq. 5.12

2

3/2CO lHete Hete l

cv,c1 cv,c2 HeteHeteB B BLrg BL

f R R1 1N = and N = α

τN τN Sk S

(5.12c)

l lHete 11 11

1 1 2ε

B 22l l

R RC d1C = , d = and d =

N dτ R R (5.12d)

Finally obtain the average saturation for the heterogeneous cores based on Eq.

5.12a:

2 Hete2 l cv,c1

Hete 1cv,c2 cv

2


CO 1 BLHete

BL

SS C e - e +S

S

(5.12a)

In summary, we provide an analytical solution to predict the brine displacement

efficiency for both homogeneous and heterogeneous cores based on the calculated

dimensionless group. For relatively homogeneous core (σlnk/lnkmean < 0.05), the average

saturation can be predicted quite well based on the homogeneous analytical solution. For

moderately to highly heterogeneous cores (σlnk/lnkmean > 0.05), the theoretical solution is

provided and can be estimated qualitatively.

85

Chapter 6

CO2 and Brine Relative Permeability in

Heterogeneous Rocks

Motivated by the multiphase flow literature regarding CO2/brine core flood experiments,

the issues described in Section 2.4 are addressed in this chapter. This chapter also

demonstrates one of the practical applications of the numerical studies described in

Chapter 3 and the analytical solutions provided in Chapters 4 and 5. The objective in this

chapter is to investigate systematically what parameters have a significant impact on

reliable drainage relative permeability measurements on rock cores. In particular, 3D

high resolution core-scale simulations are conducted to study the independent as well as

the combined effect of flow rate, capillary pressure, gravity, and rock heterogeneity, thus

allowing identification of the operational regimes under which reliable measurements of

relative permeability can be obtained using steady-state horizontal core flood

experiments. The concept of first critical capillary number, Ncv,c1, which defines the

viscous-dominated regimes introduced in Chapters 4 and 5 will be used later in this

chapter for the discussion.

In this so-called viscous-dominated regime where the average saturation of the core is

independent of capillary or gravity number, heterogeneity results in spatially varying and

lower average CO2 saturation (Figure 5. 3) as compared to that expected for a uniform

core. Consequently, the effective relative permeability for the whole core is different

than the intrinsic relative permeability of each individual voxel in the core. Saturations in

this “viscous dominated regime” vary spatially in response to the establishment of

gravity-capillary equilibrium in the core.

Therefore, for the practical interest, the concept of effective relative permeability for

86

the heterogeneous rocks is used in this chapter, and the viscous-dominated regime is

referred to the “capillary equilibrium viscous-dominated regime” or the “quasi viscous

dominated-regime” in this work.

The overall methodology for this study is illustrated in Figure 6. 1. First, we define a

set of properties for the core, including the spatial distribution of permeability (k),

porosity (φ), capillary pressure (Pc) and relative permeability curves (kr). These are then

used as input for simulations using TOUGH2 (Pruess et al., 1999) that mimic the core-

flooding procedures used for making steady-state relative permeability measurements.

Outputs from the simulation include the pressure drop across the core in both fluid

phases, as well as, the CO2 saturation distribution. These outputs are used as synthetic

“data sets” for calculating the relative permeability of the core. The influence of

flowrate, rock heterogeneity, core length, gravity, interfacial tension on the accuracy of

the calculated relative permeability curves is systematically studied by varying these

parameters over a wide range of values. Based on the comparison between the input and

calculated relative permeability curves we draw conclusions about the important sources

of error for these calculations as well as the conditions over which accurate

measurements can be obtained.

Figure 6. 1 Overview of scientific approach.

6.1 Simulation Outputs

To determine the effect of sub-core scale heterogeneity on CO2/brine multiphase flow,

three different degrees of heterogeneities including homogeneous, low contrast (Kozeny-

Carman model) and High Contrast models are simulated. In addition, to study the flow

rate effect, the injection rate is varied from 0.1 to 10 ml/min.

The core flooding experiment conducted by Perrin and Benson (2010) is simulated

for making steady-state relative permeability measurements. For the heterogeneous

Define simulation

input parameters: φ,

k, Pc, kr

Generate steady-state

synthetic data sets

(SCO2, ΔPCO2)

Calculate relative

permeability

(Steady-State Method)

87

cores, the three-dimensional porosity map (Perrin and Benson 2010) shown in Figure 6. 2

is used to generate the corresponding permeability map with the porosity-permeability

relationships as shown in Table 6.1. The Kozeny-Carman (KC) equation generates the

low contrast permeability map while the exponential function of porosity-permeability

relation generates a higher degree of heterogeneity, called High Contrast model. The

equations for the models are shown in Table 6.1 as well as their normalized standard

deviations in σlnk/ln(kmean). The permeability of each grid element is assumed to be

isotropic. These two heterogeneous models are compared to a homogeneous one to study

the effect of heterogeneity on the multiphase flow system.

Figure 6. 2 3D porosity distribution of Berea sandstone.

Table 6. 1 Synthetic input parameters for every grid in the simulations for three different

models.

σlnk/lnk Porosity Permeability (md)

Capillary

Pressure

(Pa)

Input

Relative

Permeability

Homogeneous

Model 0

Φi = Φmean

ki =kmean

Measured

Pc Curve Eq. 3.7

Kozeny-

Carman

Model

0.05

Φi

ki Φi3/(1-Φi)

2

Pc,i

i ik

(Eq. 3.6)

Eq. 3.7

High Contrast

Model 0.17

Φi

ki exp(Φi4)

Pc,i

i ik

(Eq. 3.6)

Eq. 3.7

For a given flow rate, simulations of co-injection of CO2 and brine are run until the

pressure drop and core-averaged saturation stabilize. All of the simulations have been

88

confirmed to run long enough (more than 10 pore volumes injected) to reach steady-state.

Important output parameters include grid-cell CO2 saturations, CO2 pressures and

capillary pressures. Here we also evaluate slice averaged quantities along the length of

the core such as the slice-average CO2 saturation (SCO2), slice-average pressure in the CO2

phase (PCO2), and slice-average capillary pressure (Pc). Figure 6. 3 shows a typical

simulation result, including the CO2 saturation distribution, pressure drop across the core,

and the core-averaged CO2 saturation.

Figure 6. 3 CO2 saturation distribution at steady-state for 95% fractional flow of CO2 at a

total injection flow rate 1.2 ml/min.

The pressure drops across the core are defined as the difference between the average inlet

and the outlet slice values:

2 2 2CO CO ,inlet CO ,outletΔP =P -P (6.1a)

w w,inlet w,outletΔP =P -P (6.1b)

Since Pc= PCO2–Pw, the water pressure drop can be rewritten in terms of the two output

parameters ΔPCO2 and ΔPc:

2 2 2w CO ,inlet CO ,outlet c,inlet c,outlet CO cΔP = P -P - P -P =ΔP -ΔP (6.2)

The pressure drops in each phase are used to calculate the corresponding relative

permeability values based on the simplified Darcy’s equation, shown in Eq. 6.3. As

shown in Eq. 6.2, when Pc is the same in the first and last slice of the core, the pressure

gradient drop across the core is the same in both phases.

89

6.2 Calculation of Relative Permeability

For horizontal, 1D, immiscible, two-phase flow in homogeneous and isotropic porous

media at core scale, Darcy’s law neglecting the gravity effect takes the form:

2 2

2

2

r,CO COr,w w

w CO

w CO

kk ΔPkk ΔPq = A , q = A

μ L μ L (6.3)

where qw and qCO2 are the volumetric flow rates of brine and CO2 respectively, μw and

μCO2 are the brine and CO2 viscosities, A is the core cross section, and L is the length of

the core. In this chapter, Eq. 6.3 is used to calculate the relative permeability as a

function of the average saturation in the core since the core average saturation and the

pressure drop across the core are known. Eq. 6.3 is valid once the saturation and the

pressure gradients along the core are constants. We can expect Eq. 6.3 is no longer

correct once there is a saturation gradient and hence a capillary pressure gradient across

the core. Simulations are repeated at a number of fractional flows to construct the full

relative permeability curve.

6.2.1 Relative Permeability Calculated when ΔPw=ΔPCO2

It is often assumed that the pressure drop in both phases is equal (Geffen et al. 1951;

Avraam and Payatakes 1995), which requires that the capillary pressure is constant along

the length of the core. Moreover, it is assumed that the measured pressure drop

accurately reflects the pressure drop in at least one of the phases. In this case, the relative

permeability can be calculated from:

2 2 2

2

2

CO r,CO COr,w

w CO

w CO

ΔP kk ΔPkkq = A , q = A

μ L μ L (6.4)

In this section, the relative permeability values are calculated based on the assumption

that ΔPw=ΔPCO2 and ΔPCO2 is evaluated by Eq. 6.1a. In the subsequent discussion, for

simplicity, results are presented for a range of flow rates and for two different degrees of

heterogeneity (σlnk/lnkmean=0 and 0.17). The results are later generalized in terms of

90

capillary number (Ncv) in Section 4.1.

