compositional modeling of multiphase flow and enhanced oil
TRANSCRIPT
COMPOSITIONAL MODELING OF MULTIPHASE FLOW AND
ENHANCED OIL RECOVERY PROSPECTS IN LIQUID-RICH
UNCONVENTIONAL RESERVOIRS
by
Najeeb S. Alharthy
© Copyright by Najeeb S. Alharthy, 2015
All Rights Reserved
A thesis submitted to the Faculty and the Board of Trustees of the Colorado School
of Mines in partial fulfillment of the requirements for the degree of Doctor of Philosophy
(Petroleum Engineering).
Golden, Colorado
Date
Signed:Najeeb S. Alharthy
Signed:Dr. Hossein Kazemi
Thesis Advisor
Signed:Dr. Ramona M. Graves
Thesis Advisor
Golden, Colorado
Date
Signed:Dr. Erdal Ozkan
Professor and Interim Department HeadDepartment of Petroleum Engineering
ii
ABSTRACT
Production of tight oil from shale reservoirs in North America reduces oil imports and
provides better economics than natural gas. Thus, many companies have directed their
e↵orts to liquids production from Bakken, Eagle Ford, Niobrara, etc. Bakken recoverable
reserves is estimated to be 7.4 billion barrels. Despite advances in technology, the oil recovery
factor remains low (4% to 6%) (Energy Information Administration, 2013). To produce these
liquid-rich shale reservoirs e�ciently, a thorough understanding of flow mechanisms, reservoir
properties, and rock and fluid interactions is necessary. This work will present two areas for
investigation.
First, this research work presents compositional modeling of liquid-rich unconventional
reservoirs using volume balance method. A 2D three-phase single and dual-porosity models
using volume balance method are developed and presented. Due to the explicit nature
of the phase saturation calculations, a discrepancy in the number of moles in the system
was observed and a “mole correction term” was introduced to rectify the material balance
error for the system. Since the volume balance approach uses partial molar volume as
weighting factors, a robust partial molar volume algorithm is presented and validated against
published experimental data by Wu and Ehrlich (1973). The volume balance dual-porosity
model aforementioned is used to model depletion of liquid-rich unconventional reservoir going
below saturation pressure and the model results are validated with CMG GEM compositional
simulator. Finally, the model is used to study multiphase flow regimes observed in liquid-rich
reservoirs in the field. The analysis helps decipher multiphase bilinear and multiphase linear
flow regimes using compositional flow rate–normalized pressure analysis from the volume
balance method. From the analysis, the e↵ective permeability and hydraulic fracture
permeability is calculated and presented.
iii
Second, the enhanced oil recovery potential of liquid-rich shale reservoirs was evaluated
using laboratory data from experiments conducted at Energy and Environmental Research
Center (EERC) on several Bakken core samples of di↵erent size. We present both laboratory
and numerical modeling of carbon dioxide (CO2) oil recovery from these Bakken cores. We
also evaluate the EOR potential of using produced associated gas for injection. In laboratory
experiments CO2 injection recovers higher than 90% of oil from several Middle Bakken cores
and up to 40% from Lower Bakken cores. To decipher the oil recovery mechanisms in these
experiments, a numerical compositional model was used to match laboratory results. We
concluded that CO2 injection mobilizes matrix oil by miscibility and solvent extraction –
leading to counter-current flow of oil from the matrix instead of oil displacement in the
matrix (which is the conventional EOR wisdom). Specifically, the controlling factors include
re-pressurization, di↵usion-advection mass transfer, oil swelling, and viscosity reduction.
Laboratory results were scaled to field application in a North Dakota Middle Bakken well.
The primary oil depletion period was history matched and oil production was forecasted
for 10 years, recovering 6.2% oil. Then, we devised an EOR protocol using hu↵-and-pu↵
supercritical CO2 injection and natural gas liquids (NGL). Approximately 5% additional
oil was produced by CO2 solvent and 6.25% by NGL solvent for fracture spacing of 500
feet. We believe oil recovery will increase further with closer fracture spacing.
iv
TABLE OF CONTENTS
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
LIST OF SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv
LIST OF ABBREVIATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxii
DEDICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiv
CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 BACKGROUND AND PROBLEM STATEMENT . . . . . . . . . . . . . . . . . 1
1.2 OBJECTIVES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Compositional Rate Transient Analysis in Liquid-Rich Shale Reservoirs . 4
1.2.2 Appraisal of EOR potential in Liquid-Rich Shale Reservoirs . . . . . . . 4
1.3 CONTRIBUTION OF THE STUDY . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 THE ORGANIZATION OF THE THESIS . . . . . . . . . . . . . . . . . . . . . 5
CHAPTER 2 LITERATURE REVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1 COMPOSITIONAL MODELING . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 COMPOSITIONAL RATE TRANSIENT ANALYSIS . . . . . . . . . . . . . . 8
2.3 ENHANCED OIL RECOVERY IN UNCONVENTIONAL RESERVOIRS . . . 9
CHAPTER 3 COMPOSITIONAL MODELING . . . . . . . . . . . . . . . . . . . . . . 13
3.1 VOLUME BALANCE METHOD (VBM) . . . . . . . . . . . . . . . . . . . . . 13
v
3.2 FORMULATION AND IMPLEMENTATION OF VBM INDUAL-POROSITY SYSTEMS . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.3 THERMODYNAMIC MODEL . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.3.1 Phase Equilibria and Flash . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.3.2 Equation of State (EOS) (PR 1976) . . . . . . . . . . . . . . . . . . . 19
3.4 VALIDATION OF THE THERMODYNAMIC MODEL . . . . . . . . . . . . 20
3.5 VALIDATION OF THE VBM COMPOSITIONAL MODEL . . . . . . . . . . 21
3.5.1 Model Parameters and Setup . . . . . . . . . . . . . . . . . . . . . . . 25
3.5.2 Results and Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . 25
CHAPTER 4 COMPOSITIONAL RATE TRANSIENT ANALYSIS . . . . . . . . . . 31
4.1 MODIFICATION OF RATE TRANSIENT ANALYSIS . . . . . . . . . . . . 31
4.2 RATE-NORMALIZED PRESSURE EQUATION . . . . . . . . . . . . . . . . 31
4.2.1 Single-Phase Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.2.2 Multi-phase Flow Blackoil Case . . . . . . . . . . . . . . . . . . . . . . 34
4.2.3 Multi-phase Flow Compositional Case . . . . . . . . . . . . . . . . . . 36
4.2.4 Summary of the Analytical Solutions . . . . . . . . . . . . . . . . . . . 38
4.3 LIQUID-RICH UNCONVENTIONAL RESERVOIR CASE STUDY . . . . . 39
4.3.1 Model Parameters and Setup . . . . . . . . . . . . . . . . . . . . . . . 40
4.3.2 Fluid Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.3.3 Rock-Fluid Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.4 CASE STUDY RESULTS AND ANALYSIS . . . . . . . . . . . . . . . . . . . 42
CHAPTER 5 ENHANCED OIL RECOVERY - LABORATORY AND FIELDSTUDY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.1 SUPERCRITICAL FLUID EXTRACTION . . . . . . . . . . . . . . . . . . . 49
vi
5.2 BAKKEN CO2 SOAKING EXPERIMENTS . . . . . . . . . . . . . . . . . . 50
5.2.1 Laboratory Experiments and Experimental Procedures . . . . . . . . . 50
5.2.2 Fluid System Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.2.3 Bakken Core Description . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.2.4 Laboratory Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.3 MODELING EXPERIMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.3.1 Laboratory Model: Grid System . . . . . . . . . . . . . . . . . . . . . 58
5.3.2 Laboratory Model: Fluid System . . . . . . . . . . . . . . . . . . . . . 60
5.3.3 Laboratory Model: Rock-Fluid System . . . . . . . . . . . . . . . . . . 60
5.3.4 Laboratory Model: History Matching . . . . . . . . . . . . . . . . . . 60
5.3.5 Discussion of Laboratory Results . . . . . . . . . . . . . . . . . . . . . 65
5.4 MODELING FIELD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.4.1 Field Model: Grid System . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.4.2 Field Model: Fluid System . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.4.3 Field Model: Rock-Fluid System . . . . . . . . . . . . . . . . . . . . . 70
5.4.4 Field Model: History Matching . . . . . . . . . . . . . . . . . . . . . . 73
5.4.5 Field CO2 Enhanced Oil Recovery Scheme . . . . . . . . . . . . . . . . 73
5.4.6 Field NGL Enhanced Oil Recovery Scheme . . . . . . . . . . . . . . . 77
5.4.7 Discussion of Field Results . . . . . . . . . . . . . . . . . . . . . . . . . 77
CHAPTER 6 MASS TRANSFER MECHANISMS . . . . . . . . . . . . . . . . . . . . 81
6.1 TRANSPORT MEANS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.2 ADVECTIVE FLOW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.3 MOLECULAR DIFFUSION FLUX . . . . . . . . . . . . . . . . . . . . . . . 82
vii
6.3.1 Maxwell-Stephan Model . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.3.2 Generalized Fick’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.3.3 Classical Fick’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.3.4 Di↵usion Coe�cients Correlations . . . . . . . . . . . . . . . . . . . . 86
6.3.4.1 Wilke (1950) . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.3.4.2 Wilke and Chang (1955) . . . . . . . . . . . . . . . . . . . . 87
6.3.4.3 Sigmund (1976a, 1976b) . . . . . . . . . . . . . . . . . . . . . 88
6.3.4.4 Hayduk and Minhas (1982) . . . . . . . . . . . . . . . . . . . 89
6.3.4.5 Renner (1988) . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.3.4.6 Riazi and Whitson (1993) . . . . . . . . . . . . . . . . . . . . 90
6.3.4.7 Maxwell-Stefan (MS) Multicomponent Molecular Di↵usionCoe�cients . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.3.5 Di↵usion Coe�cients Calculations (Bakken Oil) . . . . . . . . . . . . . 92
6.4 GRAVITY DRAINAGE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.5 UNDERLYING EFFECTS OF TRANSPORT PRINCIPLES . . . . . . . . . 94
6.5.1 Oil Swelling and Viscosity Reduction . . . . . . . . . . . . . . . . . . . 95
6.5.2 Reduction of Interfacial Tension (IFT) at the matrix-fractureinterface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.5.3 Better CO2 Miscibility with Lower Temperature at Matrix-FractureInterface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.5.4 Summary of Underlying Transport Principles . . . . . . . . . . . . . . 96
CHAPTER 7 CONCLUSIONS, RECOMMENDATIONS AND FUTURE WORK . . . 97
7.1 MULTIPHASE TRANSIENT ANALYSIS IN LIQUID-RICH SHALES . . . . 97
7.2 ENHANCED OIL RECOVERY IN LIQUID-RICH SHALES . . . . . . . . . . 97
viii
7.3 RECOMMENDATIONS AND FUTURE WORK . . . . . . . . . . . . . . . . 98
REFERENCES CITED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
APPENDIX A - COMPOSITIONAL MODELING USING VOLUME BALANCEAPPROACH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
A.1 Volume Balance Formulation for Single-Porosity and Dual-Porosity . . . . . 105
A.2 Derivation of Compositional Equation and Pressure Equation . . . . . . . . 105
APPENDIX B - THERMODYNAMICS . . . . . . . . . . . . . . . . . . . . . . . . . 109
B.1 Peng-Robinson Equation of State . . . . . . . . . . . . . . . . . . . . . . . . 109
B.2 Fugacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
B.3 Derivative of Fugacity with respect to Pressure and Composition . . . . . . 110
B.4 Derivative of Compressibility Factor with respect to Pressure andComposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
B.5 Partial Molar Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
B.6 Fluid Compressibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
APPENDIX C - SATURATION EQUATIONS . . . . . . . . . . . . . . . . . . . . . 113
C.1 Liquid and Vapor Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
C.2 Derivation of Saturation Equations . . . . . . . . . . . . . . . . . . . . . . . 113
ix
LIST OF FIGURES
Figure 1.1 Significance of tight oil production (EnergyInformation Administration, 2013). . . . . . . . . . . . . . . . . . . . . . . 2
Figure 3.1 General volume balance implementation. . . . . . . . . . . . . . . . . . . 18
Figure 3.2 Phase envelope for C1 = 0.70 , C4 = 0.20 ,and C10 = 0.10. . . . . . . . . . 21
Figure 3.3 Thermodynamic validation between developed routine for density and zfactor calculations with CMG PVT Package (WinProp). . . . . . . . . . . 22
Figure 3.4 Thermodynamic validation between developed routine for fugacity andviscosity calculations with CMG PVT Package (WinProp). . . . . . . . . 23
Figure 3.5 Thermodynamic validation of partial molar volume and fluidcompressibility calculations. . . . . . . . . . . . . . . . . . . . . . . . . . 24
Figure 3.6 Relative permeability curves . . . . . . . . . . . . . . . . . . . . . . . . . 26
Figure 3.7 Validation of pressure and saturation profile ( VBM vs CMG GEMsimulator). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Figure 3.8 Validation of cummulative oil and cummulative gas (VBM vs CMGGEM simulator). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Figure 3.9 Comparison of material balance error (VBM vs CMG GEM simulator). . 30
Figure 4.1 Refined gridding and well dimensions for multiphase depletion model. . . 41
Figure 4.2 Phase envelope and component specification for tight oil system. . . . . . 43
Figure 4.3 Case study validation for multiphase flow depletion run. . . . . . . . . . . 45
Figure 4.4 Di↵erent flow regimes in stimulated horizontal well. . . . . . . . . . . . . 46
Figure 4.5 Deciphered flow regimes and dual porosity feature. . . . . . . . . . . . . . 47
Figure 5.1 Enhanced oil recovery experiments on Bakken Cores (performed atEERC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
x
Figure 5.2 Compositions of separator samples and produced streams for MiddleBakken. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Figure 5.3 Produced composition stream for Lower Bakken core. . . . . . . . . . . . 54
Figure 5.4 Thin sections for Middle Bakken core at di↵erent resolutions,mineralogy composed of abundant monocrystalline quartz grains (whitecolor) with non-skeletal calcerous grains, minor calcite and Fe-Dol (tanand brown color), and some K-spar, Plagioclase, and Pyrite (blackcolor). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Figure 5.5 Thin sections for Lower Bakken core at di↵erent resolution, mineralogycomposed of quartz and calcite dominated (white and tan color), minoramount of clays such as illite (dark brown color), and kerogen patches(black color). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Figure 5.6 Oil recovery factor for Middle Bakken core soaking experiment. . . . . . . 57
Figure 5.7 Oil recovery for Lower Bakken core soaking experiment. . . . . . . . . . . 58
Figure 5.8 Single-porosity radial grid system used in Bakken core CO2 soakingexperiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Figure 5.9 Middle Bakken synthetic lumped fluid composition and phase envelope. . 61
Figure 5.10 Lower Bakken synthetic lumped fluid composition and phase envelope. . . 62
Figure 5.11 Relative permeability curves. . . . . . . . . . . . . . . . . . . . . . . . . 63
Figure 5.12 Fracture relative permeability curves. . . . . . . . . . . . . . . . . . . . . 64
Figure 5.13 History match results for Middle Bakken and Lower Bakken CO2 coreflooding experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
Figure 5.14 Reservoir dimensions (single-stage HF) for a North Dakota Bakken wellmodel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Figure 5.15 Middle Bakken reservoir fluid description. . . . . . . . . . . . . . . . . . . 69
Figure 5.16 Equation of state (EOS) model tuning of Gas-Oil Ratio (GOR) and oildensity with PVT laboratory data. . . . . . . . . . . . . . . . . . . . . . . 71
Figure 5.17 Equation of State (EOS) model tuning of oil viscosity and swellingfactor with PVT laboratory data. . . . . . . . . . . . . . . . . . . . . . . 72
xi
Figure 5.18 History matching process with oil rates control . . . . . . . . . . . . . . 73
Figure 5.19 History match of bottom hole pressure and gas rates . . . . . . . . . . . 74
Figure 5.20 Bottom hole pressure and oil rates during EOR scheme1. . . . . . . . . . 76
Figure 5.21 Gas rates and comparison of all EOR schemes. . . . . . . . . . . . . . . . 78
Figure 5.22 Comparison of two solvent types and e↵ect of molecular di↵usion . . . . 79
Figure 6.1 Flowchart for calculating molecular di↵usion coe�cients usingLeahy-Dios and Firoozabadi (2007) approach. . . . . . . . . . . . . . . . . 93
xii
LIST OF TABLES
Table 3.1 Three-component fluid system used for thermodynamic validation . . . . . 21
Table 3.2 Test case reservoir parameters . . . . . . . . . . . . . . . . . . . . . . . . . 25
Table 3.3 Three-component fluid system used for simulation run . . . . . . . . . . . . 27
Table 4.1 Summary of Bilinear solutions for single-phase, multi-phase black oil, andmulti-phase compositional models . . . . . . . . . . . . . . . . . . . . . . . 39
Table 4.2 Summary of Linear solutions for single-phase, multi-phase black oil, andmulti-phase compositional models . . . . . . . . . . . . . . . . . . . . . . . 39
Table 4.3 Multiphase case study reservoir parameters. . . . . . . . . . . . . . . . . . . 40
Table 4.4 Three-component fluid system used for multiphase depletion run. . . . . . . 42
Table 4.5 Bilinear multiphase flow analysis for depletion run . . . . . . . . . . . . . . 48
Table 4.6 Linear multiphase flow analysis for depletion run . . . . . . . . . . . . . . . 48
Table 5.1 XRD analysis of Middle Bakken Core . . . . . . . . . . . . . . . . . . . . . 55
Table 5.2 XRD analysis of Lower Bakken Core . . . . . . . . . . . . . . . . . . . . . . 56
Table 5.3 Radial case for Middle Bakken core . . . . . . . . . . . . . . . . . . . . . . 59
Table 5.4 North Dakota Bakken well reservoir parameters . . . . . . . . . . . . . . . . 68
Table 5.5 Lumped-component Middle Bakken fluid system used for field case . . . . . 70
Table 5.6 CO2 Enhanced oil recovery schemes . . . . . . . . . . . . . . . . . . . . . . 75
Table 5.7 NGL Enhanced oil recovery schemes . . . . . . . . . . . . . . . . . . . . . 77
Table 5.8 Summary of results for Enhanced oil recovery schemes . . . . . . . . . . . 80
Table 6.1 Molecular Di↵usion Calculations for Middle Bakken fluid system . . . . . . 94
Table 6.2 Summary of swelling tests for Middle Bakken fluid system . . . . . . . . . . 95
xiii
LIST OF SYMBOLS
A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PR-EOS coe�cient
a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . attraction parameter for EOS
B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PR-EOS coe�cient
b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . repulsive parameter for EOS
bbl
SINGLE��
[ psi
(RB/d)cp ] . . . . . . . . . y-intercept of bilinear single-phase flow equation
bl
SINGLE��
[ psi
(RB/d)cp ] . . . . . . . . . . y-intercept of linear single-phase flow equation
bbl
MULTI��
[ psi
(RB/d)cp ] . . . . . . . . . y-intercept of bilinear multi-phase flow equation
bl
MULTI��
[ psi
(RB/d)cp ] . . . . . . . . . . . . y-intercept of linear multi-phase flow equation
B [ RB
STB
] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . formation volume factor
Bo
[ RB
STB
] . . . . . . . . . . . . . . . . . . . . . . . . . . . oil formation volume factor
Bg
[ RB
STB
] . . . . . . . . . . . . . . . . . . . . . . . . . . . gas formation volume factor
Bw
[ RB
STB
] . . . . . . . . . . . . . . . . . . . . . . . . . water formation volume factor
C1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . methane
C4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . butane
C7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . heptane
C10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . decane
cv
[psi�1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . fluid compressibility
c�
[psi�1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . rock compressibility
ct
[psi�1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . total compressibility
ct,f
[psi�1] . . . . . . . . . . . . . . . . . . . . . . . . . total fracture compressibility
xiv
ct,m
[psi�1] . . . . . . . . . . . . . . . . . . . . . . . . . . total matrix compressibility
f [psi] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . fugacity
FcD
[mdft] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . fracture conductivity
h [ft] . . . . . . . . . . . . . . . . . . . . . gravity head between fracture and matrix
kc
[fraction] . . . . . . . . . . . . . . . . . . . . . . equilibrium ratio of component c
kB
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Boltzmann constant
khf
[md] . . . . . . . . . . . . . . . . . . . . . . . . . hydraulic fracture permeability
kf,eff
[md] . . . . . . . . . . . . . . . . . . . . . . . . . e↵ective fracture permeability
km
[md] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . matrix permeability
kf
[md] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . fracture permeability
krg
[fraction] . . . . . . . . . . . . . . . . . . . . . . . . . . gas relative permeability
kro
[fraction] . . . . . . . . . . . . . . . . . . . . . . . . . . oil relative permeability
krw
[fraction] . . . . . . . . . . . . . . . . . . . . . . . . water relative permeability
Lx
, Ly
, Lz
[ft] . . . . . . . . . . . . . . . . . . . . . . . fracture spacing in x,y, and z
mbl
SINGLE��
[ psi
(RB/d)cppd
] . . . . . . . . . . . slope of bilinear single-phase flow equation
ml
SINGLE��
[ psi
(RB/d)cppd
] . . . . . . . . . . . . slope of linear single-phase flow equation
mbl
MULTI��
[ psi
(RB/d)cppd
] . . . . . . . . . . . slope of bilinear multi-phase flow equation
ml
MULTI��
[ psi
(RB/d)cppd
] . . . . . . . . . . . . . slope of linear multi-phase flow equation
MoleCorr [ 1day
] . . . . . . . . . . . . . . . . . . . . . . . . . . . mole correction term
nf
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . total number of fractures
nc
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . total number of components
Ng
[lb�mole] . . . . . . . . . . . total number of moles in hydrocarbon vapor phase
xv
Ng
c
[lb�mole] . . total number of moles of component c in hydrocarbon vapor phase
No
[lb�mole] . . . . . . . . . . . . . total number of moles in hydrocarbon oil phase
No
c
[lb�mole] . . . . total number of moles of component c in hydrocarbon oil phase
Nw
[lb�mole] . . . . . . . . . . . . . . . . . total number of moles in aqueous phase
Nw
c
[lb�mole] . . . . . . . total number of moles of component c in aqueous phase
Nt
[lb�mole] . . . . . . . . . . . . . . . . . . . . . . . . . . . total number of moles
Nt
c
[lb�mole] . . . . . . . . . . . . . . . . . . total number of moles of component c
po
[psia] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . oil pressure
pcog
