relative permeability equation-of-state: the role of …
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The Pennsylvania State University
The Graduate School
RELATIVE PERMEABILITY EQUATION-OF-STATE: THE ROLE OF PHASE
CONNECTIVITY, WETTABILITY, AND CAPILLARY NUMBER
A Dissertation in
Energy and Mineral Engineering
by
Prakash Purswani
© 2021 Prakash Purswani
Submitted in Partial Fulfillment
of the Requirements
for the Degree of
Doctor of Philosophy
May 2021
ii
The dissertation of Prakash Purswani was reviewed and approved by the following:
Zuleima T. Karpyn
Dissertation Co-Adviser
Associate Dean for Graduate Education and Research, College of Earth and Mineral Sciences
Professor of Petroleum and Natural Gas Engineering, Department of Energy and Mineral Engineering
Chair of Committee
Russell T. Johns
Dissertation Co-Adviser
George E. Trimble Chair in Energy and Mineral Sciences
Professor of Petroleum and Natural Gas Engineering, Department of Energy and Mineral Engineering
Amin Mehrabian
Assistant Professor of Petroleum and Natural Gas Engineering, Department of Energy and Mineral
Engineering
Xiaolei Huang
Associate Professor of Data Science, Penn State College of Information Sciences and Technology
Turgay Ertekin
Professor Emeritus of Petroleum and Natural Gas Engineering, Department of Energy and Mineral
Engineering
Special Committee Member
Mort D. Webster
Professor of Energy Engineering
Co-Director Initiative for Sustainable Electric Power Systems
Associate Department Head for Graduate Education, Department of Energy and Mineral Engineering
iii
ABSTRACT
Relative permeability (kr) is a transport property used for characterizing the flow of multiple phases through
a porous medium. Inputs of kr are integral for reservoir simulations. Multiple parameters such as phase
saturation, wettability of the medium, fluid properties, flow characteristics, pore topology, fluid phase
topology, and fluid/fluid interfacial areas are known to affect relative permeabilities. Current kr models are
functions of phase saturation that are matched for specific flow/experimental conditions. The other
parameters affecting relative permeabilities are inherently captured through these saturation functions.
Representation of relative permeabilities only in the saturation space causes non-uniqueness and path
dependency in relative permeabilities which often cause simulations to fail because they lack generality
and are not physically based. As a result, hysteresis in relative permeabilities arises, which is a major
modeling issue for reservoir simulations.
In this dissertation, models for relative permeabilities are presented by considering functional forms
that include the effects of the key controlling parameters on relative permeabilities. The purpose of this
dissertation is twofold, to (a) understand how different parameters, specifically, phase saturation, phase
connectivity, capillary number, and wettability affect relative permeabilities; (b) propose physically-based
kr models by including the effects of these parameters.
Relative permeabilities are modeled using an equation-of-state (EOS) approach where the exact
differential for relative permeability is written in phase connectivity and saturation ( ˆ ).S − A quadratic
response-based EOS for relative permeability is modeled in the ˆ S − space. Physical limiting conditions on
the state parameters are considered to constrain the EOS model. This model is tested for different capillary
numbers ranging from one to 10-6. In addition, we calculated the partial derivatives of relative permeabilities
in the state parameters using numerical data sets generated with pore-network modeling. A response for
relative permeability is derived in the ˆ S − space following the state function approach. The locus bounded
by residual nonwetting phase connectivity and residual nonwetting phase saturation is presented for two
contact angles in the water-wet regime. Finally, we investigated the role of wettability on phase trapping
iv
also using pore-network modeling. An extended Land-based hysteresis trapping model is presented and
compared against models from the literature. In addition, models are presented to capture the trends of
residual loci for different contact angles.
Results show that a simple quadratic response for relative permeability in the ˆ S − space captures trends
across different capillary numbers. The model tuned for a capillary number in the capillary dominated
regime can show predictive capability for other capillary numbers within the same regime. The linear kr-S
paths for high capillary numbers (small Corey exponents) and nonlinear kr-S paths for low capillary
numbers (high Corey exponents) are found to occur due to fast and slow changes in phase connectivity,
respectively. Limiting constraints help in the identification of the physical region in the ˆ S − state space.
Results also show that the response derived for relative permeability from relative permeability partial
derivatives using the state function approach can predict relative permeabilities over the entire numerical
data sets, regardless of the direction of flow, thus mitigating hysteresis. Further, the analysis of the effect
of wettability shows that both phase trapping as well as the locus of residual saturation and residual phase
connectivity are sensitive to contact angle changes. For low receding phase contact angles, the residual
locus remains fairly constant, but at higher contact angles, the shape of the residual locus resembles a closed
loop. Pore structure constraint at negligible saturation is found to control the shape of the residual locus.
Phase trapping was found to reduce significantly for high contact angles owing to pore-scale mechanisms
of layer flow of the receding phase and piston-like advance of the invading phase. A newly proposed
extended Land-based model is able to capture residual saturation trends for all contact angles.
Overall, through this research endeavor, we gain insight into the different intrinsic parameters that
affect relative permeability. Through the application of pore-scale measures, these insights are further
manifested into practical models that helps describe relative permeabilities physically.
v
TABLE OF CONTENTS
LIST OF FIGURES ................................................................................................................................... viii
LIST OF TABLES ..................................................................................................................................... xiii
NOMENCLATURE ................................................................................................................................... xv
ACKNOWLEDGEMENT .......................................................................................................................... xx
CHAPTER 1. INTRODUCTION ................................................................................................................. 1
1.1. Background ........................................................................................................................................ 1
1.2. Relative permeability ......................................................................................................................... 4
1.2.1. Parameters affecting relative permeabilities ............................................................................... 8
1.2.2. Models for relative permeability ............................................................................................... 21
1.3. Research objectives .......................................................................................................................... 33
1.4. Dissertation layout ........................................................................................................................... 34
1.5. Publication list ................................................................................................................................. 36
CHAPTER 2. IMAGING AND PORE-SCALE MEASUREMENTS ....................................................... 37
2.1. Introduction ...................................................................................................................................... 39
2.2. Methodology .................................................................................................................................... 46
2.2.1. Implementation of the segmentation techniques ....................................................................... 48
2.2.2. Benchmark case ........................................................................................................................ 48
2.2.3. Test case .................................................................................................................................... 48
2.2.4. Supervised machine learning (ML)-based on Fast Random Forest algorithm .......................... 49
2.2.5. Unsupervised machine learning based on k-means and fuzzy c-means clustering ................... 50
2.3. Results and discussion ..................................................................................................................... 52
2.3.1. Bulk measurements ................................................................................................................... 56
2.3.2. Pore-scale measurements .......................................................................................................... 61
2.4. Concluding remarks ......................................................................................................................... 63
vi
CHAPTER 3. EQUATION-OF-STATE AND CAPILLARY NUMBER ................................................. 65
3.1. Introduction ...................................................................................................................................... 67
3.2. Methodology .................................................................................................................................... 71
3.2.1. Development of a state function ............................................................................................... 71
3.2.2. Phase connectivity..................................................................................................................... 73
3.2.3. Development of relative permeability EOS .............................................................................. 74
3.2.4. Comparison to the development in Khorsandi et al. (2017) ..................................................... 77
3.2.5. Estimation of the coefficients of the EOS ................................................................................. 80
3.3. Results and discussion ..................................................................................................................... 82
3.3.1. Quadratic response for relative permeability ............................................................................ 82
3.3.2. Quadratic response prediction at neighboring conditions ......................................................... 86
3.3.3. Effect of capillary number ........................................................................................................ 86
3.4. Concluding remarks ......................................................................................................................... 90
CHAPTER 4. IMPACT OF WETTABILITY ON PHASE TRAPPING ................................................... 91
4.1. Introduction ...................................................................................................................................... 93
4.2. Methodology .................................................................................................................................. 100
4.2.1. Pore-network simulations ....................................................................................................... 100
4.3. Results and discussion ................................................................................................................... 103
4.3.1. Iso-quality curves for saturation and connectivity – interpreting PNM data .......................... 103
4.3.2. Initial-residual (IR) saturation trapping curves ....................................................................... 106
4.3.3. Modeling IR saturation trapping curves .................................................................................. 108
4.3.4. Phase connectivity – phase saturation ( χ – S) paths and trapping locus ................................. 113
4.3.5. Modeling residual phase connectivity and trapping locus ...................................................... 117
4.8. Concluding remarks ....................................................................................................................... 121
vii
CHAPTER 5. DEVELOPMENT OF EQUATION-OF-STATE USING PORE-NETWORK MODELING
.................................................................................................................................................................. 123
5.1. Introduction .................................................................................................................................... 125
5.2. Methodology .................................................................................................................................. 132
5.2.1. Example implementation of the EOS ...................................................................................... 136
5.2.2. Two-phase simulations using pore-network modeling ........................................................... 138
5.3. Results and discussion ................................................................................................................... 141
5.3.1. Evolution of connectivity with saturation ............................................................................... 141
5.3.2. Fitting kr-EOS to literature data .............................................................................................. 147
5.3.3. Relative permeability scanning curves: pore-network simulations ......................................... 149
5.3.4. Estimation of relative permeability partial derivatives ........................................................... 154
5.3.5. Prediction of relative permeability .......................................................................................... 156
5.4. Concluding remarks ....................................................................................................................... 160
CHAPTER 6. CONCLUDING REMARKS AND OUTLOOK FOR FUTURE RESEARCH ................ 162
6.1. Key concluding remarks ................................................................................................................ 165
6.2. Future research ............................................................................................................................... 167
APPENDICIES ......................................................................................................................................... 169
Appendix A. Basic theory of Minkowski functionals ........................................................................... 169
Appendix B. A consistent approach for χmax determination .................................................................. 175
Appendix C. Base regression code created for analysis presented in chapter 3 ................................... 177
Appendix D. Data used in chapter 3 (adapted from Armstrong et al. 2016) ........................................ 183
Appendix E. Procedure for developing iso-quality curves discussed in chapter 4 ............................... 185
Appendix F. Residual curves for different stopping criteria and wettability alteration in PNM .......... 186
Appendix G. Base code for running PNM for generating numerical data sets for chapters 4 and 5 .... 188
REFERENCES ......................................................................................................................................... 196
viii
LIST OF FIGURES
Figure 1. 3-D visualization of fluid flow in porous media showing the endpoint states of primary drainage
and imbibition processes. (Left), (middle), and (right) show the states of the medium before primary
drainage, after the completion of primary drainage, and after the completion of imbibition, respectively.
Figure adapted from Schlüter et al. (2016). .................................................................................................. 2
Figure 2. Schematic showing water-oil relative permeabilities for a water-wet medium. Hysteresis in
relative permeabilities is also displayed. The black curves represent a primary drainage process, while the
red curves represent a primary water injection process. The solid curves are water relative permeabilities,
while the dashed curves are the oil relative permeabilities. Directions of flow are marked by arrows on the
figure. Endpoint relative permeabilities and saturations are also marked on the figure. Adapted from (Blunt
2017). ............................................................................................................................................................ 5
Figure 3. Schematic representation of a 2-D multiphase porous system showing possible phase/phase
contact lines. Left to right the wetting phase saturation increases. When considered in 3-D, phase/phase
interfacial areas would be estimated along the areal region of contact. Adapted from Dalla et al. (2002). 11
Figure 4. Segmented image of a multiphase system acquired using x-ray imaging showing (a) disconnected
nonwetting phase, and (b) more connected wetting phase. ......................................................................... 12
Figure 5. Schematic showing a water-wet solid in the presence of oil. ..................................................... 16
Figure 6. Illustrations of different types of wettability for oil/water/solid systems. From left to right the
medium’s wettability to oil increases. ......................................................................................................... 17
Figure 7. Schematic showing three types of rock wettabilities characterized qualitatively from the
visualization of two-phase relative permeability curves. Here, the two phases are oil and water and the flow
(direction marked by the arrow) represent water injection. The black, green, and red curves are for an oil-
wet, a water-wet, and a mixed-wet rock, respectively. ............................................................................... 18
Figure 8. Schematic of nonwetting phase relative permeability showing hysteresis after flow reversal from
primary drainage (black curve) to primary water injection (red curve). The initial, trapped, residual, and
flowing nonwetting phase saturations are marked on the figure. Adapted from Carlson (1981). .............. 28
Figure 9. Schematic of the laboratory setup and image acquisition system (x-ray MCT scanner). DO stands
for the distance between the detector and the object, while OS stands for the object to source distance. This
figure shows that the sample (object) is very close to the source for finer resolution, the resolution was
coarsened by moving the sample stage laterally from the source, increasing OS and decreasing DO. ...... 47
ix
Figure 10. Imaged cross-section of dry (top) and brine saturated (middle and bottom) porous glass frit at
different voxel resolutions. The brine used for saturating the porous medium was 1M NaI solution. Non-
local means was used for filtering the raw images to remove image noise. ............................................... 53
Figure 11. Histograms showing the voxel population of the different grayscale intensity values for the
corresponding scans shown in Figure 10. ................................................................................................... 54
Figure 12. Segmented cross-sectional images showing three phases (solid, brine, and air). (a) thresholding
at a resolution of 6 µm (benchmark case), (b) thresholding at a resolution of 18 µm, (c) supervised machine
learning segmentation at a resolution of 18 µm, and (d) unsupervised machine learning segmentation at a
resolution of 18 µm. Zoomed-in version of the images are displayed on the sides to highlight distinct
features of segmented images. The upper and lower regions of interest (marked inside the segmented
images) correspond to labels 1 and 2, respectively. .................................................................................... 56
Figure 13. Vertical profiles of (a) porosity, (b) brine saturation, (c) air saturation, (d) solid fraction, (e)
brine fraction, and (f) air fraction for different segmentation techniques. .................................................. 58
Figure 14. Surface areas of air, brine, and solid phases on images segmented with different techniques. 61
Figure 15. Illustration showing the EOS state approach on a real path (simulation) taken during two-phase
flow simulation in jS and ˆ
j space. ........................................................................................................... 79
Figure 16. Phase saturation, relative permeability, and normalized Euler connectivity for the nonwetting
phase for different capillary numbers used for fitting the quadratic response for relative permeability as
well as for prediction purposes. Courtesy Dr. Ryan T. Armstrong. Data from Armstrong et al. (2016). This
data is tabulated in appendix D. .................................................................................................................. 81
Figure 17. (a) Quadratic response prediction versus simulation data (b) residual between the predicted and
simulation measurements for relative permeability based on the response surface fit shown in Figure 18.
.................................................................................................................................................................... 83
Figure 18. Contour map of the response surface of relative permeability as a function of phase saturation
and normalized Euler connectivity. The capillary number (~ 10-4), wettability, and pore structure have been
kept constant. Data points shown as black dots were taken from the two-phase flow simulations presented
in Armstrong et al. (2016). Dashed line represents a limiting boundary of plausible values. .................... 84
Figure 19. Partial derivative coefficients (calculated using Eqs. (3.10) and (3.11)) expressed as a function
of (a) phase saturation and (b) normalized Euler connectivity. .................................................................. 85
Figure 20. (a) Prediction of relative permeability and (b) residual error for capillary numbers ~10-3 and
~10-5 based on the response surface fit to capillary number 10-4 described in Figure 18. .......................... 86
Figure 21. R2 error for prediction of data at different capillary numbers using sub data set at capillary
number of ~10-4 as the fitted response surface. ........................................................................................... 87
Figure 22. Quadratic response surface fits to sub data sets at different capillary numbers. ...................... 88
x
Figure 23. Phase relative permeability plots with corresponding phase connectivity value (shown in blue)
(a) high capillary number ~1 (b) low capillary number ~10-5. The red solid lines represent the fit using
Corey model with ~ exponent value of (a) 0.76 and (b) 1.29. The residual saturation in (a) was set to 0 while
computing the Corey exponent because that data point was not known. .................................................... 89
Figure 24. Schematic representation of capillary trapped CO2 by chase brine. Local heterogeneities in pore
structure, mineral complexity, and wettability are depicted. ...................................................................... 94
Figure 25. Initial-residual characteristic curves for selected literature studies. ......................................... 98
Figure 26. Steps in pore-network modeling. (a) The extracted internal structure (pore space) of the porous
medium used in this study; (b) The extracted pore-network of the porous medium in (a) represented in a
ball and stick form; (c), (d), and (e) show three saturation steps of blue phase injection in the pore-network
completely filled with the red phase. (c) shows no injection, (d) is captured after some injection, and (e) is
captured after longer injection period. ...................................................................................................... 101
Figure 27. Iso-quality trends for four wettability cases (θ1 = 180°, 120°, 60°, 0°). Plots to the left show the
iso-saturation curves, while plots to the right show the iso-connectivity curves. The phase saturation, phase
connectivity, and relative permeability are for the receding phase (phase1) during phase2 injection. The
contour lines are plotted at intervals of 0.03 units. The procedure for developing these curves is displayed
in Figure 51 in appendix E. ....................................................................................................................... 104
Figure 28. Comparison of kr-S (upper row) and – S (lower row) paths for two Si values (~ 1.0 and ~ 0.9)
for two receding contact angles cases (θ1 = 180° and 0°). ........................................................................ 106
Figure 29. Initial-residual saturation curves for different wettability cases observed using pore-network
modeling. .................................................................................................................................................. 107
Figure 30. IR saturation trapping curves for four different contact angle cases (θ1 = 180°, 120°, 60°, 0°).
The dashed lines show the fits for the trapping model. The corresponding goodness measure of the fits for
all wettability cases and R2 values are displayed on the right plots. ......................................................... 110
Figure 31. Matching parameters plotted as a function of wettability for the IR trapping model presented in
this research (Eq. (4.7)). The corresponding fits are available in Figure 30. The shaded regions mark the
95% confidence interval. .......................................................................................................................... 111
Figure 32. – S paths for the receding phase during secondary injection process for different contact angle
cases. The red open circles represent the initial condition whereas, the black open circles represent the
residual condition for each – S path. ..................................................................................................... 113
Figure 33. Relative permeability contour plots for the – S paths shown in Figure 32 ......................... 115
Figure 34. Trapping locus of phase saturation and phase connectivity for different contact angles. ...... 116
xi
Figure 35. IR phase connectivity curves for four different wettability conditions (θ1 = 180°, 120°, 60°, 0°).
The dashed lines show the match for the phase connectivity trapping model (Eq. (4.8)). The corresponding
goodness measure of the fit for all wettability cases and R2 value is displayed on the right plot. ............ 117
Figure 36. Matching parameters plotted as a function of wettability for the IR phase connectivity trapping
model. The corresponding fits are available in Figure 35. The shaded regions mark the 95% confidence
interval. ..................................................................................................................................................... 118
Figure 37. Summary of initial-residual phase saturation and phase connectivity trends for four different
contact angles (θ1 = 180°, 120°, 60°, 0°). (a) Initial connectivity versus initial saturation. Here all four
contact angles collapse to one single data set; (b) initial connectivity versus residual connectivity; (c) initial
saturation versus residual saturation; (d) residual connectivity versus residual saturation. ...................... 119
Figure 38. Ball and stick representations of a porous medium with similar number of pores but increasing
number of throats. The Euler characteristic decreases from left to right as connections (throats) are
increased. .................................................................................................................................................. 134
Figure 39. Waterflooding χ - S paths from the literature (Table 11). All cases are for NCa < 10-4. Solid curves
show the best fits to Eq. (5.14). Saturation in the experiments move from right to left as shown by the
direction of the arrow. ............................................................................................................................... 142
Figure 40. The χ - S paths for different drainage and imbibition scans using PNM for the weakly water-wet
case (o
θ ~ 50 ). All PD curves begin at zero oil saturation but terminate at So ~ 1.0 (green), So ~ 0.9 (blue),
and So ~ 0.8 (red). Next, IMB curves terminate at residual conditions (squares). Finally, all SD curves are
simulated to So ~ 1.0. ................................................................................................................................ 144
Figure 41. Fitting parameters (p and k) after matching Eq. (5.14) to PNM scanning data (see Figure 40 for
three of the scans plotted here at 1.0, 0.9, and 0.8 So). The x-axis is the nonwetting saturation at the
termination of PD. The shaded region shows the error bars calculated for 95% confidence limits using the
nlparci function in Matlab®. The contact angle averages at 50o. ............................................................. 145
Figure 42. Best fits to the literature data in Table 11 using the (a) kr-EOS with constant partial derivatives
and (b) Corey form. ................................................................................................................................... 148
Figure 43. PNM simulations of imbibition (IMB) and secondary drainage (SD). Two contact angles are
used: the left column represents a fixed contact angle of 0o, while the right column is for uniformly
distributed contact angles between 40o and 60o with an average of ~ 50o. The black solid line in (a) and (b)
is for primary drainage (oil flood). Figures (a) and (b) also give ten imbibition curves at 0.1 saturation
intervals on the PD curve, while (e) and (f) show their χ - S paths. Figures (c) and (d) are for secondary
drainage and (g) and (h) show their corresponding χ - S paths. The red open circles represent the starting point
for IMB, while the black open circles represent the residual points for each IMB scan (similar to Figure
xii
40). Arrows show the direction of saturation change (IMB points to the left while SD points to the right).
.................................................................................................................................................................. 151
Figure 44. Relative permeability contours plotted from the 200 PNM imbibition simulations for the two
contact angle cases of 0o and ~ 50o. .......................................................................................................... 152
Figure 45. PNM simulations for 200 imbibition and secondary drainage scanning curves that begin at
different PD termination saturations (spaced by 0.005 saturation units). Two contact angles are shown, 0o
(a) and ~ 50o (b). The PD curves begin at So = 0. ..................................................................................... 152
Figure 46. Locus of residual connectivity and residual saturation generated from the different scanning
curves (Figure 43). The blue data point shows the limiting value of connectivity as saturation approaches
zero (see Eq. (5.6)). ................................................................................................................................... 153
Figure 47. Relative permeability partial derivatives estimated from fitting a cluster of data to a plane.. 155
Figure 48. Contour maps of predicted relative permeability using constant partial approach (a, b) and
quadratic response (c, d) for both wettability cases. Actual versus predicted relative permeability values
for the two approaches are shown in (e) and (f). The R2 for both approaches in (e) was ~0.90, while the R2
was ~0.90 and ~0.96 for the constant partial approach and quadratic response, respectively, in (f). ....... 158
Figure 49. Schematic of capillary tube filled with wetting and nonwetting phases. A zoomed-in view of the
interface across the phases is also shown. ................................................................................................. 170
Figure 50. Schematic of different surfaces with different types of Gaussian curvatures. ........................ 172
Figure 51. Schematic showing the procedure for developing the iso-quality curves discussed in chapter 4
(Figure 27). ............................................................................................................................................... 185
Figure 52. Initial-residual saturation curves generated from pore-network modeling for different contact
angle cases measured through the receding phase. The residual saturations are obtained for different
stopping criteria and are shown by different colors. ................................................................................. 186
Figure 53. Primary drainage and water injection capillary pressure scanning curves generated using pore-
network modeling. The different colors (and the arrows mark) give different cycles of injection. These
scanning curves are generated with the endpoint of primary drainage (or starting point of water injection)
at Si = 0.9. .................................................................................................................................................. 187
xiii
LIST OF TABLES
Table 1. Empirical oil/water relative permeability functions for different rock types and wetting conditions
from the literature. ...................................................................................................................................... 25
Table 2. Characteristics of imaging techniques from various experiments of fluid flow in porous media.
.................................................................................................................................................................... 40
Table 3. Summary of percent errors calculated for the different measured properties and segmentation
techniques relative to the benchmark case. The percent errors for the bulk measurements of porosity, fluid
saturation, and phase fractions are calculated for the respective average values across the sample height.
.................................................................................................................................................................... 59
Table 4. Measurements of air-brine interfacial area and Euler number of the nonwetting (air) phase for the
different segmentation techniques. The average air saturation was 14.4 % as determined from the
benchmark case whereas, thresholding, supervised machine learning, and unsupervised machine learning,
showed average values of 11.5%, 15.0% and 11.4%, respectively. ........................................................... 62
Table 5. Physical constraints imposed on the relative permeability response by considering key limiting
conditions that affect relative permeability as a function of phase saturation and phase connectivity. The
phase is assumed nonwetting, although extensions to other phases are easily possible. ............................ 75
Table 6. Euler characteristic values estimated through 2-D extrapolation for the pore structure used during
simulations in Armstrong et al. (2016). ...................................................................................................... 81
Table 7. Model coefficients and the goodness of quadratic response surface fit to phase saturation and phase
connectivity to the data presented in (Armstrong et al. 2016) at the NCa of ~10-4. ..................................... 82
Table 8. Coefficients for the quadratic response and goodness measures for quadratic response surface fits
to sub data sets at different capillary numbers shown in Figure 22. ........................................................... 88
Table 9. IR trapping models from the literature. iS , rS , and max
rS refer to the initial, residual, and
maximum residual phase saturations, respectively, while, C, a, b, c, α, and β are model parameters. ..... 108
Table 10. Properties of the pore-network extracted from the dry micro-CT image of a Bentheimer sandstone
(Lin et al. 2019). The parameter Z (2nt/np) is the coordination number; np and nt are the number of pores and
throats, respectively. The χmin for the image data is the value V-E+F-O exacted from the pore space of the
image. The χmax for the image data is back calculated from the corresponding χmin and Z values. See
additional details for consistent estimation of χmax in appendix B. ........................................................... 139
Table 11. Summary of information for the literature data shown in Figure 39. ....................................... 142
Table 12. Values of parameters fit to Eq. (5.14) using Table 11 data. The curves from the best fits are
shown in Figure 39. The 95% confidence limits were calculated using the nlparci function in Matlab®.143
xiv
Table 13. Summary remarks for p and k values for the evolution of phase connectivity for different cycles
of injection. ............................................................................................................................................... 147
Table 14. Best fit values for the kr-EOS and Corey form shown in Figure 42. The end-point permeability
is the same for comparison purposes. Thus, only χ
α and k are used as tuning parameters for the kr-EOS and
no for Corey. R2 values are shown. ............................................................................................................ 148
Table 15. The average values of the estimated partial derivatives of relative permeability shown in Figure
47. ............................................................................................................................................................. 156
Table 16. Values of plane fitting parameters in Eq. (5.21) used for the kr response in Eq (5.22). ........... 157
Table 17. Example of different objects and their Euler numbers. ............................................................ 174
Table 18. Data used in chapter 3 for nonwetting phase saturation, connectivity, and relative permeability
(adapted from Armstrong et al. 2016). ...................................................................................................... 183
xv
NOMENCLATURE
Shorthand notation
CCUS Carbon capture utilization and storage
CK Carmen-Kozeny
CFD Computational fluid dynamics
2-D/3-D/4-D Two-dimensional/three-dimensional/4-dimensional
DL Deep learning
DO Object-to-detector
EOR Enhanced oil recovery
EOS Equation-of-state
GHG Greenhouse gas
IFT Interfacial tension, (N/m)
IMB Imbibition
IR Initial-residual
IPCC Intergovernmental panel on climate change
LBM Lattice Boltzmann methods
Micro-CT (or µCT
or MCT) Micro-computed tomography
MIA Medical image analysis
ML Machine learning
MRI Magnetic resonance imaging
OS Object-to-source
PNM Pore-network modeling
PV Pore volume (cc)
PVT Pressure-volume-temperature
PD Primary drainage
REV Representative elementary volume (cc)
SD Secondary drainage
WAG Water alternating gas
WEKA Waikato environment for knowledge analysis
xvi
Symbols
A, B, C Coefficients for planar representation of relative permeability for 2D extrapolation in
chapter 3; coefficients for model fitting in chapter 5
αi Coefficients in the quadratic expression for relative permeability, i = 0, 1, 2, 11, 22, 12
αp (αw, αnw, αs) Phase-specific surface area (specific wetting phase surface area, specific nonwetting phase
surface area, and specific solid surface area, respectively) (1/m)
αw-nw Specific fluid/fluid (wetting-nonwetting) interfacial area (1/m)
β0, β1, β2 Betti numbers 0, 1, and 2 respectively
A Fluid/fluid interfacial area (state parameter)
Ap Surface area of phase p (mm2)
C Integration constant; Land’s trapping coefficient; Model constant Burdine’s equation
C Model constant Burdine’s
Dz Elevation of depth from the reference datum (m)
dl Line element [L]
ds Surface element [L2]
δE Entire surface of object E [L2]
Δx Finite distance within the porous medium (m)
E Number of edges in a polyhedra
E Object in space
f(σ) Interfacial tension scaling factor
F Faces in a polyhedral
g Gravitational acceleration constant (m/s2)
I (e.g., cosθ) Wettability (state parameter)
I Wetting fraction of the solid
a, b, c Coefficients for plane fitting partial derivatives in chapter 3; coefficients for model fitting
in chapter 5
d, e, f Coefficients for plane fitting partial derivatives in chapter 3
K Gaussian curvature [L-2]
k Tuning parameter in the phase evolution function; base (or absolute) permeability (mD)
κ Average fluid/fluid interfacial curvature [L-1]
kabs Absolute permeability (mD)
gk Geodesic curvature [L-1]
kj Effective permeability to phase j (mD)
kr-S Relative permeability—saturation
D
rnwk Nonwetting phase relative permeability during drainage
( )D
rnwnwfk S Nonwetting phase relative permeability during drainage at flowing nonwetting saturation
xvii
( )D
rnwnwik S Nonwetting phase relative permeability during drainage at initial nonwetting saturation
I
rnwk Nonwetting phase relative permeability during imbibition
o
rk End-point relative permeability
nonwettingr
k (or krnwet) Nonwetting phase relative permeability
krj (or r
k ) Relative permeability to phase j
refr
k Reference state relative permeability
krof Oil relative permeability of fixed data point for partial derivative estimation
wettingr
k (or krwet) Wetting phase relative permeability
Lw, Lnw Lengths of wetting and nonwetting phases in capillary tube (mm)
M0, M1, M2, M3 First [L3], second [L2], third [L], and fourth [-] Minkowski functionals
M0w, M0
nw First Minkowski functionals for wetting and nonwetting phases [L3]
M1(E), M2(E),
M3(E)
Second [L2], third [L], and fourth [-] Minkowski functionals for element E under
consideration
M1w, M1
nw Second Minkowski functionals for wetting and nonwetting phases [L2]
M2nw Third Minkowski functionals for wetting/nonwetting interface [L]
µj Viscosity of phase j (cP)
N Number of state parameters
NE Number of reciprocity relations
NCa Capillary number
NP Number of phases
n1, n2, nw, no Corey exponent for phase1, phase2, water, oil; n1 also used as the exponent for the IFT
scaling factor
np Number of pores
nt Number of throats
occp
n Number of occupied pores
occt
n Number of occupied throats
occht
n Number of occupied half-throats
O Number of objects
p Tuning parameter in the phase evolution function
Pc Capillary pressure (Pa)
Pj Pressure of phase j (Pa)
r Radius of capillary tube (mm)
R (R1, R2) Principal radii of curvatures [L]
ρj Density of phase j (kg/m3)
xviii
σ Interfacial tension (N/m)
σo Original interfacial tension (N/m)
σgw, σgo, σow Interfacial tensions of gas/water, gas/oil, and oil/water systems (N/m)
Se Effective phase saturation
*
gFS Reduced free-flowing gas phase saturation
*
grS IFT scaled residual gas saturation
Sgr Residual gas saturation
Si or Snwi Initial phase saturation (or initial nonwetting phase saturation)
Srj (or Sr or Snwr) Residual phase saturation of phase j; residual nonwetting phase saturation
Sj (or S ) Saturation of phase j
Sm Minimum wetting phase saturation
Snwf Flowing nonwetting phase saturation
Snwi Initial nonwetting phase saturation
Snwt Trapped nonwetting phase saturation
Snwet Nonwetting phase saturation
Sof Oil saturation of fixed data point for partial derivative estimation
Sorg Residual oil saturation to gas
*
orgS IFT scaled residual oil saturation to gas
Sref Reference state saturation
Swet Wetting phase saturation
Swir Initial wetting phase saturation
Swirr Irreducible water saturation
Sw,imb, Sw,dr Wetting phase saturation during imbibition and drainage
*
,w imbS , *
,w drS Reduced wetting phase saturation during imbibition and drainage
yi State parameter i
Z Coordination number
S ,
,
A Partial derivatives in the EOS
λ Pore structure (state parameter); also pore-size distribution parameter
λrwet, λrnwet Wetting and nonwetting phase tortuosity ratios
λwet, λnwet Wetting and nonwetting phase tortuosity factors
θ Contact angle, °
θgw, θgo, θow Contact angles for the gas/water/solid, gas/oil/solid, and oil/water/solid systems, °
θi ith corner angle of a rough surface element, i = 1, 2, 3…n
Average contact angle
Porosity
uj Interstitial velocity of phase j (ft/d)
xix
V Number of vertices in a polyhedra
Vb Bulk volume (cc)
Euler characteristic
(δE) Euler characteristic of entire surface of object E
(E) Euler characteristic of object E
ˆ i Normalized initial phase connectivity
0 Normalized phase connectivity at zero saturation
ˆ r Normalized residual phase connectivity
ˆj
(or ) Normalized connectivity of phase j
ˆ S − Connectivity—saturation
ˆ rr S − Locus of residual phase connectivity and residual saturation
χmax Maximum disconnected Euler characteristic of the pore space in a rock
χmax-image Maximum disconnected Euler characteristic of the pore space in a rock from image data
χmax-PNM Maximum disconnected Euler characteristic of the pore space in a rock from PNM data
χmin Maximum connected Euler characteristic of the pore space in a rock
χmin-image Maximum connected Euler characteristic of the pore space in a rock from image data
χmin-PNM Maximum connected Euler characteristic of the pore space in a rock from PNM data
χnonwetting-phase Euler number for nonwetting phase
χpore-structure Euler number for the pore space
ˆref
Reference state connectivity
ˆof
Normalized oil connectivity of fixed data point for partial derivative estimation
Ф Phase distribution
Фr Residual phase distribution
Фref Reference state phase distribution
xx
ACKNOWLEDGEMENT
I would first like to express my sincere gratitude toward my advisers, Prof. Zuleima Karpyn and Prof.