Homogeneous Cores

For homogeneous cores, the effect of flow rate on brine displacement efficiency has

already shown in Figure 4. 1. Figure 6. 4 illustrates CO2 saturation as a function of the

distance from the inlet at a 95% fractional flow of CO2 over a large range of flow rates

(0.1 ml/min-6 ml/min). The saturation is uniform across the core at high flow rates

where no saturation gradient exists and hence there is no capillary pressure gradient along

the core. Decreasing the flow rate below this regime leads to a saturation gradient in the

flow direction.

As mentioned in Section 3.3.1, the flow rate dependence shown in Figure 6. 4 is not

due to the traditional capillary end effect since the outlet boundary condition does not

force the two fluids to have the same pressure. The observed saturation gradients exist at

low flow rates because gravity and capillary pressure are included in the simulation.

Gravity causes some small amount of flow in the vertical direction and consequently the

saturation of CO2 is higher near the top of the core. This in turn creates higher-than-

average factional flow of CO2 near the top of the core as the fluid moves away from the

inlet boundary. The net effect is to cause a saturation gradient along the length of the

core.

Figure 6. 4 Flow rate effect on CO2 saturation along the homogeneous core at a 95%

fractional flow of CO2 with flow rates ranging from 0.1 ml/min to 6 ml/min.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

SCO

2

x

6 ml/min

3.6 ml/min

1.2 ml/min

0.6 ml/min

0.3 ml/min

0.2 ml/min

0.1 ml/min

91

Table 6. 2 Summary of different flow rates for homogeneous cores with 95% fractional

flow of CO2. Capillary numbers are calculated based on Eq. 4.1 and Eq. 4.2,

respectively.

Flow Rate

ml/min Ca

Ncv

(Ncv,c1~887)

Saturation

Gradient

∆Sg/Sg,inlet

Regime

6 1.1E-7 35.8 0% Viscous-dominated

2.6 4.7E-8 82.5 0% Viscous-dominated

1.2 2.2E-8 178.8 0.5% Viscous-dominated

0.5 9.0E-9 429 1% at around transition

0.1 1.8E-9 2145 15% Gravity-dominated

To study the flow rate effect on relative permeability, five different injection rates are

picked to obtain the corresponding relative permeability curve. Table 6. 2 summarizes

these cases, for instance, the corresponding capillary numbers, the saturation gradients

across the core, and the flow regimes.

Figure 6. 5 compares the input relative permeability curve with the relative

permeability calculated assuming the same pressure drop for two fluids using Eq. 6.4. As

shown, the calculated relative permeability is identical to the input values when the flow

rates are close or in the viscous-dominated regime, which corresponds to the negligible

saturation gradients observed in Figure 6.4 (Table 6. 2). On the other hand, a roughly

15% saturation gradient along the flow direction results in a significant deviation of

wetting phase relative permeability (0.1 ml/min). Eq. 6.4 is no longer valid for the

wetting phase since pressure drops for the two fluids are different once saturation

gradients occur. Using the pressure gradient in the CO2 phase overestimates the pressure

drop in the water phase leading to underestimation of the water-phase relative

permeability.

92

Figure 6. 5 Relative permeability calculated by the same pressure drop (ΔPw=ΔPCO2) for

homogeneous core with 430 md permeability at different flow rates.

Heterogeneous Core (High Contrast Model)

The flow rate effect on brine displacement efficiency for the High Contrast model has

already been shown in Figure 5. 5. The High Contrast model is used to study the effects

of flow rate as well as sub-core scale heterogeneity on relative permeability curves since

its permeability distribution is generated based on the measured porosity values of the

real rock and it has a moderate degree of heterogeneity. Figure 6. 6 compares the average

CO2 saturation along the length of the core between the homogeneous and the

heterogeneous cores at the same flow rates. The general trends observed in the

homogeneous core can apply to the heterogeneous one. First, the slice-averaged

saturation is relatively uniform in the high flowrate regime (q>1.2 ml/min). The source

of saturation variation along the core is due to the core heterogeneity. Second, a large

saturation gradient across the core starts to occur once the flow rate is below the viscous-

dominated regime. Comparing the homogeneous and the heterogeneous cores, it is clear

that the core heterogeneity will enhance the flow rate dependency, decrease the average

saturation, and increase the saturation gradient.

As mentioned before, although the constant Buckley-Leverett saturation or the

intrinsic relative permeability of heterogeneous cores can be obtained once we have

reached an extreme high flow rate (>100 ml/min), it is unrealistic to use such high flow

0

0.1

0.2

0.3

0.4

0.5 0.6 0.7 0.8 0.9 1

Kr

Sw

Homogeneous Core, 430md, Same Pressure Drop

kr (6)

kr (2.6)

kr (1.2)

kr (0.5)

kr(0.1)

kr_input

93

rates in the core flood experiments. Therefore, the effective relative permeability of the

heterogeneous cores is used to compare with the homogeneous results.

Figure 6. 6 The effect of heterogeneity on CO2 saturation along the core at a fractional

flow of 95% over a wide range of flow rates.

Relative permeability curves calculated using the same pressure drop for both fluids are

illustrated in Figure 6. 7. Table 6. 3 summarizes the results of these calculations. Similar

to the homogeneous results, when saturation gradients are small, the relative permeability

is independent of the flowrate (<3%), which demonstrates that the relatively uniform

slice-averaged saturation results in the rate-independent relative permeability values

(Figure 6. 7). In addition, once large saturation gradients develop (6%, 15%, and 35%),

the wetting phase relative permeability is underestimated significantly. For the same

flow rate, the heterogeneous core results in larger saturation gradients compared to the

homogeneous core. In general, the rate-independent drainage relative permeability can

be obtained even with the highly heterogeneous core once the flow rate is high enough to

eliminate saturation gradients from one end of the core to the other. It is not required that

saturation gradients are eliminated in the middle of the core – as these results from the

heterogeneity of the rock.

On the other hand, even in the viscous-dominated regime, the CO2 relative

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

CO

2 S

atu

rati

on

X

Homo: 6

Homo: 1.2

Homo: 0.5

Homo: 0.3

Homo: 0.1

Hete: 6

Hete: 2.6

Hete: 1.2

Hete: 0.5

Hete: 0.3

Hete: 0.1

94

permeability is higher than the input value. This occurs because the effective relative

permeability for the non-wetting phase of the heterogeneous core is higher than for a

homogeneous core. This is a well known phenomenon as described by Corey and

Rathjens (1956) and Honarpour et al. (1994).

Figure 6. 7 Relative permeability calculated by the same pressure drop (ΔPw=ΔPCO2) for

heterogeneous core with 318 md permeability at different flow rates. Small picture

shows the same relative permeability curves at log scale.

Table 6. 3 Summary of different flow rates for heterogeneous cores (High Contrast

model: σlnk/lnkmean=0.168) with 95% fractional flow of CO2.

Flow Rate

ml/min Ca

Ncv

(Ncv,c1~80)

Saturation

Gradient

∆Sg/Sg,inlet

Regime

6 1.1E-7 30.7 ~0% Viscous-dominated

2.6 4.7E-8 70.9 ~3% at around transition

1.2 2.2E-8 154 ~6% Gravity-dominated



6.2.2 Relative Permeability Calculated by True Pressure Drops

(ΔPw=ΔPCO2–ΔPc)

As discussed in Sect. 6.1, both the average CO2 pressure and the average capillary

pressure along the flow direction at steady-state are known as simulation outputs, hence

95

the pressure drops of both the CO2 and water phases along the flow direction can be

determined accurately based on Eq. 6.1a and Eq. 6.2. The relative permeabilities now

can be calculated based on the following equations:

2 2 2

2

2

CO c r,CO COr,w

w CO

w CO

ΔP ΔP kk ΔPkkq = A , q = A

μ L μ L

(6.5)

Figure 6. 8 shows the drainage relative permeability data calculated based on the true

pressure drops in each phase over the same range of flowrates as shown in Figure 6. 5

and Figure 6. 7. Once the true pressure drop of the wetting phase is known and used in

the calculation, the wetting phase relative permeability collapses very nicely even with

15% saturation gradient along the core (0.1 ml/min for the homogeneous core and 0.5

ml/min for the High Contrast model). Injecting lower flow rates can still yield accurate

results as long as the true pressure drops of water and CO2 are taken into account.

Figure 6. 8 Flow rate effect on relative permeability calculated by the true pressure drops

(ΔPw ΔPCO2) for homogeneous and heterogeneous core at various flow rates.

Note that the flow rate effect is larger in the heterogeneous core, and therefore including

the capillary pressure drop when calculating relative permeability to water is not as

effective as for the homogeneous core. This method only improves the calculated

relative permeability to water over a smaller range of flow rates. Second, the effective

relative permeability to non-wetting phase is higher than for input relative permeability

curve for each grid cell even flow rate is in the viscous-dominated regime.

