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 Head Department 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,theenhancedoilrecoverypotentialofliquid-richshalereservoirswasevaluated 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)oilrecoveryfromtheseBakkencores.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
LISTOFFIGURES ...... x
LISTOFTABLES...... xiii
LISTOFSYMBOLS...... xiv
LISTOFABBREVIATIONS ...... xxi
ACKNOWLEDGMENTS ...... xxii
DEDICATION ...... xxiv
CHAPTER1 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 CONTRIBUTIONOFTHESTUDY ...... 5
1.4 THE ORGANIZATION OF THE THESIS ...... 5
CHAPTER2 LITERATUREREVIEW ...... 7
2.1 COMPOSITIONALMODELING ...... 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 IN DUAL-POROSITYSYSTEMS ...... 15
3.3 THERMODYNAMIC MODEL ...... 17
3.3.1 PhaseEquilibriaandFlash...... 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-PhaseFlow...... 32
4.2.2 Multi-phaseFlowBlackoilCase ...... 34
4.2.3 Multi-phaseFlowCompositionalCase ...... 36
4.2.4 SummaryoftheAnalyticalSolutions ...... 38
4.3 LIQUID-RICH UNCONVENTIONAL RESERVOIR CASE STUDY . . . . . 39
4.3.1 Model Parameters and Setup ...... 40
4.3.2 FluidParameters ...... 42
4.3.3 Rock-FluidParameters ...... 42
4.4 CASE STUDY RESULTS AND ANALYSIS ...... 42
CHAPTER 5 ENHANCED OIL RECOVERY - LABORATORY AND FIELD STUDY...... 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 FluidSystemProperties ...... 51
5.2.3 Bakken Core Description ...... 52
5.2.4 Laboratory Results ...... 57
5.3 MODELINGEXPERIMENTS...... 58
5.3.1 Laboratory Model: Grid System ...... 58
5.3.2 LaboratoryModel: FluidSystem ...... 60
5.3.3 LaboratoryModel: Rock-FluidSystem ...... 60
5.3.4 LaboratoryModel: HistoryMatching ...... 60
5.3.5 DiscussionofLaboratoryResults ...... 65
5.4 MODELINGFIELD ...... 65
5.4.1 FieldModel:GridSystem ...... 67
5.4.2 FieldModel:FluidSystem...... 68
5.4.3 FieldModel: Rock-FluidSystem ...... 70
5.4.4 FieldModel:HistoryMatching ...... 73
5.4.5 Field CO2 Enhanced Oil Recovery Scheme ...... 73
5.4.6 Field NGL Enhanced Oil Recovery Scheme ...... 77
5.4.7 DiscussionofFieldResults...... 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 GeneralizedFick’sLaw ...... 85
6.3.3 ClassicalFick’sLaw ...... 86
6.3.4 Di↵usion Coe cients Correlations ...... 86
6.3.4.1 Wilke(1950) ...... 87
6.3.4.2 WilkeandChang(1955) ...... 87
6.3.4.3 Sigmund (1976a, 1976b) ...... 88
6.3.4.4 HaydukandMinhas(1982) ...... 89
6.3.4.5 Renner (1988) ...... 90
6.3.4.6 RiaziandWhitson(1993) ...... 90
6.3.4.7 Maxwell-Stefan (MS) Multicomponent Molecular Di↵usion Coe cients ...... 90
6.3.5 Di↵usion Coe cients Calculations (Bakken Oil) ...... 92
6.4 GRAVITYDRAINAGE ...... 92
6.5 UNDERLYING EFFECTS OF TRANSPORT PRINCIPLES ...... 94
6.5.1 OilSwellingandViscosityReduction ...... 95
6.5.2 Reduction of Interfacial Tension (IFT) at the matrix-fracture interface ...... 95
6.5.3 Better CO2 Miscibility with Lower Temperature at Matrix-Fracture Interface ...... 96
6.5.4 SummaryofUnderlyingTransportPrinciples ...... 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
REFERENCESCITED ...... 100
APPENDIX A - COMPOSITIONAL MODELING USING VOLUME BALANCE APPROACH...... 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 and Composition ...... 111
B.5 Partial Molar Volume ...... 111
B.6 FluidCompressibility ...... 112
APPENDIX C - SATURATION EQUATIONS ...... 113
C.1 LiquidandVaporEquations ...... 113
C.2 Derivation of Saturation Equations ...... 113
ix LIST OF FIGURES
Figure 1.1 Significance of tight oil production (Energy InformationAdministration,2013)...... 2
Figure3.1 Generalvolumebalanceimplementation...... 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 z factor calculations with CMG PVT Package (WinProp)...... 22
Figure 3.4 Thermodynamic validation between developed routine for fugacity and viscosity calculations with CMG PVT Package (WinProp)...... 23
Figure 3.5 Thermodynamic validation of partial molar volume and fluid compressibility calculations...... 24
Figure3.6 Relativepermeabilitycurves ...... 26
Figure 3.7 Validation of pressure and saturation profile ( VBM vs CMG GEM simulator)...... 28
Figure 3.8 Validation of cummulative oil and cummulative gas (VBM vs CMG GEM 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
Figure4.4 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 at EERC)...... 51
x Figure 5.2 Compositions of separator samples and produced streams for Middle Bakken...... 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 (white color) with non-skeletal calcerous grains, minor calcite and Fe-Dol (tan and brown color), and some K-spar, Plagioclase, and Pyrite (black color)...... 55
Figure 5.5 Thin sections for Lower Bakken core at di↵erent resolution, mineralogy composed of quartz and calcite dominated (white and tan color), minor amount 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 soaking experiments...... 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
Figure5.11 Relativepermeabilitycurves...... 63
Figure5.12 Fracturerelativepermeabilitycurves...... 64
Figure 5.13 History match results for Middle Bakken and Lower Bakken CO2 core flooding experiments...... 66
Figure 5.14 Reservoir dimensions (single-stage HF) for a North Dakota Bakken well model...... 67
Figure5.15 MiddleBakkenreservoirfluiddescription...... 69
Figure 5.16 Equation of state (EOS) model tuning of Gas-Oil Ratio (GOR) and oil densitywithPVTlaboratorydata...... 71
Figure 5.17 Equation of State (EOS) model tuning of oil viscosity and swelling factorwithPVTlaboratorydata...... 72
xi Figure5.18 Historymatchingprocesswithoilratescontrol ...... 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
Figure5.21 GasratesandcomparisonofallEORschemes...... 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 using Leahy-DiosandFiroozabadi (2007)approach...... 93
xii LIST OF TABLES
Table 3.1 Three-component fluid system used for thermodynamic validation . . . . . 21
Table3.2 Testcasereservoirparameters ...... 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, and multi-phasecompositionalmodels ...... 39
Table 4.2 Summary of Linear solutions for single-phase, multi-phase black oil, and multi-phasecompositionalmodels ...... 39
Table4.3 Multiphasecasestudyreservoirparameters...... 40
Table 4.4 Three-component fluid system used for multiphase depletion run...... 42
Table4.5 Bilinearmultiphaseflowanalysisfordepletionrun ...... 48
Table4.6 Linearmultiphaseflowanalysisfordepletionrun ...... 48
Table5.1 XRDanalysisofMiddleBakkenCore ...... 55
Table5.2 XRDanalysisofLowerBakkenCore ...... 56
Table5.3 RadialcaseforMiddleBakkencore ...... 59
Table 5.4 NorthDakotaBakkenwellreservoirparameters...... 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
psi bblSINGLE [ ]...... y-intercept of bilinear single-phase flow equation (RB/d)cp
psi blSINGLE [ ]...... y-intercept of linear single-phase flow equation (RB/d)cp
psi bblMULTI [ ]...... y-intercept of bilinear multi-phase flow equation (RB/d)cp
psi blMULTI [ ]...... y-intercept of linear multi-phase flow equation (RB/d)cp
RB B [ STB ]...... formation volume factor
RB Bo [ STB ]...... oil formation volume factor
RB Bg [ STB ]...... gas formation volume factor
RB Bw [ STB ]...... water formation volume factor
C1 ...... methane
C4 ...... butane
C7 ...... heptane
C10 ...... decane
1 cv [psi ]...... fluid compressibility
1 c [psi ]...... rock compressibility
1 ct [psi ]...... total compressibility
1 ct,f [psi ]...... total fracture compressibility
xiv 1 ct,m [psi ]...... 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
psi mblSINGLE [ ]...... slope of bilinear single-phase flow equation (RB/d)cppd
psi mlSINGLE [ ]...... slope of linear single-phase flow equation (RB/d)cppd
psi mblMULTI [ ]...... slope of bilinear multi-phase flow equation (RB/d)cppd
psi mlMULTI [ ]...... slope of linear multi-phase flow equation (RB/d)cppd