[psia] . . . . . . . . . . . . . . . . . . . . . . . . . . oil and gas capillary pressure
pcow
[psia] . . . . . . . . . . . . . . . . . . . . . . . . oil and water capillary pressure
pc
[psia] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . critical pressure
pwf
[psia] . . . . . . . . . . . . . . . . . . . . . . . . . . flowing bottom hole pressure
pwell
[psia] . . . . . . . . . . . . . . . . . . well flowing bottom pressure (constraint)
q [ ft3
day
] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . flow rate
q̂ [ 1day
][ lb�mole
fy
3day
] . . . . . . . . . . . . . . . . . . . . reservoir flow rate per rock volume
qo
[ ft3
day
] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . oil flow rate
qg
[ ft3
day
] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . gas flow rate
qw
[ ft3
day
] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . water flow rate
R [ ft
3⇧psiao
Rlb�mole
] . . . . . . . . . . . . . . . . . . . . . . . . . . . . gas constant (10.731)
shf
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . apparent skin
sg
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . gas saturation
so
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . oil saturation
xvi
sr
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . residual saturation
sw
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . water saturation
t [day] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . time
T [oR] . . . . . . . . . . . . . . . . . . . . . . . . . . . . temperature (oF + 459.67)
Tx
[ft2md
ft
] . . . . . . . . . . . . . . . . . . Single phase transmissibility in x-direction
Tc
[oR] . . . . . . . . . . . . . . . . . . . . . . . . critical temperature (oF + 459.67)
Uc
[ lb�mole
day
] . . . . . . . . . . . . . . . . . . . . . . . net molar flux of component c
VR
[ft3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . rock volume
Vt
[ft3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . total system volume
v [ ft
3
lb�mole
] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . specific volume
vt
[ ft
3
lb�mole
] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . total specific volume
vt
c
[ ft
3
lb�mole
] . . . . . . . . . total partial molar volume with respect to component c
wc
[fraction] . . . . . . . . . . . . . mole fraction of component c in aqueous phase
whf
[ft] . . . . . . . . . . . . . . . . . . . . . hydraulic fracture width (pseudoized)
whf
original
[ft] . . . . . . . . . . . . . . . . . . . . hydraulic fracture width (original)
xc
[fraction] . . . . . . . mole fraction of component c in hydrocarbon liquid phase
yc
[fraction] . . . . . . . . mole fraction of component c in hydrocarbon vapor phase
yf
[ft] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . fracture half length
xf
[ft] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . fracture half length
z [ratio] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . deviation factor
zc
[fraction]. . . . . . . . . . overall mole fraction of component c in hydrocarbon and aqueous phases
↵ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . EOS coe�cient of attractive term
xvii
↵o
[ 1day
] . . . . . . . . . . . . . . . . . oil product term used in multiphase di↵usivity
↵g
[ 1day
] . . . . . . . . . . . . . . . . . gas product term used in multiphase di↵usivity
↵w
[ 1day
] . . . . . . . . . . . . . . . . water product term used in multiphase di↵usivity
�o
. . . . . . . . . . . . . . . . . . . . . . . . . . . . oil product term used in source term
�g
. . . . . . . . . . . . . . . . . . . . . . . . . . . gas product term used in source term
�w
. . . . . . . . . . . . . . . . . . . . . . . . . . water product term used in source term
r⇧ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . divergence operator
r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . gradient operator
4x [ft] . . . . . . . . . . . . . . . . size of the block in discrete form in x-direction
4y [ft] . . . . . . . . . . . . . . . . size of the block in discrete form in y-direction
4z [ft] . . . . . . . . . . . . . . . . size of the block in discrete form in z-direction
4pwf
[psia] . . . . . . . . . . . . . . . . . . . . . . . . . well flowing pressure change
� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dirac delta
� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kronecker delta
� [psift
] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . gravity gradient
�o
[md
cp
] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . oil phase mobility
�g
[md
cp
] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . gas phase mobility
�w
[md
cp
] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . water phase mobility
�t
[md
cp
] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . total phase mobility
µo
[cp] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . oil phase viscosity
µg
[cp] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . gas phase viscosity
µw
[cp] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . water phase viscosity
xviii
µt
[cp] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . total phase viscosity
! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . acentric factor
� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . fugacity coe�cient
� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . porosity
�f
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . fracture porosity
�m
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . matrix porosity
⇠ [ lb�mole
ft
3 ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . molar density
⇠o
[ lb�mole
ft
3 ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . oil molar density
⇠g
[ lb�mole
ft
3 ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . water molar density
� [ 1ft
2 ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . shape factor
⌧ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . tortuosity
⌧m/f
[ ft3
day
] . . . . . . . . . . . . . . . . . . . transfer rate between matrix and fracture
c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the cth component
c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . critical
f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . fracture
g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . gas phase
m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . matrix
m/f . . . . . . . . . . . . . . . . . . . . . . . . matrix or fracture based on upstream flow
o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . oil phase
r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . residual
t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . total
w . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . water phase
xix
l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . current iteration
l + 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . next iteration
n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . current timestep
n, S . . . . . . . . . . . . . . . . . . . . . . . . . . current timestep with saturation check
n+ 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . next timestep
xx
LIST OF ABBREVIATIONS
CMG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Computer Modeling Group
CO2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Carbon Dioxide
DUALPOR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dual Porosity Model
EIA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Energy Information Agency
EOS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equation of State
EERC . . . . . . . . . . . . . . . . . . . . . . Energy and Environmental Research Center
GEM . . . . CMG’s Compositional and Unconventional Oil and Gas Reservoir Simulator
LPG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquid Petroleum Gas
NGL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Natural Gas Liquids
NR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Newton Rapshon
NNR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Non Newton Rapshon
RTA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rate Transient Analysis
VBM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Volume Balance Method
WINPROP . . . . . . . . . . . . . Phase Behavior and Reservoir Fluid Property Program
xxi
ACKNOWLEDGMENTS
I would like to first and foremost thank Almighty God (Allah) for giving me the oppor-
tunity, motivation, guidance, patience, and focus towards my pursuit. The most deserving
of thanks and praise is to God (Allah), may he be glorified and exalted. He has bestowed
many great favors and blessings upon me and my family in both spiritual and worldly terms.
“My success in this task depends entirely on the help of Allah; in Him do I trust and to
Him do I turn for everything”. [Hudd 88]
I would like to express my special appreciation and profound gratitude to my advisor
Dr. Hossein Kazemi, you have been a tremendous mentor for me. I would like to thank you
for all the guidance, support, encouragement and for allowing me to grow. Your advice on
both research as well as on my career and life have been priceless. I am also thankful for
encouraging me to use shorter, clear sentences in my writings and for carefully reading and
commenting on countless revisions of this manuscript. I have been amazingly fortunate to
work with you over the years.
My co-advisor, Dr. Ramona M. Graves, I am deeply grateful for all the support (starting
in Spring 2007 onwards), your advice, and long discussions that helped me sort out the
technical details of my work. I am also thankful for reading my thesis, commenting on my
views and helping me understand and enrich my ideas. And most of all, me and my wife
would like to thank you for the box of do-si-dos peanut butter cookies. Thank you for being
there for me always.
I would also like to thank my committee members, Dr. Azra Tutuncu, Dr. Yu-Shu Wu,
my minor advisor Dr. Vaughn Gri�ths, and my chair Dr. Sonnenberg. Thank you Dr. Azra
also for your support always and letting me be part of UNGI, it was a wonderful learning
experience. Thank you all in advance for your brilliant comments and suggestions.
xxii
I would like to thank my parents, my father Mr. Salim Said Alharthy and my mother
Mrs. Jokha Hamdoon Alharthy, you both are the most amazing parents, thank you for
unconditional love, and may God (Allah) always bless you. My Siblings, Said, Farida,
Salma, Sabra, Huwayda, Samya, Nabeel, and Mohammed .... I love all of you, and thank
you for the constant encouragement and support, especially Said Alharthy, I thank you for
rekindling the dream of graduate school and of supporting it always.
Finally, last but not least, my wife Nasra, how can I thank you enough ..... you have
been so supportive and loving. You have always been there and kept strong with me, and
always support me as I was away .... You have helped me in pursuit of this milestone (it is
ours), I am so grateful for your love, support, encouragement.
xxiii
I would like to dedicate this thesis work to my Parents (Salim Alharthy and Jokha
Alharthy) and also to my newly born baby girl (Dana Najeeb Alharthy), you have added
joy to our lives - Alhamdulilah.
xxiv
CHAPTER 1
INTRODUCTION
This thesis presents development of a multiphase compositional model used for well test
analysis and assessing the potential of enhanced oil recovery in liquid-rich shale reservoirs. To
produce liquid-rich shale reservoirs e�ciently, a thorough understanding of flow mechanisms,
reservoir properties, and rock and fluid interactions is necessary. These key flow parameters
hydraulic fracture permeability (khf
) and e↵ective fracture permeability (kf,eff
) are deter-
mined from well analysis using a novel multiphase, rate-transient analysis, designed for low
permeability liquid-rich shale reservoirs. The enhanced oil recovery potential of liquid-rich
shale reservoirs was evaluated using laboratory data from experiments conducted at Energy
and Environmental Research Center (EERC) on several Bakken core samples of di↵erent
size. To evaluate field scale enhanced oil recovery, a reservoir-scale numerical model was
constructed using well data from a North Dakota Bakken well.
1.1 BACKGROUND AND PROBLEM STATEMENT
Production of light oil and gas-condensate from shale reservoirs in North America is more
economical than production of natural gas. Thus, many companies have directed their e↵orts
towards liquids production as shown in Figure 1.1(a). Eagle Ford and Bakken formations
have contributed significantly to the overall U.S. domestic production. Figure 1.1(b) shows
the forecast of U.S. oil production to year 2035 with U.S. peak production of 11 MMBbls/day
in year 2020 overtaking the historical peak production of the U.S in the 1970s.
The shift towards liquids-rich shale production is attributed to the advances in drilling
horizontal wells and multi-stage hydraulic fracturing techniques. The produced liquids are
rich in ethane, propane, and intermediate components. Kurtoglu (2013) discussed some of
the reasons why hydrocarbon production is possible in low permeability formations such as
1
(a)
(b)
Figure 1.1: Significance of tight oil production (Energy Information Administration, 2013).
2
the Bakken: 1) low viscosity hydrocarbon fluids, 2) high compressibility fluids, 3) domi-
nance of very low molecular weight components in hydrocarbon fluids, 4) abnormally high
initial reservoir pressure, 5) enhancement of natural fractures as a consequence of multi-
stage hydraulic fracturing and 6) favorable phase envelope shift of hydrocarbon mixture and
favorable ratio of gas-oil split in nanopores.
Advances in hydraulic fracturing stimulation techniques has made possible flow from a
tight matrix to a hierarchy of fracture sizes to the wellbore. Furthermore, rubblizing the
reservoir in the vicinity of hydraulic fractures creates the favorable environment of improved
drainage which is why multi-stage hydraulic fracturing is so critical in successful develop-
ment of shale reservoirs (Alharthy et al., 2013). Despite these advances in horizontal drilling
and multi-stage hydraulic fracturing, the recovery factors still remain low. For example, in
the Bakken recovery about 4% - 6% (Kurtoglu, 2013). This provides a strong motivation
to investigate and better understand ways to recover more hydrocarbons from liquid-rich
shale reservoirs such as the Bakken and Eagle-Ford. It is very important to understand
the di↵erent flow mechanisms, reservoir properties, and the controlling rock and fluid pa-
rameters necessary for production. Two issues pertaining to multiphase flow of liquid-rich
shale reservoirs were addressed in this thesis. First, a multiphase compositional model, using
volume balance method, was developed. The volume balance method is a technique that
is more amenable to rate transient analysis as it reduces the multicomponent flow equa-
tions to a single pressure equation using partial molar volumes. This method was used to
simulate and provide production data which had the desired compositional characteristics
of liquid-rich shale reservoirs. Second, the compositional model was used to evaluate the
enhanced oil recovery mechanism of liquid-rich shale reservoirs. The conventional enhanced
oil recovery through displacement alone does not apply, and it is shown that miscibility in
a narrow region near the fracture surfaces is the main mechanism of oil extraction from the
tight matrix.
3
1.2 OBJECTIVES
The thesis objective was to first develop an appropriate compositional reservoir model for
well test analysis in liquid-rich shale reservoirs. Second, was to assess oil recovery potential
in liquid-rich shale reservoirs.
1.2.1 Compositional Rate Transient Analysis in Liquid-Rich Shale Reservoirs
Typically, rate transient analysis is performed based on theory of single-phase linear
flow model. However, with production from unconventional liquid-rich shale reservoirs, the
production streams are lean and span from volatile light oils to rich gas condensate systems.
These fluid systems are highly-composition dependent and phase behavior e↵ects have to
be taken into accounted. The previous theory of single-phase flow with low and constant
parameters for compressibility and viscosity cannot be applied in liquid-rich shale systems.
Inclusion of composition dependence via volume balance method yields a more accurate
outcome while maintaining the same simplicity of a single-phase system analysis. This
presents an analytical rate analysis technique for such light oils (as well as rich gas) by
including composition-dependence directly in the conventional rate-transient analysis (RTA)
plots and analysis methods. We utilized volume balance method to perform compositional
rate-transient analysis.
1.2.2 Appraisal of EOR potential in Liquid-Rich Shale Reservoirs
The oil recovery factor from liquid-rich shale reservoirs remains low, in the range of 4% -
6% (Kurtoglu, 2013). Large volume of oil is unrecovered and this provides the motivation to
investigate the technical feasibility of performing enhanced oil recovery in liquid-rich shales.
In addition, recent lab experiments where Middle Bakken cores had undergone supercritical
gas injection using carbon dioxide (CO2), and they recovered up to 80% of oil. This provides
the impetus to pursue and further understand the mechanisms of supercritical gas injection
in liquid-rich shale reservoirs using solvents such as carbon dioxide (CO2) and natural gas
liquids (NGL) consisting of C1 to C4+. A few percent increase in oil recovery from innovative
4
enhanced oil recovery techniques could lead to millions of barrels of additional oil.
1.3 CONTRIBUTION OF THE STUDY
The first contribution of this research is development and testing of a rate-transient
analysis technique for highly composition-dependent liquid-rich shale reservoirs. A dual-
porosity architecture was used as the main framework to implement the volume balance
method and simulate well performance used for reservoir characterization in liquid-rich shale
reservoirs. Due to the explicit nature of the phase saturation calculations, a discrepancy in
number of moles in the system was observed and a new “mole correction” (MoleCorr)
term was introduced. An analytical approximation to the multi-phase solution was used to
determine hydraulic fracture permeability (khf
) and e↵ective fracture permeability (kf,eff
).
The analytical solution produced the numerical model input permeability / transmissivity
accurately.
The second contribution of this research is an understanding of the recovery mech-
anisms involved in enhanced oil recovery of liquid-rich shale reservoirs. Specifically, the
compositional modeling of supercritical gas injection deciphered the relevant oil recovery
mechanism. This understanding was scaled to field applications by considering an actual
North Dakota Middle Bakken well. In addition, di↵erent solvents in combination were used
to evaluate the e↵ects on oil recovery.
1.4 THE ORGANIZATION OF THE THESIS
This thesis has six chapters.
Chapter 1 is the introduction, which has covered background and problem statement,
objectives, and the contribution of the research work.
Chapter 2 is the literature review pertaining to general compositional modeling ap-
proaches, compositional rate transient analysis, and enhanced oil recovery in liquid-rich
shale reservoirs.
5
Chapter 3 is the compositional modeling covering volume balance method, implemen-
tation of volume balance method in single-porosity and dual-porosity systems, Equation of
State (EOS) model and validation, and volume balance model and validation.
Chapter 4 is compositional rate transient analysis, analytical multiphase solution in reser-
voir characterization, case study, and results.
Chapter 5 is enhanced oil recovery in liquid-rich shale reservoirs, supercritical gas injec-
tion using CO2 in Bakken core samples and scaling to field using CO2 and NGL solvents in
a North Dakota Bakken well.
Chapter 6 is an evaluation of recovery mechanisms for CO2 and NGL solvents in a
hu↵-n-pu↵ approach.
Chapter 7 is conclusions, recommendations, and future work discussions of the research
work.
6
CHAPTER 2
LITERATURE REVIEW
This chapter presents literature review, which includes (1) an overview of compositional
modeling in the last 40 years, (2) compositional rate transient analysis for reservoir charac-
terization and (3) enhanced oil recovery in liquid-rich shale reservoirs.
2.1 COMPOSITIONAL MODELING
Compositional modeling has significantly advanced over the last forty years. Typically,
compositional modeling is imperative during reservoir depletion when fluid system is light
and contains a great amount of light and intermediate components. Another application
of compositional modeling is to assess enhanced oil recovery with gas or solvent injection.
The gas injection could be immiscible or miscible with the reservoir oil depending on the
composition of the injected fluid, the reservoir oil, and reservoir pressure and temperature.
Examples include enriched gas drive, CO2 flooding, gas cycling in condensate reservoirs. The
simulation of these processes is complex and require proper handling of the phase behavior
thermodynamics. Young and Stephenson (1983) classify compositional models as Newton
Rapshon (NR) and non-Newton Raphson (NNR) models. These models are di↵erent in the
way the pressure and composition equations are formed. The NR methods are iterative
techniques that require larger linear solver codes, but handle larger time steps. The NNR
methods require smaller linear solvers, but they have time-step stability restrictions (Kazemi
et al., 1978) and (Ngheim et al., 1981). The pressure equations of these two models were
constructed through linear combination of the total hydrocarbon molar-conservation equa-
tion which was a result of the summation of all the conservation equations for hydrocarbon
components and the water molar-conservation equation multiplied by a weighting factors
(Wong et al., 1990). In these models, the compositional dependence of certain terms were
7
neglected in the pressure equation. The method described by Acs et al. (1985) is an ex-
ception to this, as it was the first non-Newton-Rapshon method that accurately included
compositional e↵ects. Acs’s model is a volume balance (VB) method where the reservoir
fluid volumes for all phases is forced to be equal to the pore volume. This volume balance
approach was also discussed by Kendall et al. (1985) and Watts (1986). Watts also extended
the technique to a sequential implicit saturation method. The volume balance o↵ered sev-
eral advantages over the usual newton rapshon scheme, where, the volume balance method
reduces the component equations to a single pressure equation using partial molar volumes.
2.2 COMPOSITIONAL RATE TRANSIENT ANALYSIS
In conventional well test analysis, the entire reservoir fluid system is approximated by an
equivalent single-phase fluid with constant compressibility and viscosity (Lee et al., 2003).