Russell Johns for providing their never-ending support, and sharing their valuable time, ideas, and critiques
for the betterment of my project. Prof. Karpyn has been a wonderful teacher and a true mentor. She valued
my success as her own and provided me with opportunities to continually grow as an individual and as a
professional, and my life is enriched because of those experiences. Prof. Johns always pushed me one step
further and I will always be grateful to him for the learnings that happened in those moments. He taught me
that growth is possible in moments of discomfort and that it is important to feel comfortable when
experiencing discomfort and uncertainty to become a valuable researcher.
I would like to thank Prof. Martin Blunt for hosting me at Imperial College London, and for sharing his
time, resources, and expertise that added value to my project.
I would also like to thank my dissertation committee, for their time and their critical suggestions that
brought improvements to my project.
I would like to thank my friends and the extended EME family that has been home for the past several
years. It is with their support that such projects become possible.
Lastly, I do not think I can thank my family enough, especially my parents. My education would not
have been possible without their sacrifices and their unwavering support.
Acknowledgements of financial and resource support
Part of this work was performed in support of the National Energy Technology Laboratory’s ongoing
research under the RSS contract number 89243318CFE000003. The author gratefully acknowledges
financial support from the Energi Simulation Foundation, member companies of the Enhanced Oil
Recovery Joint Industry Project at Penn State, and the John and Willie Leone Family Department of Energy
and Mineral Engineering (EME). is also gratefully acknowledged. Additional funding support from the
Holleran and Bowman scholarship from the EME Department is also kindly acknowledged. Lastly, resource
support from the EME Department and the EMS Energy Institute at Penn State, and the Department of
Earth Science Engineering at Imperial College London are gratefully acknowledged.
Prakash Purswani
March 2021
1
CHAPTER 1. INTRODUCTION
1.1. Background
Understanding fluid flow in a porous medium is at the heart of engineering disciplines such as petroleum
engineering. Single-phase flow in porous systems is less complex, but the introduction of an additional
immiscible phase creates a system with involved physics. Multiphase flow is found in a variety of technical
challenges facing society today. Some examples include sequestration of CO2 in geological formations to
mitigate greenhouse gas emissions, removal of nonaqueous chemicals for remediation of groundwater, and
secondary and tertiary recovery methods to sustain/improve energy production from hydrocarbon
reservoirs.
Simple waterflooding is a multiphase process that is deployed in the secondary phase of oil production.
The waterflooding process requires reinjection of produced water to sustain the production of oil after the
decline of high reservoir pressure. Enhanced oil recovery (EOR), however, involves the injection of an
agent such as a chemical or a gas to further oil production (Lake et al. 2014). EOR techniques are generally
employed during the tertiary phase of oil recovery when waterflooding is in its most matured stage, and oil
production is declining. One of the most common EOR method is the application of polymers to facilitate
favorable mobility ratio (close to 1) by resolving viscosity imbalances between the injecting phase, water,
and the receding phase, oil. Other common examples include the use of a surfactant to cause miscibility
between oil and water by reducing interfacial tension differences. Similarly, gases such as CO2 are often
used to improve oil recovery by oil swelling to cause the oil’s viscosity to reduce for easier recovery (Johns
and Orr 1996). Although the application of CO2 has been with the petroleum industry for decades, it is now
being increasingly favored as an important method for carbon capture purposes. This technique is referred
to as carbon capture utilization and storage (CCUS).
2
Apart from utilization, CO2 storage known as carbon sequestration is also being ramped up to reduce
carbon emissions for tackling problems of growing global temperatures. A few pilot scale efforts for this
exist in the North Sea. CO2 sequestration is a multiphase process where CO2, primarily in the supercritical
state, is injected in geological formations such as depleted hydrocarbon reservoir, or aquifers for long-term
storage. In the process of injecting CO2, water (already present in the formation) is displaced. This is known
as the primary drainage process for displacement in a water-wet media. The term ‘primary’ is used to
reflect that the external phase (CO2 in this case) is introduced in the pore space for the first time. On the
trailing end of the CO2 plume, however, the CO2 phase experiences displacement by water and this process
is typical of an imbibition process, once again, given the rock formation is primarily water-wet. A 3-D pore-
scale visualization of the endpoint states of a saturated porous medium after primary drainage and
imbibition is shown in Figure 1.
Figure 1. 3-D visualization of fluid flow in porous media showing the endpoint states of primary drainage and imbibition processes. (Left), (middle), and (right) show the states of the medium before primary drainage, after the completion of primary drainage, and after the completion of imbibition, respectively. Figure adapted from Schlüter et al. (2016).
Solid
Pore
space filled
with 100%
wetting
phase
Nonwetting
phase
3
Figure 1 is adapted from the x-ray imaging experiment by Schlüter et al. (2016). The porous medium is
a bead pack which is shown with transparency to allow for the visualization of the pore space. Figure 1
(left) represents the state just prior to primary drainage. At this point, the pore space is completely filled
with the wetting phase which is shown by the clear background. Figure 1 (middle) shows the end of the
primary drainage process, where the medium is occupied mostly by the nonwetting phase shown in green.
Empty spaces in the middle figure mark the occupancy by the wetting phase which is quantified as the
irreducible wetting phase saturation. Apart from CO2 injection into aquifers, in nature, the primary drainage
process occurs when the oil first migrates from the source rock to the oil reservoir (Blunt 2017). During the
natural density driven flow of oil, oil displaces brine which originally occupies the pore space. Figure 1
(right) shows the state of the medium at the end of the imbibition process. The majority of the medium is
now occupied by the wetting phase (clear) and the green blobs are the trapped nonwetting phase blobs,
which are quantified together as the residual nonwetting phase saturation.
Multiphase processes occur at different scales. From pore-scale (as shown in Figure 1), to core-scale
(example, laboratory corefloods), to the reservoir-scale. At the reservoir-scale, multiphase flow in
geological formations is modeled by using tools such as reservoir simulations. These are physics-based
models that take information from the field as well as laboratory experiments as inputs to make engineering
decisions. Reservoir simulations are critical as they allow for conducting ‘what-if’ scenarios to test diverse
set of conditions and thus help in mitigating risk and finding optimum engineering decisions. To capture
the physics of flow of multiple phases, simulations require transport properties such as relative
permeabilities and capillary pressures. These inputs are often estimated in the laboratory and calibrated for
use in the simulator for the process being simulated.
In this dissertation, the focus is on relative permeabilities, but discussions for capillary pressures is
provided wherever appropriate.
4
1.2. Relative permeability
Relative permeability is the transport property that helps quantify the flow of multiple phases in a porous
medium. The relative permeability (krj) to a phase (j) is the ratio of the phase’s effective permeability (kj)
to the base (or absolute) permeability (k) of the medium. The effective permeability of a phase is defined
as the permeability of a phase at less than 100% occupancy in a porous medium saturated with multiple
phases. It estimated from extensions to the Darcy’s law and is calculated as,
( )j j zj
j
j
P gDku
x
−= −
, (1.1)
where uj is the flux of phase j; the ratio kj/µj is the proportionality constant termed as the mobility ratio of
phase j; µj and ρj are the viscosity and density of phase j, respectively; the terms inside the bracket, Pj -
ρjgDz, is the potential difference; Pj is the pressure of phase j; g is gravitational acceleration constant; Δx is
the finite distance within the porous medium; Dz is the elevation or depth from the reference datum.
The effective permeability to a fluid is a complex function of a variety of different factors, such as the
phase saturation, pore structure of the medium, wettability of the medium, the flow conditions such as the
interfacial tension between the involved phases, flow rate, and fluid viscosities, and the topology of the
individual phase. Each of these factors are described in more detail in the subsequent sections of this
chapter.
Figure 2 shows a schematic of relative permeabilities to oil and water typical for primary drainage and
primary imbibition processes. This schematic is representative of a water-wet media and other wettabilities
will be considered later. Primary drainage (shown in black) begins at 100% water saturation as the oil
invades the pore space. As water recedes the porous medium, its saturation and relative permeability drops,
while oil saturation and relative permeability to oil increases. The effect of saturation on relative
permeability is intuitive. The more a phase is present in the medium, that phase’s ability to move in the
5
pore space increases. Thus, the relationship for relative permeability with saturation remains monotonic.
The curvature, the endpoint values on the relative permeability curve, and the endpoint saturations,
however, are distinct for the set of experimental conditions.
Figure 2. Schematic showing water-oil relative permeabilities for a water-wet medium. Hysteresis in relative permeabilities is also displayed. The black curves represent a primary drainage process, while the red curves represent a primary water injection process. The solid curves are water relative permeabilities, while the dashed curves are the oil relative permeabilities. Directions of flow are marked by arrows on the figure. Endpoint relative permeabilities and saturations are also marked on the figure. Adapted from (Blunt 2017).
Sw
1
0 1
k roo
Swirr
k rwo
1-Sor 551-Sor
krw (Primary drainage)kro (Primary drainage)krw (Primary imbibition)kro (Primary imbibition)
6
The endpoint of primary drainage is marked by negligible change in the oil/water saturation. At this point,
the water saturation left in the medium is termed as the irreducible water saturation (Swirr), and the endpoint
oil relative permeability (kroo) can be estimated from the flow experiment via stabilized pressure and flow
rate measurements. The reverse cycle is the imbibition process (shown in red) where now water is injected
into the medium. This is essentially waterflooding the core to extract oil. As water is injected into the
medium, water saturation rises and so does the relative permeability to water. Now, oil saturation in the
core drops as more oil is recovered, and so does the relative permeability to oil. The endpoint saturation for
oil is called residual oil saturation (or residual nonwetting phase saturation or simply residual saturation).
On the water saturation axis, this is marked as 1-Sor. At this point, the endpoint relative permeability to
water (krwo) can be measured.
The choice of residual saturation is largely subjective. Experimentalists make their own decision to
terminate the flow experiment based on the reason for the experiment. For example, some may conclude
the experiment in the first few pore volume to get trends of early oil recovery, while others may extend for
hundreds of pore volumes (or with the application of centrifuge) to drive to the limits of residual saturation
possible. The ultimate trapped phase saturation depends on the pore structure together with the wettability
of the medium and not on the stopping criterion or flow conditions. We bring more context on the issue of
phase trapping in chapter 5 of this dissertation.
The endpoint relative permeabilities and the endpoint oil/water saturations are critical inputs for
reservoir simulations. In current modeling practices, these endpoint values, measured in the laboratory, are
used for calibration of the kr-S path, specific to the experimental conditions and fed as inputs to the reservoir
simulator. These models will be described in a later section.
Figure 2 shows that the kr-S drainage paths versus imbibition paths are quite distinct for both oil and
water relative permeabilities which shows that relative permeabilities are path dependent in the saturation
space. This conveys that relative permeability models that are calibrated for the drainage cycle cannot be
used for the imbibition cycle. This path dependency or nonuniqueness in relative permeabilities is termed
hysteresis and can cause numerical problems in reservoir simulations. For example, the use of drainage
7
relative permeabilities over imbibition relative permeabilities can lead to incorrect estimation of oil
recoveries through EOR processes (Carlson 1981). Similarly, without consideration of hysteresis when
simulating carbon sequestration can lead to inaccurate estimates of CO2 migration and ultimate quantity of
trapped CO2 (Juanes et al. 2006).
One of the physical reasons considered responsible for hysteresis in relative permeabilities is associated
with phase trapping. The remaining phase at the end of one cycle of injection (for example, primary
drainage) is linked to possibilities of different reassociation of this phase when the cycle is reversed. This
results in a different flow path of the phase.
One other physical reason responsible for hysteresis in relative permeabilities is contact angle
hysteresis. Contact angle hysteresis is the difference between the contact angles measured at the three phase
(fluid/fluid/solid) contact point when the denser fluid is invading the medium (advancing contact angle)
versus when this fluid is retracting from the medium (receding contact angle). Experimental evidence shows
that advancing contact angle is greater than receding contact angle, and that both can be related to the
intrinsic contact angle, which is the contact angle measure of the static fluid/fluid/solid system on a clean
solid surface (Morrow 1975). Surface roughness is considered as the primary cause for hysteresis in contact
angle. No porous media, when considered at the microscopic scale would have perfectly clean surfaces.
These microscopic irregularities and the resultant entrapped fluid in the rough surface ridges are deemed as
the reasons for hysteresis in contact angles.
Another reason considered to cause relative permeability hysteresis, which is inherently associated with
contact angle hysteresis, is the different types of flow mechanisms that may occur when a phase is
advancing versus when the phase is receding. Piston-like advance, cooperative pore body filling, snap-off,
layer flow, and flow with bypass are some key types of flow mechanisms identified in the literature for
flow under capillary dominated regime (Lenormand and Zarcone 1984; Valvatne and Blunt 2004). Other
flow mechanisms such as drop traffic flow and ganglion dynamics are found to occur at high capillary
numbers (Avraam and Payatakes 1999; Avraam and Payatakes 1995; Rücker et al. 2015).
8
Early efforts for resolving hysteresis in multiphase flow began with the treatment of capillary pressures.
It was identified that hysteresis occurs because of representation of transport properties strictly in the
saturation space. The hypothesis was that phase saturation alone cannot represent flow and that inclusion
of other pore-scale parameters is necessary toward resolving hysteresis. This hypothesis was proven in the
works by Hassanizadeh and Gray (1993) and Reeves and Celia (1996) where application of fluid/fluid
interfacial areas was included for addressing hysteresis.
Together with phase saturation, multiple other key controlling parameters such as phase connectivity,
fluid/fluid interfacial areas, wettability, capillary number, and pore structure have been recognized in the
literature to affect relative permeabilities. In the following subsections each of these parameters is described
in some detail.
1.2.1. Parameters affecting relative permeabilities
1.2.1.1. Fluid/fluid interfacial areas1
Integral geometry provides a means to quantify the structures of geometrical entities. Researchers have
adopted this approach to quantify connectivity measures of pore structures, as well as wetting and
nonwetting phases to evaluate multiphase flow in permeable systems. There are four useful measures from
integral geometry known as Minkowski integrals that describe the shape of a 3-D geometrical structure
(Armstrong et al. 2018; Blunt 2017; Wildenschild and Sheppard 2013). The first Minkowski (M0) functional
refers to the volume of the structure. For example, pore volume of a pore structure, or saturation of fluid
phases occupying the pore space. The second Minkowski functional (M1), however, corresponds to the
1 Parts of the text presented in this sub-section are published in Purswani et al. (2020) J. Pet. Sci. Eng.
9
phase’s surface area (Ap). These, when divided by the bulk volume, result in specific surface area of the
phase (αp) (Landry et al. 2014; Landry et al. 2011),
p
p
b
A
V = . (1.2)
The use of specific surface areas is recommended over actual surface areas because it removes system
dependence. This is similar to the use of specific solid surface areas for the estimation of base permeability
of a porous medium as observed in the development of the Carmen-Kozeny (CK) equation (Lake et al.
2014). The specific surface areas are then used for estimating the specific fluid/fluid interfacial area as
(Dalla et al. 2002),
1 2
.w nw w nw s
−
= + − (1.3)
Here, αp represents a phase-specific surface area, ( , ,s w
and nw
corresponding to the specific solid surface
area, specific wetting phase surface area, and specific nonwetting phase surface area, respectively); Vb is
the bulk volume of the medium; and αw-nw represents the fluid/fluid interfacial area.
The third Minkowski functional (M2) corresponds to the average curvature at the boundary between
two objects which is commonly used for quantifying local capillary pressure from two-phase image data
(Armstrong et al. 2018; Blunt 2017). Finally, the fourth Minkowski functional (M3) represents the integral
of the total Gauss curvature of an object which is related to the Euler characteristic (χ) as,
3
4
M
= . (1.4)
10
The discussion on Euler characteristic is available in the next subsection and additional details and
examples for Minkowski functionals is available in appendix A.
Fluid/fluid interfacial areas have been an important measure to analyze multiphase flow in porous
media for a variety of applications (Culligan et al. 2004; Reeves and Celia 1996). One such application is
the nonaqueous phase liquid dissolution rates which is critical for evaluating environmental chemical
transport. Experimental attempts at measuring interfacial areas already exist, such as the oil/water and
air/water fluid/fluid interfacial area measurements in a sand pack column (Saripalli et al. 1997b; Saripalli
et al. 1997a). These measurements were made by the application of surface-reactive tracers which
selectively adsorb at the fluid/fluid interface and cause its retardation during the miscible displacement
experiment. Although such independent methods exist, the application of x-ray imaging into fluid flow
research has allowed for superior ways of quantifying fluid surface areas and interfacial areas (Blunt 2017;
Culligan et al. 2005; Culligan et al. 2004; Dalla et al. 2002; Landry et al. 2014; Landry et al. 2011).
Sophisticated visualization of the trapped fluid phases and improved algorithms for estimating surface areas
of voxelated entities have enabled the estimation of the total surface areas of the various phases inside of a
porous medium. These can then be used toward estimating interfacial areas among different phase pairs.
A schematic 2-D visualization of a saturated porous media with trapped wetting and nonwetting phases
at two different saturations is shown in Figure 3. The solid, the wetting, and the nonwetting phases are
shown by the gray, blue, and green colors, respectively. The corresponding phase pair interfacial contact
lines are also displayed. Figure 3 (right) can be understood as a snapshot following a saturation step-change
during a typical imbibition process. In a 3-D representation, these contact lines would represent the surface
areas of contact. As the respective phase saturations change, the respective total surface areas of the fluid
phases change, consequently, the fluid/fluid interfacial areas change (qualitatively expressed in Figure 3)
and thus provide a pore-scale measure of the movement of a fluid during multiphase flow.
11
Figure 3. Schematic representation of a 2-D multiphase porous system showing possible phase/phase contact lines. Left to right the wetting phase saturation increases. When considered in 3-D, phase/phase interfacial areas would be estimated along the areal region of contact. Adapted from Dalla et al. (2002).
1.2.1.2. Phase connectivity2
Figure 4 shows 3-D representations of the nonwetting (left) and wetting (right) phases inside of a saturated
porous medium. These 3-D renderings were acquired using x-ray imaging at static experimental conditions.
More details on the experiment are available in chapter 2 of this dissertation. The disconnected nature of
the nonwetting phase and the connected nature of the wetting phase can be easily visualized. The
nonwetting phase appears as isolated blobs or clusters, while the wetting phase appears more continuous.
2 Parts of the text presented in this sub-section are published in Purswani et al. (2019), Comput. Geosci.
Solid phase
Non-wetting phase
Wetting phase
Wetting phase/solid surface area
Non-wetting phase/solid surface area
Non-wetting phase/wetting phase surface area
12
Figure 4. Segmented image of a multiphase system acquired using x-ray imaging showing (a) disconnected nonwetting phase, and (b) more connected wetting phase.
Finding a unique mathematical definition for connectivity in porous media has been an active point of
research (Aydogan and Hyttinen 2013). There are a number of connectivity parameters proposed in the
literature such as the Euler characteristic (Vogel 2002), percolation theory (Hovadik and Larue 2007),
connectivity function (Allard 1993), contour tree connectivity (Aydogan and Hyttinen 2013), coordination
number, and fractal dimension (Blunt 2017). Out of these measures, the Euler characteristic () has been
the simplest and most widely used measure of connectivity in porous media (Allard 1993; Aydogan and
Hyttinen 2014).
The Euler number identifies phase connectivity by considering the number of clusters and the number
of connections for these clusters. The Euler number decreases with an increase in the number of clusters.
Euler characteristic is a topological invariant originally proposed by Leonhard Euler for a polyhedra as the
alternating sum of vertices (V), edges (E), faces (F), and objects (O) and is computed as (Richeson 2008),
V E F O = − + − . (1.5)
a b
13
Extending the concept to complex phase structures, the Euler Poincaré formula has been widely used
for quantifying connectivity of microstructures as,
0 1 2 = − + , (1.6)
where the parameters 0 1 2, , and are the zeroth, first, and second Betti numbers, respectively.
0
represents the number of clusters, 1 is the number of holes or redundant loops (the maximum number of
breaks that can be made without having the cluster split into two as explained by Herring et al. 2013), and
2 is the number of enclosed voids. 2 is usually considered to be zero for the calculation of the Euler
characteristic (connectivity) of both the wetting and the nonwetting phases. While this may be true for the
nonwetting phase, since there can be no solid grains or wetting phase globules suspended in a continuous
nonwetting phase in a consolidated porous medium, this may not be true when calculating the Euler
characteristic of the wetting phase, where suspended nonwetting phase globules can occur within a
continuous wetting phase. Euler numbers range from to − + where a highly connected phase has a large
negative value while a highly disconnected phase has a large positive value.
Recent studies conducted using x-ray micro computed tomography (micro-CT) advocate for the use of
either both fluid/fluid interfacial area as well as the Euler characteristic (Mcclure et al. 2018; Mcclure et al.
2016) or suggest use of just the Euler characteristic (Schlüter et al. 2016). In Mcclure et al. 2018 and
Mcclure et al. (2016) it was shown that by including all Minkowski functional (i.e., both Euler characteristic
and fluid/fluid interfacial areas together with saturation) nearly all of the hysteresis observed in capillary
pressure measurements could be accounted for successfully. However, the authors fail to show capillary
pressure predictions by including just phase saturation and Euler characteristic. It is likely that the majority
of the hysteresis could still be captured without the need for the interfacial area measurements. Further, for
modeling purposes, it is suitable to minimize the total number of variables involved, such that the physics
14
of the problem is reasonably captured and at the same time the overall computational complexity is
minimized. Therefore, in this work, we use Euler characteristic as the measure of phase connectivity.
Pore-scale fluid properties such as fluid/fluid interfacial areas and fluid connectivity (measured through
the Euler characteristic of a fluid phase) are measures that describe the flow of phases within the porous
medium. The porous media properties such as porosity, permeability, and tortuosity, however, help define
the representative elementary volume (REV) over which the continuum assumption holds valid. Thus, for
consistency, the size of the extracted sub volume for pore-scale analysis and property estimation should be
sufficiently large that it is equal to or greater than the porous media-defined-REV.
1.2.1.3. Capillary number
Capillary number is a dimensionless number that is described as the ratio of viscous forces to interfacial
forces. The use of capillary number allows to account for important factors that affect relative
permeabilities, namely, the interfacial tension, fluid viscosity, and the flood rate. It is usually calculated as
(Lake et al. 2014),
,Ca
uN
= (1.7)
where u is the interstitial velocity; µ is the viscosity of the injecting phase; and σ is the interfacial tension
between the flowing phases. Some researchers also include wettability (with the application of the term
cosθ) or the porosity into the definition of capillary number as multiplication factors in the denominator of
Eq. (1.7).
The importance of capillary numbers can be understood from capillary desaturation curves which show
an inverted S-shaped relationship between capillary number and the residual saturation (Lake et al. 2014).
The curve is specific to the experimental conditions of rock and fluids used. It shows that residual
saturations can be reduced significantly at very high capillary numbers. The goal for EOR processes is thus
15
to achieve high capillary numbers to attain low residual oil saturations. High capillary numbers are
accomplished through either high flood rates (within the engineering design to maintain reasonable
injectivity) or through the use of high viscosity polymers to increase viscous forces, or with the application
of surfactants to reduce interfacial tensions (and consequently reduce interfacial forces).
Two types of flow regimes, namely, capillary dominated versus viscous dominated flow can be
identified from the capillary desaturation curves. The threshold is marked around capillary numbers of 10-
4. Below this threshold, capillary dominated regime occurs which is attained with low flood rate and/or high
interfacial tension conditions. Flow near the wellbore would experience high flow rates as opposed to flow
far away from the wellbore, where the flow is capillary dominated, and thus relative permeabilities in these
regions will be different. Hence, understanding of the full range of capillary number is important. This is
critical for modeling carbon sequestration where storage in the formation is ensured via the capillary
trapping mechanism.
The kr-S paths for different capillary numbers are inherently different. From the experiments by
(Delshad et al. 1987; Fulcher et al. 1985) it was shown that relative permeability paths are straighter (x-
shaped) for high capillary numbers (low interfacial tensions) while the paths are concave for low capillary
numbers. In the work by Fulcher et al. (1985), the effect of capillary number on two-phase relative
permeabilities was investigated through steady-state experiments by considering viscosity and interfacial
tension effects independently. Both wetting and nonwetting phase relative permeabilities as well as residual
saturations were found to be significantly affected by both interfacial tensions and viscosity changes
(Fulcher et al. 1985). The overall impact of capillary number (as a group) was more significant on wetting
phase flow than on nonwetting phase flow. The impact of capillary number on multiphase flow has also
been demonstrated at the pore-scale with two-phase simulations by Armstrong et al. (2016). The
implications of capillary number on relative permeability and modeling efforts are provided in chapter 3 of
this dissertation.
16
1.2.1.4. Wettability
Wettability is defined as the ability of a solid surface to have preferential affinity to one phase in the
presence of another phase (Anderson 1986a). It is a property of the porous medium. Contact angle
measurement provide the most accurate measure of wettability. It is conventionally measured through the
denser phase. For example, for an oil/water/rock system the contact angle is measured through the water
phase. Figure 5 shows a schematic of a water-wetting solid in the presence of oil. From the Young’s
equation, the balance of interfacial forces, gives the measure of wettability (θ) as follows,
,-ws os
ow
cos
= (1.8)
where σow, σos, and, σws are the oil/water, oil/solid, and water/solid interfacial tensions, respectively.
Figure 5. Schematic showing a water-wet solid in the presence of oil.
There are different types of wettabilities. With respect to oil/water/rock system, the rock may be
characterized as water-wet (θ = 0°-75°), intermediate-wet (θ = 75°-105°), or oil-wet (θ = 105°-180°). These
rough estimates were provided through experiments by Treiber et al. (1972). Complete water-wetness and
complete oil-wetness would occur at θ = 0° and θ = 180°, respectively. Figure 6 shows a schematic of
different rock wettabilities.
Solid
OIL
WATER
17
Figure 6. Illustrations of different types of wettability for oil/water/solid systems. From left to right the medium’s wettability to oil increases.
Other important classifications of wettability include fractional-wettability and mixed-wettability. Both
these types include parts of the porous medium that may be oil-wet and other parts that may be water-wet.
The difference between the two lies in the way these wettabilities are developed in a porous medium.
Fractional wettability is often developed in unconsolidated porous media by mixing solid grains of different
wettability types (for example, plastic beads—oil-wet and glass beads—water-wet)(Klise et al. 2016;
Landry et al. 2014; Landry et al. 2011). When these grains are mixed and compressed to form one medium,
there are pores that are completely oil-wet versus pores that are completely water-wet.
On the contrary, mixed-wettability is developed in a porous medium due to the process of aging the
rock sample. As such, most oil reservoirs are naturally mixed-wet. In a laboratory, the aging process is
carried out after primary drainage when an initial oil saturation has been established in the rock. Aging
requires subjecting the rock sample to high temperatures and pressures for a period of time. These
conditions are subjective and depend on the experimentalist and are set based on the test requirements. At
initial oil saturation conditions, parts of the rock are in direct contact with the oil and yet other parts are in
contact with water which is present in irreducible amount. During aging, the portions of the rock in contact
with oil are said to become oil-wet. This leads to the generation of mixed-wettability where within the same
pore, parts can be oil-wet and parts can be water-wet (for example, corners of the pore space occupied by
water will remain water-wet). The degree of wettability alteration depends on the aging conditions. Mixed-
wettability was first coined by Salathiel (1973).
Neutral wettingWetting Nonwetting
OIL
Solid
WATER
Completely water-wet Water-wet Neutral-wet Oil-wet Completely oil-wet
18
Wettability of a medium is often characterized qualitatively using relative permeability curves. A
schematic for oil/water relative permeabilities during water injection are shown for three types of
wettability in Figure 7. The endpoint water relative permeabilities and residual oil saturations are marked
for the three scenarios.
Figure 7. Schematic showing three types of rock wettabilities characterized qualitatively from the visualization of two-phase relative permeability curves. Here, the two phases are oil and water and the flow (direction marked by the arrow) represent water injection. The black, green, and red curves are for an oil-wet, a water-wet, and a mixed-wet rock, respectively.
1
Sw
0 1
kr
flow
krwo
2
krwo
3
1-Sor1 1-Sor3
1-Sor2
krwo
1
19
The endpoint values of water relative permeabilities and residual oil saturation together with the cross-
point saturations where the oil and water relative permeability curves intersect are used as qualitative cues
for wettability assessment of the rock from a flow experiment. Typically, for a water-wet rock (green curves
in Figure 7), the cross-point saturation is greater than 0.5, and the endpoint water relative permeability is
low ~ 0.2 and can go lower than 0.05 for extremely water-wet media (Blunt 2017; Lake et al. 2014). This
is because water is wetting the surface and would consequently occupy the smaller regions of the pore space
such as the pore corners or crevices. Thus, the water conductance remains low. For an oil-wet rock (black
curves in Figure 7), however, water would occupy the centers of the pore space which leads to higher water
conductance and consequently high endpoint water relative permeabilities (~0.5). Also, for oil-wet rocks,
the cross-point saturation is typically lower than for water-wet rocks.
For mixed-wettability, endpoint water relative permeability remains higher than the water-wet case.
Interestingly, experiments have shown that the residual oil saturation for the mixed-wet case is often the
lowest (Jadhunandan and Morrow 1995; Salathiel 1973). The reasons for this observation are still in debate
in the literature, but one of the hypotheses is that mixed-wettability provides for continuous pathways for
oil to flow in the medium and if flow experiments are prolonged over long periods, oil trapping can be
minimized significantly. Discussions on wettability and its consequences to phase trapping are presented
in detail in chapter 4 of this dissertation.
1.2.1.5. Pore structure
It has always been a challenge to quantify pore structure information. No single metric exists in the
literature. Pore structure differs not just from one type of formation to another, but also from one medium
to another even from the same formation. This is because of heterogeneity that exists in natural geological
formations (Lake et al. 2014). No two naturally occurring porous media will be exactly alike. Some of the
most used porous media properties that give insight into the pore structure are the permeability of the rock,
20
which describes the ability of a porous medium to transmit fluids, and the porosity of the medium, which
describes the ability of the medium to store fluids. The ratio of the square root of permeability to porosity
is often used as a quantitative measure for characterizing pore structure information. Other measures often
used for modeling base permeabilities, are the tortuosity and specific surface areas of the solid surface.
Tortuosity, a dimensionless porous medium property, is defined as the square of the ratio of capillary tube
length to the length of the representative elementary volume (Lake et al. 2014), which is essentially the
squared ratio of path length traversed by the fluid in the pore space to the length of the porous medium.
Other quantitative measures most frequently used for characterizing pore structure information are the
distribution of pore and grain sizes. Pore-size distributions are experimentally measured though the use of
Mercury (Hg) intrusion porosimetry where a primary drainage capillary pressure curve is generated for a
Hg/air/rock system. Hg is injected as the nonwetting phase into the medium to increasingly high capillary
pressures to enter smaller sized pores. Through such experimentation, information on the pore sizes (and
average pore sizes) for the medium is extracted in the form of a frequency distribution plot. A pore-size
distribution parameter is often set as the calibrating exponent for capillary pressure and relative
permeability curves (Brooks and Corey 1964; van Genuchten 1980; Land 1968).
Measures for the pore structure such as the pore-size distribution, or the square root of permeability
over porosity are bulk (or average) measures for a porous medium. Techniques like x-ray imaging, flow in
micromodels, and pore-network extraction models provide other quantitative measures for characterizing
pore structures by taking information at the pore-scale (Blunt 2017; Fatt 1956; Lenormand and Zarcone
1984). These include the coordination number, aspect ratio, pore topology, geometric shape factor (or the
distribution of the geometric shape factors).
Coordination number and pore topology give direct information on the connectivity of the pore space.
Coordination number is defined for a pore as the average number of throats that are in direct connection to
a pore (Blunt 2017), whereas the topology of the pore space is estimated as the Euler characteristic of the
pore space which can be represented for the entire region of interest by normalizing with respect to the bulk
or pore volume of the region.
21
Individual pore element sizes play an important role in characterizing the pore structure information.
For example, the pore-body to pore-throat aspect ratio is found to cause hysteresis in relative permeabilities.
High aspect ratios are linked to increased trapping of the nonwetting phase due to increased snap-off events
during imbibition (Jerauld and Salter 1990). The geometric shape factor, however, provides information of
the shape of the pore/throat elements and is defined as the ratio of the cross-sectional area of an element to
the square of its perimeter. If all pore/throat elements of a porous medium were circular and uniform, there
would be no phase trapping in the pore space since all elements will be drained completely by the invading
phase. Therefore, for simulation techniques such as pore-network modeling the shape factors becomes
critical as it helps in attaining pore structures with noncircular (polygonal) network elements that allow for
trapping of phases in pore corners. Availability of polygonal-shaped network elements also allow layer
flow where the phase may be connected through the corners.