1 MoleCorr [ day ]...... mole correction term nf ...... total number of fractures nc ...... total number of components
N [lb mole]...... total number of moles in hydrocarbon vapor phase g
xv N [lb mole]..total number of moles of component c in hydrocarbon vapor phase gc N [lb mole]...... total number of moles in hydrocarbon oil phase o N [lb mole]....total number of moles of component c in hydrocarbon oil phase oc N [lb mole]...... total number of moles in aqueous phase w N [lb mole]...... total number of moles of component c in aqueous phase wc N [lb mole]...... total number of moles t N [lb mole]...... total number of moles of component c tc 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)
ft3 q [ day ]...... flow rate
1 lb mole qˆ [ day ][ fy 3day ]...... reservoir flow rate per rock volume
ft3 qo [ day ]...... oil flow rate
ft3 qg [ day ]...... gas flow rate
ft3 qw [ day ]...... water flow rate
3 ft ⇧psia R [ oRlb 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)
ft2md Tx [ ft ]...... Single phase transmissibility in x-direction
o o Tc [ R]...... critical temperature ( F +459.67)
lb mole Uc [ day ]...... net molar flux of component c
3 VR [ft ]...... rock volume
3 Vt [ft ]...... total system volume
ft3 v [ lb mole ]...... specific volume
ft3 vt [ lb mole ]...... total specific volume
ft3 vtc [ 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)
whforiginal [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 1 ↵o [ day ]...... oil product term used in multiphase di↵usivity
1 ↵g [ day ]...... gas product term used in multiphase di↵usivity
1 ↵w [ day ]...... 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
...... divergence operator r⇧ ...... gradient operator r x [ft]...... size of the block in discrete form in x-direction 4 y [ft]...... size of the block in discrete form in y-direction 4 z [ft]...... size of the block in discrete form in z-direction 4 p [psia]...... well flowing pressure change 4 wf ...... Dirac delta
...... Kronecker delta
psi [ ft ]...... gravity gradient
md o [ cp ]...... oil phase mobility
md g [ cp ]...... gas phase mobility
md w [ cp ]...... water phase mobility
md t [ 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 ⇠ [ ft3 ]...... molar density
lb mole ⇠o [ ft3 ]...... oil molar density
lb mole ⇠g [ ft3 ]...... water molar density
1 [ ft2 ]...... shape factor
⌧ ...... tortuosity
ft3 ⌧m/f [ 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
LP G ...... 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] Iwouldliketoexpressmyspecialappreciationandprofoundgratitudetomyadvisor 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.Thankyouforbeing 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 )ande↵ectivefracturepermeability(kf,eff)aredeter- 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)andnaturalgas liquids (NGL)consistingofC1 to C4+.Afewpercentincreaseinoilrecoveryfrominnovative
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 )ande↵ectivefracturepermeability(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)isnewand 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 alargecontactareafortheinjectedgastopenetrateanddi↵useintothelow-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 CO and natural gas liquid (NGL) comprising of C C . According to 2 2 4+ 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 °Fusingmillimeter-sizeBakkenchipsandcentimeter-diametercore 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,immisciblehydrocarbonandmisciblehydrocarbon 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+1 @p ¯n U (pn+1)+MoleCorr = V (c + c ) o (3.1) tc c o r v|zc @t c=1 X n+1 In Equation 3.1, Uc(po )isthenetmolarfluxofcomponentc 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 = Uac + Ubc (3.2)
U = (T n n⇠nxn) pn+1 + (T n n⇠nyn) pn+1 + (T n n ⇠n wn) pn+1 (3.3) ac 4x x o o c 4x o 4x x g g c 4x o 4x x w w c 4x o
n n n n n n n n n Ubc = ⇠o xc qo + ⇠g yc qg + ⇠wwc qw (3.4)
n The term ¯tc is known as the partial molar volume of a multiphase system with respect to a component c.Itisdefinedasthechangeofthesystemtotalvolumewithrespectto change of the mass ( or total number of moles ) of component c at a constant pressure, temperature, and number of moles. Mathematically,
n @Vt ¯tc = p,T,Nt,j=c (3.5) @N | 6 ✓ tc ◆ 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
1 encoded in them. Both are described in detail in Appendix A. Molecorr [ day ]isamole 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+1
yc =1 (3.9) c=1 X nc+1
xc =1 (3.10) c=1 X nc+1
wc =1 (3.11) c=1 X The hydrocarbon overall mole composition also satisfies the constraint
nc+1
zc =1 (3.12) c=1 X 3.2 FORMULATION AND IMPLEMENTATION OF VBM IN DUAL-POROSITY SYSTEMS
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+1 nc+1 n n @pof ¯t uc ¯t ⌧tc + MoleCorrf = f (c + cv zc )f (3.13) c f cf m/f | @t c=1 c=1 X X ⇣ ⌘ nc+1 n @pom ¯ ⌧t + MoleCorrm = m(c + cv z )m (3.14) tcm cm/f | c @t c=1 X ⇣ ⌘
15 The total transfer function ⌧ for each component for all the phases is defined as tcm/f
⌧ = x ⇠ ⌧ + y ⇠ ⌧ + w ⇠ ⌧ (3.15) tcm/f c o o c g g c w w
The individual phase transfer functions are described as ⌧ = k (p p )+ z [(h h ) (h h )] (3.16) o m of/m of om o wf wm of om h ⇣ ⌘ i ⌧ = k (p p )+ z [(h h ) (h h )] (3.17) g m gf/m gf gm g gf gm gf gm h ⇣ ⌘ i ⌧ = k (p p )+ z [(h h ) (h h )] (3.18) w m wf/m wf wm w wf wm wf wm h ⇣ ⌘ i 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:
1 1 1 =4 + + (3.19) L2 L2 L2 x y z Where Lx, Ly,andLz represents the dimensions of a matrix block. The height of gas, oil, and water columns in the matrix and fracture can be defined below:
S h = g L (3.20) gf/m 1 S S z ✓ wr or ◆f/m S h = g L (3.21) of/m 1 S S z ✓ wr or ◆f/m S h = g L (3.22) wf/m 1 S S z ✓ wr or ◆f/m 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”wasintroducedtorectifythematerialbalanceerrorfor the system. It appears in the compositional pressure equation as a source term. Mathemat- ically there are defined as:
16 vt n n,S MoleCorrf/m = Nt Nt (3.23) t f/m ✓4 ◆ ⇣ ⌘ ft3 n Where vt is the total specific volume [ lb mole ], Nt is the total number of moles for the n,S system (matrix or fracture) after the pressure solution at time n and Nt is the total number of moles in the system (matrix or fracture) once the saturations are computed [lb mole]. The time-step is t [days]. For implementation of volume balance method in dual-porosity 4 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+1 and matrix pressures pn+1 implicitly. Next, of om 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 ). Thefollowingsections 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
Ahydrocarbonsystemisconsideredinthermodynamicequilibriumiftheoilphaseand 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
L V are expressed through equality of fugacities (fc ,fc ). For each component in the hydro- carbon liquid phase is equal to gas phase as follows:
L V fc = fc (3.24)
Equilibrium ratio (kc)istheratioofmolefractionofcomponent(c)inthevaporphase
(yc)tothatintheliquidphase(xc), mathematically,
yc kc = (3.25) xc
Wilson (1968) proposed an empirical expression to estimate equilibrium ratio (kc)using critical pressure (pc,c), critical temperature (Tc,c), and acentric factor (!c)ofacomponentc presented below:
p T k = c,c exp 5.371 (1 + ! ) 1 c,c (3.26) c p c T ✓ ◆ 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)andofgaseousphase(yc). Also determined is the number of moles of liquid phase (No)andnumberofmolesofgaseousphase(Ng)inahydrocarbon 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 RT a p = (3.27) v b v(v + b)+b(v b) 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
ZRT molar volume v = 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
Athree-componentfluidsystemwasusedtovalidatethethermodynamicroutinesused 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
o with a bubble point pb =3600psia and a reservoir temperature Tr =180 F .Inorderto 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.