The rate-transient analysis (RTA) was developed to handle wells with varying flow rates
(Clarkson, 2013). The RTA analysis, however, adheres to the simplified underlying assump-
tions of constant rock and fluid properties. These assumptions are not valid when analyzing
liquid-rich production from shale reservoirs where the produced fluids are highly composition
dependent. If the wellbore pressure falls below the bubble or dew point pressure, two-phases
exist and the simplifying single-phase assumptions are no longer valid. Analytical modeling
of pressure and rate transient for multiphase flow in porous media is challenging because of
the nonlinearities associated with the fluid (Behmanesh et al., 2013). Various methods have
been proposed in the literature to deal with attempting to correct for multi-phase flow com-
plexities. The use of pseudo variables (Fraim and Wattenbarger, 1987) is one of the approach.
With regard to two-phase flow problem, several researchers have attempted to tackle it dif-
ferently. Single-phase solution has been proposed by di↵erent researchers (Aanonsen, 1985;
Camacho and Raghavan, 1989; Jones and Raghavan, 1988; Hamdi, 2013, and Raghavan,
1976). Other researchers have linearized the di↵usivity equation using Boltzmann transfor-
mation (Boe et al., 1989; Bratvold and Horne, 1990; and Ramey, 1970). Application of
total mobility and single-phase flow concepts when the pressure at the wellbore falls below
8
the bubble-point or the dew-point pressure were suggested by Martin (1959). A two-phase
pseudopressure was proposed to be used to determine the formation permeability from well
test data for solution-gas-drive systems by Raghavan (1976). An extension of this work was
done to introduce the concept of reservoir integral by Aanonsen (1985). The reservoir inte-
gral concept was later used to analyze drawdown and buildup responses in gas-condensate
and solution-gas-drive systems by Jones and Raghavan (1988) and Camacho and Raghavan
(1989). Two-phase pseudotime was recently introduced by Sureshjani and Gerami (2011)
for boundary-dominated flow in gas condensate reservoirs. In this work, an approximate
analytical solution to multi-phase flow which maintains the same simplicity as single-phase
system is proposed. The proposed multi-phase flow solution is used to analyze simulated well
performance generated by volume balance method outlined earlier and is able to produce
model input permeability / transmissivity very accurately.
2.3 ENHANCED OIL RECOVERY IN UNCONVENTIONAL RESERVOIRS
Enhanced oil recovery in unconventional reservoirs (ultra-low permeability) is new and
not well understood. Interest in enhanced oil recovery potential in liquid-rich shale reservoirs
stems from the fact that the primary oil production is low (<15 %) despite advances in drilling
and multistage hydraulic fracture stimulation techniques. Recovery factor for the Bakken
in Mountrail County in North Dakota was studied by Clark (2009) using three di↵erent
methods which resulted in 8.8%, 7.4%, and 7.1%. Another study by Dechongkit and Prasad
(2011) determined 9.2% for Antelope field, 14.9% for Sanish field, and 16% for Parshall fields.
The choice of enhanced oil recovery technique in liquid-rich shale reservoirs is limited to gas
flooding. Water flooding could be impractical due to ultra-low permeability of the matrix
and injectivity could be a problem. In addition, due to preferentially oil-wet and/or mix-wet
character of liquid-rich shale reservoirs, it would require a high force to overcome the high
capillary pressure.
Cyclic supercritical gas or solvent injection was proposed in the form of soaking (hu↵-n-
pu↵) as a more e↵ective approach for enhanced oil recovery in ultra-low permeability shale
9
(Chen et al., 2013). The cyclic process is described as a three step process: 1) Gas is injected
into a reservoir (injection phase). 2) The well is shut-in for a certain period (soaking phase),
it is meant for the reservoir to re-pressurize and in the process let the injected gas into the
matrix area through pressure gradient and di↵usion. As the gas penetrates into matrix,
it swells the oil and reduces the viscosity. 3) The well is put back on production and the
swelled oil with the reduced viscosity is recovered (Gamdi et al., 2013). He elaborates that
cyclic gas injection could be an e↵ective technique in unconventional reservoirs because well-
to-well connectivity is not required. In addition, multi-stage hydraulic fracturing provides
a large contact area for the injected gas to penetrate and di↵use into the low-permeability
matrix. The combination of hydraulic fractures and natural fractures provides the required
conduits for the injected gas to reach a much bigger matrix contact area. Conventional
miscible gas injection where oil displacement alone in the matrix no longer applies. It would
take a very long time for gas to propagate from the injector to the producer well in ultra-low
permeability shale reservoirs (Chen et al., 2013). The supercritical gases considered in this
thesis work are CO2 and natural gas liquid (NGL) comprising of C2 � C4+. According to
Ho↵man et al. (2014), it is estimated in January 2014, 340 million standard cubic feet per
day (MMSCF/day) was flared, which is equivalent to 125 BCF per year. He illustrates that
it is equivalent of natural gas for 2 million homes.
CO2 flooding was viewed as a promising enhancing technique for complex fracture reser-
voirs (Wang et al., 2010). The following recovery mechanisms of gas injection were listed in
naturally fractured reservoirs: gravity drainage, molecular di↵usion, and convection (Chor-
dia and Trivedi, 2010) . Through oil swelling and viscosity reduction the trapped oil is
extracted out of the matrix. Several investigators have evaluated CO2 injection through
laboratory studies. A series of experiments was conducted using low permeability (0.02 mD
to 1.3 mD) siliceous shale core samples with medium porosity (30% to 40%) (Kovscek et al.,
2009). He saturated the samples with live oil, depleted them to a lower pressure and injected
CO2 as both immiscible and near miscible modes. Two injection schemes, for each mode
10
were conducted: countercurrent flow and cocurrent flow. They used X-ray computed tomog-
raphy to help with phase flow visualization and distribution. The incremental oil recovery
from immiscible CO2 flooding was in the range of 0-10% for countercurrent flow and 18-25%
for the cocurrent flow. As for the near miscible CO2 mode, the incremental recovery was
25% for the countercurrent flow and 10% for the cocurrent flow. They concluded that near
miscible injection shows multiple contact miscibility and exhibit higher recovery factor as
compared to immiscible injection.
Gamdi et al. (2013) studied the e↵ects of cyclic gas injection on oil recovery by performing
experiments using unfractured core plugs from Barnett, Marcos, and Eagle Ford formations.
For these experiments, mineral oil (Soltrol 130) was used and the injected gas was nitrogen
instead of CO2. Also investigated were the e↵ects of cyclic time and injection pressures. The
experiments show that cyclic gas injection can increase recovery from 10-50% depending on
the injection pressure and the type of shale core.
Hawthorne et al. (2013) conducted CO2 oil extraction experiments in the laboratory
at 5,000 psi and 230 °F using millimeter-size Bakken chips and centimeter-diameter core
plugs. They concluded that oil was mobilized because of CO2 miscibility with reservoir oil,
by viscosity reduction and di↵usion mass transfer. The exposure time was up to 96 hours
for the Middle Bakken chips (clastic sediment) which resulted in near-complete hydrocarbon
recovery. However, the oil extraction experiments required smaller chips and larger exposure
time for the Upper Bakken (shale). For field applications, solvent extraction is very slow and
modest because the specific surface area of reservoir matrix blocks is very small compared
to the laboratory samples used by Hawthorne et al. (2013). Nevertheless, these experimen-
tal results provide the impetus to pursue EOR in unconventional reservoirs, and numerical
modeling becomes the tool to scale laboratory results to field. Thus, in unconventional reser-
voirs, CO2 and NGL solvents can potentially mobilize matrix oil by miscibility (promoted
by solvent extraction via condensing-vaporizing gas process) leading to counter-current oil
flow from the matrix instead of oil displacement in the matrix.
11
Shoaib and Ho↵man (2009) evaluated the impact of CO2 flooding in Elm Coulee field
through numerical simulations. They concluded that continuous horizontal injection is better
overall as it’s provides higher injection rates and also more beneficial in the long term. Their
study showed recovery factor increase to 16% after 18 years of injection. Ho↵man (2013)
later studied the impact of injecting various gases for enhanced oil recovery in shale oil
reservoirs. Gases considered were CO2, immiscible hydrocarbon and miscible hydrocarbon
gases available in the field. He concluded that recovery e�ciency is similar for both miscible
and CO2 gases and an increase of up to 20% with gas injection is possible.
Wang et al. (2010) accessed the potential of CO2 flooding for Bakken formation in
Saskatchewan. He evaluated through simulation work, the e↵ects of injection well pattern,
continuous and cyclic injection schemes, waterflooding and CO2 flooding, injected gas com-
position, and reservoir heterogeneity. He concluded CO2 flooding after primary production
is more e↵ective and promising in Saskatchewan.
Chen et al. (2013) evaluated using a compositional model the relationship between
reservoir heterogeneity and CO2 hu↵ and pu↵ technique. They concluded that recovery rate
raises to a peak rate and declines rapidly during production stage. They observed that the
peak rate decreased with increasing the hu↵ and pu↵ cycle. The use of longer shut-in does
not increase the recovery rate because CO2 penetration into the matrix is limited due to
the low permeability. Reservoir heterogeneity contributes to a faster decline in the recovery
rate.
Wan et al. (2014) evaluated using numerical modeling the potential of EOR cyclic gas
injection in stimulated shale oil reservoir. They used a dual-continuum model to attain
better characterization of matrix, fractures and fissures. They concluded cyclic gas injection
is feasible and can improve significantly incremental oil recovery.
12
CHAPTER 3
COMPOSITIONAL MODELING
In this chapter, the volume balance compositional modeling method (VBM) is presented
for single-porosity and dual-porosity reservoirs. The Peng-Robinson equation of state (PR-
EOS) is presented and an appropriate two-phase flash algorithm which constitute the heart
of phase behavior calculations. Finally, the validation of thermodynamic model is presented
by comparing with experimental data and the CMG PVT software package (WinProp). Sim-
ilarly, the pressure solution is validated against the CMG compositional simulator (GEM).
3.1 VOLUME BALANCE METHOD (VBM)
Compositional modeling using volume balance method was developed to allow accurate
sequential computation as an improvement to an earlier sequential approach by Kazemi
et al., 1978 (Acs et al., 1982; Watts, 1986; and Wong et al., 1990). The volume balance
converts mass balance to volumetric balance for the entire fluid system. Specifically, the
volume balance method reduces the component flow equations to a single pressure equation
using partial molar volumes as weighting factors. The model developed in this study is
an isothermal three phase system (oil, gas, and water) using volume balance method. The
hydrocarbon phases can be liquid, gas, or both and there is interphase mass transfer between
hydrocarbon phases. The aqueous phase is treated as a standalone and there is no interphase
mass transfer with the hydrocarbon phase. The volume balance pressure equation in single
porosity system is expressed in Equation 3.1. The complete derivation of the pressure
equation will be shown in the Appendix A.
The volume balance method is a technique that is more amenable to rate transient
analysis as it reduces the multicomponent flow equations to a single pressure equation using
partial molar volumes.
13
nc+1X
c=1
�̄n
t
c
Uc
(pn+1o
) +MoleCorr = Vr
�(c�
+ cv
|z
c
)@p
o
@t(3.1)
In Equation 3.1, Uc
(pn+1o
) is the net molar flux of component c per block volume. It is
defined below for a 1-D system as a combination of phase transmissivity terms and source
terms (no gravity and capillary e↵ects):
Uc
= Ua
c
+ Ub
c
(3.2)
Ua
c
= 4x
(T n
x
�n
o
⇠no
xn
c
)4x
pn+1o
+4x
(T n
x
�n
g
⇠ng
ync
)4x
pn+1o
+4x
(T n
x
�n
w
⇠nw
wn
c
)4x
pn+1o
(3.3)
Ub
c
= ⇠no
xn
c
qno
+ ⇠ng
ync
qng
+ ⇠nw
wn
c
qnw
(3.4)
The term �̄n
t
c
is known as the partial molar volume of a multiphase system with respect
to a component c. It is defined as the change of the system total volume with respect to
change of the mass ( or total number of moles ) of component c at a constant pressure,
temperature, and number of moles. Mathematically,
�̄n
t
c
=
✓@V
t
@Nt
c
◆|p
,
T,N
t,j 6=c
(3.5)
The terms c�
and c�
are pore and fluid compressibilities. The partial molar volume
and fluid compressibility have embedded flash and contain the phase behavior information
encoded in them. Both are described in detail in Appendix A. Molecorr [ 1day
] is a mole
correction term and will be explained in detail in section 3.2.
In this work, the hydrocarbon components exist only in the oil and gas phases and the
water component only exists in the aqueous phase as a standalone. The mole fraction of
each phase can be expressed as follows:
y = (y1, y2, . . . , ync, 0) (3.6)
x = (x1, x2, . . . , xnc
, 0) (3.7)
14
w = (0, 0, . . . , 0, 1) (3.8)
The sum of mole fractions for each phase is shown satisfy the constraint below:
nc+1X
c=1
yc
= 1 (3.9)
nc+1X
c=1
xc
= 1 (3.10)
nc+1X
c=1
wc
= 1 (3.11)
The hydrocarbon overall mole composition also satisfies the constraint
nc+1X
c=1
zc
= 1 (3.12)
3.2 FORMULATION AND IMPLEMENTATION OF VBM IN DUAL-POROSITYSYSTEMS
The liquid-rich shale reservoirs will be described using a dual-porosity model to bet-
ter capture the interaction between matrix and fractures (induced hydraulic fracture and
pre-existing natural fractures and fissures). The dual-porosity framework was built and for-
mulated on the assumption that the matrix feeds the fractures which in turn connect to the
wellbore. In addition, flow from fracture to fracture is possible, however, flow from matrix
to matrix is not.
There are two molar component flow equations, one for the fracture and other for the
matrix media. These equations are connected through the transfer function. The governing
equations are:
nc+1X
c=1
��̄n
t
c
uc
�f
�nc+1X
c=1
⇣�̄n
t
c
f
⌧t
c
m/f
⌘+MoleCorr
f
= �f
(c�
+ cv
|z
c
)f
@po
f
@t(3.13)
nc+1X
c=1
⇣�̄n
t
c
m
⌧t
c
m/f
⌘+MoleCorr
m
= �m
(c�
+ cv
|z
c
)m
@po
m
@t(3.14)
15
The total transfer function ⌧t
c
m/f
for each component for all the phases is defined as
⌧t
c
m/f
= xc
⇠o
⌧o
+ yc
⇠g
⌧g
+ wc
⇠w
⌧w
(3.15)
The individual phase transfer functions are described as
⌧o
= �km
�o
f/m
h(p
of
� pom
) +⇣�
z
�
⌘�o
[(hwf
� hwm
)� (hof
� hom
)]i
(3.16)
⌧g
= �km
�g
f/m
h(p
gf
� pgm
) +⇣�
z
�
⌘�g
[(hgf
� hgm
)� (hgf
� hgm
)]i
(3.17)
⌧w
= �km
�w
f/m
h(p
wf
� pwm
) +⇣�
z
�
⌘�w
[(hwf
� hwm
)� (hwf
� hwm
)]i
(3.18)
The shape factor (�) is a geometric factor characteristic of the geometry and boundary
conditions of the matrix block. Kazemi et al. (1976) proposed shape factor expression based
on standard seven-point finite di↵erence as:
� = 4
1
L2x
+1
L2y
+1
L2z
�(3.19)
Where Lx
, Ly
, and Lz
represents the dimensions of a matrix block. The height of gas,
oil, and water columns in the matrix and fracture can be defined below:
hg
f/m
=
✓Sg
1� Swr
� Sor
◆
f/m
Lz
(3.20)
ho
f/m
=
✓Sg
1� Swr
� Sor
◆
f/m
Lz
(3.21)
hw
f/m
=
✓Sg
1� Swr
� Sor
◆
f/m
Lz
(3.22)
The MoleCorrf
and MoleCorrm
are two correction terms that are included in the pres-
sure equations. Due to the explicit nature of the phase saturation calculations, a discrep-
ancy in the number of moles in the system was observed in both fracture and matrix
media and a “mole correction term” was introduced to rectify the material balance error for
the system. It appears in the compositional pressure equation as a source term. Mathemat-
ically there are defined as:
16
MoleCorrf/m
=
✓vt
4t
◆⇣Nn
t
�Nn,S
t
⌘
f/m
(3.23)
Where vt
is the total specific volume [ ft
3
lb�mole
], Nn
t
is the total number of moles for the
system (matrix or fracture) after the pressure solution at time n and Nn,S
t
is the total number
of moles in the system (matrix or fracture) once the saturations are computed [lb �mole].
The time-step is 4t [days]. For implementation of volume balance method in dual-porosity
systems, the algorithm decouples the hydrocarbon phase and the aqueous phase in the matrix
and fracture and is unique for its ease of clarity and implementation. This approach is
desirable for our application as the water is considered immobile. The sequential approach
starts with solving for fracture pressures pn+1o
f
and matrix pressures pn+1o
m
implicitly. Next,
the node pressures are used to solve for phase velocities, which are used, in turn, to solve
for component compositions. Finally, the new compositions are flashed in each grid cell to
determine the new state of equilibrium including phase saturations. Once phase saturations
are obtained a check is performed on the number of moles in the system (shown in yellow
color), then the appropriate MoleCorr is determined for the next time step. The general
implementation of volume balance method is shown on Figure 3.1.
3.3 THERMODYNAMIC MODEL
In compositional modeling, proper capture of phase behavior e↵ects is essential. It be-
comes important when characterizing fluids with compositional variability as in liquid-rich
shale reservoirs. In our volume balance model, the main thermodynamic parameters are the
total partial molar volume (�̄t
) and the fluid compressibility (c�
) . The following sections
will cover the phase behavior and the relevant volumetric calculations.
17
Figure 3.1: General volume balance implementation.
18
3.3.1 Phase Equilibria and Flash
A hydrocarbon system is considered in thermodynamic equilibrium if the oil phase and
gas phase coexist. In our formulation, the aqueous phase is considered as a separate phase
and does not mix with the hydrocarbon phase. The equilibrium constraints in our model
are expressed through equality of fugacities (fL
c
, fV
c
). For each component in the hydro-
carbon liquid phase is equal to gas phase as follows:
fL
c
= fV
c
(3.24)
Equilibrium ratio (kc
) is the ratio of mole fraction of component (c) in the vapor phase
(yc
) to that in the liquid phase (xc
), mathematically,
kc
=yc
xc
(3.25)
Wilson (1968) proposed an empirical expression to estimate equilibrium ratio (kc
) using
critical pressure (pc,c
), critical temperature (Tc,c
), and acentric factor (!c
) of a component c
presented below:
kc
=pc,c
pexp
5.371 (1 + !
c
)
✓1� T
c,c
T
◆�(3.26)
Flash calculations determine the split of a hydrocarbon system at a given pressure, tem-
perature, and overall mole composition. These calculations are performed to determine the
mole fraction of liquid phase (xc
) and of gaseous phase (yc
). Also determined is the number
of moles of liquid phase (No
) and number of moles of gaseous phase (Ng
) in a hydrocarbon
fluid system at given pressure and temperature. The algorithm and implementation of flash
calculations will be shown in greater detail in Appendix B.
3.3.2 Equation of State (EOS) (PR 1976)
Peng and Robinson (1976) equation of state (PR EOS) was used to accurately describe the
volumetric and phase behavior of a hydrocarbon system. Peng and Robinson (1976) proposed
a two-constant equation for improved predictions, specifically liquid-density predictions. The
PR EOS is presented as:
19
p =RT
v � b� a
v(v + b) + b(v � b)(3.27)
where p is pressure, T is the temperature, R is a gas constant, v is the specific volume, a is
’attraction’ parameter, and b is a ’repulsion’. A substitution is made by replacing specific
molar volume v = ZRT
p
then a cubic equation in terms of Z factor is obtained below:
Z3 � (1� B)Z2 +�A� 3B2 � 2B
�Z �
�AB � B2 � B3
�(3.28)
Details of the cubic EOS will shown in Appendix B.
3.4 VALIDATION OF THE THERMODYNAMIC MODEL
A three-component fluid system was used to validate the thermodynamic routines used
in our volume balance compositional model. The properties of the three-component fluid
system is shown on Table 3.1. The phase diagram for this fluid system is shown in Figure 3.2
with a bubble point pb
= 3600 psia and a reservoir temperature Tr
= 180 oF . In order to
validate the thermodynamic calculations, flash was performed at di↵erent pressure intervals
(6500 psia - 1500 psia) and this was done to check how well can the thermodynamic routine
predict phase behavior of the fluid system as it crosses to a two-phase region. The results
from the developed thermodynamic routine was compared with a commercial PVT package
software WinProp (CMG, 2013) with good agreement as shown on Figure 3.3 for density
and z factor calculations and in Figure 3.4 for fugacities and viscosity calculations.