In this dissertation, we keep pore structure information constant for the sets of simulations performed
for numerical data set generation. This is critical for the state function approach of modeling relative
permeability. Simultaneous efforts have been on going in our research group to develop state function-
based models for characterizing the base permeability of a porous medium with the knowledge of the
different pore structure metrics.
1.2.2. Models for relative permeability
1.2.2.1. Corey-type models
Initial efforts for modeling relative permeability were presented by Purcell (1949) using a bundle of
capillary tubes, and Burdine (1953) with the application of capillary pressure curves and the tortuosity
parameter. Burdine’s equations for wetting/nonwetting phase relative permeabilities were expressed as,
22
( )
2
2 0
12
0
/
/
wetS
c
rwet rwet
c
dS Pk
dS P
=
, (1.9)
( )
12
2
12
0
/
/
wet
cS
rnwet rnwet
c
dS P
k
dS P
=
, (1.10)
where krwet and krnwet are the relative permeabilities to the wetting and nonwetting phases, respectively; λrwet
(=λ/λwet) and λrnwet (=λ/λnwet) are the wetting and nonwetting phase tortuosity ratios, respectively; λ is the
porous medium tortuosity factor; λwet and λnwet are the wetting and nonwetting phase tortuosity factors,
respectively; Sm is the minimum phase saturations; and Pc is the capillary pressure.
Burdine provided simplified saturation-based expressions for the wetting and nonwetting phase
tortuosity factors as,
1 m
wer
t mwet
S S
S
−=
− (1.11)
1 m
nw
r
et nwrrnwet
nw
S S
S S
− −
−= (1.12)
where Swet and Snwet are the wetting and nonwetting phase saturations, respectively; Sm and Snwr are the
minimum wetting and residual nonwetting phase saturation, respectively.
Corey (1954) extended Burdine’s equation by approximating,
( )
2
1
0
o or o or
o orc
C S S for S S
for S SP
− =
, (1.13)
23
where ( ) / 1 ,orCC S= − C is a constant. The following relative permeability equations were presented by
Corey (1954).
4
1
o or
ro
or
S Sk
S
−=
−
, (1.14)
2 2
1 11
o or o orrg
m or or
S S S Sk
S S S
− − = − −
− −
. (1.15)
Extension to Corey’s equations were presented by Brooks and Corey (1964) in a more general form to
estimate wetting and nonwetting phase relative permeabilities as follows,
( )2 3
rwet ek S
+
= , (1.16)
( )2
21 1rnwet e ek S S
+ = − −
, (1.17)
where Se, known as the effective phase saturation, is defined as ( ) ( )/ 1r rS S S− − ; Sr is the residual phase
saturation; the parameter, λ, is the pore-size distribution index.
These Corey models were further simplified and generalized as exponential models as follows,
1
1 1
1 12 21
n
p rporp rp
rp rp
S Sk k
S S
−=
− −
, (1.18)
24
where krp1 and 1orpk are relative permeability and endpoint relative permeability to phase1; Sp1, Srp1, and Srp2
represent the saturation of phase1, endpoint saturation of phase1, and endpoint saturation of phase2,
respectively; n1 is the tuning exponent. This expression for water/oil relative permeabilities during
waterflooding then are,
1
wn
o w wirrrw rw
or wirr
S Sk k
S S
−=
− − , (1.19)
1
on
o o orro ro
or wirr
S Sk k
S S
−=
− − , (1.20)
where krw and kro are the water and oil relative permeabilities during waterflooding. orwk is the endpoint
relative permeability to water and Sor is the residual oil saturation which are determined at the end of
waterflooding, whereas orok is the endpoint relative permeability to oil and Swirr is the irreducible water
saturation which are determined at the end of oilflooding (prior to waterflooding). no and nw are the tuning
exponents. Similar to the exponential model other empirical models are available in the literature for
specific set of operating conditions (Fulcher et al. 1985; Honarpour et al. 1982). See Table 1 for empirical
expressions of oil/water relative permeabilities from the literature. For additional saturation-based models
for relative permeability the reader is referred to Honarpour et al. (1986).
Three-phase relative permeability models are not discussed in detail here, but some of the more
commonly used three-phase models include, Naar and Wygal (1961); Stone I (Stone 1970), Stone II (Stone
1973), and Land’s model (Land 1968). A comprehensive comparison of these models against three-phase
experimental data was summarized by Abder (1981).
25
Table 1. Empirical oil/water relative permeability functions for different rock types and wetting conditions from the literature.
Reference Rock type Wettability Injection type
Honarpour et al. (1982) Sandstone and conglomerate Water-wet Water injection
( )2.9
3.60.035388 0.010874 0.565561 1
w wi w orrw w w wi
wi or wi or
S S S Sk S S S
S S S S
− −= − + −
− − − − (1.21)
Honarpour et al. (1982) Sandstone and conglomerate Oil/intermediate-wet Water injection
( ) ( )( )1.91
1.5814 0.58617 1.2484 11 1
w wi w orrw w wi wi w wi
wi wi or
S S S Sk S S S S S
S S S
− −= − − − − −
− − − (1.22)
Honarpour et al. (1982) Sandstone and conglomerate All Water injection
( )( )
1.8
21
0.76067 2.6318 11 1
oor
wi o orro or o or
or wi or
SS
S S Sk S S S
S S S
−
− − = + − − − − −
(1.23)
Honarpour et al. (1982) Limestone and dolomite Water-wet Water injection
( )0.43
2.15
10.0020525 0.051371w wi
rw w wia
S Sk S S
k
−= − −
(1.24)
Honarpour et al. (1982) Limestone and dolomite Oil/intermediate-wet Water injection
( )2 4
0.29986 0.32797 0.4132591 1 1
w wi w or w wirw w wi
wi wi or wi or
S S S S S Sk S S
S S S S S
− − −= − − +
− − − − − (1.25)
Honarpour et al. (1982) Limestone and dolomite All Water injection
2
1.26241 1
o or o orro
or wi or
S S S Sk
S S S
− −=
− − − (1.26)
26
Fulcher et al. (1985) Berea sandstone Water-wet Oil injection
( )( )
D
B C ln woro dr
o
k AS
+ =
(1.27)
( )
11
w
o
B D ln
w wirrrw dr
wirr
S Sk A
S
+
−=
− (1.28)
( )
( )
21
caB D ln N
w wirrrw dr
wirr
S Sk A
S
+ −=
− (1.29)
Fulcher et al. (1985) Berea sandstone Water-wet Water injection
( )
1
w
o
B C ln D ln
o orro im
or
S Sk A
S
+ +
−=
− (1.30)
( )
11
w
o
B D ln
w wirrrw im
wirr
S Sk A
S
+
−=
− (1.31)
( )
( )
21
caB D ln N
w wirrrw im
wirr
S Sk A
S
+ −=
− (1.32)
Residual saturations, endpoint relative permeabilities, and Corey exponents are used for tuning a
specific kr-S path. These empirical models provide good match in most cases, but the challenge with such
representation of relative permeabilities is that the information about the pore structure, wettability, and
capillary number are all incorporated into the tuning exponents. Each kr-S path would thus be distinct and
not generalizable. As such, hysteresis in relative permeability is not resolved. For this, different researchers
have attempted different modeling solutions. Some of the commonly known hysteresis models are
described next.
Table 1. (Continued)
27
1.2.2.2. Land-type models
Figure 8 shows a schematic displaying hysteresis in nonwetting phase relative permeabilities. Hysteresis in
relative permeability, as described previously, is the path dependency of relative permeability in the
saturation space. The black curves show nonwetting phase relative permeability during primary drainage
while the red curve shows the nonwetting phase relative permeability during imbibition. The imbibition
process is begun at the initial nonwetting phase saturation (Snwi) and ends at the residual nonwetting phase
saturation (Snwr). This set of primary drainage and imbibition forms one set of scanning relative permeability
curves. Other such sets of scanning curves can be generated experimentally, each with its own starting and
ending nonwetting phase saturation.
28
Figure 8. Schematic of nonwetting phase relative permeability showing hysteresis after flow reversal from primary drainage (black curve) to primary water injection (red curve). The initial, trapped, residual, and flowing nonwetting phase saturations are marked on the figure. Adapted from Carlson (1981).
Naar and Henderson (1961) proposed a model for imbibition relative permeability by considering the
trapped nonwetting phase saturation during the imbibition process. They developed the following
relationship between the imbibition and drainage saturations for the same value of nonwetting phase relative
permeability.
*, , ,
* * 20.5( ),w imb w dr w drS S S= − (1.33)
Snwt
1
Snw
0 1
krn
w
Snwi
Snwr Snw
Snwf
29
where *,, ) / ( ),( 1w imbi wi iw wmbS S S S= − − is the reduced wetting phase saturation during an imbibition process;
Swi is the initial wetting phase saturation (or the irreducible wetting phase saturation) prior to the imbibition
process; Sw,imb is the wetting phase saturation during imbibition; *
,, ( 1) / ( ),w dr w dr wi wiS S S S= − − is the reduced
wetting phase saturation during a drainage process; Sw,dr is the wetting phase saturation during
drainage. Further, by using Eq. (1.33), the following model for nonwetting phase imbibition relative
permeabilities ( Irnwk ) could be established by using the saturation information of the prior drainage
process,
( ), * 2,
*
0.5,
0.51I
rnww imb
w imb
Sk S
− = −
(1.34)
Land (1968) observed trends of characteristic initial-residual (IR) saturation curves and proposed a
relationship between Snwr and Snwi as follows,
1 1
nwr nwi
CS S
− = . (1.35)
The nonwetting phase relative permeability during imbibition is estimated by extracting information of
flowing (Snwf) and trapped saturation (Snwt). On any point on the krnw-Snw path (see Figure 8), following
relationship exist among the different saturations,
nw nwt nwfS S S= + . (1.36)
Using Eq. (1.35) and Eq. (1.36), Snwf is estimated as follows (derivation available in Carlson 1981,
originally equation was presented in Land 1968),
30
( ) ( ) ( )21 4
2
nwf nw nwr nw nwr nw nwrS S S S S S SC
= − + − + −
. (1.37)
Land (1968) then followed similar treatment as that of Corey (1954) model to propose the following relation
for gas relative permeability with the use of the flowing gas saturation,
( ) ( )2
2* *1 1g grg F Fk S S
+ = − −
, (1.38)
where *gFS is the free-flowing gas phase saturation which is normalized to the effective pore space,
* / ( )1gF wirrgFS S S−= ; and λ is the pore-size distribution index.
Killough (1976) proposed an interpolation-based approach for estimating imbibition relative
permeabilities based on drainage relative permeabilities as follows,
( ) ( ) I D nw nwrrnw nw rnw nwi
nwi nwr
S Sk S k S
S S
−
= −
, (1.39)
where ( )Irnw nwk S is the imbibition relative permeability and ( )D
rnw nwik S drainage relative permeability at
initial nonwetting phase saturation. This proposed form satisfies the limiting conditions for imbibition
relative permeabilities, where ( ) ( )I Drnw nwi rnw nwik S k S= and ( ) 0I
rnw nwrk S = . Carlson further simplified
Killough’s approach and proposed that ( )Irnw nwk S can be estimated from the corresponding drainage
nonwetting phase relative permeability (Drnwk ) with the knowledge of the flowing nonwetting phase
saturation,
31
( ) ( ) I Drnw nw rnw nwfk S k S= . (1.40)
Both Killough’s and Carlson’s model require the knowledge of the Land’s trapping coefficient.
Thus far, most relative permeability models are developed for water-wet media. The trapping model by
Spiteri et al. (2008) was an improvement over Land’s model. It was the first model to be used for different
wettabilities. It was developed using initial-residual (IR) trapping data sets generated using pore-network
modeling. Their model was given as,
2
r i iS S S = − , (1.41)
where α and β are model parameters. Similar to the treatments by Carlson (1981), based on the information
of the trapped versus flowing saturation and the previous primary drainage curves, relative permeabilities
for the waterflooding cycle for different contact angles can be calculated.
1.2.2.3. Limitations of relative permeability models
Some limitations of the models described for relative permeability in the previous subsections are as
follows.
• The major limitation of Land-based models is that the path dependency for relative permeability is
not resolved. Based on the information of the trapped phase, the relative permeabilities for the water
injection cycle are predicted which inherently depends on the particular scanning curve. Each set
of scanning curve will consequently have its own initial and residual saturation and its own path to
be traced. And each path will require tuning of its own Land’s trapping coefficient, for example,
32
for the set of flow conditions that process is subjected to. This leads to ad hoc combinations of
Corey model with Land’s trapping coefficient for calculating relative permeabilities.
• These models are empirical and lack pore-scale physics, despite the understanding of the different
factors that affect relative permeabilities. These models are therefore less predictive away from the
conditions under which they are developed.
• Most Land-based models are developed for water-wet systems and are primarily used for predicting
imbibition relative permeabilities.
• These models are deficient because they require knowledge of previous drainage curve to calculate
imbibition relative permeabilities. In addition, they require knowledge of different inputs such as
the initial and residual nonwetting phase saturations.
33
1.3. Research objectives
Current relative permeability (kr) models are functions of phase saturations that are matched for specific
flow/experimental conditions. However, as examined from the literature, together with phase saturation,
multiple parameters affect relative permeabilities such as the wettability of the medium, capillary number,
pore structure, fluid phase topology, and fluid/fluid interfacial areas. These other parameters affecting
relative permeabilities are inherently captured through the empirical saturation functions. Representation
of relative permeabilities only in the saturation space causes non-uniqueness and path dependency in
relative permeabilities which often cause simulations to fail because they lack generality and are not
physically based. As a result, hysteresis in relative permeabilities arises, which is a major modeling issue
for reservoir simulations.
Efforts have been presented in this dissertation to model relative permeabilities by considering
functional forms that include the effects of the key controlling parameters on relative permeabilities. The
purpose of this dissertation is twofold, to
(a) understand how different parameters, specifically, phase saturation, phase connectivity, capillary
number, and wettability affect relative permeabilities;
(b) propose physically-based kr models by including the effects of these parameters.
34
1.4. Dissertation layout
There are five additional chapters in this dissertation after the introductory chapter. For chapters 2 to 5, an
abstract, relevant literature survey, methodology, results and discussions, and conclusions are presented.
Below a brief summary for each chapter is presented.
In chapter 2, a static x-ray imaging experiment of a multiphase system is discussed to quantify
measurement-based errors due to image segmentation. A high-resolution (6 µm) and a low-resolution (18
µm) x-ray scan of the same system was acquired. The high-resolution scan was used as ground truth while
the low-resolution scan was used to test different image segmentation methods and quantify errors in pore-
scale measurements. It was found that pore-scale measures of phase topology and fluid/fluid interfacial
areas are highly sensitive to image analyses procedures such as that of image segmentation. To mitigate
these errors, images with high-resolution should be acquired and these should be obtained in steps to
improve the accuracy of image segmentation. In addition, supervised machine learning based-algorithm
was found to provide the closest pore-scale measures to the ground truth. From this work, the need to
supplement experimental data sets with numerical data sets was identified.
In chapter 3, a state function-based approach for relative permeabilities is discussed. A relative
permeability equation-of-state (kr-EOS) is forced as a quadratic response for kr in the phase connectivity-
phase saturation space ( ˆ ).S − The EOS is constrained to limiting conditions in the ˆ S − space. Although
the model is built for fixed capillary number conditions, it is tested for different capillary numbers, ranging
from one to 10-6. The dependence of phase connectivity on capillary number is also explored. It was found
that a quadratic response for relative permeabilities work across different capillary numbers. The linear kr-
S paths for high capillary numbers (small Corey exponents) and nonlinear kr-S paths for low capillary
numbers (high Corey exponents) were found to occur due to fast and slow changes in phase connectivity,
respectively. From this work, the need for large numerical data sets to calculate relative permeability partial
derivates for the EOS development was identified.
35
In chapter 4, numerical data sets of phase saturation and phase connectivity are generated using pore-
network simulations to study the effect of wettability on phase trapping. During primary drainage, the
contact angle was set at zero degrees. However, during secondary injection process, the contact angles were
changed from 0° to 180°. Trends of residual phase saturation and residual phase connectivity are analyzed
for different contact angles. Hysteresis trapping models are presented to capture the residual trends and
comparison is presented against models from the literature. It was found that wettability significantly affects
receding phase trapping and that pore-scale mechanisms of layer flow and piston-like advance of the
invading phase become critical when the receding phase is wetting to the surface.
In chapter 5, the workflow of pore-network simulations from chapter 4 is utilized to generate numerical
data sets of nonwetting phase relative permeability, saturation, and connectivity. Here, capillary number
and pore structures were kept fixed, and two wettability cases were considered both in the water-wet regime.
Through hundreds of simulations, the kr, S, and data sets are analyzed to estimate partial derivates of kr
in the ˆ S − space. These partial derivatives are then utilized for the development of an EOS response for
relative permeability. It is found that the EOS predicts kr for the entire data set, regardless of the direction
of flow, thus resolving hysteresis in relative permeabilities.
In chapter 6, key concluding remarks from this study and outlook for future research efforts that can be
built from this dissertation are presented.
36
1.5. Publication list
Peer reviewed publications
• Purswani P., Johns R.T., Karpyn Z.T., (2021), Relationship between Residual Saturations and
Wettability using Pore-Network Modeling, SPEJ, (in preparation)
• Purswani P., Johns R.T., Karpyn Z.T., (2021), Impact of Wettability on Capillary Phase Trapping
using Pore-Network Modeling, Water Resour. Res., (in preparation)
• Purswani P., Johns R.T., Karpyn Z.T., Blunt M.J., (2021), Predictive Modeling of Relative
Permeability using a Generalized Equation-of-State, SPEJ, (26), 191-205,
https://doi.org/10.2118/200410-PA
• Purswani P., Karpyn Z.T., Khaled E., Yuan X., Xiaolei H., (2021), Evaluation of Image
Segmentation Techniques for Image-Based Rock Property Estimation, J. Pet. Sc. Eng., (195),
https://doi.org/10.1016/j.petrol.2020.107890
• Purswani P., Tawfik, M.T., Karpyn Z.T., Johns R.T., (2019), On the Development of a Relative
Permeability Equation of State, Comput Geosci., (24), 807-818, https://doi.org/10.1007/s10596-
019-9824-2
Conference and talks
• Purswani P., Johns R.T., Karpyn Z.T., (2021), Relationship between Residual Saturations and
Wettability using Pore-Network Modeling, SPE ATCE, Virtual, (in preparation)
• Purswani P., Johns R.T., Karpyn Z.T., Blunt M.J., (2020), Predictive Modeling of Relative
Permeability using a Generalized Equation-of-State, SPE IOR Conference, 31st Aug-4th Sept,
Virtual
• Purswani P., Tawfik, M.T., Karpyn Z.T., Johns R.T., (2018), On the Development of a Relative
Permeability Equation of State, EME Research Showcase, State College, Pennsylvania
• Purswani P., Tawfik, M.T., Karpyn Z.T., Johns R.T., (2018), On the Development of a Relative
Permeability Equation of State, ECMOR XVI, 16th European Conference on the Mathematics of
Oil Recovery, 3-6 September, Barcelona, Spain
• Purswani P., Karpyn Z.T., Johns R.T., (2018), Correlating Transport Parameters Impacting
Multiphase Flow through Permeable Media, Gordon Research Conference, 8-13th July, Maine,
USA
37
CHAPTER 2. IMAGING AND PORE-SCALE
MEASUREMENTS
Preface
The contents of this chapter were originally published in the Journal of Petroleum Science and Engineering
and are referenced as,
Purswani P., Karpyn Z.T., Khaled E., Yuan X., Xiaolei H., (2020)
Evaluation of Image Segmentation Techniques for Image-Based Rock Property Estimation, J. Pet. Sc. Eng,
(195), https://doi.org/10.1016/j.petrol.2020.107890
Author contributions: Purswani P. and Karpyn Z.T. conceptualized the experiment. Purswani P. and Enab
K. performed the experiments and wrote the original draft in consultation with Karpyn Z.T. All coauthors
contributed toward analyzing the data and updating the manuscript.
Abstract
Accurate characterization of rock and fluid properties in porous media using x-ray imaging techniques
depends on reliable identification and segmentation of the involved phases. Segmentation is critical for the
estimation of porosity, fluid saturations, fluid and rock topology, and pore connectivity, among other pore-
scale properties. Therefore, the purpose of this study was to compare the effectiveness of different image
segmentation techniques when applied to image data analysis in porous media. Two machine learning based
segmentation techniques – a supervised ML technique called Fast Random Forest, and an unsupervised
method combining k-means and fuzzy c-means clustering algorithms – were compared using an
38
experimental data set. Comparisons are also presented against traditional thresholding segmentation. In
addition, we discuss the potential and limitations of applying deep learning-based segmentation algorithms.
The performance of the segmentation techniques was compared on estimates of porosity, saturation, and
surface area, as well as pore-scale estimates such as fluid/fluid interfacial areas, and Euler characteristic.
X-ray micro-computed tomography images for a sintered glass frit, saturated with two-phases (air and
brine), were acquired at two different voxel resolutions. The high-resolution images (6 µm) were used as
the benchmark case, while the low-resolution images (18 µm) were segmented by three segmentation
techniques: Fast Random Forest, clustering, and thresholding. The results for porosity and phase saturation
from thresholding and from the supervised ML method (i.e., Fast Random Forest) were found to be close
to the benchmark case. Segmentation results from the unsupervised ML method (i.e., clustering) were
largely unsatisfactory, except for total surface area measurements. The supervised ML segmentation results
provided better measurements for air-brine interfacial areas by capturing three-phase interfacial regions.
Also, all segmentation techniques resulted in similar measurements for air-phase Euler characteristic
confirming poor connectivity of the trapped air phase, although the closest results were obtained by the
supervised ML method. Finally, despite the supervised ML segmentation technique being more
computationally intensive, it was found to require less user intervention and its implementation was more
straightforward. In summary, this work provides insights into different segmentation techniques, their
implementation, as well as advantages and limitations with regards to quantitative analysis of pore-scale
properties in saturated porous media.
39
2.1. Introduction
High-quality, non-destructive imaging is at the heart of innovative science in a variety of disciplines. It
provides researchers with the ability to examine objects at small length scales, which enables the estimation
of a variety of structural and topological properties. In the geosciences, applications of non-destructive
imaging include characterization of rock heterogeneities, pore-network properties, roughness, fluid
distributions, and transport in porous systems (Blunt 2017; Lai et al. 2015; Noiriel et al. 2004; Wildenschild
and Sheppard 2013). X-ray micro-computed tomography (µCT) is one such imaging technique that
generates a three-dimensional (3-D) mapping of linear attenuation coefficients acquired by a digital x-ray
detector. These attenuation coefficients are distinct for each material phase in the object (Cnudde and Boone
2013). As such, an x-ray image provides both quantitative and qualitative information about the elements
constituting the object scanned. To draw meaningful information from these digital images, a series of
image processing steps are necessary. These steps help improve the visual appearance of digital x-ray
images, as well as prepare them for feature and property analyses.
There are three main steps of image processing, namely, pre-processing, segmentation, and post-
processing. Image pre-processing consists of steps to reduce the impact of image artifacts such as noise,
image blur, beam hardening, ring effects, and bright spots (Huda and Abrahams 2015). This is achieved by
the application of image filters like median (Bernstein 1987), mean, non-local mean (Buades et al. 2005),
and edge detecting filters (Sheppard et al. 2004) that help improve the quality of reconstructed raw images
and prepare them for image segmentation. Image segmentation is the process of categorizing (or labeling)
each voxel to a specified class or phase in the object. This labeling step assigns a characteristic number to
all voxels belonging to the same phase. This assists in quantitative analysis on the images, for example,
voxel counting is used for porosity and phase saturation measurements. Lastly, image post-processing is
the operation of fixing any misrepresentation of phases in the segmented image. All image processing steps
are crucial for consistent and accurate feature measurements. The purpose of this work is to evaluate various
image segmentation techniques for image-based rock property estimation.
40
Due to the growing access of x-ray µCT scanners to numerous researchers in Earth sciences, there has
been an increase in the number of studies that use this technique for studying fluid flow in porous media.
Segmented images are used to measure phase characteristics to provide the observational basis for
understanding different processes such as multiphase fluid flow, structural morphology, pore connectivity,
fluid/fluid, and fluid/solid interfaces (Blunt 2017). These characteristics are subsequently used to quantify
fluid transport through estimations of flow properties such as relative permeabilities and capillary pressures
(Khorsandi et al. 2017). Carefully segmented x-ray images are often used as a starting point for simulating
fluid flow by using techniques such as Lattice Boltzmann simulations (Armstrong et al. 2016; Landry et al.
2014; Liu et al. 2018; Mcclure et al. 2018) or pore-network modeling (Dong and Blunt 2009; Joekar-Niasar
et al. 2010; Joekar-Niasar et al. 2008; Reeves and Celia 1996; Valvatne and Blunt 2004). Table 2 lists a
few of the experimental studies performed over the past two decades. It can be inferred from Table 2 that a
variety of porous systems spanning natural and synthetic media have been studied. It can also be inferred
that, in general, over time, the scanning resolutions have improved as technology advances. Further, there
is a general acceptance for using non-local means filtering technique for pre-processing purposes.
Table 2. Characteristics of imaging techniques from various experiments of fluid flow in porous media.
Reference Porous
media
Image
filtration
Voxel
resolution
(um)
Segmentation
technique Post processing
(Culligan et al.
2004)
Glass bead
pack Median filter 18 k-means clustering -
(Culligan et al.
2005)
Soda lime
beads Median filter 17 k-means clustering -
(Porter and
Wildenschild 2010) Bead pack
Anisotropic
diffusion filter 5.9; 11.8 k-means clustering -
(Karpyn et al. 2010) Glass bead
pack - ~26 Thresholding -
(Landry et al. 2011) Acrylic bead
pack Median filter ~26 Thresholding
Smoothing of
surfaces for area
measurements
41
(Herring et al. 2013) Bentheimer
sandstone - 10 Indicator Kriging
Removal of
nonwetting clusters
smaller than 100
voxels
(Celauro et al. 2014) Coated glass
bead packs - ~27
Gauss curve fitting
to gray value
histograms
-
(Harper 2013;
Herring et al. 2013;
Joekar-Niasar et al.
2013; Porter and
Wildenschild 2010)
Crushed tuff Anisotropic
diffusion filter 17.5 k-means clustering -
Sintered
glass bead
pack
Median filter 13 Thresholding -
(Herring et al. 2015) Bentheimer
sandstone Median filter 5.8 Thresholding
Removal of air
clusters smaller
than 125 voxels
(Rücker et al. 2015) Gildehauser
sandstone
Non-local
means 2.2 Watershed -
(Berg et al. 2016) Gildehauser
sandstone
Non-local
means 2.2 Watershed
Segmented phases
were cleaned using
morphological
operations
(Schlüter et al.
2016)
Sintered soda
lime bead
pack
Non-local
means and
total variation
denoising filter
8.4 Markov random
field technique -
Sintered
glass bead
pack
Median filter 2.2 Watershed
Removal of
clusters smaller
than 125 voxels
(Gao et al. 2017) Bentheimer
sandstone
Non-local
means 6 Thresholding -
(Singh et al. 2017) Ketton
limestone
Non-local
means 3.28
Seeded watershed
algorithm and
thresholding
Dilation of rock
phase for curvature
analysis
(Lin et al. 2018) Bentheimer
sandstone
Non-local
means 3.58
Seeded watershed
algorithm and
thresholding
Boundary
smoothing for
curvature analysis
(Rücker et al. 2019) Ketton
limestone
Non-local
means 3
Watershed
algorithm and
thresholding
-
Table 2. (Continued)
42
(Lin et al. 2019) Bentheimer
sandstone
Non-local
means 3.58
Seeded watershed
algorithm and
thresholding
Boundary
smoothing for
curvature analysis
This work
Sintered
glass bead
pack
Non-local
means 6; 18
Thresholding; k
and c means
clustering; and
supervised machine
learning
Removal of small
nonwetting phase
clusters for Euler
number analysis
Pore-scale measurements such as porosity, phase saturation, fluid topology, and fluid/fluid interfacial
areas can be extremely sensitive to the results of image processing steps, in particular, image segmentation.
Segmentation methods can largely be categorized into two groups, global methods and local adaptive
methods (Iassonov et al. 2009). Global methods, such as intensity-based thresholding, work by identifying
valley points on the voxel population histogram of the filtered images. A threshold gray value is set to
classify the voxels, such that gray values above the threshold are identified as one phase, while the voxels
below the threshold are identified as the other phases. This method worked reasonably well for a multiphase
system with sufficient contrast in the gray-levels of each phase, which makes the identification of the valley
points in the histogram easier. Because of its ease of application, intensity-based thresholding continues to
be a common method of segmentation in the digital rocks community (Prodanovic et al. 2015).
Locally adaptive segmentation refers to the segmentation methods that make segmentation decisions
for each voxel in the image. There have been numerous developments on this type of segmentation to
achieve more refined results. Watershed segmentation (Vincent and Soille 1991), converging active contour
method (Sheppard et al. 2004), Markov random field segmentation (Kulkarni et al. 2012), and indicator
kriging (Oh and Lindquist 1999) are a few examples of locally adaptive methods. A comprehensive review
of the implementation and comparison of these locally adaptive methods is available in Schluter et al.
(2014). Machine learning techniques such as fuzzy c-means (Pham and Prince 1999), a combination of k-
means and fuzzy c-means (Dunmore et al. 2018), and supervised machine learning are other examples of
locally adaptive methods of segmentation.
Table 2. (Continued)
43
The capability of machine learning (ML) approaches in solving classification problems has enabled the
utilization of such techniques for generating segmentation algorithms. Traditional supervised ML
algorithms work as feedback methods by learning from annotated voxel labels of some part of an image in
order to predict the class distribution of each voxel in the whole image. Such an ML model is learnt in a
training process that extracts a vector of features that influence voxel class labels based on feedback from
annotated labels of some voxels (Kotsiantis 2007). After training, the resulting ML model is used to assign
a class label to each voxel in the entire image based on voxel feature values. Support Vector Machines,
Neural Networks (Multilayer Perceptron), Decision Trees, Random Forest, and Fast Random Forest are
examples of supervised ML algorithms that can be used to generate classification models for image
segmentation purposes.
Unsupervised ML approaches, unlike supervised methods, do not need annotations for part of data, but
operate by grouping voxels based on similarities. Clustering, also known as cluster analysis, is one of the
most common types of unsupervised ML methods and it is widely used for classification purposes in data
analysis and data mining. K-means clustering, and fuzzy clustering (c-means or soft k-means clustering)
are two common methods for clustering. Both k-means and c-means clustering are iterative methods that
operate by identifying the similarity of an element in the population to different groups of elements. The
assignment of an element to a particular group is probabilistic in c-means as opposed to deterministic in k-
means.
In the past decade or so, deep learning (DL) methods based on multi-layer artificial neural networks
have produced state-of-the-art results in many fields including computer vision, speech recognition,
medical image analysis, and material inspection. In the area of image segmentation, DL has also achieved
success, including in the segmentation of µCT images. Through training with a large number of fully
annotated images, DL models can extract meaningful visual features automatically and use them to infer
segmentation maps. For 2-D image segmentation, U-net (Ronneberger et al. 2015), Deeplab series (Chen
et al. 2018; Chen et al. 2017a; Chen et al. 2017b), Mask R-CNN (He et al. 2017) among others (Oktay et
al. 2018; Xue et al. 2018; Zhou et al. 2018) cover a variety of different applications from natural images to
44
medical images. For 3-D image segmentation, prior works (Çiçek et al. 2016; Milletari et al. 2016; Xue et
al. 2019) mainly focus on medical applications such as 3-D magnetic resonance imaging (MRI) or CT
scans.
For rock image segmentation, Wang et al. (2020) introduced a novel 3-D µCT segmentation method
built on U-net (Ronneberger et al. 2015) and ResNet (He et al. 2016) in their recent work. Niu et al. (2020)
and Karimpouli and Tahmasebi (2019) used Convolutional Neural Network (CNN)-based algorithms for
segmenting sandstone data sets. It was found that CNN algorithms can minimize the need for user-defined
inputs (Niu et al. 2020). Although DL methods can achieve promising segmentation results, their feature
learning capacity heavily relies on the large amount of training images as well as high-quality manual
annotations. Moreover, unlike other types of visual recognition tasks such as image classification which
only require image level annotations, the annotation of 3-D µCT rock images for segmentation purposes
requires labeling at a voxel-by-voxel level for the entire image, which can be very expensive and
impractical. Further, because the mineral composition and structural features differ in porous media, voxel-
label annotations obtained for one rock system may not be useful as training data to train the segmentation
model for other rock systems. Thus, DL methods may not be a suitable choice for image segmentation
unless diverse saturated porous media image data are available as training data sets.
In summary, the literature presents a variety of image segmentation techniques including both global
and local techniques. However, newer ML techniques are less commonly used in the porous media
community. Therefore, the purpose of this study is to compare the effectiveness of two ML-based
segmentation techniques – a supervised ML technique called Fast Random Forest, and an unsupervised
method combining k-means and fuzzy c-means clustering algorithms – for segmenting a saturated
multiphase porous medium. Comparisons are also presented against a thresholding-based segmentation
technique. We investigate the segmentation methods on a small-scale dataset where DL models can easily
overfit to training samples which makes them less generalizable. For this reason, we do not include deep
learning-based methods in our comparison. Our goal is to compare feasible segmentation methods for
identification of fluid and solid phases in saturated porous media. We provide a quantitative analysis of the
45
segmented phases to estimate physical characteristics of porosity, phase saturations, phase surface areas,
interfacial areas, and phase connectivity to demonstrate and compare the capabilities and limitations of each
segmentation technique with recommendations for each when applied to saturated porous media. In this
way, this research provides the readers with insight into emerging machine learning-based image
segmentation techniques, their implementations, their comparative advantages, as well as limitations with
regards to applications in porous media research.
