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A study of diffusive mass transfer in tight dual-porosity systems (unconventional)

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A study of diffusive mass transfer in tight dual-porosity systems (unconventional)

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Content A Study of Diffusive Mass Transfer in Tight Dual-Porosity Systems (Unconventional)
by Saeed Alahmari
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirement for the Degree
DOCTOR OF PHILOSOPHY
(PETROLEUM ENGINEERING)
August 2024
Copyright 2024 Saeed Alahmari

ii
Dedication
To my parents may their souls rest in peace.

iii
Acknowledgements
I would like to convey my deepest appreciation to anyone who has encouraged and supported me
throughout my graduate school journey: My everlasting gratitude goes to my parents may their
souls rest in peace.
I express my sincerest thanks to my PhD advisor, Professor, Kristian Jessen, for his guidance,
mentorship, communication, and encouragement. His impact on me reaches far beyond my PhD
work as I have learned from him so many great life lessons during our frequent meetings and
discussions. Special thanks to Dr. Ghaithan A. Al-muntasheri who paved the way for me to conduct
experiments at Aramco Americas Research Center in Houston.
I also extend my appreciation to Professors Iraj Ershaghi, Donald Paul, Felipe de Barros, Birendra
Jha and Anuj Gupta who helped me with their valuable comments and recommendations that
significantly improved the quality of my doctoral dissertation.
I am grateful to my friends in the greater Los Angeles area with whom I have enjoyed amazing
memories. My special thanks go to Mohammed Raslan and Ye Lyu for their constant support and
encouragement.
My PhD work could not have been possible without the outstanding support from my employer,
Saudi Aramco: I am extremely thankful and proud to be part of this company that is leading the
energy industry.
Last but not least, my eternal thanks to my wife, Maureen, who has always been beside me emitting
infinite love, patience, and support.

iv
Table of Contents
Dedication....................................................................................................................................... ii
Acknowledgements........................................................................................................................iii
List of Tables................................................................................................................................. vii
List of Figures.............................................................................................................................. viii
Abstract........................................................................................................................................ viii
Chapter 1 Introduction .................................................................................................................... 1
1.1 Background ........................................................................................................................... 1
1.2 Motivation and Efforts.......................................................................................................... 3
1.3 Manuscript Organization....................................................................................................... 6
Chapter 2 Literature Review........................................................................................................... 8
2.1 Diffusive Mass Transfer in Unconventional Reservoirs ....................................................... 8
2.2 Measurement of Diffusion Coefficients...............................................................................11
2.3 Correlation of Diffusion Coefficients.................................................................................. 13
2.3.1 Infinite Dilution Diffusion Coefficients ....................................................................... 14
2.3.2 Binary Diffusion Coefficients....................................................................................... 15
2.3.3 Multicomponent Diffusion Coefficients....................................................................... 16
Chapter 3 Experimental Investigation of Diffusive Mass Transfer in Tight Dual-Porosity
Systems......................................................................................................................................... 20
3.1 Introduction ......................................................................................................................... 20
3.2 Mesoporous Dual-Porosity System and Analog Fluid ........................................................ 21
3.3 Evaluation of Molecular Diffusion Coefficients................................................................. 23
3.3.1 Infinite Dilution Diffusion Coefficient......................................................................... 24
3.3.2 Binary Diffusion Coefficients....................................................................................... 28
3.3.3 Multicomponent Diffusion Coefficients....................................................................... 33

v
3.4 Displacement Experiments.................................................................................................. 34
3.5 Simulation of Displacement Experiments........................................................................... 39
3.6 Summary and Conclusions.................................................................................................. 43
Chapter 4 CO2 Huff-n-Puff: Experimental Investigation of Diffusive Mass Transfer and
Recovery from Shale Cores.......................................................................................................... 45
4.1 Introduction ......................................................................................................................... 45
4.2 Experimental Approach....................................................................................................... 47
4.2.1 Shale Core Samples and Fluid System......................................................................... 47
4.2.2 HnP Experimental Setup and Procedure....................................................................... 48
4.3 Results and Analysis............................................................................................................ 52
4.3.1 Experimental Observations and Evaluation ................................................................. 52
4.3.2 Interpretation of Experimental Observations via Simulation....................................... 60
4.4 Summary and Conclusions.................................................................................................. 66
Chapter 5 Diffusive Mass Transfer and Recovery During CO2 Huff-n-Puff in an Eagle Ford
Shale Core..................................................................................................................................... 68
5.1 Introduction ......................................................................................................................... 68
5.2 Experimental Approach....................................................................................................... 69
5.2.1 Shale Core and Fluid System........................................................................................ 69
5.2.2 Experimental Setup and Procedure............................................................................... 70
5.3 Results and Analysis............................................................................................................ 75
5.3.1 Experimental Observations and Evaluation ................................................................. 75
5.3.2 Interpretation of Experimental Observation via Numerical Modeling......................... 88
5.4 Summary and Conclusion ................................................................................................. 103
Chapter 6 Summary and Future Research Direction .................................................................. 106
Appendix......................................................................................................................................110
Nomenclature.............................................................................................................................. 120

vi
References................................................................................................................................... 122

vii
List of Tables
Table 3-1: Pure component properties for the analog fluid at 25 °C and 14.7 psi........................ 22
Table 3-2: Dimensions of the diffusion coefficients experimental setup...................................... 26
Table 3-3: Infinite dilution diffusion coefficients (new experiments vs. literature). .................... 28
Table 4-1: Core dimensions. ......................................................................................................... 47
Table 4-2: Fluid system properties and composition. ................................................................... 48
Table 4-3: Injection pressure in psi per cycle used for the two cores. .......................................... 50
Table 4-4: Binary interaction parameters for CO2-nCx................................................................. 51
Table 4-5: Porosity and permeability for the two shale cores....................................................... 53
Table 4-6: Oil composition (Mole %) in cores after completion of 1st cycle................................ 57
Table 4-7: Oil composition (Mole %) in cores after completion of 2nd cycle............................... 58
Table 4-8: Parameters used to set-up the CMG-GEM model. ...................................................... 61
Table 4-9: Tuned volume shift parameters used in CMG-GEM................................................... 62
Table 5-1: EF-3 dimensions. ......................................................................................................... 70
Table 5-2: Injection pressure in psi per cycle. .............................................................................. 71
Table 5-3: Porosity and permeability for EF-3. ............................................................................ 75
Table 5-4: Initial conditions for pressure pulses experiments....................................................... 76
Table 5-5: Oil remaining in the core after HnP cycles.................................................................. 81
Table 5-6: Multipliers to correct for the infinite dilution coefficients obtained from Hayduk
and Minhas (1982)........................................................................................................................ 96

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List of Figures
Figure 1-1: U.S. tight oil production............................................................................................... 2
Figure 3-1: Silica gel mesoporous material. ................................................................................. 22
Figure 3-2: Quaternary phase diagrams (mass fractions). Left: CO2-CH4-nC4-nC12 at 2280 psi
and 100 °C (PR EOS). Right: Water-MeOH-IPA-iC8 at 68 °F and 14.7 psi (UNIQUAC model)
from Rastegar and Jessen (2011). ................................................................................................. 23
Figure 3-3: Experimental setup for the measurement of infinite dilution diffusion coefficient. .. 26
Figure 3-4: Absorbance vs. time for IPA infinitely diluted in H2O............................................... 27
Figure 3-5: Absorbance vs. time for MeOH infinitely diluted in H2O. ........................................ 27
Figure 3-6: Modeling of binary diffusion coefficients for IPA-H2O mixture at 30°C.................. 31
Figure 3-7: Modeling of binary diffusion coefficients for MeOH-H2O mixture at 30°C............. 31
Figure 3-8: Modeling of binary diffusion coefficients for IPA-iC8 mixture at 30°C.................... 32
Figure 3-9: Modeling of binary diffusion coefficients for IPA-MeOH mixture at 30°C.............. 32
Figure 3-10: Modeling of binary diffusion coefficients for MeOH-iC8 mixture at 30°C............. 33
Figure 3-11: Packed columns used for displacement experiments. .............................................. 35
Figure 3-12: Experimental setup for displacement experiments. ................................................. 36
Figure 3-13: Temperature program for GC oven. ......................................................................... 36
Figure 3-14: GC calibration curve for MeOH. ............................................................................. 37
Figure 3-15: GC calibration curve for IPA. .................................................................................. 37
Figure 3-16: GC calibration curve for iC8. ................................................................................... 38
Figure 3-17: Effluents’ mass fraction (experimental and simulated) for the 1st displacement
experiment..................................................................................................................................... 40
Figure 3-18: Effluents’ mass fraction (experimental and simulated) for the 2nd displacement
experiment..................................................................................................................................... 41
Figure 3-19: A comparison between 2 approaches to compute Deff and 1 case with no diffusive
mass transfer.................................................................................................................................. 43
Figure 4-1: Cores used in this work: 4HC and 3HA..................................................................... 47

ix
Figure 4-2: HnP experimental setup. ............................................................................................ 49
Figure 4-3: FCMP for CO2-synthetic oil....................................................................................... 51
Figure 4-4: Experimental and simulated VLE for EOS tuning..................................................... 52
Figure 4-5: Pressure and temperature variations for 4HC and 3HA during HnP cycles. ............. 53
Figure 4-6: Oil recovery for 4HC and 3HA.................................................................................. 54
Figure 4-7: Produced oil composition for 4HC and 3HA............................................................. 55
Figure 4-8: Pressure change during soaking for 3HA and 4HC (1st Cycle).................................. 56
Figure 4-9: Pressure change during soaking for 3HA and 4HC (2nd Cycle)................................. 58
Figure 4-10: Pressure change during soaking for 3HA and 4HC (3rd Cycle). .............................. 59
Figure 4-11: CO2 FCMP (max Psat) for synthetic oil and oil in 3HA after 2nd cycle. ................... 60
Figure 4-12: Radial single porosity model of extraction cell and core......................................... 61
Figure 4-13: Tuned and measured mixture density data for EOS tuning. .................................... 62
Figure 4-14: Experimental vs. CMG-GEM results for oil recovery. ............................................ 63
Figure 4-15: Produced oil composition (4HC) - experimental vs. calculated. ............................. 64
Figure 4-16: Produced oil composition (3HA) - experimental vs. calculated. ............................. 65
Figure 4-17: Pressure variation during soaking – experiments and calculation via CMG-GEM
(1st Cycle)...................................................................................................................................... 66
Figure 5-1: Eagle Ford core used in this work.............................................................................. 70
Figure 5-2: Experimental setup for pressure pulse decay and flow-through experiments. .......... 72
Figure 5-3: Upstream and downstream pressure data from the 2nd pulse..................................... 77
Figure 5-4: ln (PD) vs. time from the 2nd pulse. ............................................................................ 77
Figure 5-5: Calculated permeability from the pressure pulse decay measurements..................... 77
Figure 5-6: Permeability estimated from gas flow-through measurements: black symbols
represent the upstream pressure, blue symbols depict the downstream pressure, and red
symbols show the estimated permeability .................................................................................... 78
Figure 5-7: Calculated permeability from the flow-through measurements................................. 79

x
Figure 5-8: Pressure and temperature profiles for EF-3 HnP experiment: The pressure is shown
by blue symbols while green symbols represent the temperature................................................. 79
Figure 5-9: Oil recovery from EF-3 (Mass fraction of original oil in place)................................ 80
Figure 5-10: Produced liquid composition from EF-3.................................................................. 82
Figure 5-11: Observed pressure-drop during soaking for all cycles............................................. 83
Figure 5-12: Estimated core pore volume from He pulse-decay experiments.............................. 86
Figure 5-13: Dimensionless gas amount vs. time for He and CO2 pressure-pulse experiments.
....................................................................................................................................................... 87
Figure 5-14: Excess and absolute CO2 adsorption on EF-3.......................................................... 88
Figure 5-15: Different views of the numerical model describing the shale core: Left panel
shows the whole model, middle panel shows the shale core, and the right panel shows the
horizontal cross-section from top of the extraction cell................................................................ 91
Figure 5-16: Oil recovery from experimental data and simulation results. .................................. 93
Figure 5-17: To the left is the comparison between the interpolated infinite dilution
coefficients for CO2-oil and the calculated values from each correlation. To the right shows
the interpolated values as a function of carbon number................................................................ 95
Figure 5-18: The experimental infinite dilution coefficients for oil in CO2 and the calculated
values from selected correlations.................................................................................................. 96
Figure 5-19: Oil recovery when multipliers are applied to adjust Hayduk and Minhas (1982)
infinite dilution coefficients.......................................................................................................... 97
Figure 5-20: Updated recovery plots with the addition of the modified diffusion models........... 98
Figure 5-21: Comparison of produced oil composition (experimental vs simulation). The first
column reports simulation results where diffusion is neglected, columns 2-3 report
calculations with the classical Fick’s law, while columns 4-5 provide results from the
generalized Fick’s law................................................................................................................... 99
Figure 5-22: Comparison of oil recovery for two different relative permeability cases............. 100
Figure 5-23: Comparison of produced oil composition for two different relative permeability
cases. ........................................................................................................................................... 101
Figure 5-24: Pressure variation during soaking period of the first cycle (experimental vs
simulation). ................................................................................................................................. 103

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Abstract
The recovery factors reported from unconventional reservoirs are, in general, extremely low, e.g.,
Hawthorne et al. (2013) reports that primary oil recovery from the Bakken ranges from 4 to 6%.
Accordingly, enhanced oil recovery (EOR) techniques such as CO2 injection, are needed to unlock
additional production. During gas injection, mass transfer mechanisms including molecular
diffusion and convective mixing often play a critical role in the oil recovery process.
Compositional simulators are generally used to quantify the potential incremental hydrocarbon
recovery from such processes. Our work studies diffusive mass transfer during gas injection in
tight fractured reservoirs and aims to promote accurate prediction of incremental recovery through
improving the representation of diffusive mass transfer in compositional simulation tools.
In the 1st part of this research, we aim to improve the understanding of multicomponent
diffusive mass transfer between matrix and fracture segments through experimental and modeling
work. Displacement experiments were carried out using analog fluids and a synthetic mesoporous
medium to effectively isolate and study the relevant physical mechanisms at play. The experiments
were performed in packed columns utilizing silica-gel particles with an internal porosity. The
silica-gel particles (40-70 micron) include an internal porosity with a narrow pore-size-distribution
(PSD) centered at 6 nm that makes up approximately 50% of the overall porosity.
A quaternary analog fluids system consisting of Water, Methanol, Isopropanol, and
Isooctane, was used because it mimics at ambient conditions the phase behavior of CO2, Methane,
Butane and Dodecane mixtures at 2,280 psi and 100°C. Our selection of the analog fluid system
and porous medium allowed us to investigate matrix-fracture fluid exchange of relevance to
enhanced recovery operations in ultra-tight fractured systems. The effluents from these
displacement experiments served as the basis for our analysis of the role of diffusive mass transfer.

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The role of molecular diffusion in the displacement experiments was investigated by first
performing separate diffusion experiments to obtain diffusion coefficients for all relevant binary
mixtures. Infinite dilution diffusion coefficients were measured for all binary mixtures and then
used to model binary and multicomponent diffusion coefficients over the entire relevant
composition range. The accuracy of this approach was investigated and documented by performing
additional binary diffusion experiments over a broader range of compositions.
The displacement experiments were performed and simulated using an in-house simulator
and an excellent agreement was obtained: The extensive experimental/modeling work related to
the diffusion coefficients of the analog fluid system was used in interpreting the diffusive mass
transfer between the matrix (stagnant) and fracture (flowing) domains via a 1D linear dual-porosity
model. The uniqueness of this framework pertains to the combined experimental and modeling
workflow used to obtain the multicomponent and effective diffusion coefficients. By obtaining
such data, we were able to predict, with reasonable accuracy, the displacement experiments without
the need to perform any additional parameter estimation/adjustments.
In the 2
nd part of this research, we study diffusive mass transfer in shale cores by conducting
lab experiments and simulation of CO2 Huff-n-Puff (HnP) at high pressure and temperature. The
aim of this effort was to further elucidate the role of diffusive mass transfer during EOR in tight
fractured rocks. Two cores from a formation in the Middle East were evacuated and then saturated
at 3500 psi and 50°C with a well-characterized normal alkane mixture consisting of Decane (nC10),
Dodecane (nC12), Tetradecane (nC14) and Hexadecane (nC16). The normal alkane system was
selected to facilitate EOS modeling of the fluid behavior and hence reduce uncertainty related to
the fluid behaviors. We performed multiple HnP cycles at varying injection conditions: 2900-4000
psi and 70 °C. Diffusive mass transfer was then investigated via (1) evaluating the effect of

xiii
injection pressure on oil recovery, (2) analyzing produced oil compositions, and (3) by studying
the pressure decline during the soaking period.
Our experimental observations demonstrate that a higher oil recovery is achieved when
CO2 is injected at a higher pressure to facilitate development of first-contact miscibility. We also
observe that molecular diffusion acts as a dominant recovery mechanism in these HnP
experiments, as evident from analyzing the produced oil composition and from examining the
pressure behavior versus time during the soaking periods: The observed decline rate of the pressure
during soaking demonstrates that molecular diffusion dictates the mass transfer during the HnP
experiments. Additionally, we note that miscibility conditions can change from one HnP cycle to
another, as the injected gas mixes with an oil composition that changes between cycles.
We used a commercial compositional simulator (CMG-GEM) to interpret the results from
the HnP experiments. When multicomponent diffusion coefficients were computed using the
correlation of Sigmund (1976), the simulator is unable to provide a reasonable prediction of oil
recovery and produced oil compositions. This indicates that compositional modeling of HnP
processes may be in large error unless additional information is available to calibrate diffusion
modeling based on Sigmund’s correlation. Such calibration or modification of either the diffusion
coefficients or the diffusion models are, however, not possible with most commercial simulators.
Consequently, we adapt an open-source simulator Matlab Reservoir Simulation Toolbox (MRST)
that provides for additional flexibility in the modeling of diffusion coefficients and application of
different diffusion models.
In the 3rd part of this research, the objective was to further demonstrate the importance of
diffusive mass transfer and recovery in fractured media during EOR operations and how simulation
tools should be modified to provide for more accurate predictions. To accomplish this, we combine

xiv
experimental and modeling work: Four CO2 HnP cycles were performed on an Eagle Ford core
with a porosity of 11.89% and permeability of 2.5 µD. The core was initially saturated with a
synthetic oil composed of nC10, nC12, nC14 and nC16. These experiments were carried out at
reservoir conditions (3,900-4,400 psi and 70 ºC) under first-contact miscibility (FCM) conditions.
Based on the pressure drop during soaking as well as the compositional analysis of the produced
oil, molecular diffusion was again observed to be the central recovery mechanism during the HnP
experiments.
We also note that the pressure variations during soaking periods indicate that CO2
adsorption appears to contribute to the observed pressure-drop. Therefore, to understand CO2
affinity to adsorb on the Eagle Ford core, pressure pulse decay (PPD) experiments were conducted
and confirmed CO2 adsorption with an excess adsorption of 0.32 mol/kg at 4,400 psi. An additional
set of PPD experiments, as well as gas flow-through measurements, were carried out to further
characterize the shale core and facilitate modeling of the experimental observations.
To interpret the experimental observations, a compositional simulation study was
performed using MRST. The governing equations of MRST were modified to include molecular
diffusion represented by a) the classical Fick’s law (using effective diffusivity and independent
concentration gradients as a driving force) as well as b) the generalized Fick’s law (using a full
diffusivity matrix and coupled concentration gradients as a driving force). Using both diffusion
models, numerical calculations show a very good agreement with the lab data (for all cycles) in
terms of the recovery and produced composition, with the former model demonstrating higher
accuracy and computational efficiency: Contrary to the findings of some researchers, the classical
Fick’s law is, after minor adjustments, very accurate in predicting the experimental data. This is

xv
promising given the favorable (reduced) computational requirement associated with this approach
compared to the generalized Fick’s law, which entails additional layers of calculations.

Chapter 1 Introduction
1.1 Background
The global demand for crude oil has increased from ~84 million barrels per day (MBD) in 2005 to
approximately ~102 MBD in 2023. The energy demand is forecasted to rise further in the coming
years with additional supply emerging from renewable sources (solar, wind, hydro, geothermal
and ocean). These renewable sources contribute around 5% of the total energy supply and is set to
reach 17% in 2030 (U.S. Energy Information Administration, 2023). Oil and natural gas will
consequently continue to supply most of the energy until the renewable energy sector is
substantially expanded and capable of providing the desired sustainable and affordable energy.
One of the main challenges in the oil and gas industry is related to the reserves depletion
rate in conventional reservoirs. Currently, oil fields outside OPEC (The Organization of the
Petroleum Exporting Countries) have an annual depletion rate of 6% and oil fields within OPEC
report a 2% depletion rate. To overcome this challenge and ensure sustainable supply, two main
strategies have been implemented: (1) deploying enhanced recovery methods and (2) development
of unconventional resources. The enhanced recovery methods include thermal (steam flooding and
in-situ combustion), chemical (polymer and surfactant flooding) and gas injection (methane CH4,
ethane C2H6, carbon dioxide CO2 and nitrogen N2) processes. A major development of
unconventional oil and gas plays has taken place in the United States since 2007. Oil production
from the U.S. shale plays has increased from 0.4 MBD in 2007 to 8.6 MBD in 2022 as depicted in
Figure 1-1. Oil production from the U.S. shale provided 66% of the total US crude oil production
in 2022 (U.S. Energy Information Administration, 2023).