Since the heart of the volume balance method is partial molar volume (�̄n
t
c
) and fluid
compressibility (c�
) calculations, the developed algorithm for partial molar volume (�̄n
t
c
) was
validated against published experimental data of Wu and Ehrlich (1973) where a known
mixture of 95 g-moles C2 and 5 g-moles of nC7 at 80oC and 74.5 atm were used. Later 2
g-moles of nC7 was added at a time to the mixture while holding pressure and tempera-
ture constant and the new volume of the mixture was measured. The experimental results
are compared against partial molar volume routine derived from thermodynamic principles.
Fluid compressibility calculations were compared with WinProp PVT data. Figure 3.5 shows
20
the validation with very good agreement. Derivations of partial molar volume �̄n
t
c
and the
fluid compressibility c�
is presented in detail in Appendix B.
Table 3.1: Three-component fluid system used for thermodynamic validationFluid Characterization
Components Critical Critical Acentric Molecular MolePressure Temperature Factor Weight Fraction(psia) ( oR ) (��) (lbm/lbmmol) (frac)
C1 667.19 343.08 0.008 16.043 0.70C4 551.10 765.36 0.193 58.124 0.20C10 367.55 1119.78 0.443774 134 0.10
Figure 3.2: Phase envelope for C1 = 0.70 , C4 = 0.20 ,and C10 = 0.10.
3.5 VALIDATION OF THE VBM COMPOSITIONAL MODEL
For validation purposes, the model parameters, setup and results are shown on the next
two subsections.
21
(a)
(b)
Figure 3.3: Thermodynamic validation between developed routine for density and z factorcalculations with CMG PVT Package (WinProp).
22
(a)
(b)
Figure 3.4: Thermodynamic validation between developed routine for fugacity and viscositycalculations with CMG PVT Package (WinProp).
23
(a)
(b)
Figure 3.5: Thermodynamic validation of partial molar volume and fluid compressibilitycalculations.
24
3.5.1 Model Parameters and Setup
A simulation run was performed to check the integrity of the developed dual porosity
volume balance model with an available commercial compositional simulator (CMG GEM,
2013). Table 3.2 shows the reservoir properties used in the simulation run. Table 3.3 shows
the fluid properties used for the simulation run. This fluid system has a bubble point
pb
= 3600 psia at the reservoir temperature of Tr
= 180 oF . The well will be operated at
bottom hole pressure constraint pwell
= 1500 psia . The relative permeability curves for the
simulation were the same for the matrix and fracture as shown on Figure 3.6 and this was
done to make the case simple. The connate water saturation Swc
= 0.40, the oil residual to
water saturation Sorw
= 0.25, and gas connate saturation Sgc
= 0.05. The Corey exponents
were no = 3, nw = 3, nog = 3, and ng = 3 (Corey and Rathjens,1956).
Table 3.2: Test case reservoir parametersReservoir Dimensions and Properties
Nx
, Ny
, Nz
7x7x14x ,4y, 4z (ft) 64.285, 64.285, 40Length (ft) 500Width (ft) 500Thickness (ft) 40Depth (ft) 10000Matrix Porosity (frac) 0.05324Matrix Permeability (md) 10Fracture Porosity (frac) 0.001Fracture Permeability (md) 10Lx
, L, Lz
(ft) 3, 3, 3
3.5.2 Results and Comparison
A depletion run was done for 50 days. The producer well was located at the center of
the grid Nx
, Ny
, Nz
= 4 , 4 , 1. The well was operated with bottom-hole pressure boundary
condition pwell
= 1500 psia. The pressure profile shows the producer node goes below the
bubble point pressure after 3 days. Both the pressure and the saturation profiles for the
25
(a) Oil water relative permeability curves.
(b) Liquid-gas relative permeability curves.
Figure 3.6: Relative permeability curves
26
Table 3.3: Three-component fluid system used for simulation runFluid Characterization
Components Critical Critical Acentric Molecular MolePressure Temperature Factor Weight Fraction(psia) ( oR ) (��) (lbm/lbmmol) (frac)
C1 667.19 343.08 0.0080 16.043 0.70C4 551.10 765.36 0.1930 58.124 0.20C10 367.55 1119.78 0.4438 134.000 0.10
producer node match accurately with the commercial compositional simulator (CMG GEM,
2013) as seen in Figure 3.7. The cumulative oil and gas produced are also validated with
the commercial simulator results as shown in Figure 3.8. Furthermore, there is a good
comparison for the total fluid compressibility (cv
) parameter. Finally, two material balance
(MB) error calculations for the system were performed. The first included the mole correction
term (MoleCorr) and the other did not. As seen on Figure 3.9, inclusion of the correction
term is important, and as pointed out earlier, this was due to the explicit nature of the phase
saturation calculations. A discrepancy was discovered and the system had mass ’lost’ over
time. This discrepancy caused the MB error to increase with time (>3%). This problem
was solved however, once the known amount of missing number of moles was calculated,
it was added as a source term to the pressure equation to minimize the error over time.
Overall, the corrected volume balance method has low MB error of <1%.
The improved compositional volume balance model will used in chapter 4 to study a field
case with Bakken reservoir properties. A depletion run on a single stage hydraulic fracture
will be simulated to provide rate and pressure production data. Then this data will be used
for compositional rate transient analysis.
27
(a)
(b)
Figure 3.7: Validation of pressure and saturation profile ( VBM vs CMG GEM simulator).
28
(a)
(b)
Figure 3.8: Validation of cummulative oil and cummulative gas (VBM vs CMG GEM simu-lator).
29
(a)
(b)
Figure 3.9: Comparison of material balance error (VBM vs CMG GEM simulator).
30
CHAPTER 4
COMPOSITIONAL RATE TRANSIENT ANALYSIS
This chapter presents compositional rate transient analysis in liquid-rich shale reservoirs.
First, a brief background is presented on the need to modify and review conventional rate
transient analysis (RTA) followed by the closed form solution for rate-normalized pressures of
a single-phase stimulated horizontal well. Then, an approximate solution for blackoil multi-
phase flow model is presented. A similar approximate solution for compositional multiphase
flow model which uses volume balance method is also presented. The compositional solution
was used to analyze rate transients generated by a compositional model for a shale reservoir.
Finally, a comparison of the analytical versus numerical model results is presented.
4.1 MODIFICATION OF RATE TRANSIENT ANALYSIS
Conventional rate transient analysis (RTA) is based on solution of di↵usivity equation for
a slightly compressible fluid. This technique is accurate for engineering applications, but for
reservoir fluids, which are multi-phase and composition-dependent, the technique requires
revision. Oils produced from liquid-rich shale reservoirs are highly composition-dependent
because such oils are very light and have large solution gas-oil ratios. For shale reservoirs,
conventional RTA methods must be revised to include compositional-dependent flow. To
test the compositional RTA, a 2D, three-phase, dual-porosity simulator, developed
in Chapter 3, was used. Specifically, I used the compositional model to generate flow rate
versus time at the well for constant pressure boundary condition.
4.2 RATE-NORMALIZED PRESSURE EQUATION
Flow rate and bottom-hole flowing pressure data are used to perform rate-transient anal-
ysis. Because it is di�cult to maintain either constant bottom-hole pressure or constant
31
rate, a practical approach is to use rate-normalized pressure equation to analyze well per-
formance. As will be shown important bilinear and linear flow regime information can be
obtained using the method. Initially, a single-phase closed-form solution for bilinear flow
and linear flow regimes will be presented. Then its extension to the blackoil/volatile and
composition systems will be presented.
4.2.1 Single-Phase Flow
The single-phase flow di↵usivity equation for a slightly compressible fluid, where the
mass balance is initially honored is:
r ·✓k
µ
◆rp
o
+ q̂o
= �ct
@po
@t(4.1)
The bilinear flow equation for slightly compressible, single-phase flow from a hy-
draulic fracture in a horizontal well is:
4pwf
qB=
(44.102)µ
(hnhf
)p
whf
khf
8<
:
1
µ (�ct
)f+m
kf,eff
!1/49=
; t1/4 +141.2µ
kf,eff
hnhf
swell
hf
(4.2)
Where,
4pwf
is the well flowing pressure change (psia), q is the flow rate (STB/day), B is the
formation volume factor (RB/STB), µ is the oil viscosity (cp), kf,eff
e↵ective permeability
(md), h is formation thickness (ft), L is the horizontal well length (ft), � is the porosity
(fracture/matrix), and ct
is the total compressibility (fracture/matrix). nhf
is the number
of hydraulic fracture stages, whf
is the width of the hydraulic fracture in (ft), and khf
is the
hydraulic fracture permeability (md). The skin factor for the hydraulic fracture is swell
hf
.
The hierarchy of flow is from stimulated macro-fractures to the hydraulic fractures to
the wellbore. The slope for the bilinear log-log plot of 4p
wf
qB
versus t, is 14 . The slope of
Cartesian plot of 4p
wf
qB
versus t1/4 during the bilinear flow period is used to estimate
hydraulic fracture permeability (khf
). From the equation 4.2 it takes the following form:
32
4pwf
qB= m
bl
SINGLE��
t1/4 + bbl
SINGLE��
(4.3)
Where,
the slope mbl
SINGLE��
is in [ psi
(RB/d)cppd
] and the y-intercept bbl
SINGLE��
is in [ psi
(RB/d)cp ] are:
mbl
SINGLE��
=(44.102)µ
(hnhf
)pw
hf
khf
8<
:
1
µ (�ct
)f+m
kf,eff
!1/49=
; (4.4)
bbl
SINGLE��
=141.2µ
kf,eff
hnhf
swell
hf
(4.5)
The numerical value for kf,eff
is obtained from the linear time plot.
The linear flow equation for slightly compressible, single-phase flow in the hori-
zontal well is:
4pwf
qB=
(4.064)�⇡
2
�µ
(hnhf
yf
)p
kf,eff
8<
:
1
µ (�ct
)f+m
!1/29=
; t1/2 +141.2µ
kf,eff
hnhf
swell
hf
(4.6)
Where,
yf
is the fracture half-length (ft) for a single transverse hydraulic fracture in a multistage
completion. The slope of log-log plot of 4p
wf
qB
versus t, is 12 . Furthermore,
4pwf
qB= m
l
SINGLE��
pt+ b
l
SINGLE��
(4.7)
Where,
the slope ml
SINGLE��
is in [ psi
(RB/d)cppd
] and bl
SINGLE��
is in [ psi
(RB/d)cp ]
ml
SINGLE��
=(4.064)
�⇡
2
�µ
(hnhf
yf
)pkf,eff
8<
:
1
µ (�ct
)f+m
!1/29=
; (4.8)
bl
SINGLE��
=141.2µ
kf,eff
hnhf
swell
hf
(4.9)
The slope of Cartesian plot of 4p
wf
qB
versus t1/2 yields the e↵ective fracture permeability
(kf,eff
).
33
4.2.2 Multi-phase Flow Blackoil Case
The multi-phase flow di↵usivity equation for a slightly compressible fluid is used to
derive the approximate solution in blackoil model.
r · [k (�w
+ �o
+ �g
)rpo
+ (q̂o
Bo
+ q̂g
Bg
+ q̂w
Bw
) = �ct
@po
@t(4.10)
r · [k (�t
)rpo
+ q̂t
= �ct
@po
@t(4.11)
Where,
q̂t
= q̂o
Bo
+ q̂g
Bg
+ q̂w
Bw
is the total flow rate per unit volume ((RB/day)/volume) and
�t
= �o
+ �g
+ �w
is total phase mobility. Individual phase mobility can be expressed as
�↵
= k
r↵
µ
↵
and B↵
is the phase formulation volume factor (RB/STB), where ↵ is the phase
either gas, oil, or water.
The approximate analytical solution of multi-phase flow in the bilinear flow regime
can be extracted using the same frame-work as the single-phase flow case. The bilinear regime
represents flow in the hydraulic fractures as it feeds the horizontal well. It can be presented
as:
4pwf
qt
=(44.102)
(hnhf
)p
whf
khf
�t
8<
:
�t
(�ct
)f+m
kf,eff
!1/49=
; t1/4 +141.2
kf,eff
hnhf
�t
swell
hf
(4.12)
where,
qt
= qo
Bo
+ qg
Bg
+ qw
Bw
is the total flow rate for blackoil system (RB/day). q↵
is
the phase flow rate (STB/day) and B↵
is the phase formulation volume factor (RB/STB),
where ↵ is the gas, oil, or water phase.
The slope for the bilinear log-log plot of 4p
wf
q
t
versus t, is 14 . The slope of Cartesian plot
of 4p
wf
q
t
versus t1/4 during the bilinear flow period is used to estimate hydraulic fracture
permeability (khf
). From the equation 4.12 it takes the following form
4pwf
qt
= mbl
MULTI��
t1/4 + bbl
MULTI��
(4.13)
34
Where,
the slope mbl
MULTI��
[ psi
(RB/d)cppd
] and the y-intercept bbl
MULTI��
[ psi
(RB/d)cp ] are:
mbl
MULTI��
=(44.102)
(hnhf
)p
whf
khf
�t
8<
:
�t
(�ct
)f+m
kf,eff
!1/49=
; (4.14)
bbl
MULTI��
=141.2
kf,eff
hnhf
�t
swell
hf
(4.15)
The linear flow equation at the hydraulic fracture face in a horizontal well can be
written as:
4pwf
qt
=(4.064)
�⇡
2
�
(hnhf
yf
)p
kf,eff
�t
8<
:
�t
(�ct
)f+m
!1/29=
; t1/2 +141.2
kf,eff
hnhf
�t
swell
hf
(4.16)
Where,
yf
is the fracture half-length (ft) for a single transverse hydraulic fracture in a multistage
completion. From a diagnostic log-log plot the 4p
wf
q
t
versus t, the slope is 12 for the linear
region.
From the equation 4.16 it takes the following form:
4pwf
qt
= ml
MULTI��
pt+ b
l
MULTI��
(4.17)
Where,
the slope ml
MULTI��
[ psi
(RB/d)cppd
] and bl
MULTI��
[ psi
(RB/d)cp ] are:
ml
MULTI��
=(4.064)
�⇡
2
�
(hnhf
yf
)pkf,eff
�t
8<
:
�t
(�ct
)f+m
!1/29=
; (4.18)
bl
MULTI��
=141.2
kf,eff
hnhf
�t
swell
hf
(4.19)
The slope of Cartesian plot of 4p
wf
q
t
versus t1/2 is used to estimate e↵ective fracture
permeability (kf,eff
).
35
4.2.3 Multi-phase Flow Compositional Case
The multi-phase flow pressure equation in compositional model using volume balance
method is:
nc+1X
c=1
�̄n
t
c
uc
(pn+1o
) = �(c�
+ cv
|z
c
)@p
o
@t(4.20)
where the molar flux is described as:
uc
= u↵
+ u�
= �(c�
+ cv
|z
c
)@p
o
@t(4.21)
u↵
= r ·⇥(⇠
o
xc
k�o
) +�⇠g
yc
k�n
g
�+ (⇠
w
wc
k�n
w
)⇤(rp
o
) (4.22)
u�
= (⇠o
xc
q̂o
) + (⇠g
yc
q̂g
) + (⇠w
wc
q̂w
) (4.23)
which carries a similar form as the multiphase di↵usivity equation 4.10
r · [k (↵o
+ ↵g
+ ↵w
)] (rpo
) + (q̂o
�o
) + (q̂g
�g
) + (q̂w
�w
) = �ct
@po
@t(4.24)
Where,
↵o/g/w
with units of [1/day] and can be expressed as:
↵o
=nc+1X
c=1
�̄n
t
c
⇠o
xc
k�o
(4.25)
↵g
=nc+1X
c=1
�̄n
t
c
⇠g
yc
k�g
(4.26)
↵w
=nc+1X
c=1
�̄n
t
c
⇠w
wc
k�w
(4.27)
The �o/g/w
can be expressed as:
�o
=nc+1X
c=1
�̄n
t
c
⇠o
xc
(4.28)
�g
=nc+1X
c=1
�̄n
t
c
⇠g
yc
(4.29)
36
�w
=nc+1X
c=1
�̄n
t
c
⇠w
wc
(4.30)
The total compressibility ct
with units of (1/psi) can be expressed as:
ct
= (c�
+ cv
|z
c
) (4.31)
From analogy, the approximate analytical solution of multi-phase flow for bilinear
flow regime for compositional models is developed from an extension of single-phase
flow. It can be presented as:
4pwf
(qg
�g
+ qo
�o
+ qw
�w
)=
(44.102)
(hnhf
)pw
hf
khf
(�t
)
8<
:
�t
(� [c�
+ cv
|z
c
])f+m
kf,eff
!1/49=
; t1/4
+141.2
kf,eff
hnhf
(�t
)swell
hf
(4.32)
The bilinear slope of log-log plot 4p
wf
q
t
versus t, is 14 . The slope of Cartesian plot of
4p
wf
q
t
versus t1/4 during the bilinear flow period of the compositional model is used to
estimate hydraulic fracture permeability (khf
). From the equation 4.32 it takes the familiar
form:
4pwf
qt
= mbl
MULTI��
t1/4 + bbl
MULTI��
(4.33)
The slope mbl
MULTI��
[ psi
(RB/d)cppd
] and the y-intercept bbl
MULTI��
[ psi
(RB/d)cp ] are:
mbl
MULTI��
=(44.102)
(hnhf
)p
whf
khf
(�t
)
8<
:
�t
(� [c�
+ cv
|z
c
])f+m
kf,eff
!1/49=
; (4.34)
bbl
MULTI��
=141.2
kf,eff
hnhf
(�t
)swell
hf
(4.35)
Similarly, the approximate analytical solution of multi-phase flow for linear flow
regime for compositional models can be presented as:
37
4pwf
(qg
�g
+ qo
�o
+ qw
�w
)=
(4.064)�⇡
2
�
(hnhf
yf
)pkf,eff
(�t
)
8<
:
�t
(� [c�
+ cv
|z
c
])f+m
!1/29=
; t1/2
+141.2
kf,eff
hnhf
(�t
)swell
hf
(4.36)
From the equation 4.36, it takes the following form:
4pwf
qt
= ml
MULTI��
pt+ b
l
MULTI��
(4.37)
Where,
the slope ml
MULTI��
[ psi
(RB/d)cppd
] and bl
MULTI��
[ psi
(RB/d)cp ] are:
ml
MULTI��
=(4.064)
�⇡
2
�
(hnhf
yf
)p
kf,eff
(�t
)
8<
:
�t
(� [c�
+ cv
|z
c
])f+m
!1/29=
; (4.38)
bl
MULTI��
=141.2
kf,eff
hnhf
(�t
)swell
hf
(4.39)
4.2.4 Summary of the Analytical Solutions
In summary, a bilinear flow equation for slightly compressible, single-phase flow
was presented as shown in Equation 4.2. A linear flow equation for slightly compressible,
single-phase flow was presented as shown in Equation 4.6. An approximate analytical
solution of multi-phase flow in the bilinear flow regime for blackoil case is extracted
using the same frame-work as the single-phase flow case and is represented by Equation 4.12.
Similarly, an approximate analytical solution of multi-phase flow in the linear flow
regime for blackoil case is extracted and shown in Equation 4.16. Finally, from analogy,
the novel approximate analytical solutions of multi-phase flow for bilinear flow regime
and linear flow regime for compositional models are developed from an extension of
single-phase flow and multi-phase blackoil cases.
38
Table 4.1: Summary of Bilinear solutions for single-phase, multi-phase black oil, andmulti-phase compositional models
4p
wf
qB
= (44.102)µ
(hnhf
)p
w
hf
k
hf
⇢⇣1
µ(�ct
)f+m
k
f,eff
⌘1/4�t1/4 + 141.2µ
k
f,eff
hn
hf
swell
hf
4p
wf
(qo
B
o
+q
g
B
g
+q
w
B
w
) =(44.102)
(hnhf
)p
w
hf
k
hf
(�t
)
⇢⇣�
t
(�ct
)f+m
k
f,eff
⌘1/4�t1/4 + 141.2
k
f,eff
hn
hf
(�t
)swell
hf
4p
wf
(qg
�
g
+q
o
�
o
+q
w
�
w
) =(44.102)
(hnhf
)p
w
hf
k
hf
(�t
)
(✓�
t
(�[c�
+c
v
|z
c
])f+m
k
f,eff
◆1/4)t1/4 + 141.2
k
f,eff
hn
hf
(�t
)swell
hf
These are represented by Equation 4.32 and Equation 4.36 where the row is highlighted
in yellow. These equations will be validated and utilized in a case study in Subsection
4.3. Table 4.1 shows the summary for bilinear flow analytical solutions and Table 4.2 is the
summary for linear flow analytical solutions.