n Since the heart of the volume balance method is partial molar volume (¯ tc ) and fluid
n compressibility (c )calculations,thedevelopedalgorithmforpartialmolarvolume(¯ tc )was validated against published experimental data of Wu and Ehrlich (1973) where a known
o mixture of 95 g-moles C2 and 5 g-moles of nC7 at 80 C 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 n the validation with very good agreement. Derivations of partial molar volume ¯tc and the
fluid compressibility c is presented in detail in Appendix B.
Table 3.1: Three-component fluid system used for thermodynamic validation Fluid Characterization Components Critical Critical Acentric Molecular Mole Pressure Temperature Factor Weight Fraction (psia)(oR )()(lbm/lbm mol)(frac) C1 667.19 343.08 0.008 16.043 0.70 C4 551.10 765.36 0.193 58.124 0.20 C10 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 factor calculations with CMG PVT Package (WinProp).
22 (a)
(b)
Figure 3.4: Thermodynamic validation between developed routine for fugacity and viscosity calculations with CMG PVT Package (WinProp).
23 (a)
(b)
Figure 3.5: Thermodynamic validation of partial molar volume and fluid compressibility calculations.
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
o pb =3600psia at the reservoir temperature of Tr =180 F .Thewellwillbeoperatedat bottom hole pressure constraint pwell =1500psia .Therelativepermeabilitycurvesforthe 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,andng =3(CoreyandRathjens,1956).
Table 3.2: Test case reservoir parameters Reservoir Dimensions and Properties
Nx,Ny,Nz 7x7x1 x, y, z (ft) 64.285, 64.285, 40 4 4 4 Length (ft) 500 Width (ft) 500 Thickness (ft) 40 Depth (ft) 10000 Matrix Porosity (frac) 0.05324 Matrix Permeability (md) 10 Fracture Porosity (frac) 0.001 Fracture Permeability (md) 10 Lx,L,Lz (ft) 3, 3, 3
3.5.2 Results and Comparison
Adepletionrunwasdonefor50days.Theproducerwellwaslocatedatthecenterof the grid Nx ,Ny ,Nz =4, 4 , 1. The well was operated with bottom-hole pressure boundary condition pwell =1500psia. 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 run Fluid Characterization Components Critical Critical Acentric Molecular Mole Pressure Temperature Factor Weight Fraction (psia)(oR )()(lbm/lbm mol)(frac) C1 667.19 343.08 0.0080 16.043 0.70 C4 551.10 765.36 0.1930 58.124 0.20 C10 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,wasused.Specifically,Iusedthecompositionalmodeltogenerateflowrate 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:
k @p p +ˆq = c o (4.1) r· µ r o o t @t ✓ ◆ The bilinear flow equation for slightly compressible, single-phase flow from a hy- draulic fracture in a horizontal well is:
1/4 pwf (44.102) µ 1 1/4 141.2µ well 4 = t + shf (4.2) qB (hnhf ) whf khf 8 µ ( ct)f+m kf,eff ! 9 kf,effhnhf < = p Where, : ; p is the well flowing pressure change (psia), q is the flow rate (ST B/day), B is the 4 wf 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
well hydraulic fracture permeability (md). The skin factor for the hydraulic fracture is shf . The hierarchy of flow is from stimulated macro-fractures to the hydraulic fractures to
pwf 1 4 the wellbore. The slope for the bilinear log-log plot of qB versus t,is4 .Theslopeof
p 1 4 wf /4 Cartesian plot of qB versus t during the bilinear flow period is used to estimate hydraulic fracture permeability (khf ). From the equation 4.2 it takes the following form:
32 pwf 1/4 4 = mblSINGLE t + bblSINGLE (4.3) qB Where,
psi psi the slope mblSINGLE is in [ ]andthey-interceptbblSINGLE is in [ ]are: (RB/d)cppd (RB/d)cp
1/4 (44.102) µ 1 mblSINGLE = (4.4) (hnhf ) whf khf 8 µ ( ct)f+m kf,eff ! 9 < = p 141:.2µ well ; bblSINGLE = shf (4.5) kf,effhnhf
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:
1/2 ⇡ pwf (4.064) 2 µ 1 1/2 141.2µ well 4 = t + shf (4.6) qB (hnhf yf ) kf,eff 8 µ ( ct)f+m ! 9 kf,effhnhf < = p Where, : ;
yf is the fracture half-length (ft)forasingletransversehydraulicfractureinamultistage
pwf 1 4 completion. The slope of log-log plot of qB versus t,is 2 .Furthermore, p wf p 4 = mlSINGLE t + blSINGLE (4.7) qB Where,
psi psi the slope mlSINGLE is in [ ]andblSINGLE is in [ ] (RB/d)cppd (RB/d)cp
1/2 ⇡ (4.064) 2 µ 1 mlSINGLE = (4.8) (hnhf yf ) kf,eff 8 µ ( ct)f+m ! 9 < = p 141.2:µ well ; blSINGLE = shf (4.9) kf,effhnhf
p 1 4 wf /2 The slope of Cartesian plot of qB versus t 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. @p [k ( + + ) p +(ˆq B +ˆq B +ˆq B )= c o (4.10) r· w o g r o o o g g w w t @t
@p [k ( ) p +ˆq = c o (4.11) r· t r o t t @t Where,
qˆt =ˆqoBo +ˆqgBg +ˆqwBw 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
= kr↵ and B is the phase formulation volume factor (RB/STB), where ↵ is the phase ↵ µ↵ ↵ 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:
1/4 pwf (44.102) t 1/4 141.2 well 4 = t + shf (4.12) qt (hnhf ) whf khf t 8 ( ct)f+m kf,eff ! 9 kf,effhnhf t < = p where, : ;
qt = qoBo + qgBg + qwBw is the total flow rate for blackoil system (RB/day). q↵ is the phase flow rate (ST B/day)andB↵ is the phase formulation volume factor (RB/STB), where ↵ is the gas, oil, or water phase.