As shown above, to obtain more accurate relative permeability at lower flow rates,

capillary pressure gradient (ΔPc) needs to be included in the calculation (Eq. 6.5). This

96

concept may apply to core flood experiments and give us a more reliable relative

permeability. However, capillary pressure gradients in general are not measured in the

experiment. It is possible to estimate capillary pressure gradients based on the average

saturation values at the inlet and outlet slices of the core. Once saturations at the ends of

the core are measured (e.g. using X-Ray CT scanning), the corresponding capillary

pressure values can be estimated from independently measured capillary pressure curves:

c,inlet c inlet c,outlet c outlet P =P S , P =P S (6.6)

Therefore

c c inlet c outletΔP =P S P S (6.7)

Comparing to the relative permeability calculated using the same pressure drop for both

fluids (Figure 6.7 or LHS of Figure 6.9), it is observed that including this corrected

capillary pressure drop in the pressure drop of water, the accuracy of relative

permeability to water for the heterogeneous core at 1.2, 0.5, and 0.1 ml/min flowrates can

be increased (Figure 6.9). The deviations for these lower flowrate cases are smaller if the

corrected capillary pressure drop is considered. Once the saturations at the ends of the

core are known, the application of this methodology can improve the accuracy of the

measured relative permeability curves even at lower flowrates and in the presence of

saturation gradients.

Figure 6. 9 (LHS) Relative permeability calculated using the same pressure drop for both

fluids (Figure 6.7); (RHS) relative permeability calculated by the corrected pressure

drops for High Contrast models at various flow rates.

0

0.1

0.2

0.3

0.4

0.5 0.6 0.7 0.8 0.9 1

Kr

Sw

High Contrast Model, 318md, Corrected Pressure Drop

kr (6)

kr (2.6)

kr (1.2)_Correction

kr (0.5)_Correction

kr (0.1)_Correction

kr_input

97

6.3 Sensitivity Studies for Different Core

Properties

In Chapters 4 and 5, we investigated the sensitivity of the average saturation of the core

over a wide range of sub-core scale heterogeneity, core dimensions, interfacial tension

and absolute permeability. In the rest of this chapter, we provide support for the

conclusion that the effective relative permeability of a core can be measured accurately in

the viscous-dominated regime or even in the gravity-dominated regime, if the correct

pressure gradients are used in each phase. The results will be combined with information

derived from the 2D semi-analytical solutions to establish a “rules of thumb” for making

accurate effective relative permeability measurements even in heterogeneous cores using

horizontal core floods.

6.3.1 Effects of Heterogeneity

The core-averaged saturations with different degrees of heterogeneity have already been

illustrated in Figure 5. 3 with capillary numbers Ncv ranging from 10 to 105. It can be

easily concluded from the figure that once the capillary number is small enough to be in

the viscous-dominated regime, the brine displacement efficiency is independent of flow

rate, and is expected to have a nearly uniform slice-averaged saturation profile even with

large capillary heterogeneity (Figure 5. 1 and Figure 5. 2). Based on the previous results,

it is reasonable to hypothesize that reliable relative permeability can be obtained as long

as the saturation is relatively uniform.

To validate this conclusion, we use the homogeneous core and the four heterogeneous

models to obtain the relative permeability curves calculated based on the true pressure

drops in each phase (Figure 6. 10). The injection flow rates required for the cores to

reach the viscous dominated regime depend on the degree and nature of heterogeneity.

For example, the higher flowrate or the smaller Ncv is required to reach viscous-

dominated regime for the higher degree of heterogeneity (larger σlnk/lnkmean). Table 6.4

lists all the relevant details of these five cases.

The effective relative permeability depends on the degree of heterogeneity. Figure 6.

98

10 illustrates that the effective relative permeability to water krw is not as sensitive as the

krg to the small scale heterogeneity, for example, the water relative permeability krw is

almost identical when the heterogeneity factor σlnk/lnkmean < 0.17. The effective relative

permeability to the gas phase calculated based on Darcy’s law is larger for more

heterogeneous cores while the effective water relative permeability is smaller for the

largest degree of heterogeneity (Random3).

Table 6. 4 Summary of different flow rates for different heterogeneous cores:

Homogeneous, Random 2, Kozeny-Carman, High Contrast and Random 3 models.

q,

ml/min σlnk/lnkmean Ca Ncv Regime

Homo 0.5 0 9.0E-9 429 Viscous-dominated

Random2 0.5 0.0419 9.0E-9 429 Viscous-dominated

KC 1.2 0.0455 2.2E-8 180 Viscous-dominated

HC 2.6 0.1680 4.7E-8 70 Viscous-dominated

Random3 6 0.2343 1.1E-7 33 Viscous-dominated

Figure 6. 10: Relative permeability calculated by the true pressure drops for five different

heterogeneous cores in the viscous-dominated regimes: homogeneous and the Random 2

cores (q=0.5 ml/min), Kozeny-Carman models (q=1.2 ml/min), High Contrast models

(q=2.6 ml/min), the Random 3 cores (q=6 ml/min) and the input relative permeability

curves.

0.001

0.01

0.1

1

0.5 0.6 0.7 0.8 0.9 1

Kr

Sw

True Pressure Drop

σ/lnk=0

σ/lnk=0.04

σ/lnk=0.05

σ/lnk=0.17

σ/lnk=0.23

kr_input

99

6.3.2 Effects of Core Length (15.24-45.72 cm)

In order to assess whether or not the flow rate dependency observed in the previous

results depends on the length of the core, the average CO2 saturations as a function of

capillary numbers for the three different core lengths are shown in the LHS of Figure 6.

11. The aspect ratios Rl are 3.14, 6.29, and 9.43 respectively. The RHS illustrates the

corresponding relative permeability with 0.1 ml/min injection flow rate. Table 6.5

summarizes the information of these three cases. Simulation results show that even with

up to 15% saturation gradient, we can still obtain the intrinsic relative permeability for

different lengths of homogeneous cores.

Figure 6. 11 (LHS) Brine displacement efficiencies for three different lengths of

homogeneous core with capillary number ranging from 10 to 107; (RHS) Relative

permeability calculated by the true pressure drops for homogeneous cores at 0.1 ml/min

flow rates.

Table 6. 5 Summary of different lengths of heterogeneous cores (High Contrast model:

σlnk/lnkmean=0.168).

Homo Flow Rate

ml/min Ca Ncv

Saturation

Gradient

∆Sg/Sg,inlet

Regime

L 0.1 1.8E-9 2145 15% Gravity-dominated

2L 0.1 1.8E-9 4290 10% Gravity-dominated

3L 0.1 1.8E-9 6435 7% Gravity-dominated

6.3.3 Effects of Absolute Permeability (31.8-3180 md)

Figure 6. 12 (RHS) is an example of relative permeability calculated from the true

pressure drops at 6 ml/min flow rate for the highly heterogeneous cores with 31.8, 318,

and 3180 md permeabilities. The corresponding capillary numbers Ncv are around 9.72,

100

30.7, and 97.2 respectively. The brine displacements for 31.8 md and 318 md cases are

in the viscous-dominated regime while the displacement for 3180 md case slightly

deviates from the viscous-dominated regime (shown in the LHS of Figure 6. 12). The

heterogeneity not only increases the flow rate dependency but also reduces the sensitivity

of average saturation on permeability.

Nevertheless, the relative permeability sets for the three different permeability cores

are expected to be almost identical since their saturation gradients are small. Figure 6. 12

(RHS) shows that reliable relative permeability data can be obtained even for the

heterogeneous cores with a wide range of core permeability once the displacement is

within or near the viscous-dominated regime.

Figure 6. 12 (LHS) Brine displacement efficiencies for three different permeability values

with capillary number Ncv ranging from 10 to 105; (RHS) Permeability effects on relative

permeability calculated in the true pressure drops for heterogeneous core (High Contrast

Model) at 6 ml/min flow rates.

6.3.4 Effects of Interfacial Tension (7.49-67.41 mN/m)

Different pressure, temperature and water salinity will result in different interfacial

tension. The effect of interfacial tension on relative permeability of CO2/brine systems

has been reported in the literature (Fulcher et al. 1985; Bachu and Bennion, 2007).

Therefore, the sensitivity study of interfacial tension over the same range of capillary

number has also been simulated. As we mentioned earlier, the variation of interfacial

tension does not account for varying temperature and pressure conditions. The wide

0.001

0.01

0.1

1

0.5 0.6 0.7 0.8 0.9 1

Kr

Sw

True Pressure Drop

HC(6):10K

HC(6):K

HC(6):0.1K

kr_input

101

range of IFT values is purely hypothetical and used solely to explore the sensitivity of the

IFT values.

Figure 6. 13 (LHS) compares the average saturation of three different interfacial

tensions (7.49 mN/m, 22.47 mN/m, and 67.41 mN/m) between the homogeneous model

and the High Contrast model. It is observed that the range of interfacial tension effects

on the average saturation is smaller for the large degree of heterogeneity. If the core

sample is very heterogeneous, then it is always safer to use the smaller capillary numbers

Ncv or larger flow rates to enter the viscous-dominated regime.