46
2.2. Methodology
Trusted segmented data are required as a benchmark for comparing the effectiveness of the different
segmentation techniques examined in this research. For this, a saturated porous medium was prepared for
this study and scanned at two different voxel resolutions. The first, benchmark scan was acquired at a voxel
resolution of 6 µm. The second test scan was acquired at a voxel resolution of 18 µm which was used to
test and compare different segmentation techniques. Although 18 µm represents low resolution in this
research, it is still typical of x-ray micro-tomographic studies (Table 2).
The experimental set-up (Figure 9) used in this research is a static set-up consisting of a sintered glass
frit (pore sizes between 100-160 µm) saturated with brine and air, representing wetting and nonwetting
phase, respectively. The x-ray scanner used was GE v | tome | x L300 system with a 300kV x-ray tube. The
sintered glass frit was a specific type of borosilicate glass filter (Robu) procured from Adam and Chittenden
Scientific Glass, California, USA. It is a glass filter widely used for water filtration purposes, 10 mm in
diameter, 2.8 mm long, with ~18% porosity. The sintering allows for the porous medium to be rigid and
maintain its pore structure during handling. The brine phase used in this experiment was a solution of 1M
sodium iodide (NaI). Doping the brine with 1M NaI helps to attenuate more x-rays such that enough contrast
can be achieved to isolate the three phases. This particular concentration of NaI was found to be optimum
and was achieved after multiple trials to minimize imaging artifacts and maximize contrast.
47
Figure 9. Schematic of the laboratory setup and image acquisition system (x-ray MCT scanner). DO stands for the distance between the detector and the object, while OS stands for the object to source distance. This figure shows that the sample (object) is very close to the source for finer resolution, the resolution was coarsened by moving the sample stage laterally from the source, increasing OS and decreasing DO.
The porous glass frit was held fixed inside a thin (~ 1mm wall thickness) plastic tubing, open at the top,
closed at the bottom and secured to the scanner’s rotating sample mount. The setup was placed within a
few millimeters of the x-ray source to maximize the image resolution to 6 µm (Figure 9). At this position,
a scan for the dry frit was acquired. Next, a pipette filled with the brine solution was used to drop a couple
of droplets into the porous glass frit. After waiting for 20 minutes for the liquid to saturate the glass frit, the
benchmark x-ray scan was acquired. At this stage, the scanned system consisted of three phases (solid glass,
brine, and trapped air). Upon completion of this scan, the sample mount was moved laterally by increasing
the object-to-source (OS) distance and decreasing object-to-detector (DO) distance from the x-ray source
to acquire the exact same scan at a resolution of 18 µm. This was termed as the test scan.
X-ray source
Detector
Glass frit
X-ray chamber
Plastic tubing
Sample mount
NaI dropletPipette
DO OS
48
2.2.1. Implementation of the segmentation techniques
In this section, we discuss the image processing framework used in this research. Both dry and brine
saturated raw CT images were processed through the non-local means filter to remove image noise. No
other major image artifacts were observed in the CT images. The non-local means filter was found to be
effective as compared to the median filter. The filtering step was not required for the supervised machine
learning segmentation which directly works on raw CT images. This is discussed in the subsequent section.
2.2.2. Benchmark case
To generate the benchmark segmented images, both dry and saturated images of the high-resolution scan
were used to generate reliable segmented images. First, thresholding was conducted on the filtered dry
sample to segment the solid and the pore space. This was easier to accomplish because of the significant
difference in the gray values between the air and the solid phase. Second, the segmented dry images were
subtracted from the saturated images to eliminate the solid phase. This left only the brine and trapped air
phase which were segmented once again using thresholding.
2.2.3. Test case
The comparative analysis between the supervised and unsupervised machine learning segmentation
techniques was conducted using the scanned images at a voxel resolution of 18 µm. Additional comparison
with thresholding is also presented. The implementation of these segmentation techniques is outlined below.
49
2.2.4. Supervised machine learning (ML)-based on Fast Random Forest algorithm
The supervised ML-based segmentation technique is a multi-threaded implementation of the Random
Forest algorithm, as provided in the WEKA (Waikato Environment for Knowledge Analysis) trainable
segmentation toolbox (Arganda-Carreras et al. 2017). The WEKA segmentation toolkit is implemented as
a built-in plugin in ImageJ. It works as a bridge to apply machine learning tools for image processing and
has been used in a few recent studies for segmentation purposes (Berg et al. 2018; Garfi et al. 2020). The
Random Forest algorithm (Breiman 2001) is a classification algorithm consisting of many decision trees
that operate as an ensemble. Each individual decision tree provides a vote on class prediction and, the class
with the most votes becomes the forest's class prediction. The Random Forest algorithm uses bagging and
feature randomness when building each individual tree to try to build a forest of largely uncorrelated trees
whose prediction by the "wisdom of crowds" is more accurate than that of any individual tree. Random
Forest is considered a fast classifier, but more recently, parallelized versions such as the WEKA Fast
Random Forest implementation enable one individual tree per processor core to take advantage of multi-
core processors, further reducing forest build time.
Below, we outline the general procedure for applying the Fast Random Forest segmentation method in
this research:
1. To build the training set, sample voxels representing each of the different target categories are
selected and each such sample voxel is labeled with the category to which it belongs.
2. To train the random forest classifier, a vector of image features for each voxel is used as the training
feature. The vector of features contains the CT value of the voxel in the raw image, coupled with
the CT values of the same voxel in different filtered images. The available filters provided in the
WEKA toolkit include Gaussian blur, Hessian, derivatives, structure, edges,
minimum/maximum/mean/variance/median, etc. After a trial-and-error process where different
combinations of image filters were tested to arrive at the combination that provided the best
performance, we finally chose the minimum, maximum, mean, and variance filters. Note that these
50
filters are applied at a voxel-level: the voxels within a small radius (e.g., within a 3x3x3 small
neighborhood) from the target voxel are subjected to the pertinent operation (min, max, mean, or
variance) and the target voxel is set to that value in the filtered image. Once the training features
for all sample voxels (with ground-truth labels) are computed, the random forest classifier is trained
using the (feature-vector, label) pairs for the sample voxels.
3. After the random forest classifier is trained, it is then used to classify every voxel in the entire
image, thus achieving full segmentation of the whole image.
During training of the classifier (step 2 above), the sample dataset of labeled voxels is automatically
divided into three subsets: training, validation, and testing. The subset for training consists of ~ 80% of the
sample data and is used to build the classifier. The subset for validation consists of ~ 10% of the sample
data, and it is used to adjust parameters of the classifier and choose the combination of image filters to use;
that is, many classifiers can be trained with different parameter values (e.g., number of trees in the forest)
and different combinations of image filters, and then the optimal values and filters are chosen based on
which ones give the best performance on the validation subset. Lastly, the subset for testing consists of the
remaining 10% of the sample data and is used to test and report the accuracy of the final chosen classifier.
2.2.5. Unsupervised machine learning based on k-means and fuzzy c-means clustering
The unsupervised technique selected for this study is the medical image analysis (MIA) – clustering
technique, which is an open-source algorithm useful for multiphase image segmentation (Dunmore et al.
2018; Wollny et al. 2013). This segmentation technique requires denoised images as input, so we apply
Gaussian filtering to a raw image before applying the technique, which combines two unsupervised
clustering methods, k-means and fuzzy c-means. K-means clustering is an iterative scheme that places each
member (e.g., a voxel) in a given data set (e.g., all voxels in a CT image) into different clusters representing
different classes. The iterative process starts by defining the centroids (or means) of each cluster arbitrarily.
51
Then, each data point is grouped to a data cluster as a function of the Euclidean distance between the data
point and a cluster centroid. Next, the cluster centroid values are updated until the difference between the
previous and the updated centroid values meet a user-defined tolerance. C-means clustering is similar to k-
means clustering with one difference which lies in the flexibility of allowing a data point to probabilistically
belong to more than one cluster (Bezdek, James C; Ehrlich, Robert; Full 1984; Dunn 1973). The
probabilistic nature of this approach is enabled by the inclusion of a membership function (with value
between zero and one) and a term called the fuzzifier (a real number between one and two). The membership
function governs the degree to which a particular data point belongs to a particular cluster, whereas the
fuzzifier determines the fuzziness level of a cluster. Larger values of the fuzzifier lead to smaller values of
the membership function and vice versa.
The MIA-clustering algorithm requires little user intervention. Two main input parameters supplied by
the user are the number of classes that exist in the image and the grid-size used for partitioning the image
into overlapping cubes so that segmentation can be refined locally within the cubes. In the first part of the
algorithm, the K-means algorithm clusters all voxels, based on voxel intensity in the denoised CT image,
into the number of classes specified by the user. Subsequently the fuzzy c-means algorithm is applied to
iteratively estimate all class membership probabilities for each voxel, expressed as a vector. Then, by
assigning each voxel to a class based on its highest membership probability, the whole image is clustered
into distinct classes representing structures. However, this global segmentation may miss some fine details
because of intensity inhomogeneities in the input image. Therefore, in the second part of the MIA-clustering
algorithm, fuzzy c-means is applied locally. The whole image volume is subdivided into overlapping small
cubes based on the grid-size parameter. In a cube, the sum of membership probabilities of all voxels for
each class is calculated; if the sum for a class falls below a threshold, then that class is not considered for
the local, refined c-means clustering in the cube. After the local refinement is done for all cubes, class
probabilities for each voxel in overlapping cubes are merged, and once again, voxels are assigned to the
class for which they have the highest membership probability, producing the whole segmented image. More
details about the MIA-clustering algorithm can be found in Dunmore et al. (2018).
52
2.3. Results and discussion
In this section, results from the implementation and comparison of the machine learning (supervised and
unsupervised) segmentation techniques are discussed. Additional comparisons are presented against
thresholding segmentation. For quantitative analysis, we present bulk measurements of porosity, fluid
saturations, phase fractions, and phase surface areas, as well as pore-scale measurements of phase
connectivity (measured as the Euler characteristic) and fluid/fluid interfacial areas.
The imaged cross-sections of the porous glass frit (dry and brine saturated) are shown in Figure 10, and
the corresponding grayscale intensity histograms are shown in Figure 11. In Figure 10 (top), the brighter
region corresponds to the solid (sintered glass) whereas, the darker region corresponds to the pore (air)
space. The solid, being denser, attenuates more x-rays and appears bright. Figure 10 (top) and Figure 11
(top) show that the quality of the acquired dry x-ray scan is excellent as evidenced from the histogram of
the raw image which is well resolved between the pore and solid space even before applying any
enhancement filters. Minimal x-ray imaging artifacts are observed. Upon the application of the non-local
means filter, the difference in the voxel populations of the solid phase and the pore space becomes clearer.
This assists in the segmentation of two phases by thresholding. We note here that the parameters set for the
non-local means image filtration were kept uniform across all image data sets, irrespective of the resolution,
or whether the data sets were dry, or brine saturated to maintain a common pre-processing procedure for
all images to be segmented.
In Figure 10 (middle) and Figure 10 (bottom), light-gray regions correspond to the brine phase, middle-
gray regions represent the solid phase, and the darker isolated regions represents the air phase. Notice that
the application of the non-local means filter removes image noise (Figure 10 right column). This is more
prominent for the scan at a voxel resolution of 6 µm as opposed to the lower quality scan.
53
Figure 10. Imaged cross-section of dry (top) and brine saturated (middle and bottom) porous glass frit at different voxel resolutions. The brine used for saturating the porous medium was 1M NaI solution. Non-local means was used for filtering the raw images to remove image noise.
Filtered scan
Dry
Bri
ne
sat
ura
ted
Raw scan
Vo
xel r
eso
luti
on
= 6
µm
Vo
xel r
eso
luti
on
= 1
8 µ
m
54
Figure 11. Histograms showing the voxel population of the different grayscale intensity values for the corresponding scans shown in Figure 10.
Figure 12 shows a comparison of segmented top view ortho slices using the machine learning and
thresholding segmentation techniques, against the benchmark case. For thresholding, the average of two
distinct attempts was considered. Each attempt was carried out by manually adjusting the threshold mark
between the phases. We see that the air phase, represented by the darker isolated regions, is easier to
recognize and is consequently successfully segmented by all techniques. Segmentation differences amongst
the various approaches are most evident in the identification of brine and solid phases, as shown in Figure
0
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Measure of CT intensity
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Measure of CT intensity
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xel r
eso
luti
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= 6
µm
Vox
el r
eso
luti
on
= 1
8 µ
m
55
12. The benchmark case (Figure 12a) shows that the brine phase has fragmented clusters making the phase
appear more disconnected in the two-dimensional space, although the phase may be connected in the three-
dimensional pore space. This continues to be seen in the other segmented cases; however, relatively bigger
clusters seem to be apparent for the unsupervised machine learning (Figure 12d). Further, it can be noticed
that the benchmark case shows a clear interfacial contact among the air, brine, and the solid phases. When
comparing the interfacial contact detected by the three segmentation techniques, it is observed that all the
segmentation techniques closely detect the interfacial contact between the air and the solid phases.
However, for the interfacial contact between the air and the brine (direct fluid/fluid contact), supervised
machine learning segmented image showed slightly better performance. Due to the missing three-phase
contact in the unsupervised machine learning and thresholding cases (Figure 12b and Figure 12d), we
observe that the air phase appears to be isolated. Oil-brine interfaces are being misidentified as part of the
solid phase during segmentation, thus potentially leading to the loss of fluid/fluid contact areas. This is
problematic because inaccurate estimations of the three-phase contacts can lead to erroneous contact angle
measurements (Alhammadi et al. 2017; Klise et al. 2016; Scanziani et al. 2017). These preliminary
observations are substantiated quantitatively through bulk and pore-scale measures in the following
sections.
56
Figure 12. Segmented cross-sectional images showing three phases (solid, brine, and air). (a) thresholding at a resolution of 6 µm (benchmark case), (b) thresholding at a resolution of 18 µm, (c) supervised machine learning segmentation at a resolution of 18 µm, and (d) unsupervised machine learning segmentation at a resolution of 18 µm. Zoomed-in version of the images are displayed on the sides to highlight distinct features of segmented images. The upper and lower regions of interest (marked inside the segmented images) correspond to labels 1 and 2, respectively.
2.3.1. Bulk measurements
Bulk measurements of porosity, phase saturations, and fluid surface areas are critical measures for
understanding fluid flow in porous media. Porosity and saturation measures help quantify the amount of oil
and gas reserves present in an oil reservoir (Lake et al. 2014); whereas, area measurements are often used
by hydrologists to quantify the extent of a chemical (non-aqueous phase liquids) spill for groundwater
remediation purposes (Culligan et al. 2005). These measurements are provided for all segmentation
techniques used in this research.
a b
c dc1
c2
b1
b2
d1
d2
a1
a2
57
2.3.1.1. Porosity, fluid saturations, and phase fractions
Figure 13 displays the bulk measures of porosity, fluid (air/brine) saturation, and individual phase
(air/brine/solid) fractions as a function of the height of the sample calculated using the segmented images
from each technique, while Table 3 presents the corresponding percent errors calculated against the
benchmark case for each average bulk measure. It is observed that for the benchmark case, porosity ranges
between (~15 to 21%) across the height of the sample and decreases slightly at the top part of the sample.
This could be due to the manufacturing artifact of the glass frit or due to slight wear of the glass frit on the
edges as it was pushed in the plastic tubing for acquiring the x-ray scans. It is also observed that all
segmentation techniques were able to capture the porosity trend. However, thresholding and supervised
machine learning performed better as opposed to the unsupervised machine learning (see Table 3 for
comparing % errors calculated from the benchmark case). The average porosity of the scanned medium for
the benchmark case, thresholding, unsupervised machine learning, and supervised machine learning to be
around 17.8%, 17.3%, 22.3%, and 19.1%, respectively.
58
Figure 13. Vertical profiles of (a) porosity, (b) brine saturation, (c) air saturation, (d) solid fraction, (e) brine fraction, and (f) air fraction for different segmentation techniques.
0
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f
59
Table 3. Summary of percent errors calculated for the different measured properties and segmentation techniques relative to the benchmark case. The percent errors for the bulk measurements of porosity, fluid saturation, and phase fractions are calculated for the respective average values across the sample height.
Measured property Phase Thresholding
(% error)
Unsupervised machine
learning
(% error)
Supervised
machine learning
(% error)
Sample porosity - 0.06 24.87 6.83
Fluid saturations Air 20.45 20.61 4.33
Brine 3.44 3.47 0.73
Phase fractions
Air 20.37 1.62 11.61
Brine 3.54 29.56 5.98
Solid 0.01 5.40 1.48
Phase surface areas
Air 27.27 13.51 6.02
Brine 31.84 14.90 29.01
Solid 26.22 8.45 22.85
Phase saturation is a measure of the relative volumes of the fluids occupying the pore space. Figure 13b
displays the brine phase saturation and Figure 13c displays the air phase saturation as a function of height
for each segmented image. We observe a mirror image plot for the two fluids with the wetting phase being
more abundant. It is noticeable that air saturation increases near the bottom of the sample. This is intuitive
since the droplets of brine were introduced from the top of the sample and less brine percolates to the bottom
of the sample – trapping more air at the bottom.
The average saturation of the wetting phase calculated using the segmented images of the benchmark
case, thresholding, unsupervised machine learning, and supervised machine learning were 85.6%, 88.6%,
88.6%, and 85%, respectively, showing that supervised machine learning provided the closest estimates to
the benchmark case with a percent error of just 0.73 % (see Table 3).
Successful segmentation is most challenging for phases that have relatively close gray-level intensities
(brine and solid in this case). We present results for the individual fractions of the solid, brine, and air
phases in Figure 13d, Figure 13e, and Figure 13f. It is found that the air phase makes a small portion of the
total bulk volume at an average fraction of ~ 0.025 when calculated using the images of the benchmark
case. The solid makes a large portion at an average fraction of ~ 0.8 (Figure 13d), while the brine phase
makes the remaining fraction at 0.175. The overall trends from different segmentation techniques continue
60
to give similar findings as to the benchmark case. As observed for the case of porosity, we see that
unsupervised machine learning differs the most from the benchmark case. It appears that this technique
estimates a larger brine fraction (consequently, lower solid fraction – Figure 13d and Figure 13e). This is
evident when comparing the benchmark case (Table 3) for the air fraction and air saturation, which shows
that despite capturing air fraction fairly, there is a large error for air saturation owing to misrepresented
brine fraction.
2.3.1.2. Surface areas
The surface areas for the different segmentation techniques are plotted in Figure 14 while the corresponding
percent errors calculated against the benchmark cases are presented in Table 3. The algorithm used to
measure the surface area uses a modified marching cube technique to generate a polygonal surface that
wraps around the three-dimensional object (the phases of interest in this case) (Landry et al. 2011). Each
face of the polygonal surface represents a triangle of fixed dimensions based on voxel size. The total surface
area of these triangles represents the total surface area of that phase. All segmented image sets were
imported into Avizo® Fire 9.43 for surface area measurements. Only the interior phase areas are reported
for meaningful comparisons.
Figure 14 shows that the total surface area of air (~ 35 mm2) is the lowest, due to the very small amount
of the air phase. We notice that the three segmentation techniques provide close measures for the total
surface area for the air phase owing to the convenient segmentation of the air phase. We also see that the
total surface areas for the brine and the solid phases are comparable, which is attributed to the fact that the
majority of the solid surface is in contact with the brine phase. Further, it is observed that the unsupervised
machine learning segmentation approach provides close estimates to the benchmark case, as opposed to the
other two segmentation techniques. This may appear to be counter-intuitive since for the other bulk
3 https://www.fei.com/software/amira-avizo
61
measurements unsupervised machine learning showed poor results. However, this suggests that it is difficult
to compare the different segmentation techniques consistently only on bulk measurements. Therefore, next
we present pore-scale measures of fluid/fluid interfacial areas and Euler characteristic.
Figure 14. Surface areas of air, brine, and solid phases on images segmented with different techniques.
2.3.2. Pore-scale measurements
The measurements of the fluid/fluid (air/brine) interfacial areas and the Euler characteristic (discussed in
section 3.2.2) of the air phase are shown in Table 4. Both these measurements were made using Avizo®
Fire 9.4. We observe large differences in the interfacial area measurements for the different segmentation
techniques when compared to the benchmark case. One reason for the higher surface area and interfacial
area measurement for the benchmark case could be attributed to the coastline paradox which is inherent as
resolution improves (Mandelbrot 1982). Despite this, the closest results are observed for the supervised
machine learning case, whereas unsupervised machine learning and thresholding present quite different
results (two to three orders of magnitude). High fluid/fluid interfacial area for supervised machine learning
0
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300
Air Brine Solid
Are
a (m
m2)
Supervised Machine Learining
Unsupervised Machine Learining
Thresholding
Benchmark Case
62
(close to the benchmark case) suggests that the air/solid interfacial area is less, such that the air clusters are
less dispersed in the solid phase but rather form continuous contact with the brine and solid phases. These
results complement observations made in Figure 12. This is an important demonstration of the success of
the supervised machine learning technique.
Table 4. Measurements of air-brine interfacial area and Euler number of the nonwetting (air) phase for the different segmentation techniques. The average air saturation was 14.4 % as determined from the benchmark case whereas, thresholding, supervised machine learning, and unsupervised machine learning, showed average values of 11.5%, 15.0% and 11.4%, respectively.
Technique Air-brine interfacial
area (mm2)
Euler number
(air phase)
Supervised machine learning 3.272 298
Unsupervised machine learning 0.013 274
Thresholding 0.003 209
Benchmark case 11.757 534
Unlike the fluid/fluid interfacial area, the Euler characteristic identifies the connectivity of a phase by
considering the number of clusters of the phase and the connection of those clusters. For the Euler
characteristic of the air phase, it is seen that all segmentation techniques, together with the benchmark case
show positive numbers – suggesting a disconnected phase. This is true since the air phase was trapped and
remained largely disconnected upon introduction of the brine phase. Further, it is observed that all three
segmentation techniques gave relatively close results, which can be attributed to the more consistent
segmentation of the air phase. Lower value for the test case as opposed to the benchmark case could also
be attributed to partial volume effects associated with the low-resolution scan which could blur out tiny,
isolated air clusters. To avoid this artifact, usually post-processing of segmented images is carried out, such
as, smoothing, erosion, dilation for removal of very small-sized clusters. However, for the results shown in
Table 4 no post-processing steps were carried out for a one-to-one comparison of each segmentation
technique with the benchmark case. Nevertheless, the closest results were observed for supervised machine
learning, although differences were slight.
63
2.4. Concluding remarks
In this research, supervised and unsupervised machine learning segmentation techniques were compared to
evaluate their strengths and limitations in segmenting multiple phases in saturated porous media. Additional
comparisons were made against thresholding segmentation technique. For this, a synthetic porous medium
with three phases (solid, brine, and air) was scanned using x-ray micro-CT at two resolutions, a high voxel
resolution of 6 µm and a low voxel resolution of 18 µm. The segmented high-resolution scan was used as
the benchmark case while the low-resolution images were segmented by the three segmentation techniques
to evaluate their performance in estimating the physical properties of the multiphase system. The properties
compared included both bulk measures such as porosity, phase saturations, and phase surface areas, as well
as pore-scale measures such as fluid/fluid interfacial areas and Euler characteristic of the nonwetting phase.
It was found that segmented images by thresholding and supervised machine learning techniques
provided close estimates of porosity, saturation, and phase fractions as that of the benchmark case.
Supervised machine learning provided better measures of fluid/fluid interfacial areas by more appropriately
capturing the three-phase contact regions. All segmentation techniques provide relatively similar estimates
for the air phase connectivity measured through the Euler characteristic, although the closest estimates were
found for the supervised machine learning case. Even though the performance of thresholding and
supervised machine learning segmentation were found to be close, supervised machine learning delivered
as a better segmentation technique for the following reasons. First, supervised machine learning showed
much better capability in capturing fluid/fluid interfacial areas compared to the other techniques. Second,
supervised machine learning required little user-dependent intervention (no image preprocessing required)
and therefore, the user-dependent time consumed for segmentation was less as compared to thresholding.
This is a significant advantage because avoiding prior denoising steps are known to damage image
structures - particularly when it is a 2-D method applied slice wise, as is the case for the implementation of
the non-local means filter. One limitation when applying supervised machine learning would be the longer
64
computational time needed to train the classifier and then apply classification voxel-by-voxel in order to
segment the whole image.
The segmentation techniques presented and compared in this research are general and can be transferred
within the scope of segmenting porous media grayscale images. For each image, a separate training set of
labeled sample voxels should be collected for the supervised machine learning technique to achieve good
performance. Parameter values such as the number of classes and grid size should be adjusted for different
images when using the unsupervised machine learning technique. The voxel intensity threshold value needs
to be chosen carefully when applying the thresholding-based segmentation. The analysis presented in this
research showed rock property estimates with varying degree of sensitivity to the choice of segmentation
technique. Therefore, such selection has to be informed by the intended application and analysis of rock
properties. We recommend the utilization of the supervised machine learning segmentation techniques to
unfiltered raw images when the speed of the segmentation process is not a challenge and the accuracy of
both bulk and pore-scale measurements is a priority. Unsupervised machine learning, as well as
thresholding segmentation techniques, are recommended for use only on bulk measurements when the
speed of the segmentation or computational resources are a challenge.
65
CHAPTER 3. EQUATION-OF-STATE AND CAPILLARY
NUMBER
Preface
The contents of this chapter were originally presented at ECMOR XVI, 16th European Conference on the
Mathematics of Oil Recovery, 3-6 September, Barcelona, Spain, 2018. The manuscript was accepted for
publication in the Computational Geoscience Journal and are referenced as,
Purswani P., Tawfik, M.S., Karpyn Z.T., Johns R.T. (2019)
On the Development of a Relative Permeability Equation of State, Comput. Geosci., (24), 807-818,
https://doi.org/10.1007/s10596-019-9824-2
Author contributions: Johns R.T. conceptualized the state function approach. Purswani P. processed the
data set used in the model. Purswani P. and Tawfik M. S. developed the modeling efforts and wrote the
original draft in consultation with Johns R.T. and Karpyn Z.T. All coauthors contributed toward analyzing
the data and updating the manuscript.
Abstract
Standard compositional simulators use composition-dependent cubic equations-of-state (EOS), but
saturation-dependent relative permeability and capillary pressure. This discrepancy causes discontinuities,
increasing computational time and reduced accuracy. In addition, commonly used relative permeability
models such as the Corey model, are empirical functions of phase saturation, where the effect of other pore-
66
scale phenomena and rock-fluid interaction is incorporated into the tuning parameters. To rectify this
problem, relative permeability has been recently defined as a state function, so that it becomes
compositional dependent and single valued. Such a form of the relative permeability EOS can significantly
improve the convergence in compositional simulation for both two and three-phase flow.
This chapter revisits the recently developed EOS for relative permeability by defining relevant state
variables and deriving functional forms of the partial derivatives in the state function. The state variables
include phase saturation, phase connectivity, wettability index, capillary number, and pore topology. The
developed EOS is constrained to key physical boundary conditions. The model coefficients are estimated
through linear regression on data collected from a pore-scale simulation study that estimates relative
permeability based on micro-CT image analysis. The results show that a simple quadratic expression with
few calibration coefficients gives an excellent match to two-phase flow simulation measurements from the
literature. The goodness of fit, represented by the coefficient of determination (R2) value is 0.97 for relative
permeability at variable phase saturation and phase connectivity, and constant wettability, pore structure,
and capillary number (~10-4). The quadratic response for relative permeability also shows excellent
predictive capabilities.
67
3.1. Introduction
Multiphase flow in porous media is of great interest in a wide array of applications including hydrocarbon
recovery, groundwater resource utilization (Nadafpour and Rasaei 2014; Parker 1989), CO2 storage (Bachu
and Bennion 2008), aquifer remediation (Chang et al. 2009; Gerhard and Kueper 2003), and two-phase
flow in proton-exchange membrane fuel cells (Akhgar et al. 2017; Lamanna et al. 2014). Each of these
applications is governed by a multitude of underlying pore-scale phenomena, such as Haines jumps (Berg
et al. 2013; Haines 1930), snap-offs (Roof 1970; Singh et al. 2017), corner flow (Mohanty et al. 1987),
capillary and viscous fingering (Nadafpour and Rasaei 2014), diffusion and dispersion (Sahimi and Imdakm
1988), dissolution/precipitation (Al-Khulaifi et al. 2017; Menke et al. 2017), and wettability alteration
(Purswani et al. 2017; Purswani and Karpyn 2019; Zhang and Austad 2006).
Considering the breadth of multiphase flow literature, studies can be classified into two categories:
macroscopic and microscopic. In macroscopic studies, averaged transport properties, such as relative
permeabilities and capillary pressure are measured on core samples to capture the effect of these flow
properties on macro-scale properties of interest, such as oil recovery. The averaged transport properties at
the core scale act as a proxy to pore-scale processes, which govern the multiphase flow process.
Standard compositional simulation employs the principles of continuum mechanics to model
multiphase flow in complex porous media, using averaged transport properties. The most commonly used
two-phase relative permeability model in commercial compositional simulators are the Corey-type models,
which are saturation-based relations that empirically carry the information of other key controlling
parameters that affect relative permeability. As explained in Khorsandi et al. (2018), this has several
limitations, including the necessity of phase labeling, which causes discontinuities as a phase disappears or
changes to another phase, causing serious convergence and stability problems (Khorsandi et al. 2018). In
addition, in order to include the effect of hysteresis, modification of the Corey model is used (Land 1968).
The modified model, however, assumes that phase saturation follows a particular path. Saturation history
is not sufficient to consider the effect of phase distribution on multiphase flow (Majid Hassanizadeh and
68
Gray 1993). Khorsandi et al. (2017) proposed a new functional form for compositionally dependent relative
permeability based on the state function concept that eliminates the need for phase labeling (Khorsandi et
al. 2017). They demonstrated excellent predictive capability even for complex hysteretic flow. Khorsandi
et al. (2018) then proposed a new compositional simulation approach that eliminated the inconsistencies
caused by saturation-based transport properties, demonstrating significant improvement compared to
current commercial simulators, resolving stability and convergence issues, as well as increased robustness
and accuracy (Khorsandi et al. 2018).
With the advancement in x-ray micro computed tomography (CT), the technology is utilized in several
microscopic multiphase flow studies, allowing for visualization and quantification of previously theorized
pore-scale processes (Berg et al. 2016; Celauro et al. 2014; Herring et al. 2017; Herring et al. 2016; Karpyn
et al. 2010; Landry et al. 2014; Pak et al. 2015). Fast synchrotron-based x-ray tomography allows for real-
time visualization of phenomena like cooperative pore filling, corner filling, droplet fragmentation, snap-
off and (Armstrong et al. 2016; Avraam and Payatakes 1999; Berg et al. 2016). Avraam and Payatakes
(1999) were among the first to propose four main flow regimes observed during multiphase flow through
porous media via 2D micromodel experiments. These included small and large ganglion dynamics, drop
traffic flow, and connected pathway flow. It was observed that as the flow rate of the wetting phase
increased, the flow regime shifted from large ganglion dynamics to small ganglion dynamics to drop traffic
flow to connected pathway flow. A simultaneous increase in relative permeabilities was observed (Avraam
and Payatakes 1999). These findings have been recently corroborated through experiments and simulation
studies by Armstrong et al. (2016).
In more recent studies by Pak et al. (2015) and Khishvand et al. (2016), the authors conducted micro-
CT experiments to visualize and quantify the trapped nonwetting phase structures at variable capillary
numbers. In addition, it was observed during drainage cycles that at higher flow rates, the number of
individual nonwetting phase clusters increases. The authors also noted the importance of the pore structure
by observing that droplet fragmentation was not severe in homogenous rocks like sandstones compared to
the more complex carbonate pore structures (Pak et al. 2015). In such studies, researchers have adopted
69
various quantitative approaches for characterizing phase connectivity, including Euler characteristic,
coordination number, percolation threshold, fractal dimension, (Blunt 2017) and specific fluid/fluid and
fluid/solid interfacial areas (Landry et al. 2014; Landry et al. 2011).
Micromodel experiments were performed by Osei-Bonsu et al. (2020) to generate phase connectivity
maps of three-phase flow systems useful for water alternating gas (WAG) enhanced oil recovery (EOR)
processes. Two WAG cycles were compared by considering oils of different viscosities and another WAG
injection was performed with surfactant added to the aqueous phase. Different measures for oil/water/gas
phase connectivity were compared, namely, Euler characteristic, normalized Euler characteristic ( ˆ ) ,
specific surface areas of the fluid phases, and wetted fractions of the solid (I). They found that the majority
of oil was recovered during the water injection step. The gas injection step, instead, assisted in fragmenting
the oil phase, which was then subsequently recovered by the water flow in a double-drainage fashion. The
trapped gas phase also prevented the formation of water channeling. Visualization maps of gas phase
connectivity corresponded well with the numerical measure of normalized Euler characteristic and therefore
this metric was found to be a better measure of phase connectivity. The gas phase showed smaller
saturations but higher connectivity due to larger-sized ganglia than that of oil or water. It was also found
that the wetting fraction of the solid (I) was useful toward describing the state space (S, , I) of the phases
involved which may be useful for resolving phase labeling issues during compositional simulations.