pwf 1 The slope for the bilinear log-log plot of 4 versus t,is .TheslopeofCartesianplot qt 4
p 1 of 4 wf versus t /4 during the bilinear flow period is used to estimate hydraulic fracture qt permeability (khf ). From the equation 4.12 it takes the following form
pwf 1/4 4 = mblMULTI t + bblMULTI (4.13) qt
34 Where,
psi psi the slope mblMULTI [ ]andthey-interceptbblMULTI [ ]are: (RB/d)cppd (RB/d)cp
1/4 (44.102) t mblMULTI = (4.14) (hnhf ) whf khf t 8 ( ct)f+m kf,eff ! 9 < = p 141:.2 well ; bblMULTI = shf (4.15) kf,effhnhf t The linear flow equation atthehydraulicfracturefaceinahorizontalwellcanbe written as:
1/2 ⇡ pwf (4.064) 2 t 1/2 141.2 well 4 = t + shf (4.16) qt (hnhf yf ) kf,eff t 8 ( ct)f+m ! 9 kf,effhnhf t < = p Where, : ;
yf is the fracture half-length (ft)forasingletransversehydraulicfractureinamultistage
pwf 1 completion. From a diagnostic log-log plot the 4 versus t,theslopeis for the linear qt 2 region. From the equation 4.16 it takes the following form: p wf p 4 = mlMULTI t + blMULTI (4.17) qt Where,
psi psi the slope mlMULTI [ ]andblMULTI [ ]are: (RB/d)cppd (RB/d)cp
1/2 ⇡ (4.064) 2 t mlMULTI = (4.18) (hnhf yf ) kf,eff t 8 ( ct)f+m ! 9 < = p 141.2: well ; blMULTI = shf (4.19) kf,effhnhf t
p 1 The slope of Cartesian plot of 4 wf versus t /2 is used to estimate e↵ective fracture qt 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+1 @p ¯n u (pn+1)= (c + c ) o (4.20) tc c o v|zc @t c=1 X where the molar flux is described as: @p u = u + u = (c + c ) o (4.21) c ↵ v|zc @t
u = (⇠ x k )+ ⇠ y k n +(⇠ w k n ) ( p )(4.22) ↵ r· o c o g c g w c w r o ⇥ ⇤ u =(⇠oxcqˆo)+(⇠gycqˆg)+(⇠wwcqˆw)(4.23)
which carries a similar form as the multiphase di↵usivity equation 4.10
@p [k (↵ + ↵ + ↵ )] ( p )+(ˆq )+(ˆq )+(ˆq )= c o (4.24) r· o g w r o o o g g w w t @t Where,
1 ↵o/g/w with units of [ /day]andcanbeexpressedas:
nc+1 n ↵o = ¯tc ⇠oxck o (4.25) c=1 X nc+1 n ↵g = ¯tc ⇠gyck g (4.26) c=1 X nc+1 n ↵w = ¯tc ⇠wwck w (4.27) c=1 X The o/g/w can be expressed as:
nc+1 n o = ¯tc ⇠oxc (4.28) c=1 X nc+1 n g = ¯tc ⇠gyc (4.29) c=1 X
36 nc+1 n w = ¯tc ⇠wwc (4.30) c=1 X 1 The total compressibility ct with units of ( /psi)canbeexpressedas:
c =(c + c )(4.31) t v|zc 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:
1/4 p (44.102) 4 wf = t t1/4 (qg g + qo o + qw w) (hnhf ) whf khf ( t) 8 ( [c + cv zc ])f+m kf,eff ! 9 < | = p 141.2 well : ; + shf (4.32) kf,effhnhf ( t)
pwf 1 The bilinear slope of log-log plot 4 versus t,is .TheslopeofCartesianplotof qt 4
p 1 4 wf versus t /4 during the bilinear flow period of the compositional model is used to qt estimate hydraulic fracture permeability (khf ). From the equation 4.32 it takes the familiar form:
pwf 1/4 4 = mblMULTI t + bblMULTI (4.33) qt psi psi The slope mblMULTI [ ]andthey-interceptbblMULTI [ ]are: (RB/d)cppd (RB/d)cp
1/4 (44.102) t mblMULTI = (4.34) (hnhf ) whf khf ( t) 8 ( [c + cv zc ])f+m kf,eff ! 9 < | = p : 141.2 well ; bblMULTI = shf (4.35) kf,effhnhf ( t) Similarly, the approximate analytical solution of multi-phase flow for linear flow regime for compositional models can be presented as:
37 1/2 p (4.064) ⇡ 4 wf = 2 t t1/2 (qg g + qo o + qw w) (hnhf yf ) kf,eff ( t) 8 ( [c + cv zc ])f+m ! 9 < | = p 141.2 well : ; + shf (4.36) kf,effhnhf ( t)
From the equation 4.36, it takes the following form: p wf p 4 = mlMULTI t + blMULTI (4.37) qt Where,
psi psi the slope mlMULTI [ ]andblMULTI [ ]are: (RB/d)cppd (RB/d)cp
1/2 ⇡ (4.064) 2 t mlMULTI = (4.38) (hnhf yf ) kf,eff ( t) 8 ( [c + cv zc ])f+m ! 9 < | = p 141:.2 well ; blMULTI = shf (4.39) kf,effhnhf ( t)
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, and multi-phase compositional models 1/4 pwf (44.102)µ 1 1 4 141.2µ 4 / well qB = µ( c ) k t + k hn shf (hnhf )pwhf khf t f+m f,eff f,eff hf ⇢⇣ ⌘
1/4 pwf (44.102) 1 4 141.2 4 t / well (q B +q B +q B ) = ( c ) k t + k hn ( ) shf o o g g w w (hnhf )pwhf khf ( t) t f+m f,eff f,eff hf t ⇢⇣ ⌘
1/4 pwf (44.102) 1 4 141.2 4 t / well (q +q +q ) = t + k hn ( ) shf g g o o w w (hnhf )pwhf khf ( t) ( [c +cv zc ]) kf,eff f,eff hf t (✓ | f+m ◆ )
These are represented by Equation 4.32 and Equation 4.36 where the row is highlighted in yellow.TheseequationswillbevalidatedandutilizedinacasestudyinSubsection 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 ⇡ 1/2 pwf (4.064)( 2 )µ 1 1 2 141.2µ 4 / well qB = µ( c ) t + k hn shf (hnhf yf )pkf,eff t f+m f,eff hf ⇢⇣ ⌘
⇡ 1/2 pwf (4.064)( 2 ) 1 2 141.2 4 t / well (q B +q B +q B ) = ( c ) t + k hn shf o o g g w w (hnhf yf )pkf,eff t t f+m f,eff hf t ⇢⇣ ⌘
1 2 ⇡ / pwf (4.064)( 2 ) 1 2 141.2 4 t / well (q +q +q ) = t + k hn ( ) shf g g o o w w (hnhf yf )pkf,eff ( t) ( [c +cv zc ]) f,eff hf t (✓ | f+m ◆ )
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)usedas 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 )calculatedusinganalyticalmethodwascomparedtothenumericalmodelinput,and for the linear region, the e↵ective permeability (kf,eff)wasalsocalculatedandcompared.