Figure 6. 13 (RHS) is an example of relative permeability calculated from the true

pressure drops at 6 ml/min flow rate for the highly heterogeneous cores (High Contrast

Model) with 7.49, 22.47, and 67.41 mN/m IFT values. The corresponding capillary

numbers Ncv are around 10, 30.7, and 92 respectively. The brine displacements for 7.49

and 22.47 mN/m cases are in the viscous-dominated regime while the displacement for

67.41 mN/m case slightly deviates from the viscous-dominated regime. Similarly, the

relative permeability sets for the three different interfacial tension values are expected to

be almost identical since their saturation gradients are small. Reliable relative

permeability data can be obtained even for the heterogeneous cores with a wide range of

IFT values once the displacement is within or near the viscous-dominated regime.

Figure 6. 13 (LHS) Average CO2 saturation as a function of capillary number Ncv for

homogeneous and High Contrast models with three different values of interfacial

tensions; (RHS) Interfacial tension effects on relative permeability calculated in the true

pressure drops for heterogeneous core (High Contrast Model) at 6 ml/min flow rates.

0.001

0.01

0.1

1

0.5 0.6 0.7 0.8 0.9 1

Kr

Sw

True Pressure Drop

HC(6):σ/3

HC(6):σ

HC(6):3σ

kr_input

102

6.3.5 Effects of Gravity

Figure 6. 14 shows a sensitivity study on the effect of gravity for the homogeneous and

the two heterogeneous cores. The dashed lines represent simulations without considering

gravity (setting g=0) while the solid lines represent the simulations with gravity (setting

g=9.8 m/s2). It is shown that gravity override is eliminated even with horizontal

displacements when the capillary numbers Ncv are smaller than the critical values.

Furthermore it is shown that moderately heterogeneous cores are only minimally affected

by horizontal displacements. It is verified that the effect of gravity due to the density

difference between two fluids and the long core is small in the viscous-dominated regime,

as mentioned before. However, even without considering gravity in the simulation, flow

rate dependency is observed in the heterogeneous cores. In this condition, only capillary

force and viscous forces are competing with each other in our system. The transition is

much more abrupt than when gravity is included (Figure 6.14).

Figure 6. 14 Average CO2 saturation as a function of capillary number Ncv for

homogeneous, Kozeny-Carman (small heterogeneity) and high contrast (large

heterogeneity) models with and without gravity (1G/0G).

103

6.4 Discussion of Relative Permeability

Measurements

6.4.1 Observations from the Numerical and Semi-Analytical Models

Numerical simulations and semi-analytical studies of brine displacement efficiency in

homogeneous and heterogeneous cores have been presented in Chapter 4 and Chapter 5

respectively. The critical gravity numbers Ngv,c1 and Ngv,c2 define the transitions between

flow regimes, first from the viscous to gravity dominated regime and next, from the

gravity to capillary dominated regime. The different flow regimes are observed for both

the homogeneous and the heterogeneous models. Simulation results show that when the

capillary number Ncv is small enough (below the critical value), viscous forces dominate

in this regime. It is clearly shown in these chapters is that in the viscous-dominated

regime, the viscous force is much greater than gravity and capillary forces. Gravity

segregation can be neglected in this regime even with horizontal core flooding.

Consequently, the calculated effective relative permeability is independent of flow rate,

gravity and Bond number in this regime.

In the gravity-dominated regime, buoyancy of CO2 causes lower displacement

efficiency and results in a vertical saturation gradient, which leads to the deviations of

relative permeability values observed in Figure 6. 5 and Figure 6. 7. In this regime,

gravity not only causes the inaccuracy of relative permeability values but also results in

large flow rate dependency. Capillary heterogeneity will increase this flow rate

dependency.

For the cores with high permeability, low interfacial tension or a smaller degree of

heterogeneity, the two-phase flow displacement will encounter a stronger gravity effect

while the gravity effect is irrelevant for the cores with low permeability, high interfacial

tension or larger degree of heterogeneity. In addition, the highly heterogeneous cores

require the smaller Ncv to reach viscous-dominated regime and hence to obtain the

reliable relative permeability data (Table 6.6).

104

Table 6. 6 Summary of the first critical capillary number Ncv,c1 for different heterogeneity

cores.

Random 3 High Contrast Homogeneous

σlnk/lnkmean 0.234 0.168 0

Ncv,c1 (L) 168 222 887

Ncv,c1 (2L) - 419 1774

It is consistent with previous studies that indicate that once a saturation gradient develops

along the core, the relative permeability calculated based on Darcy’s law is no longer

valid (Avraam and Payatakes 1995). However, there are several methods to obtain the

reliable relative permeability. First, we can increase flow rates to minimize the saturation

gradients. Second, we can use true pressure drops for two fluids to get more reliable

relative permeability values even with 15% saturation gradient exists. Finally, if the

saturations at the inlet and outlet are known, we can increase the accuracy of relative

permeability to water by including the corresponding capillary pressure drop.

6.4.2 General Rule of Thumb for Reliable Relative Permeability

Measurements

In Chapter 4, it was shown that when the effect of gravity is important for the multiphase

flow system, we should use gravity number Ngv (Eq. 4.3) to non-dimensionalize the

saturation data, for example, for the homogeneous and mildly heterogeneous cores. On

the other hand, when the capillary heterogeneity is taken into account, the impact of

gravity is much smaller and capillary number Ncv (Eq. 4.2) is a better dimensionless

number to characterize our system (Chapter 5). The advantage of using appropriate

dimensionless numbers can be easily seen from Figure 6.16.

Since the critical capillary number depends on the rock heterogeneity, the higher the

degree of heterogeneity in the core, the smaller the capillary number Ncv required to get

the reliable relative permeability data.

105

Figure 6. 15 Average CO2 saturation as a function of alternative capillary number Ncv,

and alternative gravity number Ngv for homogeneous and High Contrast models.

In principle, we can calculate the first critical number based on Eq. 5.12c

2CO lHete

cv,c1 HeteB rg BL

f R1N =

τN k S

(5.12c)

where the Bond number is defined in Eq. 4.4

gv

B *

cv c

N ΔρgHN =

N p (4.4)

pc* is the characteristic capillary pressure of the medium, chosen as a so-called

displacement capillary pressure. The displacement capillary pressure is a capillary

pressure value at the brine saturation Sw equal to 1, and it is tangent to the major part of

the capillary pressure data. The pc* value for our core is about 3000 Pa, shown in Figure

6. 16.

Figure 6. 16 Laboratory capillary pressure data.

106

However, during the core-flood experiment, relative permeability evaluated at SBLHete

is

unavailable before we perform the relative permeability measurements. The parameter τ

is also dependent on σlnk/ln(kmean), which required detailed information on the rock.

Therefore, another practical method is proposed based on the simulation results for the

heterogeneous models. According to the simulation results (Figure 5.3, Figure 6.13 and

Figure 6.15), if the critical capillary number Ncv,c1 is chosen to be 15, relatively uniform

saturation profiles will be obtained for the most types of heterogeneity cores

(σlnk/ln(kmean)<0.5).

If the core is known to be a relatively homogeneous, a larger critical number can

result in the viscous-dominated regime, where we can expect to get a uniform saturation

and an intrinsic relative permeability data. The more information we know about the

core, the larger the Ncv (smaller flow rate) can be tolerated to obtain reliable relative

permeability data.

In summary, the accurate whole-core relative permeability measurements can be

achieved when

2

*

c

cv 2

Tco

kLp AN = 15

qH μ

(6.17)

which assures that the injection flowrate is in the viscous-dominated regime for a wide

range of heterogeneity with the aspect ratio Rl equals to 3.1445. Although additional

work is needed to establish the bounds for reliable relative permeability measurements

for a wide range of Rl, most of the laboratory cores have similar size of Rl, so it is

reasonable to use Eq. 6.17 to predict the first critical number qualitatively.

A hypothesis based on analytical and numerical results in Chapter 4, 5 and 6 is

provided as follows: reliable relative permeability measurements can be achieved once

we satisfy the conditions below. First, if the core is known as relatively homogeneous:

2

2

*CO lmean c

cv cv,c12

T B rg BLco

f Rk Lp A 1N = N

q N k SH μ

(6.18)

Second, if it is very heterogeneous:

107

2CO lHete

cv cv,c1 HeteB rg BL

f R1N N =

τN k S

(6.19)

For practical interest, we establish the relation of parameter τ in terms of the normalized

standard deviation σlnk/ln(kmean) based on all the sensitivity cases shown in this work

(Figure 6.17):

lnk

mean

στ 1+35

ln k

(6.20)

Eq. 6. 20 gives us an approximate estimation of τ, which can be used to estimate the first

critical number roughly based on Eq. 6.19.

Figure 6. 17 Heterogeneous parameter τ in terms of normalized standard deviation.