2
Despite the substantial contribution to the overall production being realized from
unconventional resources, a significant challenge exists with respect to realized recovery. The
reported recovery factors from these resources are extremely low. The primary oil recovery from
the Bakken ranges from 4 to 6% of the oil in place (Hawthorne et al., 2013). Gas injection has
been demonstrated to be an effective approach to enhance recovery from these formations. The
open literature offers a large collection of gas injection studies and projects, employing CO2, CH4,
and other gases, and reports a considerable improvement in oil recovery over primary production:
Miscible gas injection can increase the recovery factor to > 20% (Hoffman, 2012). During gas
injection, mass transfer mechanisms including molecular diffusion and convective mixing often
play a critical role in the oil recovery process. This explains the growing interest in studying
diffusive mass transfer during CO2 injection to delineate the sequestration potential in concert with
enhanced oil recovery from unconventional resources. There are several factors that favor the CO2
over other gases:
1. The miscibility conditions for CO2 can be achieved at lower pressures compared to CH4
and N2: This results in lower capital investments for injection projects.
2. CO2 injection results in a superior reduction of oil viscosity and interfacial tension.
Figure 1-1: U.S. tight oil production.

3
3. Oil swelling due to dissolution of CO2 exceeds that of other gases (Torabi et al., 2011).
1.2 Motivation and Efforts
The capital and operational investment required to design, execute, and operate CO2 injection
projects, for recovery enhancement purposes, is a multi-billion dollars endeavor and hence calls
for a meticulous economic analysis. One of the most important factors in economic analysis is the
potential incremental recovery that is commonly predicted using compositional reservoir
simulators in addition to laboratory studies.
Several investigators have dedicated their efforts over the past decades and presented
excellent observations and outstanding results of EOR operations in unconventional settings (see
Chapter 2). They confirm the significant role of diffusive mass transfer in enhancing recovery from
unconventional rocks. However, many fundamental questions pertaining to the physics of
multicomponent flow and transport are still left unanswered. A common approach applied in the
gas injection literature is to match, and not predict, the experimental results using compositional
simulators. In general, correlations are used to estimate molecular diffusion coefficients without
examining the accuracy of such correlations by comparison to independent experimental
observations. Furthermore, diffusion coefficients are often used as adjustable parameters to
provide for a better agreement between calculations and experimental observations. The lack of a
proper representation of diffusion coefficients, and diffusive mass transfer, in compositional
simulators therefore leads to tedious history matching exercises to reproduce experimental
recovery/composition data. This has motivated us to conduct integrated experimental and
modeling work to study diffusive mass transfer during gas injection and to facilitate prediction of
recovery within a reasonable accuracy. We started our work on analog systems near standard

4
conditions (Chapter 3) and then conducted additional work on cores, extracted from tight fractured
reservoirs, at high pressure and temperature (Chapters 4 & 5).
In Chapter 3, we present displacement experiments with analog fluids in synthetic
mesoporous materials to effectively isolate and study the relevant physical mechanisms at play.
The experiments were performed in packed columns utilizing silica-gel particles that have internal
porosity. The particle size is 40-70 micron with a highly controlled internal pore size of 6 nm that
makes up approximately 50% of the overall porosity. The effluents from these displacement
experiments served as the basis for our analysis of diffusive mass transfer. The role of molecular
diffusion in the displacement experiments was investigated by first performing separate diffusion
experiments to obtain diffusion coefficients for all relevant binary mixtures. Infinite dilution
diffusion coefficients (D
∞
) were measured for all binary mixtures and then used to model binary
and multicomponent diffusion coefficients over the entire relevant composition range. The
accuracy of this approach was studied by performing additional binary diffusion experiments over
a broader range of compositions. The displacement experiments were then simulated using an inhouse simulator and excellent agreement was obtained: The extensive experimental/modeling
work related to the diffusion coefficients of the analog fluid system was used in interpreting the
diffusive mass transfer between the matrix (stagnant) and fracture (flowing) domains.
In Chapter 4, we study diffusive mass transfer in shale cores by conducting and simulating
CO2 Huff-n-Puff (HnP) experiments at high pressure and temperature. We performed multiple HnP
cycles at varying injection conditions: 2900-4000 psi and 70 °C. Diffusive mass transfer was
investigated via (1) evaluating the effect of injection pressure on oil recovery, (2) analyzing
produced oil compositions, and (3) studying the pressure decline during the soaking period. We
demonstrate that higher oil recovery was achieved by injecting at a higher pressure (facilitating

5
development of first-contact miscibility). We also discuss and illustrate the potential change in
miscibility conditions between HnP cycles and how this impacts the oil recovery. Furthermore, we
analyze the pressure behavior during soaking to identify the role of diffusive mass transfer. We
used the CMG-GEM compositional simulator to interpret the HnP experimental results. The
general trends were captured by the model, e.g., the composition of oil in the cores versus cycle.
However, a quantitative agreement between model and experiments was not obtained:
Compositional modeling of HnP processes may accordingly be in large error unless additional
work is performed to support accurate representation of diffusive mass transfer.
In Chapter 5, the objective was to further demonstrate the importance of diffusive mass
transfer on recovery in fractured media during EOR operations and to investigate proper
representation of relevant physics in simulation tools. Four CO2 HnP cycles were performed on an
Eagle Ford core with a porosity of 11.89% and permeability of 2.5 µD. These experiments were
carried out at reservoir conditions (at 3,900-4,400 psi and 70 ºC) under FCM conditions. Based on
the pressure drop during soaking as well as the compositional analysis of the produced oil,
molecular diffusion is observed to be the dominant recovery mechanism in the performed HnP
experiments.
We note that the pressure variations during soaking periods indicate that CO2 adsorption
appears to contribute to the observed pressure-drop. Therefore, pressure pulse decay (PPD)
experiments were conducted and confirmed CO2 adsorption with an excess adsorption of 0.32
mol/kg at 4,400 psi. An additional set of PPD experiments as well as gas flow-through
measurements were carried out for further characterization of the core and to facilitate modeling.
To interpret the experimental observations, a compositional simulation study was
performed using the open-source Matlab Reservoir Simulation Toolbox (MRST). The governing

6
equations of MRST were modified to represent molecular diffusion that is described by a) the
Classical Fick’s law (using effective diffusivity and independent composition gradients) as well as
b) the Generalized Fick’s law (using a full diffusivity matrix and coupled concentration gradients).
Using both diffusion models, following moderate adjustments, our calculation results show a very
good agreement with the lab data (for all cycles) in terms of the recovery and produced
composition, with the former model demonstrating higher accuracy and computational efficiency:
Contrary to the findings of some researchers, the classical Fick’s law was demonstrated to be very
accurate in predicting the experimental data. This is promising given the favorable (reduced)
computational requirement associated with this approach compared to the generalized Fick’s law,
which entails additional layers of calculations.
1.3 Manuscript Organization
The remainder of this manuscript is organized as follows. In Chapter 2, we present a literature
review on gas injection experimental and simulation studies to enhance recovery from
unconventional formations. We also provide an overview of experimental methods to obtain
diffusion coefficients as well as a discussion of the widely used models to predict these
coefficients. In Chapter 3, we report our experimental and simulation approach to study diffusive
mass transfer and predict recovery using analog fluids and a synthetic porous medium. In Chapter
4, we study diffusive mass transfer in shale cores from the Middle East by conducting and
simulating CO2 Huff-n-Puff (HnP) experiments at high pressure and temperature. We highlight
that compositional modeling of HnP processes may be in large error unless additional information
is available to calibrate diffusion modeling based on Sigmund’s correlation. In Chapter 5, we
present an additional CO2 HnP experiment (4 cycles) conducted on an Eagle Ford core. We
highlight the criticality of accurate multicomponent diffusion coefficients that are often

7
inadequately modeled, by default, in most commercial simulators. Accordingly, these fundamental
observations were integrated in the open-source MRST simulator and demonstrated to result in an
improved prediction of the HnP experiment. Finally, a summary of the thesis and suggestions for
future work is provided in Chapter 6.

8
Chapter 2 Literature Review
2.1 Diffusive Mass Transfer in Unconventional Reservoirs
Unconventional resources are found as continuous or quasi-continuous hydrocarbon bearing
geological formations that cannot be exploited using conventional development methods. Because
of the extremely low matrix’s permeability, horizontal wells coupled with multistage hydraulic
fracturing are used to produce hydrocarbons from these resources. The primary recovery factor is
reported to be within 5-10% and hence, improved/enhanced recovery techniques such as gas
injection are applied to increase the ultimate recovery. During gas injection processes, molecular
diffusion can play a central role in the recovery from fractured unconventional resources because
of the matrix’s extreme low-permeability and the large surface area (matrix/fracture interface)
available for diffusion. Several investigators have studied the role of molecular diffusion, and other
mechanisms, in enhancing hydrocarbon recovery from fractured unconventional geological
systems. In the following we provide a review of the main contributions in this area.
Darvish et al., (2006) conducted miscible CO2 injection experiments on cores with a
permeability of 4 mD to study mass transfer between matrix and fracture subsystems. They
confirmed a dominant contribution of molecular diffusion on recovery but were unable to predict
the experimental results using a commercial simulator. This was considered due to limitations in
modeling of diffusion across gas/liquid interfaces and thus, an artificial two-phase zone between
the matrix and fracture was introduced. This interface composition was then adjusted until a
reasonable match was obtained between experimental observations and modeling. Hoteit and
Firoozabadi (2009) studied diffusion in fractured rocks numerically; they determined that at
pressures below the minimum miscibility pressure (MMP), molecular diffusion can enhance

9
recovery. They also concluded that condensate recovery in fractured reservoirs might be enhanced
significantly through gas recycling because of diffusion. Similarly, modeling of continuous gas
injection was performed by Shoaib et al. (2009) and Wang et al. (2010) who reported an
incremental recovery of 10-20% of the original oil in place. Vega et al. (2010) used a siliceous
low-permeability core (1.3 mD) with artificial fractures to conduct miscible CO2 injection
experiments with 3 different boundary conditions to mimic relevant field scenarios. They
illustrated the significance of molecular diffusion as well as convection/dispersion on oil
displacement and recovery. A commercial simulator was used to predict the experimental data, but
an exact match was not obtained: A reasonable match was achieved for the total oil recovery with
a noticeable mismatch in timing. Alharthy et al. (2015) presented laboratory and modeling work
on CO2, CH4- C2H6 and N2 gas injection performed on Bakken cores. They report, for a middle
Bakken core, the highest recovery was obtained from CO2 injection while the lowest recovery was
observed from N2 injection. Based on their simulation work, they were able to match the
experimental data through representation of 3 recovery mechanisms: gravity drainage, Darcy flow
and molecular diffusion. They concluded that molecular diffusion and advective mass transfer are
the main mechanisms for the observed incremental oil recovery. Ghasemi et al., (2016) studied
mass transfer between matrix and fracture in a composite chalk core during CO2 injection. The
core was saturated with stock-tank oil and water was used to control the confining pressure. A
commercial simulator was used to simulate the experimental results, and a reasonable match was
achieved by tuning oil and gas diffusion coefficients. Li et al. (2017) performed HnP gas injection
on Wolfcamp cores using 3 gases: CO2, CH4 and N2 at 2000 psi and 40 °C. CO2 injection provided
the best response in terms of recovery followed by N2 and CH4. The experimental conditions were
simulated with a commercial simulator that did not provide for a consistent agreement with the

10
observed oil recovery: A significant mismatch was observed for injection of N2 and CH4, while a
reasonable agreement was obtained for injection of CO2. The reported calculation results were
limited to oil recovery and a comparison between the experimental and simulated results in terms
of produced oil composition was not provided. Liberty Resources LLC partnered with Energy and
Environmental Research Center (EERC) and conducted extensive work on gas injection in the
Bakken formation including experiments, simulation, and field activities from 2017 to 2020
(Pospisil et al., 2020). The reservoir pressure (at or higher than MMP) is most significant to the
success of EOR as cited in the study’s key observations. The study furthermore provides several
recommendations for future projects including:
• Evaluate water alternating gas (WAG) injection to improve conformance.
• Conduct HnP cycles at bottom hole pressures that are higher than the MMP.
• Study geological complexity and consider it during site selection and project design.
Tran et al., (2021) studied enhanced recovery with a core from the Montney tight-oil formation
and performed HnP by injecting CH4 and a mixture of CH4 and C2H6. The core plug was coated
by silicone to allow gas and oil transport from one end-face only. Two experiments were performed
and the best response in terms of recovery was achieved from injection of the CH4/C2H6 mixture.
They observed that advection-dominated flow serves as the main mechanism at early times, while
molecular diffusion dominates at later times. Observation: A common approach used in the
HnP/gas injection literature is to match the experimental results using compositional simulators.
Diffusion coefficients are often used as adjustable parameters to provide for a better agreement
between calculations and experimental observations.
To quantify the flux due to molecular diffusion, we consider the molecular diffusion
potential. The proportionality factor between the two is the diffusion coefficient. The diffusion flux

11
can be defined using molar, mass, volume, and solvent-based average reference velocity (Cussler
2009; Leahy-Dios and Firoozabadi 2007). Molar-based diffusion coefficients are used throughout
this study. The diffusive mass transfer flux is commonly characterized via 3 widely used models:
The classical Fick’s law, the generalized Fick’s law, and the Maxwell-Stefan (M-S) model. In the
classical Fick’s model, the diffusion flux of every component depends only on its concentration
gradient and ignores the interaction with gradients of other species within the mixture. The
generalized Fick’s takes into consideration the interaction with gradients in the other components.
The diffusion fluxes in the M-S model depend on chemical potential gradient and component
interactions. Under specific conditions outlined by Taylor and Krishna (1993), the generalized
Fick’s and M-S model are equivalent. One of the most critical parameters in describing diffusive
mass transfer is the diffusion coefficients and thus, obtaining accurate coefficients is paramount.
These diffusion coefficients can be obtained either experimentally or by using correlations and this
is the subject of the next section.
2.2 Measurement of Diffusion Coefficients
The most frequently used measurement techniques include the diaphragm cell (DC), constant
volume (CVD) or constant pressure (CPD) diffusion experiments, Taylor dispersion (TD)
techniques, capillary methods (CM) and interferometery (IF).
• DC is a pseudosteady state method and consists of two compartments that are separated by
a porous diaphragm (Barnes, 1934). The solution in each compartment is under agitation
to ensure uniform concentrations. The diffusion coefficient is then calculated from
measurements of the solution concentrations. Medvedev (2005) highlighted that the
obtained diffusion coefficient is an effective value that is influenced by the membrane’s

12
porosity and tortuosity. Most of the DC measurements reported in the literature are at
atmospheric pressure with the highest temperature reported at 65 °C.
• CVD is an unsteady state method where two unequilibrated phases are brought into contact
in a fixed volume cell under constant temperature (e.g. Christoffersen, 1992). The pressure
in the system decreases with time during the measurement, and some investigators refer to
this method as a pressure decay measurement (Riazi, 1996). The variation of the system
pressure and liquid height are fitted either numerically, semi-analytically or analytically
and diffusion coefficients are determined indirectly. CVD measurements can be conducted
at elevated pressure and temperature.
• CPD is another unsteady state method and measures two-phase diffusion in a PVT cell at
constant temperature and pressure. The pressure is kept constant by gradual loading of the
diffusing gas into the PVT cell. The diffusion coefficient is calculated based on the injected
(loaded) gas moles (Reamer and Sage, 1956). CPD can also be performed at high pressure
and temperature.
• TD is a dynamic method where a solvent flows through a capillary coiled tube and a sharp
solute’s pulse is injected at the inlet. The sharp concentration profile develops and becomes
Gaussian-shaped which is captured by e.g., a differential refractometer or a UV-Vis at the
outlet (Taylor, 1953 and Aris, 1956). The observed solute profile is due to the combined
effects of radial and axial diffusion and axial advection. The main assumption in TD is that
the axial diffusion is negligible compared to the axial advection. TD measurements
reported in literature have been performed at very high pressure and temperature conditions
(10,000 psi in Cadogan et al., 2016 and 294 °C in Matthews et al., 1978).

13
• CM uses a capillary tube to minimize the effects of convection. Grogan et al. (1988)
constructed a high-pressure capillary tube to measure gas diffusion in liquid: A gas bubble
is trapped by liquid and the change in the bubble length with time is tracked and used to
calculate the diffusion coefficient. Other investigators used the same CM principle but
designed different experimental setups (Stefan 1871; Malik and Hayduk 1968;
McManamey and Wollen 1973; Witherspoon and Saraf 1965). Most of the CM
measurements in literature are at atmospheric pressure with the highest temperature at 50
°C.
• IF tracks the evolution of the refractive index with time and uses that to calculate the
diffusion coefficients (Kegeles and Gosting 1947; Hampe et al., 1991). It is performed
under static fluid conditions and takes many experimental configurations. The most
common one is where a layer of solvent is placed inside a temperature-controlled cavity
with solute above. Alizadeh and Wakeham (1982) highlighted that IF generates high
accuracy results near ambient conditions but seems extremely difficult to use at other
conditions.
The applicability of the above listed methods differs based on the fluid system to be studied (gasgas diffusion, gas-liquid diffusion, and liquid-liquid diffusion). It also depends on the desired
pressure and temperature conditions. Furthermore, some methods are suitable for concentrationdependent diffusion coefficients while others are for infinite dilution diffusion coefficients.
2.3 Correlation of Diffusion Coefficients
Based on the type of diffusion coefficients that are needed, correlations can be divided into three
categories:
1. Infinite dilution diffusion coefficient (D
∞
)

14
2. Binary diffusion coefficient
3. Multicomponent diffusion coefficient
We review each of these in more detail below.
2.3.1 Infinite Dilution Diffusion Coefficients
Infinite dilution diffusion coefficients, D
∞
, are commonly used to predict diffusivities in
concentrated mixtures. There are multiple theories used to develop correlations for D
∞
including
kinetic theory, activated jump, free volume and friction. The main difference between these
theories pertains to how the solvents molecules surrounding the solute are treated. Popular
correlations used in literature include Wilke-Chang (1955), Hayduk-Minhas (1982), and Wong and
Hayduk (1990) that are based on the friction theory: These correlations are empirical modification
of Stokes-Einstein’s equation. Wilke & Chang correlated D
∞ with temperature, solute molar
volume at the normal boiling point, solvent molecular weight, and solvent viscosity: They obtained
an absolute average deviation (AAD) of 10% for 285 data points. Hayduk-Minhas developed
correlations specific for certain mixtures (alkanes, nonpolar and mixtures with water as solvent).
For normal alkanes, they correlated D
∞ with temperature, solute molar volume at the normal
boiling point and solvent viscosity and obtained an AAD of 3.4% for 58 data points. Wong and
Hayduk generated correlations for normal alkanes, dissolved gases in organic solvents and
dissolved gases in water. D
∞ was correlated to temperature, critical volume, and molecular weight
of solute and solvent in addition to solvent viscosity. The AAD is comparable to that of HaydukMinhas with a slight improvement. Another widely used and relatively recent approach was
presented by Leahy-Dios and Firoozabadi (2007): They expressed the ratio of the densitydiffusivity product as a function of viscosity ratio, reduced pressure and temperature and the
acentric factor. The correlation was developed using gas, liquid and supercritical nonpolar

15
mixtures and resulted in an overall AAD of 12.1%. Based on our review and the work of Yang et
al., (2022), we did not find a single correlation that provides consistently accurate prediction of
D
∞
for all mixtures. However, it is possible and recommended to identify the most suitable
correlation(s) which can be superior to others in modeling specific mixtures.
2.3.2 Binary Diffusion Coefficients
Sigmund (1976) developed a correlation in which he expressed the ratio of molar densitydiffusivity product as a third-order polynomial function of pseudo-reduced molar density. He used
a Loschmidt diffusion cell to study three binary gas mixtures: nitrogen-methane, methane-propane
and methane-n-butane at high pressure and temperature. The results from his diffusion
experiments, in addition to published experimental data for gas and liquid mixtures, were used to
develop the correlation. The AAD for gas mixtures is 10% while it is 40% for liquid mixtures. This
significant difference in the AAD is commonly observed between diffusion in gases and liquids.
The Sigmund correlation is widely used in process and reservoir simulation tools, but it is critical
to note that it is valid only up to a reduced density of <3. da Silva and Belery (1989) modified
Sigmund’s correlation because they noticed that it predicts negative diffusion coefficients
whenever the reduced density exceeds ~3. However, they did not provide a theoretical basis for
their modification. Shi et al. (2022) measured and modeled CH4 diffusion in normal alkanes
mixtures: They demonstrated that diffusion coefficients obtained from Sigmund’s correlation
(Sigmund, 1976; da Silva and Belery, 1989) deviated from the experimental data and that a
correction factor, unique to each binary mixture, is needed to obtain a good agreement.
Another approach to estimate binary diffusion coefficients is through an interpolation
between the two diffusivities at infinite dilution. The simplest model was developed by Darken
(1948), in which he assumed the MS-diffusivities to be a linear function of composition. Hartley