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 =0.0001 md). The fracture conductivity is (FCD =10mdft), from
fracture width of (whforiginal =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)andwidth(whf =2ft), 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 21x11x1 x, y, z (ft) VARI, 36.364, 40 4 4 4 Length (ft) 400 Width (ft) 400 Thickness (ft) 40 Depth (ft) 10000 Matrix Porosity (frac) 0.05324 Matrix Permeability (md) 0.0001 Fracture Porosity (frac) 0.001 E↵ective Fracture Permeability (md) 0.005 Hydraulic Fracture Width (ft) 0.001 Fracture Half Length (ft) 200 Lx,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
o fluid system has a bubble point pb =3350psia at the reservoir temperature of Tr =240 F as shown on Figure 4.2. The well will be operated at bottom hole pressure constraint pwell =2000psia .
Table 4.4: Three-component fluid system used for multiphase depletion run. Fluid Characterization Components Critical Critical Acentric Molecular Mole Pressure Temperature Factor Weight Fraction (psia)(oR )()(lbm/lbm mol)(frac) C1 667.19 343.08 0.00800 16.043 0.60 C7 455.13 977.76 0.308301 96.000 0.30 C10 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(CoreyandRathjens,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,andboundary 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 )usinganalytical model versus numerical model is very good (error within 2.5%). Table4.6showspa- 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 after a 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 flow regimes.
(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 run Bilinear Flow Regime
Viscosity oil (µo) 0.0830 Viscosity gas (µg) 0.000 Rel perm oil (kro) 0.48 Rel perm gas (krg) 0.000 t 5.7831 1 t 0.1729 Parameters Analytical Model Numerical Model Slope (bilinear) 9.34 khf whf t 59 whf 2 2 khf 5.118 5.000 khf whf 10.2 10
Table 4.6: Linear multiphase flow analysis for depletion run Linear Flow Regime
Viscosity oil (µo) 0.0768 Viscosity gas (µg) 0.0230 Rel perm oil (kro) 0.27 Rel perm gas (krg) 0.001786 t 3.5933 1 t 0.2783 Parameters Analytical Model Numerical Model Slope (linear) 3.39 kf,eff t 0.04 kf,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)ispresented.
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↵)forenhancedoilrecoveryinliquidrichshalereservoirsby 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 °Fandcritical 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 °Fusingmillimeter-sizeBakkenchipsandcentimeter- 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 Core Mineralogy Content Middle Bakken Core Chlorite 0 Kaolinite 0 Clays Total 5 Illite 5 MxIS 0 Calcite 21 Dolomite 0 Carbonates Total 38 Fe-Dol 17 Siderite 0 Quartz 42 K-spar 7 Plagioclase 6 Other Minerals Total 57 Pyrite 2 Zeolite 0 Barite 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 calcerous grains, minor calcite and Fe-Dol (tan and brown color), and some K-spar, Plagioclase, and Pyrite (black color).
55 Table 5.2: XRD analysis of Lower Bakken Core Mineralogy Content Lower Bakken Core Chlorite 1 Kaolinite 0 Clays Total 7 Illite 6 MxIS 0 Calcite 42 Dolomite 2 Carbonates Total 44 Fe-Dol 0 Siderite 0 Quartz 42 K-spar 2 Plagioclase 2 Other Minerals Total 49 Pyrite 3 Zeolite 0 Barite 0
Figure 5.5: Thin sections for Lower Bakken core at di↵erent resolution, mineralogy composed of 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’hasane↵ectontherecoveryof 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 adominantmineralogyandcanimpedeflowofhydrocarbons.
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.Theporosityofthecoreis8%.Sixlayers are used for the length of the extraction vessel with z =0.95cm.Eightradialringsare 4 used for the diameter, the outer ring represents the fracture with dimensions x =0.185 4 cm on each side.
Table 5.3: Radial case for Middle Bakken core Reservoir Dimensions and Properties
ni,nj,nk (Radial Grid) 8x1x6 Core Length (cm) 3.68 Core Diameter (cm) 1.13 Extraction Vessel Length (cm) 5.7 Extraction Vessel Diameter (cm) 1.5 Matrix Porosity (frac) 0.08 Matrix Permeability (md) 0.043 Fracture 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 (C C ) as shown in Figure 5.3. Fluid initialization is an unknown, however, the pres- 7 10 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,andng = 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,andconcentration 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,andconcentration 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 flooding experiments.
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 whforiginal =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 =2ft,tomaintainthesamefracture 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 parameters Reservoir Dimensions and Properties
Nx,Ny,Nz 11x27x1 x, y, z (ft) 45.455, VARJ, 50 4 4 4 Length (ft) 500 Width (ft) 2640 Thickness (ft) 50 Depth (ft) 10000 Matrix Porosity ( m)(frac) 0.0560 Matrix Permeability (km) (USRV) (md) 0.0005 Matrix Permeability (km) (SRV) (md) 0.0008 Fracture Porosity ( f )(frac) 0.0022 E↵ective Fracture Permeability (kf,eff) (USRV) (md) 0.005 E↵ective Fracture Permeability (kf,eff) (SRV) (md) 0.05 Hydraulic Fracture Width (whf )(ft) 0.001 Fracture Half Length (yf )(ft) 180 Lx,Ly,Lz (USRV) (ft) 50, 50, 50 Lx,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 case Fluid Characterization Hydrocarbon Mole Critical Critical Acentric Molecular Components Fraction Pressure Temperature Factor Weight (frac) (atm) (K) ( ) (lbm/lbm mol) N 2 0.0159 33.50 126.2 0.040 28.01 CO2 0.0038 72.80 304.2 0.225 44.01 CH4 0.3519 45.40 190.6 0.008 16.04 C2H6 0.1448 48.20 305.4 0.098 30.07 C3H8 0.0932 41.90 369.8 0.152 44.10 IC NC 0.0574 37.17 421.5 0.189 58.12 4 4 IC NC 0.0359 33.33 466.4 0.242 72.15 5 5 FC6 0.0280 32.46 507.5 0.275 86.00 C C 0.1726 26.29 604.5 0.406 125.04 7 13 C C 0.0625 16.45 747.4 0.720 235.57 14 22 C C 0.0342 10.67 803.9 1.242 441.99 23 30 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,andng =3(CoreyandRathjens,1956).
70 (a)
(b)
Figure 5.16: Equation of state (EOS) model tuning of Gas-Oil Ratio (GOR) and oil density with PVT laboratory data.
71 (a)
(b)
Figure 5.17: Equation of State (EOS) model tuning of oil viscosity and swelling factor with PVT 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,andsolvents types.ThefourEORschemes
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 =2500psia.
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 schemes EOR Schemes Solvent Scheme Injection Soaking Production Number Type Description (days) (days) (days) Cycles Scheme 1 (200 MSCF/day) 15 15 120 31 Scheme 2 (200 MSCF/day) 15 30 120 31 CO 2 Scheme 3 (400 MSCF/day) 15 15 120 31 Scheme 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 schemes EOR Schemes Solvent Scheme Injection Soaking Production Number Types Description (days) (days) (days) Cycles Scheme 1 (200 MSCF/day) 15 15 120 31 Scheme 2 (200 MSCF/day) 15 30 120 31 NGL Scheme 3 (400 MSCF/day) 15 15 120 31 Scheme 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,thesoakingtimesweredoubled(30 days) and the injection rates were the same (400 MSCF/day), the incremental recovery was almost 5%.Fromtheseresults,onecanconcludethatlonger 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 schemes EOR Scheme Injection Rate Soaking Incremental RF Solvent Utilization Solvent Type (MSCF/day) (days) (%) (MSCF/STB) CO2 Scheme 200 15 3.32 6.80 CO2 Scheme 400 15 5.00 10.15 NGL Scheme 200 15 4.00 5.90 NGL 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%.InjectionofNGL 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,theincremental 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, auniquecharacteristicofsupercriticalCO2 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,andgravity 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.Equation6.2isa
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.