Table 4.2: Summary of Linear solutions for single-phase, multi-phase black oil, and multi-phase compositional models
4p
wf
qB
=(4.064)(⇡
2 )µ(hn
hf
y
f
)p
k
f,eff
⇢⇣1
µ(�ct
)f+m
⌘1/2�t1/2 + 141.2µ
k
f,eff
hn
hf
swell
hf
4p
wf
(qo
B
o
+q
g
B
g
+q
w
B
w
) =(4.064)(⇡
2 )(hn
hf
y
f
)p
k
f,eff
�
t
⇢⇣�
t
(�ct
)f+m
⌘1/2�t1/2 + 141.2
k
f,eff
hn
hf
�
t
swell
hf
4p
wf
(qg
�
g
+q
o
�
o
+q
w
�
w
) =(4.064)(⇡
2 )(hn
hf
y
f
)p
k
f,eff
(�t
)
(✓�
t
(�[c�
+c
v
|z
c
])f+m
◆1/2)t1/2 + 141.2
k
f,eff
hn
hf
(�t
)swell
hf
4.3 LIQUID-RICH UNCONVENTIONAL RESERVOIR CASE STUDY
To validate the compositional RTA (Equation 4.32 and Equation 4.36), a 2D, three-phase,
dual-porosity simulator that uses volume balance method developed in Chapter 3 was used
to generate production data. Specifically, flow rate versus time data was generated for a
single-stage hydraulic fracture in liquid-rich unconventional reservoir. Since the permeability
39
was low (0.0001 md), refined gridding was used to capture transient e↵ects. The model
results were validated with GEM compositional simulator. This model was used to study
multiphase flow regimes observed in liquid-rich unconventional reservoirs. From the analysis
of multiphase bilinear and linear flow regimes, the flow parameters (khf
and kf,eff
) used as
inputs in the numerical model were back calculated using analytical solutions Equation 4.32
and Equation 4.36. Specifically, in the bilinear region, the hydraulic fracture permeability
(khf
) calculated using analytical method was compared to the numerical model input, and
for the linear region, the e↵ective permeability (kf,eff
) was also calculated and compared.
4.3.1 Model Parameters and Setup
The model input parameters for the depletion run are shown in Table 4.3. The matrix
permeability is low (km
= 0.0001md). The fracture conductivity is (FCD
= 10mdft), from
fracture width of (whf
original
= 0.001 ft) and the hydraulic permeability (khf
= 10000md).
Due to computational problems caused by small grids, the hydraulic fracture was pseudoized
to have permeability (khf
= 5md) and width (whf
= 2 ft), to maintain the same fracture
conductivity. The well schematic and dimensions are shown on figure Figure 4.1(a).
Table 4.3: Multiphase case study reservoir parameters.Reservoir Dimensions and Properties
Nx
, Ny
, Nz
21x11x14x ,4y, 4z (ft) VARI, 36.364, 40Length (ft) 400Width (ft) 400Thickness (ft) 40Depth (ft) 10000Matrix Porosity (frac) 0.05324Matrix Permeability (md) 0.0001Fracture Porosity (frac) 0.001E↵ective Fracture Permeability (md) 0.005Hydraulic Fracture Width (ft) 0.001Fracture Half Length (ft) 200Lx
, L, Lz
(ft) 5, 5, 5
40
(a)
(b)
Figure 4.1: Refined gridding and well dimensions for multiphase depletion model.
41
4.3.2 Fluid Parameters
Table 4.4 shows the fluid properties used for the multiphase depletion run. The tight oil
fluid system has a bubble point pb
= 3350 psia at the reservoir temperature of Tr
= 240 oF
as shown on Figure 4.2. The well will be operated at bottom hole pressure constraint
pwell
= 2000 psia .
Table 4.4: Three-component fluid system used for multiphase depletion run.Fluid Characterization
Components Critical Critical Acentric Molecular MolePressure Temperature Factor Weight Fraction(psia) ( oR ) (��) (lbm/lbmmol) (frac)
C1 667.19 343.08 0.00800 16.043 0.60C7 455.13 977.76 0.308301 96.000 0.30C10 367.55 1119.78 0.443774 134.000 0.10
4.3.3 Rock-Fluid Parameters
The relative permeability curves for the simulation were the same as ones used in chapter
3 Figure 3.6, the matrix and fracture as same and this was done to make the case simple.
The connate water saturation Swc
= 0.40, the oil residual to water saturation Sorw
= 0.25,
and gas connate saturation Sgc
= 0.05. The Corey exponents were no = 3, nw = 3, nog = 3,
and ng = 3 (Corey and Rathjens, 1956).
4.4 CASE STUDY RESULTS AND ANALYSIS
In this section, results are presented for the liquid-rich unconventional reservoir case
study shown in Table 4.4. The depletion run was for 365 days, and the horizontal producer
well was located in the i-direction of the grid Ny
, Nz
= 6 , 1. The pressure profile shows the
producer node going below the bubble point pressure after 50 days. The node pressure and
oil rate are validated with the CMG GEM simulator shown in Figure 4.3. The simulated
production data is used to perform multiphase rate transient analysis. As noted earlier the
compositional model was constructed using dual porosity architecture where the matrix feeds
42
(a)
(b)
Figure 4.2: Phase envelope and component specification for tight oil system.
43
the hierarchy of local fractures which in turn feed the hydraulic fractures and subsequently
connect to the wellbore. The local fractures are classified similar to how pores are defined in
the literature. Instead of pore diameter, fracture width is used. The following classification
is adopted, microfractures is less than 2 nm in fracture width, between 2 and 50 nm as
mesofractures and larger than 50 nm as macro-fractures (Alharthy et al., 2012).
There are three anticipated flow regions: 1) very early linear flow in the hydraulic
fractures as shown in Figure 4.4(a) - it is di�cult to observe 2) bilinear where flow is in
the micro, meso and macro-fractures as shown in Figure 4.4(b), and 3) linear and bound-
ary dominated where flow is influenced by boundary e↵ects after a long time as shown
in Figure 4.4(c). Figure 4.4 shows the di↵erent flow regimes encountered in a stimulated
horizontal well. A diagnostic log-log plot of rate-normalized pressure versus time is as
shown in Figure 4.5. This diagnostic log-log plot of rate-normalized pressure versus time
is used to identify these flow regimes . Bilinear region has a slope of 1/4, linear region
has a slope of 1/2, and boundary dominated flow has slope of 1 as shown on Figure 4.5(a).
In the very early times, hydraulic fracture storage e↵ect is shown. Figure 4.5(b) shows the
derivative plot with a dual porosity v-shape signature.
Table 4.5 shows parameters used for bilinear flow analysis. For the bilinear flow regime,
the comparison between calculated hydraulic fracture permeability (khf
) using analytical
model versus numerical model is very good (error within 2.5%) . Table 4.6 shows pa-
rameters used for linear flow analysis. In the linear flow analysis, the calculated (kf,eff
)
using analytical model is very close to the numerical model (error within 5%). An ana-
lytical approximation to the multi-phase solution was used successfully to perform reservoir
characterization and it was able to produce the numerical model input results.
44
(a)
(b)
Figure 4.3: Case study validation for multiphase flow depletion run.
45
(a) Extremely early linear flow in hydraulic fractures.
(b) Bilinear regime where flow is in micro, meso and macro-fractures.
(c) Linear and Boundary Dominated regime where flow aftera long time is influenced by boundary e↵ects.
Figure 4.4: Di↵erent flow regimes in stimulated horizontal well.
46
(a) Diagnostic plot showing bilinear, linear, and boundary dominated flowregimes.
(b) Dual porosity signature in a linear model.
Figure 4.5: Deciphered flow regimes and dual porosity feature.
47
Table 4.5: Bilinear multiphase flow analysis for depletion runBilinear Flow Regime
Viscosity oil (µo
) 0.0830Viscosity gas (µ
g
) 0.000Rel perm oil (k
ro
) 0.48Rel perm gas (k
rg
) 0.000�t
5.7831��1t
0.1729Parameters Analytical Model Numerical ModelSlope (bilinear) 9.34khf
whf
�t
59w
hf
2 2khf
5.118 5.000khf
whf
10.2 10
Table 4.6: Linear multiphase flow analysis for depletion runLinear Flow Regime
Viscosity oil (µo
) 0.0768Viscosity gas (µ
g
) 0.0230Rel perm oil (k
ro
) 0.27Rel perm gas (k
rg
) 0.001786�t
3.5933��1t
0.2783Parameters Analytical Model Numerical ModelSlope (linear) 3.39kf,eff
�t
0.04kf,eff
0.01053 0.010
48
CHAPTER 5
ENHANCED OIL RECOVERY - LABORATORY AND FIELD STUDY
This chapter presents enhanced oil recovery in liquid rich shales. First, the concept of
cyclic supercritical fluid extraction using solvents like carbon dioxide (CO2) is presented.
Second, laboratory data from CO2 cyclic soaking experiments conducted at Energy & En-
vironmental Research Center (EERC) on Bakken cores is presented. Then, history match
of the laboratory data using compositional model is presented. Finally a field case from a
North Dakota Bakken well is first history matched and then enhanced oil recovery scheme
is performed on it and incremental oil is presented.
5.1 SUPERCRITICAL FLUID EXTRACTION
Supercritical Fluid Extraction (SFE) is the process of separating one component from
another using supercritical fluids such as Carbon Dioxide (CO2) as the extracting solvent.
SFE has been widely used in the food and pharmaceuticals industries to extract unwanted
materials or collect desired materials (Hawthorne et al., 2013). Cyclic SFE was proposed in
the form of soaking (hu↵-n-pu↵) for enhanced oil recovery in liquid rich shale reservoirs by
Chen et al. (2013). The process involves extracting fluid hydrocarbons in a cyclic manner
from a tight shale matrix using CO2 as the extracting solvent. The cyclic process is described
as a three step process: 1) injection phase, 2) soaking phase, and 3) production phase.
Liquid-rich shales consists of lean hydrocarbon production streams and would favorably be
extracted by supercritical solvents. Solvents such as CO2 and Natural Gas Liquids (NGL)
can potentially mobilize matrix oil by miscibility through soaking (hu↵-n-pu↵). Extraction
conditions for supercritical CO2 are above the critical temperature of 87.8 °F and critical
pressure of 1073 psia. This extraction process is completely di↵erent from oil mobilization
in conventional reservoirs, where the injected fluids mobilize oil to form an oil bank ahead
of the injected fluid and, then, push the oil bank through the matrix pores to an eventual
49
outlet. The cyclic supercritical fluid extraction is an advective-di↵usive-based process, with
the solvent required to di↵use into the matrix, and the extracted hydrocarbon to di↵use
out of the matrix. Hawthorne et al. (2013) conducted CO2 oil extraction experiments in
the laboratory at 5,000 psi and 230 °F using millimeter-size Bakken chips and centimeter-
diameter core plugs. They concluded that oil was mobilized because of CO2 miscibility with
reservoir oil, by viscosity reduction and di↵usion mass transfer. The exposure time was up
to 96 hours for the Middle Bakken chips (clastic sediments) which resulted in near-complete
hydrocarbon recovery. For Lower and Upper Bakken Shale, the oil extraction experiments
required smaller chips and larger exposure time. For field applications, solvent extraction is
modest because the specific surface area of reservoir matrix blocks is very small compared to
the laboratory samples used by Hawthorne et al. (2013). Nevertheless, these experimental
results provide the impetus to pursue EOR in unconventional reservoirs, and numerical
modeling becomes the tool to scale laboratory results to field.
5.2 BAKKEN CO2 SOAKING EXPERIMENTS
This section will present the experiment setup, the Bakken cores sizes, the procedure,
and results.
5.2.1 Laboratory Experiments and Experimental Procedures
Cylindrical Middle and Lower Bakken cores that are 3-4 cm long with 1 cm in diameter
are placed in 10 mL extraction vessel (one at a time). For the Middle Bakken cores, the
porosity range is 4.5% to 8.1% and the permeability range is 0.002 to 0.04 md (Kurtoglu,
2013). The permeability for the Lower Bakken cores is orders of magnitude lower (Hawthorne
et al., 2013). An ISCO pump injects CO2 at 5000 psi at the inlet valve of the extraction
vessel and maintains a constant delivery at that pressure. The extraction vessel is inside a
heat flow restrictor to maintain the temperature at 230 °F. There is space between the inside
of the extraction vessel wall and the cylindrical core, and CO2 flushes around the core sample
as opposed to being forced through the core sample (Hawthorne et al., 2013). The space is
50
similar to fracture surrounding a core matrix. During the injection phase, the outlet valve
is closed, and CO2 stays inside the extraction vessel to soak the core for a certain period
(50 minutes). After that, the outlet valve is opened for 10 minutes only, while the injection
pressure at the inlet valve is maintained at 5000 psia. This flushes the CO2 with extracted
oil phase from the core matrix to the collection vessel where it is analyzed using capillary
gas chromatography coupled with a flame ionization detector (GC/FID) (Hawthorne et al.,
2013). This hu↵ and pu↵ process is repeated up to 96 hours, where almost complete recovery
of 95% is achievable for Middle Bakken cores and up to 40% for Lower Bakken cores. The
cores are later crushed and soaked multiple times until no more significant hydrocarbon
recovery can be extracted. The experimental setup and process is shown in Figure 5.1.
Figure 5.1: Enhanced oil recovery experiments on Bakken Cores (performed at EERC)
5.2.2 Fluid System Properties
The Middle Bakken samples were cored and not properly preserved. The lighter com-
ponents were lost prior to the beginning of the CO2 soaking experiments. For modeling
purposes, the system fluid initialization composition is an unknown, however, the presence
of five field separator samples and laboratory produced stream samples can help reduce this
uncertainty. Five oil separator samples from Middle Bakken were analyzed, they all contain
51
intermediate components C11 up to C36+only and no lighter components as shown in Fig-
ure 5.2(a). From the CO2 soaking experiments, recovered streams were analyzed and also
confirm presence of intermediate hydrocarbons only (C11 to C29+) as shown in Figure 5.2(b).
The Lower Bakken sample retains more of the lighter hydrocarbon components starting from
C7 onwards as seen from produced streams in Figure 5.3.
5.2.3 Bakken Core Description
The Bakken formation is in the Williston Basin and it covers parts of Montana, North
Dakota, and South Dakota (Clarkson, 2011). The Bakken formation overlies the Upper De-
vonian Three Forks formation and underlies the Lower Mississippian Lodgepole formation.
The Bakken has three distinct members, the upper member (Upper Bakken), the middle
member (Middle Bakken), and the Lower member (Lower Bakken) as discussed by Kurtoglu
(2013). The Upper Bakken is organic-rich pyritic with fissile features and it is approximately
8 to 12 ft (Shoaib and Ho↵man, 2009). The total organic content (TOC) ranges from 12 to
36% weight, averaging 25 to 28% weight over large parts of the basin. It is considered as
the source rock for the Bakken formation. The Middle Bakken is organic-poor with TOC
of 0.1 to 0.3% weight (Price, 1999) and is the main reservoir. The Middle Bakken lithology
varies from clastics (including silts and sandstone) to carbonates (silty dolomites), with five
distinct lithofacies identified in North Dakota portion of the Williston Basin. The thickness
of Middle Bakken formation is 6 - 15 ft with porosity of 6 - 8% and permeability of 10 - 40
microdarcies (Shoaib and Ho↵man, 2009). The Lower Bakken is brownish, noncalcareous,
organic mudstone with an organic content up to 21%. It is approximately 0 - 6 ft thick and
very tight (Chen et al., 2013). Both the Upper and Lower Bakken contain a high concentra-
tion of Type II kerogen and are the source rocks for petroleum in Bakken formation. The
Middle Bakken cores used were obtained from depth 10848.50 ft. The air permeability is
0.038 md and porosity is 5.7%. Mineralogy analysis using XRD data is shown in Table 5.1.
Below are thin sections shown for this sample at di↵erent size magnification. As seen on
Figure 5.4, 400X (Plate 14C) is with plain transmitted light, and it shows monocrystalline
52
(a)
(b)
Figure 5.2: Compositions of separator samples and produced streams for Middle Bakken.
53
Figure 5.3: Produced composition stream for Lower Bakken core.
quartz grains are abundant (white color) with non-skeletal calcareous grains. Minor calcite
and Fe-Dol (tan and brown color), and some K-spar, Plagioclase, and Pyrite (black color)
are also observed. The 400X (Plate 14D) is with epiflourescent lighting technique (light
blue) and was used to observe micropores. The Lower Bakken cores used were obtained
from depth 10885.45 ft. There is no permeability and porosity data available, however, we
believe it is orders of magnitude lower than Middle Bakken core. Mineralogy analysis using
XRD data is shown in Table 5.2. Below are thin sections shown for this sample at di↵erent
size magnification. As seen on Figure 5.5, 400X (Plate 14C) is with plain transmitted light,
and it shows abundance of organic matter and suggests anoxic conditions that deposited the
Lower Bakken (amorphous - sapropelic OM) (Theloy and Sonnenberg, 2012) . Thin sections
and XRD analysis show that it is calcite and quartz dominated. Minor amount of clays such
as illite are also observed. The 400X (Plate 14D) is with epiflourescent lighting technique
(light blue) and was used to observe micropores. Pyrolysis analysis data shows Tmax of
443 oC within oil generating window, the hydrogen index (HI) 326 and oxygen index (OI)
5.199. The kerogen is Type II considered as marine deposit (Kurtoglu,2013).
54
Table 5.1: XRD analysis of Middle Bakken CoreMineralogy Content Middle Bakken Core
Clays
Chlorite 0
Total 5Kaolinite 0Illite 5MxIS 0
Carbonates
Calcite 21
Total 38Dolomite 0Fe-Dol 17Siderite 0
Other Minerals
Quartz 42
Total 57
K-spar 7Plagioclase 6
Pyrite 2Zeolite 0Barite 0
Figure 5.4: Thin sections for Middle Bakken core at di↵erent resolutions, mineralogy com-posed of abundant monocrystalline quartz grains (white color) with non-skeletal calcerousgrains, minor calcite and Fe-Dol (tan and brown color), and some K-spar, Plagioclase, andPyrite (black color).
55
Table 5.2: XRD analysis of Lower Bakken CoreMineralogy Content Lower Bakken Core
Clays
Chlorite 1
Total 7Kaolinite 0Illite 6MxIS 0
Carbonates
Calcite 42
Total 44Dolomite 2Fe-Dol 0Siderite 0
Other Minerals
Quartz 42
Total 49
K-spar 2Plagioclase 2
Pyrite 3Zeolite 0Barite 0
Figure 5.5: Thin sections for Lower Bakken core at di↵erent resolution, mineralogy composedof quartz and calcite dominated (white and tan color), minor amount of clays such as illite(dark brown color), and kerogen patches (black color).
56
5.2.4 Laboratory Results
The CO2 soaking experiment results for Middle Bakken core are presented in Figure 5.6.
The oil recovery factor for Middle Bakken core is up to 80% in 7 hours. Initially, before
the CO2 injection, the core is at atmospheric pressure in the extraction vessel. The pump
injects continuous supply of CO2 at 5000 psia and it takes less than 10 minutes to fill up
the extraction vessel. This ’initial repressurization’ has an e↵ect on the recovery of
hydrocarbons as empty pore space is quickly filled by CO2 and is seen at the beginning of
the first cycle of recovery in Figure 5.6. The amount of CO2 used is high and it quickly soaks
the core and is able to recover the hydrocarbons. The CO2 soaking experiment results for
Lower Bakken core are presented in Figure 5.7. The oil recovery factor for Lower Bakken
core is 19% in 7 hours and is much lower compared to Middle Bakken core. The permeability
is considered to be orders of magnitude smaller. The thin sections show presence of large
amounts of organic matter in a form of kerogen. This is considered as the source rock and
possibly immature and amount of moveable oil is much less. The calcite cement content is
a dominant mineralogy and can impede flow of hydrocarbons.
Figure 5.6: Oil recovery factor for Middle Bakken core soaking experiment.
57
Figure 5.7: Oil recovery for Lower Bakken core soaking experiment.
5.3 MODELING EXPERIMENTS
The following section will present numerical modeling of Bakken cores CO2 soaking ex-
periments. First. the gridding of the numerical model is presented, then the system fluid
initialization for non preserved cores is shown. Finally, the historical match between labo-
ratory oil recovery data compared to numerical model is presented.
5.3.1 Laboratory Model: Grid System
A single-porosity radial model was developed and used for the Bakken CO2 soaking
experiments. The radial grid was used to represent the extraction vessel and the cylindrical
cores. The dimensions of the Middle Bakken cylindrical cores are length 3.68 cm and diameter
1.13 cm. The extraction vessel dimensions used to store the core had dimensions of length
5.7 cm and diameter 1.5 cm. Figure 5.8 shows the radial grid system with dimensions. The
58
total space inside the extraction vessel wall and the core is 0.37 cm and each side is 0.185
cm. This space is meant to resemble a fracture surrounding the matrix.
