INVESTIGATION OF NANOPORE CONFINEMENT EFFECTS ON CONVECTIVE AND
DIFFUSIVE MULTICOMPONENT MULTIPHASE FLUID TRANSPORT IN SHALE USING
IN-HOUSE SIMULATION MODELS
Fengshuang Du
Dissertation submitted to the faculty of the Virginia Polytechnic Institute and State University in
partial fulfillment of the requirements for the degree of
Doctor of Philosophy
In
Mining Engineering Department
Bahareh Nojabaei, Chair
Nino Ripepi
Cheng Chen
Bagus Muljadi
August 10, 2020
Blacksburg, Virginia
Keywords: multicomponent phase behavior, flow in porous media, nanopore confinement
effects, large gas-oil capillary pressure, critical property shift, shale reservoirs, molecular
diffusion, multi-phase flow
© 2020 Fengshuang Du
Investigation of Nanopore Confinement Effects on Convective and Diffusive Multicomponent
Multiphase Fluid Transport in Shale Using In-house Simulation Models
Fengshuang Du
ABSTRACT
Extremely small pore size, low porosity, and ultra-low permeability are among the characteristics
of shale rocks. In tight shale reservoirs, the nano-confinement effects that include large gas-oil
capillary pressure and critical property shifts could alter the phase behaviors, thereby affecting the
oil or gas production. In this research, two in-house simulation models, i.e., a compositionally
extended black-oil model and a fully composition model are developed to examine the nano-pore
confinement effects on convective and diffusive multicomponent multiphase fluid transport.
Meanwhile, the effect of nano-confinement and rock intrinsic properties (porosity and tortuosity
factor) on predicting effective diffusion coefficient are investigated.
First, a previously developed compositionally extended black-oil simulation approach is
modified, and extended, to include the effect of large gas-oil capillary pressure for modeling first
contact miscible (FCM), and immiscible gas injection. The simulation methodology is applied to
gas flooding in both high and very low permeability reservoirs. For a high permeability
conventional reservoir, simulations use a five-spot pattern with different reservoir pressures to
mimic both FCM and immiscible displacements. For a tight oil-rich reservoir, primary depletion
and huff-n-puff gas injection are simulated including the effect of large gas-oil capillary pressure
in flow and in flash calculation on recovery estimations. A dynamic gas-oil relative permeability
correlation that accounts for the compositional changes owing to the produced gas injection is
introduced and applied to correct for changes in interfacial tension (IFT), and its effect on oil
recovery is examined. The results show that the simple modified black-oil approach can model
well both immiscible and miscible floods, as long as the minimum miscibility pressure (MMP) is
matched. It provides a fast and robust alternative for large-scale reservoir simulation with the
purpose of flaring/venting reduction through reinjecting the produced gas into the reservoir for
EOR.
Molecular diffusion plays an important role in oil and gas migration in tight shale formations.
However, there are insufficient reference data in the literature to specify the diffusion coefficients
within porous media. Another objective of this research is to estimate the diffusion coefficients of
shale gas, shale condensate, and shale oil at reservoir conditions with CO2 injection for EOR/EGR.
The large nano-confinement effects including large gas-oil capillary pressure and critical property
shifts could alter the phase behaviors. This study estimates the diffusivities of shale fluids in
nanometer-scale shale rock from two perspectives: 1) examining the shift of diffusivity caused by
nanopore confinement effects from phase change (phase composition and fluid property)
perspective, and 2) calculating the effective diffusion coefficient in porous media by incorporating
rock intrinsic properties (porosity and tortuosity factor). The tortuosity is obtained by using
tortuosity-porosity relations as well as the measured tortuosity of shale from 3D imaging
techniques. The results indicated that nano-confinement effects could affect the diffusion
coefficient through altering the phase properties, such as phase compositions and densities.
Compared to bulk phase diffusivity, the effective diffusion coefficient in porous shale rock is
reduced by 102 to 104 times as porosity decreases from 0.1 to 0.03.
Finally, a fully compositional model is developed, which enables us to process multi-
component multi-phase fluid flow in shale nano-porous media. The validation results for primary
depletion, water injection, and gas injection show a good match with the results of a commercial
software (CMG, GEM). The nano-confinement effects (capillary pressure effect and critical
property shifts) are incorporated in the flash calculation and flow equations, and their effects on
Bakken oil production and Marcellus shale gas production are examined. The results show that
including oil-gas capillary pressure effect could increase the oil production but decrease the gas
production. Inclusion of critical property shift could increase the oil production but decrease the
gas production very slightly. The effect of molecular diffusion on Bakken oil and Marcellus shale
gas production are also examined. The effect of diffusion coefficient calculated by using Sigmund
correlation is negligible on the production from both Bakken oil and Marcellus shale gas huff-n-
puff. Noticeable increase in oil and gas production happens only after the diffusion coefficient is
multiplied by 10 or 100 times.
Investigation of Nanopore Confinement Effects on Convective and Diffusive Multicomponent
Multiphase Fluid Transport in Shale Using In-house Simulation Models
Fengshuang Du
GENERAL AUDIENCE ABSTRACT
Shale reservoir is one type of unconventional reservoir and it has extremely small pore size,
low porosity, and ultra-low permeability. In tight shale reservoirs, the pore size is in nanometer
scale and the oil-gas capillary pressure reaches hundreds of psi. In addition, the critical properties
(such as critical pressure and critical temperature) of hydrocarbon components will be altered in
those nano-sized pores. In this research, two in-house reservoir simulation models, i.e., a
compositionally extended black-oil model and a fully composition model are developed to
examine the nano-pore confinement effects on convective and diffusive multicomponent
multiphase fluid transport. The large nano-confinement effects (large gas-oil capillary pressure
and critical property shifts) on oil or gas production behaviors will be investigated. Meanwhile,
the nano-confinement effects and rock intrinsic properties (porosity and tortuosity factor) on
predicting effective diffusion coefficient are also studied.
vi
ACKNOWLEGEMENTS
I would like to thank the following individules and organizations, without whom I would not
have been able to complete this dissertation! First, I would like to express my deepest apprication
to Dr. Bahareh Nojabaei, my academic advisor, for her excellent guidance, strong support, and
consistant encouragement during my entire Ph.d research career. She is such a nice lady and is
always with patience. She gave me so much valuable advice and guidance throughout the duration
of my reaserch, which is also very meaningful for my future research and life. In addition, I am
grateful to other committee members: Dr. Nino Ripepi, Dr. Cheng Chen, and Dr. Bagus Muljadi,
for their valueble time and effort to serve on my committee and for their very helpful suggestions
and feedback throughout my Ph.D study.
I would like to thank my research group memebers: Mr. Kaiyi Zhang and Mr. Deraldo de
Carvalho, for their helpful technical discussions and assistance during my Ph.d study. I would also
like to thank Dr. Cigdem Keles, for her helpful suggestions.
I would like to acknowlege the financial assistance provided by the US. Department of Energy
through the National Energy Technology Laboratory’s Program.
Finally, I would like to express my deepest gratitude to my husband, Jingwei Huang, who has
stood by me through all my travails, and gave me encouragement, love, and support. I thank my
parents for their unconditional love and constant support during my life.
vii
TABLE OF CONTENTS
LIST OF FIGURES ....................................................................................................................x
LIST OF TABLES .................................................................................................................... xv
Chapter 1 Introduction ................................................................................................................1
1.1 Background .......................................................................................................................1
1.2 Objectives and Scope of This Study ...................................................................................4
1.3 Outline of Dissertation .......................................................................................................5
Chapter 2 Literature Review........................................................................................................6
2.1 Gas injection approaches....................................................................................................6
2.1.1 Huff-n-Puff gas injection and mechanisms ..................................................................7
2.1.2 Gas flooding and mechanisms ................................................................................... 15
2.2 Nano-confinement effects in phase behaviors................................................................... 16
2.3 Diffusion coefficient in shale rock ................................................................................... 18
2.4 Greenhouse gas control .................................................................................................... 22
Chapter 3 Compositionally Extended Black Oil Simulation Model ............................................ 26
3.1. Methodology .................................................................................................................. 26
3.1.1 Flash calculations ...................................................................................................... 26
3.1.2 Slim-tube simulation ................................................................................................. 28
3.1.3 Compositionally extended black oil model ................................................................ 29
3.1.4 IFT-dependent relative permeability curves ............................................................... 30
3.2 Bakken oil properties in nanopores .................................................................................. 31
3.3 Miscible and immiscible gas flooding in conventional reservoir ....................................... 35
3.4 Huff-n-puff in an oil-rich tight reservoir........................................................................... 38
3.5 Effect of IFT-dependent relative permeability on recovery ............................................... 46
viii
Chapter 4 Diffusion Coefficient with Nano-confinement Effects ............................................... 50
4.1 Methodology ................................................................................................................... 50
4.2 Validation of empirical correlations ................................................................................. 53
4.3 Diffusivity of shale fluids without confinement effects .................................................... 56
4.4 Nano-confinement effects on diffusivity .......................................................................... 60
4.4.1 Critical Property Shift................................................................................................ 60
4.4.2 Gas-oil Capillary pressure ......................................................................................... 61
4.4.3 Diffusion with confinement effect on shale oil production ......................................... 72
4.5 Effective Diffusion Coefficient in Porous Media .............................................................. 74
4.5.1 Methodology ............................................................................................................. 74
4.5.2 Effective molecular diffusivity in porous media ......................................................... 76
Chapter 5 Compositional Simulation Model .............................................................................. 79
5.1 Mathematical Formulation ............................................................................................... 79
5.1.1 Material Balance Equaitons ....................................................................................... 79
5.1.2 Source or sink term.................................................................................................... 80
5.1.3 Numerical Solution.................................................................................................... 82
5.1.4. Relative permeability................................................................................................ 85
5.2 Phase Behavior Model ..................................................................................................... 86
5.2.1 Equation of state ........................................................................................................ 86
5.2.2 Vapor-Liquid Equilibrium ......................................................................................... 86
5.2.3 Phase properties ........................................................................................................ 88
5.3Validation results .............................................................................................................. 89
5.4 Nano-confinement effects ................................................................................................ 98
5.4.1 Critical property shift ................................................................................................ 98
5.4.2 Oil–gas capillary pressure .......................................................................................... 98
5.4.3 Simulation results .................................................................................................... 100
ix
5.5 Molecular diffusion effect .............................................................................................. 110
5.5.1 Molecular diffusion in Bakken oil ........................................................................... 112
5.5.2 Molecular diffusion in Marcellus shale gas .............................................................. 118
Chapter 6 Conclusions ............................................................................................................ 120
6.1 Summary and conclusions.............................................................................................. 120
6.2 Future research .............................................................................................................. 123
6.2.1 Slim tube simulation to estimate MMP as a function of permeability and fluid
compositions. ................................................................................................................... 123
6.2.2 Inclusion of adsorption behavior in the compositional model to investigate CO2
injection in shale gas in nano-sized pores. ........................................................................ 123
6.2.3 To develop an Embedded Discrete Fracture Model (EDFM). ................................... 123
APPENDIX A EQUATION OF STATE ................................................................................. 124
APPENDIX B VAPOR-LIQUID EQUILIBRIUM .................................................................. 126
APPENDIX C PHASE PROPERTIES .................................................................................... 128
C.1 Molecular weight .......................................................................................................... 128
C.2 Oil and gas densities ...................................................................................................... 128
C.3 Oil and gas viscosity ..................................................................................................... 128
C.3.1 Viscosity of gas phase ............................................................................................. 128
C.3.2 Viscosity of liquid phase ......................................................................................... 129
C.4 Saturation ...................................................................................................................... 130
C.5 Water properties ............................................................................................................ 130
References .............................................................................................................................. 131
x
LIST OF FIGURES
Figure 2.1 Incremental oil recovery factor of huff-n-puff and gas flooding from simulation studies
(Table 2.2), for the range of matrix permeability from 0.1 to 100 µd. Different colors represent
different simulation studies (Du and Nojabaei, 2019). ............................................................... 14
Figure 2.2 (a) Total flared/vented natural gas in United States; (b) Produced, marketed, and flared/
vented natural gas in the state of North Dakota. ......................................................................... 23
Figure 3.1 Pressure‒composition plot for Bakken oil for three different effective pore sizes (a) by
using the same overall compositions at infinitely large pore sizes (Eq. 3.6) and (b) by using
different overall compositions at different pore sizes (Eq. 3.7). ................................................. 32
Figure 3.2 Extrapolation of oil and gas densities for three different effective pore sizes (a) by using
the overall compositions from Eq. 3.6 and (b) by using the overall compositions from Eq. 3.7. . 33
Figure 3.3 Oil component recovery as a function of pressure of 1.0 PVI by using the extrapolation
of black oil properties (a) getting from Eq. 3.6 and (b) from Eq. 3.7. ......................................... 34
Figure 3.4 Cumulative oil production of gas floods at different initial reservoir pressures and
production pressures.................................................................................................................. 36
Figure 3.5 Gas composition at production well block for gas injection in conventional reservoirs
at different initial reservoir pressures and production pressures. ................................................ 37
Figure 3.6 Gas saturation distribution in the conventional reservoir at different times (Pi=3000 psia
and Pw =2900 psia). ................................................................................................................... 38
Figure 3.7 Gas saturation distribution in the conventional reservoir at different times (Pi=4500 psia
and Pw =4400 psia). ................................................................................................................... 38
Figure 3.8 Reservoir permeability map for tight oil-rich reservoir. ............................................ 40
Figure 3.9 Cumulative oil recovery for primary depletion and huff-n-puff gas injection with and
without considering capillary pressure in tight oil-rich reservoirs at at Pi=5000 psia and (a) Pw
=1000 psia, (b) Pw =2000 psia, (c) Pw =3000 psia, 4000 psia, and 4400 psia. ............................. 42
Figure 3.10 Oil pressure at the well block for primary depletion and huff-n-puff gas injection with
and without considering capillary pressure in tight oil-rich reservoirs at Pi=5000 psia and Pw=1000,
2000, 3000, 4000, and 4500 psia, respectively. .......................................................................... 42
xi
Figure 3.11 Cumulative gas recovery for primary depletion and huff-n-puff gas injection at Pw =
1000, 2000, 3000, 4000, and 4500 psia, respectively accounting for the capillary pressure both in
flow and flash............................................................................................................................ 45
Figure 3.12 Relative permeability curves for case 3, adopted from Yu et al. (2014). .................. 47
Figure 3.13 Oil-gas IFTs of Bakken black oil at different reservoir pressures. ........................... 47
Figure 3.14 Cumulative oil production for huff-n-puff gas injection in tight oil-rich reservoirs
using IFT-dependent relative permeability curves and base relative permeability curves,
respectively, at Pi=5000 psia and (a) Pw =2000 psia, (b) Pw =3000 psia, and (c) Pw =4000 psia. . 49
Figure 4.1 The diffusion coefficient of CH4 in C1/C3 mixtures by using empirical correlations and
comparison with experimental data (Sigmund, 1976a) at (a) 160 °F and 3000 psia; (b) 160 °F and
2000 psia; (c) 100 °F and 2000 psia; (d) 220 °F and 1000 psia. ................................................. 54
Figure 4.2 The diffusion coefficient of CH4 in C1/C10 mixtures by using empirical correlations and
comparison with experimental data (Dysthe and Hafskjold, 1995) at (a) 86 °F and 40 MPa; (b)
86 °F and 50 MPa. .................................................................................................................... 55
Figure 4.3 The diffusivities of components of Marcellus shale gas in gas phase at the reservoir
temperature. .............................................................................................................................. 57
Figure 4.4 The diffusivities of components of Marcellus shale condensate in gas and liquid phases
at the reservoir temperature. ...................................................................................................... 58
Figure 4.5 The diffusivities of components in (a) Bakken oil in gas phase and (b) Bakken oil in
liquid phase at reservoir temperatures. ....................................................................................... 59
Figure 4.6 P–T phase envelops of Bakken shale oil, Bakken shale oil with CO2 injection at 20%
and 50%, Marcellus shale condensate, condensate with CO2 injection at 20%, 50% and 80%, and
Marcellus shale gas. .................................................................................................................. 62
Figure 4.7 For Marcellus shale gas, the diffusivities of (a) CO2 and (b) CH4 in gas phase versus
CO2 mole percentage at different pressures and pore sizes with and without considering critical
property shifts. .......................................................................................................................... 63
Figure 4.8 For Lower Huron, the diffusivities of (a) N2 and (b) CH4 in gas phase versus N2 mole
percentage at different pressures and pore sizes with and without considering critical property
shifts. ........................................................................................................................................ 65
xii
Figure 4.9 The gas molar density of (a) Marcellus shale gas versus CO2 mole percentage and (b)
Lower Huron shale gas versus N2 mole percentage at different pressures and pore sizes with and
without considering critical property shifts. ............................................................................... 66
Figure 4.10 For Marcellus shale condensate, the diffusivities of CO2 in gas phase versus CO2 mole
percentage at different pressures and pore sizes with and without considering nano-confinement
effects. ...................................................................................................................................... 68
Figure 4.11 Original Bakken oil with and without nano-confinement effect (capillary pressure,
critical property shifts) at pore size of 10 nm. ............................................................................ 69
Figure 4.12 For Bakken shale oil, the diffusivities of (a) CO2 and (b) C5C6 in gas phase versus CO2
mole percentage at 1500 psia and at pore size of 10 nm with and without considering nano-
confinement effects. .................................................................................................................. 70
Figure 4.13 For Bakken shale oil, the diffusivities of CO2 and C5C6 in liquid phase versus CO2
mole percentage at at pore size of 10 nm with and without considering nano-confinement effects,
including (a) CO2 at 1500 psia; (b) C5C6 at 1500 psia; (c) CO2 at 3000 psia; and (d) C5C6 at 3000
psia. .......................................................................................................................................... 72
Figure 4.14 Reservoir permeability map for tight oil-rich reservoir. .......................................... 73
Figure 4.15 Cumulative oil production of primary depletion and huff-n-puff gas injection with and
without molecular diffusion....................................................................................................... 73
Figure 4.16 Reservoir permeability map for tight oil-rich reservoir. .......................................... 74
Figure 5.1 The relative permeability data for validation tests. .................................................... 90
Figure 5.2 Validation of primary depletion. For the producer well block: (a) well block pressure;
(b) oil production rate; and (c) gas production rate; and (d) water production rate...................... 91
Figure 5.3 Validation of water injection at constant injection rate of 50 bbl/day. For the producer
well block: (a) well block pressure; (b)water production rate; (c) oil production rate; and (d) gas
production rate. ......................................................................................................................... 92
Figure 5.4 Validation of water injection at constant injection rate of 50 bbl/day. For injector well
block: (a) well block pressure; (b) water injection rate ............................................................... 93
Figure 5.5 Water saturation distributions at different times at constant injection rate of 50 bbl/day.
................................................................................................................................................. 93
xiii
Figure 5.6 Validation of water injection at constant injection pressure of 5000 pisa. For the
producer well block: (a) well block pressure; (b)water production rate; (c) oil production rate; and
(d) gas production rate. .............................................................................................................. 94
Figure 5.7 Validation of water injection at constant injection pressure of 5000 psia. For injector
well block: (a) well block pressure and (b) water injection rate. ................................................. 94
Figure 5.8 Water saturation distributions at different times at constant injection pressure of 5000
psia. .......................................................................................................................................... 94
Figure 5.9 Validation of gas injection at constant injection rate of 5 Mscf/day. For the producer
well block: (a) well block pressure; (b)water production rate; (c) oil production rate; and (d) gas
production rate. ......................................................................................................................... 95
Figure 5.10 Validation of gas injection at constant injection rate of 5 Mscf/day. For injector well
block: (a) well block pressure; (b) gas injection rate .................................................................. 96
Figure 5.11 Gas saturation distributions at different times at constant gas injection rate of 5
Mscf/day. .................................................................................................................................. 96
Figure 5.12 Validation of gas injection at constant injection pressure of 2000 psia. For the producer
well block: (a) well block pressure; (b)water production rate; (c) oil production rate; and (d) gas
production rate. ......................................................................................................................... 97
Figure 5.13 Validation of gas injection at constant injection pressure of 2000 psia. For injector
well block: (a) well block pressure; (b) gas injection rate .......................................................... 97
Figure 5.14 Pressure distributions at different times at constant gas injection pressure of 2000 psia.