16
and Crank (1948) introduced another model, based on the Stokes-Einstein relation, that has an
inverse proportionality relationship between diffusivity and viscosity. Darken’s model was
modified by Vignes (1966), by using a logarithmic relationship between diffusivities and
composition. Leffler and Cullinan (1970) developed a model similar to that of Hartley and Crank
(1948), but with a logarithmic, instead of linear, relationship. Bosse and Bart (2006) used the basis
of Eyring’s theory and developed a new model, which is an extension of Vignes (1966) model and
includes an additional term for excess Gibbs energy that allows for an explicit description of
thermodynamic nonidealities. Some of these models provide great predictive capabilities when
applied to ideal mixtures but fail for nonideal and/or associating mixtures, while others work for
nonideal mixtures but fail for associating mixtures. Furthermore, the binary diffusion coefficient
models discussed above, that utilize diffusivities at infinite dilution, require the calculation of MS diffusivities. To apply these models to represent experimentally measured data points, a
transformation to Fickian diffusivities is required. This transformation is generally performed by
using thermodynamic models (for the liquid phase) such as Margules, Van Laar, Wilson, NRTL,
UNIQUAC, and UNIFAC (Taylor and Krishna,1993). We note that the transformation process may
induce additional uncertainties in the predictive capabilities of models for binary diffusion
coefficients.
2.3.3 Multicomponent Diffusion Coefficients
Diffusion processes of relevance to the oil/gas industry often involve multicomponent mixtures,
rather than binary mixtures, and hence, representation of multicomponent diffusion coefficients is
imperative. Many investigators have studied multicomponent diffusion coefficients and developed
several models that are mostly extensions of the techniques applied for binary mixtures
(Wesselingh and Krishna, 1990; Koojiman and Taylor, 1991; Rutten, 1992; Rehfeldt and

17
Stichlmair, 2010). To this end, Vignes (1966) equation has been generalized (Wesselingh and
Krishna, 1990; Koojiman and Taylor, 1991) as follows:
Dij = (Dij
i→0
)
xj
∙ (Dij
j→0
)
xi∏(Dij
xk→1
)
xk
n
k=1
k≠i,j
2.1
where 𝐷𝑖𝑗
𝑖→0
is the diffusion coefficient of component i infinitely diluted in component j, 𝑥𝑖
is the
mole fraction of component i and n is the number of components. The first two terms on the righthand side are equal to the binary diffusivities at infinite dilution and can be obtained from
experiments. The third term 𝐷𝑖𝑗
𝑥𝑘→1
represents the limiting value of the M-S diffusivity in a mixture
where component k exists in substantial excess: This term is hard to measure experimentally since
it is the diffusivity of component i in j where both mole fractions are infinitely small. Accordingly,
this term is often represented by an assumed model: Wesselingh and Krishna (1990) used the below
geometrical rule to approximate that term.
Dij
k→1 = √Dij
i→1
∙ Dij
j→1 2.2
Koojiman and Taylor (1991) developed another model, as shown in equation 2.3, based on
an investigation of different mixing rules.
Dij = (Dij
xj→1
)
xj
∙ (Dij
xi→1
)
xi
∙ (Dik
xk→1
∙ Djk
xk→1
)
xk/2
2.3
They compared the results of their model to the work of Wesselingh and Krishna, after conversion
to the Fickian frame, with experimental data for several ternary mixtures including Acetone- (1)
Butanol- (1) Propanol, Methanol- (1) Butanol- (1) Propanol, and Acetone-Water-(1) Propanol.
Based on their work, some models are observed to perform well for certain mixtures but none of

18
the models exhibits consistent and accurate predictions for all the mixtures. Moreover, the model
of Koojiman and Taylor provides for good predictions in a strongly non-ideal mixture consisting
of (1) Propanol- (1) Chlorobutane – n Heptane.
Rutten (1992) extended the model of Wesselingh and Krishna by introducing a viscositybased correction term as shown in equation 2.4 where 𝜂𝑖 corresponds to the viscosity of component
i. He highlights that the model of Wesselingh and Krishna appears inaccurate for mixtures with
components that have very different diffusion coefficients.
Dij
k→1 = √
ηi
ηk
∙ Dij
i→1
∙
ηj
ηk
∙ Dij
j→1 2.4
Rehfeldt and Stichlmair (2010) conducted binary and ternary diffusion experiments using
holographic laser-interferometry and developed a new model as presented in equation 2.5.
Dij
k→1 = ( Dij
i→1
∙ Dij
j→1
∙ Dik
k→1
∙ Djk
k→1
)
xk
4 2.5
They compared the performance of their model with the three models discussed above (2-2 to 2-
4). Based on their analysis, some of these models provide for great predictive capabilities in certain
mixtures. However, none of the models exhibit good predictions for all the examined ternary
mixtures.
Another approach to compute multicomponent diffusion coefficients is through an
effective diffusivity (Wilke, 1950; and Bird et al., 1960). Wilke (1950) developed a model for gas
mixtures by simplifying the M-S approach. The approach computes the effective diffusion
coefficient for component i in a mixture as a function of mole fractions and binary diffusion
coefficients for all pairs in the mixture as shown in equation 2.6. Other investigators have used
Wilke’s equation for liquid mixtures (e.g. Shi et al. 2022).

19
Di,eff = [
(1 − xi
)
∑
xj
Dij
n
j=1,j≠i
] 2.6
We apply a subset of these approaches to model diffusion in binary, ternary and quaternary
mixtures in the following chapters.

20
Chapter 3 Experimental Investigation of Diffusive Mass Transfer in Tight
Dual-Porosity Systems1
3.1 Introduction
In this chapter, we present a framework for studying mass transfer between matrix and fracture
blocks by using analog fluids and a mesoporous dual-porosity system to perform and model
displacement experiments. Moreover, molecular diffusion is characterized in detail by conducting
tandem diffusion experiments to obtain the necessary diffusion coefficients. The uniqueness of this
framework is the combined experimental and modeling workflow to obtain the multicomponent
and effective diffusion coefficients. By obtaining such data, we were able to predict the
displacement experiments without the need to perform additional parameter adjustments or
estimation.
The displacement experiments were performed in packed columns utilizing silica-gel
particles with internal porosity. The particle size was 40-70 micron with an average internal pore
size of 6 nm makes up approximately 50% of the overall porosity in the packed column. The
quaternary analog fluid system consists of Water (H2O), Methanol (MeOH), Isopropanol (IPA),
and Isooctane (iC8), was used because it mimics (at ambient conditions) the phase behavior of
CO2, Methane, Butane and Dodecane mixtures at 2,280 psi and 100°C (Rastegar and Jessen, 2011).
The effluents from these displacement experiments served as the basis for our analysis of the
diffusive mass transfer.
1 Most of the results in this chapter have been presented in the SPE Annual Technical Conference and Exhibition:
Alahmari, S., & Jessen, K. (2021, September). An Experimental Investigation of Mass Transfer in Tight Dual-Porosity
Systems. In SPE Annual Technical Conference and Exhibition.

21
The role of molecular diffusion in the displacement experiments was investigated by first
performing separate diffusion experiments to obtain diffusion coefficients for all relevant binary
mixtures. Infinite dilution diffusion coefficients were measured for all binary mixtures and then
used to model binary and multicomponent diffusion coefficients over the relevant composition
range. The accuracy of this approach was determined by performing additional binary diffusion
experiments over a broader range of compositions.
The displacement experiments were simulated using an in-house simulator and excellent
agreement was obtained: The extensive experimental/modeling work related to the diffusion
coefficients of the analog fluid system was used in interpreting the diffusive mass transfer between
the matrix (stagnant) and fracture (flowing) domains via a 1D dual-porosity simulation model.
We start by presenting the mesoporous medium and analog fluid system. We then present
our experimental and modeling work for evaluating molecular diffusion coefficients. Results from
displacement experiments are then presented alongside modeling, simulation, and interpretation.
We conclude the chapter with a summary of our findings and a set of recommendations for
modeling EOR in tight fractured systems.
3.2 Mesoporous Dual-Porosity System and Analog Fluid
Silica-gel with internal porosity (from Sigma-Aldrich) was used with a particle size of 40-75
micron and an average internal pore diameter of 6 nm. The pore space between particles serves as
the fracture porosity, while the internal pore space in the silica-gel particles represents the matrix
porosity. The matrix porosity makes up approximately 50% of the total porosity. The silica-gel is
a powder-like material as shown in Figure 3.1 (left). Under the microscope, the particles appear
irregular in shape as depicted in Figure 3.1 (right) but can be reasonably approximated as spheres.

22
Figure 3-1: Silica gel mesoporous material.
We performed a compatibility test between the silica-gel and our analog fluid system and
observed no chemical interaction; the silica gel was exposed to different mixtures in separate vials
and the vials were placed inside an oven at 30°C (the same temperature as our displacement
experiments) for 3 days. During that time, we did not observe any change in the fluids’ level and
subsequently, we were able to recover the silica gel.
The analog fluid system consists of H2O, MeOH, IPA and iC8. Pure component properties
are summarized in Table 3.1. The analog fluid phase behavior at ambient conditions is comparable
to the high-pressure CO2-hydrocarbon system (Methane, Butane and Dodecane) as depicted in
Figure 3.2. A detailed description of the analog fluid system including phase behavior, density,
viscosity, and interfacial tension can be found in Rastegar and Jessen (2011).
Table 3-1: Pure component properties for the analog fluid at 25 °C and 14.7 psi.
Component Density
(g/cc)
Viscosity
(cp)
Molecular weight
(g/mole)
Critical Pressure
(psi)
Critical Temperature
(K)
MeOH 0.787 0.544 32.04 1174.21 512.64
IPA 0.781 1.96 60.10 691.00 508.29
iC8 0.687 0.473 114.23 373.28 543.96
H2O 0.997 0.890 18.01 3199.31 647.13

23
Figure 3-2: Quaternary phase diagrams (mass fractions). Left: CO2-CH4-nC4-nC12 at 2280 psi and 100 °C (PR EOS). Right:
Water-MeOH-IPA-iC8 at 68 °F and 14.7 psi (UNIQUAC model) from Rastegar and Jessen (2011).
3.3 Evaluation of Molecular Diffusion Coefficients
Several mechanisms can contribute to mass transfer in porous media, including viscous flow,
gravity, diffusion, and capillarity. In fractured unconventional reservoirs, molecular diffusion plays
a central role because of the ultra-low permeability of the matrix, combined with the relatively
large surface between fractures and matrix. One of the most critical parameters used for
interpreting diffusive mass transfer in tight systems is the diffusive flux, and hence the
multicomponent diffusion coefficients. Accordingly, obtaining accurate diffusion coefficients for
our analog fluid system is of paramount importance. The representation of diffusion coefficients
has traditionally been simplified (or ignored) in the literature. Some researchers treat diffusion
coefficients as history-matching parameters to match experimental observations, an exercise that
does not directly provide valuable insight towards understanding (and predicting) the mechanism
of diffusive mass transfer. To accurately measure/calculate the necessary diffusion coefficients
needed to predict the behavior of our displacement experiments, we used the following three-step
approach:

24
1. We measured infinite dilution diffusion coefficients (D∞
) experimentally for all binary
mixtures using the approach of Taylor (1953) and Aris (1956).
2. We then model binary diffusion coefficients over relevant composition ranges using the
models of Vignes (1966) and Bosse & Bart (2006). This step includes the calculation of
thermodynamic correction (nonideality) factors for all relevant binary mixtures using the
UNIFAC (UNIQUAC Functional-group Activity Coefficient) model and comparing
predicted binary diffusion coefficients with experimentally measured data either from the
literature or from separate in-house measurements.
3. We then calculate the effective diffusion coefficients following the approach of Wilke
(1950) via two approaches to model the multicomponent diffusion coefficients: The binary
M-S diffusion coefficients, and the relation by Koojiman and Taylor (1991). The results
were then compared with our displacement experiments to examine the quality of these
approaches.
3.3.1 Infinite Dilution Diffusion Coefficient
In this work, we use the method proposed first by Taylor (1953) and modified by Aris (1956) to
experimentally determine D
∞
for our analog fluid system. The basis of the theory is that the velocity
profile of the laminar flow inside a capillary tube is a parabolic function of the tube’s radius; the
velocity profile reaches its maximum at the center and zero at the wall. Sample molecules move
with different velocities depending on their positions inside the tube and get re-distributed because
of the combined effects of radial/axial diffusion and axial advection. The concentration profile
inside the tube develops into a Gaussian shape, short time after pulse injection, and can be
described at the detection point as:

25
C =
M
2 ∙ π
2
3 ∙ Rc
2
∙ √K ∙ tR
exp
(−
(t−tR)
2
2∙σ2
)
3.1
where C represents the solute concentration, Rc is the tube radius, tR is the mean residence time, M
is the amount of solute in the injected sample, σ
2
is the variance of the concentration profile and K
is the dispersion coefficient that is related to tR and σ
2
as demonstrated by Bello et al. (1994):
tR =
L
u
+
2 ∙ K
u
2
3.2
σ
2 =
2 ∙ K ∙ L
u
3 +
8 ∙ K
2
u
4
3.3
Here u denotes the mean fluid velocity and L is the length of tube before the detection point. The
relationship between dispersion K and D
∞ was developed analytically by Aris (1956):
K = D
∞ +
Rc
2
48 ∙ D∞
3.4
The diffusivity of the solute can then be computed from a simpler equation when Taylor’s
condition is satisfied:
D
∞ ≪
Rc
2
∙ u
2
48 ∙ D∞
3.5
If the requirement of equation (3.5) is satisfied; the diffusivity of the solute can be calculated as:
D
∞ =
Rc
2
24 ∙ σ
2
∙ tR 3.6
The experimental setup, used in this work, is shown in Figure 3.3 and consists of a solvent
reservoir, a high precision dosing pump (Chemyx-Fusion 4000), an auto-injector switching valve
(MXP-7900, IDEX), a stainless-steel coiled tube, a UV-Vis (Prostar 325), a computer to process
the data and a waste container. The capillary tube was coiled and placed in a water bath to control

26
the desired temperature within 0.03 °C. The coiling radius was designed to meet the conditions of
Janssen (1976) and Alizadeh et al. (1980) to avoid any secondary flow effects. The dimensions for
the diffusion column are reported in Table 3.2.
Figure 3-3: Experimental setup for the measurement of infinite dilution diffusion coefficient.
Table 3-2: Dimensions of the diffusion coefficients experimental setup.
Capillary tube radius, m 0.0003
Capillary tube length, m 10.00
Capillary tube coiling radius, m 0.0850
Ratio of coiling to capillary radius 283.3
A solution pulse with a slightly different concentration (a difference of 2-3% by mole) from
solvent was injected via a 50 µL syringe into the auto-injector. The 5 µL sample loop that is
attached to the auto-injector was filled and the remaining injected volume was used to flush the
connections and ports to ensure complete filling of the sample loop. The absorbance vs. time at
the outlet of the diffusion column was detected via a UV-Vis unit and used to calculate the infinite
dilution diffusion coefficients.
Each experiment was repeated at least 3 times to ensure accuracy and reproducibility of
the results. The recorded data of absorbance vs. time were fitted to a Gaussian curve and equation
3.6 was used to compute D∞
. To test/validate our experimental setup, we started by focusing on

27
mixtures within our analog fluid system that has previously been studied in the literature. Thus,
we performed diffusion coefficients experiments on 2 mixtures: IPA in H2O and MeOH in H2O.
Our results show excellent agreement with the literature data (max. deviation of 4.2%). After
validation, we proceeded with the remaining experiments. Table 3.3 summarizes the results and
compares them to the literature data, when available. It is worth noting that the other 7 datapoints
were not compared to any literature values as, to the best of our knowledge, they were not
measured previously. Figures 3.4 and 3.5 report the absorbance vs. time for IPA in H2O and MeOH
in H2O.
Figure 3-4: Absorbance vs. time for IPA infinitely diluted in H2O.
Figure 3-5: Absorbance vs. time for MeOH infinitely diluted in H2O.

28
Table 3-3: Infinite dilution diffusion coefficients (new experiments vs. literature).
Binary Mixture Experiments D∞, m2
/s Literature D∞, m2
/s % Deviation from literature
Methanol in Water 1.89E-09 1.83E-09 (1) 3.3
Water in Methanol 2.18E-09 2.22E-09 (3) 1.8
Isopropanol in Water 1.37E-09 1.43E-09 (2) 4.2
Water in Isopropanol 1.04E-09 N/A N/A
Methanol in Isooctane 1.14E-09 N/A N/A
Isooctane in Methanol 1.83E-09 N/A N/A
Isopropanol in Isooctane 2.72E-09 N/A N/A
Isooctane in Isopropanol 4.90E-10 N/A N/A
Methanol in Isopropanol 9.25E-10 N/A N/A
Isopropanol in Methanol 2.05E-09 N/A N/A
(1) and (2) Matthews and Akgerman (1983) and (3) Lee and Li (1991)
From the infinite dilution diffusion coefficients, we proceed to model the binary diffusion
coefficients for all relevant mixtures within our analog fluid systems. The modeling and validation
of binary diffusion coefficients is discussed in the next section.
3.3.2 Binary Diffusion Coefficients
The diffusivity of a binary mixture can be estimated using the relevant values of D
∞
for that
mixture. Numerous models exist in the literature for the estimation of binary diffusion coefficient
from the limiting diffusivities (e.g., Darken,1948; Hartley and Crank, 1948; Vignes, 1966; Leffler
and Cullinan, 1970; Rutten, 1992; Bosse and Bart, 2006). Most of these were summarized in the
literature section (Chapter 2). Our analog fluid system presents many challenges in terms of strong
nonideality and association that is evident from the viscosity measurements performed by Rastegar
(2010). Thus, it is imperative to screen available models and to select the most accurate model(s).
It is worth emphasizing that experimentally measured diffusivities are Fickian diffusivities.
However, for modeling purposes, the Maxwell-Stefan (M-S) approach is used, and therefore, a

29
transformation between these types of diffusivities is required. A thermodynamic correction factor
is needed to transform between the two types of diffusivities, as described in equations 3.7 and 3.8
according to Taylor and Krishna (1993).
Dij = Γ ∙ Dij ,
where
3.7
Γ = 1 + xi
∂lnγi
∂xi
|
T,P,x𝑗
.
3.8
𝐷𝑖𝑗 and 𝐷𝑖𝑗 are the Fickian and M-S binary diffusion coefficients, respectively, Γ is the
thermodynamic correction factor, xi and 𝛾𝑖 are the mole fraction and activity-coefficient of
component i, T is the temperature and P is the pressure.
We reviewed the literature and found experimental data for two mixtures of relevance to
our analog fluid system. These mixtures are MeOH-H2O (Lee and Li, 1991) and IPA-H2O (Pratt
and Wakeham, 1975). We compared the experimental data of MeOH-H2O and IPA-H2O against
Vignes (1966), Rutten (1992) and Bosse & Bart (2006) models to examine their predictive
capabilities and to select the most suitable model(s). Prior to presenting the results of the
comparison, we first provide a summary of the three models and readers are advised to refer to
original papers for a detailed explanation of theory and derivation. Vignes modified the work of
Darken and developed a logarithmic relationship between D∞
and mole fractions to predict the
binary diffusion coefficient 𝐷𝑖𝑗 as shown in equation 3.9.
Dij = (Dij
i→0
)
xj
∙ (Dij
j→0
)
xi
3.9
Dij
i,j→0
is the infinite dilution diffusion coefficient and xi (or j), is the mole fraction. Rutten expanded
on the work of Hartley and Crank and added viscosity correction parameters: Pure and mixture
viscosity data are needed.

30
Dij =
η
ηj
(
xj
Dij
i→0) +
η
ηi
(
xi
Dij
j→0) , 3.10
where η is the mixture viscosity and ηi (or j)
is the pure component viscosity. Bosse and Bart
applied Eyring’s theory and developed a new model that is an extension of Vignes that includes an
additional term for the excess Gibbs energy (See Eq. 3.11).
Dij = (Dij
i→0
)
xj
∙ (Dij
j→0
)
xi
exp
(−
g
E
R∙T
)
3.11
where g
E
is the excess Gibbs energy and R is the gas constant. This additional term provides an
explicit representation of thermodynamic nonidealities for the mixture, at the cost of the additional
effort to compute the excess Gibbs energy term via the activity-coefficients of components within
the mixture.
There are several models available to calculate the excess Gibbs energy, via activitycoefficients, as summarized in Taylor and Krishna (1993). The UNIFAC model, originally
developed by Fredenslund et al. (1977), was used to calculate the excess Gibbs energy and the
thermodynamic correction factor in this work.
Based on our comparison of the experimental data of MeOH-H2O and IPA-H2O versus
Vignes, Rutten and Bosse and Bart models in Figures 3.6 and 3.7, we make several observations:
• The models of Vignes and Bosse & Bart provide for accurate predictions, especially
in the case of the IPA-H2O mixture where a clear minimum exists.
• The largest deviation from the experimental data occurs at or near the minimum
point for the IPA-H2O mixture.