Figure 5.8: Single-porosity radial grid system used in Bakken core CO2 soaking experiments.
Table 5.3 shows the matrix and fracture core properties, specifically the matrix perme-
ability is 0.043 md and the fracture is 750 md. The porosity of the core is 8%. Six layers
are used for the length of the extraction vessel with 4z = 0.95 cm. Eight radial rings are
used for the diameter, the outer ring represents the fracture with dimensions 4x = 0.185
cm on each side.
Table 5.3: Radial case for Middle Bakken coreReservoir Dimensions and Properties
ni
, nj
, nk
(Radial Grid) 8x1x6Core Length (cm) 3.68Core Diameter (cm) 1.13Extraction Vessel Length (cm) 5.7Extraction Vessel Diameter (cm) 1.5Matrix Porosity (frac) 0.08Matrix Permeability (md) 0.043Fracture Permeability (md) 750
59
5.3.2 Laboratory Model: Fluid System
As previously noted, the cores were not properly preserved and most of the lighter com-
ponents were lost. The Lower Bakken, however, retained some of the lighter components
(C7 � C10) as shown in Figure 5.3. Fluid initialization is an unknown, however, the pres-
ence of five Middle Bakken field separator samples shown in Figure 5.2(a) and laboratory
produced stream samples from CO2 soaking experiments, shown in Figure 5.2(b), can help
reduce this uncertainty. A synthetic Middle Bakken fluid composition was proposed and
used with the numerical model. The compositions are lumped in order to help with simula-
tion run time. Figure 5.9 shows the fluid composition and the phase envelope. Similarly for
the Lower Bakken a synthetic fluid composition that contains some lighter components is
proposed and used with the numerical model. Figure 5.10 shows the fluid composition and
the phase envelope.
5.3.3 Laboratory Model: Rock-Fluid System
Two sets of relative permeability curves were used for the numerical model to distinguish
the matrix and fracture mediums. The Bakken core is represented by the matrix relative
permeability curves. The matrix connate water saturation is Swc
= 0.25, the oil residual to
water saturation Sorw
= 0.25, and gas connate saturation Sgc
= 0.05. The Corey exponents
were no = 3, nw = 3, nog = 3, and ng = 3. Figure 5.11 shows the matrix medium relative
permeability curves. The space inside the extraction vessel is regarded as open fracture.
The fracture connate water saturation is Swc
= 0.01, the oil residual to water saturation
Sorw
= 0.01, and gas connate saturation Sgc
= 0.01. The Corey exponents were no = 1.2,
nw = 1.2, nog = 1.2, and ng = 1.2. Figure 5.12 shows the relative permeability curves for
fracture medium which is shown in yellow.
5.3.4 Laboratory Model: History Matching
To capture the underlying oil recovery mechanisms, the fluxes induced by pressure
gradient, gravity gradient, and concentration gradient (molecular di↵usion) were used
60
(a)
(b)
Figure 5.9: Middle Bakken synthetic lumped fluid composition and phase envelope.
61
(a)
(b)
Figure 5.10: Lower Bakken synthetic lumped fluid composition and phase envelope.
62
(a) Matrix oil-water relative permeability curves.
(b) Matrix gas-liquid relative permeability curves.
Figure 5.11: Relative permeability curves.
63
(a) Fracture oil-water relative permeability curves.
(b) Fracture gas-liquid relative permeability curves.
Figure 5.12: Fracture relative permeability curves.
64
in the history matching process. For the pressure gradient, the injector node of the numerical
model was maintained at 5000 psi (constant pressure injection) and the producer node was
operated at a lower value to create the potential gradient needed for flow during the collection
cycle. Similarly the gravity gradient was included in the flow equation, however, the e↵ects
are minor as the length of the core is small. Concentration gradient also was used to model
movement of CO2 from higher concentration (in fracture medium) to lower concentration (in
the matrix medium). All relevant phase behavior e↵ects were taken into account. Chapter
6 will discuss all the underlying recovery mechanisms in detail. Figure 5.13 shows Middle
Bakken and Lower Bakken history match with e↵ects of di↵erent gradients.
5.3.5 Discussion of Laboratory Results
In order to capture the underlying oil recovery mechanisms, the fluxes induced by pres-
sure gradient, gravity gradient, and concentration gradient (molecular di↵usion) were
used in the history matching process. For the pressure gradient, the injector node of the
numerical model was maintained at 5000 psi (constant pressure injection) and the producer
node was operated at a lower value to create the potential gradient needed for flow during
the collection cycle. Similarly the gravity gradient was included in the flow equation, how-
ever, the e↵ects are minor as the length of the core is small. Concentration gradient also was
used to model movement of CO2 from higher concentration (in fracture medium) to lower
concentration (in the matrix medium). All relevant phase behavior e↵ects were taken into
account. Chapter 6 will discuss all the underlying recovery mechanisms in detail. Figure 5.13
shows Middle Bakken and Lower Bakken history match with e↵ects of di↵erent gradients.
5.4 MODELING FIELD
The following section will present numerical modeling of enhanced oil recovery CO2
soaking performed in a North Dakota Bakken well. First, the gridding of the numerical
model is presented, then a reservoir fluid sample was tuned to all the pressure-volume-
temperature (PVT) laboratory experiments. Then historical match for the well production
65
(a)
(b)
Figure 5.13: History match results for Middle Bakken and Lower Bakken CO2 core floodingexperiments.
66
data is done and finally an enhanced oil recovery scheme using CO2 and other types of
solvents is performed and incremental oil is presented.
5.4.1 Field Model: Grid System
A dual-porosity cartesian model was used for the North Dakota Bakken well. A single
stage hydraulic fracture was modeled, and the dimensions of the reservoir model were length
500 ft, width 2640 ft and thickness 50 ft as shown in Figure 5.14. The matrix and fracture
is divided into stimulated reservoir region (SRV) closer to the hydraulic fracture and and
unstimulated reservoir region (USRV) further away from the hydraulic fracture. The model
input parameters are shown in Table 5.4. The fracture conductivity is FCD
= 100mdft, from
fracture width of whf
original
= 0.001 ft and the hydraulic permeability khf
= 100000md. Due
to computational problems caused by small grids, the hydraulic fracture was pseudoized
to have permeability khf
= 50md and width whf
= 2 ft, to maintain the same fracture
conductivity.
Figure 5.14: Reservoir dimensions (single-stage HF) for a North Dakota Bakken well model.
67
Table 5.4: North Dakota Bakken well reservoir parametersReservoir Dimensions and Properties
Nx
, Ny
, Nz
11x27x14x ,4y, 4z (ft) 45.455, VARJ, 50Length (ft) 500Width (ft) 2640Thickness (ft) 50Depth (ft) 10000Matrix Porosity (�
m
) (frac) 0.0560Matrix Permeability (k
m
) (USRV) (md) 0.0005Matrix Permeability (k
m
) (SRV) (md) 0.0008Fracture Porosity (�
f
) (frac) 0.0022E↵ective Fracture Permeability (k
f,eff
) (USRV) (md) 0.005E↵ective Fracture Permeability (k
f,eff
) (SRV) (md) 0.05Hydraulic Fracture Width (w
hf
) (ft) 0.001Fracture Half Length (y
f
) (ft) 180Lx
, Ly
, Lz
(USRV) (ft) 50, 50, 50Lx
, Ly
, Lz
((SRV) (ft) 5, 5, 5
5.4.2 Field Model: Fluid System
A Middle Bakken PVT report was used for the field case North Dakota Bakken well. As
shown on Figure 5.15(a), the fluid composition is mainly dominated by lighter components
and phase envelope is shown in Figure 5.15(b) with saturation pressure of 2870 psia. The
reservoir temperature is 237 oF and the well will be operated by honoring the production
data and predicting the flowing bottom hole pressure.
The fluid model was lumped from 30 component system to a 10 component system in
order to reduce the model run time. Table 5.5 shows the fluid properties used for the
simulation run.
The equation of state (EOS) model was tunned using available laboratory PVT data.
Figure 5.16(a) shows comparison of GOR data between the created EOS model (numerical)
and the available laboratory data (experimental). Initially the GOR is 1900 SCF/STB,
and after the saturation pressure of 2870 psia the GOR goes down as the gas comes out of
solution. Figure 5.17(b) below shows oil density comparison between the EOS model and
68
(a) Middle Bakken reservoir fluid composition.
(b) Middle Bakken reservoir fluid phase envelope.
Figure 5.15: Middle Bakken reservoir fluid description.
69
Table 5.5: Lumped-component Middle Bakken fluid system used for field caseFluid Characterization
Hydrocarbon Mole Critical Critical Acentric MolecularComponents Fraction Pressure Temperature Factor Weight
(frac) (atm) (K) (��) (lbm/lbmmol)N2 0.0159 33.50 126.2 0.040 28.01CO2 0.0038 72.80 304.2 0.225 44.01CH4 0.3519 45.40 190.6 0.008 16.04C2H6 0.1448 48.20 305.4 0.098 30.07C3H8 0.0932 41.90 369.8 0.152 44.10IC4 �NC4 0.0574 37.17 421.5 0.189 58.12IC5 �NC5 0.0359 33.33 466.4 0.242 72.15FC6 0.0280 32.46 507.5 0.275 86.00C7 � C13 0.1726 26.29 604.5 0.406 125.04C14 � C22 0.0625 16.45 747.4 0.720 235.57C23 � C30 0.0342 10.67 803.9 1.242 441.99
the laboratory data, it is seen that the mixture density becomes denser as the gas in solution
escapes below the saturation pressure. Figure 5.17(a) shows the oil viscosity comparison, as
the gas in solution escapes, the leftover mixture is more viscous. Figure 5.17(b) shows the
swelling factor comparison and due to dominant light components in the oil, it is able to
swell the oil and expand to almost two times its original volume with addition of only 50%
CO2 mole percent. The implication of this is enormous during enhanced oil recovery and
will be discussed later in chapter 6 as part of the main recovery mechanisms.
5.4.3 Field Model: Rock-Fluid System
The relative permeability curves for the simulation were the same for the matrix and
fracture as shown on Figure 3.6 in chapter 3. The connate water saturation Swc
= 0.40, the
oil residual to water saturation Sorw
= 0.25, and gas connate saturation Sgc
= 0.05. The
Corey exponents were no = 3, nw = 3, nog = 3, and ng = 3 (Corey and Rathjens, 1956).
70
(a)
(b)
Figure 5.16: Equation of state (EOS) model tuning of Gas-Oil Ratio (GOR) and oil densitywith PVT laboratory data.
71
(a)
(b)
Figure 5.17: Equation of State (EOS) model tuning of oil viscosity and swelling factor withPVT laboratory data.
72
5.4.4 Field Model: History Matching
Production data for a period of 1.2 years was available for a Middle Bakken horizontal
well with 15 hydraulic fracture stages in Reunion Bay. The base model was built for a single
stage hydraulic fracture and used to history match the scaled down (factor of 15) production
data. The base model was run with oil rate control and used to predict the bottom hole
flowing pressure and the gas rates in Figure 5.19(a) and Figure 5.19(b). Figure 5.18 shows
the scale down oil rates during the history match process.
Figure 5.18: History matching process with oil rates control
5.4.5 Field CO2 Enhanced Oil Recovery Scheme
After the initial history match, the base model was produced for 10 years with bottom
hole pressure constraint pwell
= 2500 psia. After the primary production, four enhanced
oil recovery (EOR) schemes using CO2 and NGL solvents were undertaken to better
understand cyclic solvent soaking mechanisms. Three main parameters were chosen for study,
these were soaking times, injection rates, and solvents types. The four EOR schemes
73
(a)
(b)
Figure 5.19: History match of bottom hole pressure and gas rates
74
are described below and are shown in Table 5.6. Injection rate of 200 MSCF/day per stage
was chosen in order to simulate a more realistic field case where the amount of total injected
gas would be practical to obtain. Both injection rates and soaking times were doubled in
order to see the e↵ects on incremental oil recovery. Two solvents were used and their e↵ect
was investigated. During the EOR schemes, the well was operated at pwell
= 2500 psia.
1. EOR scheme 1 involves injection of CO2 or NGL at a rate of 200 MSCF/day for 15
days, soaking for 15 days, and production for 120 days.
2. EOR scheme 2 involves injection of CO2 or NGL at a rate of 200 MSCF/day for 15
days, soaking for 30 days, and production for 120 days.
3. EOR scheme 3 involves injection of CO2 or NGL at a rate of 400 MSCF/day for 15
days, soaking for 15 days, and finally production for 120 days.
4. EOR scheme 4 involves injection of CO2 or NGL at a rate of 400 MSCF/day for 15
days, soaking for 30 days, and finally production for 120 days.
Table 5.6: CO2 Enhanced oil recovery schemesEOR Schemes
Solvent Scheme Injection Soaking Production NumberType Description (days) (days) (days) Cycles
CO2
Scheme 1 (200 MSCF/day) 15 15 120 31Scheme 2 (200 MSCF/day) 15 30 120 31Scheme 3 (400 MSCF/day) 15 15 120 31Scheme 4 (400 MSCF/day) 15 30 120 31
The minimum miscibility pressure (MMP) using CO2 as an injection gas for the Bakken
oil composition (Table 5.5) is 2572 psia. The injection rate is set to 200 MSCF/day and 400
MSCF/day and maximum injection pressure of 5000 psia.
Figure 5.20(a) shows results for the bottom hole pressure and Figure 5.21(b) show results
of the oil rates with EOR scheme 1.
75
(a)
(b)
Figure 5.20: Bottom hole pressure and oil rates during EOR scheme1.
76
Figure 5.21(a) shows results for gas rates with EOR scheme 1. Figure 5.21(b) compares
all four EOR schemes for CO2 injection.
5.4.6 Field NGL Enhanced Oil Recovery Scheme
NGL solvent consisting of composition C1 = 0.56, C2 = 0.24, C3 = 0.13, and C4 = 0.07
is used for similar EOR schemes as above and were undertaken to better understand
cyclic solvent soaking mechanisms. The minimum miscibility pressure for NGL is 2717 psia.
Table 5.7 shows the di↵erent EOR schemes.
Table 5.7: NGL Enhanced oil recovery schemesEOR Schemes
Solvent Scheme Injection Soaking Production NumberTypes Description (days) (days) (days) Cycles
NGL
Scheme 1 (200 MSCF/day) 15 15 120 31Scheme 2 (200 MSCF/day) 15 30 120 31Scheme 3 (400 MSCF/day) 15 15 120 31Scheme 4 (400 MSCF/day) 15 30 120 31
Figure 5.22(a) shows the comparison between CO2 and NGL solvent during EOR scheme
1. Figure 5.22(b) shows the comparison of solvents with and without molecular di↵usion
e↵ects.
5.4.7 Discussion of Field Results
Injection of CO2 solvent in EOR scheme 1 yields an incremental recovery of 3.32%
for the period of 31 cycles (almost 12 years). For EOR scheme 2, the soaking time
was doubled (15 days) and injection rates were the same (200 MSCF/day), the marginal
incremental recovery of 3.33% was observed. For EOR scheme 3, the rates were doubled
(400 MSCF/day) and the soaking time was the same (15 days), the incremental recovery
of almost 5% was observed. For EOR scheme 4, the soaking times were doubled (30
days) and the injection rates were the same (400 MSCF/day), the incremental recovery was
almost 5%. From these results, one can conclude that longer soaking times yield similar
77
(a)
(b)
Figure 5.21: Gas rates and comparison of all EOR schemes.
78
(a)
(b)
Figure 5.22: Comparison of two solvent types and e↵ect of molecular di↵usion
79
amounts of recovery and this is attributed to the low permeability of the matrix. Also,
one can deduce that increasing the rate of injection has increased the amount of oil recovered,
This is attributed to the amount of solvent CO2 which helps with the miscibility. As seen
in Figure 5.20(b), the oil recovery e�ciencies decrease with the number of soaking
cycles and this is due to the low permeability of the matrix.
Table 5.8: Summary of results for Enhanced oil recovery schemesEOR Scheme Injection Rate Soaking Incremental RF Solvent UtilizationSolvent Type (MSCF/day) (days) (%) (MSCF/STB)CO2 Scheme 200 15 3.32 6.80CO2 Scheme 400 15 5.00 10.15NGL Scheme 200 15 4.00 5.90NGL Scheme 400 15 6.18 8.50
Injection of NGL solvent in EOR scheme 1 yields an incremental recovery of 4.00% for
the period of 31 cycles (almost 12 years). For EOR scheme 2, the incremental recovery
is marginal at 4.10%. For EOR scheme 3, the incremental recovery is 6.18%. Finally for
EOR scheme 4, the incremental recovery is 6.25%. Injection of NGL produces more
EOR oil. Solvent utilization calculations are shown in Table 5.8, and overall hu↵-n-pu↵
approach is considered e�cient. Furthermore, NGL is slightly more e�cient than CO2 as a
solvent.
Figure 5.22(b) shows inclusion of molecular di↵usion e↵ects for both solvents. It is ob-
served that for CO2, the incremental oil recovery is 0.27% and for NGL, the incremental
recovery for including molecular di↵usion e↵ect is 0.77%. Overall, the impact of molecu-
lar di↵usion is modest on a field scale, however, with closer fracture spacing, the surface
area of the matrix per unit volume increases and molecular di↵usion can become more ap-
parent.
80
CHAPTER 6
MASS TRANSFER MECHANISMS
This chapter presents an evaluation of the mass transfer mechanisms involved in cyclic
supercritical fluid soaking for enhanced oil recovery in liquid rich shales. First, various
transport means and literature encompassing correlations and models for moving solvents
closer to miscibility in the reservoir are presented. Then, an outline will be presented for
the controlling parameters of the underlying transport principles for solvents using both
the laboratory data from CO2 cyclic supercritical fluid extraction experiments conducted at
Energy & Environmental Research Center (EERC) on Bakken cores and field scale simulation
work in North Dakota Bakken well presented in Chapter 5.
6.1 TRANSPORT MEANS
When CO2 is compressed and heated, its physical properties change and it becomes a
supercritical fluid. Under these conditions, it has the solvating power of a liquid and the
di↵usivity of a gas. It’s density becomes closer to a liquid (i.e closer to oil density) and this
increases the interaction between CO2 and the oil, similar to a liquid solvent. Moreover,
a unique characteristic of supercritical CO2 is it has low viscosity similar to gases and
zero surface tension, which in turn allow for relative penetration into tight matrix pores
to extract oil. Cyclic supercritical CO2 can help mobilize matrix oil by miscibility at the
matrix-fracture interface. The following are three main transport means for moving CO2
closer to miscibility: advective flow , molecular di↵usion , and gravity drainage .
6.2 ADVECTIVE FLOW
Advection is a transport mechanism that is based on pressure and gravity gradients.
Advective flow is responsible for mechanically moving supercritical CO2 as fluid’s bulk mo-
tion from fractures to the matrix by pressure and gravity gradients. Equation 6.2 is a
81
hydrocarbon component mass balance equation for a three-phase hydrocarbon system. The
bracketed terms are the contribution from the pressure gradient. Increase in pressure as a
result of increased injection rates and also as a result of oil swelling can a↵ect the viscous
flow and reduce oil viscosity with promotes interaction, miscibility and oil mobility.
�r ·h�
�
⌧
So
��!J
o,c
+��
⌧
Sg
��!J
g,c
+��
⌧
Sw
��!J
w,c
i+
r ·⇠o
xc
¯k�
o
⇣rp
o
� �o
rD⌘+⇠
g
yc
¯k�
g
⇣rp
g
� �g
rD⌘�
+
r ·
¯⇠w
wc
¯k�
w
⇣rp
w
� �w
rD⌘�
+
⇠o
xc
q̂o
+ ⇠g
yc
q̂g
+ ⇠w
wc
q̂w
= @
@t
[�zc
(⇠o
So
+ ⇠g
Sg
+ ⇠w
Sw
)]
(6.1)
Where,
⌧ is tortuosity, � porosity,�!J
↵,c
phase molecular di↵usion flux of component c, ⇠↵
is the
molar density, xc
, yc
, and wc
are the liquid, vapor, and water mole fractions of component c,
k permeability, �↵
phase mobility, rp↵
phase pressure gradient, �↵
phase gamma, rD depth
gradient, q̂↵
phase flow rate per volume, zc
overall mole fraction, and S↵
phase saturation.
↵ phase (oil, gas, or water).