................................................................................................................................................. 97
Figure 5.15 Reservoir permeability map for tight oil-rich reservoir. ........................................ 101
Figure 5.16 Reservoir pore size map for tight oil-rich reservoir. (a) constant pore size ( nmrp 19 )
and (b) pore size proportional to permeability. ........................................................................ 102
Figure 5.17 (a) Cumulative oil and (b) cumulative gas production without confinement effect and
with capillary pressure effect by using constant radius ( nmrp 19 ) and using the radius that is
proportional to the permeability. ............................................................................................. 103
Figure 5.18 Oil-gas capillary pressure distributions at different production times by using (a)
constant pore size ( nmrp 19 ) and (b) pore size proportional to permeability. ........................ 104
xiv
Figure 5.19 Oil-gas interfacial tension at different grid blocks at different production times by
using (a) constant pore size ( nmrp 19 ) and (b) pore size proportional to permeability. .......... 105
Figure 5.20 (a) Cumulative oil and (b) cumulative gas production without confinement effect and
with critical property shift effect by using constant pore size ( nmrp 19 ) and using the pore size
that is proportional to the permeability. ................................................................................... 106
Figure 5.21 (a) Cumulative oil and (b) cumulative gas production without confinement effect and
with both confinement effects by using constant pore size ( nmrp 19 ) and using the pore size that
is proportional to the permeability. .......................................................................................... 107
Figure 5.22 Reservoir permeability map with matrix permeability as 0.00005 md. .................. 109
Figure 5.23 Cumulative gas production without confinement effect and with critical property shift
effect by using constant pore size ( nmrp 19 ) and using the pore size that is proportional to the
permeability. ........................................................................................................................... 110
Figure 5.24 Reservoir permeability map with matrix permeability as 0.001 md. ...................... 113
Figure 5.25 (a) Cumulative oil and (b) cumulative gas production without molecular diffusion and
with molecular diffusion by multiplying diffusion coefficient by 1, 10 and 100 times when the
matrix permeability is 0.001 md. ............................................................................................. 114
Figure 5.26 Reservoir permeability map with matrix permeability as 0.00005 md. .................. 115
Figure 5.27 (a) Cumulative oil and (b) cumulative gas production without molecular diffusion and
with molecular diffusion by multiplying diffusion coefficient by 1, 10 and 100 times when the
matrix permeability is 0.00005 md. ......................................................................................... 116
Figure 5.28 (a) Cumulative oil production, (b) cumulative gas production, and (c) well block
pressure with two huff-n-puff circles without molecular diffusion and with molecular diffusion by
multiplying diffusion coefficient by 1, 10 and 100 times when the matrix permeability is 0.00005
md. .......................................................................................................................................... 117
Figure 5.29 Cumulative gas production without molecular diffusion and with molecular diffusion
by multiplying diffusion coefficient by 1, 10 and 100 times when the matrix permeability is
0.00005 md. ............................................................................................................................ 118
xv
LIST OF TABLES
Table 2.1 Experimental studies about different gas injection approaches in shale reservoir for EOR.
...................................................................................................................................................9
Table 2.2 Simulation studies about different gas injection approaches in shale reservoirs for EOR.
................................................................................................................................................. 11
Table 3.1 Reservoir and fluid properties for conventional reservoir. .......................................... 35
Table 3.2 Reservoir and fluid properties for tight oil-rich reservoir. ........................................... 39
Table 3.3 Gas production of huff-n-puff at Pw = 4400 psia and 2000 psia using two different gas
adding approaches. .................................................................................................................... 45
Table 4.1 Compositions of Marcellus shale gas, Lower Huron shale gas, Marcellus shale
condensate, and Bakken shale oil (unit: mole fraction). ............................................................. 56
Table 4.2 Measured tortuosity and tortuosity factor of different shale samples using 3D
tomographic imaging techniques. .............................................................................................. 76
Table 4.3 Calculated tortuosity factor and the ratio of effective diffusivity to bulk diffusivity at
different porosities (φ = 0.03, 0.05, and 0.10) by using tortuosity-porosity relations and measured
tortuosity (or tortuosity factor) from tomographic imaging techniques....................................... 77
Table 5.1 The properties of hydrocarbon components for validation tests. ................................. 89
Table 5.2 The binary interaction parameters of hydrocarbon components. ................................. 89
Table 5.3 The reservoir conditions and production modeling design for validation tests. ........... 90
Table 5.4 Compositions and parameters of Bakken oil ............................................................ 100
Table 5.5 Binary interaction coefficients of Bakken oil ........................................................... 100
Table 5.6 Cumulative oil and gas production without confinement effects and with confinement
effects by using constant pore size ( nmrp 19 ) and using the pore size that is proportional to the
permeability. ........................................................................................................................... 108
Table 5.7 Binary interaction coefficients of Marcellus shale gas .............................................. 108
Table 5.8 Binary interaction coefficient of Bakken oil with CO2. ............................................ 112
Table 5.9 The increased percentage of oil and gas production of Bakken oil after considering
molecular diffusion. ................................................................................................................ 117
Table 5.10 The increased percentage of gas production of Marcellus gas after considering
molecular diffusion. ................................................................................................................ 119
1
Chapter 1 Introduction
1.1 Background
Fossil fuels, including petroleum, natural gas, and coal, are the primary source of energy in the
United States (total of 81% in 2016) (EIA, 2017). To meet the expanding demand for petroleum
and natural gas, great attention has been given to the development of unconventional oil and gas
reservoirs. Generally, unconventional reservoirs can be categorized into the tight and shale
reservoirs, coalbed methane reservoirs, gas hydrates, heavy oil, and tar sands, among others. Shale
reservoirs worldwide are associated with high total organic carbon (TOC), with an estimated
reserve that is equivalent to 345 billion barrels of oil and 7299 trillion cubic feet of gas (EIA, 2013).
Based on the initial fluid properties and phases at the reservoir condition, as well as the phase
behavior changes during the production process, shale reservoirs are grouped into three categories:
shale oil reservoirs, shale gas reservoirs, and shale gas-condensate reservoirs. However, until
recently, it was challenging to unlock shale oil or gas because of the extremely small pore size,
low porosity, and ultra-low permeability of shale. Over the last decade, two advanced
technologies—horizontal drilling and multistage hydraulic fracturing—have been successfully
applied in shales and made it profitable to boost oil or gas production from such tight formations.
In 2015, oil and gas production from unconventional shale oil and gas plays was 4.89 million
barrels per day and 37.4 billion cubic feet per day, respectively, which accounted for
approximately half of the total U.S. crude oil and natural gas production (EIA, 2016). By using
intensive horizontal drilling and hydraulic fracturing techniques, oil or gas escapes from the tight
matrix to the hydraulic fractures through primary depletion under the reservoir depressurization or
by gas expansion drives, boosting a tremendous increase in production. Nevertheless, field
2
production data invariably indicated, after a few years of production, a sharp decline in oil or gas
production rate was observed, followed by a prolonged low-production rate period. Only less than
10% of oil was recovered from the unconventional formations during this primary depletion period
(Hoffman and Evans, 2016), resulting in an enormous unrecovered oil bank remaining in the
reservoir.
Simulation with a black-oil model is fast and more robust compared to compositional
simulation. Generally, black oil properties such as density or formation volume factor, viscosity,
solution gas-oil, and volatile oil-gas ratio, as a function of pressure are determined prior to
simulation and used as tabular input. These pressure-dependent properties are typically obtained
from experiments, or through flash calculations with a compositional equation-of-state (EOS).
Black oil simulation today, however, is incapable of describing reservoir behavior during gas
injection under miscible and near-miscible conditions owing to significant changes in reservoir
fluid composition. Black-oil simulation has been modified in the past by including a fourth
component (e.g. Todd and Longstaff, 1972; Shoaib and Hoffman, 2009; and Dong and Hoffman,
2012), but these modifications are only valid for first-contact (FCM) floods.
Nojabaei and Johns (2016) developed an approach to calculate black-oil fluid properties to be
used in a compositionally-extended black oil model, which can be used to model both miscible
and immiscible gas injection. They constructed a binary gas-oil PX diagram with a critical point
by feeding a small fraction of the equilibrium gas at the current bubble point pressure to reach a
new bubble point pressure until the critical point was achieved. The approach uses both a volatile
and solution gas-oil ratio and gives a continuous bubble-point and dew-point curve.
Small pores can also impact phase behavior and the production during both primary and
enhanced recovery. Nojabaei et al. (2013) examined the effect of capillary pressure on phase
3
behavior of hydrocarbon fluids in nano-sized pores. They concluded that capillary pressure
affected both the bubble-point and dew-point curves, and corresponding two-phase fluid properties.
This in turn impacted primary recovery. Wang et al. (2016) developed a Parachor model to
investigate the confinement effect on interfacial tensions (IFTs). They used their model to estimate
the MMP of CO2 for Middle Bakken oil. They found that for pores larger than 10 nm, MMP was
independent of pore width. For a pore width of 3 nm, the IFT and MMP decreased by 67.5% and
23.5%, respectively. Zhang et al. (2017) calculated MMPs for CO2 flooding in Bakken by using
the Vanishing Interfacial Tension (VIT) method and concluded that the MMP was decreased by
5% due to the confinement.
Gas-oil relative permeability can change significantly as the composition of reservoir fluid
changes. Kalla et al. (2015) measured the gas-condensate relative permeability curves with
variable IFT at reservoir conditions and concluded that an increase in relative permeability at fixed
saturation was observed for both gas and liquid phases when IFT was decreased. Among the
different methods to capture the effect of IFT on relative permeability curves, the following two
methods have been tested by Blom and Hagoort (1998): 1) Corey functions by determining the
Corey coefficients; and 2) interpolation between immiscible and miscible relative permeability
curves.
In tight shales, the nano-confinement effects, including the large gas-oil capillary pressure and
critical property shifts are significant at extremely small pore sizes and alter the fluid properties,
such as phase compositions, density, viscosity, and saturation pressure to some extent (Nojabaei
et al., 2013; Teklu et al., 2014; Huang et al., 2019a). Molecular diffusion, as a random Brownian
motion of molecules caused by concentration gradient, is highly associated with pressure,
temperature, and fluid properties as well. Yet, the effect of nano-pore confinement that occurs in
4
ultra-tight shale formations on the molecular diffusion have not been investigated. In many gas
injection studies of shale reservoirs, different attempts have been made to examine the role of
molecular diffusion in gas injection process for EOR/EGR. The results revealed that the molecular
diffusion effect on improving shale oil and gas recovery is highly sensitive to the employed
diffusion coefficient (Du and Nojabaei, 2019).
(The introduction part are adopted from the introduction sections of my three published journal
papers: https://doi.org/10.3390/en12122355, https://doi.org/10.1016/j.petrol.2020.107362, and
https://doi.org/10.1016/j.fuel.2019.116680)
1.2 Objectives and Scope of This Study
The primary objectives of this research study are to:
Extrapolate Bakken oil properties with considering nano-confinement effects by using
two different gas adding approaches;
Perform immiscible and miscible gas injection simulation for both conventional and
tight oil-rich reservoirs using compositionally extended black-oil simulation model;
Examine the effect of IFT-dependent relative permeability on recovery in tight
reservoirs;
Predict diffusion coefficients of shale fluids by incorporating nano-confinement effects;
Estimate the effective diffusivity by including the rock intrinsic properties;
Examine the confinement effects on shale oil and shale gas production; and
Examine the diffusion behavior on shale oil and shale gas huff-n-puff gas injection.
5
1.3 Outline of Dissertation
This dissertation consists of seven chapters. Chapter 1 is the introduction and states the objective
and scope of this study. Chapter 2 provides an updated literature review on gas injection methods
in shale reservoirs, nano-confinement effects in phase behavior, and diffusion coefficient in shale
rock. Chapter 3 introduces the calculation of black oil properties in nano pores and performance
of miscible and immiscible gas injection in both conventional and tight oil-rich reservoirs using
compositionally extended black oil model simulation. In Chapter 4 the diffusion coefficients of
shale fluids (shale oil, shale condensate, and shale oil) by incorporating nano-confinement effects
are calculated, and the effective diffusivity in shale rock porous media by using the tortuosity
factor from measurements and empirical correlations are estimated. Chapter 5 introduces the
development of a fully compositional model and the model application to examine the confinement
effects and molecular diffusion behaviors in shale oil and gas production. Chapter 6 summarizes
the conclusions and describes the future research plan.
6
Chapter 2 Literature Review
(This chapter was published in Energies, https://doi.org/10.3390/en12122355. The title is “A
Review of Gas Injection in Shale Reservoirs: Enhanced Oil/Gas Recovery Approaches and
Greenhouse Gas Control.” Section 2.3 was published in Journal of petroleum Science Engineering,
https://doi.org/10.1016/j.petrol.2020.107362. The title is “Estimating diffusion coefficients of
shale oil, gas, and condensate with nano-confinement effect”)
2.1 Gas injection approaches
Recently, gas injection enhanced shale oil/gas recovery methods, including huff-n-puff gas
injection (or cyclic gas injection) and gas flooding, have been experimentally studied at the
laboratory scale or conducted in field, and numerically examined through simulation by many
researchers (Chen et al., 2014; Meng et al., 2017; Gamadi, et al., 2014; Hoffman, 2012; Yu et al.,
2017; Jin et al., 2017; Fathi and Akkutlu, 2014). Generally, the injected gas could be carbon
dioxide, nitrogen, flue gases (N2 + CO2), and produced gas, depending on shale fluids with unique
characteristics at specific reservoir conditions. Injected CO2 in shale reservoirs not only could be
permanently sequestered within the small pores in an adsorbed state, but also could participate in
enhancing recovery of oil or natural gas through maintaining pressure, multi-contact miscible
displacement (Jin et al., 2017a;b), molecular diffusion (Fathi and Akkutlu, 2012;2014; Yu et al.,
2015), or desorption of methane (Fathi and Akkutlu, 2012;2014; Sun et al., 2013; Kalantari-
Dahaghi 2010). N2, as an economic and eco-friendly alternative, could displace oil mostly through
an immiscible displacement approach because of the high minimum miscibility pressure (MMP).
Owing to the low viscosity of N2, viscous fingering may occur during the displacement process.
Flue gas, as the mixture of CO2 and N2, is also deemed as a potential injection gas resource for
shale reservoirs and has been successfully injected in other unconventional reservoirs (coalbed
methane and gas hydrate); however, not so many research studies have been carried out yet
7
regarding flue gas injection in shale reservoirs. Moreover, substantial produced gas associated with
oil production is flared or vented into the air during oil recovery, which is not only energy waste,
but also hazardous to the environment (EPA, 2013; Prenni et al., 2016; Ratner and Tiemann, 2018).
In order to reduce gas flaring or venting and compensate for the oil production decline, produced
gas could be effectively used for recycled gas enhanced oil recovery (EOR).
2.1.1 Huff-n-Puff gas injection and mechanisms
The experimental study of huff-n-puff gas injection or cyclic gas injection in tight rock samples
have been conducted in multiple publications (Yu et al., 2017; Jin et al., 2017b; Wan et al., 2015).
The commonly used injection gases or solvents are N2, CO2, CH4, C2H6, andCH4/C2H6 mixture.
The core plug samples were mostly selected from Eagle Ford (Yu et al., 2017; Wan et al., 2015),
Bakken (Jin et al., 2017a;2017b; Song and Yang, 2017; Yang et al., 2015), and Barnett and Marcos
(Gamadi et al., 2013). Table 2.1 summarized experimental studies about different gas injection
approaches in shale reservoir for EOR. In addition to experimental studies, a number of simulation
work has been performed using in-house simulation approaches or commercial software tools to
study field-scale huff-n-puff injection in tight formation. Sensitivity analysis is conducted along
with experiments or simulations to examine effects of various operation parameters (injection
pressure and rate, initial injection time, gas injection duration, soaking time, number of cycles, and
heterogeneity) on recovery performance and will be discussed in the following section. Table 2.2
summarized the simulation studies of different gas injection approaches in shale reservoirs for
EOR.
The effect of gas injection pressure on oil recovery in the huff-n-puff scheme has been
investigated in many literature works. A general conclusion was that recovery factor was increased
with increasing injection pressure. Some authors concluded that re-pressurization is the primary
8
oil recovery mechanism for the huff-n-puff process (Gamadi et al., 2013;2014a;2014b). Further
investigations (Song and Yang, 2017; Gamadi et al., 2014b) indicated that increasing pressure only
resulted in a good recovery performance at immiscible condition. When the injection pressure was
above the MMP, a further increase in injection pressure could not result in a significant increase
in recovery factor. The experimental results from one study (Song and Yang, 2017) showed that
near-miscible and miscible CO2 huff-n-puff injection could effectively enhance crude oil recovery
up to 63.0% and 61.0% respectively, while water flooding and immiscible CO2 huff-n-puff would
result in final recovery factor of 42.8% and 51.5%, respectively. They concluded that dominant
mechanisms for the huff-n-puff process in shale oil formations included viscosity and interfacial
tension reduction, oil swelling effect, light-components extraction, and solution gas drive. It should
be noted that the Bakken rock samples used in their study is not ultra-tight, but tight (permeability
in 10-1 md). This conclusion may well explain the mechanisms of huff-n-puff in conventional or
tight formation, where gas is comparatively easier to dissolve into the matrix; but further analysis
may be required to better understand the mechanisms of huff-n-puff gas injection in ultra-tight
formation, where oil is trapped in nanosized pores and gas is more difficult to get in contact with
oil. Recently, one study (Adel et al., 2018) used CT scanning technology to monitor the saturation
change with time in an organic-rich Eagle Ford core plug. The core sample was placed in a high-
pressure CO2 core holder, below and above MMP, and they observed that when injection pressure
was above MMP, the recovery was still increasing with increasing pressure.
Gas injection rate is one of the most important parameters in huff-n-puff gas injection EOR.
Yu et al. (2014) conducted a series of sensitivity analysis and concluded that gas injection rate was
the most important parameter to enhance oil recovery in comparison to other factors, such as
injection time and number of cycles. It was also concluded that a higher injection rate resulted in
9
Table 2.1 Experimental studies about different gas injection approaches in shale reservoir for EOR.
10
a higher oil recovery factor (Sun et al., 2016; Yu et al., 2014; Zhang et al., 2018). Other studies
examined the effect of CO2 injection rate on oil recovery factor by using the injection rate of 500
and 5000 Mscf/day (Sun et al., 2016) and 100, 1000, and 10,000 Mscf/day (Zhang et al., 2018),and
found out that the recovery factor was increased by 1.0%–5.4%, correspondingly. The result is not
a total surprise as higher injection rates ensure more gas to be injected into the reservoir in one
cycle, keeping the reservoir pressure high. On the other hand, higher injection rate also means
more capital investment, especially when the injection rate is increased by one or two orders, much
more CO2 would be injected into the reservoir. From a profitability standpoint, it is not reasonable
to inject a large amount of CO2, and economic evaluation should be conducted to optimize the
injection rate.
The initial gas injection time and injection duration are also two key parameters in gas injection
process. Sun et al. (2016) found that delaying the initial gas injection time from 1000 days to 2000
days could increase the oil recovery by 2.47%. Sanchez-Rivera et al. (2015) investigated the initial
gas injection time by adopting 30, 200, 400, 500, and 1000 days of primary depletion. They also
concluded that delaying the start of huff-n-puff injection (from 30 to 400 days) yielded an
increased recovery; however, when the gas injection was started at a later time (400 to 1000 days)
oil recovery was not enhanced effectively. Similar to cycle numbers and gas injection rate, longer
gas injection time is beneficial to oil recovery because larger volume of gas would be injected into
the formation and maintain a high reservoir pressure. However, from a cash-flow perspective, gas
injection duration should be optimized.
Soaking time, as another important operation parameter in the huff-n-puff process, is normally
examined along with cycle numbers. Long soaking time enabled the injection gas to better mix
with oil through dissolution, thereby improving the efficient recovery per mole of CO2. However,
11
Table 2.2 Simulation studies about different gas injection approaches in shale reservoirs for EOR.
12
a long shut-in period would result in a shorter production time. The optimum soaking time can be
determined by calculating the gross/net gas utilization (Gamadi et al., 2014), as well as associating
the cycle numbers and pressure distribution (Atan et al., 2018). Some experimental and simulation
results indicated that at miscible CO2 injection condition, a longer soaking period allowed gas to
diffuse further into the matrix, leading to a higher accumulative recovery (Gamadi et al.,
2013;2014a;2014b). Some studies reported that in a fixed duration of time, shortening the soaking
time and allowing for more cycle numbers was more effective than a long soaking time with fewer
cycles (Yu et al., 2017; Sun et al., 2016; Gamadi et al., 2014b; Chen et al., 2013). Chen et al. (2014)
realized that the cumulative recovery after a certain period of time for CO2 huff-n-puff injection
was lower than that of the primary depletion. They explained that for the huff-n-puff process, the
injection and soaking periods resulted in a shorter production time and caused uncompensated
production loss. Sheng (2015) used an in-house model to repeat the case and verified the simulation
results. The author explained that the low final recovery factor for huff-n-puff injection in the
former publication was a result of the low injection pressure of 4000 psi, which should have been
higher than the initial reservoir pressure of 6840 psi. In another study by Sun et al. (2016), it was
concluded that soaking time (1, 15, 100 days) had zero effect on the recovery performance. It is
worth noting that, in this sensitivity analysis, only one cycle of gas injection was performed after
1000 days of primary depletion while the total production time was 5000 days and the soaking
period was far shorter compared to the production time.
The effect of heterogeneity of reservoir formation on huff-n-puff or cyclic natural gas injection
efficiency has also been investigated (Chen et al., 2014; Gamadi et al., 2014a; Yu et al., 2015; Yu
et al., 2014). The common conclusion that was drawn by different authors was that the recovery
factor for a heterogeneous reservoir with low-permeability region outperformed homogenous
13
reservoirs, since, for the latter one, CO2 migrates into the deeper formation without playing the
role of increasing the reservoir pressure and carrying oil back to the well. Reservoir heterogeneity
could effectively prevent injected gas moving to the deeper formation and contribute to
maintaining a relatively-high near-well reservoir pressure.