As mentioned, krg(SBLHete

) is unavailable before we perform the relative permeability

measurements, an iterative approach can be used to obtain a reliable input relative

permeability values by checking that Eqs. 6.18 and 6.19 are satisfied after the initial

measurements are made. However, Eq. 6.17 is an empirical approach and only suitable

for the aspect ratio Rl=3.1445. Finally, if there is no information about the core

properties, measurements should be made for a capillary number as small as possible (Eq.

6.17). Therefore, further investigation on the aspect ratio sensitivity studies for different

degrees of heterogeneity is required to generalize the results to a wide range of

conditions.

108

Chapter 7

Conclusions and Future Work

This dissertation addressed fundamental studies of multiphase flow of CO2 and brine in

heterogeneous porous media at the core-scale both numerically and analytically. The

combined influence of gravity, flow rate and small scale heterogeneity on core-scale

multiphase flow of CO2 and brine is an active and important research area due to the need

for storing CO2 emissions in deep saline aquifers. We now summarize this work and

make recommendations for future research directions.

7.1 Summary and Conclusions of the Present

Work

In this work, we developed and investigated new analytical techniques to study the

balance of three forces as well as the sub-core heterogeneity in multiphase flow system.

The following conclusions can be drawn from this study of the semi-analytical models:

1. A new semi-analytical solution has been developed to predict the influence of

gravity and capillary numbers on the average saturation expected during

multiphase flow experiments. The new solution provided here is a quick and easy

way to estimate the flow regimes for horizontal core floods. A general solution

and a specific solution at core-scale were provided for further development and

investigation.

2. Having a semi-analytical solution provides a useful tool for investigating

multiphase fluid displacement efficiency over a wide parameter space of practical

interest. The new semi-analytical solution can be used to estimate the average

109

saturation over a wide range of conditions in terms of several important

dimensionless numbers such as mobility ratio, average saturations in the viscous-

dominated regime (SBLHere

), relative permeability to gas evaluated at SBLHere

,

normalized standard deviation (σlnk/lnkmean), aspect ratio Rl, Bond number NB and

capillary number Ncv.

3. A summary of how to use analytical solution was provided. The semi-analytical

solution can be used to predict displacements in cores with small degrees of

heterogeneity.

4. For cores with a greater degree of heterogeneity, a modification of the solution is

provided that is capable of predicting average saturations even with a high degree

of heterogeneity.

5. Practical applications include helping to design core flood experiments, including

assuring that relative permeability measurements are made in the viscous

dominated regime, evaluating potential flow rate dependence, influence of core-

dimension on a multiphase flow experiments, influence of fluid properties on the

experiments, and influence of sub-core scale heterogeneity.

Potential applications of the analytical results include establishing the bounds over which

relative permeability can be accurately measured in horizontal core-flood experiments.

In this work, we investigated sensitivity studies on volume-averaged (up-scaled) relative

permeability that accounts for the role of sub-core scale heterogeneity on multi-phase

flow in CO2/brine systems. We now summarize this part of the work:

1. Increasing flow rate results in minimizing the saturation gradient caused by the

combined effects of capillary, viscous and gravity forces; hence, the relative

permeability approaches the maximum value asymptotically and stabilizes when

the uniform saturation is achieved.

2. Simulation results shown here indicate that the flow-rate dependent saturation

occur not only in the heterogeneous core but also in homogeneous cores. In

addition, we show that the heterogeneity will increase the flow-rate dependency,

but this flow-rate dependent behavior is mainly due to the complex interaction

between gravity, viscous, and capillary forces.

110

3. Correct procedures for calculating relative permeability are discussed.

4. Despite the complexity of heterogeneity, it is possible to obtain the accurate

relative permeability measurements for heterogeneous cores. The incomplete

fluid displacement is primarily due to the heterogeneity and unfavorable mobility

ratio, not gravity segregation. A flow rate-independent relative permeability can

be obtained even with the horizontal core orientation as long as the alternative

capillary number is small enough, in most of the cases, smaller than 15.

5. If the permeability of the core or the core heterogeneity is unknown, it is always

safer to choose a smaller alternative capillary number in order to get the rate-

independent relative permeability.

7.2 Directions for Future Research

Some possibilities and suggestions for improving the present study as well as for future

research directions are outlined below.

Accurate Critical Numbers: From the homogeneous studies, we can determine the

two critical numbers by comparing the pressure gradients associated with the

viscous, gravity and capillary forces as a function of gravity numbers. However,

it is not obvious how to determine the first critical number of the heterogeneous

cores using this methodology because the viscous pressure gradients are always

greater than the buoyancy pressure gradients (Figure 5.5). Based on Figure 6.13,

we can clearly identify the gravity-dominated regime by comparing simulations

with and without gravity. This approach can identify flow regimes more

accurately for the heterogeneous cores. Further investigation on different degrees

of heterogeneity is needed to give us insight into the two critical numbers.

Derive Fitting Parameters: It is difficult but also important to derive several fitting

parameters from fundamental principles to increase the robustness of the model.

Fourth Flow Regime: On the other hand, Figure 6.13 also shows a very interesting

result for the homogeneous cores. Even without including the gravity effect in the

111

simulation, a slight flow rate dependency occurs when Ncv > 4x105. This could be

evidence for a fourth flow regime or it might simply be due to the implemented

boundary condition. If this additional flow regime indeed exists, it is necessary to

include this in the analytical analysis. Further investigation is needed to resolve

this issue.

Correlation lengths: The heterogeneous porous medium created in this work is

either based on a one specific rock or using a random log-normal distribution. A

systematic study on cores with various correlation lengths can generalize and

extend the analytical solution to a wider range of rock types.

Anisotropic Porous Medium: It is important and realistic to study the influence of

anisotropic properties in both numerical simulations and the semi-analytical

solutions. The same strategy used to obtain the semi-analytical solution could be

applied provided constraints are provided by the simulations.

Different Aspect Ratios in Heterogeneous Porous Medium: Although

homogeneous semi-analytical solution predicts numerical results very well over a

wide range of aspect ratios, validation of heterogeneous semi-analytical solution

with different core dimensions is important and needed to test the 2D

heterogeneous model.

Upscaling: Investigation of upscaling strategies in the transition between the

viscous, gravity and capillary dominated regimes is very useful for increasing the

computational speed of high-resolution reservoir simulations. Additionally, the

results have relevance for understanding reservoir-scale processes, particularly at

the sub-grid scale, where intra-grid block processes may have an influence on

flow and transport parameterizations.

Compressibility: The effect of compressibility is included in the simulations.

Considering the compressibility in the analytical model may improve the

predictions.

Experimental Evaluation at Core Scale: Applying the semi-analytical solution to

the experimental systems for studying multiphase flows for a wide range of fluid

pairs, geometric configurations and rock properties is another way to test the

model.

112

Predict Pressure Drops: it will be very useful to generalize the semi-analytical

solution to the predict pressure gradients across the core. Once we can also

predict the pressure drops analytically, relative permeability can be calculated for

heterogeneous rocks based on the average saturation and the corresponding

pressure drop across the core.

The results shown in this work were mostly achieved with a specific core-shaped

geometry and a limited number of permeability distribution. The limitations inherent in

the results are discussed in the following. First, if there are two systems with the same

aspect ratio but the physical dimensions of the porous medium are meters instead of

centimeters, the dimensionless plots which show the saturations are expected to be still

valid. However, this needs to be confirmed with the large scale simulations.

Additionally, specific distributions of permeability and a specific measure of

heterogeneity are used in this study. The results from our work should be qualitatively

applied to a system which is encountered different heterogeneity than what examined

here. Also, to quantitatively predict the average saturation with different heterogeneity

distributions, a systematic investigation is needed to generalize results.

The most sensitive inputs of the results are the second critical number due to the

empirical definition obtained from the sensitivity studies. Although this definition works

reasonably well for most of the sensitivity studies, it is not as accurate as the first critical

number; therefore the errors will propagate from the homogeneous cores to the

heterogeneous cores.

The conclusions drawn from the relative permeability studies should apply to

unsteady-state relative permeability measurements, but further studies are needed to

confirm this. On the other hand, note that this work cannot be applied to the miscible

multiphase flow systems since all the simulations and all the theoretical work consider

the capillary pressure. In addition, the results shown here have only studied the physical

behavior of multiphase flow system, there is no chemical reactions considered in the

simulation or the theoretical analysis. Therefore, the results drawn from here may not

apply to carbonate systems, which have significant chemical reactions between the fluids

and rock.