31
• Careful selection of functional groups for each component in the calculation of the
thermodynamic correction factor is crucial to achieving the best match between
experimental data and model (Reid, R. C. ,1987).
Figure 3-6: Modeling of binary diffusion coefficients for IPA-H2O mixture at 30°C.
Figure 3-7: Modeling of binary diffusion coefficients for MeOH-H2O mixture at 30°C.
0
0.25
0.5
0.75
1
1.25
1.5
0 0.2 0.4 0.6 0.8 1 Diffusion coefficient, m2/s (10
-9
)
IPA Mole Fraction
Vignes Bosse and Bart
Rutten Pratt and Wakeham (1975)
This work
0
0.5
1
1.5
2
2.5
0 0.2 0.4 0.6 0.8 1 Diffusion coefficient, m2/s (10
-9
)
MeOH Mole Fraction
Lee and Li (1991) This work
Vignes Bosse and Bart
Rutten

32
Subsequently, we modeled the binary diffusion coefficients for all the remaining mixtures using
the two models. Furthermore, the accuracy of our approach was validated by performing additional
binary diffusion experiments over a broader range of compositions as shown in Figures 3.8-3.10.
Figure 3-8: Modeling of binary diffusion coefficients for IPA-iC8 mixture at 30°C.
Figure 3-9: Modeling of binary diffusion coefficients for IPA-MeOH mixture at 30°C.
0
0.4
0.8
1.2
1.6
2
2.4
2.8
3.2
0 0.2 0.4 0.6 0.8 1 Diffusion coefficient, m2/s (10
-9
)
IPA Mole Fraction
This work Vignes Bosse and Bart
0
0.5
1
1.5
2
2.5
0 0.2 0.4 0.6 0.8 1 Diffusion coefficient, m2/s (10
-9
)
IPA Mole Fraction
This work Vignes Bosse and Bart

33
Figure 3-10: Modeling of binary diffusion coefficients for MeOH-iC8 mixture at 30°C.
We observe that two phases form for the Methanol-Isooctane mixture at certain
concentrations (see Figure 3.10), where the Fickian diffusion coefficients approach zero near the
borders of the two-phase region (spinodal points) and assume negative values for two-phase
mixtures. A similar variation of the diffusion coefficients was reported by Rutten (1992) for
mixtures of Methanol and n-Hexane.
3.3.3 Multicomponent Diffusion Coefficients
We use the approach by Wilke (1950) to calculate the effective diffusion coefficient for each
component 𝐷𝑖,𝑒𝑓𝑓, as shown by equation 3.12.
Di,eff = [
(1−xi
)
∑
xj
Dij
n
j=1,j≠i
] 3.12
We examined two methods to compute the multicomponent diffusion coefficients, Dij, in equation
3.12:
a) The binary diffusion coefficients (M-S)
b) The approach of Kooijman and Taylor (1991) model as described by the below equation.
0
0.5
1
1.5
2
0 0.2 0.4 0.6 0.8 1 Diffusion coefficient, m2/s (10
-9
)
MeOH Mole Fraction
This work Vignes Bosse and Bart
2-Phase Region

34
Dij = (Dij
xj→1
)
xj
∙ (Dij
xi→1
)
xi
∙ (Dik
xk→1
∙ Djk
xk→1
)
xk/2
3.13
In the 1st method (M-S binary), the multicomponent diffusion coefficients were calculated using
the binary diffusion coefficients that do not include the effect of the 3rd component and mole
fractions were normalized to reflect the binary mixture. In the 2nd method, the Koojiman and Taylor
model was used, and it includes the effect of the 3rd component on the diffusivity of component
(1) and (2). The results of these methods will be discussed later in the simulation section. We used
the effective diffusivities in our simulation work to interpret the diffusive mass transfer in the
displacement experiments.
Prior to choosing the model of Kooijman and Taylor, we reviewed several models including
Wesselingh and Krishna (1990), Rutten (1992) and Rehfeldt and Stichlmair (2010). In our review
process, we assessed the performance of each model against the published experimental diffusivity
data for several ternary mixtures: “Acetone - (1) Butanol - (1) Propanol”, “Methanol - (1) Butanol
- (1) Propanol”, “Acetone - Water - (1) Propanol” and “(1) Propanol - (1) Chlorobutane – n
Heptane”. We select the model of Kooijman and Taylor over the others because it provides better
prediction for non-ideal mixtures such “Acetone -Water - (1) Propanol” and “(1) Propanol - (1)
Chlorobutane – n Heptane” as referenced in Rehfeldt and Stichlmair (2010).
3.4 Displacement Experiments
Displacement experiments were performed on packed columns with an inner diameter of 0.18”
(0.46 cm) and a length of 5.91” (15 cm). The packed column, shown in Figure 3.11, was fitted
with frits (2 µm) at the column ends to prevent migration of the particles. The inner diameter is
sufficiently large to minimize any wall effects. We packed the columns using a slurry packing
method to avoid any trapped air in the internal porosity. The column’s porosity was computed
gravimetrically; the total porosity is 0.4 that is approximately split evenly between flowing (inter-

35
particle) and stagnant (intra-particle) porosities. The stagnant porosity fraction was distinguished
from the flowing by using the silica-gel manufacture specifications.
Figure 3-11: Packed columns used for displacement experiments.
We used Darcy’s law to estimate the permeability of the packed columns. Isooctane was injected
at a constant rate and the pressure drop across the column was monitored. After achieving a stable
pressure drop, the permeability was calculated to be 3 Darcy.
The setup for our displacement experiments is shown in Figure 3.12 and consists of a high
precision dosing pump (Chemyx-Fusion 4000) with two channels for displacing and displaced
fluids, a 3-way valve, a water bath, a stainless-steel packed column, and a sampling point. The
experimental procedure is as follows:
a) We pack the column using the slurry packing method; a mixture with known amounts of
the silica-gel and the desired fluid(s) are mixed and introduced to the column.
b) The column is connected to the setup and immersed in the water bath.
c) We pre-flush the column with 8 pore volumes of the displaced fluids(s) to ensure that there
is no trapped air.
d) We start the experiments by injecting the desired displacing fluid(s) at constant and stable
rate and collect effluent samples every 4 minutes at the outlet.

36
e) The effluent samples were analyzed using Gas Chromatography (GC). We used a Varian
3800 GC equipped with a capillary column (WCOT fused silica 50MX 0.53 MM ID
coating CP-SIL 5CB) and TCD (Thermal Conductivity Detector). Ethanol was selected as
solvent to dilute the collected samples and a GC method was developed to ensure a good
separation between peaks: The method is comparable to the one developed by Rastegar
(2010). Helium was used as a carrier gas with a flowrate of 5 mL/min and 1 µL of the
samples was injected through spilt/spiltless injector with a ratio of 10. The injector
temperature was set at 220 °C and the column’s oven temperature was programmed as
shown in Figure 3.13. To analyze the displacement effluent, we created a set of calibration
curves as shown in Figures 3.14-3.16.
Figure 3-12: Experimental setup for displacement experiments.
Figure 3-13: Temperature program for GC oven.

37
Figure 3-14: GC calibration curve for MeOH.
Figure 3-15: GC calibration curve for IPA.
R² = 0.9992
0
0.05
0.1
0.15
0.2
0.25
0.3
0 500 1000 1500
Mass Fraction
Area under curve, mV.min
Methanol
R² = 0.9994
0
0.05
0.1
0.15
0.2
0 500 1000 1500 2000
Mass Fraction
Area under curve, mV.min
Isopropanol

38
Figure 3-16: GC calibration curve for iC8.
We performed two displacement experiments, that are unconditionally stable for the setup
used, at 30°C which is the same temperature that we conducted our diffusion coefficients
experiments/modeling at:
1. 100% IPA was used to displace a mixture of iC8 (87% mass fraction) and IPA (13% mass
fraction).
2. A mixture of IPA (74% mass fraction) and MeOH (26% mass fraction) was used to displace
100% iC8.
Prior to presenting and analyzing the results, we first introduce the simulation model used to
interpret the results.
R² = 0.9994
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 500 1000 1500
Mass Fraction
Area under curve, mV.min
Isooctane

39
3.5 Simulation of Displacement Experiments
We used an in-house one-dimensional dual-porosity model that is based on the work of Coats and
Smith (1964). The continuity equations for multicomponent two-phase flow in the packed column
are written for the flowing and stagnant domains as described in Shojaei and Jessen (2014):
f ∙ ∅ ∙
∂
∂t
(∑ xij ∙ ρj
∙ Sj
np
j=1
)+
∂
∂z
∑ (xij ∙ ρj
∙ vj − ∅ ∙ Kij ∙ Sj
∂xijρj
∂z
)+ qi
np
j=1 = 0 𝑖 = 1, … , 𝑛𝑐
3.14
(1 − f) ∅ ∙
∂
∂t
(∑ xij ∙ ρj
∙ Sj
np
j=1
) − qi = 0 3.15
where f represents the flowing fraction, Ø is the porosity, xi,j is the mole fraction of component i in
phase j, ρj is the molar density of phase j, Sj is the saturation of phase j, vj is the velocity of phase
j, Kij is the combined effect of dispersion and diffusion, t represents time, z denotes the direction
along the packed column, np is the number of phases, nc is the number of components, and qi
represents the mass transfer between the flowing and stagnant domains. For the performed
displacement experiments, we have a fully miscible flow and by assuming an ideal mixing, we can
re-write equation 3.14 as:
f ∙ ∅ ∙
∂Ci
∂t
+ v ∙
∂Ci
∂z
− ∅ ∙ Ki
∂
2Ci
∂z
2 + θi
∙ (Ci
f − Ci
s
) = 0 3.16
θi = σ ∙ Di,eff 3.17
Ci is the concentration, and qi is expanded into 𝜃𝑖 and the concentration gradient between the
flowing and stagnant domains. 𝜃𝑖
in equation 3.17 is a representation of a shape factor σ (for the
silica-gel particles) multiplied by an effective diffusion coefficient 𝐷𝑖,𝑒𝑓𝑓, as explained in the
discussion of molecular diffusion provided previously. The key element of our model involves
evaluation of 𝜃𝑖 and this is accomplished by utilizing our experimental/modeling work to obtain

40
effective diffusion coefficients alongside a reasonable assumption of a spherical shape factor for
the silica-gel.
We start by interpretation of the 1st displacement experiment where 100% IPA was injected
at a rate of 0.02 mL/min to displace a mixture of iC8 and IPA (87% and 13% mass fraction,
respectively). The below chart, Figure 3.17, reports the experimental and simulated effluent mass
fractions where symbols correspond to experimental data while lines represent the calculated
behavior.
Figure 3-17: Effluents’ mass fraction (experimental and simulated) for the 1st displacement experiment.
More than 98% of the displaced fluids were recovered after 1.28 pore volume injected (PVI). We
observe a good match between the experimental and simulated data in Figure 3.17 and this match
was not a result of a history-matching exercise; theta was computed based on the effective diffusion
coefficient and the shape factor, resulting in the observed agreement. We do see a less-accurate
agreement at ~ 1.2 PVI, and this could be attributed to either heterogeneities in our packed column
or error in analysis of the effluent composition.
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5
Mass Fraction
PVI
IPA Exp iC8 Exp IPA iC8

41
A 2nd displacement experiment was performed to test our approach further. In this
experiment, 100% iC8 was displaced by a mixture of IPA and MeOH (74% and 26% mass fraction,
respectively) at a flow rate of 0.03 mL/min. We chose to use a ternary mixture to be able to
calculate multiple theta values (nc -1 needed) based on the effective diffusion coefficients for each
component. We computed the effective diffusion coefficients based on 2 approaches: The binary
diffusion coefficients (M-S) and the model of Koojiman and Taylor. In Figure 3.18, we compared
the results of Koojiman and Taylor to our displacement experiment and observed a good agreement
between the experimental data and the calculated behavior.
Figure 3-18: Effluents’ mass fraction (experimental and simulated) for the 2nd displacement experiment.
The figure reports the experimental and simulated effluent mass fractions where symbols
correspond to experimental data while the lines represent the calculated behavior. 100% recovery
of the displaced fluid was achieved after 1.4 PVI and we observe that the calculations predict an
earlier breakthrough as compared to the experimental data, with a difference of around 0.09 PVI.
The experimental data show a sharp displacement front, that mimics a piston-like displacement,
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2 Mass Fraction
PVI
iC8 Exp IPA Exp MeOH Exp
IPA iC8 MeOH

42
which suggests an extremely fast mass transfer process. We did not attempt to match the data
further as this is not the intent of our work; since our prediction (no matching) is reasonably
accurate, and obtained from computing theta for each component, additional parameter estimation
was not pursued.
In Figure 3.19, we compared the displacement experimental results with the two models
(The binary diffusion coefficients (M-S) and the model of Koojiman and Taylor): The model by
Koojiman and Taylor provides the best agreement, and the results demonstrate that the diffusivities
of all components are critical to accurately model the diffusive mass transfer in ultra-tight fractured
systems. Utilizing the M-S binary diffusion coefficient approach, inclusive of binary coefficients
and renormalized mole fractions, provides for less accurate modeling of the diffusive mass transfer
as shown in Figure 3.19 where we can see that the approach of Koojiman and Taylor leads to a
better agreement.

43
Figure 3-19: A comparison between 2 approaches to compute Deff and 1 case with no diffusive mass transfer.
We also investigated the importance of molecular diffusion as a recovery mechanism in the
performed displacement experiment and simulated a scenario where we turn off the diffusive mass
transfer and compare the simulated effluent with our displacement experiment to examine the
dominant recovery mechanism. As expected, we observed in Figure 3.19 a faster breakthrough,
that is a result of sweeping only the flowing fraction of the pore volume, which illustrates that the
molecular diffusion is the dominant recovery mechanism and must be included to model the
displacements.
3.6 Summary and Conclusions
In this chapter, we have presented experimental and simulation work to highlight the critical data
needed to effectively characterize the diffusive mass transfer in (analog) ultra-tight fractured
systems. The uniqueness of this work is the combined experimental and modeling workflow to
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2 iC8 Mass Fraction
PVI
Experimental Data
Deff calculated from Koojiman and Taylor
Deff calculated from Binary M-S
No Diffusive Mass Transfer

44
obtain the multicomponent and effective diffusion coefficients. By obtaining such data, we were
able to predict, with reasonable accuracy, the displacement experiments without the need to
perform an additional parameter estimation. Based on the results and interpretations presented in
this chapter, we arrive at the following conclusions and recommendations:
• Molecular diffusion can play a critical role in recovery processes from ultra-tight fractured
systems.
• An accurate estimation of the multicomponent diffusion coefficients is required to
characterize and predict diffusive mass transfer. This crucial step is simplified or ignored
in many papers where correlations, with no experimental validation, were used to obtain
the relevant transport coefficients.
• The diffusive mass transfer process in the performed displacement experiments was
successfully characterized by calculating the effective diffusion coefficients for each
component.
• The model of Koojiman and Taylor provides for the best agreement with experimental
observations by integrating the diffusivities of all components. In contrast, the M-S binary
diffusion coefficient approach, that does not include the effect of the 3rd component and
was found to be less accurate.
• The performed experiments used analog fluids and mesoporous medium, but the workflow
is applicable to reservoir fluids, cores, at other conditions.

45
Chapter 4 CO2 Huff-n-Puff: Experimental Investigation of Diffusive Mass
Transfer and Recovery from Shale Cores2
4.1 Introduction
Gas injection has been demonstrated to be an effective approach to enhance recovery from ultratight fractured reservoirs where the role of molecular diffusion often becomes significant. The
open literature offers a large collection of work concerned with gas injection studies and projects,
employing CO2, CH4 and other gases, and reports a considerable improvement in oil recovery over
primary production (see Chapter 2). CO2 injection has an additional advantage over other gases
through the potential for geological sequestration. This explains the growing interest in studying
diffusive mass transfer during CO2 injection to delineate the sequestration potential in concert with
enhanced oil recovery from unconventional resources. However, additional work is needed to
arrive at a comprehensive understanding and representation of diffusive mass transfer in ultra-tight
fractured formations.
In this chapter, we study diffusive mass transfer in shale cores by conducting and
simulating CO2 Huff-n-Puff (HnP) experiments at high pressure and temperature. Two cores from
a formation in the Middle East were evacuated and then saturated at 3200 psi and 50°C with a
synthetic oil consisting of decane (nC10), dodecane (nC12), tetradecane (nC14) and hexadecane
(nC16). We performed multiple HnP cycles at varying injection conditions: 2900-4000 psi and 70
°C. Diffusive mass transfer was then investigated via (1) evaluating the effect of injection pressure
2 Most of the results in this chapter have been presented in the SPE Middle East Oil and Gas Show and Conference:
Alahmari, S., Raslan, M., Khodaparast, P., Gupta, A., Duncan, J., Althaus, S., & Jessen, K. (2023, March). CO2
Huff-n-Puff: An Experimental and Modeling Approach to Delineate Mass Transfer and Recovery from Shale Cores.

46
on oil recovery, (2) analyzing produced oil compositions, and (3) studying the pressure decline
during the soaking period.
Our experimental observations show that a higher oil recovery is achieved when injecting
at a higher pressure. We also observe that molecular diffusion acts as a dominant recovery
mechanism in the HnP experiments, as evident from analyzing the produced oil composition and
from examining the pressure behavior versus time during the soaking periods: The observed
decline rate in the pressure during soaking signifies that molecular diffusion dictates the mass
transfer during the HnP experiments. Additionally, we note that miscibility conditions will change
from one HnP cycle to another, as the injected gas mixes with an oil composition that changes
between cycles.
The CMG-GEM compositional simulator was used to interpret the HnP experimental
results. When multicomponent diffusion coefficients were computed using the correlation of
Sigmund (1976), the simulator is unable to provide a reasonable prediction of oil recovery and
produced oil compositions: The general trends were captured by the model, e.g., the composition
of oil in the cores versus cycle. However, a quantitative agreement between model and experiments
was not obtained. This indicates that compositional modeling of CO2 HnP processes may be in
large error unless additional experimental and modeling work is available to calibrate diffusion
modeling based on Sigmund’s correlation: An observation that is consistent with the findings of
Shi et. al., (2022).
The remainder of this chapter is arranged as follows: We start by introducing the
experimental work including the core samples, the fluid system, the experimental setup, and the
experimental procedures. We then present our HnP experimental results alongside simulation and

47
interpretation. We conclude the chapter with a summary of our findings and a set of
recommendations for future gas HnP studies.
4.2 Experimental Approach
In this section, we present our experimental work to study the role of diffusive mass transfer on
two shale cores, 4HC and 3HA, including a description of the fluid system, preparation of the
cores, and the HnP experimental set up and procedures.
4.2.1 Shale Core Samples and Fluid System
We worked with two cores from a formation in the Middle East (4HC and 3HA). The cores are
shown in Figure 4.1 and their dimensions are listed in Table 4.1.
Figure 4-1: Cores used in this work: 4HC and 3HA.
Table 4-1: Core dimensions.
Core Diameter, in Length, in
3HA 1.485 2.548
4HC 1.487 2.488
We used a synthetic oil that consists of nC10, nC12, nC14 and nC16) The properties of our
fluid system including molecular weight, critical parameters and composition is reported in Table
4.2. The composition was confirmed via Gas Chromatography (GC). The synthetic oil was selected

48
to facilitate the interpretation of CO2 HnP experiments through available experimental data from
literature on the PVT behavior of binary CO2-nCx mixtures.
Table 4-2: Fluid system properties and composition.
Component Molecular Weight, g/mole Pc, psi Tc, K Omega Mole, %
nC10 142.3 305.68 617.6 0.49 7.9
nC12 170.3 264.53 658.2 0.56 19.9
nC14 198.4 235.14 694.0 0.68 30.8
nC16 226.5 205.74 717.0 0.74 41.4
The cores were evacuated at 110 °C for approximately three weeks, during which the change in
weight was monitored. When the observed change was less than 0.01 g, the dry weight was
recorded. The cores were then saturated in a high-pressure vessel for 3 months at 3200 psi and 50
°C. The saturated weight of the core was then recorded (providing an estimate of the porosity).
4.2.2 HnP Experimental Setup and Procedure
HnP experiments were performed on the cores using an advanced gas injection/recycling system.
A schematic of the HnP setup is shown in Figure 4.2 and consists of a CO2 source, a booster pump
to provide the selected injection pressure, two extraction cells that are enclosed in an oven with
high-precision temperature control, upstream and downstream valves to control gas injection from
the top and production from the bottom, pressure sensors to record/log pressures in the extraction
cells, separators for each extraction cell, a wet meter to measure gas rate and effluent collection
ports to extract the produced liquids.