For huff-n-puff gas injection in shale oil reservoirs, re-pressurization is one of the most
important mechanisms for EOR and could be achieved by using high injection pressure (Song and
Yang, 2017; Gamadi et al., 2013; Adel et al., 2018), by increasing the injection rate (Sun et al.,
2016; Yu et al., 2014), by extending the injection duration, and by increasing the cycle numbers
(Gamadi et al., 2014b; Chen et al., 2013). It is necessary to optimize these operational parameters
of a huff-n-puff injection process from profit-motive and cash flow perspectives. Another
important mechanism is that the injected solvents (CO2, CH4, C2H6, or produced gas) could extract
the light components from the oil through a multi-contact miscible process. Meanwhile, those
solvents dissolve into the oil, leading to a viscosity and interfacial tension reduction and the
swollen-diluted oil is much easier to be recovered. The above mentioned mechanisms may play
important roles in tight (e.g., Middle Bakken formation) or conventional reservoirs, where gas is
relatively easier to diffuse into the matrix and to make contact with oil. Recent studies visualized
the gas sweep volume in ultra-tight shale plugs by using CT images (Adel et al., 2018; Li et al.,
2019), indicating that gas could make contact with the oil that is trapped in nanosized pores.
Furthermore, the nanoconfinement effect may influence the estimations of MMP and alter the fluid
properties, so the inclusion of capillary pressure effect and the shift in critical properties results in
more accurate recovery prediction (Zhang et al., 2017; Nojabaei and Johns, 2016). In addition, the
mechanism of molecular diffusion in shale reservoirs is controversial in the literature. The effect
of molecular diffusion on recovery performance is highly related to the diffusion coefficient and
14
soaking time. Nevertheless, laboratory measurements of gas diffusion coefficient in oil-saturated
tight porous media is limited. A more reliable diffusivity is crucial for accurately evaluating the
role of molecular diffusion in huff-n-puff gas injection. The effect of matrix permeability on EOR
is also evaluated by plotting the increased oil recovery factor versus matrix permeability in Figure
2.1. Different colors represent different simulation works in Table 2.2. Huff-n-puff shows a
promising performance on EOR at a wide range of permeability. The various results attribute to
the variety of simulation models with different incorporations of effects. Generally, a dual porosity
dual permeability system with developed natural fractures (Zuloaga et al., 2017; Sun et al., 2019;
Wang and Yu, 2019; Yu et al., 2018), that included nanoconfinement effect, and molecular
diffusion by employing a higher diffusivity, and adopted optimized huff-n-puff parameters (cycles,
injection time, etc.), could achieve a better recovery performance.
Figure 2.1 Incremental oil recovery factor of huff-n-puff and gas flooding from simulation studies
(Table 2.2), for the range of matrix permeability from 0.1 to 100 µd. Different colors represent
different simulation studies (Du and Nojabaei, 2019).
15
2.1.2 Gas flooding and mechanisms
In the literature, experimental and simulation studies of gas flooding in shale reservoirs are limited
compared to huff-n-puff, probably owing to the low injectivity of tight shale rock. Yu et al. (Yu et
al., 2017) experimentally compared N2 flooding to N2 huff-n-puff by using Eagle Ford shale core
plugs (with permeability of 85‒400 nd). In the gas flooding scheme, the production rate was
decreased after N2 breakthrough. The huff-n-puff production scheme maintained a relatively
longer effective recovery performance owing to the continuous favorable pressure gradient in each
cycle. It should be noted that the experimental conditions (Pinj = 1000 psia, T = 72 °F) failed to
reflect the real reservoir pressure and temperature. Yang et al. (2015) experimentally examined the
CO2 WAG (water-alternating-gas) injection in tight Bakken formation cores (with permeability of
250‒440 µd) at reservoir temperature of 140 °F. The results indicated that shorter water slug size
or a longer CO2 slug size was beneficial for improving fluid injectivity, but resulted in a decrease
in recovery efficiency because of early gas breakthrough. Similarly, an increase in cycle time
during water injection period led to a decrease in the fluid injectivity. However, after the fluid
injectivity was decreased to a threshold value, it became sensitive to CO2 slug size instead.
Among the simulation studies, Sheng and Chen (2014) evaluated and compared natural gas
injection and water injection methods in hydraulically-fractured shale oil reservoirs (with
permeability of 0.1 µd). A small model was used to simulate gas flooding between two lateral
hydraulic fractures of a horizontal well. They concluded that the gas flooding method resulted in
a slightly higher oil recovery than cyclic gas injection method; however, the former required a
much greater amount of injection gas than the latter. Water injection performance was not as good
as gas injection because of the low water injectivity in the shale reservoir. Hoffman (2012)
performed a numerical simulation model to examine gas flooding at both miscible and immiscible
16
conditions in shale oil reservoirs at the Elm Coulee Field. The results indicated that significant oil
recovery could be achieved regardless of injection gas types at both miscible and immiscible
conditions. Hydrocarbon gas as an alternative injection gas performed as well as CO2 injection at
miscible condition. At immiscible condition, hydrocarbon injection could also result in favorable
recovery.
In the ultra-tight shale matrix, gas flooding was less effective compared to huff-n-puff gas injection
in shale reservoirs because of the low gas injectivity. It would take a much longer time for the
injection gas to migrate from the injection well to the production well. A closed pair of injection
and production wells (e.g., 200 ft apart in (Sheng and Chen, 2014)) and highly developed natural
fractures or effective hydraulic fractures could alleviate this issue to some extent. At relatively
high-permeability shales, the performance of gas flooding is improved and surpasses huff-n-puff
over a turning point of permeability, as shown in Figure 2.2 (Zuloaga et al., 2017). In addition,
solvent (CO2, CH4, or produced gas) flooding still outperformed pure water flooding in tight (and
not ultra-tight) formations, since solvent could be miscible with oil, reduce oil viscosity, and lead
to a larger volume of contacted oil compared to water. CO2 WAG injection, as an alternative for
EOR in tight formations, combines the advantages of water flooding and CO2 continuous flooding,
leading to an improved macroscopic sweeping efficiency and an enhanced microscopic
displacement efficiency.
2.2 Nano-confinement effects in phase behaviors
Multiple phase behavior research studies have been conducted recently investigating the gas
injection characteristics of oil shale reservoirs influenced by confinement effect in nanopores.
Teklu et al. (2014) used the multiple mixing cell method (MMC) to calculate MMP of Bakken oil
during injection of CO2 and mixtures of CO2 and CH4 while critical pressure and temperature of
17
the fluids were shifted due to confinement effects. They recognized MMP reduction of 600 psi due
to the shift in critical properties; however, they concluded that the large gas–-oil capillary pressure
owing to nanopores did not influence MMPs. Zhang et al. (2018) used method of characteristics
(MOC), multiple mixing cells, and slim tube simulation approaches to examine capillary pressure
effect on MMP. For CO2 injection, inclusion of high capillary pressure would enhance the recovery
of heavy oil components for around 10% in the immiscible pressure region. In addition, capillarity
effect might change the MMP and this change varied for different fluid compositions. For a ternary
mixture, this influence would decrease MMP; for the Bakken fluid, MMP increased with high
capillary pressure, and for the Eagle Ford fluid, no significant change of MMP was observed. In a
similar study, Zhang et al. (2017) calculated MMPs for CO2 floods in Bakken and concluded that
the MMP was reduced by 5% due to the confinement effects of nanopores, including both large
capillary pressures and the shift in critical properties. It should be noted that in this study and
another similar study (Jin et al., 2017a), the MMP was measured by using the vanishing interfacial
tension (VIT) method, which has been shown to have significant limitations even for conventional
reservoirs (Jessen and Orr, 2008). Nojabaei and Johns (2016) studied the effect of large gas–-oil
capillary pressure on fluid properties and saturation pressures when the produced gas was injected
to enhance oil recovery. They showed that as the original oil mixed with the injection gas, the
effect of capillary pressure on recoveries would get smaller. They did not recognize any change in
the MMP of produced gas with the original oil due to large gas–-oil capillary pressure. One reason
for not recognizing a change in MMP can be that they used a compositionally-extended black oil
approach with two oil and gas pseudo-components. The MMP would be the same as the critical
point of this pseudo-binary mixture, at which interfacial tension (IFT), and subsequently gas–-oil
capillary pressure would be zero. Wang et al. (2016) developed a Parachor model to account for
18
the effect of confinement on interfacial tensions (IFTs). They used their model to calculate CO2
MMP of Bakken oil. They concluded that for the pores larger than 10 nm, MMP is independent of
pore width. For a pore width of 3 nm, they observed 67.5% and 23.5% decrease in IFT and MMP,
respectively. Huang et al. (2019a) proposed that including capillary pressure effect could reduce
oil and gas recovery, meanwhile, alter the compositions of residuals. Du et al. (2018) used a black-
oil simulation approach to examine the capillary pressure effect in the huff-n-puff gas injection
process in a tight formation. Inclusion of the capillary pressure effect in phase behavior could
increase the oil recovery at a lower production pressure. However, at miscible or near-miscible
conditions, the influence of capillary pressure on reservoir performance was decreased owing to
the reduced IFT between oil and gas phases.
2.3 Diffusion coefficient in shale rock
In the past, a variety of experiments has been conducted to measure diffusivities by direct/system-
intrusive or indirect/non-intrusive approaches. The former method requires to take fluid samples
from the system directly and perform compositional analyses (Sigmund, 1976a; Dysthe and
Hafskjold, 1995), which is straightforward but system-intrusive; the latter uses new techniques,
such as nuclear magnetic resonance (NMR) (Gottwald et al., 2005) and computed tomography
(CT) scanning (Song et al., 2010) techniques to obtain the concentration profiles, which are non-
intrusive to the system. Meanwhile, a number of empirical correlations have been derived over the
past decades to predict the diffusivities. The widely used empirical correlations include Wilke‒
Chang (Wilke and Change, 1955), Hayduk–Minhas (Hayduk and Minhas, 1982), Sigmund
(Sigmund, 1976a; 1976b), etc. The expressions of the empirical correlations are in terms of
temperature and fluid properties, such as phase compositions, density and viscosity. Both Wilke‒
Chang and Sigmund have been incorporated in commercial software tools (GEM, CMG). Wilke–
19
Chang and Hayduk–Minhas correlations are developed for low-pressure liquid systems. The
Sigmund correlation was proposed to predict the binary diffusion coefficients for high-pressure
gas and liquid mixtures.
In tight shales, the nano-confinement effects, including the large gas-oil capillary pressure and
critical property shifts are significant at extremely small pore sizes and alter the fluid properties,
such as phase compositions, density, viscosity, and saturation pressure to some extent (Nojabaei
et al., 2013; Teklu et al., 2014; Huang et al., 2019a). Molecular diffusion, as a random Brownian
motion of molecules caused by concentration gradient, is highly associated with pressure,
temperature, and fluid properties as well. Yet, the effect of nano-pore confinement that occurs in
ultra-tight shale formations on the molecular diffusion have not been investigated. In many gas
injection studies of shale reservoirs, different attempts have been made to examine the role of
molecular diffusion in gas injection process for EOR/EGR. The results revealed that the molecular
diffusion effect on improving shale oil and gas recovery is highly sensitive to the employed
diffusion coefficient (Du and Nojabaei, 2019). Owing to the lack of reference data, most studies
assumed a diffusivity based on the literature or calculated the diffusivities using empirical
correlations without considering the confinement effects. Yu et al. (2015) examined the effect of
molecular diffusion in CO2 huff-n-puff injection in Middle Bakken formation (with permeability
of 10 µD). The oil recovery for the huff-n-puff scheme was increased by 0.10‒3.25% with
molecular diffusions ranging from 10-4 to 10-2 cm2/s. Sun et al. (2016) investigated a CO2 huff-n-
puff EOR process in a tight matrix (with permeability of 100 nD) with complex fracture networks.
The CO2 diffusion coefficients obtained from the core-scale simulation were in the range of 10-7‒
10-9 cm2/s. They concluded that the small diffusion coefficients and short duration of huff-n-puff
(30-day injection plus 15-day soaking, compared to 5000-day production) made the effect of
20
molecular diffusion negligible. Fathi and Akkutlu (2014) proposed a triple-porosity single-
permeability simulation model to study gas transport from the organic micro-pores to the inorganic
macro-pores and fractures in a shale gas reservoir. Both molecular diffusion (diffusivity in the
order of 10-5 cm2/s) and surface diffusion (diffusivity in the order of 10-2 cm2/s) of the absorbed
molecules in the micro-pores are incorporated in the governing equation. Jiang and Younis (2016)
examined the molecular diffusion in shale condensate reservoir. They claimed that the diffusion
coefficient in the liquid phase was orders of magnitude smaller than in the gas phase and assumed
that molecular diffusion only took place in the gas phase.
In addition, molecular diffusions in porous media are different from those in a bulk phase. Most
laboratory measurements of diffusion coefficients were in the bulk phase. Only a few papers
experimentally predicted the diffusion coefficients within porous media by matching the
experimental pressure decline curves with mathematical model (Li and Dong, 2009) or simulation
model (Jia et al., 2019). They used Berea sandstone (160–263mD, porosity 18.2%–19.7%) and
concluded that the diffusion coefficient was reduced by one or two orders in porous media than in
a bulk phase. Since the presence of matrix makes the diffusivity measurements difficult, an
effective diffusion coefficient is suggested by including two intrinsic rock properties, i.e.,
tortuosity factor and porosity, to characterize the diffusion behavior in porous media. It should be
noted that tortuosity factor is different from tortuosity, although both characterize the
interconnected paths and the geometry of a porous solid (Epstein, 1989; Tjaden et al., 2016;
Backeberg et al., 2017). Tortuosity is defined as the ratio of the actual flow path length to the
geometrical length of the sample (Epstein, 1989; Matyka et al., 2008), while tortuosity factor
quantifies the apparent decrease in diffusive transport resulting from convolutions of the flow paths
through porous media (Tjaden et al., 2016; Cooper et al., 2016). For a porous media where the
21
cross-sectional area normalized by flow path is fixed, tortuosity factor is equal to the square of
tortuosity (Epstein, 1989). Both tortuosity and tortuosity factor approach to one as the flow paths
tend to be straight in the flow direction (Cooper et al., 2016).
Tortuosity factor has been the focus of a wide range of disciplines over a century; however,
direct access to tortuosity factor is difficult (Cooper et al., 2016). Different types of empirical and
theoretical tortuosity-porosity relations have been used (Huang et al., 2019b) and summarized in
a review paper (Shen and Chen, 2007). Recently, 3D tomographic imaging techniques, such as X-
ray computed tomography (CT) and focused ion beam-scanning electron microscopy (FIB-SEM),
create the potential for quantifying the tortuosity (Shabro et al., 2013, Cooper et al., 2014, 2016)
or tortuosity factor (Backeberg et al., 2017) directly from complex and heterogeneous
microstructure by using different simulation approaches. In general, the lamination-perpendicular
direction with the lowest permeability yields the largest tortuosity/tortuosity factor (Chen et al.,
2013, Peng et al., 2015, Backeberg et al., 2017), indicating poor geometric interconnectivity and
transport potential perpendicular to the bedding planes. For some of the studies that used higher-
porosity shale samples (porosity >10%) (Shabro et al., 2013, Chen et al., 2013, Sun et al., 2017),
the tortuosity falls within a similar range with a relatively small value, i.e., 1.6 2.9 .
Backeberg et al. (2017) computed the tortuosity factor (tortuosity squared) directly from nano-CT
and micro-CT tomographic data by using TauFactor. The smaller porous phase (pores plus organic
matter) volume percentage (3% or 5%) gave a poor interconnectivity, resulting in a larger or
infinite tortuosity factor. The larger porous phase volume percentage (10% and 20%) provided
tortuosity factors ( 2 ) up to 9 and 39, respectively, within 16 µm geometrical length of the
sample.
22
2.4 Greenhouse gas control
Carbon dioxide is a powerful greenhouse gas and has long residence time in the atmosphere.
Anthropogenic carbon dioxide emissions have been greatly accelerated as our energy needs
strongly depend on fossil fuels. It was reported that the average growth rate of CO2 emissions
increased from 1.1% per year for 1990–1999 up to 3% per year for 2000–2004 (Raupach et al.,
2007). Some options have been suggested for geological storage of carbon dioxide, such as deep
saline aquifers, depleted oil and gas fields, unmineable coalbeds, and deep oceans (Barrufet et al.,
2010). Methane, as another greenhouse gas, is associated with a greater global warming potential
compared to carbon dioxide in a short time scale (Howarth et al., 2011). In petroleum and natural
gas industry, natural gas emission from the gas-bearing strata to surface occurs over a lifetime of
a well during both well completion and production stage. In most of oil fields, natural gas is
concurrently produced with oil during primary production; under reservoir conditions it is
dissolved in the oil but as the oil is extracted and pressure drops, it is released from solution. The
produced natural gas can be either recovered, reinjected to the reservoir, flared, or vented.
Considering the low price of natural gas, it is not economically efficient to sell the produced gas
especially where there is limited infrastructure for gas transportation available. Therefore,
reservoir engineers typically flare or vent the excess produced gas—polluting the environment in
the process. Flaring and venting are common practices in many oil production operations. Figure
8a shows that the amount of gas flared/vented in the U.S. has substantially increased with the
development of shale resources since 2001. Based on U.S. Energy Information Administration,
more than 270 billion cubic feet of natural gas was flared or vented in 2015 (EIA 2015). North
Dakota, where Bakken oil shale is located, contributed to more than one third of this total.
23
Before 2005, there were less than 100 active producing wells in Bakken; however, this number
has increased to more than 2500 wells by 2016 (Ghaderi et al., 2017). As Figure 2.2 shows, Bakken
oil (and the associated gas) production increased significantly since 2005, as did natural gas flaring
and venting (EIA 2019). The Energy Information Administration (EIA) reports indicated that
approximately 12.85% of the produced gas from Bakken shale (equivalent to 88 billion cubic feet)
was flared or vented in 2017 and no gas has been reinjected since the start of Bakken development.
Flaring not only unproductively wastes energy but also emits carbon dioxide and other hazardous
gases, such as CO, SO2, NOX, owing to incomplete combustion, as well as volatile organic
compounds, impacting regional air quality (Prenni et al., 2016).
Figure 2.2 (a) Total flared/vented natural gas in United States; (b) Produced, marketed, and flared/
vented natural gas in the state of North Dakota.
(a)
(b)
24
The growing increase of greenhouse gases in the atmosphere could change the climate and
induce possible biological consequences (Prenni et al., 2016). It is necessary to take steps to control
greenhouse gas emissions in oil industries, including reducing natural gas flaring or venting,
capturing the flue gases from potential sources such as power plants, cement plants, and oil
refineries, and reinjecting the produced gases and flue gases into reservoirs for enhancing oil or
gas recovery (EOR/EGR) and simultaneously storing greenhouse gases in the reservoir formation
permanently. Clearly flaring needs to stop and reinjection of produced gas back in the shale
formation is a viable solution. However, initiating a gas-reinjection and/or CO2 storage global
revolution will face financial constraints and challenges especially for smaller operators. Flaring
reduction and underground CO2 storage will be effectively possible only if it is cost effective and
can create markets. To create wide-scale uptake of gas storage research studies in shale reservoirs,
one needs to provide convincing evidence that reinjecting the produced gas and/or CO2 injection
not only reduces emissions, but also will help improve oil production, thereby underpinning the
economic argument.
In consideration of the subsurface geological sequestration of anthropogenic carbon dioxide,
some necessary characteristics for gas storage should be examined, such as gas storage capacity,
trapping mechanism, gas migration and possible leakage, and infrastructure for gas transport from
the surface to underground (Kang et al., 2011). Although shale reservoirs are currently in primary
depletion, many favorable experimental and simulation results and a number of successful gas
injection EOR pilots indicated there is potential for gas injections in shale reservoirs for EOR and
EGR. Gas injection not only could enhance oil or gas recovery under the mechanisms of re-
pressurization, diffusion, re-vaporization, and desorption, but the significant amount of carbon
dioxide could also be trapped in kerogen-rich shale reservoirs, which have a large storage capacity
25
with tremendous nanopores acting as molecular sieves to safely and permanently store CO2 in an
adsorbed state (Hughes, 2000).
Aside from above-mentioned targets for CO2 sequestration in unconventional reservoirs, a CO2
fracturing technique that participates in commercial-scale tight shale oil and gas production could
also contribute to CO2 storage to some degree (Pu et al., 2017; Middleton et al., 2014). In
comparison with water-based fracturing, CO2 fracturing significantly reduces or completely
eliminates the water usage (Harris et al., 1984), resulting in a low-water saturation environment
near the wellbore, which is favorable for oil or gas mobility in tight shale formation.
26
Chapter 3 Compositionally Extended Black Oil Simulation Model
(This chapter is adopted from our journal paper published in Fuel,
https://doi.org/10.1016/j.fuel.2019.116680. The title is “A black-oil approach to model produced
gas injection in both conventional and tight oil-rich reservoirs to enhance oil recovery.”)
3.1. Methodology
In this section, we first briefly illustrate the approach to determine oil fluid properties as a function
of the pore size (confinement effect). Then, the compositionally-extended black oil equations and
unknowns in our simulation model are presented. IFT-dependent relative permeability curves are
also presented.