There is enough evidence in the literature that suggests that multiphase flow in porous media is affected
by rock properties: rock mineralogy, surface roughness, pore geometry, pore topology, heterogeneity; fluid
properties: viscosity, density; and rock-fluid and fluid/fluid interactions: wettability, adsorption,
precipitation, chemical dissolution and interfacial tension (Blunt 2017). Avraam and Payatakes (1995),
recognized the importance of the dependence of relative permeability on various parameters, in addition to
fluid saturation, such as capillary number, viscosity ratio between injected and displaced fluid, bond
number, advancing and receding contact angles, coalescence factor, pore geometrical and topological
70
factor, and the history of flow. Within reason, a number of these parameters were varied to evaluate their
impact on fluid distribution and relative permeabilities (Avraam and Payatakes 1995).
Khorsandi et al., (2017) proposed an equation-of-state (EOS) approach for modeling relative
permeabilities as a state function. The main advantage of this approach is that it is physically based and
ensures a single valued solution for relative permeability. The contributing parameters that affect relative
permeabilities were saturation of the fluid phases, phase connectivity, capillary number, wettability of the
medium, and the pore structure of the medium. The authors evaluated the importance of phase saturation
and connectivity on relative permeabilities and found a good match against experimentally published data
(Khorsandi et al. 2017). However, there was no discussion about the verification of the EOS being a valid
state function nor its validity at limiting boundaries of the state variables.
In this research, we present a structured workflow for the development of an EOS for relative
permeability using a response surface modeling approach. We define relevant boundary conditions to
physically constrain the EOS under limiting conditions and derive functional forms for the partial
derivatives. For this development we implement similar state variables proposed by Khorsandi et al. (2017).
The calibrating parameters in the final form of the EOS are determined through linear regression on the
data presented in the recent literature that presents measurements of phase saturation, phase connectivity,
and relative permeabilities. In the following sections we outline the development of the model, provide the
description of the boundary conditions, and present the results that show the fit of the model to the literature
data.
71
3.2. Methodology
3.2.1. Development of a state function
A state function, by definition, is a property whose value depends only on the condition or state of the
system irrespective of the path taken to reach that state (Cleveland and Morris 2014; Sandler 1989). This
implies that for relative permeability to be a state function, it must only have one value at a given set of the
variables considered. To satisfy this condition, the developed relative permeability function must be an
exact differential. For an exact differential, the partial derivative coefficients must satisfy the Euler
reciprocity relation (Osborne 1908). In thermodynamics, for a property that is a function of n independent
variables to be an exact differential, it must satisfy EN reciprocity relations, where
EN is the number of
conditions given by:
1
1
.n
E
i
N i−
=
= (3.1)
For a state function 1 1 2 2 n ndQ f dx f dx f dx= + + , the conditions will be of the form:
,,
kk
ji
j i x k ix k j
ff
x x
=
, (3.2)
for i 1: 1n − ; j 1:i n + ; k 1: n .
Another condition contributing to the validity of a state function is whether the state variables
considered are independent. Properties are considered independent of each other when one property can be
72
varied while all other input properties are held constant. On analyzing microscopic multiphase flow
experimental studies in the literature, it is observed that saturation and phase connectivity, are independent
variables except when saturation is exactly one (e.g., Schlüter et al. (2016)). At the same saturation value,
multiple fluid configurations can exist leading to widely different phase connectivity.
As discussed previously, there are numerous pore-scale variables that may contribute to changes in
relative permeability. Hence, given the complex nature of the problem, there are many degrees of freedom
that can be specified to fully define relative permeability as an EOS. Including all the parameters that
contribute to changes in transport properties in the state function would theoretically result in an exact
match of the literature data. Practically, however, a very complex model would be required to account for
all state variables that affect relative permeability. Therefore, in this research, we use the minimum number
of variables that sufficiently define the state of the system, exhibiting a good match with literature data, and
allowing for reliable relative permeability predictions, with an acceptable degree of accuracy.
Khorsandi et al. (2017) proposed an equation-of-state approach (Eq. (3.3) and Eq. (3.4)) to calculate
the change in relative permeability as a function of five measurable, pore-scale state variables (Khorsandi
et al. 2017).
( )ˆ, , , ,r j j j j cak f S I N = . (3.3)
Expressing Eq. (3.3) in exact differential form,
ˆ ˆ
r j r j r j r j r j
r j j j j ca
j j j ca
k k k k kdk dS d dI dN d
S I N
= + + + +
, (3.4)
73
where Sj represents saturation of phase j; ˆj represents connectivity in terms of the normalized Euler
characteristic of phase j; Ij is the wettability index of phase j; NCa is the capillary number; and is the pore
structure.
At constant wettability (dIj = 0), constant pore structure (dλ = 0), and constant flow rate and fluid
properties (i.e.: dNCa = 0), Eq. (3.4) reduces to,
ˆ
ˆˆ
j j
r j r j
r j j j
j j S
k kdk dS d
S
= +
. (3.5)
For the simplified relative permeability state function defined in Eq. (3.5), only one reciprocity condition
(see Eq. (3.1)) must be honored as,
ˆ
ˆ ˆj j
r j r j
j j j j S
k k
S S
=
. (3.6)
By forcing relative permeability to satisfy Eq. (3.5), we ensure that there is only one value of relative
permeability as a function of two variables, while simultaneously capturing the essential physics. The error
introduced by this approach is minimized by tuning to literature data.
3.2.2. Phase connectivity
The Euler characteristic depends on saturation, saturation history, pore topology, and the scale of the
measurement. To allow for better comparison based on phase connectivity only, we must normalize the
Euler characteristic to eliminate such effects. Herring et al. (2013) developed a normalization scheme that
74
eliminated the effect of pore structure by dividing χnon-wetting phase by χpore-structure, which is equivalent to the
Euler characteristic at 100% phase saturation (Herring et al. 2013). However, this only sets an upper bound
to the value of the normalized Euler characteristic, , where ˆ 1− . Khorsandi et al. (2017) modified
this normalization scheme to eliminate the effect of measurement scale and phase saturation, as well as to
set a lower bound to the value of (Khorsandi et al. 2017). In this chapter, we use a simpler expression
for the normalization of the Euler characteristic (Eq. (3.7)) such that is bounded between zero and one.
,ˆ
max
min max
−=
− (3.7)
where χmax represents the limiting case for a completely dispersed phase which is expected to occur when
all the pores in the porous medium are filled with the phase, but no throats are filled to connect the pores;
and χmin represents the case where a phase is fully connected, occupying 100% of the pore space. χmin can
be easily estimated as the Euler number of the pore space from its micro-CT image, whereas χmax can be
estimated as the number of pores from the extracted pore-network of the pore space, or from the
coordination number of the rock type and its χmin value. These minimum and maximum values of the Euler
characteristic are independent of the fluid type.
Equation (3.7) defines an intrinsic connectivity parameter for a homogeneous medium that is no longer
dependent on the pore volume considered. A value of zero for the dimensionless phase connectivity means
the phase is disconnected completely, while a value of one is perfect connectivity.
3.2.3. Development of relative permeability EOS
For developing the EOS (i) must satisfy the reciprocity relation shown in Eq. (3.6), (ii) must honor physical
boundary conditions, and (iii) must be the simplest functional form possible to minimize overfitting of test
75
data. Therefore, we consider that the relative permeability state function takes a simple form, such that the
partial differential coefficients are linear in ˆj and Sj. Thus, we make the relative permeability state function
a quadratic response to Sj and ˆj expressed by,
2 2
0 1 2 11 22 12ˆ ˆ ˆ r j j j j j j jk S S S = + + + + + . (3.8)
Next, we describe the principle limiting conditions presented in Table 5 to constrain the EOS. The first
constraint is that for both saturation and phase connectivity equal to 1.0 the relative permeability must be
1.0. At phase saturation just below 1.0, the phase connectivity can theoretically vary over its entire range,
although physically only a small range is likely for a given set of variables.
Table 5. Physical constraints imposed on the relative permeability response by considering key limiting conditions that affect relative permeability as a function of phase saturation and phase connectivity. The phase is assumed nonwetting, although extensions to other phases are easily possible.
We set the phase connectivity to be 1.0 and saturation to be 0.0 for the second constraint. At low phase
saturation, in general, the phase connectivity should be low, however, for a wetting phase under extreme
wetting conditions the connectivity could be high as well. Also, for a nonwetting phase (say oil), the region
near ˆj = 1 and Sj = 0 is a plausible physical region for cases such as film drainage when two other phases
are present (say gas and water). Otherwise, it is unlikely to achieve flow near this region.
Physical Constraint Remarks
1) 1r jk = at 1 jS = and ˆ 1j = Relative permeability at complete saturation must be 1
2) 0r jk = at 0 jS = and ˆ 1j = Relative permeability at low phase saturation should be
negligible
3) 0
ˆ0
j
r j
j S
k
=
= at ˆ 1j = The change of relative permeability with phase connectivity
should be negligible near full phase connectivity
4) ˆ 0
0
j
r j
j
k
S =
=
at 1jS = The change of relative permeability with phase saturation
should be negligible near full phase saturation
Remarks
Relative permeability at 100% saturation must be 1
Relative permeability at low phase saturation should be
negligible
The change in relative permeability with full phase
connectivity should be negligible near full phase connectivity
The change in relative permeability with full phase saturation
should be negligible near 100% saturation
76
The third and fourth constraints are set to ensure that the partial derivatives are positive over the entire
ˆj and Sj space. These constraints could be removed if more experimental data is available to improve the
values of relative permeability in regions near these limits. We found it necessary to include these
constraints for the data examined in this chapter.
We did not constrain the relative permeability function at Sj = 1.0 and ˆ 0j = , as it is not physical to
reach this value of connectivity. That is, at exactly a saturation of 1.0 the phase connectivity must be 1.0 in
that it is no longer independent, but at a saturation of 0.99 and ˆ 0j = the relative permeability should be
zero. We omitted this constraint from the fitting procedure based on the recognition that complex porous
media would likely never have values near this region. It is likely that there is a limiting value of ˆj as a
function of saturation based on pore morphology and other state variables.
Upon implementing these physical constraints, we obtained the final form of the model shown below.
( ) ( )ˆ ˆ ˆ ˆ2 2
r j 11 j j 22 j j j j j jk = α 1- 2χ + χ +α -2S +S + χ S + χ S . (3.9)
The coefficients 11 and
22 are determined through linear regression on measured data. Evaluation of
the partial derivatives of Eq. (3.9) gives,
( )11 22 11ˆ 12 2
ˆj
r j
j j
j S
kS
= + −
+
, (3.10)
( ) 222 22
ˆ
2ˆ 1 2 2
j
r j
j j
j
kS
S
+= + −
, (3.11)
77
Euler reciprocity shows that these derivatives define a state function (Eq. (3.6). That is,
ˆ
221ˆ ˆ
j j
r j r j
j j j j S
k k
S S
= =
+
. (3.12)
The exact differential form (Eq. (3.5)) then becomes from Eqs. (3.10) and (3.11),
( ) ( ) 222 211 22 1 2 21ˆ ˆ ˆ12 2 12 2 r j jj j j j jd S d S dk S = + − + + − + + . (3.13)
3.2.4. Comparison to the development in Khorsandi et al. (2017)
The relative permeability EOS proposed by Khorsandi et al. (2017) is shown below (Khorsandi et al. 2017),
( ) kn
r j k j r jΦk C Φ= − , (3.14)
where jΦ is the phase distribution term defined as
jjS
+ ;
r jΦ is the residual phase distribution of phase
j; Ck, αφ, and nk are tuning parameters. It was assumed in this formulation that the ratio of the two partial
differential coefficients was constant. The form of Eq. (3.14) allowed for direct use of the Corey model,
while also making the equation simple. Although Eq. (3.14) satisfies reciprocity, it does not satisfy all
boundary conditions in Table 5 and likely is not reliable except near the tuned experimental data.
We set the value of nk to 2 in Eq. (3.14) to get,
( )2
2 2 2 2 2 2j j jr j k j r j j j r j r jk S Φ ΦS ΦC S = + + + − − . (3.15)
78
Comparing our development in Eq. (3.9) to Eq. (3.15) we see,
22kC = , (3.16)
11
22
= , (3.17)
11
22
r jΦ
= . (3.18)
More complicated cubic or higher order polynomial equations could also be used, but the simplest form
that reasonably matches experimental data and boundary conditions is preferred to avoid over-fitting. Most
importantly, the EOS developed in this chapter honors the physical boundary constraints presented in Table
5. The response surface formulation provides justification for Eq. (3.14), although future research could
define a form of Eq. (3.14) that honors physical constraints like those in Table 5.
79
Figure 15. Illustration showing the EOS state approach on a real path (simulation) taken during two-phase flow simulation in
jS and ˆj space.
In Figure 15, we illustrate that once the EOS is determined, the real path from the initial to the final
state can be separated into a constant saturation path followed by a constant phase connectivity path or vice
versa to arrive at the same final state. In this way, relative permeability can be calculated by integrating the
individual partial differential coefficients to arrive at the final state’s relative permeability. The shortcoming
of such an approach is that it requires phase connectivity values at the initial and the final state, which is
not readily available unless sophisticated techniques such as x-ray micro-CT are implemented. One way of
overcoming this shortcoming is to determine a functionality between phase connectivity and phase
saturation so as to bypass the dependence of relative permeability on phase connectivity. This provides an
avenue for future research. Khorsandi et al. (2017) solved this problem by assuming that the change in
connectivity with saturation is constant for any drainage path, and similarly for any imbibition path. They
used simple but different models for drainage and imbibition and tuned them to available data. From these
fixed tuned models, they could predict hysteretic scans that began at different saturations.
Simulation
path
( )11 22 11ˆ 12 2
ˆj
r j
j j
j S
kS
= + −
+
( ) 222 22
ˆ
2ˆ 1 2 2
j
r j
j j
j
kS
S
+= + −
80
3.2.5. Estimation of the coefficients of the EOS
The data set used for estimating the coefficients of the model in Eq. (3.9) is from (Armstrong et al. 2016).
In their paper, the authors coupled experimental research with simulations to study the effect of phase
topology on macroscopic system behavior during two-phase flow in a porous medium. Micro-flow
experiments were conducted in a sintered glass sample with chemically doped water as the wetting phase
and decane as the nonwetting phase. The two phases were co-injected at different fractions maintaining
steady state conditions, and three different flow rates were tested to represent three different capillary
numbers, namely, 10-4,10-5, and 10-6. From the segmented images acquired during micro-flow experiments,
11 different fluid configurations (each representing a different fluid saturation arrangement) were used as
the initial condition for two-phase flow simulations to determine relative permeabilities for a wider range
of capillary numbers obtained by varying fluid properties during these simulations. A 4-D connected
component algorithm was implemented to track the fluid ganglion during simulations for estimating phase
connectivity.
The full data set is displayed in Figure 16. Figure 16a shows the data for phase relative permeability
while Figure 16b shows the data for phase connectivity, which is measured as normalized Euler
characteristic using Eq. (3.7). The phase shown here is the nonwetting phase because the Euler
characteristic for the wetting phase was not reported.
To calculate the normalized Euler characteristic values ˆj , χmax and χmin need to be known. Because the
number of pores and the Euler number of the pore structure was not reported, a 2D extrapolation was carried
out on the data for NCa = 1, which showed the maximum and minimum values for the Euler number. Large
capillary numbers imply very low interfacial tension, which may explain the largest and smallest
connectivity values observed.
81
Figure 16. Phase saturation, relative permeability, and normalized Euler connectivity for the nonwetting phase for different capillary numbers used for fitting the quadratic response for relative permeability as well as for prediction purposes. Courtesy Dr. Ryan T. Armstrong. Data from Armstrong et al. (2016). This data is tabulated in appendix D.
We assumed a planar relationship among krj, Sj, and ˆj for the 2D extrapolation,
.ˆjr j jk AS B C= + + (3.19)
Three point-extrapolation was carried out such that the extrapolated values were near the actual data as
opposed to extrapolation on the entire data, which could lead to errors in estimation. The first, three data
points of the Euler number were used for determining χmax, where krj was set to zero, while the last three
data points were used for determining χmin, where krj was set as one. Simultaneously, the same three data
points on either end were used for fitting lines through Sj and ˆj for fixed NCa, which were then intersected
with the plane (Eq. (3.19)) to estimate χmax and χmin values. The extrapolated values for the Euler
characteristic of the pore structure are shown in Table 6.
Table 6. Euler characteristic values estimated through 2-D extrapolation for the pore structure used during simulations in Armstrong et al. (2016).
a b
c
max 5788
min -10,704
82
3.3. Results and discussion
In this section, we present the results for the fitted quadratic response for relative permeability. The sub
data set of capillary number ~10-4 was used for response surface fitting. This fit was used to predict data
sets at different capillary numbers. The goodness of fit is evaluated using residual error and R2 values.
Further, we present the partial derivative coefficient to the exact differential of relative permeability as a
function of phase saturation and phase connectivity. Finally, we compare response surfaces generated for
different capillary numbers using linear regression on the individual sub data sets to evaluate the impact of
capillary number.
3.3.1. Quadratic response for relative permeability
We used linear regression with Matlab® to find the coefficients in the proposed model described in Eq.
(3.9) that best fits the data set at the fixed capillary number of ~10-4. The base code is provided in appendix
C. Table 7 provides the information for the fitting parameters and the goodness measure of the fit.
Table 7. Model coefficients and the goodness of quadratic response surface fit to phase saturation and phase connectivity to the data presented in (Armstrong et al. 2016) at the NCa of ~10-4.
The contour map of the response surface fit is shown in Figure 18. Contour map of the response surface
of relative permeability as a function of phase saturation and normalized Euler connectivity. The capillary
number (~ 10-4), wettability, and pore structure have been kept constant. Data points shown as black dots
were taken from the two-phase flow simulations presented in Armstrong et al. (2016). Dashed line
11 -0.229
22 -0.589
R2 0.971
Root mean squared error 0.147
83
represents a limiting boundary of plausible values. The dots represent the data points at the fixed capillary
number of ~10-4 used for estimating the coefficients for the response surface. The corresponding plot for
the residual error is shown in Figure 17. As shown, the quadratic response gives small residual error values
scattered around zero with a mean of -0.009, showing little systematic error.
As shown in Figure 18, the general trends of relative permeability versus saturation and relative
permeability versus phase connectivity are honored. The contour map gives the relative permeability value
for a known value of normalized phase connectivity and its corresponding value of phase saturation,
irrespective of the path/direction a particular experiment/simulation may take. The contour map is also
independent of the phase label (gas, oil, or water, for example). This is valid for the fixed wettability, pore
structure, and capillary number used in the development of this quadratic response.
Figure 17. (a) Quadratic response prediction versus simulation data (b) residual between the predicted and simulation measurements for relative permeability based on the response surface fit shown in Figure 18.
a b
84
Figure 18. Contour map of the response surface of relative permeability as a function of phase saturation and normalized Euler connectivity. The capillary number (~ 10-4), wettability, and pore structure have been kept constant. Data points shown as black dots were taken from the two-phase flow simulations presented in Armstrong et al. (2016). Dashed line represents a limiting boundary of plausible values.
We present a notional boundary on the contour plot in Figure 18, which represent the limits of possible
physical experimental/simulation conditions. The region below and to the right of the curve are extreme
cases controlled by the topology of the rock structure itself, as well as other variables such as wettability.
This region suggests that even at very high phase saturation, the phase remains extremely disconnected.
Such a case would be highly unusual to occur, especially in real porous media. It may occur in theoretical
porous media with a highly disconnected pore structure, or a pore structure with a very large aspect ratio
between the pore and connecting throats so that very high capillary forces are required for the phase to pass
through. Since there is insufficient data in this region, the prediction from our quadratic response fit may
not lead to conclusive results for this region. A similar “unrealistic” region could be present in the upper
left corner of the contour map, although this is not shown in Figure 18. To achieve low saturations with
high connectivity would require very thin wetting films or spreading of an intermediate wetting phase (in
three-phase systems), where even at very low phase saturation a phase remains highly connected. Although
85
the phase would remain highly connected, the relative permeability would be small in this region of jS
and ˆj space owing to small saturation. More experimental studies should be conducted under extreme
conditions and varying wetting states and pore topology of the medium to acquire more complete data sets
to enhance predictive capabilities of these cases and to capture the loci of zero relative permeabilities in Sj
and ˆj space.
In Figure 19, the partial differential coefficients expressed in the final form of the exact differential
(Eq. (3.13)) are shown as a function of phase saturation (Figure 19a) and phase connectivity (Figure 19b).
Both partial derivatives with respect to relative permeability are always positive, suggesting that relative
permeability increases with an increase in Sj as well as ˆj . We further observe that the rate of increase in
relative permeability decreases with increasing saturation whereas, the rate of increase of the relative
permeability with increasing phase connectivity increases, although this increase is minor and plateaus near
~ 0.55ˆj . This suggests that the effect of an increase in phase saturation on relative permeability slowly
declines, while the effect of phase connectivity grows. This is consistent with our understanding that for a
phase to be sufficiently connected in the porous medium some phase saturation should exist.
Figure 19. Partial derivative coefficients (calculated using Eqs. (3.10) and (3.11)) expressed as a function of (a) phase saturation and (b) normalized Euler connectivity.
a b
86
3.3.2. Quadratic response prediction at neighboring conditions
We now present the predictive capability of the fitted response surface to neighboring conditions. We use
the same data set by Armstrong et al. (2016) but at capillary numbers of ~10-3 and ~10-5. This ensures that
the pore structure and wetting conditions remain the same between the fitted and the predicted cases. The
plots for the response and the corresponding residual errors are shown in Figure 20.
Figure 20. (a) Prediction of relative permeability and (b) residual error for capillary numbers ~10-3 and ~10-5 based on the response surface fit to capillary number 10-4 described in Figure 18.
Figure 20 shows that the predicted response fits the data well. The residual errors between the predicted
and actual relative permeability values show little systematic errors and R2 values near ~0.94 for both
capillary numbers. It is likely that capillary number impacts these values, as is discussed in the next section
in more detail.
3.3.3. Effect of capillary number
To capture the effect of capillary number on relative permeability we use the surface fits to predict relative
permeability at different capillary numbers ranging from ~ one to ~10-6. The goodness measure of these
prediction cases is shown in Figure 21.
a b
87
Figure 21. R2 error for prediction of data at different capillary numbers using sub data set at capillary number of ~10-4 as the fitted response surface.
From Figure 21 we see that the R2 value showing the goodness of fit is the maximum for capillary
number ~10-4 marked by the dashed red line. This is because the regression was carried out using this
capillary number sub data set. As stated earlier, the R2 values in the neighborhood of the fitted response are
excellent at about 0.94, however, as we move two to three orders of magnitude away from the original
capillary number, the prediction with NCa ~10-4 leads to erroneous values. This clearly suggests the
importance of capillary number as a parameter that affects relative permeability. This is also shown in
Figure 22 for the fitted quadratic responses to individual sub data sets at different capillary numbers (see
Table 8 for fitting parameters). As capillary number increases the response surface becomes more planar
showing that the dependence of relative permeability on phase connectivity is reduced significantly while
relative permeability becomes more sensitive to the change in saturation. The occurrence of these
observations can be quantitatively observed in Table 8, where the magnitude of α11 term decreases
significantly at higher capillary number. A sharp decrease in the magnitude of this term explains the reduced
effect of phase connectivity (see Eq. (3.9)). Overall, these observations are consistent with those observed
by Armstrong et al. (2016), where connectivity was found to be larger at greater capillary numbers and
88
therefore relative permeability becomes more strongly dependent on phase saturation (Armstrong et al.
2016).
Figure 22. Quadratic response surface fits to sub data sets at different capillary numbers.
Table 8. Coefficients for the quadratic response and goodness measures for quadratic response surface fits to sub data sets at different capillary numbers shown in Figure 22.
Nca = ~10-6
Nca = ~10-4
Nca = ~100
Coefficients for the
quadratic response and
goodness measure
Capillary number
~100 ~10-4 ~10-6
11 -0.009 -0.229 -0.517
22 -0.806 -0.589 -0.667
R2 0.994 0.971 0.863
Root mean squared error 0.064 0.147 0.249
89
Figure 23. Phase relative permeability plots with corresponding phase connectivity value (shown in blue) (a) high capillary number ~1 (b) low capillary number ~10-5. The red solid lines represent the fit using Corey model with ~ exponent value of (a) 0.76 and (b) 1.29. The residual saturation in (a) was set to 0 while computing the Corey exponent because that data point was not known.
In Figure 23, we show typical Corey fit to the data at extreme capillary numbers. The fit to the exponent
was < 1 for the high capillary number case and > 1 for the low capillary number case. These exponents
govern the curvature of the relative permeability change with respect to saturation, which causes a slower
or faster increase in relative permeability as saturation changes. These plots reveal the inherent importance
of phase connectivity implicitly assumed in Corey’s approach for fitting relative permeability. The
slower/faster change in relative permeability as seen in Figure 23 is the result of the changes in phase
connectivity. Such relative permeability curves are also observed during microemulsion/excess oil and
microemulsion/excess brine relative permeability measurements conducted by Delshad et al. (1987) where
the microemulsion phase relative permeability were observed to increase sharply (Delshad et al. 1987).
This increase was attributed to the wettability and low interfacial tension of the microemulsion phase, which
in principle improved connectivity to increase relative permeability.
Under limiting conditions of NCa approaching ∞, for example, at negligible interfacial tension between
the phases, the two phases will collapse to a single phase. At this point, S and will approach the limiting
value of one and relative permeability would also approach one.
a b
90
3.4. Concluding remarks
In this chapter, we present the development of a physically-based quadratic state function for relative
permeability in phase saturation and connectivity. The coefficients of the EOS are determined through
linear regression on two-phase flow simulation data from the literature. The following conclusions can be
drawn under the assumptions in which this study is conducted.
• A simple quadratic response for relative permeability gives an excellent fit to simulation data at
fixed capillary numbers.
• The quadratic response fit acquired from one data set shows excellent predictive capabilities at
similar flow conditions. However, away from original conditions, the predictive capability of the
kr response surface decreases owing to the dependence on capillary number.
• Connectivity increases faster at low saturations than at high saturations for high capillary numbers.
This explains the small Corey exponents obtained for ultra-low interfacial tension corefloods. The
reverse is true for small capillary number where capillary effects dominate.
Although the model presented in this chapter may not be the only solution, our approach was designed
to seek the simplest EOS that honors key limiting physical constraints and provides a reasonable fit to the
data from the literature. The presented approach is mathematically simple, which is advantageous for
potential use in compositional simulation. The assumption of a quadratic response surface may not fit all
relative permeability data, where higher-order polynomials may be needed. Nevertheless, the response
surface approach developed here justifies the relative permeability functionality presented by Khorsandi et
al. (2017) for an exponent of 2.0. In addition, one of the main outcomes of using the developed EOS is being
able to eliminate discontinuities resulting from using saturation-dependent relative permeability
correlations in compositional simulators, which increases computational time and reduces accuracy. This
would be especially useful to account for complex processes such as mass transfer and dynamic changes
that often occur during transport of multiple phases in real porous media.
91
CHAPTER 4. IMPACT OF WETTABILITY ON PHASE
TRAPPING
Abstract
Capillary trapping is an important carbon capture strategy for mitigating greenhouse gas emissions.
Experimental investigations suggest that multiple factors like pore structure, surface roughness, wettability,
and CO2 dissolution can impact capillary trapping. Among these, reservoir wettability is of great
significance as it impacts phase distribution and consequently phase trapping. We use pore-network
modeling to investigate the effect of wettability on phase mobility and capillary trapping. For this, we
recreate strong wetting and nonwetting conditions during the secondary flooding cycles. Different initial
phase saturations are investigated, and the trends of residual phase saturation are studied for each wettability
case. We also present residual phase connectivity trends to explore the impact of wettability on the locus
of residual phase connectivity and phase saturation. Finally, we propose simple models to capture all
residual trends.
Simulations show that when the receding phase is the wetting phase, there is reduced trapping of that
phase which occurs due to combinations of layer flow of the wetting phase and piston-like advance by the
nonwetting phase. Such flow regimes cause nonlinearity in the characteristic initial-residual (IR) saturation
curve with a maximum trapped (residual) saturation of ~0.5 corresponding to initial saturation ranging
between 0.7-0.95. A new extended Land-based model matches the IR trends for all wettabilities.
Simulations also show that the loci of residual phase saturation and phase connectivity, are a strong function
of wettability. At strongly nonwetting conditions, the locus remains fairly constant in phase connectivity,
while at increasingly wetting conditions, the locus shows trends that resemble a closed-loop and the size of
the locus changes. The limiting values of the residual locus are found to be dependent on the pore topology.
92
The residual locus of phase connectivity and saturation provide a true limiting condition useful for reservoir
modeling. Through network-modeling, this research discusses the impact of wettability on residual trapping
of a phase, which is critical for evaluating the success of long-term CO2 storage projects.
DISCLAIMER
This work was funded by the Department of Energy, National Energy Technology Laboratory, an agency
of the United States Government, through a support contract with Leidos Research Support Team (Neither
the United States Government nor any agency thereof, nor any of their employees, nor LRST, nor any of
their employees, makes any warranty, expressed or implied, or assumes any legal liability or responsibility
for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed,
or represents that its use would not infringe privately owned rights Reference herein to any specific
commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not
necessarily constitute or imply its endorsement, recommendation, or favoring by the United States
Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily
state or reflect those of the United States Government or any agency thereof.
93
4.1. Introduction
If global warming were to persist at current rates, global temperatures may rise by 1.5 °C by 2035 and 2052.
Further elevated levels of warming may cause irreversible damage to the environment (IPCC 2018).
Greenhouse gas (GHG) emissions, such as those of anthropogenic carbon from industrial and energy
sources contribute to global warming. Therefore, long-term sequestration of carbon dioxide in deep
subsurface formations will be an important mitigation strategy for combating emissions of GHGs (Orr
2009; Schrag 2009) In order to sustain energy demand, implementation of carbon capture strategies as an
integral part of the evolving energy mix can ensure energy security for future generations (Dammel et al.
2011; Szulczewski et al. 2012).
Different mechanisms of CO2 sequestration have been identified (Iglauer et al. 2015; IPCC 2005;
Krevor et al. 2015). These include structural trapping of buoyant CO2 plumes under low permeability
caprocks; residual trapping and local capillary trapping caused as entrapment of blobs/ganglia of CO2
because of capillary forces (Pentland et al. 2010; Pentland et al. 2011); dissolution trapping of CO2 in saline
aquifers (Iglauer et al. 2011a; Iglauer et al. 2011b); or mineral trapping of CO2 caused as carbonate mineral
precipitation. Dissolution and mineral trapping are slow processes and would occur over prolonged time
scales; but residual trapping can be achieved faster and thus will be an effective strategy for decades to
come (IPCC 2005). Residual trapping is also attractive due to the availability of the necessary infrastructure
in the energy industry as well as the storage capacity to scale carbon capture. A 2-D schematic
representation of capillary trapping is shown in Figure 24. The schematic depicts heterogeneity in the
porous medium in terms of local pore/grain size variations, mineral complexities, and local variations in
wettability. The moving front of the flowing chase brine leaves capillary trapped globules of CO2 behind.
94
Figure 24. Schematic representation of capillary trapped CO2 by chase brine. Local heterogeneities in pore structure, mineral complexity, and wettability are depicted.
Laplace’s equation on capillary pressure across the interface between two immiscible fluids, in a
capillary tube geometry, shows that interfacial forces acting on the fluid/fluid/solid system and the
capillary’s cross-section govern the entry pressure through the opening. This implies that both wettability
and pore structure are critical in dictating pore occupancy by the CO2 phase and thus the amount of trapped
CO2. A variety of other factors have also been recognized to impact phase trapping. These include, the
initial CO2 saturation before brine flooding; pore-throat aspect ratio, pore and grain size distribution
(Jerauld and Salter 1990), heterogeneity in porosity and permeability (Krevor et al. 2011; Ni et al. 2019;
Perrin and Benson 2010) rock and fluid interactions such as rock wettability; chemical interactions such as
wettability alteration (Anderson 1987; Salathiel 1973; Tanino and Blunt 2013), rock dissolution, salt
precipitation, or dissolution of CO2 into the brine; as well as mineral heterogeneity of the rock (Iglauer et
al. 2015). Of these factors, the impact of rock wettability on capillary trapping remains a topic of great
interest in the porous media research community.
One reason for the importance of wettability is that it directly affects phase distribution (Blunt et al.
2020; Sun et al. 2019) which can consequently impact capillary trapping. This can have implications for
capillary trapping
CO2 plume
Chase brineLeading end of the CO2 plumeTrapped CO2
Trailing end of the CO2 plume
1
95
the evaluation of the ultimate quantity of carbon that can be successfully sequestered. Thus, wettability can
govern the suitability of storage sites like aquifers or depleted hydrocarbon reservoirs (Levine et al. 2014).
It is likely that the more nonwetting the formation is to CO2, the larger the trapping capacity, given that
nonwetting phase occupies the centers of pores (Blunt 2017). In addition, wettability is critical for
preserving caprock integrity and CO2 leakage because if caprock minerals become altered to a CO2 wetting
state, the sealing capacity as measured by the entry capillary pressure will be reduced. For example, Chiquet
et al. (2007) found that the wettability to brine in a CO2/brine/mineral system reduced under typical
reservoir pressures. More substantial wettability alteration was observed on mica as opposed to quartz. In
addition, spontaneous imbibition could give rise to preferential leakage pathways for the CO2 to escape.