S !J + S !J + S !J + r · ⌧ o o,c ⌧ g g,c ⌧ w w,c h ¯ ¯ i ⇠oxck o p o o D + ⇠gyck g p g g D + r· r r r r (6.1) ⇣ ⌘ ¯ ¯ ⇣ ⌘ ⇠ w k p D + r· w c w r w wr ⇣ @ ⌘ ⇠oxcqˆo + ⇠gycqˆg + ⇠wwcqˆw = @t [ zc (⇠oSo + ⇠gSg + ⇠wSw)] Where,
⌧ is tortuosity, porosity, !J ↵,c phase molecular di↵usion flux of component c, ⇠↵ is the molar density, xc, yc,andwc are the liquid, vapor, and water mole fractions of component c, k permeability, phase mobility, p phase pressure gradient, phase gamma, D depth ↵ r ↵ ↵ r gradient,q ˆ↵ 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.
S !J + S !J + S !J + r · ⌧ o o,c ⌧ g g,c ⌧ w w,c ¯ ¯ ⇠ox ck o ( po o D)+ ⇠gyck g ( pg g D) + (6.2) r· r r ¯ ¯ r r ⇥ ⇠wwck w ( pw w D) + ⇤ r· r@ r ⇠oxcqˆo + ⇠gycqˆg +h ⇠wwcqˆw = @t [ zc (⇠oSo +i⇠gSg + ⇠wSw)]
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).
x nc x x (u u ) i µ = i j i j (6.3) RT rT,p i j=1 Dij j=i X6 where xi is mole fraction of component i, R is gas constant, T is temperature, µi is the chemical potential, D , i, j =1,...,nc (i = j)aretheMSdi↵usioncoe cients,which ij 6 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 n (n 1) /2MSdi↵usioncoe cients.Atconstant c c
83 temperature, T ,andpressure,p,duetoGibbs-Duhemequation
nc x µ =0 (6.4) irT,p i i=1 X The chemical potential gradient can be written in terms of the fugacity, fi and the composition gradient as follows (Firoozabadi, 1999).
nc 1 @ ln (f ) µ = RT i x (6.5) rT,p i @x r j j=1 j X With substitution, Equation 6.3 can be written as
nc 1 nc @ ln (f ) 1 (x J x J ) x i x = j i i j (6.6) i @x r j ⇠ j=1 j j=1 Dij j=i X X6 In matrix form, Equation 6.6 can be written as
BJ = ⇠ x (6.7) r Where, ⇠ is overall molar density and the di↵usion coe cients (B)canbeexpressed as:
xi + 1 nc xk i = j Dinc ⇠ j=1 Dik j=i Bij = 6 (6.8) 8 1 1 xi P i = j < Dij Dinc 6 ⇣ ⌘ : B =[B ] (6.9) ij i,j=1,....,nc 1 and the non-ideality correction factor ( ) is expressed as:
@ ln (fi) ij = xi (6.10) @xj Where,
=[ ] (6.11) ij i,j=1,....,nc 1
x =[ xi]i=1,....,nc 1 (6.12) r r
J =[ Ji]i=1,....,nc 1 (6.13) r
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
1 B as:
1 J = ⇠B x (6.14) r 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:
nc 1 J = c D x (6.15) i ijr j j=1 X Where, i =1,....,nc 1. In the above Equation 6.15, only (n 1) independent di↵usion c fluxes appear. The last di↵usion coe cient can be calculated from the sum of total di↵usion flux below:
nc
Ji =0 (6.16) i=1 X Equation 6.15 can be written in a matrix form as
J = cD x (6.17) r Where, D is a (n 1) x (n 1) matrix known as the Fickian di↵usion coe cient matrix. c c 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 D = D ; i = j. Comparing fluxes from Equation 6.14 and Equation 6.17 leads to the i,j 6 j,i 6 following relationship:
1 D = B (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:
J = cDeff x (6.19) i i r i eff Where, Di 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.
1 y D = i (6.20) i,m nc 1 yj j=1 Dij j=1 6 P 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
8 0 7.40 10 MWi,m T D = ⇥ (6.21) i,m q 0.6 µm (vbi )
Where, µm is the mixture viscosity in cp, vbi 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
nc j=i yj,mMWj MW0 = 6 (6.22) i,m 1 y P i,m
and the partial molar volume, vbi ,isestimatedusingTyneandCalusmethodfromcritical
3 volume, Vc (cm /mol), as follows (Reid et al., 1987):
1.048 vbi =0.285Vc (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:
⇠mDij 2 3 0 =0.99589 + 0.096016⇠r 0.22035⇠r +0.03287⇠r (6.24) (⇠mDij) 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-
0 Robinson EOS for each phase, ⇠r is the mixture reduced molar density, (⇠mDij) is the low pressure molar density-di↵usivity product.
The Reduced mixture, ⇠r,canbecalculatedusingEquation6.25
nc 5/3 ⇠ yi,mVci ⇠ = m = ⇠ j=1 (6.25) r m nc 2/3 ⇠c "Pj=1 yi,mVci # Where, P
⇠c is the critical molar density, and Vci is the critical molar volume of component i . 0 The low-pressure molar density-di↵usivity product, (⇠mDij) is calculated using Reid et al., 1987 as:
1 0.0018583T 1/2 1 1 /2 (⇠ D )0 = + (6.26) m ij 2 ⌦ R M M ij ij ✓ i j ◆ 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 + = i j (6.27) ij 2
T 1/3 =(2.3551 0.087! ) ci (6.28) i i p ✓ ci ◆
88 Where, !i is acentric factor of component i, Tci is the critical temperature of component i,andpci is the critical pressure of component i .
The collision integral of the Lennard-Jones potential ⌦ij can be expressed as:
1.06036 0.193 1.03587 1.76474 ⌦ij = 0.1561 + + + (6.29) Tij exp (0.47635Tij) exp (1.52996Tij) exp (3.89411Tij)
kBT Tij = (6.30) "ij
"ij = p"i"j (6.31)
"i = kB (0.7915 + 0.1963!i) Tci (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
⇠mDij 0 =0.18839 exp (3 ⇠r) (6.33) (⇠mDij)
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:
10.2/v 0.791 8 1.47 ( bi ) 0.71 Di,m =13.3 10 T µm v (6.34) ⇥ bi Where,
vbi 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:
9 0.4562 0.6898 1.706 1.831 4.524 Di,m =10 µm MWi vi p T (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)
⇠mDij Riazi and Whitson (1993) correlate the 0 similar to Sigmund (1976) to reduced (⇠mDij ) molar density and also to viscosity ratio performs better. Their correlation is shown in 6.36.
p ( 0.27 0.3!m)+( 0.05+0.1!m) ⇠ D µ pc,m m ij =1.07 m (6.36) 0 µ0 (⇠mDij) ✓ m ◆ Where,
0 ⇠m and µm are the mixture viscosity at conditions of the system, µm is viscosity of the mixture at low pressure. And for a binary system components A and B with molar fraction xA and xB,pseudocriticalpressureandpseudoacentricfactorofthemixturearegivenas follows:
pc,m = xApc,A + xBpc,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.