3.1.1 Flash calculations
Flash calculations that incorporate the effect of pore size and the resulting high capillary pressure
are used to calculate the fluid properties as a function of oil pressure. These input properties include
gas and oil densities, viscosities, solution gas-oil ratio, volatile oil-gas ratio, and interfacial tension
as a function of oil pressure. Here, oil pressure is assumed to be the reference phase pressure, while
water is assumed to be the most wetting phase, oil the intermediate wetting, and gas is the not-
wetting phase. Gas pressure is therefore calculated using the Laplace equation as given below:
rPP o
2g (3.1)
At equilibrium, component fugacities in the gas and liquid phases are equal, although the phase
pressures are not, i.e.,
),...,,,(),...,,,( 2121 CC Ng
g
iNo
o
i yyyPTfxxxPTf (3.2)
Equation 2 assumes that fluid properties are continuous and can be written macroscopically.
Fugacity coefficient of component i for both oil and gas phase is defined as:
27
i
oo i
i o
f
x P (3.3a)
i
gg i
i g
f
y P (3.3b)
The equilibrium constant (Ki), defined as the /i i iK y x , in the successive substitution method
(SSM) is written as:
1
n no
gn n ii i g
i o
PfK K
f P
(3.4)
where n denotes the time step.
2
141 10on
i
gi i
f
f
(3.5)
Black oil properties above the original bubble-point pressure at different pore sizes are
estimated by adding small fractions of equilibrium gas composition at the current bubble-point
pressure to reach a new larger bubble-point pressure, until the critical point is achieved. Here, two
different approaches are used and compared, while extending the properties: using the same
bubble-point gas composition ( )old pZ r and equilibrium gas ( )old py r at infinitely large pore sizes
for varied effective pore sizes, i.e., Eq. 3.6, which was proposed in previous research (Nojabaei
and Johns, 2016); and using different bubble-point gas compositions at varied pore sizes (pr ), i.e.
Eq. 3.7. In the first approach, the overall compositions at different pores sizes are the same. In the
second approach, the overall composition is a function of pore size.
( ) (1 ) ( ) ( )new p old p old pZ r Z r y r . (3.6)
( ) (1 ) ( ) ( )new p old p old pZ r Z r y r . (3.7)
28
where Z is the overall composition, α is the selected dilution factor, and y is the equilibrium gas
composition at current bubble-point pressure. Smaller dilution factors yield more reliable
saturation pressures.
During primary depletion or gas injection, gas-oil interfacial tension changes with pressure and
reservoir fluid composition. Interfacial tension is calculated by using the Macleod and Sugden
correlation (developed by Macleoad (1923) and Sugden (1924), and modified for multicomponent
mixtures by Weinaug and Katz (1943); Pederson and Christensen (2007)) as a function of phase
composition and densities:
4
)(
cN
i
V
i
L
ii yx (3.8)
Using the Peng-Robinson cubic EoS (Peng and Robinson (1976)), we pre-calculate interfacial
tensions as a function of pressure and use this data as input to the simulator. Capillary pressure is
not only a function of IFT, but also pore size. The correlation among effective pore size, capillary
pressure, and saturation is derived from the Leverett J-function (Nojabaei et al. 2016). In this way,
fluid properties can be expressed as tabular functions of capillary pressure, saturation and pore
sizes.
3.1.2 Slim-tube simulation
To estimate MMPs by using the newly-proposed black oil fluid properties extrapolation approach,
slim-tube simulation is performed by injecting 1.0 PV (pore volume) produced gas into a 1-D slim-
tube with 100 grids saturated with Bakken oil. The black oil fluid properties of the pseudo-binary
mixture are pre-calculated. The dimensionless mass conservation equation for pseudo-component
i is expressed as:
29
0i i
D D
C F
t x
(3.9)
where iC is the volumetric composition of i and iF is the flux term.
1
pN
i ij i
j
C C S
(3.10)
1
pN
i ij i
j
F C f
(3.11)
The fractional flow is defined as:
1 1
/
/p p
j rj j
i N N
j rj j
j j
kf
k
(3.12)
3.1.3 Compositionally extended black oil model
In this study, our objective is to perform gas injection simulation for a variety of schemes in
conventional and tight oil-rich reservoir by using the compositionally-extended black oil model.
This black oil model is a special case of a standard compositional model with three components
(pseudo-oil, pseudo-gas, and water) and three phases (oleic, vapor, and aqueous). The fluid
properties as a function of pressure and pore-size dependent capillary pressure have been derived
from flash calculations prior to conducting simulation. We solve the following mass balance
equations for the principle unknowns of oil pressure Po, water saturation Sw, and overall pseudo-
gas mass composition Zg.
1 1 1
P P PN N Nrj
ij j j j ij j j ij
j j jj
kkP g z M S
t
(3.13)
30
where ij is the mass fraction of component i in phase j, and it is calculated as a function of
solution gas-oil ratio soR and volatile oil-gas ratio vR as described in a previous paper (Nojabaei et
al., 2016). Mij is the well mass flow term; and for water component we use the following equation:
w
www
ww
rw
B
S
tqP
B
kk
(3.14)
3.1.4 IFT-dependent relative permeability curves
We use the model by Coats (1980), that is a method to capture the effect of variable oil-gas IFT
changes on relative permeability curves by using a weighted sum of the base curve (immiscible
condition) relative permeability at capillary pressure-dominated flow and straight line (miscible
condition) relative permeability at viscous dominated flow. The weighting functions to calculate
gas and oil relative permeabilities are given below:
( ) 1 ( )gn
g grg rgcwk k f S f S (3.15)
( ) 1 ( )ogn
o orog rogcgk k f S f S (3.16)
1
g grg
wir gr
S SS
S S
(3.17)
1
1
g wir orgo
wir org
S S SS
S S
(3.18)
( )gr grS f S (3.19)
( )org orgS f S (3.20)
31
and
11/
( )
n
o
f
(3.21)
where gn
gS and ogn
oS are the base gas and oil relative permeability curves from experimental data
at immiscible conditions, gn and on are two tuning exponents to match with base curves; gS and
oS are two straight lines that represent relative permeabilities at miscible conditions; o is the
reference value of IFT corresponding to the base curves; and 1n is a tuning number in the range
of 4 to 10. We slightly modified the oil relative permeability by multiplying it by rogcgk (oil
relative permeability at connate gas saturation), as this term was absent in Coats’s method.
Correspondingly, oil relative permeability in three-phase condition is calculated by applying
Stones’ second method (Stone, 1973), i.e.,
( )( )ro rocw row rw rog rg rw rgk k k k k k k k . (3.22)
3.2 Bakken oil properties in nanopores
The pseudo-oil (dead oil) and pseudo-gas (surface gas) compositions were determined by
performing flash calculations at standard pressure and temperature. The pressure‒composition
diagrams of Bakken oil using two different gas adding approaches (Eq. 3.6 and Eq. 3.7) are shown
in Figure 3.1a and 3.1b, where Zg is the pseudo-gas composition. By using the same overall
compositions (at infinitely large pore sizes) for three different effective pore sizes (Eq. 3.6), the
critical point pressure or the first contact minimum miscibility pressure (MMP) converges at 4325
psia (in Figure 3.1a). Meanwhile, the oil and gas densities with pressure at three different pore
sizes converge at the critical point pressure, as shown in Figure 3.2a. By using different overall
32
compositions at different pore sizes (Eq. 3.7), the bubble-point curves of three pore sizes converge
at the pressure (P = 3689 psia at Zg = 0.34) lower than the critical point pressure.
Figure 3.1 Pressure‒composition plot for Bakken oil for three different effective pore sizes (a) by
using the same overall compositions at infinitely large pore sizes (Eq. 3.6) and (b) by using
different overall compositions at different pore sizes (Eq. 3.7).
The pseudo-oil densities at different pore sizes also converge at the same pressure, as shown in
Figure 3.2b.The stronger the nano-confinment is, the higher critical point pressure is achieved, i.e.,
33
4325 psia, 4371 psia, and 4464 psia for pr =∞,
pr =30 nm, and pr =10 nm, respectively. It should
be noted that we only use the black oil properties obtained from Eq. 3.6 to perform simulation of
conventional reservoir gas flooding in section 3.2 and primary depletion and huff-n-puff gas
injection in section 3.3 and 3.4. We also examine the second gas adding approach (Eq. 3.7) by
comparing the huff-n-puff simulation and primary depletion results with the first approach at Pw
= 4400 psia (near critical point pressure) and Pw = 2000 psia in section 3.3.
Figure 3.2 Extrapolation of oil and gas densities for three different effective pore sizes (a) by using
the overall compositions from Eq. 3.6 and (b) by using the overall compositions from Eq. 3.7.
34
Figure 3.3a and b shows the oil component recovery as a function of pressure of 1.0 PVI by
using the extrapolated black oil properties obtained from Eq. 3.6 and from Eq. 3.7, and the turning
point of the curve to a horizontal line gives the MMP. The MMP by use of the first gas adding
approach (Eq. 3.6) is 4325 psia, while the second gas adding approach (Eq. 3.7), gives the MMPs
of 4325 psia, 4371 psia, and 4464 psia for pr =∞,
pr =30 nm, and pr =10 nm, respectively.
Figure 3.3 Oil component recovery as a function of pressure of 1.0 PVI by using the extrapolation
of black oil properties (a) getting from Eq. 3.6 and (b) from Eq. 3.7.
35
3.3 Miscible and immiscible gas flooding in conventional reservoir
The reservoir properties are tabulated in Table 3.1. The reservoir is a 2-D square domain with 225
(15 by 15) grid blocks. The reservoir fluid is Bakken oil with a bubble point pressure of Pb = 2860
psia at zero capillary pressure and reservoir temperature is 240 °F. The initial water saturation is
0.25.
Table 3.1 Reservoir and fluid properties for conventional reservoir.
Reservoir size, 2D 750 ft × 750 ft
Uniform matrix grid block size 50 ft × 50 ft
Matrix permeability 1000 md
Matrix porosity 10%
Reservoir thickness 10 ft
Fluid type Bakken fluid
Temperature 240 oF
For gas flooding, a homogeneous quarter of a five-spot pattern is used. The injector and
producer are located diagonally at the corner of the reservoir domain. The injected gas is surface
gas, which is also defined as the composition of the pseudo-gas component (see Nojabaei and
Johns (2016)). Gas floods at three different initial reservoir pressures of Pi = 2500, 3000, and 4500
psia are simulated. The corresponding bottomhole pressures for the production wells are 100 psi
lower than the initial reservoir pressures, i.e., Pw = 2400, 2900, and 4400 psia. The surface gas is
injected continuously from time zero with a constant injection rate of 1000 Mscf/day. The total
production time is 560 days and the cumulative oil recoveries from the compositionally-extended
black oil model are plotted in Figure 3.4.
At Pi = 2500 psia and Pw = 2400 psia, the reservoir is initially saturated (Pi < Pb) and the injected
gas displaces the oil immiscibly. Gas break-through occurs early at 12.8 days, which corresponds
to 0.14 PVI (pore volume injected). At Pi = 3000 psia and Pw = 2900 psia, the reservoir is initially
undersaturated (Pi > Pb) and the gas flood remains immiscible, and gas break-through occurs at
36
26.1 days (PVI = 0.23), slightly later than the previous case. At Pi = 4500 psia and Pw = 4400 psia,
the reservoir is initially undersaturated (Pi >Pb) and the gas flood is miscible because the BHP is
larger than the critical pressure or MMP. Gas break-through occurs at 51.1 days, which
corresponds to 0.25 PVI. By comparing the cumulative oil recovery curves, it is observed that
before gas breaks through, the oil recovery rate is the same for three scenarios regardless of being
miscible or immiscible. After gas breaks through, immiscible flooding at higher pressures results
in a larger oil recovery rate. Additionally, miscible flooding maintains a constant high recovery
rate and reaches the maximum recovery in a shorter period of time.
Figure 3.4 Cumulative oil production of gas floods at different initial reservoir pressures and
production pressures.
The gas composition in the production well block for the three scenarios are shown in Figure
3.5. For the case of Pi = 2500 psia, Zg is initially the same as the original pseudo-gas composition
at 0.25. As gas is injected and breaks-through the production well, the pseudo-gas composition
increases gradually while the reservoir fluid remains in the two-phase region. At later time the gas
composition suddenly increases to 1.0, showing that oil has been completely vaporized. For Pi =
37
3000 psia, the increase is similar to the previous case although the two-phase period is shorter. For
Pi = 4500 psia, there is no two-phase period as this corresponds to a completely miscible first-
contact flood, i.e. above the MMP and flow would be piston-like with a very sharp front if more
grid blocks had been used. Numerical dispersion is smearing this front since there is no self-
sharpening behavior above the MMP.
Figure 3.5 Gas composition at production well block for gas injection in conventional reservoirs
at different initial reservoir pressures and production pressures.
The distribution of gas saturation at different displacing times for cases of Pi = 3000 psia and
Pw = 2900 psia, and Pi = 4500 psia and Pw = 4400 psia (first-contact miscible flooding) are plotted
in Figures 3.6 and 3.7, respectively. For the lower pressure case, the gas saturation increases at the
production well after 30 days. The two-phase (oil-gas) region appears near time zero, and
propagates throughout the entire reservoir for continous gas injection. At later time, oil saturation
at both injection and production well is zero. However, for the misicible flood case, the
dispalcement is piston-like. A distinct single displacing front is clearly observed from the
38
beginning of the simulation to the end. Mobile oil in the reservoir is completely recovered in 150
days, which is also verified by the cumulative oil recovery curve in Figure 3.7.
Figure 3.6 Gas saturation distribution in the conventional reservoir at different times (Pi=3000 psia
and Pw =2900 psia).
Figure 3.7 Gas saturation distribution in the conventional reservoir at different times (Pi=4500 psia
and Pw =4400 psia).
3.4 Huff-n-puff in an oil-rich tight reservoir
For this case, the reservoir size and fluid properties are the same as case 1 (given in Table 3.1).
However, reservoir permeability for case 2 is different, which will be discussed next. The reservoir
39
fluid is Bakken oil with a bubble point pressure of Pb = 2860 psia when capillary pressure is zero
and Pb = 2750 psia when capillary pressure in flash calculations is taken into account and pore size
is 15 nm. The initial reservoir pressure is 5000 psia and the initial water saturation is 0.15. Middle
Bakken, the formation considered here, is not ultra-tight, but only tight (permeability about 0.05
md).
Table 3.2 Reservoir and fluid properties for tight oil-rich reservoir.
Reservoir size, 2D 750 ft × 750 ft
Uniform matrix grid block size 50 ft × 50 ft
Matrix porosity 10%
Reservoir thickness 10 ft
Fluid type Bakken fluid
Initial reservoir pressure 5000 psia
Temperature 240oF
Instead of creating hydraulic fractures, we increase the matrix permeability near the production
well to account for the effect of increased conductivity in the reservoir caused by fractures. The
reservoir permeability map and the location of the horizontal well are shown in Figure 3.8. The
increased permeability changes the physics to completely advection-dominated mass transfer. For
the Middle Bakken, as the formation is not ultra-tight, mass transfer is likely advection-dominated
anyway, and diffusion should play a small role.
40
Figure 3.8 Reservoir permeability map for tight oil-rich reservoir.
In this section, we consider four BHPs at Pw = 1000, 2000, 3000, and 4000 psia to study the
production performances for both primary depletion and huff-n-puff gas injection schemes. For
the huff-n-puff gas injection, after producing for 50 days through primary depletion, produced gas
(surface gas) is injected at constant rate of 25 Mscf for 30 days, followed by a 20-day soaking
period. For all schemes, the total production time is 800 days. Unlike conventional reservoirs,
nano-pore size-induced large capillary pressure in tight formations strongly affects the fluid phase
behavior (2013), and may further affect the oil recovery prediction. In order to investigate the
capillary pressure effect on production performance, we design three scenarios: without capillary
pressure in flow and flash calculations, with capillary pressure in flow but without capillary
pressure in flash calculations, and with capillary pressure both in flow and flash calculations. We
considered an average pore size of 15nm for the cases that capillary pressure influences flash
calculations. To incorporate the effect of capillary pressure on fluid properties through flash
calculations, we multiplied the interfacial tension calculated by using Mcleod and Sugden
correlation by a factor of three, as the calculated interfacial tensions are underestimated as stated
41
by Nojabaei et al. (2013). We perform the primary and huff-n-puff gas injection schemes at four
different BHPs under three different scenarios. All the simulation results are plotted in Figure
3.9a–c. The corresponding oil pressure at the well block at different times are plotted in Figure
3.10.
42
Figure 3.9 Cumulative oil recovery for primary depletion and huff-n-puff gas injection with and
without considering capillary pressure in tight oil-rich reservoirs at at Pi=5000 psia and (a) Pw
=1000 psia, (b) Pw =2000 psia, (c) Pw =3000 psia, 4000 psia, and 4400 psia.
Figure 3.10 Oil pressure at the well block for primary depletion and huff-n-puff gas injection with
and without considering capillary pressure in tight oil-rich reservoirs at Pi=5000 psia and Pw=1000,
2000, 3000, 4000, and 4500 psia, respectively.
43
It can be seen from Figure 3.9a–c that a lower BHP contributes to a higher cumulative oil
recovery for both primary depletion and huff-n-puff gas injection schemes. This is because a larger
pressure difference between the reservoir pressure and BHP is beneficial for fluid flow.
Furthermore, cumulative oil recoveries for primary depletion without considering capillary
pressure at Pw = 1000 and 2000 psia are around 8000 STB, which is 1-2 times larger than 2800
STB at Pw = 3000 psia and 4-5 times larger than 1400 STB at Pw = 4000 psia. The large recovery
at the lower BHPs is also owing to the effective solution gas drive, while the dissolved gas releases
from oil for reservoir pressures below the bubble point pressure (2860 psia). Comparing the
cumulative oil production of primary depletion to huff-n-puff gas injection scheme at the same
BHP, it is observed that when the production pressure (Pw = 1000 and 2000 psia) is below bubble
point pressure, both schemes achieve almost the same final oil recovery. However, when the
production bottomhole pressures (Pw = 3000, 4000, and 4500 psia) are above bubble point
pressure, huff-n-puff results in a larger final oil recovery compared to primary depletion with an
increase of 7.22%, 12.86%, and 26.75%, respectively. This is because huff-n-puff gas injection is
more efficient at miscible or near-miscible conditions (also concluded by Song and Yang (2017)).
The viscosity and interfacial tension of original oil could be effectively reduced as gas is dissolved
in oil; meanwhile, the swollen diluted-oil is easy to be displaced and recovered (Du et al., 2019).
From previous research, we know that large gas-oil capillary pressure due to the nano-
confinement significantly affects the recovery performance (Nojabaei et al., 2016). The results in
Figure 3.9a–c indicate that inclusion of capillary pressure in flash could significantly increase oil
recovery at a lower BHP (Pw = 1000 psia). However, when BHP pressure is increased, the impact
of capillary pressure in flash on oil recovery weakens and almost disappears at Pw = 4000 psia.
The reason is that the IFT between oil and gas phases reduces as the reservoir pressure increases
44
and becomes zero at MMP (4325 psia), resulting in a diminished capillary pressure. Nevertheless,
inclusion of capillary pressure in flow does not contribute to oil recovery at higher BHPs and even
decreases the recovery to a small extend at a lower BHP. Inclusion of capillary pressure in flash
calculation influences primary recoveries for lower pressure cases, however, it doesn’t influence
the enhanced oil recovery as gas injection enhances recoveries only at higher pressures where the
effect of capillary pressure becomes negligible.
Here, we also plot the cumulative gas recoveries for primary depletion and huff-n-puff gas
injection for the case with capillary pressure in both flow and flash at five different BHPs, as shown
in Figure 3.11. This plot indicates that the cumulative produced gas for huff-n-puff is always higher
than that for primary depletion. The total injected gas in the huff period is 750 Mscf. The enhanced
cumulative gas recoveries for Pw = 1000, 2000, 3000, 4000 and 4500 psia are 508, 570, 485, 390,
and 389 Mscf, respectively. Most of the injected gas is produced for lower pressures (Pw = 1000
and 2000 psia) (production is still going on). It can be concluded that if gas is injected when
reservoir is already saturated (Pw = 1000 and 2000 psia), the injected gas will flow back with oil
instead of dissolving into crude oil and increasing the oil recovery. However, when the reservoir
pressure is at miscible or near-miscible condition (Pw = 3000, 4000 and 4500 psia), most of the
injected gas dissolves into the crude oil and improves the oil recovery.
45
Figure 3.11 Cumulative gas recovery for primary depletion and huff-n-puff gas injection at Pw =
1000, 2000, 3000, 4000, and 4500 psia, respectively accounting for the capillary pressure both in
flow and flash.
In this section, we also examine the second gas adding approach (Eq. 3.7) by comparing the
huff-n-puff gas injection with the first approach (Eq. 3.6) at production pressure of Pw = 4400 psia
(near critical point pressure) and Pw = 2000 psia. The results are tabulated in Table 3.3. Adopting
different black oil properties by using different gas adding approaches would make a difference
on recovery when Pc in both flash and flow is included, although the difference is not significant.
A decrease in recovery is observed at a higher production pressure (4400 psia) and an increase in
recovery is observed at a lower pressure by using second approach owing to the black oil properties
distortion below and above P = 3689 psia.
Table 3.3 Gas production of huff-n-puff at Pw = 4400 psia and 2000 psia using two different gas
adding approaches.