Although such leakage is deemed to less concerning in comparison to those that may occur because of over-
pressurized faults or abandoned wells (Iglauer et al. 2015).
Extensive research efforts have shown that wettability of CO2/brine/solid systems depends on different
factors such as CO2 phase pressure which can influence the amount of dissolved CO2 in the brine phase
(Chiquet et al. 2007; Espinoza and Santamarina 2010). Other factors such as the brine phase salinity, the
system temperature, mineral heterogeneity, and surface roughness of the rock can all have significant
impact on the wetting preference of the rock. For detailed reviews on the topic of wettability of
CO2/brine/rock systems, the reader is referred to the works of Arif et al. 2019; Iglauer et al. 2015; and
Yekeen et al. 2020. It is largely concluded that an increase in the phase pressure of CO2 results in weakening
water-wetting for the CO2/brine/rock systems.
Bachu and Bennion (2008), and Espinoza and Santamarina (2010) observed a drop in the CO2/brine
interfacial tension with increasing CO2 pressure. This was ascribed to the increase in CO2 concentration at
the fluid/fluid interface because of an increase in the dissolved CO2. Espinoza and Santamarina (2010) also
found oil-wet surfaces to be either intermediate or nonwetting to brine in the presence of CO2. Saraji et al.
(2013) found wettability alteration to weakly water-wet conditions on quartz surface at pressures
corresponding to supercritical CO2 conditions. High temperature was also found to cause an increase in
water advancing contact angles. Despite an increase of up to 10° in contact angles, all measurements
96
remained primarily under water-wet conditions likely due to the application of cleaned surfaces (Iglauer et
al. 2015; Saraji et al. 2013). Other researchers, however, have found more significant changes. For example,
Kim et al. (2012) showed through micromodel studies conducted with silica grains that wettability of a
CO2/brine/silica system can be altered to weakly water-wet conditions within few hours of CO2 injection.
They observed contact angles as high as 80° measured through the brine phase. The alteration in wettability
was attributed to the dissolution of CO2 into the brine and the associated pH drop to acidic conditions.
Similarly, weakly water-wet trends were observed for higher brine salinity which the authors ascribed to
the thinning of water films at high salinity (Kim et al. 2012). Still, wettability data of CO2/brine/rock
systems for varying salinity and temperature conditions remains sparse and less conclusive in the literature
(Arif et al. 2019; Iglauer et al. 2015).
Some researchers have qualitatively assessed the wettability of CO2/brine/rock system through relative
permeability measurements. For example, Levine et al. (2014) found the endpoint relative permeability to
CO2 to be clustered between 0.35 and 0.4, which is half that of typical water-wet reservoirs, suggesting for
weakly-water wetting conditions. This may result in decreased CO2 injectivity and consequently affect the
disposal capacity of reservoirs that may be prone to leakage at high pressures (Levine et al. 2014). Yet
others, for example, Akbarabadi and Piri (2013) have found supercritical CO2 (scCO2) as nonwetting with
an endpoint drainage relative permeability of 0.19 and cross-over brine saturation of >0.75. However, when
comparing the trapped CO2 saturation for a given initial CO2 saturation, through unsteady-state flow
experiments, the amount of gaseous CO2 that was trapped was greater than scCO2. This was attributed to
brine being more wetting in the presence of gaseous CO2 (Akbarabadi and Piri 2013).
Trapping behavior is characterized through initial-residual (IR) saturation trapping curves. This was
first proposed by Land (1968) to resolve relative permeability hysteresis by linking imbibition relative
permeability to drainage relative permeability (Carlson 1981; Killough 1976; Land 1968). IR characteristic
curves are extremely useful in reservoir engineering from the perspective of phase trapping for carbon
sequestration, and for the resolution of relative permeability hysteresis (Juanes et al. 2006; Spiteri et al.
2008). The IR curves describe the relationship between the initial phase saturation at the start of the water
97
injection process (example: waterflooding) to the remaining (and/or residual) saturation of the receding
phase and are considered specific to the rock being flooded (Lake et al. 2014). This is due to the dependence
of IR curves on the pore structure and wettability of the rock. Most studies in the literature with water-wet
behavior find similar IR trends, i.e., as initial saturation increases (~ up to 0.5), the residual saturation
increase because of higher amount of nonwetting phase available for trapping, but at initial nonwetting
phase saturation of >0.5, the residual phase saturation either increase slightly or remain constant (Alyafei
and Blunt 2016; Li et al. 2015; Pentland et al. 2010; Pentland et al. 2011).
Most two-phase flooding experiments, however, are unable to capture the region of very high initial
phase saturation (>0.8) (see Fig. 10 in Krevor et al. 2015) likely due to experimental limitation of achieving
very high capillary pressures to attain such high initial nonwetting phase saturations. Rocks that are
nonwetting to brine, show a broader spread in the IR saturation curve than those of water (brine) wet (see
Figure 2 in Alyafei and Blunt 2016). One observation between water-wet and non-water-wet IR curves is
that the residual saturations following brine injection is lower for the non-water-wet systems (Alyafei and
Blunt 2016). Few researchers have observed a nonlinear trend in the IR saturation curve with an apex near
high initial saturations. For example, see experiment 4 in Pentland et al. (2010), experiment with Mt. Simon
rock in Krevor et al. (2012) and Krevor et al. (2011), water-wet Estiallades in Alyafei and Blunt (2016),
Bentheimer sandstone experiments Herring et al. (2015), pore-network simulations by Spiteri et al. (2008).
Figure 25 shows the IR curves for these experiments.
98
Figure 25. Initial-residual characteristic curves for selected literature studies.
From the literature discussed here, it is evident that wettability is a critical parameter that impacts phase
trapping. Secondly, experimental works show that a range of wetting conditions are possible for the
CO2/brine/rock system. Thirdly, flow experiments provide a good basis for the trapping trends but are
unable to cover the full initial saturation and wettability ranges. Thus, limiting the ability of currently used
models in capturing trapping trends over the full saturation range. Moreover, multiple factors (CO2 pressure/
temperature/ brine salinity/ surface roughness/ mineral heterogeneity) can impact wettability
simultaneously when investigated through experiments.
Only fewer studies have focused on phase connectivity (measured as the Euler characteristic) toward
phase trapping. Herring et al. (2015) and Herring et al. (2013) identified the importance of initial phase
topology toward the residual saturation of the nonwetting phase. However, the medium studied was only
water-wetting, and also the trends of residual phase connectivity were not discussed. In the works by
Khorsandi et al. (2017), Purswani et al. (2020), Purswani et al. (2019), an equation-of-state model for
relative permeability was discussed by including phase connectivity was developed. Experimental and
numerical data sets using pore-network modeling were used to find the functional form of relative
0
0.2
0.4
0.6
0.8
0 0.2 0.4 0.6 0.8 1
Res
idu
al s
atu
rati
on
(-)
Initial saturation (-)
Krevor et al. (2011) - Mt. SimonHerring et al. (2015) - Nca = 10^-8.6Herring et al. (2015) - Nca = 10^-5.6Pentland et al. (2010) - Exp #4Alyafei and Blunt (2015) - EstialladesSpiteri et al. (2008) - Theta = 60 degSpiteri et al. (2008) - Theta = 100 deg
No productionunit slope
99
permeability. Purswani et al. (2020) showed through the estimation of partial derivatives of relative
permeability that a response for relative permeability could be developed in the phase connectivity-phase
saturation space. They also discussed the locus of residual phase saturation and phase connectivity.
However, only two wettability cases, both under the water-wet regime were studied. Thus, the full spectrum
of wettability was not covered. Spiteri et al. (2008) covered the full spectrum of wettability through pore-
network modeling; however, the importance of phase connectivity was not discussed. Also, the quadratic
trapping model studied seems less adaptable to the more commonly known Land-based models.
In this chapter, we build on the works of Purswani et al. (2020) and Spiteri et al. (2008) to investigate
the importance of wettability on phase mobility and capillary trapping. We use pore-network modeling to
establish strict control on wettability. We change the contact angles during the water injection cycles to
cover the full range of wettability. Different initial phase saturations are investigated, and the trends of
residual phase saturation of the receding phase are studied for each wettability case. We also present
residual phase connectivity trends to explore the impact of wettability on the locus of residual phase
connectivity and phase saturation. We present appropriate functions for modeling the IR saturation trends
as well as the trends of the residual locus. Comparison with commonly used trapping models is also
provided.
We first provide the description of the numerical data sets generated using pore-network simulations
for studying capillary trapping. Next, we provide a description of the commonly known trapping models of
initial-residual (IR) saturation curves. We then introduce a new trapping model for the IR curves. We also
provide a model for the residual phase connectivity and thus model the locus of residual phase saturation
and phase connectivity for all wettability cases.
100
4.2. Methodology
4.2.1. Pore-network simulations
We used pore-network modeling (PNM) to generate numerical data sets of phase saturation, phase
connectivity, and relative permeability for different wettability conditions. The details of the pore-network
extraction model and two-phase flow simulation model used can be found in our previous work (Purswani
et al. 2020). We used an x-ray computed tomography scan of Bentheimer sandstone (Lin et al. 2019) for
network extraction using the approach of Dong and Blunt (2009). The porosity and permeability of the
pore-network were 22% and 2500mD, respectively, while the number of pores, number of throats, and
coordination were, 16850, 42637, and 5.061, respectively. The network simulations were carried out under
capillary dominated regime (NCa < 10-4), where a saturation step change is guided by capillary entry pressure
at pore throats which governs each network element occupancy by the invading and receding phases. Figure
26 shows a sequence of steps involved in pore-network modeling.
Scanning curves of primary phase1 injection and primary phase2 injection were conducted similar to
Spiteri et al. (2008). Here, we denote the two immiscible phases as phase1 and phase2, and avoid the terms
such as wetting, nonwetting, drainage, or imbibition since wettability of the medium is variable. We also
refer to phase1 as the receding phase because this phase is removed during secondary injection. All initial
and residual saturation or phase connectivity terms discussed in subsequent sections are in the context of
the receding phase (phase1). Similarly, all contact angles in this work are also reported through phase1 for
consistency. The contact angle during phase1 injection was kept at 180° (0° for the phase2) to mimic
completely water-wetting condition, similar to the injection of CO2 in brine filled reservoirs, or oil
migration that occurs in oil fields over geological times. For the phase2 injection, we change the receding
phase contact angle from 180° to 0° (advancing contact angle from 0° to 180°), (mimicking a range of
wettability alteration scenarios, from no alteration (strongly phase2-wet) to strong alteration (strongly
phase1-wet). Most hydrocarbon reservoirs are mixed-wet because of the prolonged contact of oil with the
101
rock which renders the rock surface to be oil-wet (Jadhunandan and Morrow 1995; Salathiel 1973). We
input the same physical properties (density of 1000 kg/m3 and viscosity of 1 cP) for both phases during
network simulations, although no change would occur in the simulations given the capillary pressure rule-
based execution of the pore-network model.
Figure 26. Steps in pore-network modeling. (a) The extracted internal structure (pore space) of the porous medium used in this study; (b) The extracted pore-network of the porous medium in (a) represented in a ball and stick form; (c), (d), and (e) show three saturation steps of blue phase injection in the pore-network completely filled with the red phase. (c) shows no injection, (d) is captured after some injection, and (e) is captured after longer injection period.
The advantage of using PNM is that strict control on wettability can be achieved. For receding phase
contact angles (θ1) less than 90°, the pore-network’s fluid occupancy is rearranged such that the nonwetting
phase (if occupying the corners), the arc menisci bulge outward for pore corners. In addition, the nonwetting
ba
c d e
102
phase also occupies the pore centers, while the wetting phase remains sandwiched as wetting layers
(Valvatne and Blunt 2004).
For each θ1, scanning curve pairs of phase1 and phase2 injection sequences were conducted with varying
initial phase1 saturation (~0.05 to 1.0 in intervals of 0.05). This was achieved by terminating the phase1
injection process at the required saturation step. The residual phase1 saturation was achieved for each initial
phase1 saturation by driving phase2 injection until a high magnitude of capillary pressure (magnitude of 105
Pa). Phase1 connectivity was estimated for each saturation step by calculation of Euler characteristic as
(Purswani et al. 2020),
,occ occ occ
p t htn n n = + − (4.1)
where ,occp
n ,occt
n and occht
n are the number of pores, throats, and half throats occupied at each saturation step.
The phase connectivity ( ˆ ) is defined as the normalized Euler characteristic which is calculated as
(Purswani et al. 2019),
,ˆ max
min max
−=
− (4.2)
where χmin = np – nt and χmax = np; np and nt refer to the number of pores and number of throats in the pore-
network, respectively. Therefore, in this study, χmin = -25787 and χmin = 16850. The base code to extract
phase connectivity data from PNM is provided in appendix G.
103
4.3. Results and discussion
4.3.1. Iso-quality curves for saturation and connectivity – interpreting PNM data
Trends of phase saturation, phase connectivity, and relative permeability for the nonwetting phase in water-
wet media have been discussed in Purswani et al. (2020). Here, the discussion is extended over the full
range of wettability. To investigate these trends more effectively, we analyze relative permeability iso-
quality curves in the saturation and connectivity space for the receding phase (phase1) during phase2
injection, similar to how isotherms and iso-volume curves are analyzed in a pressure-temperature-volume
(PVT) phase diagram. For example, the iso-connectivity curves convey the behavior of relative
permeabilities with a change in saturation at fixed phase connectivity. These are shown in Figure 27. To
generate these plots, we perform 2-D linear interpolations on the PNM data to arrive at relative permeability
values at equidistant points (intervals of 0.01 units) in the connectivity-saturation space.
104
Figure 27. Iso-quality trends for four wettability cases (θ1 = 180°, 120°, 60°, 0°). Plots to the left show the iso-saturation curves, while plots to the right show the iso-connectivity curves. The phase saturation, phase connectivity, and relative permeability are for the receding phase (phase1) during phase2 injection. The contour lines are plotted at intervals of 0.03 units. The procedure for developing these curves is displayed in Figure 51 in appendix E.
θ1
= 1
80
o
θ1
= 1
20
oθ
1=
60
oθ
1=
0o
Iso-saturation Iso-connectivity
105
For θ1 = 180° and 120°, mostly flat iso-saturation contours are observed at high saturations (>0.5) (left
column of Figure 27), showing that relative permeability is less dependent on phase connectivity at high
saturation. On the contrary, for these high receding phase contact angle cases, from the iso-connectivity
plots (right plots), the importance of phase connectivity on relative permeability can be observed at
intermediate and low saturation (0.5 and below). Similar trends are discussed in chapter 5 through partial
derivative calculations. For the high receding phase contact angle cases, snap-off of the receding
(nonwetting) phase governs trapping, and at low and intermediate saturation, connectivity becomes
important because a critical snap-off event can restrict flow (see iso-saturation plots for θ1 = 180° and 120°
in Figure 27).
As contact angle decreases, θ1 = 60° and 0°, the receding phase becomes the wetting phase. Now, iso-
saturation curves show that at high saturations (>0.7), the contours are no longer flat. Instead, relative
permeability decreases as connectivity increases for high and intermediate saturations. Saturation, however,
continues to be important toward relative permeability as seen for the high receding phase contact angle
cases. With a small change in saturation, kr changes significantly for fixed connectivity. This is especially
true at high and intermediate saturations.
The unexpected trend of increasing connectivity but decreasing relative permeability is more clearly
visible from Figure 28 (also see Figure 33) where two extreme receding contact angle cases (θ1 = 180° and
θ1 = 0°) are compared. Here, the kr-S and ˆ S − paths are compared for Si ~ 1.0 and Si ~ 0.9. From Figure
28, it is seen that the kr-S path for Si ~ 1.0 is below that of Si ~ 0.9 for high saturations (~0.5 – 0.9) when
the receding phase is wetting (θ1 = 0°). At these high saturations, however, the ˆ S − paths for the same
contact angle, is higher for the Si ~ 1.0 path. This suggests that as one moves vertically (at fixed S) across
the two paths from Si ~ 1.0 to Si ~ 0.9, one will encounter that the phase connectivity decreases, but the
relative permeability increases. This is not observed for θ1 = 180°. The anomaly observed for θ1 = 0° arises
because kr decreases more sharply along the Si ~ 1.0 path because of more efficient displacement of the
106
initially well-connected phase, while the decrease in connectivity is gradual. This demonstrates the
independent nature of the dependence of relative permeability on phase connectivity and relative
permeability. Finally, along the same path (Si ~ 0.9 or Si ~ 1.0), however, the trends between kr-S and ˆ S −
change monotonically as would be expected.
Figure 28. Comparison of kr-S (upper row) and – S (lower row) paths for two Si values (~ 1.0 and ~
0.9) for two receding contact angles cases (θ1 = 180° and 0°).
4.3.2. Initial-residual (IR) saturation trapping curves
We establish phase trapping when the magnitude of capillary pressure (Pc) reaches a very high value
(magnitude of 105 Pa). Other residual saturations for different kr stopping criteria are discussed in appendix
F. For cases where the receding phase (phase1) is nonwetting, Pc remains positive (showing spontaneous
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
k r
S
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
S
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
S
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
S
θ1 = 0o θ1 = 180o
Si ~ 1.0
Si ~ 0.9
107
imbibition of phase2), whereas when phase1 is wetting, Pc becomes negative and forced injection of phase2
is required.
Figure 29 shows the IR saturation curves for the different wettability cases. The trends are similar to
those seen previously in Spiteri et al. (2008), in that the typical nature of IR curves holds until very high
saturations (~0.8). With increasing initial saturation, the residual saturation increases because more phase
trapping becomes possible with more initial phase. Trapping occurs because of snap-off, which is more
dominant for the high receding phase contact angle cases. For decreasing contact angles (wetting phase1
conditions), we see a decrease in the residual saturation, due to a decrease in snap-off events. For these
contact angles, flow of the receding phase occurs through piston-like displacement by the advancing phase,
as well as through layer flow. Thus, the residual saturation values are driven to low values. This occurs
more strongly at very high initial saturation, which shows that if the medium is initially completely filled
with a particular phase, its initial connectivity may help in driving the residual saturations to very low
values. This is similar to the trends observed experimentally by Herring et al. (2015). For complete initial
phase1 saturation (Si ~ 1.0) and medium completely wetting to phase1 (receding phase), the residual phase1
saturation goes to ~ zero which occurs due to layer flow of phase1. Although such flow scenario may not
be possible through typical core flooding experiment.
Figure 29. Initial-residual saturation curves for different wettability cases observed using pore-network modeling.
108
4.3.3. Modeling IR saturation trapping curves
Different trapping models have been proposed in the literature. A list of some of the most commonly used
models is presented in Table 9. We implement these models on the PNM data sets and compare these
against a new proposed model.
Table 9. IR trapping models from the literature. iS , rS , and max
rS refer to the initial, residual, and
maximum residual phase saturations, respectively, while, C, a, b, c, α, and β are model parameters. Reference IR trapping model Remarks
Land (1968)
1
ir
i
SS
CS=
+ (4.3)
First trapping model developed for
mitigating relative permeability hysteresis.
Developed from observations in water-wet
media. C is known as the Land’s trapping
coefficient.
Ma and Youngren
(1994) 1
ir b
i
SS
aS=
+ (4.4)
This model is a modification to the Land’s
model when b → 1.0.
Jerauld (1997) ( )( )1 / 111 1
max max
r r
ir
bS S
imax
r
SS
SS
+ −
=
+ −
(4.5) This model was derived from the Land’s
model. The trapping coefficient is replaced
by a calculation from the maximum residual
saturation.
Spiteri et al. (2008) 2
r i iS S S = − (4.6)
This quadratic model was presented based
on PNM observations where IR curves were
found to show a maximum.
This work 1
ir c
i
aSS
bS=
+ (4.7)
Our model collapses to the Ma and
Youngren model when a → 1.0; and to the
Land’s model when both a and c → 1.0.
Land’s model and other Land-based models are developed on water-wet media with data sets that are
limited in very high initial phase saturation conditions and thus may not appropriately capture the IR
characteristic curves completely. The quadratic model by Spiteri et al. (2008), however, was a substantial
improvement to the currently used Land-based models. However, this model appears restrictive because it
assumes a quadratic nature for all IR curves, which may not be true for water-wet systems. Thus, the Spiteri
et al’s model fails to collapse into the known Land-based models under limiting conditions. Therefore, we
propose a new IR trapping model with a slight adjustment to the Ma and Youngren model (Table 9).
109
Figure 30 shows the IR saturation trapping curves for four different contact angle cases as well as the
corresponding matches shown by dashed lines for the models presented in Table 9. The goodness measure
of these models for all wettability cases is shown in the right plots in Figure 30, and the matching parameter
information for the model presented in this study (Eq. (4.7)) is provided in Figure 31.
110
Figure 30. IR saturation trapping curves for four different contact angle cases (θ1 = 180°, 120°, 60°, 0°). The dashed lines show the fits for the trapping model. The corresponding goodness measure of the fits for all wettability cases and R2 values are displayed on the right plots.
Lan
dM
a an
d Y
ou
ngr
enJe
rau
ldSp
iter
i et
al.
This
wo
rk
θ1 = 180o
θ1 = 120o
θ1 = 60o
θ1 = 0o
111
Figure 31. Matching parameters plotted as a function of wettability for the IR trapping model presented in this research (Eq. (4.7)). The corresponding fits are available in Figure 30. The shaded regions mark the 95% confidence interval.
From Figure 30 we see that Land and other Land-based models (Ma and Youngren, and Jerauld) match
well for high receding phase contact angles (θ1 = 180° and 120°). However, the performance by these
models suffers for lower receding phase contact angles, showing their limitation for water-wet media.
Spiteri et al’s model, however, performs better (R2 = 0.91) than Land-based models. Still, this model does
not match well for cases when receding phase is wetting (θ1 = 60° and 0°). This may likely be because the
curves do not seem typically quadratic. With the model presented in this work (Eq. (4.7) in Table 9),
substantial improvement (R2 ~ 0.99) is found in capturing the trapping trends for all contact angles. We find
that the additional parameter in the numerator, a, is a strong function of wettability and lies between 0 and
1 (Figure 31). It helps achieve control on the nonlinearity of the IR curves. Similar to water-wet conditions,
for high receding phase contact angles, the value of a is closer to one and decreases as the receding phase
contact angles decrease. The other two parameters also change with wettability, but the change is relatively
less.
112
The abrupt change in the parameters near contact angle of 80° occurs likely as the flow regimes shift
due to shift in wettability. For θ1 = 70° and for Si = 1.0, there is sharp decrease in the Sr value (see the green
curve in Figure 29) which is likely causing this jump in the fitting parameters because of the nature of this
Si-Sr curve. The reason for this is that both wettability and initial saturation govern the nature of the Sr curve.
At θ1 = 70°, there is now a shift in the wettability and that wettability effect is more pronounced when Si =
1, that is when the medium is completely filled with the receding phase.
113
4.3.4. Phase connectivity – phase saturation ( χ – S) paths and trapping locus
The – S paths for the receding phase (phase1) for four different contact angle cases are displayed in Figure
32. For high receding phase contact angles (θ1 = 180° and 120°), the ˆ S − paths are linear due to gradual
snap-off of the receding phase. The locus bounded by the residual phase1 saturation and phase1 connectivity
( ˆr – Sr) shown by black open circles, remains fairly constant (~ 0.395) for θ1 = 180°. Here, we find
nonlinearity in the locus because of lower Sr values for very high initial phase1 saturation cases owing to
better displacement which is more evident for θ1 = 180°.
Figure 32. – S paths for the receding phase during secondary injection process for different contact
angle cases. The red open circles represent the initial condition whereas, the black open circles represent the residual condition for each – S path.
θ1 = 180o
θ1 = 60o
θ1 = 120o
θ1 = 0o
114
For lower contact angles (θ1 = 60° and 0°), we continue to find linear ˆ S − trends for higher saturations,
but for lower saturations, goes down to ~ zero (the most disconnected state) at saturation of ~0.15. This
corresponds to the total number of network elements occupied by the receding phase to be roughly equal
to the total number of pores in the network. Further reduction in saturation, causes an increase in , which
occurs due the pore structure constraints and the definition of (Eq. (4.2)). As smaller number of network
elements get occupied, χ < χmax thus, increases and ultimately reaches the limiting condition of ~ 0.395
[–χmax/(χmin– χmax)] as the Euler number reaches a very small number. Thus, the saturation of ~0.15 which
marks the most disconnected state for the receding phase could be referred to as the percolation threshold
for the wetting phase.
Figure 33 shows the relative permeability contours for the ˆ S − paths shown in Figure 32. The
monotonic behavior of kr versus both and S is visible when the phase is nonwetting to the medium (θ1 =
180° and θ1 = 120°). Relative permeability continues to be monotonic with S for cases where the receding
phase is wetting to the solid. Here, the dependence of kr on S is even stronger with sharper change in kr as
S changes slightly (see high saturation regions for θ1 = 60° and θ1 = 0°). Finally, the inverse trends between
kr and for fixed S, discussed previously in Figure 27 and Figure 28, can also be seen from the kr contours
for θ1 = 60° and θ1 = 0° where the receding phase is wetting to the solid.
115
Figure 33. Relative permeability contour plots for the – S paths shown in Figure 32
In Figure 32, this is the first time that ˆ S − trends for the wetting phase are discussed using network
modeling. In general, the data for wetting phase connectivity remains limited in the literature due to the
difficulty in resolving the wetting phase and accurately calculating the Euler number through image data.
Therefore, more high-resolution experimental data is needed to corroborate these trends.
The ˆ S − path for Si = 1 for the θ1 = 120° crosses the residual locus. However, the relative
permeabilities on this ˆ S − path (near the residual locus) are close to zero (for example, kr = 0.004 at ,S
= 0.360, 0.300 (point near the residual locus for θ1 = 120° in Figure 32)). This shows that kr =f( ,S) may
lose uniqueness near the extreme regions of kr. Thus, consideration of additional parameters such as the
fluid/fluid interfacial may be needed to get a more exact solution for the kr. Nevertheless, the error with the
θ1 = 180o
θ1 = 60o
θ1 = 120o
θ1 = 0o
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consideration of kr =f( ,S) is not very significant. Figure 34 shows a compilation of the trapping loci for
the different contact angles.
Figure 34. Trapping locus of phase saturation and phase connectivity for different contact angles.
Few observations can be made from Figure 34. There is a finite value of the minimum trapped phase
connectivity, despite ˆ S − paths showing a ~ zero value of . This is because increases again due to
layer flow (see Figure 32). Next, the shape of the locus opens as contact angle decreases (180° to 80°) and
shrinks back as contact angle decreases further (70° to 0°), showing a strong influence of wettability. This
occurs because as receding phase contact angle decreases, the residual saturation decreases, leaving a more
disconnected state. But, as contact angle decreases further where the receding phase becomes the wetting
phase, the trapped saturation becomes very low, increasing again. The pore structure limiting condition
influences the locus strongly. The shrinking nature of the locus depends on the reducing residual saturation
which occurs at low contact angles because of efficient displacement of the receding phase.
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4.3.5. Modeling residual phase connectivity and trapping locus
We model residual phase connectivity ( ˆ )r as a function of the initial phase connectivity similar to how IR
saturation is modeled. We propose a simple quadratic function as,
2 2
(ˆ ˆ ˆ ˆ ˆ ˆ( ,) )r i o i o o
A B = + +− − (4.8)
where A and B are model parameters; ˆo
is the limiting phase connectivity at negligible phase saturation
[–χmax/(χmin– χmax)]; ˆi
and ˆr
are the initial and residual phase connectivity, respectively. The combination
of modeling the residual phase saturation and residual phase connectivity allows us to model the residual
locus. Figure 35 shows the IR phase connectivity trapping curve for four different contact angle cases as
well as the corresponding match shown by dashed lines for the model for residual phase connectivity. The
goodness measure of the model for all wettability cases is shown in the right plot in Figure 35, and the
matching parameter information for the model presented is provided in Figure 36.
Figure 35. IR phase connectivity curves for four different wettability conditions (θ1 = 180°, 120°, 60°, 0°). The dashed lines show the match for the phase connectivity trapping model (Eq. (4.8)). The corresponding goodness measure of the fit for all wettability cases and R2 value is displayed on the right plot.
θ1 = 180o
θ1 = 120o
θ1 = 60o
θ1 = 0o
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Figure 36. Matching parameters plotted as a function of wettability for the IR phase connectivity trapping model. The corresponding fits are available in Figure 35. The shaded regions mark the 95% confidence interval.
Few observations can be made from Figure 35. As discussed previously, the residual phase connectivity
remains fairly constant for high receding phase contact angles. However, as contact angle decreases, the
trends become parabolic in nature. This occurs because of a couple of reasons. First, the topological
constraint of the pore structure. For example, ˆi
cannot go lower than ~0.395 which constraints the curve
on one end. This is because during the primary (phase1) injection, the receding contact angle is set at 180°.
Thus, phase1 is nonwetting to the surface and layer flow does occur to reduce ˆi
. Second, ˆr
for lower
contact angles reduces below 0.395 due to efficient displacement of the receding phase. Finally, ˆr
becomes once again constrained at very high ˆi
because here the residual saturation almost goes to zero
(see Figure 29 and Figure 30). The proposed function (Eq. (4.8)) captures the trends for residual phase
connectivity across all wettability cases (R2 ~ 0.88).
On the basis of Eq. (4.7) and (4.8), the residual phase saturation and residual phase connectivity trends
can be captured. A summary of the initial-residual phase saturation and phase connectivity trends for
different four different contact angles are presented in Figure 37.
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Figure 37. Summary of initial-residual phase saturation and phase connectivity trends for four different contact angles (θ1 = 180°, 120°, 60°, 0°). (a) Initial connectivity versus initial saturation. Here all four contact angles collapse to one single data set; (b) initial connectivity versus residual connectivity; (c) initial saturation versus residual saturation; (d) residual connectivity versus residual saturation.
In Figure 37a, initial phase connectivity is shown as a function of initial phase saturation. All four
contact angles collapse to a single data set (shown by blue open circles). This is because the initial condition
prior to secondary phase2 injection, represent the primary phase1 injection at which point all flow conditions
including contact angle of 180° degree are the same. The dashed line in Figure 37a is a match to the
following power law function,
( ) ,ˆ 'ˆ 'k kp S S − = − (4.9)
θ1 = 180o
θ1 = 120o
θ1 = 60o
θ1 = 0o
a) b)
c) d)
120
where ˆ ' and 'S are points on the ˆ S − path; and p and k are matching parameters. For the match in
Figure 37a, we constrain the model on the ˆ S − path at the first and the last points, namely, (0.395,0) and
(1,1). Figure 37b and Figure 37c were discussed in Figure 30 and Figure 35, respectively. Here, the axes
on Figure 37b are flipped for seamless representation of initial-residual curves. Combination of Figure 37b
and Figure 37c helps develop the residual locus (Figure 37d).
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4.8. Concluding remarks
Pore-network simulations show that very high initial phase saturations can lead to reduced trapping and
that an optimum trapping saturation exist for initial saturations in the range of 0.7-0.95 which depends on
the wettability. This is an important finding which suggests that the primary CO2 injection should be
designed such that this range of initial CO2 saturation is targeted to maximize trapping. The influence of
initial phase saturation becomes more significant for cases where the receding phase is strongly wetting.
Thus, the combination of the higher initially connected CO2 phase if is also wetting to the surface, can
reduce the trapping capacity significantly. The most suitable trapping scenario (trapped saturation ~50%)
is found for intermediate wettability conditions and an initial saturation of between 0.6-0.8.
Initial-residual (IR) saturation trends may be inaccurately captured by the application of models that
are built for water-wet media. If the receding phase is wetting to the surface, flow caused by layer flow and
piston-like advance of the invading phase are important physical displacements regimes that can
significantly affect trapping. This has also been seen in experimental studies. For example, Iglauer et al.
(2011a) discuss the presence of sandwiched oil layers which drive low residual oil saturations. They found
Sor in oil-wet medium to be 0.18 as opposed to 0.35 for water-wet medium, with fewer larger-sized oil
clusters. Larger-sized oil clusters were found for the water-wet case by Landry et al. (2011). The general
trapping model presented in this chapter provides a simple and elegant solution to accurately represent
trapping trends across all wettability conditions of the system. In addition, the model easily collapses to the
traditionally known models.
A few researchers have observed a more complex increasing-decreasing-increasing IR trend at mixed-
wetting conditions (Salathiel 1973; Tanino and Blunt 2013). The quadratic concave-up profile as described
in Tanino and Blunt (2013) was not observed through PNM simulations. The difference likely occurs
because as initial saturation increases, more oil contacts the rock surface and low residual oil saturations
are observed due to layer flow. However, at higher initial oil saturations, layer flow continues to exist in
PNM, but experiments show a slight decrease due to oil being forced into the corners of the larger pores or
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into micropores that can’t be displaced as effectively and Sor increases further in those experiments (Tanino
and Blunt 2013). However, these trends have not been observed for a CO2/brine/rock system that most flow
experiments are unable to reach such large initial oil saturation values (>0.9). Thus, more experiments
should be designed to corroborate this trend and future network models should be built to incorporate these
trends.
Recent literature has suggested the importance of phase connectivity and by including this term helps
resolve hysteresis in relative permeability (Khorsandi et al. 2017; Purswani et al. 2020; Purswani et al.