1 x1 x2 D1,2 =(D121) (D211) (6.39)
Where,
D121 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:
xj xi nc xk/2 Di,j = Dij1 Dji1 k=1 Dik1Djk1 (6.40) k=i,j 6 Q Where, i = j =1,.....,nc ; i = j, D is Maxwell-Stefan (MS) of the binary pair i j, 6 i,j
Dij1 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)andacentric factor (!)asshowninEquation6.41:
⇠ D µ T p A1 µ [A2(!1,!2)+A3(pr,Tr)] 1 211 = f 1 ,T ,p ,! = A r,1 r,2 1 (6.41) 0 µ0 r r 0 T p µ0 (⇠1D21) ✓ 12 ◆ ✓ r,2 r,1 ◆ ✓ 12 ◆ Where,
A0, A1, A2 and A3 are constants given by:
A =exp( 0.0472) (6.42) 0
A1 =0.103 (6.43)
91 A = 0.0147 (1 + 10! ! +10! ! )(6.44) 2 1 2 1 2
(0.337 3) 0.337 1.852 0.1852 Tr,1pr,2 A3 = 0.0053 pr, 1 ⇥ 6pr, 2 +6Tr 0.1914Tr, 1 +0.0103 (6.45) Tr,2pr,1 ⇣ ⌘ ✓ ◆ Where,
0 (⇠1D21) is the dilute gas density-di↵usion coe cient product (mol/ms)andiscalculated using the Fuller et al. (1969) as shown in Equation 6.46:
1/2 0.00101 T 0.75 1 + 1 0 ⇥ M1 M2 (⇠1D21) = 2 (6.46) 1/3 ⇣ 1/3 ⌘ R ( v1) +( v2) The flowchart for calculating molecularh diP↵usion coeP cientsi 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 and Firoozabadi (2007) approach.
93 Table 6.1: Molecular Di↵usion Calculations for Middle Bakken fluid system Molecular Di↵usion Coe cients Components Wilke and Chang Sigmund Hayduk-Minhas Leahy Dios-Firoozabadi 2 2 2 2 (cm /sec) (cm /sec) (cm /sec) (cm /sec) N 2 9.40E-05 3.82E-06 5.47E-04 4.43E-05 CO2 9.06E-05 2.64E-06 5.64E-04 4.04E-05 CH4 1.06E-04 3.41E-06 5.92E-04 2.12E-05 C2H6 7.31E-05 2.56E-06 6.60E-04 2.86E-05 C3H8 5.81E-05 2.02E-06 6.55E-04 2.62E-05 IC NC 4.93E-05 1.69E-06 6.26E-04 2.34E-05 4 4 IC NC 4.38E-05 1.45E-06 5.95E-04 2.09E-05 5 5 FC6 4.02E-05 1.30E-06 5.72E-04 1.89E-05 C C 3.57E-05 1.09E-06 4.97E-04 1.08E-05 7 13 C C 2.32E-05 7.24E-07 3.48E-04 7.50E-06 14 22 C C 2.10E-05 5.75E-07 2.49E-04 —– 23 30 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.
S !J + S !J + S !J + r · ⌧ o o,c ⌧ g g,c ⌧ w w,c h ¯ ¯ i ⇠oxck o p o o D + ⇠gyck g pg g D + r· r r r r (6.47) ⇣ ⌘ ¯ ¯ ⇣ ⌘ ⇠ w k p D + r· w c w r w wr ⇣ @ ⌘ ⇠oxcqˆo + ⇠gycqˆg + ⇠wwcqˆw = @t [ zc (⇠oSo + ⇠gSg + ⇠wSw)]
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 system Laboratory Data Solvent Bubble Point Density Viscosity FVF Solution-Gas-Ratio Swelling V SCF mole (%) Pressure(psia) (g/cc) (cp) ( R/VS ) ( /STB) Factor 0.00 2530 0.647 0.383 1.615 867.34 1.000 13.09 2663 0.652 0.305 1.736 1091.35 1.068 20.96 2783 0.655 0.257 1.845 1342.56 1.110 45.20 3403 0.669 0.153 2.271 2198.93 1.326 54.60 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 )ande↵ective fracture permeability (kf,eff), were determined using anovelcompositionalrate-transientanalysis,designedforlowpermeabilityliquid-richshale 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.Theformulationhasamole 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)andhydraulic
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 anarrowregionnearthefracture-matrixinterface.
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.
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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+1 @p ¯n U (pn+1)+MoleCorr = V (c + c ) o (A.1) tc c o r v|zc @t c=1 X For dual-porosity system, the compositional volume balance formulation is
nc+1 nc+1 n n @pof ¯t uc ¯t ⌧tc + MoleCorrf = f (c + cv zc )f (A.2) c f cf m/f | @t c=1 c=1 X X ⇣ ⌘ nc+1 n @pom ¯ ⌧t + MoleCorrm = m(c + cv z )m (A.3) tcm cm/f | c @t c=1 X ⇣ ⌘ The total transfer function ⌧ for each component for all the phases is defined as tcm/f
⌧ = x ⇠ ⌧ + y ⇠ ⌧ + w ⇠ ⌧ (A.4) tcm/f c o o c g g c w w
The individual phase transfer functions are described as ⌧ = k (p p )+ z [(h h ) (h h )] (A.5) o m of/m of om o wf wm of om h ⇣ ⌘ i ⌧ = k (p p )+ z [(h h ) (h h )] (A.6) g m gf/m gf gm g gf gm gf gm h ⇣ ⌘ i ⌧ = k (p p )+ z [(h h ) (h h )] (A.7) w m wf/m wf wm w wf wm wf wm h ⇣ ⌘ i A.2 Derivation of Compositional Equation and Pressure Equation
The component pressure equation is shown as
@ z u = c (A.8) c @t v ✓ t ◆
105 The component net molar flux is
u = k x ( p D)+ k y ( p D)+ c r· o c r o or r· g c r g gr k w ( p D)+⇠nxnqn + ⇠nynqn + ⇠n wnqn (A.9) r· w c r w wr o c o g c g w c w
The total pressure equation is
@ u = (A.10) t @t v ✓ t ◆ Multiply by A.10 by zc and subtract it from A.8 @ z @ u z u = c z (A.11) c c t @t v c @t v ✓ t ◆ ✓ t ◆ @ @z @ @z u z u = z + c z = c (A.12) c c t c @t v v @t c @t v v @t ✓ t ◆ t ✓ t ◆ t The compositional equation is
v @z t (u z u )= c (A.13) c c t @t From A.8, expand the right hand side
@ z @z @ u = c = c + z (A.14) c @t v v @t c @t v ✓ t ◆ t ✓ t ◆ Summing on all the components
nc+1 @ 1 @ @ 1 u = = + (A.15) c @t v v @t @t v c=1 t t t X ✓ ◆ ✓ ◆ nc+1 1 @ 1 @v u = + t (A.16) c v @t v2 @t c=1 t t X nc+1 1 @ @p 1 @v @p nc+1 @v @z u = t + t c (A.17) c v @p @t v @p @t @z @t c=1 t " t c=1 c !# X X ✓ ◆ This expression can be rewritten in terms of pore compressibility (c )andfluidcompress- ibility (cv)