46
Pw (psia)
Pc in flow Pc in flash
Huff-n-huff
Production/Mscf
(using Eq. 3.6)
Production/Mscf
(using Eq. 3.7)
4400 Yes Yes 1048.25 1047.53
2000 Yes Yes 9122.98 9132.59
3.5 Effect of IFT-dependent relative permeability on recovery
IFT-dependent oil-gas relative permeability data that accounts for the pressure and
compositional change are incorporated in this compositionally-extended simulation model.
Simulation studies are performed in a tight oil-rich reservoir (the reservoir model of Case 2 is used)
to examine the effect of dynamic relative permeability on recovery in primary production and huff-
n-puff gas injection schemes. The initial reservoir pressure is 5000 psia and the BHPs are set to be
2000, 3000, and 4000 psia, respectively and the simulation results are compared to those with fixed
base relative permeability curves. According to Coats’s method, base gas and oil relative
permeability curves from experimental data at immiscible conditions are required. In this study,
the base relative permeability curves for the Bakken reservoir are adopted from Yu et al. (2014)
as shown in Figure 3.12. The IFT-dependent relative permeability curves are then calculated using
weighted base (immiscible) relative permeability curves and weighted straight line (miscible)
relative permeability curves (dotted line in Figure 3.12). The IFT change with composition is pre-
calculated as a function of pressure and the data are plotted in Figure 3.13. We assume that the
reference value of IFT ( o in Eq. 3.21), corresponding to the base oil-gas relative permeability
curves, is 4.736 dyne/cm (at 2000 psia).
47
Figure 3.12 Relative permeability curves for case 3, adopted from Yu et al. (2014).
Figure 3.13 Oil-gas IFTs of Bakken black oil at different reservoir pressures.
The oil recovery simulation results for primary depletion and huff-n-puff gas injection schemes
at different BHPs are plotted in Figure 3.14a–c. It should be noted that for all the curves, capillary
pressure in both flow and flash calculations are included. At a lower BHP (Pw = 2000 psia), an
obvious increase in the cumulative oil recovery is observed for both primary and huff-n-puff gas
48
injection schemes when the IFT-dependent relative permeability curves are used in the simulation
model. However, at higher BHPs (Pw = 3000 and 4000 psia), no distinct change in oil recovery is
observed compared to the base relative permeability case. The reason is that for the primary
depletion schemes, only single phase oil exists at high BHPs (Pw > Pb) and IFT is not defined.
Therefore, fluid flow in reservoir is exactly the same as the case using base relative permeability
curves. For the huff-n-puff gas injection schemes, in spite that a small fraction of injected gas
appears near the production well, oil saturation is still very high and oil phase flow is dominant in
the reservoir. It can be concluded that IFT-dependent relative permeability significantly impacts
oil recovery only for a reservoir with a dominant mobile oil-gas two-phase region.
49
Figure 3.14 Cumulative oil production for huff-n-puff gas injection in tight oil-rich reservoirs
using IFT-dependent relative permeability curves and base relative permeability curves,
respectively, at Pi=5000 psia and (a) Pw =2000 psia, (b) Pw =3000 psia, and (c) Pw =4000 psia.
50
Chapter 4 Diffusion Coefficient with Nano-confinement Effects
(This chapter of the dissertation is adopted from our published paper in the Journal of Petroleum
Science Engineering, https://doi.org/10.1016/j.petrol.2020.107362. The title is “Estimating
diffusion coefficients of shale oil, gas, and condensate with nano-confinement effect”)
4.1 Methodology
Sigmund correlation is one of the most extensively used correlations to calculate diffusion
coefficients; however, the unit systems used were not well documented and varied in different
studies (Nghiem et al., 2001; Yu et al., 2015; Alharthy et al., 2018). Applying Sigmund correlation
with uncertain units may result in incorrect diffusivities. For example, one simulation work
(Alharthy et al., 2018) used Sigmund correlation and tried to match the calculated diffusion
coefficients with the experimental data selected from Sigmund’s paper but the results were
unsatisfactory (underestimated diffusivities by one or two orders). Here, Sigmund correlation
(Sigmund, 1976a; 1976b) and complementary formulas by later researchers (da Silva and
Belery,1989; Reid et al., 1987) are summarized. The definitions of parameters with the appropriate
units are given in the “Nomenclature” section. According to Sigmund correlation, the binary
diffusion coefficient, ijD, between components i and j, is calculated as:
0 0
2 30.99589 0.096016 0.22035 0.032874k ij
ij kr kr kr
k
DD
(4.1)
da Silva and Belery (1989) claimed that Eq. 4.1 gave a negative ijD at 3.7kr and suggested
to use the following expression for 3.7kr :
0 0
0.18839exp(3 )k ij
ij kr
k
DD
(4.2)
51
0 0
k ijD is the zero pressure limit of the density-diffusion coefficient product in phase k,
calculated by:
1/21/2
0 0
2
0.0018583 1 1k ij
ij ij i j
TD
R M M
(4.3)
5/3
1
2/3
1
c
c
n
ik ciikr k n
ik cii
y v
y v
(4.4)
where, ij is the Lennard Jones size parameter (collision diameter), and
ij is the collision integral
of the Lennard-Jones potential, both are associated with the component critical properties (Reid et
al., 1987):
2
i j
ij
(4.5)
ij i j (4.6)
1/3
(2.3551 0.087 ) cii i
ci
T
P
(4.7)
(0.7915 0.1963 )i B i cik T (4.8)
0.15610
1.06036 0.19300 1.03587 1.76474
( ) exp(0.47635 ) exp(1.52996 ) exp(3.89411 )ij
ij ij ij ijT T T T (4.9)
Bij
ij
kT T
(4.10)
where Bk is the Boltzmann’s constant (=1.3805E-16 ergs/K) and ij is the characteristic Lennard-
Jones energy.
Sigmund (1976b) also defined the diffusion coefficient of component i in phase k of a
multicomponent mixture based on Wilke formula (Wilke, 1950), as given in Eq. 4.11.
52
1 ikik
jk
i jij
yD
y
D
(4.11)
In Wilke‒Chang correlation (Wilke and Chang, 1955), the diffusion coefficient of component
i in phase k in mixture, is expressed as:
8 1/2
0.6
7.4 10 ( )ikik
k bi
M TD
v
(4.12)
with
1
jk jj i
ik
jk
y MM
y
(4.13)
where biv is the partial molar volume of component i at the boiling point and is calculated as:
1.0480.285bi cv v (4.14)
Hayduk and Minhas (1982) proposed three empirical correlations for water, n-paraffins (C5 to
C32), and nonaqueous mixtures as the solutes, respectively. In this study, the correlations for
nonaqueous mixtures (Eq. 4.15) and for n-paraffins (Eq. 4.18) are used to predict the diffusivities
in gas or oil phase.
8 1.29 0.5
0.92 0.23 0.42
1.55 10 ( )
( )
ikik
k ik i
T PD
v P
(4.15)
with
1
jk jj i
ik
ik
y PP
y
(4.16)
1
jk bjj i
ik
ik
y vv
y
(4.17)
where jP is the parachor of component j.
53
0.791(10.2/ )8 1.47
0.71
1.33 10
( )
bi
kik
bi
TD
v
(4.18)
4.2 Validation of empirical correlations
First, we apply the three empirical correlations (Wilke‒Chang, Hayduk–Minhas, and Sigmund) to
calculate diffusion coefficients and compare them with the laboratory measurements, aiming at
screening out the correlations that are suitable for shale fluids at reservoir conditions. Here,
experimental data for two mixtures at high pressures are selected from the literature, i.e., C1/C3
mixture (Sigmund 1976a) and C1/C10 mixture (Dysthe and Hafskjold, 1995). The first set (C1/C3
mixture) contains light hydrocarbons, with similar properties as shale gas and condensate. The
second set (C1/C10 mixture) is light hydrocarbon in decane, similar to a mixture of injected gas in
crude oil. The experimental data and fitting curves are plotted in Figures 4.1 and 4.2.
54
Figure 4.1 The diffusion coefficient of CH4 in C1/C3 mixtures by using empirical correlations and
comparison with experimental data (Sigmund, 1976a) at (a) 160 °F and 3000 psia; (b) 160 °F and
2000 psia; (c) 100 °F and 2000 psia; (d) 220 °F and 1000 psia.
55
Figure 4.2 The diffusion coefficient of CH4 in C1/C10 mixtures by using empirical correlations and
comparison with experimental data (Dysthe and Hafskjold, 1995) at (a) 86 °F and 40 MPa; (b)
86 °F and 50 MPa.
In Figure 4.1, diffusion coefficients are calculated and compared to the four groups of
measurements for the C1/C3 mixture at high pressures and temperatures sets: (a) 160 °F and 3000
psia, (b) 160 °F and 2000 psia, (c) 100 °F and 2000 psia, and (d) 220 °F and 1000 psia. As was
expected, employing Sigmund correlation always resulted in a good match at different pressures,
considering that Sigmund used these experimental data to develop the correlation. Wilke‒Chang
56
and Hayduk‒Minhas correlations for nonaqueous mixtures (Eq. 4.15) overestimate the
diffusivities at high pressures (2000 and 3000 psia), but underestimate the diffusivities at relatively
lower pressure of 1000 psia. Hayduk‒Minhas correlation for n-paraffins (Eq. 4.18) underestimate
the diffusivities at all given pressures. In Figure 4.2, two groups of calculations and comparisons
were performed for C1/C10 mixture at following high pressures: (a) 86 °F and 40 MPa and (b) 86
°F and 50 MPa. Sigmund correlation is still superior to the other three correlations and offers the
best fitting. One reason to explain the unsatisfactory prediction of diffusivity using Wilke‒Chang
and Hayduk‒Minhas correlations is that these correlations were derived from and suitable for low-
pressure liquid systems. Based on the validation results, Sigmund correlation is selected to predict
the diffusivities in shale gas and liquid systems in the following sections.
4.3 Diffusivity of shale fluids without confinement effects
The compositions of Marcellus shale gas (Bullin et al., 2008), Lower Huron shale gas, Marcellus
shale condensate (Elamin, 2013), and Bakken shale oil (Nojabaei and Johns, 2016) are given in
Table 4.1. The corresponding reservoir temperatures are 140 °F, 110 °F, 150 °F, and 240 °F,
respectively.
Table 4.1 Compositions of Marcellus shale gas, Lower Huron shale gas, Marcellus shale
condensate, and Bakken shale oil (unit: mole fraction).
Marcellus Shale Gas
CO2 C1 C2 C3 N2
0.003 0.955 0.03 0.01 0.002
Lower Huron Shale Gas
CO2 C1 C2 C3 N2
0.0023 0.972 0.0153 0.0004 0.01
Marcellus Shale Condensate
CO2 N2 C1 C2 C3 IC4 NC4 IC5 NC5 C6 C7
0.0124 0.0034 0.7967 0.0547 0.0396 0.011 0.0152 0.0079 0.0063 0.0113 0.0415
Bakken Shale Oil
C1 C2 C3 C4 C5C6 C7C12 C13C21 C22C80
57
0.36736 0.14885 0.09334 0.05751 0.06406 0.15854 0.0733 0.03704
Sigmund correlation is used to calculate the diffusivities of three types of shale fluids at
reservoir temperatures. For Marcellus shale gas (Figure 4.3), the diffusivity is decreased
dramatically with increasing pressure at lower pressures, but the decline reaches a plateau at higher
pressures. Lighter components have relatively higher diffusivities compared to heavier
components.
Figure 4.3 The diffusivities of components of Marcellus shale gas in gas phase at the reservoir
temperature.
For Marcellus shale condensate, the diffusivities of CO2, CH4, and n-C4H10 in gas and liquid
phases are plotted in one figure (Figure 4.4). At lower pressures, the diffusivities of components
in gas phase are larger than in liquid phase by almost one order of magnitude. For example, at
1500 psia, the diffusivity of CH4 in gas phase is 1.17×10-3 cm2/s and in liquid phase, it is 1.00×10-
4 cm2/s. As pressure increases, the diffusivities in gas phase decline dramatically. Meanwhile, the
diffusivities in liquid phase increase smoothly and converge to diffusivities in gas phase with a
58
sudden rise at P = 2570 psia. This is because the liquid phase disappears at the dew-point pressure.
Above the dew-point pressure, the diffusivities in gas phase keep decreasing slowly with pressure.
Figure 4.4 The diffusivities of components of Marcellus shale condensate in gas and liquid phases
at the reservoir temperature.
For Bakken shale oil, the diffusivities of all components in gas phase are plotted in Figure 4.5a.
Similar to Marcellus shale gas, lighter components yield a higher diffusivity. At 1000 psia, the
diffusivity of CH4 in gas phase is almost one order lager than that of C22C80. As pressure increases,
the diffusivities decrease and the gap between light and heavier components is narrowed. In the
liquid phase (Figure 4.5b), the diffusivities are decreasing slightly with pressure, which is opposite
to the case in shale condensate. After reaching bubble-point pressure (Pb = 2860 psia), an almost
linear declining of diffusivities with pressure is observed. Comparing Figure 4.5a and 4.5b, the
diffusivities in gas phase are almost two orders larger than in liquid phase at the same pressure.
59
Figure 4.5 The diffusivities of components in (a) Bakken oil in gas phase and (b) Bakken oil in
liquid phase at reservoir temperatures.
60
4.4 Nano-confinement effects on diffusivity
In this section, two methods are introduced to consider the nano-confinement effects, i.e., critical
property shift and large gas-oil capillary pressure. Both methods can alter the phase properties in
the nano-pores with their specific performance.
4.4.1 Critical Property Shift
Morishige et al. (1997) observed a significant reduction in critical temperature of argon nitrogen,
oxygen, ethylene, and CO2 in nano-scale porous materials. Zarragoicoechea and Kuz (2004)
developed a model to account for the critical temperature shift based on van der Waals equation
of state (Eq. 4.19) and reported a good match with the experimental data by Morishige et al. (1997).
They also developed an equation for the critical pressure shift (Eq. 4.20), even though the equation
has not been experimentally verified owing to the difficulty in operation (Zarragoicoechea and
Kuz, 2004).
2
0.9409 0.2415c cp ij ij
c
c p p
T TT
T r r
(4.19)
2
0.9409 0.2415c cp ij ij
c
c p p
P PP
P r r
(4.20)
30.244 cij
c
T
P (4.21)
where cT and cP are the bulk critical temperature and pressure, and cpT and
cpP are the pore critical
temperature and pressure, respectively.
61
4.4.2 Gas-oil Capillary pressure
Owing to the large gas-oil capillary pressure in nano-sized pores, there is a phase pressure
difference in as oil and gas phases co-exist. Gas pressure is calculated using the Laplace equation
as given below:
g
2o
p
P Pr
(4.22)
where, is the interfacial tension (IFT) between oil and gas phases and is calculated by using the
Macleod and Sugden correlation (developed by Macleoad, 1923, and Sugden, 1924, and modified
for multicomponent mixtures by Weinaug and Katz, 1943; Pedersen and Christensen, 2007) as a
function of phase composition and densities:
4
)(
cN
i
V
i
L
ii yx (4.23)
At equilibrium, component fugacities in the gas and liquid phases are equal, although the phase
pressures are not, i.e.,
),...,,,(),...,,,( 2121 CC Ng
g
iNo
o
i yyyPTfxxxPTf . (4.24)
4.4.3 Nano-confinement effects on diffusivity
62
Figure 4.6 P–T phase envelops of Bakken shale oil, Bakken shale oil with CO2 injection at 20%
and 50%, Marcellus shale condensate, condensate with CO2 injection at 20%, 50% and 80%, and
Marcellus shale gas.
For Marcellus shale gas, the reservoir fluid is only single gas phase at the reservoir temperature,
as shown in Figure 4.6. The gas-oil capillary pressure is zero and only the critical property shifts
nano-confinement effect is taken into consideration. The diffusivities of CO2 and CH4 in gas phase
at variable CO2 mole percentage and at different pressures and pore sizes are plotted in Figure 4.7a
and b. Both CO2 and CH4 diffusivities in gas phase decrease slightly with continuous CO2 injection.
Lower pressure yields a higher diffusivity. At higher pressures, the declining of diffusivity with
pressure slows down. Including critical property shifts could increase the diffusion coefficients;
the smaller the pore size is, the more significant the effect is. For example, at 80% CO2 mole
63
percentage, compared to the base case (without nano-confinements), including critical property
shifts increases the CO2 diffusivity from 4.8% to 14.1% for 10nmpr and from 9.4% to 28.0%
for 5nmpr as pressure ranges from 1000 psia to 3000 psia.
Figure 4.7 For Marcellus shale gas, the diffusivities of (a) CO2 and (b) CH4 in gas phase versus
CO2 mole percentage at different pressures and pore sizes with and without considering critical
property shifts.
64
Here, we calculate the diffusivity of N2 and CH4 for Lower Huron shale gas at varied N2 mole
percentage to examine the diffusion behavior when nitrogen is used as the fracking fluid, and the
results are plotted in Figure 4.8a and b. It should be noted that we also calculated the diffusivity
of CO2 and CH4 for Lower Huron shale gas at varied CO2 mole percentage, but the results are not
showing here because the data are very close to but only slightly smaller than those in Marcellus
shale gas (Figure 7a and b) due to its lower reservoir temperature. Comparing Figure 4.7a and b to
Figure 4.8a and b, it is found that the diffusivities of N2 and CH4 increases with N2 concentration,
in contrast with the diffusivities of CO2 and CH4 in CO2 environment. This is because that the
molar density of shale gas is increased with continuous injection of CO2 (Figure 4.9a), but is
declined with an increase in N2 concentration, as shown in (Figure 4.9b). In addition, including
critical property shifts reduces the molar density (Figure 4.9a and b), resulting in an increased
diffusivity for both CO2 and N2 cases.
65
Figure 4.8 For Lower Huron, the diffusivities of (a) N2 and (b) CH4 in gas phase versus N2 mole
percentage at different pressures and pore sizes with and without considering critical property
shifts.
66
Figure 4.9 The gas molar density of (a) Marcellus shale gas versus CO2 mole percentage and (b)
Lower Huron shale gas versus N2 mole percentage at different pressures and pore sizes with and
without considering critical property shifts.
For Marcellus shale condensate, CO2 diffusivities in gas phase at variable CO2 percentage with
and without nano-confinements are plotted in Figure 4.10, including (a) P = 1000 psia, 30nmpr
; (b) P = 1000 psia, 15nmpr ; (c) P = 3000 psia, 30nmpr ; and (d) P = 3000 psia, 15nmpr .
At 1000 psia (Figure 10a and b), compared with the base case, including critical property shifts
decreases the diffusivity in gas phase at CO2 percentage lower than 50% (two phases co-exist in
the system), but increases the diffusivities in gas phase at CO2 percentage larger than 60% (single
gas phase exists in the system). The smaller the pore size is, the stronger the effect becomes; for
example, at 40% CO2, including critical property shifts reduces the CO2 diffusivity about 1.0%
and 2.3% at 30nmpr and 15 nm respectively. At 80% CO2, including critical property shifts
increases the CO2 diffusivity about 1.9% and 3.8% at 30nmpr and 15 nm, respectively. The gas-
67
oil capillary pressure, as another nano-confinement effect, exists only in a two-phase system. In
this study, the interfacial tension is increased by three times (multiplying Eq. 4.23 by three) based
on a previous study (Ayirala and Rao, 2006) indicating that measured IFTs were two or three times
larger than those from Macleod and Sugden correlation (Eq. 4.23). As shown in Figure 10a and
b, including gas-oil capillary pressure effect alters the diffusion behaviors by decreasing the
diffusivities before in the two phase region (at CO2 <80% for 30nmpr and 70% for 15nmpr )
and after the single gas phase appears, its influence on diffusivity diminishes. Again, the smaller
the pore size is, the more significant the effect is; for example, at 40% of CO2, including gas-oil
capillary pressure effect reduces the CO2 diffusivity about 13.1% and 19.1% at 30nmpr and 15
nm, respectively. When both effects are incorporated, capillary pressure effect is more pronounced
compared to the critical property shifts.
68
Figure 4.10 For Marcellus shale condensate, the diffusivities of CO2 in gas phase versus CO2 mole
percentage at different pressures and pore sizes with and without considering nano-confinement
effects.
69
For Bakken shale oil, Figure 4.11 shows the oil with and without nano-confinement effect
(capillary pressure, critical property shifts) at pore size of 10 nm. Including capillary pressure
effect reduces the bubble-point-pressure curve but slightly increases the dew-point-pressure curve.
The entire phase envelope shrinks if the critical property shifts is considered and the shrinkage
becomes severer at higher temperatures.
Figure 4.11 Original Bakken oil with and without nano-confinement effect (capillary pressure,
critical property shifts) at pore size of 10 nm.
The diffusivities of CO2 and C5C6 in gas phase versus CO2 mole percentage at 1500 psia and
10nmpr with and without nano-confinement effects are plotted in Figure 4.12a and b. Including
capillary pressure effect reduces the diffusivity of CO2 and C5C6 in gas phase, by 16.2% and 17.4%
at 40% CO2. Including critical property shifts reduces the diffusivity of CO2 and C5C6 at lower
CO2 percentage slightly, and increases the diffusivity of C5C6 at higher CO2 percentage. When
70
both effects are included, the effect of capillary pressure to reduce the diffusivity in gas phase is
dominant.