113

Appendix A

2D Analytical Derivation for the

Homogeneous Model

A.1 Detail of 2D General Solution

2D mass conservation equations for incompressible flow with constant porosity and

permeability are:

j j j,x j,z

j

S S u uφ u =φ =0

t t x z

(4.11)

Pressure equation can be obtained with the condition Sw+2COS =1:

w,x g,x w,z g,z

t

u +u u +uu 0

x z

(4.12)

Therefore uw,x+ug,x=constant1 and uw,z+ug,z=constant2 where uj,x and uj,z are Darcy

velocity of phase j in the x and z direction, respectively. Darcy flow velocities for both

phases are given by

j j rj

j j j j,x j,z j

j

p p ku =-λ k , +ρ g u , u where λ =

x z μ

(4.13)

where uj, λj, pj, ρj and μj are Darcy flow velocity, relative mobility, pressure, density, and

viscosity of phase j, respectively. At the top and the bottom boundaries, uw,z=ug,z=0

hence results constan2=0. The total velocity in the z direction is vanished, therefore

uw,z=-ug,z. In the flow direction x, total volumetric flow rate is sum of water and gas

114

flow, uw,x+ug,x=ut. We can obtain the capillary pressure in the x and z directions in terms

of Darcy’s flow velocities:

w

w,x wg w g,xw,xc

g w g

g,x g

pu =-λ k

p -p uupx= -

p x x kλ kλu =-λ k

x

(A-1a)

w

w,z w w

g w g,zw,zc

w g

g w g

g,z g g

pu =-λ k +ρ g

p -pz uup= - + ρ -ρ g

p z z kλ kλu =-λ k +ρ g

z

(A-1b)

Apply two boundary conditions and rearrange we can obtain

g gt c c

g,x g,z

w

Mkλ Mkλq p pu = - and u = Δρg-

Akλ x 1+M z 1+M

(A-2)

Therefore we can solve velocity of gas phase ug

g t c c

g g,x g,z

w

Mkλ q p pˆ ˆ ˆ ˆu =u x+u z= - x+ Δρg- z

1+M Akλ x z

(A-3)

where M= λw/λg is mobility ratio

w rw w

w g

g rg g

λ k /μM= = , Δρ ρ -ρ

λ k /μ (A-4)

The mass conservation of gas phase now becomes:

g g gt c c

w



(4.16)

To non-dimensionlise the equation, we define xD=x/L, zD=z/H, tD=tut/φL, and pc(Sw)

=pc*J (Sw). Substituting all the defined terms into Eq. 4.16 yields the dimensionless mass

conservation equation for gas phase is

2

g rg rg rg

gv cv

D D D l D D D D

S Mk Mk Mk1 1 J J+ +N -N + =0

t x 1+M z 1+M R x x 1+M z z 1+M

(4.17)

where Ncv and Ngv are shown in Eq. 4.2 and Eq. 4.3, and Rl =L/H is the shape factor or

115

the so-called the aspect ratio. Since M, J, and krg all depend on saturations, we can

rewrite the dimensionless equation as

3 31 2

g g rg g rg g rg gcv

gv cv2

D g D g D D w D D w Dl

F FF F

S S Mk S Mk S Mk SNd 1 d dJ dJ+ +N + +N =0

t dS 1+M x dS 1+M z x dS 1+M x z dS 1+M zR

(A-5)

Or a more compact form

g g g g gcv

1 3 gv 2 cv 32

D D D D D D Dl

S S S S SN+F + F +N F +N F =0

t x x x z z zR

(A-6)

where F1, F2, and F3 are functions of saturations. At steady-state,

g

g D D

D

S=0 S SG x ,z

t

(A-7)

Assume Ncv and Ngv are independent of xD and zD, and then we can rewrite equations in

terms of xD and zD separately:

cv 3 cv 3

1 2 gv2

D 1 D D gv 2 Dl

N F N FSG SGF SG+ +F N SG+ =0

x F x z N F zR

(A-8)

Since xD and zD are independent, we assume that the dependence of the steady-state

solution SG on xD and zD is separated, that is:

D D D DSG x ,z =X(x )Z(z ) (A-9)

Substituting saturation back into the equation, and defining a=F1/F3, b=F2/F3, and Bond

number

gv

B *

cv c

N ΔρgHN =

N p (4.4)

Then Eq. (A-8) becomes

2 gvcv1

2

D D D B Dl

F NNF d 1 dX d 1 1 dZX+ + Z+ =0

X dx a dx Z dz N b dzR

(A-10)

116

To find a solution that satisfies Eq. (A-10), then

2l

Dcv

B D

cvR a1 D2 - xNDl

D 1 D

-N bz

D 2 D2 D

B D

N dΧΧ+ =const c z

dxR aΧ=A z e +c z

1 dΖΖ=B x e +c xΖ+ =const c x

N b dz

(A-11)

Substituting Eq. (A-11) back to Eq. (A-9) yields

2 2l l

D B D Dcv cv B D

R a R a- x -N bz - x

N N -N bz

1 D D 2 D D 3 D D 4 D DSG=C x , z e +C x , z e +C x , z e +C x , z (A-12)

where C1, C2, C3 and C4 are functions of xD and zD.

A.2 Derivation of Parameters a and b

From Eq. (A-5), F1, F2, and F3 are defined as

1 g 2

g g

d 1 1 dMF S = =-

dS 1+M dS1+M

(A-13a)

rg rg rg

2 g 2

g g g

Mk k dkd dM MF S = =

dS 1+M dS 1+M dS1+M

(A-13b)

rg rgw

3 g

w

Mk MkdJ SF S = =J'

dS 1+M 1+M (A-13c)

Assuming the derivative of relative permeability is proportional to itself,

rgrw

rw w rw rg g rg

w w

dkdkk '= =a k and k '= =-a k

dS dS (A-14)

Using different forms of relative permeability results in different aw and ag. In this paper,

aw and ag are as follows:

gw

w g

w wr wr

nna = and a =

S -S 1-S (A-15)

117

Since the mobility ratio M= λw/λg = μg krw/μw krg, the derivative of M with respect to Sw is

g -1 -2

rw rg rw rg rg w g

w w

μdMM'= = k 'k -k k k ' = a +a M

dS μ (A-16)

Substituting Eq. (A-14) and (A-16) back into (A-13) yields

1 g w g2 2

g g

d 1 1 dM MF S = =- = a +a

dS 1+M dS1+M 1+M

(A-17a)

rg g w rg

2 g

g

Mk Ma -a MkdF S = =

dS 1+M 1+M 1+M

(A-17b)

Therefore, we can derive the two variables a and b based on the definitions in Appendix

A.1:

w g1

3 rg

a +aF 1a= =

F J'k 1+M (A-18a)

g w2

3

a M-aF 1b= =

F J' 1+M (A-18b)

Since fCO2=1/(1+M), then a and b becomes

2

2

w g CO g w

CO

rg

a +a f a M-aa and b f

J' k J'

(A-19)

Therefore a and b are proportional to fCO2/krg and fCO2, respectively.

118

Appendix B

2D Analytical Derivation for the

Heterogeneous Model

B.1 General Solution

Substituting xD=x/L, zD=z/H, tD=tut/φL, τ(xD, zD)=sqrt(k/kmean) and pc(Sw) =pc*J (Sw)/τ

into Eq.(5.3), the dimensionless mass conservation equation at steady-state becomes

2

2 2 2cv

CO gv w rg w rg cv w rg2

D D D D D Dl

N J τ J τf +N f k τ = f k τ +N f k τ

x z x x z zR

(B-1)

Assuming all the variables are continuously differentiable. Differentiate the capillary

terms

2 2

D D D D D D

J/τ SG J' J τ J/τ SG J' J τ and

x x τ x z τ zτ τz

(B-2)

where J’=dJ/dSw. Substituting Eq. (B-2) into (B-1) and using the definition of Ms=fwkrg,

H=-fwkrgJ’=-MsJ’ and G=fwkrgJ= MsJ, then we can rewrite the steady-state mass balance

as follows, which is already shown in Eq. (5.6):

2

2cv cv

CO cv B rg w cv2 2

D D D D D D D Dl l

N Nτ τ SG SGf G +N N k f τ +G = Hτ +N Hτ

x x z z x x z zR R

(5.6)

The gradients of heterogeneity are defined as follows:

119

x z

D D

τ ττ and τ

x z

(B-3)

Substituting back into Eq.(5.6), and the left hand side becomes

2

2cv

CO x cv B S z2

D Dl

NSG d SG dLHS= f + Gτ +N N M τ Gτ

x dSG dSGR z

(B-4)

Apply the definition of F1, F2, and F3 used in Kuo and Benson (2012) and also provided

in Appendix A.2, then

2cv

1 x cv B 2 z2

D Dl

NSG d SG dLHS= F + G τ +N N F τ G τ

x dSG dSGR z

(B-5)

Differentiate the function G

S S 2 2

d dG= JM M J' JF H JF

dSG dSG (B-6)

Combine LHS and RHS of Eq.(5.6) into x-dependent and z-dependent terms:

2

2cv l

1 2 x cv B 2 2 z2

D cv D D D D Dl

N RSG SG SG SGF + H JF τ - Hτ +N N F τ H JF τ - Hτ 0

x N x x zR z z

(B-7)

Define

2

l

1 1 2 x

cv

RG = F + H JF τ

N (B-8)