49
Figure 4-2: HnP experimental setup.
After saturating the cores and determining the effective porosity (from material balance and fluid
density), the HnP experiments were performed in the following five steps:
a) Thermal equilibration - The cores were placed in the extraction cells and the system was
allowed to reach and stabilize at 70 °C. At this step, all the upstream and downstream valves
were closed.
b) CO2 injection - After thermal stabilization was achieved, the upstream valves were opened
to charge the cells with CO2. The upstream valves were then closed when the desired
pressure was reached.
c) Soaking - The cores were soaked for about 100 hours to provide sufficient time for CO2 to
transport into the cores. The determination of the soaking time depends upon several
parameters including core dimensions, core physical properties (porosity, permeability, and
tortuosity), injected gas (CO2, N2, CH4, C2H6), oil saturating the core and miscibility
conditions. Gamadi et al., (2013) conducted gas HnP experiments on several shale cores
(Barnett, Marcos, and Eagle Ford) at near-miscible conditions and studied the effect of
soaking time on ultimate recovery. They reported that a soaking time of around 5 days is
70 C
Pressure
70 C
Pressure
CO2
Supply
Booster
Pump
Valves Valves
Oven
Cell-1 Cell-2
Effluent
Collection
Effluent
Collection

50
sufficient. Other investigators (Yu and Sheng, 2015; and Sheng, 2017) have reported the
optimum soaking time for their respective studies as 1 to 8 days.
d) Pressure depletion/production – The extraction cells were then gradually depressurized to
atmospheric conditions. The depressurization was performed through the downstream
valves (bottom) of the extraction cells, and during this step, produced gas was measured
via a wet meter while effluent liquid was collected in sealed vials. The depressurization
stage was completed in three stages and took approximately 4.0 hours per extraction cell.
e) Data collection and analysis - After the depressurization step was completed, the weight of
the cores was measured, the volume and weight of the collected liquid was recorded, and
the composition of the produced liquid was determined by GC analysis.
The above process was repeated for multiple cycles and Table 4.3 reports the injection pressure
used for each core and each cycle.
Table 4-3: Injection pressure in psi per cycle used for the two cores.
4HC 3HA
1
st Cycle 3200 2926
2
nd Cycle 3980 3580
3
rd Cycle 3730 2906
A minimum injection pressure of 2900 psi was selected to ensure first contact miscibility between
CO2 and the initial oil in place, based on preliminary phase behavior modeling with the fluid
system. The calculated first-contact miscibility pressure (FCMP) for CO2-syntethic oil at 70 °C is
around 2600 psi as shown in Figure 4.3: This was evaluated using the Peng-Robinson equationof-state (PR78) with binary interaction parameters tuned to match the vapor-liquid equilibrium
(VLE) experimental data relevant to all binary mixtures (Inomata et al., 1986; Camacho-Camacho

51
et al., 2011; Gasem et al., 1989; and Latsky et al., 2020). The binary interaction parameters are
listed in Table 4.4 while the experimental and simulated VLE data are shown in Figure 4.4.
Figure 4-3: FCMP for CO2-synthetic oil.
Since 4HC and 3HA originated from the same well, HnP cycles were performed at different
injection pressures to examine the impact of the injection pressure on the oil recovery (Table 4.3).
The difference in injection pressure between the cycles was increased from 280 psi in the 1st cycle
to 830 psi in the last cycle.
Table 4-4: Binary interaction parameters for CO2-nCx.
Mixture Binary interaction
CO2-nC10 0.09694
CO2-nC12 0.08947
CO2-nC14 0.08777
CO2-nC16 0.07676
0
500
1000
1500
2000
2500
3000
0 0.2 0.4 0.6 0.8 1 Saturation Pressure, psi
CO2 Mole Fraction

52
Figure 4-4: Experimental and simulated VLE for EOS tuning.
4.3 Results and Analysis
In this section, we present: 1) the experimental observations and related analysis, and 2) a
comparison between experimental observations and numerical calculations using CMG-GEM.
4.3.1 Experimental Observations and Evaluation
The porosity was evaluated from the weight of the core before and after saturation (gravimetrical
approach), while the permeability was measured via the Gas Research Institute (GRI) technique
(Guidry et al., 1996): The GRI method is performed on crushed samples and evaluates permeability
by using the propagation of high-pressure gas from a reference cell to a test cell, that contains the
sample, while monitoring the pressure decay. Relevant values of porosity and permeability are
listed in Table 4.5.
0
500
1000
1500
2000
2500
3000
0 0.25 0.5 0.75 1 Pressure, psi
CO2 Mole Fraction
CO2
- nC16
0
500
1000
1500
2000
2500
3000
0 0.25 0.5 0.75 1 Pressure, psi
CO2 Mole Fraction
CO2
-nC14
0
500
1000
1500
2000
2500
3000
0 0.25 0.5 0.75 1 Pressure, psi
CO2 Mole Fraction
CO2
-nC10
0
500
1000
1500
2000
2500
3000
0 0.25 0.5 0.75 1 Pressure, psi
CO2 Mole Fraction
CO2
-nC12
Inomata et al., 1986
Camacho et al., 2011
Gasem et al., 1989
Latsky et al., 2020

53
Table 4-5: Porosity and permeability for the two shale cores.
Core Porosity, % Permeability, μD
3HA 6.32 0.798
4HC 5.25 0.923
Next, we consider the experimental conditions during the HnP process for the two cores.
The pressure and temperature variations observed during the three HnP cycles are reported in
Figure 4.5: The average injection pressure is shown by the red dotted line. In these experiments,
the average injection pressure for 4HC was higher than that of 3HA by design to investigate the
impact of pressure on recovery. We note that the injection pressures used in the HnP sequence (of
both cores) were above the FCMP of the initial oil/CO2.
Figure 4-5: Pressure and temperature variations for 4HC and 3HA during HnP cycles.
We start by analyzing the experimental results in terms of oil recovery and proceed by analyzing
the produced oil composition and the pressure behavior during soaking. Oil recovery was
evaluated from a material balance on the cores, with cores weights measured after each cycle, and
3636
3138

54
are reported in Figure 4.6: We observe that the recovery from 4HC is higher than for 3HA for all
HnP cycles.
Figure 4-6: Oil recovery for 4HC and 3HA.
The difference in the recovery factors is moderate after the 1st cycle (~2%) but increases steadily
to ~12% after the 3rd cycle. The trend in differences between recovery factors mirror the differences
in the injection pressure used for the two cores: The difference in the injection pressure is modest
during the 1st cycle (280 psi) and increases to 830 psi for the 3rd cycle. Accordingly, the selected
injection schedule for two cores resulted in an 12% higher recovery from 4HC than from 3HA,
despite the fact that all injection pressures exceed the FCMP for the initial oil. From our material
balance, we arrive at a recovery in excess of 100% for 4HC - 106% to be exact. This implies that
we have produced all the synthetic oil and extracted additional hydrocarbon that was potentially
not extracted fully prior to the saturation of the core with the synthetic oil used in this study. We
note that the two cores were prepared following the same protocol, and that additional hydrocarbon
may have been extracted from 3HA (albeit still at a recovery less than 100% after the 3rd cycle).
0
20
40
60
80
100
1 2 3 Oil Reocery, %
Cycle
3HA 4HC

55
The composition of the extracted oil was determined by GC, and the mole fractions of each
component in the produced oil, along with the initial values in synthetic oil are reported in Figure
4.7.
Figure 4-7: Produced oil composition for 4HC and 3HA.
The variations in compositions observed in Figure 4.7 indicate that diffusion is actively
contributing to the recovery mechanism, as the mole fractions of the lighter components (nC10 and
nC12) in the oil produced from the 1st cycle are higher than in the initial synthetic oil composition.
We also observe that the relative change in produced oil composition between cycles is larger for
the lighter normal alkanes as compared to the heavier components. Additionally, we find that more
0.02
0.04
0.06
0.08
0.1
0.12
1 2 3 Mole Fraction
Cycle
nC10
3HA 4HC Synthetic Oil
0.17
0.19
0.21
0.23
0.25
1 2 3 Mole Fraction
Cycle
nC12
3HA 4HC Synthetic Oil
0.29
0.31
0.33
0.35
1 2 3 Mole Fraction
Cycle
nC14
3HA 4HC Synthetic Oil
0.29
0.33
0.37
0.41
0.45
1 2 3 Mole Fraction
Cycle
nC16
3HA 4HC Synthetic Oil

56
nC16 was produced from 4HC relative to 3HA and we infer that this is due to the difference in the
injection pressure (see further discussion below).
The pressure variation, recorded during the soaking periods, allows us to further examine
the role of diffusive mass transfer. For each cycle, we review and compare the pressure behavior
during soaking of 3HA vs. 4HC. At the onset of the 1st soaking period, CO2 is invading the cores
that are saturated with the synthetic oil: Both cores contain oil of the same composition. For the
first cycle, the injection pressure applied to 4HC was 280 psi higher than that applied to 3HA. If
we assume that 280 psi difference in injection pressure is relatively small, we then expect a larger
pressure drop and, potentially, a faster diffusive mass transfer in 3HA, attributed to a larger
porosity in comparison to 4HC. This was indeed the case, as seen from Figure 4.8 (left), where the
pressure drop for 3HA exceeds that for 4HC.
Figure 4-8: Pressure change during soaking for 3HA and 4HC (1st Cycle).
Figure 4.8 (right) compares the pressure-drop vs. √t for the two cures. In diffusiondominated transport, one could expect the pressure drop to be approximately linear with respect to

57
√t. We observe a deviation from linearity at early times, a behavior that has been discussed by
many researchers (Reamer et al., 1956; Caskey and Michelsen, 1973; Chukwuma, 1983; Renner,
1988; Tan and Thorpe, 1992): e.g., Chukwuma correlated this to an “incubation” period for
equilibrium to be established at the gas-oil contact area. After the early time departure, we observe
a linear trend for both cores in support of diffusion-dominated transport in our HnP experiments.
We note that in addition to the incubation idea, discussed briefly above, oil swelling out of the core
may also contribute to the departure from √t behavior at early times.
Using the GC compositional data for the produced oil from the 1st cycle combined with
material balance calculations, we evaluate the composition of oil remaining in cores after the 1st
cycle as reported in Table 4.6. We find that the remaining oil in 4HC is heavier than the oil in 3HA:
The remaining nC10 and nC12 combined mole fraction in 4HC is 0.198 compared to 0.227 in 3HA.
Table 4-6: Oil composition (Mole %) in cores after completion of 1st cycle.
Component 3HA 4HC
nC10 4.88 4.04
nC12 17.84 15.73
nC14 31.83 31.61
nC16 45.45 48.62
In the 2nd
cycle, the injection pressure applied to 4HC was 400 psi higher than that applied
to 3HA. Combined with the oil compositions reported in Table 4.6, two predictions can be made
regarding the expected pressure drop during soaking: 1) A larger pressure drop is expected for 3HA
since CO2 is diffusing into a lighter oil, assuming that a difference of 400 psi is negligible, or 2) A
larger pressure drop is observed for 4HC due to a higher injection pressure. Figure 4.9 reports the
actual variation in pressures during the 2nd soak period. We observe from Figure 4.9 that a slightly

58
larger pressure drop is recorded for 4HA even though CO2 is diffusing into a heavier oil relative
to 3HA. Similarly, to the observation for the 1st cycle, we observe a linear trend in the pressure
drop vs. √t for both cores after an incubation period that is shorter than for the first cycle.
Figure 4-9: Pressure change during soaking for 3HA and 4HC (2nd Cycle).
The oil composition remaining in the cores following the 2nd HnP cycle was estimated as
discussed above and is reported in Table 4.7. We find that the oil remaining in 4HC is notably
heavier than the oil remaining in 3HA.
Table 4-7: Oil composition (Mole %) in cores after completion of 2nd cycle.
Component 3HA 4HC
nC10 3.15 1.20
nC12 15.08 8.70
nC14 31.92 30.76
nC16 49.85 59.34

59
In the 3rd cycle, the injection pressure applied to 4HC was ~830 psi higher than that applied
to 3HA. Figure 4.10 compares the pressure drop observed for the two cores during the 3rd cycle,
and we observe that the pressure drop for 4HC is larger even though CO2 was diffusing into a
heavier oil relative to 3HA.
Figure 4-10: Pressure change during soaking for 3HA and 4HC (3rd Cycle).
Another interesting observation is that 3HA does not exhibit any significant pressure drop
after 40 hours of soaking: It appears that the molecular diffusion process in 3HA was considerably
reduced. One possible explanation for this noticeable reduction in the diffusive mass transfer
process is that CO2 did not diffuse under FCM conditions. To investigate this, we used the
remaining oil composition in 3HA after the 2nd cycle to calculate the corresponding FCMP. A
comparison of the calculated FCMP (2972 psi) and the injection pressure (2906 psi) indicates that
CO2 did not diffuse under FCMP as illustrated in Figure 4.11 and therefore, the diffusive mass
transfer process was progressing by gas/liquid interface transport.

60
Figure 4-11: CO2 FCMP (max Psat) for synthetic oil and oil in 3HA after 2nd cycle.
4.3.2 Interpretation of Experimental Observations via Simulation
We used CMG-GEM (The Computer Modeling Group) to build a single porosity compositional
model for each core to simulate the HnP experiments. We used a radial grid to represent the
extraction cell and the core. The internal diameter and length of the extraction cell are 1.65” and
9.30”, respectively. The model has a total of 2,534 cells with ~60% of that representing the core.
The model includes 12 layers in the radial direction and 14 in the vertical direction as shown in
Figure 4.12: 3-D view (left), 3-D core view (middle) and 2-D top view of core and cell (right). An
injection well and a production well were perforated in (1,1,1) and (1,1,14), respectively. We
assumed the cores were 100% saturated with oil and isotropic in porosity and permeability.
Additional parameters used to set up the model(s) are listed in Table 4.8.
0
500
1000
1500
2000
2500
3000
3500
0 0.2 0.4 0.6 0.8 1 Saturation Pressure, psi
CO2 Mole Fraction
3HA_after 2nd Cycle Synthetic Oil

61
Figure 4-12: Radial single porosity model of extraction cell and core.
Table 4-8: Parameters used to set-up the CMG-GEM model.
Property Value
Core porosity same as in Table 4.5
Core permeability same as in Table 4.5
Cell permeability, md 500
Cell porosity, % 99
Initial oil saturation, % 100
Initial core pressure, psi 14.7
Injection pressure same as in Table 4.3
Production pressure, psi 14.7
Cell/core temperature, °C 70
We used the PR78 EOS to model the phase behavior of our fluids system. The binary
interaction (kij) and dimensionless volume shift parameters (Si) were obtained by matching
experimental vapor-liquid equilibria (VLE) and density data. The results for kij were presented
earlier in Table 4.4 and Figure 4.4. Volume translation was applied, and shift parameters were
estimated by matching available density data. The results are shown below in Table 4.9 and Figure

62
4.13 where we observe a good agreement between experimental data (Song et al., 2012; Zhang et
al., 2014; Zhang et al., 2015; Yang et al., 2019) and EOS calculations.
Table 4-9: Tuned volume shift parameters used in CMG-GEM
Component Volume shift
CO2 0.12279
nC10 0.06478
nC12 0.10308
nC14 0.15413
nC16 0.17669
Figure 4-13: Tuned and measured mixture density data for EOS tuning.
0.65
0.7
0.75
0.8
0.85
0.65 0.7 0.75 0.8 0.85
Tuned Density, g/cc
Measured Density, g/cc
CO2_nC10 CO2_nC12 CO2_nC14 CO2_nC16

63
In multicomponent mass transfer, there are three widely used models in compositional
simulators to compute the diffusive fluxes: the classical Fick’s law, the generalized Fick’s law, and
the Maxwell-Stefan (M-S) model. CMG-GEM calculates the diffusive flux using the classical
Fick’s law. In our efforts to model the HnP experiments, we considered 3 cases:
a) No diffusion: Molecular diffusion is turned off.
b) Default diffusion: The binary diffusion coefficients are calculated using the approach of
Sigmund (1976). Then, the effective diffusion coefficient for each component is then
computed using the equation of Wilke (1950).
c) Adjusted diffusion: Effective diffusion coefficients, per component, are assigned as input
parameters into CMG-GEM and are assumed constant throughout the calculations
We compare the experimental observations with calculations, for each core, in terms of oil
recovery, produced oil composition and pressure variation during soaking. The experimental oil
recovery is compared to the calculated values in Figure 4.14.
Figure 4-14: Experimental vs. CMG-GEM results for oil recovery.
We observe a substantial difference between experimental and simulation results when molecular
diffusion is not activated. In contrast, the default diffusion model (Sigmund + Wilke) overestimates
0
20
40
60
80
100
1 2 3 Oil Recovery, %
Cycle
4HC
Exp
Sim-No Diffusion
Sim. (Diffusion-Sigmund Based)
Sim. (Diffusion-Adjusted)
0
20
40
60
80
100
1 2 3 Oil Recovery, %
Cycle
3HA
Exp
Sim. (no Diffusion)
Sim. (Diffusion-Sigmund Based)
Sim. (Diffusion-Adjusted)

64
the oil recovery: An observation that is consistent with the finding of Shi et al. (2022). They
measured and modeled CH4 diffusion in normal alkane mixtures and demonstrated that the
diffusion coefficients, estimated from Sigmund’s correlation, require correction factors to
accurately model multicomponent diffusion coefficients. Therefore, in our 3
rd calculations case,
we obtained the binary diffusion coefficients, from Sigmund correlation, based on 50% CO2 and
50% synthetic oil (by mole), and then calculated the corresponding effective diffusion coefficients.
Finally, we then adjusted these constant effective diffusion coefficients by a multiplier of 0.05 to
improve agreement between calculations and data.
We compare the produced oil composition from our experiments with calculations, based
on the constant diffusivities (case 3) in Figures 4.15 and 4.16. We observe that the general trend in
composition, between HnP cycles, is captured by the calculations. However, we did not attempt to
obtain a better agreement by further adjusting diffusivities in this work.
Figure 4-15: Produced oil composition (4HC) - experimental vs. calculated.
0
0.04
0.08
0.12
1 2 3 Mole Fraction
Cycle
nC10
Exp Sim. (Diffusion-Adjusted)
0.15
0.19
0.23
0.27
1 2 3 Mole Fraction
Cycle
nC12
Exp Sim. (Diffusion-Adjusted)
0.3
0.32
0.34
1 2 3 Mole Fraction
Cycle
nC14
Exp Sim. (Diffusion-Adjusted)
0.34
0.38
0.42
0.46
1 2 3 Mole Fraction
Cycle
nC16
Exp Sim. (Diffusion-Adjusted)

65
Figure 4-16: Produced oil composition (3HA) - experimental vs. calculated.
Finally, we compare the pressure drop during soaking for the 1st cycle (4HC and 3HA) in
Figure 4.17. The blue symbols represent the experimental data, gray dashed line corresponds to
the default diffusion case and green dashed line shows the adjusted diffusion case. Both simulated
curves underestimate the pressure drop observed during soaking. This observed mismatch between
experiments and calculations may be attributed to several factors: (1) an inaccurate representation
of the multicomponent diffusion process in CMG-GEM and (2) the potential for CO2 adsorption
on the shale cores that is not considered in this work.
0.04
0.06
0.08
0.1
0.12
1 2 3 Mole Fraction
Cycle
nC10
Exp Sim. (Diffusion-Adjusted)
0.17
0.19
0.21
0.23
1 2 3 Mole Fraction
Cycle
nC12
Exp Sim. (Diffusion-Adjusted)
0.28
0.3
0.32
0.34
1 2 3 Mole Fraction
Cycle
nC14
Exp Sim. (Diffusion-Adjusted)
0.35
0.39
0.43
0.47
1 2 3 Mole Fraction
Cycle
nC16
Exp Sim. (Diffusion-Adjusted)

66
Figure 4-17: Pressure variation during soaking – experiments and calculation via CMG-GEM (1st Cycle).
4.4 Summary and Conclusions
In this chapter, we presented experimental observations and simulation work to interpret/analyze
HnP experiments conducted on 2 shale cores. The emphasis was to further elucidate the role of
diffusive mass transfer during EOR in tight rocks. We analyzed the effect of injection pressure on
oil recovery by performing tandem experiments with cores at different injection pressures.
Produced oil compositions were analyzed, via GC, and allowed for an evaluation of the
composition of remaining oil in place. The pressure variations during the soaking periods were
also investigated and discussed in the context of compositional changes between cycles. A model
of the experimental setup was developed in CMG-GEM and used to predict the HnP experiments
based on Sigmund’s correlation. The general trends were captured by the model, e.g., the
composition of oil in the cores versus cycle. However, a quantitative agreement between model
and experiments was not obtained. Based on the observations and interpretations provided in this
chapter, we arrive at the following conclusions and recommendations:

67
• Higher oil recovery was realized by injecting at a higher pressure. The calculated oil
recovery for 4HC, which was exposed to higher injection pressures, was approximately
12% higher than for 3HA. This observation is important for field applications where capital
investment for injection requirements and expected incremental recovery factor must be
balanced.
• Molecular diffusion was observed to be the dominant recovery mechanism in the
performed HnP experiments, and this is supported by: a) The compositional analysis of the
produced oil when compared to the original synthetic oil, a larger fraction of lighter
components was produced initially, while larger fractions of heavier components are
produced in later cycles. b) The pressure behavior during the soaking time exhibits a linear
relationship with respect to √t after an early time incubation, which imply a diffusiondominated process.
• During HnP/gas injection experiments, miscibility conditions can change from one cycle
to another as the injected gas is diffusing into a different oil composition relative to the
initial oil. This was observed for 3HA where CO2 appeared to be diffusing under first
contact miscibility conditions in the 1st and 2nd cycles but not in the 3rd cycle. The potential
change in miscibility conditions should be thoroughly studied during design of laboratory
gas injection experiments or field pilot tests as it can have a significant impact on recovery.
• Pressure behavior during soaking provides important data that can be interpreted to identify
the role of diffusive mass transfer.
• Compositional modeling of HnP experiments may be in large error unless additional
information is available to calibrate diffusion modeling based on Sigmund’s correlation
(see Chapter 5).