Figure 4.12 For Bakken shale oil, the diffusivities of (a) CO2 and (b) C5C6 in gas phase versus CO2
mole percentage at 1500 psia and at pore size of 10 nm with and without considering nano-
confinement effects.
The diffusivities of CO2 and C5C6 in liquid phase versus CO2 mole percentage at 10nmpr
with and without nano-confinement effects are plotted in Figure 4.13, including (a) CO2 at 1500
71
psia, (b) C5C6 at 1500 psia, (c) CO2 at 3000 psia, and (d) C5C6 at 3000 psia. At a lower pressure
i.e., 1500 psia, gas-oil capillary pressure effect decreases the diffusivities with an increase in CO2
percentage, whereas, critical property shifts increase the diffusivities with a decrease in CO2
percentage. At a higher pressure, i.e., 3000 psia, both capillary pressure effect and critical property
shifts increase the diffusivities in liquid phase. A combination of both confinement effects
increases the diffusivities in liquid phase by 25% at lower CO2 percentage, but this effect
diminishes with continuous CO2 injection.
72
Figure 4.13 For Bakken shale oil, the diffusivities of CO2 and C5C6 in liquid phase versus CO2
mole percentage at at pore size of 10 nm with and without considering nano-confinement effects,
including (a) CO2 at 1500 psia; (b) C5C6 at 1500 psia; (c) CO2 at 3000 psia; and (d) C5C6 at 3000
psia.
4.4.3 Diffusion with confinement effect on shale oil production
As discussed in section 4.4.3, the confinement effect will affect the diffusion coefficient in
nano-pore shale reservoirs. In this section, we will examine the diffusion with confinement effect
73
on Bakken shale oil production during huff-n-puff produced gas injection process. The
permeability of the shale matrix is shown in Figure 4.14.
Figure 4.14 Reservoir permeability map for tight oil-rich reservoir.
Figure 4.15 Cumulative oil production of primary depletion and huff-n-puff gas injection with and
without molecular diffusion.
The gas is injected at constant injection rate of 10 Mscf/day. Two circles of huff-n-puff is
performed, including 55 days of gas injection and 37 days of soaking period for each circle. First,
we examine the molecular diffusion effect on oil production. The cumulative oil production of
primary depletion and huff-n-puff gas injection with and without molecular diffusion are plotted
74
in Figure 4.15. The results show that using huff-n-puff gas injection approach could increase the
oil production by 11.91%. Moreover, inclusion of molecular diffusion effect could increase the oil
production by almost 2% compared to one without including molecular diffusion effect.
Figure 4.16 Reservoir permeability map for tight oil-rich reservoir.
We also examine the diffusion with confinement effect on production behavior. The results are
plotted in Figure 4.16. It shows that including huff-n-puff could slight increase the oil production
during the huff-n-puff process but decrease oil production at a later time. Overall, the effect is not
very significant.
4.5 Effective Diffusion Coefficient in Porous Media
4.5.1 Methodology
An effective diffusion coefficient is suggested to characterize the diffusion behavior in a porous
media by the following expression (Petersen, 1958; Epstein, 1989; Cooper et al., 2016; Backeberg,
2017):
effD
D
(4.25)
75
2
2 eL
L
(4.26)
where effD is the effective diffusion coefficient in porous media, is the porosity,
is the
tortuosity factor, and is tortuosity.
Shen and Chen (2007) summarized three types of empirical tortuosity-porosity relations,
including the power correlation (Eq. 4.27) (Lerman, 1979; Ullman and Aller 1982), linear
correlation (Eq. 4.28) (Iversen and Jorgensen, 1993; Low, 1981), and logarithmic correlation (Eq.
4.29) (Boudreau, 1996; Weissberg, 1963):
1n
mA (4.27)
1 (1 )B (4.28)
1 lnC (4.29)
where A, m, n, B, and C are lithology-dependent parameters and the values vary for different types
of rock. Marcellus shale and Bakken shale are clay-abundant formation types with clay contents
reaching to 43% (Chalmers et al., 2012) and 50-60% (Steptoe and Carr, 2011), respectively. For
clay-silt sediments, the above adjustable parameters were suggested as A = 1, n = 1, and m = 2.5‒
5.4 (Atkins and Smith, 1961), B = 3 (Iversen and Jorgensen, 1993), and C = 2 (for an universal
type of rock) (Boudreau et al., 1996).
Among the proposed numerous theoretical tortuosity-porosity relations, the Bruggeman
equation (Bruggeman, 1935) probably is the most commonly used equation:
(4.30)
with 0.5 for spheres and 1 for cylinders (Tjaden et al., 2016). However, Chen et al. (2015)
claimed that the tortuosity factor of shales was much larger than that commonly applied in
76
Bruggeman equation. Based on scanning electron microscopy (SEM) images of four shale samples
and Markov chain Monte Carlo (MCMC) reconstruction technique, they suggested a range of
1.33 1.65 in the Bruggeman equation by using Lattice Boltzmann (LB) simulation.
It is worthwhile to note that the anisotropic nature and geometric complexity of porous media
are not reflected by the empirical and theoretical relations. We will also use the shale tortuosity
obtained from 3D tomographic imaging techniques to calculate the effective diffusivity and make
a comparison with the results by using empirical and theoretical correlations.
4.5.2 Effective molecular diffusivity in porous media
The computed shale tortuosity ( ) or tortuosity factor ( ) from literature by using the recent 3D
tomographic imaging techniques are given in Table 4.2. Owing to the heterogeneity inherent, the
tortuosity varies from sample to sample.
Table 4.2 Measured tortuosity and tortuosity factor of different shale samples using 3D
tomographic imaging techniques.
Sample (%) 2( )
Shabro et al. (2013) Eagle Ford 11.10 2.01 –
Chen et al. (2013) –
29.9
(intrakerogen)
1.84, 2.54, 2.65
(x,y,z directions)
–
Sun et al. (2017)
Silurian
Longmaxi
13.0–25.9 1.61–2.91 –
Peng et al. (2015) Barnett 3.25
2.6, 3.0, 4.2
(x,y,z directions)
–
Tahmasebi (2016) – 1.5–3.5 4.01, 3.76 –
Backeberg et
al.(2017)
– 2.2–5.6 – 9–39 (parallel)
100–1000 (perpendicular)
77
The effective molecular diffusivity within porous media is calculated by applying Eq. 4.25.
Three types of empirical tortuosity-porosity relations (Eqs. 4.27–4.28) with suggested parameters
for clay-rich type rocks (Atkins and Smith, 1961, Iversen and Jorgensen, 1993, Boudreau et al.,
1996) and Bruggeman theoretical equation (Eq. 4.30) with suggested value by Chen et al.
(2015) are used to estimate the tortuosity factor. The results are compared with the measured shale
tortuosity factor
Table 4.3 Calculated tortuosity factor and the ratio of effective diffusivity to bulk diffusivity at
different porosities (φ = 0.03, 0.05, and 0.10) by using tortuosity-porosity relations and measured
tortuosity (or tortuosity factor) from tomographic imaging techniques.
Tortuosity-porosity relations Measured or
1
nmA 1 (1 )B 1 lnC
at 03 0. 192–5×106 3.91 8.01 106–325 6.76–39
3 at 0.0effD
D
6×10-9–1.56×10-4 7.7×10-3 3.75×10-3 9.23×10-5–2.83×10-4
7.69×10-4–4.44×10-3
at 05 0. 89–5×105 3.85 6.99 54–140 9–39
5 at 0.0effD
D
6×10-8–5.62×10-4 1.30×10-2 7.15×10-3 3.57×10-4–9.26×10-4
7.69×10-4–5.56×10-3
at 10 0. 32–2.5×104 3.70 5.61 21–45 2.59–8.47
0 at 0.1effD
D
4×10-6–3.13×10-3 2.70×10-2 8.91×10-2 2.22×10-3–4.76×10-3
1.18×10-2–3.86×10-2
from 3D tomographic data. The calculated tortuosity factor and the ratio of effective diffusivity to
bulk diffusivity ( /effD D ) at three porosities ( ? .05,燼nd?.100.03, ) are tabulated in Table 4.3.
At the entire porosity range, power correlation (Eq.4.27) and the Bruggeman equation (Eq. 4.30)
overestimate the tortuosity factor and offer an underestimated /effD D . The linear correlation (Eq.
4.28) is insensitive to the porosity, yielding a smaller tortuosity factor and an overestimated
/effD D . The tortuosity factor from logarithmic correlation (Eq. 4.29) falls within the experimental
range but lies close to the lower limit at smaller porosities (0.03 and 0.05). The results indicate that
78
these tortuosity-porosity relations may characterize the clay-rich rock in some extent, owing to the
geometric heterogeneity and complexity of shales; however, more decent fitting parameters are
required while applying them to a specific shale-type rock. In view of the laboratory measurements
from 3D tomographic data, the larger matrix porosity generally yields a smaller measured
tortuosity. Due to inadequate shale samples and the heterogeneity of natural shales, it seems
impossible to derive a universal tortuosity-porosity correlation for all type of shales. Nevertheless,
we can still make a conclusion based on the selected shale samples in this study, i.e., the effective
diffusion coefficient in a porous shale rock is reduced by 102 to 104 times, as porosity decreased
from 0.1 to 0.03.
79
Chapter 5 Compositional Simulation Model
5.1 Mathematical Formulation
In this section, the mathematical models of the fully compositional simulator will be introduced,
including the law of mass conservation, the treatment of the source or sink term, the numerical
solutions in solving the multicomponent multiphase flow equations, and solver that is used to solve
the unknowns. The development of a compositional simulation model has also been described
Siripatrachai’s dissertation (Siripatrachai, 2016).
5.1.1 Material Balance Equaitons
The mole balance equation for component i is:
ppp N
j
jijj
N
j
ij
N
j
j
j
rj
jij St
Qkk
111
~~
(5.1)
where ij is the molar fraction of component i in phase j; j~ is the molar density of phase j
(lbmol/ft3); rjk is the relative permeability of phase j; j is the viscosity of phase j (cp); jS is the
saturation of phase j (cp); ijQ is the production/injection rate source term (lbmol/(ft3‧day)).
For water volume balance equation is:
w
w
b
ww
ww
rw
B
S
tV
q
B
kk
(5.2)
where rwk is the relative permeability of water phase; w is the viscosity of water phase (cp); wS is
the saturation of water phase (cp); wq is the water production/injection rate of the source term
(ft3/day), and bV is the bulk volume (ft3).
80
5.1.2 Source or sink term
Source or sink terms are used to represent the injection or production fluid from the grid block.
Here, Peaceman’s well model (1983) are considered and incorporated in the model.
5.1.2.1 Specification of production pressure
For a specified production pressure, the phase volumetric production rate for the perforated well
block is:
wfjjjijij PPWIq ~ (5.3)
The total production rate of component i in both oil and gas phases is:
pN
j
wfjjjiji PPWIQ1
~ (5.4)
where, ij is the molar fraction of component i in phase j. Well index (WI) accounts for the
geometric characteristics of the well and the reservoir properties around the well, and is used to
relate the well bottom-hole pressure (Pwf) and the well block pressure ( jP ).
For water:
wfwww PPWIq (5.5)
For a vertical well, the well index is defined as:
sr
r
zkWI
w
e
ave
ln
2 (5.6)
where, the geometric mean permeability ( avek ) is used to for anisotropic well block properties:
yxave kkk (5.7)
Based on Peaceman’s model, an equivalent well block radius ( er ) that is suitable for non-squares
well-block with anisotropic permeability is given by:
81
4/14/1
2
2/1
2
2/1
28.0
y
x
x
y
y
x
x
y
e
k
k
k
k
yk
kx
k
k
r (5.8)
It should be noted that the mobility of the fluid in the grid block is different for producer and
injector (Chappelear and Williamson, 1979), as given by:
injector or
producer or
,,block well
block well
fu
k
fu
k
wgoj j
rj
j
rj
j (5.9)
5.1.2.2 Specification of injection pressure
When gas is injected into the grid block at a specified injection pressure, the injection rate of
component i at reservoir condition is:
wfgg
inj
g
inj
i
inj
i PPWIyQ ~ (5.10)
where, inj
iy is the composition of the injected gas and is specified by the researchers, inj
g~ is the
molar density of the injection gas at the grid block pressure and reservoir temperature, g is the
mobility of the total phases from eq.5.9, and wfP is the well-block pressure.
For water, the injection rate at reservoir condition is:
wfww
w
inj
w PPWIB
q 1
(5.11)
Similarly, w is the mobility of the total phases from eq.5.9.
82
5.1.2.2 Specification of production rate
When the production rate of a phase is specified, the molar production of component i at surface
condition is:
pN
j
jjiji qQ1
~ (5.12)
The water production rate at surface condition is:
w
wscw
B
qq ,
(5.13)
5.1.2.4 Specification of injection rate
When the gas injection rate at surface condition is specified, the injection rate of component i at
reservoir condition is calculated as:
RT
Pq
yQ
sc
scinj
scg
inj
iinj
i ,615.5
(5.14)
5.1.3 Numerical Solution
The finite difference approach is applied to solve the mole balance equations for multi-components
mass balance (eq.5.1) and water volume balance (eq. 5.2). Transmissibility terms are used here to
measure how much fluid flows into or out of the grid block. A Jacobian matrix is established with
entries of partial derivatives of residuals with respect to the principal unknowns and linear solver
to solve the Jacobian matrix is used.
5.1.3.1 Finite difference approximation
The backward time central space (BTCS) finite difference method is used. For component i, the
mole balance equation can be expressed as:
83
pp
p
p
N
j
n
zyxjij
n
zyxjijb
N
j
ij
N
j
n
zyx
n
zyxzyx
rjzz
j
jj
c
in
zyx
n
zyxzyx
rjzz
j
jj
c
ij
n
zyx
n
zyxzyx
rjyy
j
jj
c
in
zyx
n
zyxzyx
rjyy
j
jj
c
ij
n
zyx
n
zyxzyx
rjxx
j
jj
c
in
zyx
n
zyxzyx
rjxx
j
jj
c
ij
N
j
n
zyxj
n
zyxjzyx
rjzz
j
j
ij
n
zyxj
n
zyxjzyx
rjzz
j
j
ij
n
zyxj
n
zyxjzyx
rjyy
j
j
ij
n
zyxj
n
zyxjzyx
rjyy
j
j
ij
n
zyxj
n
zyxjzyx
rjxx
j
j
ij
n
zyxj
n
zyxj
n
zyx
rjxx
j
j
ij
SSV
M
GGz
kkA
ug
gGG
z
kkA
ug
g
GGy
kkA
ug
gGG
y
kkA
ug
g
GGx
kkA
ug
gGG
x
kkA
ug
g
PPz
kkA
uPP
z
kkA
u
PPy
kkA
uPP
y
kkA
u
PPx
kkA
uPP
x
kkA
u
1
,,
1
,,
1
1
1
1,,
1
,,
2
1,,
1
,,
1
1,,
2
1,,
1
,1,
1
,,,
2
1,
1
,,
1
,1,,
2
1,
1
,,1
1
,,,,
2
1
1
,,
1
,,1,,
2
1
1
1
1,,
1
,,
2
1,,
1
,,
1
1,,
2
1,,
1
,1,
1
,,,
2
1,
1
,,
1
,1,,
2
1,
1
,,1
1
,,,,
2
1
1
,,
1
,,1
1
,,2
1
~~
615.5
~
144
~
144
~
144
~
144
~
144
~
144
~~
~~
~~
(5.15)
The convective transmissibility is extracted from eq. 5.15 and is used to describe the amount of
fluid exchange from the grid block that is driven by convective flow. The expressions of
transmissibility for x and y directions are:
1
,,2
1,,
2
1
1
,,2
1
1
,,2
1,
~
n
zyxrjij
zyx
xxn
zyxj
jn
zyxxij k
x
kA
uT
(5.16)
1
,2
1,,
2
1,
1
,2
1,
1
,2
1,
,
~
n
zyxrjij
zyx
yyn
zyxj
jn
zyxyij k
y
kA
uT
(5.17)
1
2
1,,
2
1,,
1
2
1,,
1
2
1,,
,
~
n
zyxrjij
zyx
yyn
zyxj
jn
zyxzij k
y
kA
uT
(5.18)
1
,,2
1,
1
,,2
1
1
,,2
1,144
1
n
zyxxij
n
zyxj
c
n
zyxxijG T
g
gT
84
By using the finite difference approximation, the water volume balance equation can be express
as:
n
zyx
w
on
zyx
w
obwell
w
n
zyx
n
zyxzyx
rwzz
ww
w
c
n
zyx
n
zyxzyx
rwzz
ww
w
c
n
zyx
n
zyxzyx
rwyy
ww
w
c
n
zyx
n
zyxzyx
rwyy
ww
w
c
n
zyx
n
zyxzyx
rwxx
ww
w
c
n
zyx
n
zyxzyx
rwxx
ww
w
c
n
zyxw
n
zyxwzyx
rwzz
ww
n
zyxw
n
zyxwzyx
rwzz
ww
n
zyxw
n
zyxwzyx
rwyy
ww
n
zyxw
n
zyxwzyx
rwyy
ww
n
zyxw
n
zyxwzyx
rwxx
ww
n
zyxw
n
zyxw
n
zyx
rwxx
ww
B
S
B
SVq
GGz
kkA
Bug
gGG
z
kkA
Bug
g
GGy
kkA
Bug
gGG
y
kkA
Bug
g
GGx
kkA
Bug
gGG
x
kkA
Bug
g
PPz
kkA
BuPP
z
kkA
Bu
PPy
kkA
BuPP
y
kkA
Bu
PPx
kkA
BuPP
x
kkA
Bu
,,
1
,,
1
1,,
1
,,
2
1,,
1
,,
1
1,,
2
1,,
1
,1,
1
,,,
2
1,
1
,,
1
,1,,
2
1,
1
,,1
1
,,,,
2
1
1
,,
1
,,1,,
2
1
1
1,,
1
,,
2
1,,
1
,,
1
1,,
2
1,,
1
,1,
1
,,,
2
1,
1
,,
1
,1,,
2
1,
1
,,1
1
,,,,
2
1
1
,,
1
,,1
1
,,2
1
615.5
144
1
144
1
144
1
144
1
144
1
144
1
11
11
11
(5.19)
5.1.3.2 Jacobian matrix
Flow equations for matrix domain is solved based on fully implicit formulation using the Newton-
Raphson method. Eq. 5.20 represents system of equations solved at every iteration level in matrix
domain. R represents residual. The principle unknowns are oil pressure (Po), water saturation (Sw),
and overall compositions of water and Nc‒1 other non-aqueous components. The total number of
unknowns are Nc+1.
RXJ (5.20)
The Jacobian is calculated as:
85
121
121
1
3
2
3
1
333
1
2
2
2
1
222
1
1
2
1
1
111
...
...
..................
...
...
...
Nc
www
w
w
o
w
Nc
NcNcNc
w
Nc
o
Nc
Ncwo
Ncwo
Ncwo
dZ
dR
dZ
dR
dZ
dR
dS
dR
dP
dR
dZ
dR
dZ
dR
dZ
dR
dS
dR
dP
dR
dZ
dR
dZ
dR
dZ
dR
dS
dR
dP
dR
dZ
dR
dZ
dR
dZ
dR
dS
dR
dP
dR
dZ
dR
dZ
dR
dZ
dR
dS
dR
dP
dR
J
(5.21)
The unknown vector of X is arranged as:
121 ,...,,, Ncwo ZZZSPX (5.22)
R is the vector of residuals of conservation equations and is expressed as:
wNc RRRRRR ,...,, 1321 (5.23)
The elements in the Jacobian matrix and residual vector is calculated by using numerical
differentiation.
5.1.3.3 Linear solver
In this study, the generalized minimal residual method (GMRES) as an iterative method is used to
get the numerical solution. GMRES is designed for nonsymmetrical linear systems. The solver
was developed by Lili Ju and John Burkardt, University of South Carolina. The incomplete LU
decomposition approach is implemented as a preprocessor.
5.1.4. Relative permeability
Stone’s Model II (Stone, 1973) is used as a predictor to estimate the relative permeabilities.
rgrwrg
rocw
rog
rw
rocw
row
rocw
ro kkkk
kk
k
k
k
k
(5.24)
86
wrw Sfk , grg Sfk , and woro SSfk , (5.25)
5.2 Phase Behavior Model
Predicting the behaviors of hydrocarbon in gas and liquid phase at reservoir condition is
required when modeling the compositional behavior.
5.2.1 Equation of state
The Peng-Robison Equation of State (PR-EOS) (Peng and Robinson, 1976)) is chosen as the
phase behavior model and is used in this in-house simulator. The purpose is to get the Z
(compressibility factor) of vapor or liquid phase by solving the cubic equation as follows.
0)]1([)]1()([]1)1[( 2
2121
2
21
2
21
3 BBmmABZBBmmBmmAZBmmZ
(5.26)
The detailed formulations about PR-EOS can be found in Appendix A.