2

2 B 2 2 zG =N F τ H JF τ (B-9)

We can rewrite Eq. (B-7) into a simpler form:

cv

1 cv 22

D 1 D D D 2 D Dl

N SG 1 SG SG 1 SGG - Hτ +N G - Hτ 0

x G x x G zR z z

(B-10)

Since

120

1 D D D 1 D D 1 D

2 D D D 2 D D 2 D

1 SG Hτ SG 1 SGHτ Hτ

G x x x G x x G x

1 SG Hτ SG 1 SGHτ Hτ

G z G z G zz z z

(B-11)

Assuming F1, F2, H, τx and τz are independent of xD and zD respectively, then

1 2 x 2 x

2 2 2

D 1 D D D1 1 1

2

2 B 2 2 z z B 2 2 z

2 2 2 2 2

D 2 D D D D2 2 2 2 2

G F τ F τ J'1 1 J SG

x G x x xG G G

G N F F τ 2ττ N F F τ J'1 1 τ J SG

z G z z zG G G G Gz

(B-12)

Eq. (B-12) is neglected since they are secondary effects, then

cv

1 cv 22

D 1 D D 2 Dl

N Hτ SG Hτ SGG SG- +N G SG- 0

x G x G zR z

(B-13)

We assume that the dependence of the steady-state solution SG on xD and zD is separated,

that is:

D D D DSG x ,z =X(x )Z(z ) (B-14)

cv cv 21

2

D 1 D D 2 Dl

N N GG Hτ dX Hτ dZX- + Z- 0

X x G dx Z G dzR z

(B-15)

To satisfy Eq. (B-15), which implies

1D

2D

Gx

1 D HτD 1 D1

Gz

Hτ2 D D 2 D

2

HτX- X'=const c z

Χ=A z e +c zG

HτZ- Z'=const c x Ζ=B x e +c x

G

(B-16)

where A(zD) and c1(zD), B(xD) and c2(xD) are parameters dependent on zD and xD

respectively. Substituting Eq. (B-16) into Eq. (B-14) yields the steady-state saturation as

2 1 2

D D D

G G Gz x z

Hτ Hτ Hτ1 D D 2 D D 3 D D 4 D DSG= C x , z e C x , z e C x , z e C x , z

(B-17)

121

C1, C2, C3 and C4 are functions of xD and zD. Since H=-F3, a=F1/ F3, and b= F2/ F3, and

assume ε=Jb-1, then we can obtain

2 2

l l1 2 x x 2

cv cv1 l x

3 cv

R RF + H+JF τ a+ -1+Jb τ

N NG R τa- -ε

Hτ -F τ -τ N τ τ (B-18)

2 2

B 2 2 z B z2 z

B

3

N F τ + H+JF τ N bτ + -1+Jb τG τ= =-N bτ-ε

Hτ -F τ -τ τ (B-19)

Use the previous definitions (Eq. B-3) and define the two heterogeneity terms

x z

x z

D D D D

τ τ1 τ ln τ 1 τ ln τlnτ and lnτ

τ τ x x τ τ z z

(B-20)

Therefore, we can rewrite the solution Eq. (B-17) as

2l

DD D D Dcvz x B D x z B D

R a- x

-ε lnτ z -ε lnτ x -ε lnτ x -ε lnτ zN τ-τN bz -τN bz

1 2 3 4SG= C e e e C e e C e e C

(B-21)

or

2l

DcvB D B D

R a- xN τ-τN bz -τN bzHete Hete Hete

1 2 3 4SG= C e C e C e C (B-22)

with coefficients

D Dz x-ε lnτ z -ε lnτ xHete

1 1 D DC C x , z e e (B-23a)

Dx-ε lnτ xHete

2 2 D DC C x , z e (B-23b)

Dz-ε lnτ zHete

3 3 D DC C x , z e (B-23c)

C1, C2, C3 and C4 are functions of xD and zD. Based on this solution, the time-

independent CO2 saturation SG depends on its position xD and zD, M, as well as the

dimensionless numbers Rl, Ncv, NB and τ.

122

B.2 Analytical Solution with Simulation

Constraints

To eliminate some of unknown coefficients, we apply the constraints from simulations

(Eq. 5.9 and 5.10) to the general equation. First, the average CO2 saturations (Eq. 5.8)

are independent of capillary number Ncv when Ncv ≤ Ncv,c1.

2l

0,D2 cv

R1ax

CO N

cv

S0 e 0

N

(B-24)

Therefore, in the viscous-dominated regime, Ngv ≤ Ngv,cl, the core average saturation

becomes

0,D B 0,DB D z

2

-ε lnτ z bN zN bzHete

CO 3 4 3 4S C e +C C e e +C

. (B-25)

Second, the average saturation in this regime is also independent of Bond number, which

leads to

2 0,D B 0,DzCO -ε lnτ z bN z3 4

0,D 3

B B B

S C C- bz C e e + 0

N N N

. (B-26)

Solving for

B 0,D B 0,DbN z bN z

3 3,0 4 4,0C =const e C e and C =const=C

. (B-27)

C3,0 and C4,0 are constants. Therefore, the average saturation in the viscous-dominated

regime (Eq. B-25) is simplified to

0,Dz

2

-ε lnτ z

CO 3,0 4,0S C e +C (B-28)

For the homogeneous core, τ=1, (lnτ)x=(lnτ)z=0, the average saturation in the viscous-

dominated regime is Buckley-Leverett solution, which results in C3,0+C4,0=SBL. For

mathematical convenience, assume C4,0=0, then C3,0=SBL. We can obtain the average

CO2 saturation for the heterogeneous core in the viscous-dominated regime:

0,Dz-ε lnτ zHete

BL 1 BL 1S h τ S where h τ e (B-29)

123

Since h1(τ)<1, the average saturation for the heterogeneous cores is lower than the

Buckley-Leverett solution SBL, which is consistent with the simulation results. The

deviation of saturation from Buckley-Leverett solution is dependent on the degree of

heterogeneity, (lnτ)z, mainly the gradient of heterogeneity in the vertical direction. Eq.

B-29 also implies that strong heterogeneity results in lower CO2 saturation. Now we can

rewrite Eq. (B-21) as

2l

0,DcvB D

2

aR1x

N-τN bzHete Hete Hete

CO 1 2 BLS C e C e +S

(B-30)

For the homogeneous cores, Eq. (B-30) can be reduced to

2l

0,DB 0,D cv

2

aRx

bN z N

CO 1 2 BLS C e +C e +S

(B-31)

When Bond number equals to zero (g=0), the average saturations are observed to be a

constant SBL (Kuo et al. 2010), which results in C2= -C1. Putting this constraint back to

the coefficients defined in Eq. (B-23) yields:

0,D 0,D 0,Dx z x-ε lnτ x -ε lnτ z -ε lnτ xHete Hete

1 1 2 1C C e e , C =-C e (B-32)

Define

0,Dx-ε lnτ x

2h τ =e (B-33)

Substituting h1, h2, and Eq. (B-32) back to Eq. (B-30), then we can obtain

2l

0,DB 0,D cv

2

R1ax

N bz N Hete

CO 1 2 1 BLS C h τ h τ e -1 e +S

(B-34)

Eq. (B-34) can be reduced to the homogeneous results when τ=1:

2

l0,D

B 0,D cv

2

Rax

N bz N

CO 1 BLS C e -1 e +S

(B-35a)

where

2CO11 BL

1 0,D 11 0,D

B rg BL B 22l

fC S1 1C = , ax = d and bz =

N k S N αdR

(B-35b)

and hence give us the expressions for the coefficients C1, ax0,D and bz0,D.where C11, d11, α

and d22 are constant parameters determined by curve matching the semi-analytical

124

solution with the simulation results for homogeneous cores. Substituting Eq. (B-35b)

into Eq. (B-34), the final equation for evaluating the average CO2 saturation can be

written as follows:

2l

0,DB 0,D cv

2

R1ax

N bz N Hete11

CO 2 1 BL

B 1

C 1S h h e - e +S

N h

(B-36a)

2CO0,D

11 HeteBrg BLl

fax 1 1 1= d

τ τ Nk SR

(B-36b)

Hete

BL

0,D

22

Sτbz =τ

αd (B-36c)

A more compact form can be rearranged:

2l

0,DB 0,D cv

2

R1- ax

-τN bz τ NHete HeteBL

CO 1 BLHete

BL

SS =C e - e +S

S

(B-37a)

where

0,Dz-ε lnτ zHete

BL 1 BL 1S h τ S where h τ e (B-37b)

0,Dx-ε lnτ x11 2 1Hete

1 2

B

C h τ h τC where h τ e

N (B-37c)

or finally in terms of critical numbers:

2 Hete2 l cv,c1

Hete 1cv,c2 cv

2


CO 1 BLHete

BL

SS C e - e +S

S

(B-38a)

2

Homo Hete

gv,c1 BL CO lHete

cv,c1 HeteB B rg BL

N S f R1 1N = =

τ N τN k S

(B-38b)