nc+1 @p 1 nc+1 @v @z u = [c + c ] t c (A.18) c v v|zc @t v @z @t c=1 t ( t c=1 c ) X X ✓ ◆
106 By definition,
nc+1 @v ¯ = t (A.19) tc @z c=1 c X ✓ ◆ Therefore,
@p nc+1 @z u = [c + c ] ¯ c (A.20) t v v|zc @t v2 tc @t t t c=1 X From the definition of partial molar volume,
V ¯ = 4 t (A.21) tc N ✓4 c ◆p,T,N Nc 0 4 ! @V ¯ = t (A.22) tc @N ✓ c ◆p,T,N Divide by total number of moles N
@Vt/N ¯tc = (A.23) @Nc/N p,T,Nn=c ✓ ◆ 6 @v ¯ = t (A.24) tc @z c p,T,Nn=c ✓ ◆ 6 From A.20, substitute the definition of compositional equation A.13,
@p nc+1 v u = [c + c ] ¯ t (u z u ) (A.25) t v v|zc @t v2 tc c c t t t c=1 X @p 1 nc+1 v u = [c + c ] ¯ t (u z u ) (A.26) t v v|zc @t v v tc c c t t t ( t c=1 ) X @p 1 nc+1 u = [c + c ] [¯ u (¯ z ) u ] (A.27) t v v|zc @t v tc c tc c t t t ( c=1 ) X From,
nc+1
vt = ¯tc zc (A.28) c=1 X Substitute A.28 into A.27
@p 1 nc+1 u = [c + c ] ¯ u v u (A.29) t v v|zc @t v tc c t t t t ( c=1 ) X
107 @p 1 nc+1 [c + c ] = ¯ u (A.30) v v|zc @t v tc c t t c=1 X Rearranging, the pressure equation is
nc+1 @p ¯ u = [c + c ] (A.31) tc c v|zc @t c=1 X
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:
RT a 2bRT a bRT v3 b v2 + 3b2 v b b2 =0 (B.1) p p p p p ✓ ◆ ✓ ◆ ✓ ◆ 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
ZRT molar volume v = 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,
nc nc
a = amnxmxn (B.3) m=1 n=1 X X a =(1 ) a1/2a1/2 (B.4) mn mn m n
1 R2T 2 /2 a1/2 = ⌦ cm 1+ 1 T 1/2 (B.5) m a p m rm cm ⇥ ⇤ =0.37464 + 1.54226! 0.26992!2 (B.6) m m m
nc
b = bmxm (B.7) m=1 X RTcm bm =⌦b (B.8) pcm
nc nc p A = a x x (B.9) mn m n R2T 2 m=1 n=1 ! X X
109 nc p B = b x (B.10) m m RT m=1 ! X 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 p ⇠ = = (B.11) v zRT
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
fm m = (B.12) xmp The natural log of fugacity coe cient of phase (↵ = o, g)isdefinedas
↵ bm ln m = (z↵ 1) ln (z↵ B↵) b↵ nc 1 A 2 x a b z↵ + p2+1 B↵ ↵ n=1 n nm m ln (B.13) 2p2 (B↵ a↵ b↵ "z↵ p2 1 B↵ #) ✓ P ◆ B.3 Derivative of Fugacity with respect to Pressure and Composition
The derivative of fugacity of phase (↵ = o, g)withrespecttopressureis
↵ @ ln m = bm @z↵ 1 @z↵ @B↵ @p b↵ @p (z↵ B↵) @p @p nc 1 2 n=1 xnanm ⇣ bm ⌘ 2p2 a↵ b↵ P @hz↵ +(p⇣2+1) @B↵ @z↵ ⌘i(p2 1) @B↵ A↵ @p @p @p @p (B.14) B↵ z↵+ p2+1 B↵ z↵ p2 1 B↵ ( ) ( ) ⇢⇣ ⌘ ✓ ◆ The derivative of fugacity of phase (↵ = o, g)withrespecttocompositionis
110 ↵ @ ln m bm @z↵ @b↵ 1 @z↵ @B↵ = 2 b (z 1) @x b ↵ @x ↵ @x (z↵ B↵) @x @x k,↵ ↵ k,↵ k,↵ k,↵ k,↵ h @A↵ i@B↵ ⇣ ⌘ B↵ A↵ nc z↵+ p2+1 B↵ @x @x 1 ( ) k,↵ k,↵ 2 n=1 xnanm bm ln ✓ ◆ 2 ✓ ◆ + 2p2 z↵ (p2 1)B↵ B↵ a↵ b↵ ( " # P !) ⇣ ⌘ nc @a↵ 2 a↵akm xnanm n=1 @xk,↵ 1 A↵ bm @b↵ p B P a2 + b2 @x + 2 2 ( ↵ " ↵ ↵ k,↵ #) ⇣ ⌘ @z @B @z @B nc ↵ + p2+1 ↵ ↵ p2 1 ↵ 2 x a @x ( ) @x @x ( ) @x 1 A↵ n=1 n nm bm k,↵ k,↵ k,↵ k,↵ (B.15) 2p2 B↵ a↵ b↵ z↵+ p2+1 B↵ z↵ p2 1 B↵ P ( ) ( ) ⇢⇣ ⌘⇣ ⌘ ✓ ◆ 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
(B z) a @z = @p 3z2 2(1 B) z +(A 2B 3B2) R2T 2 (z2 2z 6Bz (A 2B 3B2)) ⇣ b ⌘ (B.16) 3z2 2(1 B) z +(A 2B 3B2) RT ✓ ◆ The derivative of compressibility factor (z factor) with respect to composition is
(B z) nc (a + a ) x p @z = n=1 nm mn n @xm 3z2 2(1 B) z +(A 2B 3B2) R2T 2 P (z2 2z 6Bz (A 2B 3B2)) ⇣ b p ⌘ m (B.17) 3z2 2(1 B) z +(A 2B 3B2) RT ✓ ◆ B.5 Partial Molar Volume
The partial molar volume per component m is defined as
@V ¯ = t (B.18) tm @N ✓ m ◆p,T,N Where,
Vt = voNo + vgNg (B.19)
The derivative of total volume with respect to total number of moles
111 @V @ (v N ) @ (v N ) t = o o + g g (B.20) @N @N @N t,m t,m t,m With manipulation,
@ (v N ) z RT @N RT @z o o = o o + N o (B.21) @N p @N p o @N t,m o t,m o t,m With further manipulation and substitution, the end result
nc nc @ (voNo) 1 @zo @No,k = vo 1+ [ kf xf ] (B.22) @Nt,m (" zo @xf # @Nt,m ) Xk=1 Xf=1 ✓ ◆ ✓ ◆ Similarly for gas phase,
nc nc @ (vgNg) 1 @zg @Ng,k = vg 1+ [ kf yf ] (B.23) @Nt,m (" zg @yf # @Nt,m ) Xk=1 Xf=1 ✓ ◆ ✓ ◆ Therefore, the total partial molar volume per component m can be represented as
nc nc 1 @zo @No,k ¯tm = vo 1+ [ kf xf ] + (B.24) (" zo @xf # @Nt,m ) Xk=1 Xf=1 ✓ ◆ ✓ ◆ nc nc 1 @zg @Ng,k vg 1+ [ kf yf ] (B.25) (" zg @yf # @Nt,m ) Xk=1 Xf=1 ✓ ◆ ✓ ◆ B.6 Fluid Compressibility
The fluid compressibility can be expressed as
1 @vt cvt = (B.26) vt @po ✓ ◆T,zm where the derivative is expressed as
@v N @v v nc @N N @v v nc @N t = o o + o o,k + g g + g g,k @po Nt @p Nt @po Nt @p Nt @po ✓ ◆T,zm " ✓ ◆T,zm k=1 # " ✓ ◆T,zm k=1 # X X (B.27)
112 APPENDIX C - SATURATION EQUATIONS
C.1 Liquid and Vapor Equations
The liquid (L)andvapor(V )relationshipsareshownbelow S ⇠ L = o o (C.1) So⇠o + Sg⇠g S ⇠ V = g g (C.2) So⇠o + Sg⇠g
C.2 Derivation of Saturation Equations
The saturation equation can be derived using liquid (L)andvapor(V )relationships below
(1 Sw) L⇠g So = (C.3) V⇠o + L⇠g
(1 Sw) V⇠o Sg = (C.4) V⇠o + L⇠g
113