5.2.2 Vapor-Liquid Equilibrium
The equilibrium ratios (Ki) is defined as the ratio of the mole fraction of component i in the
gas phase (yi) to the mole fraction of component i in the liquid phase (xi), given as:
i
ii
x
yK (5.27)
For a two-phase (oil and gas phases) system at equilibrium state, Rachford and Rice (1952)
suggested the following equation:
0)1(1
)1()(
1
cn
i ing
iing
Kf
Kcfg (5.28)
The above Rachford-Rice Objective Function is used to solve the gas mole fraction in the
mixture ( ngf ), given that the K-values are known. Generally, Newton Raphson iterative approach
87
is used to solve the gas fraction (ngf ). Newton Raphson method is fast but may lead to an
unphysically acceptable range. In that case, the Bisection method is used. The detailed procedure
of two methods are discussed in Appendix B.
Here, we use the Wilson’s correlation to provide an approximate value for initial Ki.
)
11)(1(37.5
1
ri
i
ri
iT
EXPP
K (5.29)
Since Wilson’s correlation only gives an approximate prediction for equilibrium ratios, more
reliable K-values are required by applying the thermodynamic equilibrium. At thermodynamic
equilibrium state, the chemical potentials for all the phases should be the same. In other words, the
fugacities of component i in oil and gas phases are equal, i.e.:
gili ff (5.30)
where, lif is the fugacity of component i in the liquid phase, and lif is the fugacity of component
i in the gas phase. To calculate the fugacity, the fugacity coefficient is introduced, which is defined
as the ratio of the fugacity of a component to its partial pressure. The fugacity coefficient of
component i for gas and liquid phases are given as:
Py
f
i
gi
gi (5.31)
Px
f
i
lili (5.32)
For a generalized cubic equation of state, the fugacity coefficient of component i for one phase is
calculated as:
88
)1(ln
2
)()ln(ln
1
21
21
ZB
B
BmZ
BmZ
B
B
A
cA
Bmm
ABZ ii
n
j
jij
i
c
(5.33)
where, Z-factor is the for the liquid and gas phases, and jc is the corresponding phase composition
To achieve the thermodynamic equilibrium, the successive substitution method (SSM) is
applied. The K-value is related to the fugacity coefficient of li and gi .
gi
li
i
i
gi
lii
f
f
x
yK
(5.34)
For SSM, we updated the K-values as follows:
n
gi
li
n
i
in
if
f
x
yK
1 (5.35)
n
gi
lin
i
n
if
fKK
1 (5.36)
The convergence is achieved when:
10
2
101
n
i gi
li
f
f (5.37)
5.2.3 Phase properties
Once the vapor fraction ( ngf ) and K-values are determined from SSM, the phase compositions
are calculated by using the following equations:
)1(1
ing
ii
Kf
cx (5.38)
)1(1
ing
iii
Kf
cKy (5.39)
89
We can calculate more important phase properties by applying the obtained phase compositions.
The calculations of molecular weight, density, viscosity, saturation of oil and gas phases are shown
in Appendix C.
5.3Validation results
In this section, we validate the developed in-house compositional simulator through cross-
checking the results CMG (GEM). The hydrocarbon components that were used include CH4, NC4,
NC7, and NC10, and their properties are tabulated in Table 5.1. The binary interaction parameters
are given in Table 5.2.
Table 5.1 The properties of hydrocarbon components for validation tests.
Mole
fraction Pc (psia) Tc (R)
Acentric
factor
Mw
(lb/lbmol)
Critical
volume
(ft3/lbmol) Parachor
CH4 0.2 667.1961 343.08 0.008 16.043 1.586 77
NC4 0.3 551.0981 765.36 0.193 58.124 4.085 189.9
NC7 0.3 396.791 972.36 0.351 100.205 6.921 312.5
NC10 0.2 305.6757 1111.68 0.49 142.286 9.66 433.5
Table 5.2 The binary interaction parameters of hydrocarbon components.
CH4 NC4 NC7 NC10
CH4 0 0 0 0
NC4 0 0 0 0
NC7 0 0 0 0
NC10 0 0 0 0
The relative permeability curves for validation tests are plotted in Figure 5.1.
90
Figure 5.1 The relative permeability data for validation tests.
The results from primary depletion, water injection and gas injection modeling are validated.
For water and gas injection schemes, two cases are tested, i.e., constant injection pressure and
constant injection rate. The reservoir conditions and production modeling design are given in Table
5.3.
Table 5.3 The reservoir conditions and production modeling design for validation tests.
Parameters value unit
reservoir dimensions 750 x 750 x100 ft
Reservoir temperature 100 ̊F
formation thickness 10 ft
porosity 0.1 fraction
water saturation 0.3 fraction
Primary depletion
initial reservoir pressure 3000 psia
production pressure 500 psia
reservoir permeability 5 md
Water Injection
Case 1: constant injection rate 50 bbl/day
Case 2: constant injection pressure 5000 psia
Gas Injection
initial reservoir pressure 2000 psia
production pressure 500 psia
Case 1: constant injection rate 5 Mscf/day
reservoir permeability 5 md
91
Case 2: constant injection pressure 2000 psia
reservoir permeability 0.5 md
For the primary depletion scheme, the reservoir is initially in single‒oil phase and
experiences phase change during the production process. The producer well block pressure, oil
production rate, gas production rate, and water production rate versus time are plotted in Figure
5.2.
Figure 5.2 Validation of primary depletion. For the producer well block: (a) well block pressure;
(b) oil production rate; and (c) gas production rate; and (d) water production rate.
For water injection scheme, the constant injection rate (50 bbl/day) and constant injection
pressure (5000 psia) are separately validated. The results for producer and injector are plotted in
92
Figure –Figure. We also plotted the water saturation distributions at different times in Figure and
Figure. From the plot, the water breaks through at 920 days and 910 days respectively.
Figure 5.3 Validation of water injection at constant injection rate of 50 bbl/day. For the producer
well block: (a) well block pressure; (b)water production rate; (c) oil production rate; and (d) gas
production rate.
93
Figure 5.4 Validation of water injection at constant injection rate of 50 bbl/day. For injector well
block: (a) well block pressure; (b) water injection rate
1 day 100 days 200 days 500 days 1000 days
Figure 5.5 Water saturation distributions at different times at constant injection rate of 50 bbl/day.
Sw
94
Figure 5.6 Validation of water injection at constant injection pressure of 5000 pisa. For the
producer well block: (a) well block pressure; (b)water production rate; (c) oil production rate; and
(d) gas production rate.
Figure 5.7 Validation of water injection at constant injection pressure of 5000 psia. For injector
well block: (a) well block pressure and (b) water injection rate.
1 day 100 days 200 days 500 days 1000 days
Figure 5.8 Water saturation distributions at different times at constant injection pressure of 5000
psia.
For gas injection scheme, the constant injection rate (5 Mscf/day) and constant injection
pressure (2000 psia) are separately validated. The results for producer and injector are plotted in
Figure5.9 –Figure 5.10. We also plotted the gas saturation distributions at different times for
constant gas injection rate and pressure distributions at different times for constant injection
Sw
95
pressure in Figure and Figure. From the plot, the gas breaks through at 2430 days for the scheme
of constant gas injection rate.
Figure 5.9 Validation of gas injection at constant injection rate of 5 Mscf/day. For the producer
well block: (a) well block pressure; (b)water production rate; (c) oil production rate; and (d) gas
production rate.
96
Figure 5.10 Validation of gas injection at constant injection rate of 5 Mscf/day. For injector well
block: (a) well block pressure; (b) gas injection rate
1 day 100 days 300 days 500 days 1000 days
1500 days 2000 days 2430 days
Figure 5.11 Gas saturation distributions at different times at constant gas injection rate of 5
Mscf/day.
Sg
97
Figure 5.12 Validation of gas injection at constant injection pressure of 2000 psia. For the producer
well block: (a) well block pressure; (b)water production rate; (c) oil production rate; and (d) gas
production rate.
Figure 5.13 Validation of gas injection at constant injection pressure of 2000 psia. For injector
well block: (a) well block pressure; (b) gas injection rate
1 day 100 days 500 days 800 days 1000 days
Figure 5.14 Pressure distributions at different times at constant gas injection pressure of 2000 psia.
Overall, the validation results for primary depletion, water injection, and gas injection show a
good match with commercial software.
P (psia)
98
5.4 Nano-confinement effects
In this section, the nano-confinement effects (critical property shift and capillary pressure effect)
on shale oil and shale gas production would be examined by using the developed fully
compositional simulator. The matrix permeability in the stimulated reservoir volume (SRV) is
increased caused by the fracking. The corresponding pore size should also be increased. However,
in previous studies, the enlargement of pore size in SRV is not considered while examine the nano-
confinement effects. In this study, the pore size variation in SRV will be considered by correlating
the pore size with permeability.
5.4.1 Critical property shift
In nanoscale porous media, a significant reduction of critical temperatures of several substances
have been observed in the experimental work of Morishige et al. (1997). Zarragoicoechea and Kuz
(2004) developed models to account for the shift of both critical pressure and critical temperature.
The effect of critical property shift on diffusion coefficient has been discussed in Section 4.4.1.
The equations of critical property shifts are given in Section 4.4.1, as shown in Eq.4.19-4.21. In
this section, the critical property shift as a function of pore radius will be reflected in the flash
calculation.
5.4.2 Oil–gas capillary pressure
The oil-gas capillary pressure effect on estimation of diffusion coefficient has been discussed
in Section 4.4.2. Meanwhile, the oil-gas capillary pressure effect on production has also been
investigated in Section 3.4 by using a reduced compositionally black-oil type model. By using
black oil model, the fluid properties with and without considering capillary pressure need to be
calculated prior to performing the reservoir simulation. The advantage is that we only need to run
the flash calculation once. The disadvantage is that for a heterogeneous reservoir with varied pore
99
size, the fluid properties as a function of all possible pore sizes are required to calculated prior to
simulation. Therefore, much more preliminary work is required compared to single pore size
scheme. In this section, a fully compositional model is used to examine the oil-gas capillary
pressure. The fluid properties as a function of pore size and oil-gas interfacial tension is updated
at each iteration. The oil-gas capillary pressure in both flash calculation and in fluid flow are
incorporated. The pore size is correlated to the permeability. The pore throat aperture as a function
of permeability that developed by Gao and Hu (2013) is used:
214.0log225.2log prk (5.40)
The wetting phase is assumed to be the wetting phase. Oil pressure is taken as the reference
phase pressure. The expressions of gas pressure and interfacial tension as a function of phase molar
density and phase compositions have been given in Eq. 4.22 and 4.23.
At thermodynamic equilibrium state, the fugacities of component i in the vapor and liquid
phases are identical, despite the fact that the phase pressures are not, i.e.,
NcggiNcoli yyyPTfxxxPTf ,...,,,,...,,, 2121 (5.41)
The fugacity coefficients of component i for liquid and vapor phases are defined as:
oi
lili
Px
f (5.42)
gi
gi
giPy
f (5.43)
The equilibrium constant K-value, defined as the 𝐾𝑖 = 𝑦𝑖/𝑥𝑖, in the successive substitution method
(SSM) is rewritten as:
n
o
g
n
gi
lin
i
n
o
g
n
gi
li
n
i
in
iP
P
f
fK
P
P
f
f
x
yK
1 (5.44)
100
An thermodynamic phase equilibrium state is reached until
14
2
101
n
i gi
li
f
f (5.45)
5.4.3 Simulation results
In this study, the Bakken crude oil (Nojabaei et al., 2013) and Marcellus shale gas (Bullin et
al., 2008) are used. For Marcellus shale gas, the reservoir fluid is single gas phase at the reservoir
temperature of 140 ̊F, as shown in Figure 4.6. The gas-oil capillary pressure is zero and only the
critical property shifts nano-confinement effect is taken into consideration.
5.4.3.1 Confinement effects on shale oil
For Bakken oil, the original eight components are lumped into five pseudo-components. The
reservoir temperature is 240 °F. The molar fraction of the five components at reservoir conditions
and the critical properties used in Peng-Robinson EOS are given in Table 5.4. The binary-
interaction parameters (BIP) of the five pseudo-components are recalculated use weight-based
grouping approach. Bubble point pressure is 2768 psia. The results are tabulated in Table 5.5.
Table 5.4 Compositions and parameters of Bakken oil.
Component Mole
fraction
Critical
Pressure (psia)
Critical
temperature (oR)
Acentric
factor
Mole
weight Parachor
C1 0.36736 655.02 335.34 0.01020 16.54 74.8
C2-C3 0.24219 681.05 594.68 0.12176 35.70 124.7
C4-C6 0.12157 501.58 820.48 0.23103 68.75 221.5
C7-C12 0.15854 363.34 1053.25 0.42910 120.56 350.2
C13-C80 0.11034 229.64 1504.10 0.81953 295.51 800.4
Table 5.5 Binary interaction coefficients of Bakken oil.
C1 C2-C3 C4-C6 C7-C12 C13-C80
C1 0 0.0044 0.0036 0.0033 0.0033
C2-C3 0.0044 0 0.0019 0.0016 0.0016
101
C4-C6 0.0036 0.0019 0 0 0
C7-C12 0.0033 0.0016 0 0 0
C13-C80 0.0033 0.0016 0 0 0
A reservoir domain with 15 by 15 grid blocks is used. The producer is located at the center of
the reservoir domain. The initial reservoir pressure is 2500 psia and the production pressure is 500
psia. The initial water saturation is 0.25. The total production time is 1000 days. We increase the
matrix permeability near the production well to account for the stimulated reservoir volume (SRV)
in the reservoir that is caused by hydraulic fractures. The reservoir permeability map is shown in
Figure 5.15. Here, the matrix permeability is 0.05 md. The Middle Bakken formation is considered
as the prototype, and the reservoir is not ultra-tight, but only tight. The mass transfer is completely
convection-dominated as the permeability is relatively large.
Figure 5.15 Reservoir permeability map for tight oil-rich reservoir.
Two schemes of reservoir pore size are implemented, as plotted in Figure 5.16 (a) and (b). In
Figure 5.16 (a), a constant pore size is used for the entire reservoir, and this is also how we treated
the confinement effects in a reservoir domain by using the black oil type model. The fixed pore
throat size of 19 nm is from a Bakken sample rock (Nojabaei et al., 2013). In Figure 5.16 (b), the
102
pore size in the SRV is increased with the increased corresponding matrix permeability. According
to the correlation between the pore size and matrix permeability as given in Eq. 5.40, the pore size
is proportional to the matrix permeability.
(a) (b)
Figure 5.16 Reservoir pore size map for tight oil-rich reservoir. (a) constant pore size ( nmrp 19 )
and (b) pore size proportional to permeability.
First, the capillary pressure effect on Bakken oil production is examined. The Cumulative oil
and cumulative gas production without confinement effect and with capillary pressure effect by
using constant radius (blue dotted line) and using the radius that is proportional to the permeability
(red dash-dotted-dotted line) are plotted in Figure 5.17 (a) and (b), respectively.
103
Figure 5.17 (a) Cumulative oil and (b) cumulative gas production without confinement effect and
with capillary pressure effect by using constant radius ( nmrp 19 ) and using the radius that is
proportional to the permeability.
As shown in Figure 5.17, inclusion of capillary pressure effect significantly increases the oil
production, but suppresses the gas production no matter which pore size scheme is used. This result
is consistent with the finding from the literature (Du et al., 2020). For the scheme that uses fixed
pore size, the effect of capillary pressure on increasing the oil production or decreasing the gas
production is more obvious. As is expected, for the scheme of the pore throat radius that is
104
proportional to the permeability, the effect of capillary pressure on production is much reduced,
especially in the early period where oil or gas is mainly produced from the high permeable SRV.
1 day 100 days 200 days 500 days 1000 days
(a)
1 day 100 days 200 days 500 days 1000 days
(b)
Figure 5.18 Oil-gas capillary pressure distributions at different production times by using (a)
constant pore size ( nmrp 19 ) and (b) pore size proportional to permeability.
We visualized the distribution of oil-gas capillary pressure in the reservoir domain at different
production times for both pore size schemes, as shown in Figure 5.18. For fixed pore size, i.e.,
Figure 5.18 (a), the oil-gas capillary pressure in the SRV could reach up to 400 psi. In Figure 5.18
(b), the oil-gas capillary pressure is very small owing to the very large pore size in the SRV.
1 day 100 days 200 days 500 days 1000 days
Pc (psia)
105
(a)
1 day 100 days 200 days 500 days 1000 days
(b)
Figure 5.19 Oil-gas interfacial tension at different grid blocks at different production times by
using (a) constant pore size ( nmrp 19 ) and (b) pore size proportional to permeability.
We also visualized the oil-gas interfacial tension for both pore size schemes, as plotted in
Figure 5.19. For fixed pore size, the oil-gas interfacial tension becomes larger as the pressure
decreases with time. This explains the increasing oil-gas capillary pressure for fixed pore size
scheme. For the pore size that is proportional to the permeability, the oil-gas interfacial tension is
also increasing as pressure decreases, but the oil-gas capillary pressure in the SRV is still very
small since it still cannot compensate loss caused by the very large pore size.
IFT
(dyne/cm)
106
Figure 5.20 (a) Cumulative oil and (b) cumulative gas production without confinement effect and
with critical property shift effect by using constant pore size ( nmrp 19 ) and using the pore size
that is proportional to the permeability.
In this section, we also examine the effect of critical property shift on Bakken oil production.
In Figure 5.20 (a) and (b), the cumulative oil production and gas production without confinement
effect and with critical property shift effect by using two pore sizes schemes are plotted,
respectively. Oil production is significantly increased with considering critical property shift.
Similarly, the rise of oil production using varied pore size lags behind the scheme using fixed pore
size. This is because the effect of critical property shift is negligible in early production period
when the fluid is produced from the SRV with large pore size. In figure 5.20 (b), including critical
property shift decreases the gas production slightly for the constant pore size scheme. The effect
of critical property shifts on gas production is very small and almost negligible for the scheme of
using pore size proportional to permeability.
107
Figure 5.21 (a) Cumulative oil and (b) cumulative gas production without confinement effect and
with both confinement effects by using constant pore size ( nmrp 19 ) and using the pore size that
is proportional to the permeability.
Finally, both capillary pressure effect and critical property shift are considered in flash and in
flow. The nano-confinement effects on oil and gas production are examined at two pore size
schemes. The results are plotted in Figure 5.21. A combination effect of capillary pressure effect
and critical property shift is reflected on oil and gas recovery.
108
Table 5.6 Cumulative oil and gas production without confinement effects and with confinement
effects by using constant pore size ( nmrp 19 ) and using the pore size that is proportional to the
permeability.
Confinement effects Qo
(bbl)
increased
Qo (%)
Qg
(Mscf)
increased
Qg (%)
constant
radius
no confinement 1155.05 – 8482.66 –
shift 1517.44 31.37 8532.95 0.59
Pc 2003.31 73.44 7549.82 -11.00
shift+Pc 2469.60 113.81 7836.82 -7.61
radius
proportional
to k
no confinement 1155.05 – 8482.66 –
shift 1426.64 23.51 8379.06 -1.22
Pc 1474.36 27.64 7922.28 -6.61
shift+Pc 1850.46 60.21 7955.46 -6.22
The increased or decreased percentage of oil and gas production after considering confinement
effects are calculated and tabulated in Table 5.6. In summary, both critical property shift and
capillary pressure effect increases the oil production. For the case of considering capillary pressure,
when the pore size is proportion to the permeability in the SRV, the increased oil production is
half of the increased oil production while using constant pore size. However, for the case of
including critical property shift, the increased oil production differs not that much for both pore
size schemes.
5.4.3.1 Confinement effects on shale gas production
For Marcellus shale gas, the compositions have been given in Table 4.1. The binary interaction
coefficients are given in Table 5.7. The reservoir permeability map is shown in Figure 5.22 and
the matrix permeability is 0.00005 md. Again, the matrix permeability near the production well is
increased to account for the stimulated reservoir volume (SRV) in the reservoir that is caused by
hydraulic fractures.
Table 5.7 Binary interaction coefficients of Marcellus shale gas
109
CO2 C1 C2 C3 N2
CO2 0 0.1 0.13 0.135 -0.02
C1 0.1 0 0 0 0.036
C2 0.13 0 0 0 0.05
C3 0.135 0 0 0 0.08
N2 -0.02 0.036 0.05 0.08 0
Figure 5.22 Reservoir permeability map with matrix permeability as 0.00005 md.
For Marcellus shale gas, the reservoir fluid is in single gas phase. The gas-oil capillary pressure
is zero and only the critical property shifts is taken into consideration. In Figure 5.23, the
cumulative gas production with critical property shift is examined under two pore size schemes.
Similar to the gas production of Bakken oil, including critical property shift reduces the gas
production very slightly.
110
Figure 5.23 Cumulative gas production without confinement effect and with critical property shift
effect by using constant pore size ( nmrp 19 ) and using the pore size that is proportional to the
permeability.
5.5 Molecular diffusion effect
A recent molecular simulation study indicated that, at pore size of 10 nm or even 5 nm,
molecular diffusion still occurs in those nano-pores (Wang et al., 2016). In such ultra-tight low
permeability reservoirs, molecular diffusion is comparable to the convection viscous flow and
mass transfer is expected to be diffusion-dominated (Cronin et al., 2019). In many gas injection
studies of shale reservoirs, different attempts have been made to examine the role of molecular
diffusion in gas injection process for EOR/EGR. A review study revealed that the molecular
diffusion effect on improving shale oil and gas recovery is highly sensitive to the employed
diffusion coefficient (Du and Nojabaei, 2019). Owing to the lack of reference data, most studies
assumed a diffusivity based on the literature. In this section, we will examine the role of molecular
diffusion in huff-n-puff Bakken oil production and Marcellus shale gas production. The molecular
diffusion coefficient is calculated by using the Sigmund correlation (Sigmund et al., 1976a;1976b).