Homo HeteHete 3/2gv,c2 BLgv,c2Hete l

cv,c2 Hete

B B B BL

N SN αR1N = = =

N τN τN S (B-38c)

125

B.3 Heterogeneous Factor, σlnk/ln(kmean)

In this section, we want to rewrite all the heterogeneous terms such as τ, h1(τ) and h2(τ) in

terms of the known factor, σlnk/ln(kmean). Assuming the heterogeneity changed in the x

and z direction, (lnτ)x and (lnτ)z are proportional to the σlnk/ln(kmean) respectively with

proportional factors β1 and β2:

lnk lnk

2 1z x

D mean D mean

σ σln τ ln τlnτ β and lnτ β

z ln k x ln k

(B-39)

Specifically, for random fields, the change rate of heterogeneity in both x and z direction

should be the same, therefore the proportional factor β1 is equal to β2. Since for the

homogeneous core, σlnk =0, τ=1, and (lnτ)x=(lnτ)z=0, we can solve lnτ and τ in terms of

the function σlnk/ln(kmean):

lnk lnk

2 D 1 D

mean mean

σ σln τ β z β x

ln k ln k (B-40a)

lnk lnk 1 D lnk

2 D 1 D 2 Dmean mean 2 D mean

σ σ β x σβ z β x 1+ β z

ln k ln k β z ln kτ=e =e

(B-40b)

Both Eq. (B-39) and Eq. (B-40) satisfy the homogeneous requirement. Substituting Eq.

(B-38) into the definitions of h1(τ) and h2(τ) yields

lnk

2 0,D0,D meanz

σ-εβ z

-ε lnτ z ln k

1h τ e =e (B-41a)

lnk

1 0,D0,D meanx

σ-εβ x

-ε lnτ x ln k

2h τ e =e (B-41b)

Therefore we can rewrite C1Hete

in terms of τ:

11 2 1Hete -ε11

1

B B

C h τ h τ CC τ

N N (B-42)

126

Appendix C

Flow Regimes

C.1 Critical Gravity Numbers

Figure C.1 and Figure C.2 illustrate the two critical numbers (dashed lines) and the

pressure gradients associated with the viscous, capillary and gravity forces for different

lengths of the cores as well as different interfacial tension and permeability values.

Figure C.1: Pressure gradients for viscous, capillary and gravity forces for different

aspect ratios.

127


interfacial tension and permeability values.


permeability values for High Contrast model.

128

Appendix D

Parameter Tables for Heterogeneous

Cores

Table D.1: Summary of Sensitivity Studies for High Contrast Models (HC).

Rl fCO2 kmean(mD) σ(mN/m) σlnk/ln(kmean) ω ε

Base(k, σ) 3.1445 0.95 318 22.47 0.1667 132 2.9

10k 3.1445 0.95 3180 22.47 0.1191 108 3.145

0.1k 3.1445 0.95 31.8 22.47 0.2776 145 3.153

σ/3 3.1445 0.95 318 7.49 0.1667 108 3.147

3σ 3.1445 0.95 318 67.41 0.1667 154 3.147

0.79f 3.1445 0.79 318 22.47 0.1667 78 3.2

0.51f 3.1445 0.51 318 22.47 0.1667 67 3.4

0.34f 3.1445 0.34 318 22.47 0.1667 65 3.5

2L 6.2890 0.34 318 22.47 0.1679 170 3.135

Table D.2: Summary of Sensitivity Studies for Random 3 Models.

Rl fCO2 kmean(mD) σ(mN/m) σlnk/ln(kmean) ω ε

Base(k, σ) 3.1445 0.95 366 22.47 0.2317 54 3.2

10k 3.1445 0.95 3660 22.47 0.1667 46 4.7

0.1k 3.1445 0.95 36.6 22.47 0.3740 62 2.6

σ/3 3.1445 0.95 366 7.49 0.2317 48 4.3

Table D.3: Summary of Sensitivity Studies for Different Heterogeneous Models.

Rl kmean(mD) σlnk/ln(kmean) Ncv,c1 Ncv,c2 ω ε

Homo 3.1445 430 0 886 3621 - -

Random2 3.1445 430 0.0419 850 3387 420 50

K-C 3.1445 430 0.0455 604 2283 183 10

H-C 3.1445 31.8 0.1667 173 618 132 2.9

Random3 3.1445 318 0.2317 170 464 54 3.2

Random4 3.1445 318 0.4793 131 213 31.5 2.7

129

Appendix E

Validation of TOUGH2 Simulations

E.1 Relative Convergence Tolerance ε1=10-5

A much smaller relative convergence tolerance ε1=10-5

is used to compare with the one

used in the simulations, ε1=10-2

. Figure E.1 shows the typical CO2 saturations as a

function of pore volume injected (PVI) for homogeneous cores with different flow rates.

It is observed that even with the large relative convergence tolerance, the results of

saturation are almost identical to the results with ε1=10-5

. Once capillary heterogeneity is

introduced into the simulations, it is extremely difficult to reach the steady-state

condition by using such smaller relative convergence tolerance to solve three-

dimensional mass conservation equations with both gravity and capillarity. Therefore,

the high convergence tolerance is acceptable due to the accuracy of the results and the

feasibility runoff simulating highly heterogeneous cores.

130

Figure E. 1 CO2 saturation as a function of pore volume injected (PVI) for different flow

rates.

E.2 GPRS

In order to validate the TOUGH2 simulation results, an alternative reservoir simulation

software, GPRS (General Purpose Research Simulator), is used for comparison to the

TOUGH2 results (Cao 2002; Jiang 2007; Li 2011). Figure E. 2 compares the grid

saturations of certain slices between GPRS and TOUGH2. The GPRS results are almost

identical to the TOUGH2 results (Li 2011).

Figure E. 2 GPRS vs TOUGH2 results for slice x=16 (LHS) and x=21 (RHS)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03

CO

2 S

atu

rati

on

PVI

Homogeneous: 0.1 ml/min

ε1=0.01

ε1=1E-50.00

0.02

0.04

0.06

0.08

0.10

1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02

CO

2 S

atu

rati

on

PVI


ε1=0.01

ε1=1E-5

0.00

0.02

0.04

0.06

0.08

0.10

1.E-03 1.E-02 1.E-01 1.E+00 1.E+01

CO

2 S

atu

rati

on

PVI


ε1=0.01

ε1=1E-5

0.00

0.02

0.04

0.06

0.08

0.10

1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02

CO

2 S

atu

rati

on

PVI


ε1=0.01

ε1=1E-5

0.00

0.02

0.04

0.06

0.08

0.10

1.E-03 1.E-02 1.E-01 1.E+00 1.E+01

CO

2 S

atu

rati

on

PVI


ε1=0.01

ε1=1E-5

131

Nomenclature

A cross section area of the core

A1 J-function fitting parameter

B1 J-function fitting parameter

C11 fitting parameter of analytical solutions for homogeneous cores

Ca traditional capillary number, μgut/σ

H height of the core [m]

J Leverett’s J function

L length of the core [m]

M mobility ratio, λw/ λg

NB Bond number, ΔρgH/pc*

Ncv alternative capillary number, kLpc*/H

2μg ut

Ngv gravity number, ΔρgkL/Hμg ut

Rl aspect ratio, L/H

S average saturation or scaling factor of Kozeny-Carman equation

SG steady-state CO2 saturation

Sp J-function fitting parameter

S* normalized brine saturation

a/b functions depending on water or CO2 saturation, fluid properties, and the

exponents of relative permeability functions, Eq. A-19

d11,22 fitting parameters of analytical solutions for homogeneous cores

f fractional flow, 1/1+M

g acceleration [m/s2]

h1 heterogeneous factor, Eq. 5.12b

k average permeability [mD]

132

kr relative permeability [mD]

n functional exponent of relative permeability

p pressure [Pa]

pc* characteristic capillary pressure [Pa]

t time

u Darcy velocity [N/m]

x x-coordinate

z z-coordinate

GREEKS

α proportional factor of second critical gravity number, 12.79737

β1,2 proportionality factors, which measure the degrees of heterogeneity in the x- and

z- directions, respectively, Eq. 5.11b

ε fitting parameter of analytical solutions for heterogeneous cores

λ1,2 J-function fitting parameter

λ relative mobility, kr/μ

μ viscosity [cp]

φ porosity

ΔP pressure difference between the average inlet and the outlet slice values

Δρ density difference between CO2 and brine [kg/m3]

σ CO2-brine interfacial tension [N/m] or standard deviation

θ contact angle, 0˚

τ heterogeneity parameter

ω fitting parameter of analytical solutions for heterogeneous cores

133

SUBSCRIPTS

c capillary

c1 critical number of transition 1

c2 critical number of transition 2

g gas (displacing fluid)

j phase

t total

w water (displaced fluid)

wr residual

BL Buckley-Leverret

CO2 CO2 (displacing fluid)

D dimensionless

134

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