68
Chapter 5 Diffusive Mass Transfer and Recovery During CO2 Huff-n-Puff in
an Eagle Ford Shale Core3
5.1 Introduction
In this chapter, we present an experimental and modeling study of a carbon dioxide (CO2) Huff-nPuff (HnP) process in an Eagle Ford shale core (EF-3). The integrated work addresses critical
findings from Chapter 3 and 4 pertaining to multicomponent diffusion coefficients and diffusion
models used for studying diffusive mass transfer during EOR. Furthermore, we document the
importance of diffusive mass transfer on recovery in fractured media during EOR operations and
investigate proper representations in simulation tools.
Four CO2 HnP cycles were performed on EF-3 (with porosity of 11.89% and permeability
of 2.5 µD) saturated with a synthetic oil composed of nC10, nC12, nC14 and nC16. These experiments
were carried out at reservoir conditions (3,900-4,400 psi and 70 ºC) under FCM conditions. Based
on the pressure drop during soaking as well as the compositional analysis of the produced oil,
molecular diffusion is found to be an active recovery mechanism in the HnP experiments.
Pressure pulse-decay experiments and gas flow-through measurements were carried out to
further characterize the core sample and provide input to the numerical interpretation of the HnP
experiments. The experimental observations indicate that the lamination, visually observed, in the
shale core contributes to an anisotropic core permeability. We also note that the magnitude of the
pressure reduction during the soaking periods indicates that CO2 adsorption cannot be ignored in
the interpretation of the observed pressure response. Therefore, pressure pulse decay (PPD)
3 A joint work with Mohammed Raslan and Ye Lyu that is currently in preparation for publication.

69
experiments were conducted, after the HnP cycles, and confirmed CO2 adsorption with an excess
adsorption of 0.32 mol/kg at 4,400 psi.
To interpret the experimental observations, a compositional simulation study was
performed using the open-source Matlab Reservoir Simulation Toolbox (MRST). The MRST
modeling work was mainly performed by Mohammed Raslan and the reader is referred to Raslan
(2024) for additional details pertaining to MRST modeling work.
The remainder of this chapter is arranged as follows: We start by introducing the
experimental work including the core sample, the fluid system, the experimental setup, and the
experimental procedures. We then present our PPD, gas flow-through and HnP experimental
results. We then proceed to carry out compositional simulation to interpret the experimental
observations. To improve the simulation results, we present three separate methods to alleviate the
inaccuracy of the existing approaches in modeling diffusive mass transfer. We conclude with a
summary of our findings.
5.2 Experimental Approach
In this section, we report on the experimental work, including a description of the core sample, the
fluid system, the preparation of the core, and the experimental setups and procedures used in the
relevant experiments.
5.2.1 Shale Core and Fluid System
The shale core from the Eagle Ford outcrop, used in this work, is shown in Figure 5.1 while the
relevant dimensions are listed in Table 5.1. We observe from Figure 5.1 that the EF-3 core has
laminations along its entire length, most likely attributed to calcite deposits.

70
Figure 5-1: Eagle Ford core used in this work.
Table 5-1: EF-3 dimensions.
Core Diameter, in Length, in Core Weight, g
EF-3 1.000 3.278 91.900
To develop a more detailed characterization of the core’s permeability, we performed PPD and gas
flow-through measurements to delineate the effects of the observed laminations (as discussed in
more detail in the results section below). We used the same hydrocarbon fluid system as previously
described in Chapter 4, with fluid properties and compositions reported in Table 4.2.
The core was evacuated at 110 °C for approximately three weeks, during which the change
in weight was monitored. When the observed change was less than 0.01 g, the dry core weight was
recorded. The core was then saturated in a high-pressure vessel for 3 months at 3000 psi and 50
°C. The saturated weight of the core was then recorded to provide an estimate of the porosity.
5.2.2 Experimental Setup and Procedure
In this section, we present the experimental setup and procedures for HnP, PPD and gas flowthrough measurements.
Huff-n-Puff
We used the same experimental setup and procedure as described in Chapter 4. We conducted four
HnP cycles on the EF-3 where each cycle starts with injection followed by soaking and subsequent

71
pressure depletion at a constant temperature of 70 °C. Table 5.2 reports the injection pressure for
each cycle. We used a soaking time of approximately 100 hours for all cycles. The pressure was
gradually decreased during the depletion in three stages until reaching 14.7 psi: The depletion rate
was around 170 psi/minute in the 1st stage and 12 psi/minute in the 2nd and 3rd stages. The same
depletion rates were applied in all the cycles.
Table 5-2: Injection pressure in psi per cycle.
Cycle Injection Pressure, psi
1
st 4160
2
nd 4400
3
rd 3990
4
th 3900
We planned to use an injection pressure of ~4000 psi for each cycle and were able to achieve that,
within +/- 200 psi, except for the 2nd cycle where we had some technical issues with a backpressure regulator. We selected this injection pressure to conduct our HnP experiments under FCM
conditions during all the cycles. As reported in Chapter 4, the calculated first-contact miscibility
pressure (FCMP) for CO2-syntethic oil at 70 °C is ~ 2600 psi. However, we note that the miscibility
conditions might change from one cycle to the next, due to changes in oil composition between
cycles, as discussed for the HnP experiments conducted on the 3HA core in Chapter 4. Therefore,
to avoid such scenario and to ensure that all HnP cycles were performed under FCM conditions,
we assumed an extreme scenario where EF-3 is fully saturated with pure nC16 and calculated the
FCMP. For such setting, the FCMP is evaluated to ~ 3,700 psi, and hence, we expect all the HnP
cycles to be at FCM conditions.

72
Pressure Pulse Decay and Gas Flow-Through Experiments
To characterize the core permeability, we performed pressure pulse decay (PPD) and gas flowthrough measurements using helium (He), in part, to delineate the effects of the laminations (see
Figure 5.1). For these measurements, we used the setup shown in Figure 5.2 that includes a gas
source, a hydraulic pump to control confining pressure, a constant temperature oven, two cells
(upstream and downstream), a core holder, a vacuum pump, two pressure gauges, and a data
acquisition system.
Figure 5-2: Experimental setup for pressure pulse decay and flow-through experiments.
We used the same setup for gas flow-through measurements with the addition of a mass flow meter
after the downstream cell to measure the flow rate. It is worth noting that both PPD and gas flowthrough measurements were conducted on EF-3 following the HnP experiments. We performed
the PPD measurements following the below procedure:
a) The core was initially wrapped in shrink-tubing, as shown in the lower left part of Figure
5-2, and then placed in the biaxial core holder.
b) The oven temperature was set at 50°C and the system was monitored until thermal
equilibrium was established.
Downstream Cell

73
c) A confining pressure was applied on the entire system and was evacuated for 2 days.
d) After the evacuation process was completed, the reference cell was isolated from the core
holder and charged with gas at a selected pressure (see additional information below).
e) Then gas was allowed to flow from the reference cell (upstream) through the core to the
downstream cell while the cell pressures (P1 and P2) were recorded.
A total of six PPD measurements were performed and the observed pressure decay was used to
evaluate the core permeability via a modified version of the method developed by Brace et al.
(1968) and later extended by Dicker & Smits (1988) and Jones (1997): The gas, helium in our
measurements, in the pore space of the core is initially at the same pressure as in the downstream
vessel (vacuum for first pulse). The gas pressure in the upstream vessel is set slightly higher than
the core pressure (Brace et al., 1968). Upon opening the valve connecting the upstream cell to the
core holder, the upstream pressure decays as the gas flows from the upstream cell through the core
and into the downstream cell. The observed pressure response versus time is then used to calculate
the core permeability as detailed below.
A general solution for the dimensionless pressure difference, as a function of dimensionless
time, during a PPD experiment was presented by Dicker and Smits (1988):
∆𝑃𝐷 = 2 ∑
𝑎(𝑏
2 + 𝜃𝑚
2
) − (−1)𝑚𝑏√(𝑎
2 + 𝜃𝑚
2
)(𝑏
2 + 𝜃𝑚
2
)
𝜃𝑚
2
(𝜃𝑚
2 + 𝑎 + 𝑎
2 + 𝑏 + 𝑏
2) + 𝑎 ∙ 𝑏(𝑎 + 𝑏 + 𝑎𝑏)
∙ 𝑒
(−𝜃𝑚
2
𝑡𝐷)
∞
𝑚=1
5.1
Here 𝜃𝑚 are roots of equation 5.2, while a and b are the ratios of the core pore volume (Vp) to the
upstream (V1) and downstream (V2) cells, respectively:
tan 𝜃 =
(𝑎 + 𝑏)𝜃
𝜃
2 − 𝑎 ∙ 𝑏
5.2

74
𝑎 =
𝑉𝑝
𝑉1
5.3
𝑏 =
𝑉𝑝
𝑉2
5.4
∆𝑃𝐷 is the differential dimensionless pressure and tD is the dimensionless time given by:
∆𝑃𝐷 =
𝑃1
(𝑡)
2 − 𝑃2
(𝑡)
2
𝑃1
(0)
2 − 𝑃2
(0)
2 5.5
𝑡𝐷 =
6.805 ∙ (10−5
) ∙ 𝑘 ∙ 𝑡
𝑐 ∙ 𝜇 ∙ ∅ ∙ 𝐿
2 5.6
Here, 𝑃 is the pressure, 𝑡 is the time, 𝑘 is the core permeability, 𝜙 is the core porosity, and 𝐿 is the
core length. If early time dynamics are ignored, equation 5.1 reduces to the below equation:
𝑙𝑛[∆𝑃𝐷] = ln[𝑓0
] − [
6.805 ∙ (10−5
)𝑘 ∙ 𝜃1
2
𝜇𝑔 ∙ 𝑐𝑔 ∙ ∅ ∙ 𝐿
2
]𝑡
5.7
where 𝜇𝑔 is the gas viscosity, 𝑐𝑔 is the gas compressibility, 𝜃1
2
can be approximated as in shown
in equation 5.8 (Dicker and Smits, 1988), while 𝑓0 is obtained from Eq. 5.9:
𝜃1
2 = (𝑎 + 𝑏 + 𝑎 ∙ 𝑏) −
1
3
(𝑎 + 𝑏 + 0.4132𝑎 ∙ 𝑏)
2 + 0.0744 (𝑎 + 𝑏 + 0.0578𝑎 ∙ 𝑏)
3 5.8
𝑓0 =
2 [𝑎(𝑏
2 + 𝜃1
2
) + 𝑏√(𝑎
2 + 𝜃1
2
)(𝑏
2 + 𝜃1
2
)]
𝜃1
2
(𝜃1
2 + 𝑎 + 𝑎
2 + 𝑏 + 𝑏
2) + 𝑎𝑏(𝑎 + 𝑏 + 𝑎 ∙ 𝑏)
5.9
The core compressibility is negligible when compared to the gas compressibility and hence, was
ignored in the interpretation. We see from equation 5.7 why the pressure pulse must be small: Gas
viscosity and compressibility vary with pressure and if the upstream and downstream pressures
are substantially different, at time zero, then equation 5.7 would provide a coarse approximation.

75
5.3 Results and Analysis
In this section, we present: 1) the experimental observations and related analysis, and 2) a
comparison between experimental observations and numerical calculations.
5.3.1 Experimental Observations and Evaluation
Core Porosity and Permeability Estimates
The core porosity was estimated from the weight of the core pre and post saturation (gravimetrical
method): The dry core weight was 91.90 g, the saturated weight was 95.64 g, and the fluid density
was 0.7616 g/cc resulting in a pore volume of 4.9 cc and a porosity of 11.9 %. The permeability
was measured using the Gas Research Institute (GRI) technique (Guidry et al. 1996) on a ground
sample obtained from a different plug that was cut from the same core. Relevant values of porosity
and permeability are listed in Table 5.3.
Table 5-3: Porosity and permeability for EF-3.
Core Porosity, % Permeability, μD
EF-3 11.89 2.50
A total of six (6) PPD measurements were performed and analyzed to further evaluate the
permeability of the core. Upstream and downstream pressures used in the PPD measurements are
listed in Table 5.4.

76
Table 5-4: Initial conditions for pressure pulses experiments.
No. P1, psia P2, psia
1 500 0
2 498 450
3 560 493
4 615 555
5 712 608
6 815 700
We present an example of the experimental data from the 2
nd pulse in Figure 5.3 where blue circles
represent the upstream pressure and green circles correspond to the downstream pressure. The data
from each pulse was interpreted, as outlined above, by plotting ln (PD) versus time (see example
with 2
nd pulse in Figure 5.4) to evaluate the matrix permeability by the method of Brace et al.
(1968), Dicker and Smits (1988), and Jones (1997). The resulting values of permeability, from all
pulses, are plotted versus the inverse of average core pressure in Figure 5.5, and a corrected
permeability of 2.79 μD is estimated. Compared to the GRI method (2.5 μD), the two estimates of
the matrix permeability are consistent. The data for the remaining pulses are included in AppendixA.

77
Figure 5-3: Upstream and downstream pressure data from the 2
nd pulse.
Figure 5-4: ln (PD) vs. time from the 2
nd pulse.
Figure 5-5: Calculated permeability from the pressure pulse decay measurements.
y = 685.19x + 2.7926
R² = 0.9978
1
2
3
4
5
0 0.0005 0.001 0.0015 0.002 0.0025 0.003
Permeability (uD)
1/Pressure (psia-1)

78
After obtaining an estimation of the matrix permeability from PPD, gas flow-through
measurements along the core’s bedding plane were performed to analyze the effects of lamination.
A total of 4 experiments were performed and the results from one experiment are reported in Figure
5.6, where the black symbols represent the upstream pressure, blue symbols depict the downstream
pressure, and red symbols show the estimated permeability. The results of the other experiments
are included in Appendix-A. The interpreted results of the four experiments are shown in Figure
5.7 where permeability is plotted versus the inverse of average core pressure: We observe that the
permeability in the axial direction is approximately 3.88 μD. In comparison to the PPD and GRI
estimates (2.79/2.50 μD), that primarily probe the matrix permeability, the difference indicates an
increased connectivity in the core along the bedding plane (laminations) by a factor of 1.39.
Figure 5-6: Permeability estimated from gas flow-through measurements: black symbols represent the upstream
pressure, blue symbols depict the downstream pressure, and red symbols show the estimated permeability

79
Figure 5-7: Calculated permeability from the flow-through measurements.
Recovery, Produced Composition, and Pressure Behavior
For the HnP experiment, the pressure and temperature profiles, as observed during the 4 cycles,
are reported in Figure 5.8: The pressure is shown by blue symbols while green symbols represent
the temperature. Each cycle starts with an injection stage where the pressure is increased to the
desired setpoint. The injected gas is then allowed to soak the core for ~ 100 hours, before the
pressure is depleted over a period of ~ 4 hours.
Figure 5-8: Pressure and temperature profiles for EF-3 HnP experiment: The pressure is shown by blue symbols while
green symbols represent the temperature

80
In the following, we analyze the experimental results by 1) evaluation of the oil recovery, 2)
analysis of the produced oil composition and 3) interpretation of the pressure decline during the
soaking phase.
Oil recovery (by mass) was evaluated from a material balance calculation using
measurements of the core weight after each cycle and is reported in Fig. 5-9. From Figure 5.9, we
observe that the largest incremental oil recovery was attained during the 1st cycle (58.2%) with an
addition of 19.5% in the 2nd and nearly complete recovery after the 3rd cycle (96%). The additional
recovery from the 4th cycle was only ~2%.
The produced oil composition, reported alongside the initial synthetic oil composition (see
Figure 5.10) indicates that diffusion plays a central role in the recovery process: We observe from
the 1st cycle that the produced oil composition is very close to the synthetic oil composition. After
the 1st cycle, we notice that the mole fractions of nC10 and nC12 continuously decrease while nC14
and nC16 continue to increase. Furthermore, we do not observe any nC10 production from the 4th
cycle.
Figure 5-9: Oil recovery from EF-3 (Mass fraction of original oil in place).

81
As the estimated additional recovery from the 4th cycle is comparatively small - around 2%, we
used the available GC compositional data for the produced liquids to estimate the remaining oil
composition residing in the core after the 3rd cycle: The combined mole fraction of nC14 and nC16
remaining in the core is approximately 0.89 representing a significant change relative to the
original synthetic oil (0.72). The estimated oil composition left in the core after each cycle is
reported in Table 5.5: The oil remaining in the core becomes heavier after each cycle which
highlights the role of diffusion that contributes to producing lighter component first followed by
increasingly heavier components.
Table 5-5: Oil remaining in the core after HnP cycles.
Post 1st Cycle Post 2nd Cycle Post 3rd Cycle Post 4th Cycle
nC10 0.0708 0.0598 0.0428 0.0119
nC12 0.1914 0.1798 0.1584 0.1016
nC14 0.3102 0.3152 0.3320 0.3560
nC16 0.4276 0.4452 0.4669 0.5306

82
Figure 5-10: Produced liquid composition from EF-3.
Next, we explore the pressure variation during the soaking phase to identify the relevant modes of
mass transfer. The pressure-drop vs time (and square root of time) during all soaking stages is
reported in Figure 5.11. During the injection phase, the oil initially in the core is compressed from
atmospheric pressure to the injection pressure. From calculations with bulk fluid properties, this
compression amounts to approximately 3% of the pore volume. Subsequently, the injected CO2
invades the core further and causes the extraction cell pressure to decrease over time. The pressure
decline should eventually stop upon reaching an equilibrium state (if the soaking period is
sufficiently long).

83
Figure 5-11: Observed pressure-drop during soaking for all cycles.

84
For the first cycle (upper row-right panel in Figure 5.11), we observe a linear relationship between
the pressure-drop and √t , suggesting a diffusion-dominated process. As we have observed and
discussed in Chapter 4, there is a deviation from linearity at early times, a behavior that has been
discussed by many researchers (Reamer et al., 1956; Caskey and Michelsen, 1973; Chukwuma,
1983; Renner, 1988; Tan and Thorpe, 1992): e.g., Chukwuma attributed this to an “incubation”
period for equilibrium to be established at the gas-oil contact area. We also observe from Figure
5.11 (left panel) that the pressure during soaking does not equilibrate during the experiment.
Subsequent cycles (2-4) depart from an exponential pressure decay with respect to time,
indicating the presence of a minor leak and/or the impact of adsorption of CO2 in the core. The
potential for adsorption is discussed in more detail below. While any leak from the extraction cell
would invalidate the interpretation of the pressure responses, the overall pressure drop during
soaking is relatively small (<60 psi) and is not expected to affect the extraction process.
Accordingly, we restrict further analysis of the pressure decline to consider only the first cycle.
To further investigate the pressure variation during the soaking in the 1
st cycle, we compare
the total pressure-drop to the equilibrium pressure obtained from a Volume-Temperature (VT) flash
calculation (equilibrium calculation for fixed V, T and molar feed composition). We used PR EOS
(1978), with tuned binary interaction coefficients and volume shift parameters (see Chapter 4), to
calculate the equilibrium pressure of a mixture of CO2 and synthetic oil that corresponds to the
moles of initial oil in place, and the moles of CO2 injected into the cell. The CO2 moles were
estimated using CO2 density (at pressure and temperature conditions relevant to the HnP) and total
system volume (volume of the extraction cell minus the core’s bulk volume) The final pressure
recorded at the end of soaking during the 1st cycle is 4107 psi, while the equilibrium pressure from
the VT flash calculation is 4150 psi: The difference might be attributed to other mechanisms such

85
as CO2 adsorption that causes the pressure during soaking to drop below the calculated equilibrium
pressure.
In order examine the adsorption potential of CO2 in the shale core, additional PPD
experiments were performed using He and CO2 in tandem. The experiments were performed in
setup shown in Figure 5.2 combined with a floating-piston accumulator and an ISCO-pump used
to deliver gases at high pressure to the upstream cell. A total of 12 pulses (6 for each gas), were
performed at 500-4500 psi and 70 °C corresponding to the conditions of the HnP experiments. We
started by performing He pulse experiments to evaluate the pore volume of EF-3 as depicted in
Figure 5.12. The error bars represent projected uncertainties in pressure readings (0.08% of the
full scale) and system dead volume (+/- 0.5%). It is important to highlight the observed difference
in the estimated pore volume using core’s weight, NMR and PPD: The total pore volume using
core’s weight is 4.9 cc, while a value of 4.4 cc is obtained from NMR. The PPD results estimate a
maximum pore volume of 6.6 cc. The pore volume from the 1
st two methods, core weight and
NMR, appear consistent with around ~10% difference. However, the PPD results indicate a
significant increase in pore volume that amounts to approximately 35% when compared to the
estimation using the core weight. One possible explanation for this increase in pore volume is
attributed to gas (Helium) accessing smaller pores within the core than what is accessible by the
synthetic oil. Another possible explanation relates to handling and transportation of the core when
shipped from the HnP lab (Houston) to the PPD lab (Los Angeles): The core was visually inspected
upon arrival and did not show any fracture or/and cracks.