The governing equation of Eq. 5.1 is modified by adding the diffusive term, as shown below:
111
ppp p N
j
jijj
N
j
ij
N
j
N
j
ijj
j
rj
jij St
QJkk
111 1
~~
(5.46)
The Fick’s classical model is used and the diffusion flux is expressed as:
jijijjij DSJ ~ (5.47)
The backward time central space (BTCS) finite difference method is used. For component i,
the mole balance equation with considering molecular diffusion can be expanded as:
pp
p
p
p
N
j
n
zyxjij
n
zyxjijb
N
j
ij
N
j
n
zyx
n
zyxzyx
rjzz
j
jj
c
in
zyx
n
zyxzyx
rjzz
j
jj
c
ij
n
zyx
n
zyxzyx
rjyy
j
jj
c
in
zyx
n
zyxzyx
rjyy
j
jj
c
ij
n
zyx
n
zyxzyx
rjxx
j
jj
c
in
zyx
n
zyxzyx
rjxx
j
jj
c
ij
N
j
n
zyxjij
n
zyxjijzyx
zjijn
zyxjij
n
zyxjij
n
zyx
zjij
n
zyxjij
n
zyxjijzyx
yjijn
zyxjij
n
zyxjij
n
zyx
yjij
n
zyxjij
n
zyxjijzyx
xjijn
zyxjij
n
zyxjij
n
zyx
xjij
N
j
n
zyxj
n
zyxjzyx
rjzz
j
j
ij
n
zyxj
n
zyxjzyx
rjzz
j
j
ij
n
zyxj
n
zyxjzyx
rjyy
j
j
ij
n
zyxj
n
zyxjzyx
rjyy
j
j
ij
n
zyxj
n
zyxjzyx
rjxx
j
j
ij
n
zyxj
n
zyxj
n
zyx
rjxx
j
j
ij
SSV
M
GGz
kkA
ug
gGG
z
kkA
ug
g
GGy
kkA
ug
gGG
y
kkA
ug
g
GGx
kkA
ug
gGG
x
kkA
ug
g
z
ASD
z
ASD
y
ASD
y
ASD
x
ASD
x
ASD
PPz
kkA
uPP
z
kkA
u
PPy
kkA
uPP
y
kkA
u
PPx
kkA
uPP
x
kkA
u
1
,,
1
,,
1
1
1
1,,
1
,,
2
1,,
1
,,
1
1,,
2
1,,
1
,1,
1
,,,
2
1,
1
,,
1
,1,,
2
1,
1
,,1
1
,,,,
2
1
1
,,
1
,,1,,
2
1
1
1
1,,
1
,,
2
1,,
1
,,
1
1,,
1
2
1,,
1
,1,
1
,,,
2
1,
1
,,
1
,1,
1
,2
1,
1
,,1
1
,,,,
2
1
1
,,
1
,,1
1
,,2
1
1
1
1,,
1
,,
2
1,,
1
,,
1
1,,
2
1,,
1
,1,
1
,,,
2
1,
1
,,
1
,1,,
2
1,
1
,,1
1
,,,,
2
1
1
,,
1
,,1
1
,,2
1
~~
615.5
~
144
~
144
~
144
~
144
~
144
~
144
~~
615.5
~~
615.5
~~
615.5
~~
615.5
~~
615.5
~~
615.5
~~
~~
~~
(5.48)
112
The diffusive transmissibility that extracted from the above equation is used to describe the
amount of fluid exchange from the grid block driven by molecular diffusion. The expressions of
diffusive transmissibility for x, y, and z directions are:
1
,,2
1
1
,,2
1
1
,,2
1,615.5
n
zyx
xn
zyxj
n
zyxij
ij
xijx
AS
DTD (5.49)
1
,2
1,
1
,2
1,
1
,2
1,
,615.5
n
zyx
yn
zyxj
n
zyxij
ij
yijy
AS
DTD (5.50)
1
2
1,,
1
2
1,,
1
2
1,,
,615.5
n
zyx
zn
zyxj
n
zyxij
ij
zijz
AS
DTD (5.51)
The huff-n-puff gas injection scheme is used to examine the diffusion effect by injecting CO2
into Bakken oil and into Marcellus shale gas.
5.5.1 Molecular diffusion in Bakken oil
The binary interaction coefficient of CO2 with Bakken oil are tabulated in Table 5.8. The
reservoir is a square domain and the fracture caused stimulated reservoir volume is in the center.
The reservoir permeability map is shown in Figure 5.24. The pore size that is correlated to
permeability is used in this section.
Table 5.8 Binary interaction coefficient of Bakken oil with CO2.
CO2 C1 C2-C3 C4-C6 C7-C12 C13-C80
CO2 0 0.1 0.13 0.125 0.1 0.08
C1 0.1 0 0.0044 0.0036 0.0033 0.0033
C2-C3 0.13 0.0044 0 0.0019 0.0016 0.0016
C4-C6 0.125 0.0036 0.0019 0 0 0
C7-C12 0.1 0.0033 0.0016 0 0 0
C13-C80 0.08 0.0033 0.0016 0 0 0
113
Figure 5.24 Reservoir permeability map with matrix permeability as 0.001 md.
The single well is located at the center. After primary depletion for 150 days, CO2 is injected
into the reservoir at injection rate of 5 Mscf/day for 30 days, followed by 15 days of shut-in period.
The cumulative oil and cumulative gas production without molecular diffusion (black line) and
with molecular diffusion (blue dotted line) are plotted in Figure 5.25. It can be seen that including
molecular diffusion almost have no influence on production. To examine the employed diffusion
coefficient on production performance, we enlarging the molecular diffusion effect by multiplying
the diffusion coefficient by 10 times (red dash-dotted line) and 100 times (yellow dash line). It can
be seen that a visible increase in gas production occurs only after the diffusion coefficient is
increased by 100 times. The results revealed that the molecular diffusion effect on improving shale
oil and gas recovery is highly sensitive to the employed diffusion coefficient.
114
Figure 5.25 (a) Cumulative oil and (b) cumulative gas production without molecular diffusion and
with molecular diffusion by multiplying diffusion coefficient by 1, 10 and 100 times when the
matrix permeability is 0.001 md.
A sensitivity analysis of matrix permeability effect on production is also performed by reducing
the matrix permeability to 0.00005 md, as shown in Figure 5.26. The cumulative oil and
cumulative gas production without molecular diffusion and with molecular diffusion by
multiplying diffusion coefficient by 1, 10 and 100 times are plotted in Figure 5.27. Again,
noticeable increase in oil and gas production happens only after the diffusion coefficient is
115
increased by 100 times. The results further revealed that even in very tight formation, convective
viscosity flow is still dominated. To get a reliable molecular diffusion coefficient is very important
on analyzing the role of diffusion in gas injection process.
Figure 5.26 Reservoir permeability map with matrix permeability as 0.00005 md.
116
Figure 5.27 (a) Cumulative oil and (b) cumulative gas production without molecular diffusion and
with molecular diffusion by multiplying diffusion coefficient by 1, 10 and 100 times when the
matrix permeability is 0.00005 md.
In addition, we also increase the number of huff-n-puff circles to investigate the importance of
molecular diffusion during huff-n-puff gas injection process. The matrix permeability of 0.00005
md is used. The cumulative oil production, cumulative gas production, and well block pressure
without molecular diffusion and with molecular diffusion by multiplying diffusion coefficient by
1, 10 and 100 times are plotted in Figure 5.28. Similarly, a significant increase in oil and gas
production happens for the case of diffusion coefficient increased by 100 times.
117
Figure 5.28 (a) Cumulative oil production, (b) cumulative gas production, and (c) well block
pressure with two huff-n-puff circles without molecular diffusion and with molecular diffusion by
multiplying diffusion coefficient by 1, 10 and 100 times when the matrix permeability is 0.00005
md.
Table 5.9 The increased percentage of oil and gas production of Bakken oil after considering
molecular diffusion.
k (md)
number
of
circles
no diffusion D*1 D*10 D*100
Qo
(bbl)
Qg
(Mscf)
increased increased increased increased increased increased
Qo (%) Qg (%) Qo (%) Qg (%) Qo (%) Qg (%)
0.001 1 373.91 2624.93 0 0.05 0.01 0.48 1.5 4.61
0.00005 1 279.74 2119.48 0.04 0.09 0.41 0.93 3.91 9.05
118
0.00005 2 260.65 2135.69 0.08 0.03 0.48 0.7 4.21 7.13
The increased percentage of oil and gas production of Bakken oil after considering molecular
diffusion are calculated. The results are tabulated in Table 5.9. As shown in this table,
incorporating molecular diffusion effect could increase the production very slightly. After
multiplying the diffusion coefficient by 10 times or 100 times, the increased production caused by
molecular diffusion is visible. Increasing the number of cycles can slight increase the oil
production.
5.5.2 Molecular diffusion in Marcellus shale gas
A huff-n-puff gas injection scheme is used to examine the diffusion effect in Marcellus shale
gas production. The reservoir permeability map in Figure 5.29 is used. After primary depletion for
150 days, CO2 is injected into the reservoir at injection rate of 5 Mscf/day for 15 days, followed
by 15 days of soaking time.
Figure 5.29 Cumulative gas production without molecular diffusion and with molecular diffusion
by multiplying diffusion coefficient by 1, 10 and 100 times when the matrix permeability is
0.00005 md.
119
The cumulative gas production without molecular diffusion and with molecular diffusion by
multiplying diffusion coefficient by 1, 10 and 100 times are plotted in Figure 5.26. The results
show that a noticeable increase in oil and gas production happens only after the diffusion
coefficient is increased by 10 times. When diffusion coefficient is enlarged by 100 times, a sharp
increase of gas production happens.
Table 5.10 The increased percentage of gas production of Marcellus gas after considering
molecular diffusion.
k (md)
no diffusion D*1 D*10 D*100
Qg (Mscf) increased
Qg (%)
increased
Qg (%)
increased
Qg (%)
0.00005 7119.23 0.56 5.20 36.17
The increased percentages of gas production of Marcellus shale gas after including molecular
diffusion are calculated in Table 5.10. Considering molecular diffusion increases the gas
production. An obvious increase in gas production occurs after the diffusion coefficient is
increased by 10 times.
120
Chapter 6 Conclusions
In this research, first, a black-oil type reservoir simulation method is used to simulate near-
miscible and immiscible produced gas injection enhanced oil recovery. The model allows for gas
injection below and above the critical pressure while black-oil fluid properties are extrapolated
above the original bubble-point pressure. This reduced compositionally black-oil method is
capable of capturing compositional changes of reservoir fluid due to gas injection, and provides a
fast and robust alternative for large-scale reservoir simulation with the purpose of flaring/venting
reduction through reinjecting the produced gas into the reservoir for EOR. Then, the diffusivities
of three types of shale fluids (gas, condensate, crude oil) at reservoir conditions are estimated using
empirical correlations. A gradual increase in CO2 composition in fluid systems accounts for the
gas injection process for EOR/EGR. For the first time, the effect of nano-pores confinement,
including large gas-oil capillary pressure and critical property shifts, on diffusivity is examined.
Meanwhile, the tortuosity factor from laboratory measurements and empirical correlations are used
to characterize the diffusion behavior in porous media. Finally, a fully compositional simulation
model is developed. Nano-confinement effects and molecular diffusion effect are examined on
Bakken oil and Marcellus shale gas production.
6.1 Summary and conclusions
The following conclusions are drawn from this study:
1. In gas flooding schemes, miscible displacement maintains a constant high recovery rate
and reaches maximum recovery in a short period of time; immiscible displacement
maintains the same oil recovery rate but it dramatically decreases after gas breaks through;
121
2. At miscible or near miscible condition, huff-n-puff gas injection is more effective because
more gas dissolves into the oil and dilutes the crude oil;
3. Huff-n-puff gas injection is not effective for the reservoirs that are saturated before gas
injection is started;
4. Using different black oil properties from different gas adding approaches only affects the
results when Pc in flash is included, and the difference is not significant;
5. In tight oil-rich reservoirs, inclusion of high capillary pressure in flash calculation can
significantly increase oil recovery; however, at miscible or near-miscible conditions, the
influence is reduced owing to the low oil-gas IFT;
6. Using IFT-dependent oil-gas relative permeability data can increase the oil recovery in
saturated reservoirs.
7. A validation of empirical correlation indicates that Sigmund correlation is superior to
Wilke‒Chang and Hayduk‒Minhas correlations in terms of predicting the diffusivity of
shale fluid systems at reservoir conditions;
8. The diffusivity of component in gas phase is almost two orders of magnitude larger than in
liquid phase for Bakken oil and is one order of magnitude larger than in liquid phase for
Marcellus shale condensate;
9. For Marcellus shale gas, there is only gas phase at reservoir condition, so no gas-oil
capillary pressure exists. Including critical property shifts could increase CO2 diffusivity
up to 28.0%;
10. For shale condensate, when two phases co-exist, large gas-oil capillary pressure could
decrease the diffusivity in gas phase up to 19.1%, but this reduction in diffusivity
diminishes at higher pressures where single gas phase exists only. Including critical
122
property shifts could slightly decrease the diffusivity in gas phase when two phases co-
exist, but increases diffusivity when single phase gas exists only;
11. For Bakken shale oil, in gas phase, as both nano-confinement effects are included, the
capillary pressure-induced reduction effect on diffusivity is more obvious. In liquid phase,
both capillary pressure effect and critical property shifts increase the diffusivity, but the
effect reduces with continuous CO2 injection;
12. The smaller the pore size is, the more significant the nano-confinement effects on
diffusivity are;
13. Owing to the geometric heterogeneity and complexity of shales, more suitable lithology-
dependent parameters are needed while applying tortuosity-porosity relations to a specific
shale;
14. Based on the shale tortuosity from 3D imaging data, the effective diffusion coefficient in
porous shale rock is reduced by 102–104 times as porosity decreases from 0.1 to 0.03;
15. Including capillary pressure effect increases the oil production but decreases the gas
production; Including critical property shift increase the oil production and slightly
decrease the gas production;
16. The molecular diffusion effect is negligible during Bakken oil or Marcellus shale gas huff-
n-puff production. An obvious increase in oil and gas production happens only after the
diffusion coefficient is multiplied by 10 or 100 times.
123
6.2 Future research
6.2.1 Slim tube simulation to estimate MMP as a function of permeability and fluid
compositions.
One future work is to do the slim tube simulation to estimate MMP as a function of permeability
and composition for different shale oil fluids. Here, the developed fully compositional simulator
will be used. In this work, different solvents (such as CO2, C1, and produced gas) will be injected
into different shale fluids. Meanwhile, the molecular diffusion will be included in the flow
equation. The importance of MMP in very tight shale formation will be investigated. The influence
of permeability, reservoir and injecting gas type, as well as the type of gas drive mass transfer will
be examined.
6.2.2 Inclusion of adsorption behavior in the compositional model to investigate CO2
injection in shale gas in nano-sized pores.
Another future work is to incorporate adsorption behavior in the fully compositional model. Then
we can evaluate the CO2 injection in shale gas reservoir for enhanced gas recovery (EGR) and
greenhouse gas control. Furthermore, confinement effects in shale gas reservoir could be examined
again while considering CO2 adsorption capability and CH4 desorption behavior.
6.2.3 To develop an Embedded Discrete Fracture Model (EDFM).
Another next step to further extend our code to solve more complicated and realistic problems is
to develop an embedded discrete fracture model (EDFM) and couple with the fully compositional
model.
124
APPENDIX A EQUATION OF STATE
Z-factor form of the generalized EOS:
0)]1([)]1()([]1)1[( 2
2121
2
21
2
21
3 BBmmABZBBmmBmmAZBmmZ
(A.1)
where:
c cn
i
n
j
ijji AccA (A.2)
5.0))(1( jiijij AAA (A.3)
2
25.00 )]1(1[ri
ririiaii
T
PTmA (A.4)
cn
i
ii BcB (A.5)
ri
rio
biiT
PB (A.6)
ci
riP
PP (A.7)
ci
riT
TT (A.8)
where ij is the binary interaction coefficient between the component i and j, P is pressure (psia),
T is temperature (R), and v is molar volume (ft3/lbmol)
For SRK EOS
2176.0574.148.0 iiim (A.9)
For PR EOS:
0.49 016666.0164423.048503.1379642.0
0.49 26992.054226.1374640.032
2
iiii
iii
iif
ifm
(A.10)
125
The definition of m1, m2,o
ai , and o
bi for different types of equation of state are is defined as:
Equation of state m1 m2 o
ai o
bi
P-R EOS 21 21 0.457235529 0.077796074
SRK EOS 0 1 0.4274802 0.08664035
126
APPENDIX B VAPOR-LIQUID EQUILIBRIUM
The Rachford-Rice Objective Function is used to solve the gas mole fraction in the mixture
( ngf ).
0)1(1
)1()(
1
cn
i ing
iing
Kf
Kcfg (B.1)
The above equation is a non-linear equation with one known. Generally, Newton Raphson
iterative approach is used to solve the gas fraction ( ngf ).
)(
)(old
ng
old
ngold
ng
new
ngfg
fgff
(B.2)
During this iterative procedure, convergence is achieved as
old
ng
new
ng ff (B.3)
It should be noted that a “negative flash” is implemented by allowing the obtained ngf to be
a negative value and the interval of ngf is:
)1(
1
)1(
1
minmax i
ng
i Kf
K
(B.4)
The compositions component i in liquid phase (xi) and in gas phase (yi) can be calculated as:
)1(1
ing
ii
Kf
cx (B.5)
)1(1
ing
iii
Kf
cKy (B.6)
Newton Raphson method is fast but may lead to an unphysically acceptable range. In that
case, the Bisection method is used. For the Bisection method, the initial upper and lower values
are:
0Ufng ; (B.7)
127
1Lfng; (B.8)
The updated ngf is
LfUff ngng
new
ng (B.9)
The updated upper and lower value becomes:
new
ngng fUf for 0new
ngfg (B.10)
new
ngng fLf for 0new
ngfg (B.11)
Similarly, the iterative procedure stops when
old
ng
new
ng ff (B.12)
128
APPENDIX C PHASE PROPERTIES
C.1 Molecular weight
The molecular weight of vapor and liquid phases are calculated by using weighted average.
n
i
iig MWyMW1
(C.1)
n
i
iil MWxMW1
(C.2)
C.2 Oil and gas densities
The density of phase ‘j’ is calculated by using its phase molecular weight and compressibility
factor ( jZ ) that is predicted from Peng-Robison equation of state.
j
jj
jZ
MW
RT
P (C.3)
C.3 Oil and gas viscosity
C.3.1 Viscosity of gas phase
The viscosity of gas phase is calculated by using the correlation by Lee, Gonzalez and Eakin
(1966), as follows:
vy
g
vvg xEXPk4.62
10 4
(C.4)
Where:
TMW
TMWk
g
g
v
19209
02.04.9 5.1
(C.5)
vv xy 2.04.2 (C.6)
129
gv MWT
x 01.0986
5.3 (C.7)
C.3.2 Viscosity of liquid phase
Lohrenz, Bray and Clark (1964) presented an empirical correlation to predict the viscosity of a
liquid hydrocarbon mixture, is given below:
444321 100093324.0040758.0058533.0023364.01023.0 rrrrml (C.8)
l is the liquid viscosity (cp); m is the viscosity at atmospheric pressure(cp-1); r is the reduced
liquid density, and is the viscosity at atmospheric pressure (cp) and is calculated by Herning
& Zipperer equation.
i
ii
i
iii
MWx
MWx
(C.9)
i is the viscosity of component i at low pressure and is suggested by Stiel and Thodos.
5.1 )67.158.4(78.17
5.1 10.34
625.05
94.05
TriforT
TriforT
i
ri
i
ri
i
(C.10)
i is the viscosity parameter of the i-th component given by:
3/2
6/14402.5
pci
pc
iPMW
T (C.11)
pcT is the pseudocritical temperature (R); pcP is the pseudocritical pressure (psia); MWl is the
molecular weight of the liquid phase (lbm/lbmol).
The reduced density of liquid mixture ( r ) is calculated from Lohrentz et al. (1969):
130
pc
l
l
pc
lr V
WM
(C.12)
pc is the pseudocritical density of the liquid phase(lbm/ft3) and Vpc is the pseudocritical volume
of the liquid per unit mole (ft3/lbmol). All mixture pseudocritical properties are calculated using
Kay’s mixing rule, as below:
ciipc TxT (C.13)
ciipc PxP (C.14)
ciipc VxV (C.15)
C.4 Saturation
Gas saturation in reservoir can be calculated from vapor and liquid mole fractions by flash
calculation. Oil saturation can be obtained based on the fact that the sum of oil, gas, and water
saturation is equal to 1.
onggng
gng
w
onogng
gng
w
og
g
wgvfvf
vfS
vfvf
vfS
VV
VSS
)1(111
(C.16)
owg SSS 1 (C.17)
C.5 Water properties
The density of water is calculated as:
)(1 ,refwww
o
ww PPC (C.18)
131
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