2019; Zhao et al. 2019) and capillary pressure (Li and Johns 2018). However, the focus has remained on
the nonwetting phase. In this chapter, we presented phase connectivity trends for receding phase under
wetting and nonwetting conditions. The residual locus provides the true limiting condition of residual phase
connectivity and residual phase saturation for the given pore structure and wettability. This can provide a
known limiting condition to calibrate multiphase models used in reservoir simulations which will be critical
in scaling carbon capture projects. Design of controlled experiments is required to capture wetting phase
connectivity trends as well as the trends of the locus of residual phase connectivity and saturation. In
addition, the limiting conditions of the pore structure topology should be experimentally confirmed.
One limitation of our work is that we have only considered capillary dominated flow which is reflective
of flow away from the wellbore, but it is well established that flow regimes like ganglion dynamics and
drop traffic flow can significantly affect phase connectivity and capillary trapping which occur at high flow
rate conditions (or high capillary numbers), therefore wettability effects in combination with high capillary
numbers will be important when investigating phase trapping (Armstrong et al. 2016; Schlüter et al. 2016).
Overall, the impact of wettability using capillary dominated pore-network simulations is studied.
Through controlled numerical experiments, trends of residual saturation and phase connectivity are
analyzed for the full range of contact angles and initial saturations. As contact angles change, the flow
regimes change, this causes the phases to distribute differently and affect trapping. The trapping models
discussed here are general and can be used across different wetting conditions. Future experiments for
limiting flow conditions of high initial saturations as well as different wettability are recommended.
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CHAPTER 5. DEVELOPMENT OF EQUATION-OF-
STATE USING PORE-NETWORK MODELING
Preface
The contents of this chapter were originally presented at the SPE IOR Conference, 31st Aug-4th Sept.,
Virtual, 2020. The manuscript was accepted for publication in the SPE Journal and is referenced as,
Purswani P., Johns R.T., Karpyn Z.T., Blunt M.J. (2020)
Predictive Modeling of Relative Permeability using a Generalized Equation-of-State, SPEJ, (26), 191-
205, https://doi.org/10.2118/200410-PA
Author contributions: Johns R.T., Karpyn Z.T., and Purswani P. conceptualized the approach. Purswani P.
processed the numerical data set used in the model, developed the modeling efforts, and wrote the original
draft in consultation with Johns R.T., Karpyn Z.T., and Blunt M.J. All coauthors contributed toward
analyzing the data and updating the manuscript.
Abstract
Reliable simulation of enhanced oil recovery processes depends on an accurate description of fluid transport
in the subsurface. Current empirical transport models of rock-fluid interactions are fit to limited
experimental data for specific rock types, fluids, and endpoint values. In this chapter, a general equation-
of-state (EOS) approach is developed for relative permeability (kr) based on a set of geometric state
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parameters: normalized Euler characteristic (connectivity) and saturation. Literature data and pore-network
modeling (PNM) simulations are used to examine the functional form of the EOS.
Our results show that the new kr-EOS matches experimental data better than the conventional Corey
form, especially for highly nonlinear relative permeabilities at low saturations. Using hundreds of PNM
simulations, relative permeability scanning curves show a locus of residual saturation and connectivity
which defines an important limit for the physical kr region. The change of this locus is also considered for
two contact angles. PNM data further allows for the estimation of the relative permeability partial
derivatives which are used as inputs in the EOS. Linear functions of these partials in the connectivity-
saturation space renders a quadratic response of kr, which shows excellent predictions. Unlike current
empirical models that are based on only one residual saturation, the state function approach allows for
dynamic residual conditions critical for capturing hysteresis in relative permeability.
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5.1. Introduction
The flow of two or more phases in a porous medium is quantified through averaged transport properties
such as relative permeability. Relative permeability is critical for accurate multiphase flow simulations of
oil and gas reservoirs, which are routinely used to estimate petroleum reserves, evaluate the success of
enhanced oil recovery (EOR) projects, and estimate the amount of CO2 that can be trapped for sequestration
purposes. Accurate description and practical modeling solutions for relative permeabilities continue to be
a long-standing concern faced by petroleum engineers and hydrologists.
Traditionally, the relative permeability of a phase is expressed as empirical functions of phase
saturations using simple functional forms (Brooks and Corey 1964; Corey 1954). However, a growing body
of literature demonstrates that there are a variety of other parameters that affect relative permeabilities
(Blunt 2017). These include the wettability of the medium, due to which the residual saturation is lower for
a mixed-wet medium than for a strongly water-wet medium (Jadhunandan and Morrow 1995). Flow
conditions such as the flow rate or the interfacial tension between the injection and displacing fluid
(captured in the capillary number) also strongly affect relative permeabilities (Armstrong et al. 2016;
Avraam and Payatakes 1995; Purswani et al. 2019). In addition, topological descriptions of the porous
medium, such as the pore/throat coordination number and aspect ratio also have an effect (Jerauld and Salter
1990).
Various flow conditions and porous medium attributes manifest themselves in different pore-scale
events that govern the movement of fluid through the pore space, thus affecting relative permeability. Pore-
scale flow experiments performed using x-ray micro-computed tomography (micro-CT) assist in the
visualization and quantification of fluid transport at spatial resolutions close to a micron. In addition to
tracking phase volumes, fluid/fluid interfacial area (Culligan et al. 2005; Culligan et al. 2004), distribution
of contact angles (AlRatrout et al. 2017; Klise et al. 2016; Scanziani et al. 2017), the topology of the
nonwetting phase (Herring et al. 2015; Herring et al. 2013; Schlüter et al. 2016) can also be estimated
accurately through such experiments. Fluid movement can in some cases be captured in real-time at
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temporal resolutions close to tens of seconds with the application of dynamic x-ray imaging facilities
(Rücker et al. 2015; Singh et al. 2017). Further, through the coupling of x-ray imaging experiments with
fast direct numerical simulations, multiphase transport can be captured at different flow conditions, such as
those at very high capillary numbers (Armstrong et al. 2016; Mcclure et al. 2018; Mcclure et al. 2016).
Pore-network modeling (PNM) is advantageous because repeat numerical experiments can be made for
different porous media topology and at different wettability while holding some parameters constant (Blunt
2017; Valvatne and Blunt 2004).
Pore-scale studies are valuable to gain a deeper appreciation of the complexity of multiphase transport
through improved visualization of displacement processes. These include piston-like displacement,
cooperative pore-filling, snap-off (choke off), Haines jumps, coalescence, drop-traffic flow, and ganglion
dynamics. These mechanisms observed in controlled micromodel experiments (Avraam and Payatakes
1999; Avraam and Payatakes 1995; Lenormand and Zarcone 1984) have been corroborated through three-
dimensional experiments and simulations (Armstrong et al. 2016; Rücker et al. 2015). These pore-scale
mechanisms often lead to phase trapping when the flow direction is reversed. In addition, surface roughness
effects (Morrow 1975) and wettability alteration (Salathiel 1973) lead to hysteresis in contact angles such
that the advancing contact angle is greater than the receding contact angle (Anderson 1986b; Blunt 2017;
Braun and Holland 1995). Phase trapping and contact angle hysteresis cause path dependence of relative
permeabilities in the saturation space, which is termed hysteresis.
Hysteresis in relative permeability continues to be a problem for accurate reservoir modeling. Large
errors manifest in the estimation of recoveries if hysteretic effects are ignored or improperly accounted for
during reservoir simulations (Carlson 1981). Estimation of relative permeability hysteresis and trapping
efficiency of the nonwetting phase is also critical for CO2 sequestration (Akbarabadi and Piri 2015; Juanes
et al. 2006). In addition, hysteretic effects become pronounced for EOR processes that involve a reversal
of flow, such as the water alternating gas (WAG) process, where gas is trapped during the water injection
cycle (Khorsandi et al. 2018; Spiteri and Juanes 2006).
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Current modeling techniques for relative permeability (Brooks and Corey 1964) are deficient as these
consider relative permeability to be empirical function of saturation and different physical effects such as
wettability and capillary number are all captured in a single parameter such as the Corey exponent. This
lack of physics in our models likely explains, in part, why relative permeability is often altered to history
match production in oil reservoirs.
Most common hysteresis models take a similar form to that of Corey. Land pioneered the estimation of
imbibition relative permeabilities by accounting for the trapped nonwetting phase. A simplified relationship
was found between the initial (or maximum) and residual nonwetting saturation during imbibition (Land
1968). A pore-size distribution parameter was required for the estimation of imbibition nonwetting phase
relative permeabilities. This was later corroborated through experiments (Land 1971).
Hysteresis models used in conventional simulators are generally extensions of Land’s model.
Killough’s model interpolates between the initial and residual nonwetting saturations for the estimation of
imbibition relative permeabilities. An estimate of the Land’s trapping coefficient is necessary to determine
the residual nonwetting saturation (Killough 1976). Carlson recognized that unlike Land’s and Killough’s
model, the need for a secondary parameter could be avoided given that one experimental data point on the
imbibition relative permeability curve is known. He suggested that the residual nonwetting phase can be
determined using Land’s formulations and the known experimental data point. More known data points on
the relative permeability curve would lead to a better average of the residual nonwetting saturation (Carlson
1981).
One limitation of these models is that they are strictly empirical and are often based on the observations
found for water-wet systems, which means that these models work under the assumption of no trapping
during secondary drainage cycles. This leads to limited prediction capability by these models, especially
for mixed-wet systems (Fatemi et al. 2012; Spiteri et al. 2008). Furthermore, for the implementation of
these models, information of the initial-residual nonwetting saturations together with the prior drainage
relative permeability is required. Spiteri et al. (2008) observed nonlinear trends in the trapping curves for
the oil phase for different wettabilities of the medium. They proposed a quadratic hysteresis model between
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the trapped and initial oil saturation and demonstrated the model’s capability in capturing hysteresis for
different wetting conditions.
Yuan and Pope (2012) proposed a Gibbs energy-based model for relative permeability with applications
to two-phase compositional processes. Relative permeabilities were made proxies of compositions, which
were determined by interpolating from a reference state of known Gibbs energy and relative permeability.
Parameters in a simplistic Corey-type model were interpolated to arrive at relative permeabilities for each
change in composition. Trapping and hysteresis in relative permeabilities were later accounted for in a
similar fashion by Neshat and Pope (2018), where parameters in an extended Land model were interpolated
using the Gibbs energy approach. These models eliminated some of the discontinuities observed in
compositional simulation owing to phase identification and labeling problems. Still, it is not clear that
Gibbs energies of each phase are always distinct and that this approach is predictive. Further, these models
depend on lots of tuning parameters and lack physical insight.
Skauge et al. (2019) presented two models to bridge immiscible and miscible WAG EOR processes. In
the first model, empirical three phase relative permeability functions were considered, and phase trapping
was incorporated using the Land formulation. These kr functions were described as functions of all three
phase saturations, the path of the saturations, and the trapped saturation of all phases. In addition, for the
consideration of the compositional effects, another empirical formulation was considered to incorporate
interfacial tension (IFT) changes. This was established using the Coats (1980) functional forms which
include the IFT effect by scaling the endpoint saturations from the original IFT (σo) as follows,
( )*gr grS f S= (5.1)
( )*org orgS f S= (5.2)
129
where ( )1
1n
of
=
; and Sgr and Sorg are the residual gas and oil saturations, respectively. Equation
(5.1) and (5.2) allow for *grS and *
orgS to be zero as interfacial tension approach zero near supercritical
conditions. The value of n1 usually remains between 4 and 10.
For the second model described by Skauge et al. (2019), wettability, pore-size distribution, and pore-
scale physics was incorporated into the compositional modeling approach. Pore-scale physics was
incorporated using pore occupancy of the phases. For this, the wetting order was based on pore sizes, for
example, the smallest pores would be occupied by the wetting phase, the largest pores would be occupied
by the nonwetting phase, and the intermediate-sized pores would occupy the phase of intermediate
wettability. They extended this idea further to consider the distribution of contact angles in the pore space
and allow for pores of different radii to have different contact angles. This was accomplished through the
Bartell-Osterhof relationship, which is given as,
.gw gw ow ow go gocos cos cos = + (5.3)
This relationship is established by considering all three Young’s relationships for the three two-phase sets
(oil/water; gas/water; and oil/gas) of interfacial tension equations. Elimination of the fluid/solid IFTs gives
Eq. (5.3).
Equation (5.3) was used to identify the pore occupancy for each phase as well as to model the transition
toward miscibility. For example, for a weakly oil-wet porous medium, water will be intermediate wetting
for an immiscible flood, but will be nonwetting (and gas will be intermediate wetting) for a near-miscible
flood due to the changes in the interfacial tension at near-miscible conditions. As a consequence, in this
example, the path followed in the three-phase displacement process will be different. For the miscible flood,
the path will be gas to oil to water, whereas for the near-miscible flood, the path will be from gas to both
oil and water.
130
The drawback of these compositional models is that they still require to track of phase labels
(oil/water/gas), which can create problems with compositional simulations because at supercritical
conditions discontinuities can occur that can cause simulations to fail or underpredict recoveries.
One proposed solution to relative permeability hysteresis is to develop a multi-parameter, physics-
based description for relative permeability, similar to capillary pressure, where hysteresis is modeled with
the inclusion of fluid/fluid interfacial area (Hassanizadeh and Gray 1993; Reeves and Celia 1996; Joekar-
Niasar 2008; 2010). However, errors are significant near high and low saturations using that approach,
because of sudden changes in the fluid/fluid interfacial areas at extreme saturations. Schlüter et al. (2016)
recognized that connectivity as quantified by the Euler characteristic is a key parameter for describing
multiphase flow. This was also found in the study by Mcclure et al. (2018), where the authors suggested
that four state variables, namely, fluid saturation (S), specific fluid/fluid interfacial area (αw-nw), fluid/fluid
average interfacial curvature (κ), and normalized fluid phase Euler characteristic ( ˆ ) describe the state of
a fluid phase in a porous medium. Based on this, κ was described as a function of the other three state
parameters, κ = f (S, αw-nw, ). Other constitutive correlations for κ were also considered, κ = f (S) and κ = f
(S, αw-nw) for comparison. The measures of these constitutive correlations were fit against two-fluid flow
simulation data in different porous media. The authors found that κ = f (S, αw-nw, ) gave the most accurate
description of the capillary pressure as opposed to the other constitutive relations considered since this was
the most complete description of the fluid phase. The most dramatic improvement was found upon the
addition of the phase connectivity term (Euler characteristic).
Johns and coworkers suggested a new conceptual framework for modeling relative permeabilities
(Khorsandi et al. 2017, 2018; Li and Johns 2018; Purswani et al. 2019). Khorsandi et al. (2017) proposed
relative permeabilities as a state function based on capturing the essential state parameters. This framework
makes relative permeability single-valued for a combination of state parameters that are fully described at
that state. This methodology can eliminate phase labeling (wetting/nonwetting, oil, water, gas) issues as
demonstrated by Khorsandi et al. (2018) where such labels cause discontinuities for supercritical fluids.
131
The EOS model was also extended to three phases and for WAG injection. Li and Johns (2018)
implemented a similar framework for proposing a coupled EOS model for capillary pressure and relative
permeability. Partial changes in capillary pressure due to saturation, connectivity, and wettability were
estimated by fitting a quadratic response of capillary pressure to literature data. Purswani et al. (2019)
similarly showed that the relative permeability EOS could be approximated as a simple response surface in
the saturation and connectivity space. Appropriate boundary conditions were set and the EOS was tested
for different capillary number conditions. Zhao et al. (2019) implemented this response surface to develop
a machine learning framework for modeling relative permeabilities.
Despite recent advances, the exact form of the EOS remains unknown and relies on significant
assumptions regarding the approximation of the partial derivatives. Therefore, in this research, we advance
the development of the state function approach for modeling relative permeability by evaluating the partial
change in relative permeability due to the changes in the state parameters considered. First, we test a
physical relationship for the state parameters involved, namely, connectivity (Euler characteristic) as a
function of saturation. This relationship is then incorporated into the exact differential form of the proposed
EOS to match measured data. We also present comparisons to the simplified Corey form of relative
permeability. Micro-computed tomography and synthetic data from PNM are used to examine these
relationships. Next, residual saturation and connectivity trends are analyzed, and values of the partial
derivatives for relative permeability are estimated using PNM. Finally, predictions of relative permeability
are made using the estimated partial derivatives.
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5.2. Methodology
In this section, we present a general development of the EOS for relative permeability. A similar
methodology could be applied to capillary pressure and other fluid-rock interaction functions. The exact
differential for relative permeability of a phase j is given as,
1
,j
j
N
i
i
r
i
r
N i
d dyy
kk
=
=
(5.4)
where yi is any state parameter affecting relative permeability, N is the number of state parameters. For NP
phases, there will be NP equations like Eq. (5.4). Although we will refer to “oil” and “water” in this chapter,
such identification is not necessary. Only the values of the state parameters, such as saturation, must be
known to estimate relative permeability.
The task described by Eq. (5.4) is then to define N state parameters that best, or nearly fully, describe
the state of the system. The degrees of freedom are large in porous media and, therefore, the goal is to find
the least number of state parameters that can capture the relative permeability function accurately. Small
errors in relative permeability estimates are acceptable if the most essential aspects of the physics are
captured with only a few input parameters. Thus, such an approach lends itself as an elegant and practical
solution for modeling relative permeabilities. Once these state parameters are known, Eq. (5.4) ensures that
relative permeability is a continuous and unique function of these state parameters.
There are two types of state parameters that can govern relative permeability. First, internal parameters
that describe the geometric state of fluid inside the porous medium, which include saturation, connectivity,
and fluid/fluid interfacial area (Mcclure et al. 2018). Second, external parameters corresponding to the flow,
which include the topology and wettability of the medium and capillary number.
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The exact differential of relative permeability of a phase j (krj) for these six parameters is then given
from Eq. (5.4) by,
, , , ,, , , , , , , ,
, , ,
ˆ
, , , , , , , , ,
ˆ
ˆ ˆ ˆ
ˆˆ
j j j
j
j j Caj Ca j Ca
j j j
j j j j Ca j j Ca
r r r
r j j
j j S I lnNA I lnN S A I lnN
r r r
Ca S I A S lnN A S lnN I
Ca
k k kdk dS d dA
S A
k k kdln dI
lnN IN
= + +
+ + +
,
A
d
(5.5)
where Sj is the saturation of phase j; ˆj
is the normalized Euler connectivity of phase j defined below; A is
the fluid/fluid interfacial area; I is the wettability of the porous medium (e.g. I = cosθ); lnNCa is the natural
logarithm of capillary number; λ is the topology of the porous medium which can be described in its most
simple form as the ratio of absolute permeability to porosity (λ = kabs/ ), although a more sophisticated
treatment of the pore topology may include, pore body/pore throat aspect ratio, coordination number, as
well as the Euler connectivity of the pore space. Equation (5.5) could also have been written in terms of the
natural logarithm of relative permeability depending on the range of relative permeability modeled. In this
chapter, we drop the subscript j, remembering that saturation, connectivity, and relative permeability can
be written for any phase, oil/water, wetting/nonwetting. Distinctions, if necessary, will be provided to help
the reader, but whether a phase is wetting or not and to what degree, is one of the input state parameters.
The normalization procedure for connectivity follows from our previous work (Purswani et al. 2019).
This is expressed as,
,ˆ max
min max
−=
− (5.6)
where χ is the Euler characteristic of the phase; χmin and χmax are the minimum and maximum values of the
Euler characteristic for a phase in a given porous medium. χmin corresponds to the completely connected
134
state of a phase, which is possible at full saturation of that phase, whereas χmax corresponds to the completely
disconnected state of a phase and therefore corresponds to the state where all pores contain the phase, but
they are not connected via throats. Consequently, χmin and χmax can be expressed by,
min p tn n = − and, ,max pn = (5.7)
where np is the number of pores and nt is the number of throats for a porous medium. A schematic
representation is presented in Figure 38. If the size of some throats were to approach that of connecting
pores, the combination will result a single pore. As long as the same number of pores and throats are
removed from the calculation, the Euler number will not change. As an example, in the second illustration
in Figure 38, if the upper two pores and the in between throat were to be combined as a single pore, the
value of χ will still be 5 (np = 9 and nt = 4).
The value of χ can also be determined from micro-CT images when pores and throats are not readily
identifiable. This can be achieved for any saturation by using the alternating sum of the numbers of vertices,
edges, faces, and objects by approximating the voxelated representation of a fluid object(s) as a regular
polyhedral or, as the alternating sum of the Betti numbers (Herring et al. 2013; Sun et al. 2019). However,
for the accurate determination of χmax an extracted pore-network of the dry image is necessary, alternatively,
information of the average coordination number (Z) of a pore structure can be used for estimating np (χmax).
Figure 38. Ball and stick representations of a porous medium with similar number of pores but increasing number of throats. The Euler characteristic decreases from left to right as connections (throats) are increased.
min
ˆ
j max
j
max
−=
−
min
ˆ
j max
j
max
−=
−
min
ˆ
j max
j
max
−=
−
• Pores• Throats
min
ˆ
j max
j
max
−=
−
135
Now, consider a set of constant external conditions, such that for a given flow experiment, wettability,
pore structure, and capillary number do not change. This reduces Eq. (5.5) to:
ˆˆ ,r S Adk dS d dA = + + (5.8)
where S
,
, and A
are the partial derivatives of relative permeability with respect to saturation,
connectivity, and fluid/fluid interfacial area, respectively. Equation (5.8) can be rewritten as,
ˆ
.ˆ A
r S
S S
dk dS d dA
= + +
(5.9)
We define a phase distribution term, Φ (Khorsandi et al. 2017), such that,
ˆ
,Φ ˆ A
S S
d dS d dA
= + + (5.10)
This simplifies the representation of the kr-EOS:
Φ.r Sdk d= (5.11)
This representation of the kr-EOS is applicable to any number of state parameters with no loss of
generality. Equation (5.11) can always be numerically integrated or in special cases integrated exactly.
Upon integrating from a reference state to the final state, we arrive at the most general form of the EOS,
136
Φ
Φ
Φ.ref
ref
r r Sk k d− = (5.12)
where in this example ( ) ( )ˆˆ/ /
S A SAS
+ + = . Normally, we take 0
refr
k = when ref r
= (one set of
residual state parameters where relative permeability is zero).
For the case when the partial derivatives are constants, Eq. (5.12) can be re-written as,
( ) ,r S rk = − (5.13)
which looks remarkably close to the Corey form of relative permeability, except that there is no exponent.
The constant coefficients for the flow function ( ) can be determined by matching available multiphase
experimental or simulation data. In theory, any path can be taken to determine the change in relative
permeability from an initial set of state parameters to the final state. However, to determine the flow
function for an actual path requires knowledge of how one state parameter changes with another along a
specific path (i.e., ˆ ( ); ( )Af S f S = = ). These dependencies give rise to nonlinear relative permeabilities as
a function of saturation even though no exponent is present in Eq. (5.13).
Equation (5.13) is the same as shown in Khorsandi et al. (2017) except that they arbitrarily included an
exponent. In addition, Eq. (5.12) generalizes the definition of to allow for non-constant partial
derivatives as well, which can further increase nonlinearities in relative permeability with saturation.
Understanding the functional form of these derivatives is one objective of this chapter.
5.2.1. Example implementation of the EOS
We assume here that changes in connectivity and saturation are sufficient to describe the state of relative
permeability (i.e., changes in other potential state parameters are either unimportant in comparison or are
137
negligible). This assumption is reasonable to first-order based on the findings by Schlüter et al. (2016), and
in any case, would be a significant improvement over current empirical model. Further, we allow for
nonlinear functionality between connectivity and saturation through a power-law expression with two
constants (p and k),
( ) ,ˆ 'ˆ 'k kp S S − = − (5.14)
where ( )ˆ ', S is a specific point that constrains the ˆ S − path. A similar expression was originally proposed
by Schlüter et al. (2016), but here we consider only two parameters for expressing the evolution of
connectivity for both drainage and imbibition. The value of ( )ˆ ', S = (1,1) for the case of drainage by a
nonwetting phase. That is, the nonwetting path must go through (1,1) for very high capillary pressure, i.e.,
when all of the wetting phase is removed. An imbibition path, however, will begin at the endpoint of the
last drainage cycle ( )ˆ ,i i
S .
Equation (5.13) can be substituted into Eq. (5.14) for the case of constant partial derivatives to yield,
( ) ( )ˆ ,k k
r S r rk S S S S = − + −
(5.15)
where ( )ˆ ˆ / Sp = . Here, it is noted that the area term (A) is not included in the development of Eq.
(5.15). Further, an additional boundary condition, o
r rk k= , at initial saturation ( )i
S , can be used to eliminate
S as,
( ) ( )'
ˆ
.
o
r
S k k
i r i r
k
S S S S
=− + −
(5.16)
138
This boundary condition is useful for appropriate comparisons to the Corey form, although one could also
use 1.0r
k = when 1S = and ˆ 1 = . As an alternative to Eq. (5.16), one could simplify Eq. (5.13) as,
.
i
o r
r r
rS
k k −
= −
(5.17)
5.2.2. Two-phase simulations using pore-network modeling
Specific connectivity-saturation ( ˆ )S − paths and other state variables are often not controllable in
experiments. For example, the pore structure of a rock can be different from one core test to another, and
parameters such as contact angle can change during hysteresis. Thus, the partial derivatives, which require
holding some parameters constant, cannot be determined consistently through experiments. Compounding
this problem is that controlled pore-scale experiments are intricate to perform. Therefore, PNM is used here
to estimate the derivatives by making numerous numerical experiments and holding state variables, for
example, contact angle, fixed.
Pore-network simulations are widely used in the prediction of reservoir transport properties based on
an idealized representation of the pore space (pores are the wider spaces connected to the narrower throats)
and a set of prescribed rules which guide the movement of multiple phases in the pore space. A capillary
dominated pore-network model ( )410
CaN
− , such as the one used in this study, works as a percolation or
an invasion percolation mechanism based on the wettability and direction of flow (Blunt 2017). These
mechanisms guide each pore filling event by the invading phase based on the occupancy of the neighboring
throats, which governs the fluid/fluid interfacial curvature and consequently the capillary pressure. When
an entry pressure to a pore is met, the invasion takes place. Saturation changes are tracked for each sequence
of filling events and relative permeability for each phase is calculated as the ratio of the flow rate of that
phase to the total flow rate in a single-phase simulation (Valvatne and Blunt 2004).
139
We used the pore-network model developed by Blunt and coworkers (Raeini et al 2018; Bultreys et al.
2018). This model is an extension of the network models generated previously (Øren et al. 1998; Øren and
Bakke 2003; Patzek 2001; Valvatne and Blunt 2004). The pore structure model was derived from a micro-
CT image of a dry Bentheimer sandstone (Lin et al. 2019). The properties of the extracted pore-network for
this Bentheimer sandstone are listed in Table 10. The network extraction was carried out using the maximal
ball approach developed by Dong and Blunt (2009) and further improved in the generalized network
extraction approach developed by Raeini et al. (2017).
Table 10. Properties of the pore-network extracted from the dry micro-CT image of a Bentheimer sandstone (Lin et al. 2019). The parameter Z (2nt/np) is the coordination number; np and nt are the number of pores and throats, respectively. The χmin for the image data is the value V-E+F-O exacted from the pore space of the image. The χmax for the image data is back calculated from the corresponding χmin and Z values. See additional details for consistent estimation of χmax in appendix B.
PNM Experimental
χmax 16850 16739
χmin -25787 -25618
np 16850 _
nt 42637 _
Z 5.061 5.061
0.22 0.24
kabs 2.493 D 2.198 D
We ran multiple PNM simulations at fixed water-wetting conditions: for one case, the contact angle
was set to exactly zero, and for the other, contact angles between 40o to 60o were uniformly distributed in
the pore-network using a Weibull distribution. For this distribution, the larger angles were assigned to larger
pores. Data for primary drainage and secondary imbibition were generated such that the nonwetting phase
injection during primary drainage was stopped at intermediate saturations, which were then used to start
the secondary imbibition process. Each imbibition process was then run until a high negative capillary
pressure (-105 Pa) was achieved at a final trapped saturation value. Lastly, all imbibition paths were
reversed as a secondary drainage process to achieve maximum oil saturation. A Matlab® sub-routine was
140
developed to automate this process which allowed for very fine saturation steps of 0.005 in between each
initial nonwetting saturation. Typically, over 40,000 data points were analyzed for each wettability case.
Normalized connectivity was calculated from the pore element occupancy of the nonwetting phase. For
this, a sub-routine was developed in Matlab®. Two-phase flow visualization files were saved for each
saturation point and information of each node (pore/throat/half-throat) occupied by the wetting/nonwetting
phase was extracted. The Euler characteristic of the nonwetting phase was then calculated using,
,occ occ occ
nw p t htn n n = + − (5.18)
where occ
pn is the number of pore nodes occupied by the nonwetting phase;
occt
n is the number of throat nodes
occupied by the nonwetting phase; and occ
htn is the number of half-throats occupied by the nonwetting phase.
A half-throat is a sub-division of a pore element incorporated in recent pore-network extraction codes to
avoid oversimplification of the pore space (Raeini et al. 2017). This helps in describing the pore-network
at a finer level.
141
5.3. Results and discussion
This section examines the evolution of connectivity as a function of saturation and hysteresis for both water-
wet and mixed-wet media. We examine literature data and synthetically generated data using pore-network
modeling (PNM). Specifically, functions are developed to describe the observed PNM paths in
connectivity-saturation ( ˆ S − ) space for hundreds of drainage and imbibition cycles and for two different
wettability values. We show the resulting residual oil saturation and connectivity locus in the ˆ S − space.
Lastly, we present estimates of the relative permeability partial derivatives using PNM data and give simple
models to predict kr.
5.3.1. Evolution of connectivity with saturation
The ability of a phase to connect or disconnect during transport in a porous medium directly impacts its
ability to flow, and consequently relative permeability. Although it is expected for the connectivity of a
phase to be proportional to its saturation, this may not always be the case. Possible exceptions include thin
film or layer flow in which wettability impacts connectivity.
We first consider example waterfloods for a variety of mixed-wet and water-wet porous media. The
selected literature data consists of similar pore structures, namely, Gildehauser sandstone (Berg et al. 2016),
Bentheimer sandstone (Gao et al. 2017; Lin et al. 2019), and a sintered glass bead pack (Armstrong et al.
2016). The segmented micro-CT images for these flow experiments (Gildehauser, and both Bentheimer)
were processed using commercial image analysis software to remove phase clusters below 125 voxels to
avoid image noise post segmentation (Herring et al. 2015). These segmented images were then used for the
estimation of the Euler characteristic (connectivity) of the oil phase and dry porous media. Next, pore-
142
networks for these dry images were extracted for the estimation of the coordination number. All porous
media properties are listed in Table 11.
Table 11. Summary of information for the literature data shown in Figure 39.
Reference Rock (Wettability) Method maxχ
minχ Z
�� at
S = 0 Remarks
Berg et al. (2016) Gildehauser sandstone
(Water-wet)
Unsteady-
state injection
/Direct
numerical
simulations
9348 -10949 4.34 0.461
Berg et al.
(2016) used
direct numerical
simulations for
the estimation
of kr
Gao et al. (2017) Bentheimer sandstone
(Water-wet, WW)
Steady-state
co-injection 11369 -11321 3.99 0.501 -
Lin et al. (2019) Bentheimer sandstone
(Mixed-wet, MW)
Steady-state
co-injection 16739 -25618 5.06 0.395 -
Armstrong et al.
(2016)
Sintered glass beads
(Water-wet)
Direct
numerical
simulations
5788 -10704 5.70 0.351
max and
min are
taken from
Purswani et al.
(2019)
Figure 39. Waterflooding χ - S paths from the literature (Table 11). All cases are for NCa < 10-4. Solid
curves show the best fits to Eq. (5.14). Saturation in the experiments move from right to left as shown by the direction of the arrow.
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
GildehauserBentheimer_WWBentheimer_MWSintered glass bead pack
143
Figure 39 shows the four waterflood paths in the ˆ S − space. The path is nearly linear for the water-wet
Gildehauser sandstone, which suggests the oil phase becomes increasingly disconnected at a fairly constant
rate with increasing water saturation. The disconnections in the oil phase result from snap-off or bypassing
by the wetting phase (water in this case). The trends for the other three floods, however, flatten at lower
saturations owing to a reduction in snap-off events. It is therefore likely that the Gildehauser flood results
would show flattening if data were available at smaller saturations.
Figure 39 also shows greater nonlinearity in the ˆ S − path for the mixed-wet Bentheimer sandstone of
Lin et al. (2019). The average contact angle for this flood is about ~ 80o, which likely resulted in a more
efficient oil displacement (piston-like) since neither phase prefers the rock surface. The number of
disconnections, therefore, change rapidly at higher oil saturations.
We fit each ˆ S − path in Figure 39 to Eq. (5.14), where the values of the fitting parameters (p and k)
are presented in Table 12. Smaller values of k are found for the water-wet cases (Gildehauser and sintered
glass bead), whereas the k value is larger for the mixed-wet case. The water-wet Bentheimer, however, also
gives a larger value of k. This latter case is less definitive as the saturation range is narrow and the data
shows an abrupt change at high oil saturation.
Table 12. Values of parameters fit to Eq. (5.14) using Table 11 data. The curves from the best fits are shown in Figure 39. The 95% confidence limits were calculated using the nlparci function in Matlab®.