6.3 MOLECULAR DIFFUSION FLUX
The molecular di↵usion flux represent transport by molecular di↵usion. It is proportional
to the concentration gradient of each molecular species. Thus, if there was concentrations
gradient for the CO2 at the fracture and matrix interface, then no di↵usive mass would be
possible. The concentration gradient is what drives the CO2 from high concentration region
(fractures) to low concentration region (matrix). Furthermore, the greater the concentration
di↵erence, the larger the imbalance of fluxes, and thus the net flux increases with the gradient.
The gas-like di↵usivities of supercritical fluids are typically one to two orders of magnitude
greater than liquids, allowing for favorable mass transfer properties.
�r ·�
�
⌧
So
� �!J
o,c
+��
⌧
Sg
� �!J
g,c
+��
⌧
Sw
� �!J
w,c
�+
r ·⇥⇠o
xc
¯k�o
(rpo
� �o
rD)+⇠g
yc
¯k�g
(rpg
� �g
rD)⇤+
r ·h
¯⇠w
wc
¯k�w
(rpw
� �w
rD)i+
⇠o
xc
q̂o
+ ⇠g
yc
q̂g
+ ⇠w
wc
q̂w
= @
@t
[�zc
(⇠o
So
+ ⇠g
Sg
+ ⇠w
Sw
)]
(6.2)
82
The proper modeling of multicomponent hydrocarbon mixtures is not a trivial task.
There are three main common models to describe molecular di↵usion flux for multicompo-
nent hydrocarbon mixtures. The most popular is based on the classical Fick’s first law,
the second is the Maxwell-Stephan (MS) model, and the third is the generalized Fick’s law
originated from the irreversible thermodynamics (Hoteit, 2011). The classical Fick’s law for
multicomponent mixtures assumes that each component in the mixture transfers indepen-
dently and does not interact with the other components (Hoteit, 2011). For classical Fick’s
law, the driving force is the self concentration gradient multiplied by the di↵usion coe�cient.
The di↵usion coe�cient is assumed to be constant and independent of composition and PVT
conditions. Hoteit (2011) discusses that the second and third molecular di↵usion flux models
are similar and can be seen as generalized of the Fick’s law. He points out that their flux
driving force is proportional to chemical potential gradient. Specifically for these models,
the thermodynamic non-ideality and the dragging e↵ect due to species interaction are taken
into account. Below is a more detailed description of the di↵usion flux models:
6.3.1 Maxwell-Stephan Model
For an isothermal system and with the absence of external forces, the generalized Maxwell-
Stephan (MS) formulation is based on the idea of two equally counterbalanced forces that
control di↵usion of a component i and can be shown in Equation 6.3 (Krishna and Taylor,
1986).
� xi
RTr
T,p
µi
=ncX
j=1j 6=i
xi
xj
(ui
� uj
)
Dij
(6.3)
where xi
is mole fraction of component i, R is gas constant, T is temperature, µi
is the
chemical potential, Dij
, i, j = 1, ..., nc (i 6= j) are the MS di↵usion coe�cients, which
represent the mutual di↵usivity for every pair of components in the mixture, and ui
and uj
is the friction velocity of component i and j . When i = j, Dij
does not exist, and Dij
are symmetric hence, there are only nc
(nc
� 1) /2 MS di↵usion coe�cients. At constant
83
temperature, T , and pressure, p, due to Gibbs-Duhem equation
ncX
i=1
xi
rT,p
µi
= 0 (6.4)
The chemical potential gradient can be written in terms of the fugacity, fi
and the
composition gradient as follows (Firoozabadi, 1999).
rT,p
µi
= RTnc�1X
j=1
@ ln (fi
)
@xj
rxj
(6.5)
With substitution, Equation 6.3 can be written as
�nc�1X
j=1
xi
@ ln (fi
)
@xj
rxj
=1
⇠
ncX
j=1j 6=i
(xj
Ji
� xi
Jj
)
Dij
(6.6)
In matrix form, Equation 6.6 can be written as
BJ = �⇠�rx (6.7)
Where, ⇠ is overallmolar density and the di↵usion coe�cients (B) can be expressed
as:
Bij
=
8<
:
x
i
D
inc
+ 1⇠
Pnc
j=1j 6=i
x
k
D
ik
i = j
�xi
⇣1
D
ij
� 1D
inc
⌘i 6= j
(6.8)
B = [Bij
]i,j=1,....,n
c
�1 (6.9)
and the non-ideality correction factor (�) is expressed as:
�ij
= xi
@ ln (fi
)
@xj
(6.10)
Where,
� = [�ij
]i,j=1,....,n
c
�1 (6.11)
rx = [rxi
]i=1,....,n
c
�1 (6.12)
J = [rJi
]i=1,....,n
c
�1 (6.13)
84
The matrix B is a function of the inverse of the MS coe�cients , and � represents
the thermodynamic non-ideality e↵ect. For ideal mixtures, � is the identity matrix. To
get an explicit expression of the flux, Equation 6.8 can be multiplied by the inverted matrix
B�1 as:
J = �⇠B�1�rx (6.14)
Where, i = j = 1, ...., nc�1. Note that in Equation 6.14, the last component was selected
as a reference and therefore di↵usion flux Jnc
is eliminated.
6.3.2 Generalized Fick’s Law
In a multicomponent non-ideal mixture, the generalized expression of the Fickian di↵usion
flux is written as:
Ji
= �cnc�1X
j=1
Dij
rxj
(6.15)
Where, i = 1, ...., nc�1. In the above Equation 6.15, only (nc
� 1) independent di↵usion
fluxes appear. The last di↵usion coe�cient can be calculated from the sum of total di↵usion
flux below:
ncX
i=1
Ji
= 0 (6.16)
Equation 6.15 can be written in a matrix form as
J = �cDrx (6.17)
Where, D is a (nc
� 1) x (nc
� 1) matrix known as the Fickian di↵usion coe�cient matrix.
The diagonal entries are the main di↵usion coe�cients and the o↵-diagonals entities are
the cross or coupling di↵usion coe�cients, which are nonzero and not symmetric - that is
Di,j
6= Dj,i
; i 6= j. Comparing fluxes from Equation 6.14 and Equation 6.17 leads to the
following relationship:
D = B�1� (6.18)
85
Using this relationship in Equation 6.18, the di↵usion coe�cients can be calculated as
will be shown later.
6.3.3 Classical Fick’s Law
The classical Fick’s is the mostly used model in reservoir engineering literature and also
in commercial simulators (Riazi and Whitson, 1993). The classical Fick’s law is used in the
context of e↵ective di↵usivity where di↵usion in multicomponent mixtures is assumed
to behave as a pseudo-binary (Da Silva and Belery, 1989). In a multicomponent mixture,
di↵usion processes of di↵erent components are assumed independent and the driving force
is the self mole fracture gradient multiplied by an e↵ective di↵usion coe�cient. The
e↵ective di↵usion coe�cient is often considered independent of composition regardless of the
thermodynamic ideality of the mixture. Even though this model is empirical, it may provide
reasonable results for many applications but has limitations which are discussed by Hoteit
(2011). The di↵usion flux is defined as:
Ji
= �cDeff
i
rxi
(6.19)
Where, Deff
i
is the e↵ective di↵usion coe�cient of component i in the mixture, i =
1, ..., nc
.
Hoteit (2011) discusses some reservations in using the classical Fick’s by pointing out that
the model neglects dragging e↵ects and that it might not honor the equimolar condition that
states that the total di↵usion flux must be zero. The generalized Fick’s law is preferred over
the classical Fick’s, and the fundamental di↵erence is in the flux driving force that is based
on the chemical potential gradient instead of the intrinsic concentration gradient
(Hoteit, 2011).
6.3.4 Di↵usion Coe�cients Correlations
There are several methods that have been proposed in the literature to predict e↵ective
molecular di↵usion coe�cients of multicomponent hydrocarbon mixtures. The e↵ec-
86
tive di↵usion coe�cients are often considered independent of composition. These methods
are summarized below:
6.3.4.1 Wilke (1950)
Molecular di↵usion coe�cient in multicomponent mixtures can be calculated using e↵ec-
tive di↵usion coe�cient of component i in mixture m using Wilke (1950) approach as shown
in Equation 6.20.
Di,m
=1� y
iPnc�1j=1j 6=1
y
j
D
ij
(6.20)
Where,
Di,m
is the e↵ective di↵usion of component i with respect to the total phase mixture m,
yi
is the mole fraction of di↵using component, and Di,j
is the binary di↵usion coe�cient of
component i with respect to component j .
6.3.4.2 Wilke and Chang (1955)
Another approach to calculate the e↵ective mixture molecular coe�cient is using di↵using
component properties and mixtures viscosities. Wilke and Chang (1955) uses this approach
as shown in Equation
Di,m
=7.40⇥ 10�8
q�MW
0i,m
�T
µm
(vb
i
)0.6(6.21)
Where, µm
is the mixture viscosity in cp, vb
i
is partial molar volume of component i at
the boiling point in (cm3/mol), T is the temperature in Kelvin. The molecular weight is
represented as
MW0
i,m
=
Pnc
j 6=i
yj,m
MWj
1� yi,m
(6.22)
and the partial molar volume, vb
i
, is estimated using Tyne and Calus method from critical
volume, Vc
(cm3/mol), as follows (Reid et al., 1987):
vb
i
= 0.285V 1.048c
(6.23)
87
6.3.4.3 Sigmund (1976a, 1976b)
The binary di↵usion coe�cients at reservoir conditions can be calculated using Sigmund
(1976a, 1976b) correlation shown in Equation 6.24:
⇠m
Dij
(⇠m
Dij
)0= 0.99589 + 0.096016⇠
r
� 0.22035⇠2r
+ 0.03287⇠3r
(6.24)
Where,
Di,j
is the binary di↵usion coe�cient of component i with respect to component j at
specified temperature and pressure, ⇠m
is the mixture molar density calculated using Peng-
Robinson EOS for each phase, ⇠r
is the mixture reduced molar density, (⇠m
Dij
)0 is the low
pressure molar density-di↵usivity product.
The Reduced mixture, ⇠r
, can be calculated using Equation 6.25
⇠r
=⇠m
⇠c
= ⇠m
"Pnc
j=1 yi,mV5/3
c
i
Pnc
j=1 yi,mV2/3
c
i
#(6.25)
Where,
⇠c
is the critical molar density, and Vc
i
is the critical molar volume of component i .
The low-pressure molar density-di↵usivity product, (⇠m
Dij
)0 is calculated using Reid
et al., 1987 as:
(⇠m
Dij
)0 =0.0018583T 1
/2
�2ij
⌦ij
R
✓1
Mi
+1
Mj
◆1/2
(6.26)
In the above Equation 6.26, the R is a gas constant in consistent units, �ij
is the collision
diameter, ⌦ij
collision integral of the Lennard-Jones potential are related to component
critical properties (Reid et al., 1987).
The collision diameter �ij
is defined as
�ij
=�i
+ �j
2(6.27)
�i
= (2.3551� 0.087!i
)
✓Tc
i
pc
i
◆1/3
(6.28)
88
Where, !i
is acentric factor of component i, Tc
i
is the critical temperature of component
i, and pc
i
is the critical pressure of component i .
The collision integral of the Lennard-Jones potential ⌦ij
can be expressed as:
⌦ij
=1.06036
T 0.1561ij
+0.193
exp (0.47635Tij
)+
1.03587
exp (1.52996Tij
)+
1.76474
exp (3.89411Tij
)(6.29)
Tij
=kB
T
"ij
(6.30)
"ij
=p"i
"j
(6.31)
"i
= kB
(0.7915 + 0.1963!i
)Tc
i
(6.32)
Where kB
is the Boltzmann’s constant (=1.3805e-16 erg/K)
Da Silva and Belery (1989) note that the Sigmund correlation does not work well for
liquid systems and propose the following extrapolation for ⇠r
> 3.7
⇠m
Dij
(⇠m
Dij
)0= 0.18839 exp (3� ⇠
r
) (6.33)
6.3.4.4 Hayduk and Minhas (1982)
Hayduk and Minhas (1982) modified Wilke and Chang (1955) correlation by dropping
the molecular weight term and modifying the exponent and coe�cients. Their viscosity-
di↵usivity correlation for para�ns was developed based on 58 experimental data and they
reported 3.4% average error compared to 13.3% average for Wilke and Chang (1955). The
correlation is shown in Equation 6.34:
Di,m
= 13.3⇥ 10�8T 1.47µ(10.2/v
b
i
�0.791)m
v�0.71b
i
(6.34)
Where,
vb
i
is partial molar volume of component i at the boiling point, T is the temperature,
and µm
is mixture viscosity.
89
6.3.4.5 Renner (1988)
Renner (1988) performed di↵usion of CO2, methane, ethane, and propane in liquid hy-
drocarbons in consolidated porous media and came up with the following correlation:
Di,m
= 10�9µ�0.4562m
MW�0.6898i
v�1.706i
p�1.831T 4.524 (6.35)
Where,
µm
is mixture viscosity (cp), MWi
molecular weight of component i, vi
is specific volume
of component i (gmmol/cm3), p is the pressure (psia), and T is temperature (K).
6.3.4.6 Riazi and Whitson (1993)
Riazi and Whitson (1993) correlate the ⇠
m
D
ij
(⇠m
D
ij
)0similar to Sigmund (1976) to reduced
molar density and also to viscosity ratio performs better. Their correlation is shown in 6.36.
⇠m
Dij
(⇠m
Dij
)0= 1.07
✓µm
µ0m
◆(�0.27�0.3!m
)+(�0.05+0.1!m
) p
p
c,m
(6.36)
Where,
⇠m
and µm
are the mixture viscosity at conditions of the system, µ0m
is viscosity of the
mixture at low pressure. And for a binary system components A and B with molar fraction
xA
and xB
, pseudocritical pressure and pseudo acentric factor of the mixture are given as
follows:
pc,m
= xA
pc,A
+ xB
pc,B
(6.37)
! = xA
!A
+ xB
!B
(6.38)
6.3.4.7 Maxwell-Stefan (MS) Multicomponent Molecular Di↵usion Coe�cients
The e↵ective multicomponent di↵usion coe�cient correlations mentioned above
consider the main di↵usion (diagonal) terms and neglect the cross-di↵usion (o↵-diagonal)
terms. Hoteit (2011) clarifies that this approach of neglecting o↵-diagonal terms is incon-
sistent and will violate equimolar balance constraint. To avoid this inconsistency, a widely
90
used MS coe�cient for binary mixtures is using infinite dilution coe�cients by Vignes (1966)
as shown in Equation 6.39 for binary component 1 and 2.
D1,2 = (D112)
1�x1 (D121)
x2 (6.39)
Where,
D112 is the molecular di↵usion coe�cients of component 1 infinitely diluted in component
2, and x1 is the mole fraction of component 1.
Kooijman and Taylor (1991) extended Vignes’s Equation 6.39 of binary MS di↵usion to
multicomponent as follows:
Di,j
=�D1
ij
�x
j
�D1
ji
�x
i
Qnc
k=1k 6=i,j
�D1
ik
D1jk
�x
k
/2(6.40)
Where, i = j = 1, ....., nc ; i 6= j, Di,j
is Maxwell-Stefan (MS) of the binary pair i � j,
D1ij
is the molecular di↵usion coe�cients of component i indefinitely diluted in component
j and xi
is the mole fraction of component i .
Leahy-Dios and Firoozabadi (2007) developed a new correlation based on 889 experi-
mental data of infinite dilution binary di↵usion coe�cient (D1). They reported a better
prediction performance in comparison with Wilke and Chang (1955), Hayduk and Minhas
(1982), and Sigmund (1976) approaches. The approach is a function of component i viscosity
(µi
), component i reduced properties (Tr,i
, pr,i
), component i molar density (⇠i
) and acentric
factor (!) as shown in Equation 6.41:
⇠1D121
(⇠1D21)0 = f
✓µ1
µ012
, Tr
, pr
,!
◆= A0
✓Tr,1pr,2
Tr,2pr,1
◆A1✓
µ1
µ012
◆[A2(!1,!2)+A3(pr,Tr
)]
(6.41)
Where,
A0, A1, A2 and A3 are constants given by:
A0 = exp (�0.0472) (6.42)
A1 = 0.103 (6.43)
91
A2 = �0.0147 (1 + 10!1 � !2 + 10!1!2) (6.44)
A3 = �0.0053⇣p�(0.337⇥3)r,1 � 6p�0.337
r,2 + 6T�1.852r
⌘�0.1914T�0.1852
r,1 +0.0103
✓Tr,1pr,2
Tr,2pr,1
◆(6.45)
Where,
(⇠1D21)0 is the dilute gas density-di↵usion coe�cient product (mol/ms) and is calculated
using the Fuller et al. (1969) as shown in Equation 6.46:
(⇠1D21)0 =
0.00101⇥ T 0.75⇣
1M1
+ 1M2
⌘1/2
Rh(P
v1)1/3 + (
Pv2)
1/3i2 (6.46)
The flowchart for calculating molecular di↵usion coe�cients using Leahy-Dios and Firooz-
abadi (2007) approach is shown by Figure 6.1 Teklu (2015) below:
6.3.5 Di↵usion Coe�cients Calculations (Bakken Oil)
The molecular di↵usion coe�cients for CO2 and NGL solvent in the oil phase were per-
formed by Teklu (2015) on a Middle Bakken oil sample. The molecular di↵usion models
used were 1) Wilke and Chang (1955), 2) Sigmund (1976), 3) Hayduk and Minhas (1982)
and finally 4) Leahy-Dios and Firoozabadi (2007). Table 6.1 shows comparison of the four
methods, and for Leahy-Dios and Firoozabadi (2007), only the diagonal terms will be pre-
sented.
6.4 GRAVITY DRAINAGE
Gravity drainage occurs when the matrix surrounded by gas flowing in the fractures,
drains the oil from the matrix as a result of density di↵erence between the gas in the fracture
and oil in matrix (Chordia and Trivedi, 2010). This drainage process is discussed by Chordia
and Trivedi (2010) to depend on several parameters such as the size and permeability of
matrix blocks, type of gas and oil, temperature and pressure, fracture size, and the rate of
gas flowing in the fracture. He elaborates that a matrix block surrounded by gas will undergo
92
Figure 6.1: Flowchart for calculating molecular di↵usion coe�cients using Leahy-Dios andFiroozabadi (2007) approach.
93
Table 6.1: Molecular Di↵usion Calculations for Middle Bakken fluid systemMolecular Di↵usion Coe�cients
Components Wilke and Chang Sigmund Hayduk-Minhas Leahy Dios-Firoozabadi(cm2/sec) (cm2/sec) (cm2/sec) (cm2/sec)
N2 9.40E-05 3.82E-06 5.47E-04 4.43E-05CO2 9.06E-05 2.64E-06 5.64E-04 4.04E-05CH4 1.06E-04 3.41E-06 5.92E-04 2.12E-05C2H6 7.31E-05 2.56E-06 6.60E-04 2.86E-05C3H8 5.81E-05 2.02E-06 6.55E-04 2.62E-05IC4 �NC4 4.93E-05 1.69E-06 6.26E-04 2.34E-05IC5 �NC5 4.38E-05 1.45E-06 5.95E-04 2.09E-05FC6 4.02E-05 1.30E-06 5.72E-04 1.89E-05C7 � C13 3.57E-05 1.09E-06 4.97E-04 1.08E-05C14 � C22 2.32E-05 7.24E-07 3.48E-04 7.50E-06C23 � C30 2.10E-05 5.75E-07 2.49E-04 —–
a gravity drainage process when the gravitational forces exceed the capillary forces, and the
e�ciency depends on the threshold height and the matrix block size. The e�ciency of CO2
gravity drainage decrease as the rock permeability decreases and the initial water saturation
increases. If fractures have su�cient vertical relief, with significant density di↵erence, CO2
injection can recover a significant amount of oil by a gravity drainage. For liquid-rich shale
reservoirs, due to low permeable matrix blocks, it is believed that gravity drainage is a minor
force. Equation 6.2 shows the flux induced through gravity gradient in bracketed terms
and can be seen to be relevant only when density di↵erence and vertical height are significant.