86
Figure 5-12: Estimated core pore volume from He pulse-decay experiments.
After evacuation, we proceeded with CO2 pulse experiments at the same pressure and temperature
conditions used in the He experiments. Figure 5.13 reports the dimensionless gas amount vs time
for one of the He and CO2 pulse-decay experiments, where blue circles represent the He data and
the grey circles correspond to CO2. The dimensionless gas amount is defined as a ratio of
concentration divided by initial concentration, C/Ci (concentration = moles / volume). We note
from Figure 5.13 that CO2 exhibits a lower dimensionless gas amount, or concentration, at the end
of the experiment when compared to the He pulse. This indicates that CO2 is adsorbing onto EF-3
assuming that the core pore volume remains constant.
6
6.25
6.5
6.75
0 1000 2000 3000 4000 5000
Core PV, cc
Core Pressure, psia

87
Figure 5-13: Dimensionless gas amount vs. time for He and CO2 pressure-pulse experiments.
Data from the He and CO2 pulses allows us to calculate the excess and absolute adsorption
of CO2 as detailed in equations 5.10 and 5.11:
𝑛𝐶𝑂2,𝑒𝑥𝑐𝑒𝑠𝑠 = 𝑛𝑡𝑜𝑡𝑎𝑙 − (𝜌𝐶𝑂2 ∗ 𝑉𝑝) 5.10
𝑛𝐶𝑂2,𝑎𝑏𝑠𝑜𝑙𝑢𝑡𝑒 = 𝑛𝐶𝑂2,𝑒𝑥𝑐𝑒𝑠𝑠/(1 −
𝜌𝐶𝑂2
𝜌𝐴𝑑𝑠.𝑝ℎ𝑎𝑠𝑒
) 5.11
where 𝑛𝐶𝑂2,𝑒𝑥𝑐𝑒𝑠𝑠 is the excess sorption (moles), 𝑛𝑡𝑜𝑡𝑎𝑙 is total moles in the system, 𝜌𝐶𝑂2 density of
CO2, 𝑉𝑝 is the core pore volume estimated from the He pluses, 𝑛𝐶𝑂2,𝑎𝑏𝑠𝑜𝑙𝑢𝑡𝑒 is the absolute
adsorption (moles), and 𝜌𝐴𝑑𝑠.𝑝ℎ𝑎𝑠𝑒 the density of the adsorbed phase (assumed here to be 1.101
g/cc = CO2 density at normal boiling point). The excess and absolute CO2 adsorption (moles per
kg of shale) vs. pressure is reported in Figure 5.14. It is evident from these measurements that CO2
absorbs onto the shale and can hence affect the pressure transient during soaking periods, and
subsequently affect the total amount of CO2 that is available for diffusion. Figure 5.14 shows that
the maximum value of the excess adsorption equals 0.32 mol/kg: Using the EF-3 skeleton density,
the maximum value is equivalent to 0.53 mol/cc. We note that the sorption measurements were

88
performed after the final cycle of the HnP experiments, and that the adsorbed amounts reported
here accordingly represent an upper bound for the role of adsorption. Furthermore, the absolute
adsorption depends on the assumed adsorbate density (see Eq. 5.11), that is still an active topic of
research.
Figure 5-14: Excess and absolute CO2 adsorption on EF-3.
5.3.2 Interpretation of Experimental Observation via Numerical Modeling
In this work, we utilize MRST, an open-source simulator developed at SINTEF Digital, for
numerical modeling of our HnP experiments. MRST provides a compositional module and is
coupled with a fluid flow module that allows for implicit compositional reservoir simulation. The
base code for compositional simulation, supporting convective flow in a single-porosity model
(SPM) representation, was previously validated against commercial and research simulators
(Moyner, 2021). The details of the MRST model used in this study are provided in Raslan (2024),
with relevant information included in this section to guide the reader. Further details can be found
in Appendix-B.
Commercial tools are frequently used to model, simulate and interpret HnP experiments,
as discussed in Chapter 4. The open-source environment of MRST provides the flexibility to

89
develop and test fit-for-purpose algorithms and investigate, in more detail, the complex
characteristics of diffusive mass transfer in unconventional resources. Building on existing MRST
modules, we developed a module to represent molecular diffusion to facilitate studies of relevance
to mass transfer during HnP in tight fractured reservoirs. The advantage of using MRST in this
study is to allow for modifications in two main aspects of the modeling of diffusive mass transfer
including: 1) evaluation of multicomponent diffusion coefficients, and 2) selection of different
representations of the diffusive fluxes.
We use PR EOS (1978) to perform equilibrium calculations, and the Lohrenz-Bray-Clark
(LBC) correlation, developed by Lohrenz et al. (1964), for evaluation of phase viscosities. In the
following, we summarize the main flow and transport equations and continue to present our
modeling of the EF-3 HnP experiments.
Mathematical Model
The isothermal multicomponent and multiphase flow can be described by a set of component
material balance equations. The mass conservation of all hydrocarbon components can be written
as follows:
𝜕
𝜕𝑡⏟(∑𝑗 𝜙𝜌𝑗𝑆𝑗𝑥𝑖𝑗)
accumulation
+ ∇ ∙ (∑ 𝜌𝑗𝑥𝑖𝑗𝑢⃗ ⏟𝑗 𝑗
convection
+ ∑ 𝐽 ⏟𝑗 𝑖,𝑗
diffusion
) = 0, 𝑖 = 1, … , 𝑛𝑐
5.12
where Ø is the porosity, xi,j is the mole fraction of component i in phase j, ρj is the molar density
of phase j, Sj is the saturation of phase j, 𝑢⃗ 𝑗
is the velocity of phase j, t represents time, Ji,j denotes
the diffusive molar flux of component i in phase j and nc is the number of components. The second
term in equation 5.12 represents the total flux due to convection and diffusion. The former is
described by Darcy’s law with an upstream weighting method. The molecular diffusion flux is
detailed in the following section.

90
Modeling of Molecular Diffusion
We extended the flux equations to include molecular diffusion using one of two options. In the
first option, the diffusive flux is calculated by the classical Fick’s law:
𝐽𝑖 = −𝜌𝐷𝑖
eff∇𝑧𝑖
, 𝑖 = 1, … , 𝑛𝑐 5.13
and in the second option, we utilize the generalized version of Fick’s law:
𝐽𝑖 = −𝜌 ∑ 𝐷𝑖𝑘
𝑛𝑐−1
𝑘=1
∇𝑧𝑘,𝑖 = 1, … , 𝑛𝑐 − 1
5.14
In equations 5.13 and 5.14, 𝐽𝑖 denotes the diffusive molar flux of component 𝑖 in bulk phase. It is
to note that 𝐷𝑖
eff in equation 5.13 represents a column vector, while 𝐷𝑖𝑘 from equation 5.14 is a
(𝑛𝑐 − 1 × 𝑛𝑐 − 1) matrix. To evaluate the diffusive flux in a porous material, we multiply
equations 5.13 and 5.14 by the ratio 𝜙𝑆𝑗/𝜏, where 𝑆𝑗
is the fluid saturation, and 𝜏 is the tortuosity.
To facilitate the calculation of the classical Fick’s law, it is common to use the extended
Sigmund correlation to obtain binary diffusion coefficients of the mixture (Sigmund, 1976a,
1976b; da Silva and Belery, 1989), combined with the formulation of Wilke (1950) to evaluate the
effective diffusivity per component.
The generalized Fick’s law is based on the Maxwell-Stefan (MS) framework, where the
MS binary diffusivities are related to the infinite dilution coefficients using the correlation of e.g.,
Kooijman and Taylor (1991). The estimation of the infinite dilution coefficients can be done from,
for example, the correlation of Leahy-Dios and Firoozabadi (2007). Finally, to convert the MS
diffusivities to Fickian diffusion coefficients, 𝐷, the following relationship is employed:

91
[𝐷] = [𝐵]
−1
[Γ] 5.15
where 𝐵 is a function of the MS diffusivities and Γ is the thermodynamic non-ideality factor. The
reader is referred to Taylor and Krishna (1993) and Hoteit (2013) for more details on the
formulation of the above diffusion models.
A compositional single-porosity radial model was built as shown in Figure 5.15, replicating
the dimensions of the experimental setup.
Figure 5-15: Different views of the numerical model describing the shale core: Left panel shows the whole model, middle panel
shows the shale core, and the right panel shows the horizontal cross-section from top of the extraction cell.
The extraction cell, representing a large fracture, is depicted in Figure 5-15 by the outer yellow
layers, and the core is shown by the inner red layers. Grid refinement, in radial and vertical
directions, was performed to reduce numerical diffusion. The extraction cell porosity was set to
99% to mimic the void space in the diffusion cell, and a ‘large’ value of permeability of 500 mD
was assigned. Matrix porosity (11.89%) and matrix permeability (𝑘𝑅 = 2.79 μD and 𝑘𝑧 =
3.88 μD) were used according to our interpretation of related experimental observations.
A value of the tortuosity factor is required to evaluate the diffusive fluxes in the porous
material, and we use here a value of 6 based on a literature review for common values reported for
the Eagle Ford formation (Zhang et al., 2019; Davudov et al., 2020).

92
The model was initialized at 14.7 psi and a constant temperature of 70 ºC. The
dimensionless volume shift parameters and the binary interaction coefficients were obtained by
matching the relevant density and solubility data from the literature. The fluid analysis was
reported in Chapter 4.
Initially, the core is saturated with 100% synthetic oil and the annulus with 100% CO2.
Furthermore, injection and production were represented by two wells completed at the top and
bottom of the (extraction cell) grid, respectively. Quadratic gas/liquid relative permeability
functions were assumed in the matrix while straight-line relative permeability functions were
assigned to the extraction cell (fracture).
With focus on proper representation of the diffusive mass transfer in modeling the HnP
experiment, we examined the performance of the two popular diffusion models: 1) The classical
Fick’s law, and 2) the generalized Fick’s law. In the classical Fick’s law, the diffusion coefficients
were evaluated using the framework of Sigmund (1976) and Wilke (1950). For the generalized
Fick’s representation, infinite dilution diffusion coefficients were calculated from the correlation
of Hayduk and Minhas (1982), and the multicomponent diffusion coefficients were estimated via
the formulation of Koojiman and Taylor (1991).
We start by exploring the calculation of oil recovery, as shown in Figure 5.16, where the
blue circles represent experimental data, the solid black line represents the result from calculations
with no diffusion, the dotted yellow line denote results from applying the classical Fick’s law, and
the purple dashed line report calculation results from using the generalized Fick’s law. We first
note the substantial difference between the no-diffusion case and the experimental data: The
difference amounts to a factor of 6 and hence demonstrates the major contribution (and
importance) of diffusion during the recovery process. Figure 5.16 also reveals that for both

93
diffusion models, diffusion contributes substantially to the recovery of oil, however, they are also
observed to overpredict the lab data.
Figure 5-16: Oil recovery from experimental data and simulation results.
From these observations, it is clear that the current (default) methods/models do not provide a
comprehensive assessment of the impact of molecular diffusion during hydrocarbon recovery. To
this end, we present and examine three approaches to address the disagreement between calculated
and observed experimental recovery.
To determine the sensitivity of the diffusion models parameters, we consider three different
approaches and assess their performance against the lab observations. It is common to use
multipliers to minimize errors that may arise in such models. Hoteit (2013) applied two multipliers
(for main- and off-diagonal diffusion coefficients) to match the results from the generalized Fick’s
law to the lab observations of Arnold and Toor (1967). Analogously, Shi et al. (2022) demonstrated
through experimental and modeling work that a single correction factor is required for each binary
when applying Sigmund’s correlation (Sigmund,1976) to model CH4 diffusion in hydrocarbon
mixtures.

94
In our first approach, we focus on the application of the generalized Fick’s law where the
calculated infinite dilution coefficients are adjusted based on lab measurements. We use
experimental data from Cadogan et al. (2016) for diffusion coefficients pertaining to CO2 infinitely
diluted in hydrocarbons, and data from Umezawa & Nagashima (1992) for hydrocarbons infinitely
diluted in CO2 as they represent measurements conducted as close as possible to the conditions of
our HnP experiment.
The procedure first entails interpolating/extrapolating the infinite dilution coefficients from
the relevant studies to 70 ºC and 4,000 psi, which then serve as the reference values for the
respective CO2-hydrocarbon binaries. These reference values are then compared to the calculated
infinite dilution coefficients (at the same conditions) from the available correlations to obtain
multipliers for each binary.
Figure 5.17 (left panel) reports a comparison of the infinite dilution coefficients for CO2 in
nC10, nC12, and nC16 obtained from interpolating the experimental work of Cadogan et al. (2016),
i.e., the reference values, with the calculated values obtained from five widely used correlations:
1. Wilke and Chang (1955)
2. Hayduk and Minhas (1982)
3. Siddiqi and Lucas (1986)
4. Wong and Hayduk (1990)
5. Leahy-Dios and Firoozabadi (2007)
We note that nC14 was not included in the experimental work conducted by Cadogan. Thus, we
developed a correlation between infinite dilution coefficient and carbon numbers using data from
Cadogan et al. (2016) and then interpolated the nC14 infinite dilution coefficient as depicted in
Figure 5.17 (right panel).

95
Figure 5-17: To the left is the comparison between the interpolated infinite dilution coefficients for CO2-oil and the calculated
values from each correlation. To the right shows the interpolated values as a function of carbon number.
Among the applied correlations, we observe that Wong and Hayduk (1990) and Hayduk and
Minhas (1982) provided the best prediction of the experimental data with average errors of 3.87
% and 10.69 %, respectively. We attribute the accuracy of the prediction from the correlation of
Wong and Hayduk (1990) to the fact that their correlation was developed using experimental data
for CO2 dissolved in organic solvents including normal alkanes. Similarly, Hayduk and Minhas
(1982) utilized experimental data for normal alkanes as solvents in developing the correlation. The
other correlations either did not use experimental data for CO2 in normal alkanes or used various
types of mixture to cover a broader range of systems (and subsequently deviate from specific
mixtures such as CO2 in normal alkanes).
A similar analysis was performed for the other concentration endpoint, i.e., hydrocarbon infinitely
diluted in CO2, with data obtained from Uzemawa and Nagashima (1992), see Figure 5-18.
However, for this data set, extrapolation to our HnP experiments was required given that the lab
data were generated at lower pressure and temperature. We note from Figure 5.18 that all
correlations deviate more than 40% from the experimental data. Hayduk and Minhas (1982),

96
shown in purple, provided the best prediction among all correlations: We attribute this to the fact
that the correlation was developed using experimental data with normal alkanes as solutes.
Figure 5-18: The experimental infinite dilution coefficients for oil in CO2 and the calculated values from selected correlations.
Based on the above analysis for the two infinite dilution limits (per binary), we note that the
correlation of Hayduk and Minhas (1982) provided the best overall prediction of the experimental
data. We proceeded to evaluate multipliers that correct for the infinite dilution coefficients
computed using the correlation of Hayduk and Minhas (1982) as shown in Table 5.6.
Table 5-6: Multipliers to correct for the infinite dilution coefficients obtained from Hayduk and Minhas (1982)
nC10 nC12 nC14 nC16
CO2-Oil 0.881 0.896 0.916 0.935
Oil-CO2 2.611 3.097 3.223 3.588
Next, the correction factors are applied to adjust the infinite dilution diffusion coefficients
predicted by Hayduk and Minhas (1982) and used in the generalized Fick’s to simulate the
recovery process and refer to this approach as the “Infinite Dilution Multiplier” approach. Figure
5.19 comparesthe experimental oil recovery from EF-3 with calculations including 1) no diffusion,
2) unmodified diffusion model, and 3) the infinite dilution multiplier approach. We observe from

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the figure that the application of correction factors does not result in a better agreement between
calculation and experimental data, with calculated recovery departing further from the
experimental observations than in the previous calculation approach.
Figure 5-19: Oil recovery when multipliers are applied to adjust Hayduk and Minhas (1982) infinite dilution coefficients.
To further investigate the gap between the experimental data and modeling results, we examine
two modifications to the formulation of diffusive mass transfer that we are applying in our
modeling. The first approach considers the uncertainty in the calculation of the multicomponent
diffusion coefficients. Measurement of multicomponent diffusion coefficients at reservoir
conditions is extremely challenging and for this reason, most published experiments are limited to
binary and ternary mixtures. The second approach explores the role of anisotropy in the core fabric,
as evident from the observed lamination in Figure 5.1, on the modeling of the HnP process. The
heterogeneity of the core potentially induces varying characteristic diffusion time with respect to
direction: This is not described by the model, thus far. Accordingly, it is reasonable to assume that
diffusive mass transfer across the observed laminations is slower than along the laminations. This
translates to a faster diffusive mass transfer in the core’s axial direction in comparison to the radial
direction.

98
The two approaches discussed above were implemented using multipliers that are applied
to reduce the overall diffusive flux into the core. In the first approach, a common multiplier is
applied to the calculated diffusion coefficients (Diffusion Case 1). In the second approach, a single
multiplier is applied on the radial diffusive flux (Diffusion Case 2). To obtain a better agreement
with the experimental oil recovery data, multipliers of 0.35 in Case 1 and 0.25 in Case 2 were
applied in calculations with the classical Fick’s law, while multipliers of 0.1 in Case 1 and 0.03 in
Case 2 were used in calculations with the generalized Fick’s law using infinite dilution coefficients
from Hayduk and Minhas (1982). The calculated oil recovery from these approaches is compared
with the experimental data in Figure 5.20.
Figure 5-20: Updated recovery plots with the addition of the modified diffusion models.
Next, we compare the produced oil composition from experiments and modeling in Figure 5.21.
We observe that in the no diffusion case, the compositions remain generally unchanged throughout
the cycles. However, for Diffusion Case 1 and 2, we note that the general trends of the produced
oil compositions are represented accurately by the numerical calculations. Additionally, we
observe that both modeling approaches, classical or generalized Fick’s, provide for a good
agreement with the lab measurements. Moreover, the application of multipliers, on either total flux

99
or in radial direction only, provides similar results. Finally, we observe that the classical Fick’s law
provides for an improved accuracy in the prediction of the produced nC12 and nC16 compared to
the generalized Fick’s law.
Figure 5-21: Comparison of produced oil composition (experimental vs simulation). The first column reports simulation results
where diffusion is neglected, columns 2-3 report calculations with the classical Fick’s law, while columns 4-5 provide results
from the generalized Fick’s law.
Discussion
Based on the above results and analysis, diffusion is observed to plays a vital role in enhancing
hydrocarbon recovery from shale during CO2 HnP processes. To highlight the contribution of
diffusion, it was illustrated how the recovery changes by a factor of 6 between a base case
simulation (with no diffusion) and the experimental observations. Moreover, for all the above
simulation calculations, quadratic gas/liquid relative permeability functions were assumed in the
matrix due to the lack of available data for shale plays in the literature. Sensitivity analysis was

100
performed on quadratic (n=2) and straight-line (n=1) relative permeability functions: No major
difference in oil recovery and produced composition results was observed as shown in Figures
5.22 and 5.23.
Figure 5-22: Comparison of oil recovery for two different relative permeability cases.

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Figure 5-23: Comparison of produced oil composition for two different relative permeability cases.
All available data pertaining to the core and fluids characterization were included during the
construction of the numerical model. While directional permeability was derived from pressure
pulse decay experiments, no available data exists to infer directional aspects of diffusional mass
transfer. To evaluate the impact of a directional diffusion flux, a multiplier was applied that allowed
us to reproduce, with reasonable accuracy, the experimental recovery. We considered two cases in
our work, one where the overall flux was reduced through a common multiplier and one where the
radial flux was adjusted by a single multiplier. While the latter approach might seem more
reasonable given the visible laminations of the core, both approaches provide for an excellent

102
agreement with the produced oil composition for all the cycles – note that the produced
compositions was not included in the selection of multipliers.
Contrary to the findings of several researchers (Hoteit, 2013; Sofla et al., 2016; Sistan et
al., 2022), we observe that the classical Fick’s law provides for accurate calculation of the produced
compositions and amounts. In fact, the classical Fick’s model performs slightly better than the
generalized Fick’s model. This is promising given the reduced computational requirement
associated with this approach compared to the generalized Fick’s law, which entails additional
layers of calculations.
As part of our work, the correlation of Leahy-Dios and Firoozabadi (2007) was also
investigated in place of the correlation of Hayduk and Minhas (1982). Both correlations provide
for similar trends in the calculated results. However, the multipliers required to match the model
using the correlation of Leahy-Dios and Firoozabadi to the lab data were less aggressive
(multipliers = 0.18 and 0.10 for Cases 1 and 2, respectively).
The focus of the above modeling work was to evaluate the recovery factors and produced
compositions. Based on the calibrated model, we then investigated the calculated pressure-drop as
observed during the soaking period. A comparison of the calculated pressure response and the lab
data for the first cycle of HnP is reported in Figure 5.24.