�r ·h�
�
⌧
So
��!J
o,c
+��
⌧
Sg
��!J
g,c
+��
⌧
Sw
��!J
w,c
i+
r ·⇠o
xc
¯k�
o
⇣rp
o
� �o
rD⌘+⇠
g
yc
¯k�
g
⇣rp
g
� �g
rD⌘�
+
r ·
¯⇠w
wc
¯k�
w
⇣rp
w
� �w
rD⌘�
+
⇠o
xc
q̂o
+ ⇠g
yc
q̂g
+ ⇠w
wc
q̂w
= @
@t
[�zc
(⇠o
So
+ ⇠g
Sg
+ ⇠w
Sw
)]
(6.47)
6.5 UNDERLYING EFFECTS OF TRANSPORT PRINCIPLES
It is concluded that hu↵ and pu↵ gas injection can help mobilize matrix oil by miscibility
(promoted by solvent extraction via condensing-vaporizing gas process) leading to counter-
94
current oil flow from the matrix instead of oil displacement in the matrix. In addition, the
conventional EOR through displacement alone does not apply, and miscibility in a narrow
region near the fracture-matrix surface interface is the main mechanism of oil extraction
from the tight oil matrix. Furthermore, underlying mechanisms such as repressurization,
viscosity reduction through oil swelling, convective flow, and di↵usion mass transfer play a
crucial role in the oil extraction process.
6.5.1 Oil Swelling and Viscosity Reduction
Table 6.2 shows Bakken oil swelling and viscosity laboratory data. As seen for example,
injection of 54.60% CO2 solvent causes the viscosity reduction of 66% and oil swelling
of 46%. The e↵ect of this is increase in matrix pore pressure hence fluids are expelled out
of the pores. In addition, due to reduction in viscosity, the oil mobility is favorable.
Table 6.2: Summary of swelling tests for Middle Bakken fluid systemLaboratory Data
Solvent Bubble Point Density Viscosity FVF Solution-Gas-Ratio Swellingmole (%) Pressure(psia) (g/cc) (cp) (VR/V
S
) (SCF/STB) Factor0.00 2530 0.647 0.383 1.615 867.34 1.00013.09 2663 0.652 0.305 1.736 1091.35 1.06820.96 2783 0.655 0.257 1.845 1342.56 1.11045.20 3403 0.669 0.153 2.271 2198.93 1.32654.60 3703 0.676 0.128 2.674 2904.42 1.463
6.5.2 Reduction of Interfacial Tension (IFT) at the matrix-fracture interface
The interfacial tension (IFT) is lower between hydrocarbon-enriched CO2 and CO2-
saturated oil. Residual oil mobilization is achievable when miscibility is possible between
injected gas and reservoir oil. The interfacial tension decreases as the pressure increase at a
fixed temperature because of CO2 solubility is higher at higher pressure.
95
6.5.3 Better CO2 Miscibility with Lower Temperature at Matrix-Fracture In-terface
Temperature has a positive e↵ect on CO2 solubility. Higher solubility is achieved (more
mixing) when the reservoir temperature is lower. Since injected CO2 is at room temperature
when injected into the formation it reduces the reservoir temperature at the fracture-matrix
interface, hence promoting favorable mixing conditions. Include graph showing e↵ect of
temperature and pressure on solubility. At lower temperature the density of CO2 is higher
(acts like a liquid), that is how it is injected on surface.
6.5.4 Summary of Underlying Transport Principles
The synergistic combination of density, viscosity, surface tension, di↵usivity, and pres-
sure and temperature dependence, allow supercritical fluids such as CO2 and NGL to have
exceptional extraction capabilities. Di↵usivities are much faster in supercritical fluids than
in liquids, and therefore extraction can occur faster. Also, there is no surface tension and
viscosities are much lower than in liquids, so the solvent can penetrate into small pores
within the matrix inaccessible to liquids. Both the higher di↵usivity and lower viscosity
significantly increase the speed of the extraction
96
CHAPTER 7
CONCLUSIONS, RECOMMENDATIONS AND FUTURE WORK
In the first part of this thesis, a compositional model, using volume balance method, was
developed and used in multiphase well test analysis where key flow parameters, (hydraulic
fracture permeability (khf
) and e↵ective fracture permeability (kf,eff
), were determined using
a novel compositional rate-transient analysis, designed for low permeability liquid-rich shale
reservoirs. The second part of this thesis, evaluated potential for enhanced oil recovery in
liquid-rich shale reservoirs both in the laboratory and field scales. The following are major
conclusions:
7.1 MULTIPHASE TRANSIENT ANALYSIS IN LIQUID-RICH SHALES
1. Developed a three-phase dual-porosity model using an improved volume balance
formulation. The formulation has a mole correction term to rectify discrepancies
in the volume balance.
2. Constructed separated analytical solution approximations in the bilinear and linear
regimes for multiphase, multicomponent systems. The analytical solutions were applied
to a model problem, which produced the reservoir permeability (kf,eff
) and hydraulic
fracture permeability (khf
).
7.2 ENHANCED OIL RECOVERY IN LIQUID-RICH SHALES
1. Modeled CO2 and NGL solvent injection into a multistage hydraulic fracture using a
hu↵-n-pu↵ scheme and determined the incremental oil recovery. For a North Dakota
Bakken well, the incremental oil recovery is approximately 5% using CO2 and 6.25%
using NGL.
97
2. Model results indicate that, in CO2 and NGL injection, the oil recovery mechanism
in the matrix pores involves re-pressurization, oil swelling, solvent extraction, and
viscosity reduction.
3. At the fracture-matrix interface, the oil recovery mass transfer mechanism includes
viscous displacement, molecular di↵usion, and gravity drainage.
4. CO2 and NGL injection mobilize matrix oil by miscibility and solvent extraction–
leading to counter-current flow of oil from the matrix.
5. Oil recovery e�ciency decreases with the number of soaking cycles. This is attributed
to low permeability of the matrix, and long soak times yield only small amounts of
additional oil recovery. This is consistent with the idea that miscibility takes place in
a narrow region near the fracture-matrix interface.
6. Injecting produced gas C1, C2, C3, and C4+ mixture (NGL), instead of CO2, produces
more EOR oil and can help reduce flaring of gas.
7. Hu↵-and-pu↵ process is more e↵ective in mobilizing oil when hydraulic fracture spacing,
in the multi-stage completion, is smaller. Closer multi-stage fracture spacing increases
the number of macro-fractures, which, in turn, increases the surface area of the matrix
per unit rock volume.
7.3 RECOMMENDATIONS AND FUTURE WORK
The following are recommendations and future work for this thesis:
1. The volume balance method developed in this thesis (Chapter 4) needs to incorpo-
rate molecular di↵usion fluxes using the various molecular di↵usion models shown in
Chapter 6.
2. The combined transport model can then be used to model extraction data sets which
will be shared by EERC from Upper Bakken to Three Forks formation.
98
3. More integrated approach needs to be undertaken by inclusion of data such as thin
sections, XRD data, core description, and pore size distribution using techniques like
mercury intrusion or nuclear magnetic resonance (NMR) in the overall modeling e↵ort
to help understand further the matrix-fracture interface and the laboratory hu↵-n-pu↵
experiments.
4. From above learnings from laboratory data sets coupled with history matching by
inclusion of molecular di↵usion flux, a protocol can be devised to extrapolate these
finding to field scale.
99
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104
APPENDIX A - COMPOSITIONAL MODELING USING VOLUME BALANCE
APPROACH
This section shows compositional formulation using volume balance method.
A.1 Volume Balance Formulation for Single-Porosity and Dual-Porosity
For single-porosity system, the compositional volume balance formulation is
nc+1X
c=1
�̄n
t
c
Uc
(pn+1o
) +MoleCorr = Vr
�(c�
+ cv
|z
c
)@p
o
@t(A.1)
For dual-porosity system, the compositional volume balance formulation is
nc+1X
c=1
��̄n
t
c
uc
�f
�nc+1X
c=1
⇣�̄n
t
c
f
⌧t
c
m/f
⌘+MoleCorr
f
= �f
(c�
+ cv
|z
c
)f
@po
f
@t(A.2)
nc+1X
c=1
⇣�̄n
t
c
m
⌧t
c
m/f
⌘+MoleCorr
m
= �m
(c�
+ cv
|z
c
)m
@po
m
@t(A.3)
The total transfer function ⌧t
c
m/f
for each component for all the phases is defined as
⌧t
c
m/f
= xc
⇠o
⌧o
+ yc
⇠g
⌧g
+ wc
⇠w
⌧w
(A.4)
The individual phase transfer functions are described as
⌧o
= �km
�o
f/m
h(p
of
� pom
) +⇣�
z
�
⌘�o
[(hwf
� hwm
)� (hof
� hom
)]i
(A.5)
⌧g
= �km
�g
f/m
h(p
gf
� pgm
) +⇣�
z
�
⌘�g
[(hgf
� hgm
)� (hgf
� hgm
)]i
(A.6)
⌧w
= �km
�w
f/m
h(p
wf
� pwm
) +⇣�
z
�
⌘�w
[(hwf
� hwm
)� (hwf
� hwm
)]i
(A.7)
A.2 Derivation of Compositional Equation and Pressure Equation
The component pressure equation is shown as
uc
=@
@t
✓�z
c
vt
◆(A.8)
105
The component net molar flux is
uc
= r · k�o
xc
(rpo
� �o
rD) +r · k�g
yc
(rpg
� �g
rD) +
r · k�w
wc
(rpw
� �w
rD) + ⇠no
xn
c
qno
+ ⇠ng
ync
qng
+ ⇠nw
wn
c
qnw
(A.9)
The total pressure equation is
ut
=@
@t
✓�
vt
◆(A.10)
Multiply by A.10 by zc
and subtract it from A.8
uc
� zc
ut
=@
@t
✓�z
c
vt
◆� z
c
@
@t
✓�
vt
◆(A.11)
uc
� zc
ut
= zc
@
@t
✓�
vt
◆+
�
vt
@zc
@t� z
c
@
@t
✓�
vt
◆=
�
vt
@zc
@t(A.12)
The compositional equation is
vt
�(u
c
� zc
ut
) =@z
c
@t(A.13)
From A.8, expand the right hand side
uc
=@
@t
✓�z
c
vt
◆=
�
vt
@zc
@t+ z
c
@
@t
✓�
vt
◆(A.14)
Summing on all the components
nc+1X
c=1
uc
=@
@t
✓�
vt
◆=
1
vt
@�
@t+ �
@
@t
✓1
vt
◆(A.15)
nc+1X
c=1
uc
=1
vt
@�
@t+ �
� 1
v2t
@vt
@t
�(A.16)
nc+1X
c=1
uc
=�
vt
"1
�
@�
@p
@p
@t� 1
vt
@v
t
@p
@p
@t+
nc+1X
c=1
✓@v
t
@zc
◆@z
c
@t
!#(A.17)
This expression can be rewritten in terms of pore compressibility (c�
) and fluid compress-
ibility (cv
)
nc+1X
c=1
uc
=�
vt
([c
�
+ cv
|z
c
]@p
@t� 1
vt
nc+1X
c=1
✓@v
t
@zc
◆@z
c
@t
)(A.18)
106
By definition,
�̄t
c
=nc+1X
c=1
✓@v
t
@zc
◆(A.19)
Therefore,
ut
=�
vt
[c�
+ cv
|z
c
]@p
@t� �
v2t
nc+1X
c=1
�̄t
c
@zc
@t(A.20)
From the definition of partial molar volume,
�̄t
c
=
✓4V
t
4Nc
◆
p,T,N
4N
c
!0
(A.21)
�̄t
c
=
✓@V
t
@Nc
◆
p,T,N
(A.22)
Divide by total number of moles N
�̄t
c
=
✓@V
t/N
@N
c/N
◆
p,T,N
n 6=c
(A.23)
�̄t
c
=
✓@v
t
@zc
◆
p,T,N
n 6=c
(A.24)
From A.20, substitute the definition of compositional equation A.13,
ut
=�
vt
[c�
+ cv
|z
c
]@p
@t� �
v2t
nc+1X
c=1
�̄t
c
vt
�(u
c
� zc
ut
) (A.25)
ut
=�
vt
[c�
+ cv
|z
c
]@p
@t� �
vt
(1
vt
nc+1X
c=1
�̄t
c
vt
�(u
c
� zc
ut
)
)(A.26)
ut
=�
vt
[c�
+ cv
|z
c
]@p
@t� 1
vt
(nc+1X
c=1
[�̄t
c
uc
� (�̄t
c
zc
) ut
]
)(A.27)
From,
vt
=nc+1X
c=1
�̄t
c
zc
(A.28)
Substitute A.28 into A.27
ut
=�
vt
[c�
+ cv
|z
c
]@p
@t� 1
vt
(nc+1X
c=1
�̄t
c
uc
� vt
ut
)(A.29)
107
�
vt
[c�
+ cv
|z
c
]@p
@t=
1
vt
nc+1X
c=1
�̄t
c
uc
(A.30)
Rearranging, the pressure equation is
nc+1X
c=1
�̄t
c
uc
= � [c�
+ cv
|z
c
]@p
@t(A.31)
108
APPENDIX B - THERMODYNAMICS
B.1 Peng-Robinson Equation of State
Peng and Robinson, (1976) equation of state (PR EOS) was used to accurately describe
the volumetric and phase behavior of a hydrocarbon system. Peng and Robinson, (1976)
proposed a two-constant equation for improved predictions, specifically liquid-density pre-
dictions. The PR cubic EOS is presented as:
v3 �✓RT
p� b
◆v2 +
✓a
p� 2bRT
p� 3b2
◆v � b
✓a
p� bRT
p� b2
◆= 0 (B.1)
where p is pressure, T is the temperature, R is a gas constant, v is the specific volume, a is
’attraction’ parameter, and b is a ’repulsion’. A substitution is made by replacing specific
molar volume v = ZRT
p
then a cubic equation in terms of Z factor is obtained below:
Z3 � (1� B)Z2 +�A� 3B2 � 2B
�Z �
�AB � B2 � B3
�(B.2)
For multicomponent system,
a =ncX
m=1
ncX
n=1
amn
xm
xn
(B.3)
amn
= (1� �mn
) a1/2m
a1/2n
(B.4)
a1/2m
=
⌦
a
R2T 2cm
pcm
�1/2 ⇥
1 + m
�1� T 1
/2rm
�⇤(B.5)
m
= 0.37464 + 1.54226!m
� 0.26992!2m
(B.6)
b =ncX
m=1
bm
xm
(B.7)
bm
= ⌦b
RTcm
pcm
(B.8)
A =
ncX
m=1
ncX
n=1
amn
xm
xn
!p
R2T 2(B.9)
109
B =
ncX
m=1
bm
xm
!p
RT(B.10)
Where, �mn
is binary interaction coe�cient for m and n components and !m
is acentric
factor for component m .
The molar density of mixture ⇠ is the reciprocal of molar volume v, thus
⇠ =1
v=
p
zRT(B.11)
B.2 Fugacity
The fugacity of component m in a mixture, fm
is defined in terms of fugacity coe�cient,
�m
and is shown as
�m
=fm
xm
p(B.12)
The natural log of fugacity coe�cient of phase (↵ = o, g) is defined as
ln�↵
m
=bm
b↵
(z↵
� 1)� ln (z↵
� B↵
)�
1
2p2
(A
↵
B↵
✓2P
nc
n=1 xn
anm
a↵
� bm
b↵
◆ln
"z↵
+�p
2 + 1�B
↵
z↵
��p
2� 1�B
↵
#)(B.13)
B.3 Derivative of Fugacity with respect to Pressure and Composition
The derivative of fugacity of phase (↵ = o, g) with respect to pressure is
@ ln�↵
m
@p
= b
m
b
↵
@z
↵
@p
� 1(z
↵
�B
↵
)
⇣@z
↵
@p
� @B
↵
@p
⌘�
h1
2p2
⇣2P
nc
n=1 xn
a
nm
a
↵
� b
m
b
↵
⌘i
⇢⇣A
↵
B
↵
⌘✓@z
↵
@p
+(p2+1) @B
↵
@p
z
↵
+(p2+1)B↵
��
@z
↵
@p
�(p2�1) @B
↵
@p
z
↵
�(p2�1)B↵
�◆�(B.14)
The derivative of fugacity of phase (↵ = o, g) with respect to composition is
110
@ ln�↵
m
@x
k,↵
= b
m
b
2↵
hb↵
@z
↵
@x
k,↵
� (z↵
� 1) @b
↵
@x
k,↵
i� 1
(z↵
�B
↵
)
⇣@z
↵
@x
k,↵
� @B
↵
@x
k,↵
⌘�
12p2
(ln
z
↵
+(p2+1)B↵
z
↵
�(p2�1)B↵
� "✓@A
↵
@x
k,↵
◆B
↵
�A
↵
✓@B
↵
@x
k,↵
◆
B
2↵
#⇣2P
nc
n=1 xn
a
nm
a
↵
� b
m
b
↵
⌘!)+
12p2
(⇣A
↵
B
↵
⌘"2
a
↵
a
km
�P
nc
n=1 xn
a
nm
@a
↵
@x
k,↵
�
a
2↵
+ b
m
b
2↵
@b
↵
@x
k,↵
#)+
12p2
⇢⇣A
↵
B
↵
⌘⇣2P
nc
n=1 xn
a
nm
a
↵
� b
m
b
↵
⌘✓ @z
↵
@x
k,↵
+(p2+1) @B
↵
@x
k,↵
z
↵
+(p2+1)B↵
��
@z
↵
@x
k,↵
�(p2�1) @B
↵
@x
k,↵
z
↵
�(p2�1)B↵
�◆�(B.15)
B.4 Derivative of Compressibility Factor with respect to Pressure and Compo-sition
The derivative of compressibility factor (z factor) with respect to pressure is
@z
@p
=(B � z)
3z2 � 2 (1� B) z + (A� 2B � 3B2)
⇣ a
R2T 2
⌘�
(z2 � 2z � 6Bz � (A� 2B � 3B2))
3z2 � 2 (1� B) z + (A� 2B � 3B2)
✓b
RT
◆(B.16)
The derivative of compressibility factor (z factor) with respect to composition is
@z
@x
m
=
(B � z)
Pnc
n=1 (anm + amn
) xn
3z2 � 2 (1� B) z + (A� 2B � 3B2)
�⇣ p
R2T 2
⌘�
(z2 � 2z � 6Bz � (A� 2B � 3B2))
3z2 � 2 (1� B) z + (A� 2B � 3B2)
�✓bm
p
RT
◆(B.17)
B.5 Partial Molar Volume
The partial molar volume per component m is defined as
�̄t
m
=
✓@V
t
@Nm
◆
p,T,N
(B.18)
Where,
Vt
= vo
No
+ vg
Ng
(B.19)
The derivative of total volume with respect to total number of moles
111
@Vt
@Nt,m
=
@ (v
o
No
)
@Nt,m
�+
@ (v
g
Ng
)
@Nt,m
�(B.20)
With manipulation,
@ (v
o
No
)
@Nt,m
�=
zo
RT
po
@No
@Nt,m
+RT
po
No
@zo
@Nt,m
(B.21)
With further manipulation and substitution, the end result
@ (v
o
No
)
@Nt,m
�= v
o
ncX
k=1
("1 +
ncX
f=1
✓1
zo
@zo
@xf
[�kf
� xf
]
◆#✓@N
o,k
@Nt,m
◆)(B.22)
Similarly for gas phase,
@ (v
g
Ng
)
@Nt,m
�= v
g
ncX
k=1
("1 +
ncX
f=1
✓1
zg
@zg
@yf
[�kf
� yf
]
◆#✓@N
g,k
@Nt,m
◆)(B.23)
Therefore, the total partial molar volume per component m can be represented as
�̄t
m
= vo
ncX
k=1
("1 +
ncX
f=1
✓1
zo
@zo
@xf
[�kf
� xf
]
◆#✓@N
o,k
@Nt,m
◆)+ (B.24)
vg
ncX
k=1
("1 +
ncX
f=1
✓1
zg
@zg
@yf
[�kf
� yf
]
◆#✓@N
g,k
@Nt,m
◆)(B.25)
B.6 Fluid Compressibility
The fluid compressibility can be expressed as
cv
t
= � 1
vt
✓@v
t
@po
◆
T,z
m
(B.26)
where the derivative is expressed as
✓@v
t
@po
◆
T,z
m
=
"N
o
Nt
✓@v
o
@p
◆
T,z
m
+vo
Nt
ncX
k=1
@No,k
@po
#+
"N
g
Nt
✓@v
g
@p
◆
T,z
m
+vg
Nt
ncX
k=1
@Ng,k
@po
#
(B.27)
112
APPENDIX C - SATURATION EQUATIONS
C.1 Liquid and Vapor Equations
The liquid (L) and vapor (V ) relationships are shown below
L =So
⇠o
So
⇠o
+ Sg
⇠g
(C.1)
V =Sg
⇠g
So
⇠o
+ Sg
⇠g
(C.2)
C.2 Derivation of Saturation Equations
The saturation equation can be derived using liquid (L) and vapor (V ) relationships
below
So
=(1� S
w
)L⇠g
V ⇠o
+ L⇠g
(C.3)
Sg
=(1� S
w
)V ⇠o
V ⇠o
+ L⇠g
(C.4)
113