103
Figure 5-24: Pressure variation during soaking period of the first cycle (experimental vs simulation).
The simulation results are consistent with conclusions made from the results presented in Figures
5.19 and 5.20, i.e., the larger the impact of the diffusion, the faster the pressure drop during
soaking, and the higher the recovery per HnP cycle. Two additional observations can be made from
Figure 5.24. The calculation with the unmodified version of the generalized Fick’s law stabilizes
at about 4,150 psi. As highlighted earlier, this represents the equilibrium pressure of the system. It
is also noticed how the simulation calculations underestimate the pressure drop relative to the
experimental data. This departure could be attributed to the adsorption of CO2 to the shale cores,
which was not considered in this simulation study.
5.4 Summary and Conclusion
We presented an integrated experimental and modeling work to investigate the role of diffusive
mass transfer during CO2 HnP in an Eagle Ford shale core. Experiments were performed at 3,900-
4,400 psi and 70 ºC on a core saturated with a well-characterized normal alkane mixture. The
pressure during soaking, recovery, and produced oil compositions were collected and interpreted.
Furthermore, PPD and gas flow-through experiments were conducted, following the 4th cycle, that

104
allowed for an estimation of critical information including the effective core permeability and
porosity.
An open-source simulation tool was modified and applied to interpret the experimental
observations. Two diffusion models were used to calculate diffusion fluxes: the classical Fick’s
law and the generalized Fick’s law.
Key findings and recommendations of this chapter can be summarized as follows:
1. The interpretation of the lab data indicates a diffusion-dominated recovery mechanism. A
larger fraction of lighter components was produced initially, while larger fractions of
heavier components are produced during later cycles. Furthermore, the pressure behavior
during the soaking phase exhibits a linear relationship with respect to √𝑡 in support of a
diffusion-dominated process.
2. PPD experiments with CO2 indicated the affinity of CO2 to adsorb onto EF-3. This
observation is expected to contribute to the additional pressure drop (beyond the phase
equilibrium pressure) during the soaking periods.
3. The mainstream diffusion models cannot reasonably predict the lab observations without
adjustments to represent the physics of the diffusion process. Therefore, the open-source
code was applied and modified to address the uncertainty of the calculated diffusion
coefficients and to account for the directionality of the diffusion flux. The modifications
result in a good agreement with the recovery and produced compositions as observed in
the lab experiments.
4. The pressure-drop during the soaking period, calculated from the calibrated model, does
not agree well with the experimental observation and additional model adjustments were

105
not attempted. We attribute the departure, in part, to adsorption that is not included in the
simulation model.

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Chapter 6 Summary and Future Research Direction
One of the most important factors in economic analysis of EOR is the potential incremental
recovery that is commonly predicted using compositional reservoir simulators. Our work aims to
enhance the prediction of incremental recovery through improving the representation of diffusion
coefficients and diffusive mass transfer in compositional simulation.
In Chapter 3, we aimed to improve the understanding of multicomponent diffusive mass
transfer between matrix and fracture segments through an integrated experimental and modeling
work. We carried out displacement experiments using analog fluids and mesoporous medium, and
the role of molecular diffusion was investigated by performing separate diffusion experiments to
obtain diffusion coefficients for all relevant binary mixtures. Infinite dilution diffusion coefficients
were used to model multicomponent diffusion coefficients. The displacement experiments were
simulated using an in-house simulator and excellent agreement was obtained: The extensive
experimental/modeling work related to the diffusion coefficients of the analog fluid system was
used in interpreting the diffusive mass transfer between the matrix (stagnant) and fracture
(flowing) domains via a 1D linear model. The uniqueness of this framework is the combined
experimental and modeling workflow to obtain the multicomponent and effective diffusion
coefficients. By obtaining such data, we were able to predict, with reasonable accuracy, the
displacement experiments without the need to perform an additional parameter estimation.
In Chapter 4, we study diffusive mass transfer in shale cores by conducting and simulating
CO2 Huff-n-Puff (HnP) experiments at high pressure and temperature. We performed multiple HnP
cycles at varying injection conditions and diffusive mass transfer was investigated via (1)
evaluating the effect of injection pressure on oil recovery, (2) analyzing produced oil compositions,
and (3) studying the pressure decline during the soaking period. We demonstrate that higher oil

107
recovery was achieved by injecting at a higher pressure. We also discuss and illustrate the potential
change in miscibility conditions between HnP cycles and how this potentially affects oil recovery.
Furthermore, we analyze the pressure behavior during soaking to identify the role of diffusive mass
transfer. We used a commercial compositional simulator (CMG-GEM) to interpret the HnP
experimental results. When multicomponent diffusion coefficients were computed using the
correlation of Sigmund (1976), the simulator is unable to provide a reasonable prediction of oil
recovery and produced oil compositions. This suggests that compositional modeling of HnP
experiments may be in large error unless additional information is available to calibrate diffusion
modeling. Such calibration or modification of either diffusion coefficients or diffusion models is
not feasible with most commercial simulators. Therefore, it became our motivation in Chapter 5
to develop an open-source simulator that provides a comprehensive flexibility in modeling
diffusion coefficients and to investigate the performance of several diffusion models.
In Chapter 5, the objective was to demonstrate the importance of diffusive mass transfer
and recovery in fractured media during EOR operations and its implementation in simulation tools.
CO2 HnP was performed on an Eagle Ford core at reservoir conditions (at 3,900-4,400 psi and 70
ºC) under first contact miscibility conditions. Based on the pressure-drop during soaking as well
as the compositional analysis of the produced oil, molecular diffusion is observed to be the
dominant recovery mechanism in the performed HnP experiments. We also note that the pressure
variations during soaking periods indicate that CO2 adsorption seems to contribute to the observed
pressure-drop. Therefore, pressure pulse decay (PPD) experiments were conducted and confirmed
CO2 adsorption with a maximum excess adsorption of 0.32 mol/kg. Additional PPD experiments
as well as gas flow-through measurements were carried out for further characterization and
revealed a heterogeneous core permeability.

108
To interpret the experimental observations, a compositional simulation study was
performed using the open-source Matlab Reservoir Simulation Toolbox (MRST). The governing
equations of MRST were modified to include molecular diffusion that is described by the classical
Fick’s law and the generalized Fick’s law. All available data pertaining to the core and fluids
characterization were included during the construction of the numerical model. While directional
permeability was derived from PPD experiments, no available data to infer directional aspects of
diffusional mass transfer. To evaluate the impact of a directional diffusion flux, a multiplier was
applied that allowed us to reproduce, with reasonable accuracy, the experimental recovery results.
We considered two cases in our work, one where the overall flux was reduced through a common
multiplier and one where the radial flux was adjusted by a single multiplier. While the latter
approach might seem more reasonable given the visible laminations of the core, both approaches
provide for an excellent agreement with the produced oil composition for all the cycles – We stress
that the produced compositions was not included in any regression.
In closing, we provide the following recommendations for future research based on the work
reported in this thesis:
1. To aid in characterizing the contribution of diffusive mass transfer to the overall recovery from
gas injection experiments (e.g., HnP) performed on shale cores, it is recommended to conduct
in parallel infinite dilution diffusion experiments pertaining to the injected gas and relevant
hydrocarbon mixture. This integrated experimental and modeling work was successfully
applied on analog system in our work (Chapter 3). This step of obtaining infinite dilution
diffusion coefficient is vital due to the lack of universal correlation(s) to accurately estimate
these coefficients. Additionally, accurate estimation of these coefficients is essential to model
multicomponent diffusion coefficients.

109
2. To improve prediction of hydrocarbon recovery from shale cores, inclusion of experimental
and modeling work of CO2 adsorption is recommended. It is more likely to overpredict
recovery if adsorption is not considered: This is due to overestimation of total CO2 amount
available for diffusion.
3. Conduct gas injection experiments shale cores using varying injection pressure above the first
contact miscibility and delineate the impact on hydrocarbon recovery. This approach was
applied to two different cores in our work (Chapter 4). This is critical for field applications
where capital investment for injection requirements and expected incremental recovery must
be studied to ensure minimal investment and maximum recovery.
4. Utilize and build on the modified open source MRST code, from this work, to perform in-depth
analysis of gas injection experiments since most available commercial simulators do not often
provide flexibility in modeling multicomponent diffusion coefficients and applying specific
diffusive mass transfer models.

110
Appendix
Appendix-A: Pressure Pulse Decay and Gas Flow-Through Measurements
Figure A-1: Pressure pulse decay - Upstream (blue circles) and downstream (green circles) pressure data from the 1st pulse.
Figure A-2: ln (PD) vs. time from the 1st pulse.

111
Figure A-3: Pressure pulse decay - Upstream (blue circles) and downstream (green circles) pressure data from the 3rd pulse.

Figure A-4: ln (PD) vs. time from the 3rd pulse.

112
Figure A-5: Pressure pulse decay - Upstream (blue circles) and downstream (green circles) pressure data from the 4th pulse.

Figure A-6: ln (PD) vs. time from the 4th pulse.

113
Figure A-7: Pressure pulse decay - Upstream (blue circles) and downstream (green circles) pressure data from the 5th pulse.
Figure A-8: ln (PD) vs. time from the 5th pulse.

114
Figure A-9: Pressure pulse decay - Upstream (blue circles) and downstream (green circles) pressure data from the 6th pulse.
Figure A-10: ln (PD) vs. time from the 6th pulse.

115
Figure A-11: Gas Flow-Through 2nd experiment: Black for upstream pressure, blue circles for downstream pressures and red
represents estimated permeability.
Figure A-12: Gas Flow-Through 3rd experiment: Black for upstream pressure, blue circles for downstream pressures and red
represents estimated permeability.

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Figure A-13: Gas Flow-Through 4th experiment: Black for upstream pressure, blue circles for downstream pressures and red
represents estimated permeability.
Appendix-B: MRST Simulation Model Validation Examples
Analytical Solution
A synthetic example is constructed to validate the diffusive flux model in MRST by comparing the
calculated response to an analytical solution (Li, 1972). The problem considers a 1D linear
diffusion process of a binary fluid containing CO2 and CH4 in a domain with length, 𝐿, of 3.28 ft
(1 m). The domain is discretized by 20 equally spaced grid blocks, with the left 10 cells initially
containing CH4 and the right 10 cells initially containing CO2 (Figure B.1 – left panel). This creates
a sharp composition change at the center of the domain where diffusion is expected to initiate in
response to a constant and mutual diffusion coefficient of 1.5e-5 ft2
/s (1.5e-6 m2
/s). The system is
initialized at standard conditions of 14.7 psi (101 kPa) and 60 F (288 K).
The diffusion process can be simulated using the classical Fick’s law assuming that the
diffusion coefficient remains constant throughout. As indicated in the right panel of Figure B.1,
which reports the CO2 mole fraction along the domain after 0.2 days, the simulation model agrees
well with the analytical solution.

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Figure B-1: The left panel shows the 1D domain. The right panel reports the CO2 mole fraction (after 0.2 day) along the domain
for the analytical and numerical solutions.
Validation Against Arnold and Toor (1967) Laboratory Experimental data
The objective here is to validate our implementation of the generalized Fick’s law, via the SPM,
using data from the laboratory experiment of Arnold and Toor (1967). The study involves diffusion
of a ternary mixture composed of CH4, argon (Ar), and hydrogen (H2) in a Loschmidt tube with
initial compositions shown in Figure B.2. The pressure and temperature of the setup were 14.7 psi
(101 kPa) and 93 F (307 K). An important observation made by the experimentalists is that the
diffusion process did not follow a Fickian behavior. Accordingly, the dragging effect is significant,
and the application of the generalized Fick’s law is warranted.
We construct a 1D grid with a total length of 2.66 ft (0.81 m) of 20 equally spaced grid
blocks. The grid is shown in the left panel of Figure B.2 where the upper 10 grid blocks
representing the upper bulb of the Loschmidt tube, and the bottom 10 grid blocks are used to
represent the lower bulb. According to the given conditions, single-phase gas diffusion is expected
to initiate at the middle of the domain.

118
Figure B-2: Schematic of Arnold and Toor (1967) experimental setup.
Figure B.3 reports the average composition, in each bulb, of CH4 (middle panel) and Ar (right
panel) for three sets of calculations: Our work with MRST, Hoteit (2013), and Taylor and Krishna
(1993). Starting with Taylor and Krishna (1993), represented by dotted lines, they approached the
multicomponent problem analytically by solving the coupled diffusion equations following the
method of Toor (1964), which assumes a constant diffusivity (refer to Sherafati and Jessen (2018)
for more details on this solution).
Hoteit and our work in MRST rely on numerical simulation to find the concentration
variation of this multicomponent problem, and the generalized Fick’s law is used in both solutions.
In order to obtain a match with the experimental data, Hoteit applied 2 multipliers to tune the
calculated diffusion coefficients. For the same objective, and following a similar approach of
Hoteit’s, we multiply the diagonal elements by 1.5 and the off-diagonal elements by 2.52. As can
be seen from the calculation results, all three solution approaches provide a reasonable agreement
with the experimental data

119
Figure B-3: Average CH4 and Ar compositions versus time in the left and right panels, respectively. The
solutions (for both panels) are in the order of lab data, MRST, Hoteit (2013), and Taylor and Krishna (1993).

120
Nomenclature
A Cross-sectional area
a Ratio of the core pore volume to the upstream vessel
B Function of the MS diffusivities
b Ratio of the core pore volume to the downstream vessel
C Solute concentration
cg Gas compressibility
D Fickian diffusion coefficient
D MS diffusion coefficient
Dij
i,j→0
Diffusion coefficient of component i infinitely diluted in component j
Di,eff Effective diffusion coefficient of component i
f Flowing fraction
g
E Excess Gibbs energy
Ji Diffusive molar flux of component i in bulk phase
K Dispersion coefficient
Kij Combined effect of dispersion and diffusion
k Core permeability
kij Binary interaction parameter
L Length
M Amount of solute in the injected sample
np Number of phases
nc Number of components
𝑛𝐶𝑂2,𝑒𝑥𝑐𝑒𝑠𝑠 Excess CO2 moles
𝑛𝑡𝑜𝑡𝑎𝑙 Total CO2 moles
𝑛𝐶𝑂2,𝑎𝑏𝑠𝑜𝑙𝑢𝑡𝑒 Absolute CO2 moles
P Pressure
∆𝑃𝐷 Differential dimensionless pressure
qi Mass transfer between the flowing and stagnant domains
R Universal gas constant
Rc Tube radius
Sj Saturation of phase j
Si Dimensionless volume shift parameter
T Temperature
t Time
tD Dimensionless time
tR Mean residence time
u Mean fluid velocity
V Vessel volume
VP Core pore volume
vj Velocity of phase j

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xi Mole fraction of component i
xij Mole fraction of component i in phase j
z Direction along the packed column
Ø porosity
θi Multiplication of shape factor by effective diffusion coefficient for component i
𝜃𝑚 Roots for equation
Γ Thermodynamic correction factor
γ Activity-coefficient
ηi Viscosity of component i
η Mixture viscosity
ρj Molar density of phase j
𝜌CO2 Density of CO2
𝜌Ads.phase Density of the adsorbed phase
𝜇𝑔 Gas viscosity
σ Shape factor
σ
2 Variance of concentration profile
𝜏 Tortuosity

122
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Abstract (if available)

Abstract The recovery factors reported from unconventional reservoirs are, in general, extremely low, e.g., Hawthorne et al. (2013) reports that primary oil recovery from the Bakken ranges from 4 to 6%. Accordingly, enhanced oil recovery (EOR) techniques such as CO2 injection, are needed to unlock additional production. During gas injection, mass transfer mechanisms including molecular diffusion and convective mixing often play a critical role in the oil recovery process. Compositional simulators are generally used to quantify the potential incremental hydrocarbon recovery from such processes. Our work studies diffusive mass transfer during gas injection in tight fractured reservoirs and aims to promote accurate prediction of incremental recovery through improving the representation of diffusive mass transfer in compositional simulation tools.
In the 1st part of this research, we aim to improve the understanding of multicomponent diffusive mass transfer between matrix and fracture segments through experimental and modeling work. Displacement experiments were carried out using analog fluids and a synthetic mesoporous medium to effectively isolate and study the relevant physical mechanisms at play. The experiments were performed in packed columns utilizing silica-gel particles with an internal porosity. The silica-gel particles (40-70 micron) include an internal porosity with a narrow pore-size-distribution (PSD) centered at 6 nm that makes up approximately 50% of the overall porosity.
A quaternary analog fluids system consisting of Water, Methanol, Isopropanol, and Isooctane, was used because it mimics at ambient conditions the phase behavior of CO2, Methane, Butane and Dodecane mixtures at 2,280 psi and 100°C. Our selection of the analog fluid system and porous medium allowed us to investigate matrix-fracture fluid exchange of relevance to enhanced recovery operations in ultra-tight fractured systems. The effluents from these displacement experiments served as the basis for our analysis of the role of diffusive mass transfer.
The role of molecular diffusion in the displacement experiments was investigated by first performing separate diffusion experiments to obtain diffusion coefficients for all relevant binary mixtures. Infinite dilution diffusion coefficients were measured for all binary mixtures and then used to model binary and multicomponent diffusion coefficients over the entire relevant composition range. The accuracy of this approach was investigated and documented by performing additional binary diffusion experiments over a broader range of compositions.
The displacement experiments were performed and simulated using an in-house simulator and an excellent agreement was obtained: The extensive experimental/modeling work related to the diffusion coefficients of the analog fluid system was used in interpreting the diffusive mass transfer between the matrix (stagnant) and fracture (flowing) domains via a 1D linear dual-porosity model. The uniqueness of this framework pertains to the combined experimental and modeling workflow used to obtain the multicomponent and effective diffusion coefficients. By obtaining such data, we were able to predict, with reasonable accuracy, the displacement experiments without the need to perform any additional parameter estimation/adjustments.
In the 2nd part of this research, we study diffusive mass transfer in shale cores by conducting lab experiments and simulation of CO2 Huff-n-Puff (HnP) at high pressure and temperature. The aim of this effort was to further elucidate the role of diffusive mass transfer during EOR in tight fractured rocks. Two cores from a formation in the Middle East were evacuated and then saturated at 3500 psi and 50°C with a well-characterized normal alkane mixture consisting of Decane (nC10), Dodecane (nC12), Tetradecane (nC14) and Hexadecane (nC16). The normal alkane system was selected to facilitate EOS modeling of the fluid behavior and hence reduce uncertainty related to the fluid behaviors. We performed multiple HnP cycles at varying injection conditions: 2900-4000 psi and 70 °C. Diffusive mass transfer was then investigated via (1) evaluating the effect of injection pressure on oil recovery, (2) analyzing produced oil compositions, and (3) by studying the pressure decline during the soaking period.
Our experimental observations demonstrate that a higher oil recovery is achieved when CO2 is injected at a higher pressure to facilitate development of first-contact miscibility. We also observe that molecular diffusion acts as a dominant recovery mechanism in these HnP experiments, as evident from analyzing the produced oil composition and from examining the pressure behavior versus time during the soaking periods: The observed decline rate of the pressure during soaking demonstrates that molecular diffusion dictates the mass transfer during the HnP experiments. Additionally, we note that miscibility conditions can change from one HnP cycle to another, as the injected gas mixes with an oil composition that changes between cycles.
We used a commercial compositional simulator (CMG-GEM) to interpret the results from the HnP experiments. When multicomponent diffusion coefficients were computed using the correlation of Sigmund (1976), the simulator is unable to provide a reasonable prediction of oil recovery and produced oil compositions. This indicates that compositional modeling of HnP processes may be in large error unless additional information is available to calibrate diffusion modeling based on Sigmund’s correlation. Such calibration or modification of either the diffusion coefficients or the diffusion models are, however, not possible with most commercial simulators. Consequently, we adapt an open-source simulator Matlab Reservoir Simulation Toolbox (MRST) that provides for additional flexibility in the modeling of diffusion coefficients and application of different diffusion models.
In the 3rd part of this research, the objective was to further demonstrate the importance of diffusive mass transfer and recovery in fractured media during EOR operations and how simulation tools should be modified to provide for more accurate predictions. To accomplish this, we combine experimental and modeling work: Four CO2 HnP cycles were performed on an Eagle Ford core with a porosity of 11.89% and permeability of 2.5 μD. The core was initially saturated with a synthetic oil composed of nC10, nC12, nC14 and nC16. These experiments were carried out at reservoir conditions (3,900-4,400 psi and 70 ºC) under first-contact miscibility (FCM) conditions. Based on the pressure drop during soaking as well as the compositional analysis of the produced oil, molecular diffusion was again observed to be the central recovery mechanism during the HnP experiments.
We also note that the pressure variations during soaking periods indicate that CO2 adsorption appears to contribute to the observed pressure-drop. Therefore, to understand CO2 affinity to adsorb on the Eagle Ford core, pressure pulse decay (PPD) experiments were conducted and confirmed CO2 adsorption with an excess adsorption of 0.32 mol/kg at 4,400 psi. An additional set of PPD experiments, as well as gas flow-through measurements, were carried out to further characterize the shale core and facilitate modeling of the experimental observations.
To interpret the experimental observations, a compositional simulation study was performed using MRST. The governing equations of MRST were modified to include molecular diffusion represented by a) the classical Fick’s law (using effective diffusivity and independent concentration gradients as a driving force) as well as b) the generalized Fick’s law (using a full diffusivity matrix and coupled concentration gradients as a driving force). Using both diffusion models, numerical calculations show a very good agreement with the lab data (for all cycles) in terms of the recovery and produced composition, with the former model demonstrating higher accuracy and computational efficiency: Contrary to the findings of some researchers, the classical Fick’s law is, after minor adjustments, very accurate in predicting the experimental data. This is promising given the favorable (reduced) computational requirement associated with this approach compared to the generalized Fick’s law, which entails additional layers of calculations.

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Creator Alahmari, Saeed (author)

Core Title A study of diffusive mass transfer in tight dual-porosity systems (unconventional)

School Viterbi School of Engineering

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Degree Program Petroleum Engineering

Degree Conferral Date 2024-08

Publication Date 09/10/2024

Defense Date 07/01/2024

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