Case p k R2
Gildehauser sandstone 0.58 ± 0.16 0.84 ± 0.44 1.00
Bentheimer (WW) sandstone 1.39 ± 0.12 4.46 ± 0.38 0.99
Bentheimer (MW) sandstone 0.66 ± 0.05 4.54 ± 0.80 0.98
Sintered glass bead pack 0.59 ± 0.04 2.65 ± 0.70 0.98
Although some trends are visible from Figure 39, it is difficult to make strong arguments for
connectivity using experimental data alone for several reasons. First, image resolution can strongly affect
the estimation of the Euler characteristic. Misidentification of fluid clusters during image segmentation can
lead to erroneous estimates of the Euler characteristic, as well as the pore space (used for normalization).
144
Second, imaging experiments are expensive and often provide limited data points. Third, these example
cases demonstrate the complexity that can occur in understanding experimental measurements of porous
media with many simultaneously varying and less controllable parameters (e.g., wettability and pore
structure). For these reasons, we study connectivity trends using pore-network modeling (PNM) next. PNM
is advantageous because simulated paths are repeatable and input parameters are largely controllable.
Figure 40 shows examples of three hysteretic displacements for the weakly water-wet case. Each cycle
begins with primary drainage (PD) (solid curves) from zero oil saturation to a specified termination
saturation (So ~ 0.8, 0.9, and 1.0), followed by imbibition (IMB) (dashed curves) to residual saturation, and
secondary drainage (SD) (dotted curves) to 100% oil saturation. The PD curve ending at So ~ 1.0 reached a
capillary pressure of 105 Pa. The imbibition scans ended when oil no longer flowed as defined by -105 Pa
capillary pressure. More extensive simulations will be presented in later sections to model kr.
Figure 40. The χ - S paths for different drainage and imbibition scans using PNM for the weakly water-
wet case ( o
θ ~ 50 ). All PD curves begin at zero oil saturation but terminate at So ~ 1.0 (green), So ~ 0.9 (blue), and So ~ 0.8 (red). Next, IMB curves terminate at residual conditions (squares). Finally, all SD curves are simulated to So ~ 1.0.
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
PD
I
SD
IMB
145
The PNM paths shown in Figure 40 are qualitatively similar to the experimental paths in Figure 39.
That is, the PNM paths are nearly linear for the imbibition paths. The drainage paths, however, are
significantly more nonlinear, likely because the nonwetting phase connects well only at higher saturations.
These results are consistent with the displacement experiments discussed by Schlüter et al. (2016).
The paths in Figure 40 were also fit to Eq. (5.14), along with six other similar scanning curves whose
PD paths terminated at smaller, but equally-spaced saturations (So ~ 0.2 – 0.7). The best fit parameters (p
and k) are presented in Figure 41.
Figure 41. Fitting parameters (p and k) after matching Eq. (5.14) to PNM scanning data (see Figure 40 for three of the scans plotted here at 1.0, 0.9, and 0.8 So). The x-axis is the nonwetting saturation at the termination of PD. The shaded region shows the error bars calculated for 95% confidence limits using the nlparci function in Matlab®. The contact angle averages at 50o.
Figure 41 shows that p is nearly constant for all drainage scans, whether primary or secondary. The
parameter, p, for drainage cycles is equal to 1 − as So approaches zero. The value of connectivity ( ) at
this limiting saturation results from only very few pores being filled by the nonwetting phase (the Euler
characteristic becomes a small positive number, such that the minimum and maximum possible values
determine the limit as So approaches zero, e.g., if only one pore were filled by the nonwetting phase, then,
Eq. (5.6) gives, ( ) ( )ˆ 1 16850 25787 16850 0.395/ = − − − = . For PD, therefore, ~ 1 0.4p − or 0.6. Further, all SD
curves that result from PD termination saturations less than about 0.7 have nearly the same p values, because
their initial ˆ S − values (residual values) lie close to the PD path. Only SD curves associated with high
146
termination saturations (> 0.7) give somewhat larger values for p because their initial ˆ S − values lie below
the PD path in the ˆ S − space.
Imbibition values for p, however, vary significantly from about 0 to 0.8 and are quite uncertain at small
saturations. This uncertainty stems from the lack of a significant change in connectivity over the imbibition
curve. Values for p do not approach 1 − at zero saturation, but rather terminate at a residual ˆ S − value.
Further, an imbibition curve given by Eq. (5.14) does not begin at (1,1), but instead at a termination value
of the PD curve. Thus, p for IMB depend on the initial and final ˆ S − values.
The parameter, k, controls the linearity of the path in ˆ S − space. Figure 41 shows that k is significantly
greater than 1.0 for drainage and ~ 1.0 for imbibition. Thus, all drainage paths curve upwards with
increasing nonwetting saturation, while all imbibition processes are nearly linear as the nonwetting
saturation decreases. The value of k varies from about 3 to 10 for the drainage cycles. The SD and PD
values vary inversely with each other in Figure 41. The k values for the PD curves at small termination
saturations only represent a small portion of the entire PD curve that goes from saturation of 0.0 to 1.0.
Thus, the variation in these values is not representative of the entire PD curve. A value of around 10 matches
well the full PD curve and any portion of that curve in between. The values of k for the SD curves vary
from about 10 at small termination saturations to about 6 at large nonwetting saturation. This result is also
consistent with the value for the PD curve because small termination saturations give a larger change in
saturation so that k approaches 10. Thus, the SD curve that begins at saturation near the PD curve (i.e., for
small termination saturation) takes a similar path as the PD curve to saturation of 1.0, resulting in the same
values of p and k as the PD curve (the SD curve limit for small termination saturation is the PD curve).
These important results could be used to predict the paths in the ˆ S − space where a value of 10 for k
can be used for all drainage processes (PD or SD), while a value of 1.0 for k is reasonably accurate for all
imbibition curves, independent of their starting ˆ S − values. Further, the value of p can be estimated from
Eq. (5.14) for all drainage curves based on the starting ˆ S − values and for all imbibition curves based on
147
the initial and residual ˆ S − values. A summary for the of p and k values for the different cycles of injection
is provided in Table 13.
Table 13. Summary remarks for p and k values for the evolution of phase connectivity for different cycles of injection.
Injection
process p k
Primary
drainage
All PD curves (if continued till the end)
appear to terminate at ˆ , S = 1,1. Therefore,
by constraining Eq. (5.14) at this point, we
can eliminate, p as (p = 1- 0 ), where 0 is
the value of at S = 0.
A value of k = 10 matches the full PD
curve regardless of where the PD curve
is terminated. This is because all ˆ , S
points lie on the full PD curve.
Imbibition
Imbibition starts and terminates at different
locations in the ˆ S − space, therefore, p, for
imbibition will be a fitting parameter for the
path under consideration
Give the linear nature of the imbibition
curves, k = 1, can be a reasonable
approximation.
Secondary
drainage
All SD curves terminate at ˆ , S = 1,1.
Therefore, by constraining Eq. (5.14) at this
point, we can eliminate, p as (p = 1- 0 ),
where 0 is the value of at S = 0.
To model the curvature observed in the
drainage curves, a high value of k for
drainage works well. Given the starting
point for SD (end of IMB) is different, k
can be a fitting parameter. However,
from observation, a value of k = 10,
works reasonably well for SD curves
5.3.2. Fitting kr-EOS to literature data
This section shows how to use the simplest form of the kr-EOS based on Eq. (5.13) (e.g., constant partial
derivatives) to fit the four data sets considered previously (see Table 11). The path in the ˆ S − space may
not be known from CT scans, so here we tune on only parameters k and
as might be done in practice
(see Eq. (5.15)). For reference, the parameter
is then calculated using the values of p in Table 12.
Figure 42 shows the best fits of the literature data using the new kr-EOS and the conventional Corey
expression. Because the Corey form uses the end-point value, the parameter S
was determined so that the
measured end-point relative permeability is obtained exactly (see Eq. (5.16)). The end-point saturations
148
were also fixed and not tuned. Thus, the only tuning parameters were
and k for the EOS, while the
conventional exponent was tuned for Corey. As shown in Table 14, the fits are excellent and are slightly
better for the new EOS (see Bentheimer MW case at low saturations) compared to the Corey curve. The
low values of s
for three of the experiments suggest that the connectivity term dominates the behavior of
relative permeability. Unlike Corey, the parameters in the kr-EOS have physical meaning.
Figure 42. Best fits to the literature data in Table 11 using the (a) kr-EOS with constant partial derivatives and (b) Corey form.
Table 14. Best fit values for the kr-EOS and Corey form shown in Figure 42. The end-point permeability is the same for comparison purposes. Thus, only
χα and k are used as tuning parameters for the kr-EOS and
no for Corey. R2 values are shown. Key parameters from literature data kr-EOS Corey
Case ( )absk D
caN
oiS
orS o
rk S
α χα k 2R
on 2
R
Gildehauser 1.5 9.1x10-7 0.77 0.30 0.49 7x10-8 1.76 2.24 1.00 1.44 1.00
Bentheimer WW 2.07 3x10-7 0.75 0.36 0.31 1x10-8 0.72 3.83 0.99 1.80 0.99
Bentheimer MW 2.198 9.9x10-7 0.89 0.11 0.93 0.14 4.06 10.50 1.00 7.24 0.99
Sintered glass bead pack 22 ~10-4 0.90 0 0.93 0.78 0.50 2.52 1.00 1.24 0.99
Although both the Corey form and kr-EOS show a reasonable match to literature data, the Corey form
has little or no predictive capability unless coupled with other models, such as an extended Land’s model
to capture hysteresis in the scanning curves. Even then Corey’s predictive capability is limited to fitting the
available data. The EOS, however, can honor expected physical trends in kr as a function of the input
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
GildehauserBentheimer MWBentheimer WWSintered glass bead pack
a b
149
parameters, i.e., , S, and θ in the examples presented here. The remainder of the paper focuses on modeling
and predicting kr using the EOS.
5.3.3. Relative permeability scanning curves: pore-network simulations
In the previous section, the partial derivatives for the kr-EOS were assumed to be constant. We use pore-
network modeling (PNM) in this section to determine the values of these partial derivatives more exactly
and also to examine how they vary in the ˆ S − space. Relative permeability can then be estimated more
accurately for a given path from these derivatives.
Figure 43 shows the results for ten scans equally spaced in saturation (terminal saturation from PD)
for two contact angles (0o and ~ 50o). Some of these scans for ~ 50o
are also given in Figure 40. The use
of different starting points in the ˆ S − space allows for increasing the region covered in that space by the
various scans. This allows for derivative calculations.
Figure 43a and Figure 43b show that the values of relative permeability intersect each other (crossover),
as has been observed previously by Spiteri et al. (2008). This is also true in the ˆ S − space (see Figure 43e
and Figure 43f). The value of kr at these intersections is nearly the same even though the processes to obtain
that value of ˆ S − were quite different. This demonstrates that for these PNM scans and S are reasonably
sufficient to represent kr, i.e., the state function (EOS) approach is reasonable. The drainage scans (Figure
43c, Figure 43d, Figure 43g, and Figure 43h) do not have significant intersections.
Figure 43 also shows slight differences in kr during primary drainage between the two wettability cases
near low and intermediate oil saturations (see Figure 43a and Figure 43b). Here, the increase in kr is
somewhat greater for the completely water-wet case. Also, there is less hysteresis in kr for the weakly water-
wet case, although the region traversed in the ˆ S − space is larger. This behavior could be attributed to the
sudden pore-filling events possible in strong water-wet medium (water can move around the oil along the
150
surfaces) as opposed to smoother more piston-like filling in the less water-wet case. Like the results
observed previously, the imbibition ˆ S − paths are significantly more linear than the drainage paths.
Furthermore, it can be noticed from Figure 43e, Figure 43f, Figure 43g, and Figure 43h that the upper
boundary of the ˆ S − space is controlled by the series of imbibition curves starting at different initial
saturations, while the lower boundary of the ˆ S − space is controlled by the secondary drainage curve
starting at the lowest ˆ S − point (one limit of the residual locus). The ˆ S − boundary which is the physical
space where relative permeability values exit depends on the wettability of the medium as well as the pore
structure constraint at S = 0, ˆ ˆ = o point (S, = 0, 0.395 for the porous medium studied here).
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Figure 43. PNM simulations of imbibition (IMB) and secondary drainage (SD). Two contact angles are used: the left column represents a fixed contact angle of 0o, while the right column is for uniformly distributed contact angles between 40o and 60o with an average of ~ 50o. The black solid line in (a) and (b) is for primary drainage (oil flood). Figures (a) and (b) also give ten imbibition curves at 0.1 saturation
intervals on the PD curve, while (e) and (f) show their χ - S paths. Figures (c) and (d) are for secondary
drainage and (g) and (h) show their corresponding χ - S paths. The red open circles represent the starting
point for IMB, while the black open circles represent the residual points for each IMB scan (similar to Figure 40). Arrows show the direction of saturation change (IMB points to the left while SD points to the right).
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f
d
h
Rel
ativ
e p
erm
eab
ility
Ph
ase
con
nec
tivi
ty
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Figure 44. Relative permeability contours plotted from the 200 PNM imbibition simulations for the two contact angle cases of 0o and ~ 50o.
Figure 45. PNM simulations for 200 imbibition and secondary drainage scanning curves that begin at different PD termination saturations (spaced by 0.005 saturation units). Two contact angles are shown, 0o (a) and ~ 50o (b). The PD curves begin at So = 0.
Figure 44 shows the nonwetting phase relative permeability contours generated using the imbibition
PNM simulations. Here, the upper boundary of the ˆ S − space is visible more clearly. Figure 45 shows
the entire PD-IMB-SD scanning curves and shows how relative permeability changes smoothly in the ˆ S −
space for 200 scans, again confirming the use of the state function concept. Further, in the saturation range
of ~ 0.45 to 0.65, ˆ S − paths cross and it is found that 0.126 0.013r
k = for 0o
= and 0.121 0.024r
k =
for ~ 50o
. This demonstrates that for these PNM simulations a similar ˆ S − value gives approximately
θ = 0o~ 50o
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a b
Zero krZero kr
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one value of kr, independent of the path taken. This powerful concept may allow for the prediction of
complex relative permeability, characteristic of natural porous media.
Relative permeability is zero everywhere in the lower-left region of the ˆ S − space. The boundary of
this region, termed as the residual locus ( ˆ )rr S − , defines an important limit for the physical kr region. The
residual loci for the two wettability cases are compiled in Figure 46.
All imbibition curves must intersect the residual locus. Traditionally, only one residual saturation is
used as determined from a few experiments and arbitrary interpolation-based schemes are implemented for
predicting relative permeability across different paths. The knowledge of the residual locus can have
important implications for the accuracy of simulations when modeling hysteresis.
Figure 46. Locus of residual connectivity and residual saturation generated from the different scanning curves (Figure 43). The blue data point shows the limiting value of connectivity as saturation approaches zero (see Eq. (5.6)).
0
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154
The shape of the locus (Figure 46) shows two values of residual connectivity over a small range in
residual saturations. This region of two values decreases for the completely water-wet case because of little
change in residual connectivity. However, for the weakly water-wet case, efficient displacements of initially
well-connected oil led to a greater decrease in connectivity (fewer trapped oil blobs). This is also seen in
the literature, for example, more oil recovery occurs for mixed-wet rocks owing to the better displacement
of oil (Jadhunandan and Morrow 1995).
5.3.4. Estimation of relative permeability partial derivatives
We now provide estimates of the relative permeability partial derivatives using the 200 PNM scanning
curves show in Figure 45. For estimation of these partials at each ˆ S − value, we fit the equation of a plane
through a cluster of data points in the vicinity of that point using,
( ) ( ) ( )ˆ
,ˆ ˆˆ
r r
r rof of of
S
k kk k S S
S
− = − + −
(5.19)
where ( ),ˆ, of of rofS k is a fixed data point. The data points in the cluster were selected based on the closest
Euclidean distance to the fixed point. The coefficients in the fitted plane then gave the partial derivatives
in Eq. (5.19) for that point. Different cluster sizes were examined, but an optimum of 700 neighboring data
points gave the best results to smooth the derivative estimates. The derivative values are shown in Figure
47.
155
Figure 47. Relative permeability partial derivatives estimated from fitting a cluster of data to a plane.
Some common trends for both wettability cases can be inferred from Figure 47. First, the values are
not constant but vary significantly. Second, the partial derivatives in the ˆ S − space were generally
positive so that kr increases with both saturation and connectivity. The partial derivative, ( )ˆ
/r
k S
, was
found to be greater for high saturation than for low saturation (varied from 0 to 2.5), while the reverse trend
was observed for ( )ˆ/r S
k . The derivative, ( )ˆ/r S
k , however, is slightly negative near the residual locus
at the smaller connectivity values. This results from the locus shape, which means that kr below the residual
curve must increase to obtain kr = 0. The average values of the partials are reported in Table 15 which will
be used in the next section for predicting kr values using a single value of partial derivatives.
The overall trends in partial derivatives convey that changes in connectivity are more important at low
and intermediate saturations, while changes in saturation control relative permeability at high saturation.
This is true for both wettabilities considered.
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Table 15. The average values of the estimated partial derivatives of relative permeability shown in Figure 47.
Wettability ( )ˆ
/r
k S
( )ˆ/r S
k
0o
= 0.43 1.31
~ 50o
0.70 0.85
5.3.5. Prediction of relative permeability
We are interested in the prediction of kr at any given point in the ˆ S − space. Here, we present two
approaches for kr prediction using the partial derivatives estimated in the previous section.
First, we assume constant partial derivatives where Eq. (5.13) is re-written as,
( ) ( )ˆ
,ˆ ˆ. .ˆref
r r
r r re
me San
r
me
f ef
an
k kk k S S
S
+= +
− −
(5.20)
The mean values of the partial derivatives in Eq. (5.20) were taken from Table 15, while the reference
values were chosen at an intermediate ˆ S − value. As such, ( ˆ ,,ref
ref ref rS k ) were set as ( )0.6, 0.439, 0.245
for 0o
= and ( )0.545, 0.417, 0.126 for ~ 50o
. These were chosen because all ˆ S − paths traversed close
to the intermediate values. Ideally, predictions using an exact EOS do not depend on the chosen reference
state.
Second, we fit planes through the partial derivatives shown in Figure 47 using,
ˆ
ˆ ˆˆ
r r
S
k kc and f
SaS b dS e
= + =+ +
+
(5.21)
A constraint, b = d, was set such that the second order derivatives of the partials are equal, i.e.,
2 2ˆ ˆ/ /r rk S k S = . This honors the condition for the exact differential. Table 16 provides the values
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of the fitted parameters shown in Eq. (5.21). No additional constraints were used in matching these plane
fits. These plane fits were then used for generating the kr response.
Table 16. Values of plane fitting parameters in Eq. (5.21) used for the kr response in Eq (5.22).
Parameter
Wettability A B C D E F
0o
= 0.97 3.28 -1.50 3.28 -7.34 2.84
~ 50o
1.78 1.21 -0.95 1.21 -2.65 1.36
Equation (5.21) gives a quadratic response for kr when integrated from some reference state to the desired
state,
( ) ( ) ( ) ( ) ( )2 2 2 2ˆ .ˆ ˆ ˆ ˆ ˆ2 2ref
r r ref ref ref ref ref ref
a ek k S b S S c S SS f − + − + − += −+ − + (5.22)
The reference values were kept the same.
Here, we have only considered simple treatments of the partials. Higher order functions such a as
quadratic response could also be considered for the partials in the ˆ S − space which would lead to the
development of a cubic kr response (as opposed to the quadratic response shown in Eq. (5.22)). Moreover,
the functional forms of the partials could also be constrained in the physical ˆ S − space.
The predicted values of kr using the two approaches are presented in Figure 48. Values of kr < 0 were
set to 0.0 and kr > 1.0 were set to 1.0 in Figure 48. Predictions are solely based on the treatment of the
estimated partial derivatives and no constraints at limiting values were used.
The contour maps in Figure 48 show that the predicted kr response is slightly better for the quadratic
response. This is because the quadratic approach gives realistic values in opposite corners of the ˆ S −
space. The corner regions, although likely inaccessible through experiments, correspond with relative
permeability near zero because either connectivity or saturation remains low. This behavior is elegantly
158
captured by the quadratic response. Also, the residual locus is captured to a reasonable extent as opposed
to a linear ˆrr S − locus given by the constant partial approach. Furthermore, as seen previously, the weakly
water-wet case shows a wider non-zero kr region owing to efficient displacement of the nonwetting phase.
Figure 48. Contour maps of predicted relative permeability using constant partial approach (a, b) and quadratic response (c, d) for both wettability cases. Actual versus predicted relative permeability values for the two approaches are shown in (e) and (f). The R2 for both approaches in (e) was ~0.90, while the R2 was ~0.90 and ~0.96 for the constant partial approach and quadratic response, respectively, in (f).
Figure 48e and Figure 48f show that excellent predictions are obtained in the region of actual data
(obtained from PNM simulations) for both methods. The likelihood of obtaining ˆ S − values outside of the
scanning region is low as evidenced by the fact that a single value for exists for zero saturation (~ 0.4 for
0
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c d
0.0
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1.0
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Constant partial
Plane fit
0.0
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1.0
0.0 0.2 0.4 0.6 0.8 1.0
Co
nto
ur
map
s
Co
nstan
t partial
Qu
adratic resp
on
se
Constant partial
Quadratic response
e f
159
the pore structure used in this study). Furthermore, even a simple EOS-based approach of constant partial
derivatives shows substantial improvement over what is traditionally done (Land-based models), where
relative permeability is fixed by Corey’s model for each kr curve, and arbitrary interpolations are used in
between each curve.
Overall, the EOS approach makes kr continuous in the space of the state parameters considered –
mitigating complex quick fixes to hysteresis as is accomplished from conventional approaches. Further, the
approach presented in this research has the ability to improve compositional simulation by avoiding phase
labeling discontinuities by incorporation of wettability into the EOS model.
160
5.4. Concluding remarks
A general equation-of-state (EOS) approach is presented for modeling relative permeabilities, where we
considered the state parameters of saturation and connectivity. Two-phase flow data from the literature and
numerical data from pore-network modeling (PNM) were used to generate key parameters for the
application and testing of the EOS. The main conclusions from this research are as follows.
• Paths in connectivity-saturation ( ˆ )S − space are more linear during imbibition compared to
drainage. During imbibition, nonwetting phase disconnects owing to snap-off events which occur
at the same rate with saturation, but during drainage, the reconnection of the nonwetting phase
occurs at high saturations. These paths are adequately described by a simple function.
• Values of relative permeability are nearly identical at the same ˆ S − value, which justifies the
state function approach for the case studied
• The new kr-EOS matches literature data well, similar to the conventional Corey form. For relative
permeability curves with significant curvature, however, the new approach matches experimental
data better than Corey, especially at low saturations.
• A residual locus exists in ˆ S − space, which is a function of wettability. Traditional hysteresis
models either ignore this possibility or offer complex interpolation-based solutions. The approach
here gives an elegant predictive methodology that is consistent with physics.
• Partial derivatives of relative permeability with respect to and S show that relative permeability
depends more strongly on connectivity at small saturations, while saturation is more important at
high saturations.
• A simple first-order approximation of the constant partial derivatives gives an EOS with reasonable
predictive capability over the region of the scanning curves considered. Partial derivatives that are
linear functions of and S give a quadratic response of relative permeabilities which show
improvements over the first-order approximation.
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PNM offers an excellent bridge for incorporating pore-scale phenomenon toward the development of a
physics-based relative permeability model. Other simulation possibilities could also be employed such
computation fluid dynamics (CFD), Lattice Boltzmann methods (LBM). CFD is generally limited to only
a few pores and smoother geometries. LBM can handle rough surfaces and could be the study of future
work. These simulation models would, however, take much longer run-times, which would greatly slow-
down the possibility of multiple numerical experiments for building very large datasets.
The generalized EOS framework presented in this research can be extended for modeling capillary
pressures (Pc), as well as for finding functional connections between kr and Pc.
162
CHAPTER 6. CONCLUDING REMARKS AND
OUTLOOK FOR FUTURE RESEARCH
Current modeling efforts for relative permeability depend on empirical functions of phase saturations.
These functions are nonunique and are therefore specific to the flow/experimental condition. As an
example, imbibition nonwetting phase relative permeabilities differ from drainage nonwetting phase
relative permeabilities. Each path then requires its own saturation function which can cause problems in
reservoir simulations of multiphase processes like CO2 sequestration or water alternating gas enhanced oil
recovery where drainage and imbibition processes often occur simultaneously. To address such issues,
research efforts have been presented in this dissertation to model relative permeabilities based on key
controlling parameters such as phase saturation, phase connectivity, capillary number, and wettability that
are known to affect relative permeabilities.
A quadratic response-based equation-of-state (EOS) for relative permeability was modeled in the phase
connectivity and phase saturation ( ˆ )S − space. Limiting conditions on the state parameters were explored
to constrain the EOS model physically. Different capillary number cases ranging from one to 10-6 were
considered in the model. Additional effort investigated the role of wettability on phase trapping using pore-
network modeling (PNM). An extended Land-based hysteresis trapping model was presented and compared
against models from the literature. In addition, models were presented to capture the trends of the loci
bounded by the residual phase connectivity and residual phase saturation for different contact angles.
Finally, numerical PNM data sets for two contact angles in the water-wet regime were used for calculating
partial derivatives of nonwetting phase relative permeabilities in the ˆ S − state parameters. A response for
relative permeability was derived using the calculated partial derivatives and compared against actual
values of relative permeabilities from PNM.
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For the implementation of the proposed EOS-based relative permeability model in reservoir
simulations, it is needed that the state parameters, namely, S, (at the least) are characterized for the flow
condition. As saturations are updated for each gridblock and for each timestep, similarly (and
simultaneously), the phase connectivity term should be updated for each gridblock and for each timestep.
Once the saturation and connectivity values are defined, the response surface can be implemented for
updating the relative permeabilities. The coefficients in the response surface should be tuned to
experimental data for the specific flow conditions (capillary number/wettability/pore structure). For
example, the flow conditions near wellbore will experience a different capillary number (higher) than far
away from the wellbore. Similarly, EOR processes of surfactant floods will have a different capillary
number than for a waterflood. Thus, typical special core analysis (SCAL) experiments should be
supplemented with pore-scale imaging/numerical simulations to extract measures of phase connectivity for
tuning the relative permeability response surfaces as is demonstrated in this work. Once tuned these will
remain fixed for the rest of the simulation sequence.
For cases where pore-scale data may not be readily available, it is advised that functional forms such
as the evolution of phase connectivity (as discussed in this dissertation) in the saturation space can be
utilized for characterizing phase connectivity at each saturation point. Here, however, the flow process such
as drainage (nonwetting phase increase) versus imbibition (nonwetting phase decrease) will be required to
accurately capture the evolution of phase connectivity. As such, these functional forms will depend on the
wettability and capillary number conditions for which they are characterized. Future work is therefore
recommended to fully characterize the functional forms of phase connectivity (and even fluid/fluid
interfacial areas) in the saturation space for different experimental conditions. This will be beneficial to
circumvent expensive (and often time-consuming) pore-scale experiments. In addition to the knowledge of
the evolution function, the characterization of the locus of the residual saturation and residual phase
connectivity which is the true limiting condition for relative permeabilities expressed in the ˆ S − space
will be important. This will enable to define the entire physical ˆ S − space where relative permeability
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will exist. A response can then be easily developed. This response will represent all scanning curves and
thus hysteresis will no longer be difficult to model.
For compositional and three-phase simulations, the role of wettability is critical. For example, in the
compositional space, properties of fluid phases can approach one another (near critical regions) which can
impact wettability at the fluid/fluid/solid contact point. To include the effect of wettability into the EOS a
simple procedure as outlined in Khorsandi et al. (2021) can be implemented. They defined the tuning
coefficients for the EOS (example, the kr response surface) individually for the “wetting” and “nonwetting”
phases instead of for each “phase label” (oil/gas/water) and proposed that the value of the tuned coefficient
for each phase label can be estimated by weighting between these wetting and nonwetting values, regardless
of the phase itself. In addition, they proposed the use of a wetting fraction for defining each phase label’s
wettability which is essentially the affinity factor of the that phase label to the solid surface.
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6.1. Key concluding remarks
The following conclusions can be made from this work.
• A simple quadratic response for relative permeability in the state parameters of phase connectivity and
phase saturation ( ˆ )S − captures trends across different capillary numbers. Model tuned for a capillary
number in the capillary dominated regime can show predictive capability for other capillary numbers
within the same regime.
• The role of phase connectivity is found to be critical for the model development. The sensitivity of kr-
S paths to capillary number is found to be implicitly caused by changes in phase connectivity. The
linear kr-S paths for high capillary numbers (small Corey exponents) and nonlinear kr-S paths for low
capillary numbers (high Corey exponents) are found to occur due to fast and slow changes in phase
connectivity, respectively.
• Limiting constraints help in the identification of physical region in the -S state space. Pore-network
simulations demonstrate that it may not be physically possible to have the state space with high phase
saturation and low phase connectivity and vice-versa.
• The analysis of the effect of wettability shows that both phase trapping as well as the locus of residual
saturation and residual phase connectivity are sensitive to contact angle changes. For low contact
angles, the residual locus remains fairly constant but, at higher contact angles, the shape of the residual
locus resembles a closed-loop due to pore structure constraints at negligible phase saturation at which
point the pore structure topology (min
and max
) govern the value of ˆ .
• Phase trapping was found to reduce significantly for low receding phase contact angles owing to pore-
scale mechanisms of layer flow of the receding phase and piston-like advance of the invading phase. A
newly developed extended Land-based hysteresis trapping model is found to capture the residual trends
for all contact angles.
166
• Application of numerical techniques like pore-network simulations can help in the development of
physics-informed transport models. Pore-network simulations allow for the generation of hundreds of
data points in the ˆ S − state space under controlled conditions which facilitates the estimation of
relative permeability partial derivatives.
• The response derived for relative permeability from the estimated partial derivatives demonstrates
predictive capability for relative permeabilities over the entire data sets, regardless of the direction of
flow, thus mitigating hysteresis.
167
6.2. Future research
The following recommendations are provided as possible extensions of future work from this dissertation.
• The modeling efforts presented in this work highlight the need for good quality and numerous
experimental data sets, especially with pore-scale measurements of Euler characteristic and fluid/fluid
interfacial areas that can help in furthering the development and testing of the models. Further, there is
a need for these data sets for different porous medium wettability, particularly, wettability conditions
representing oil-wet or CO2-wet conditions.
• Micro-flow experimental studies can be conducted for porous media with known pore structures
through the application 3-D printing techniques. This will be critical for modeling the kr-EOS where
the pore structure is constant (as was done using pore-network modeling in this study). Similarly, fixed
porous medium will be useful for repeatability of the flow measurements.
• Physics-informed simulation models like PNM can be improved with the provision of conducting
simulations for different capillary numbers. Also, these simulations can be constructed with inbuilt
options to output pore-scale measures of Euler characteristic and fluid/fluid interfacial areas directly.
• Significant errors can occur during image processing (denoising and phase segmentation) when
estimating pore-scale properties via experimental techniques such as x-ray imaging. Newer machine
learning-based tools such as Convolutional Neural Networks (CNN), U-Net, and Res-Net can be
explored for image segmentation and other image pre-processing steps.
• With the availability of numerical flow data sets such as the ones explored in this study, data driven
models such as Artificial Neural Networks (ANN) can be coupled with physical constraints for the state
parameters to develop physics-informed machine learning algorithms for transport properties such as
relative permeabilities. Such machine learning-based algorithms will depend on both the quality and
quantity of flow data which will be used for training and validation purposes. A two-step process is
envisioned. The first step will require experimental flow measurements to corroborate numerical
techniques such as PNM but for different flow conditions such as variable capillary numbers. Next,
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with the help of corroborated numerical techniques, large data sets will need to be generated to develop
and test data driven (and physics-informed) models such as ANN.
• Other polynomial forms for the kr-EOS can be studied. For example, the cubic form of kr response in
the ˆ S − space. This will allow for capturing the relative permeability partial derivatives using a
quadratic response for the partial derivatives in the ˆ S − space. The cubic form may also allow for
better representation of the residual locus in the ˆ S − space.
• Wettability indices such as the Amott index are used more traditionally to describe medium wettability.
Although these measures only provide an average wettability estimate. PNM can be used to bridge the
understanding of wettability from a local perspective (such as contact angle measurement) and
wettability indices to compare their effect on residual saturations.
• Although the focus in this work has been on modeling relative permeabilities, adaptations can be
extended to other flow properties such as modeling capillary pressures using the EOS approach
described in chapter 5. Similarly, adaptations can be developed for modeling base porous media
permeabilities by studying different characteristics that describe the structure of a porous medium such
as pore/grain size distribution, surface area of the pore space, topology of the pore space.
• A future extension to the EOS model can be on the inclusion of fluid/fluid interfacial areas into the
equation-of-state. Although this extension should be considered with implications to the final
deployment of the EOS in reservoir simulators—meaning that the improvement in the current EOS
after including fluid/fluid interfacial area should be substantial to warrant complicating the EOS model.
• Trends of fluid/fluid interfacial areas with saturations for different contact angles can be useful for
groundwater remediation applications. Simple models can also be developed to capture these trends
and predict interfacial areas for other experimental conditions. These models can also be useful for
capillary pressure hysteresis models that use fluid/fluid interfacial areas instead of Euler characteristic
for addressing hysteresis.
169
APPENDICIES
Appendix A. Basic theory of Minkowski functionals
Minkowski functionals describe the geometric measure of size for any object. For a d-dimensional object,
d+1 Minkowski functionals will be required to fully characterize the geometric shape of the object
(Armstrong et al. 2018). For example, a sphere will require four Minkowski functionals, whereas a surface
element such as a triangle or a line segment will require three Minkowski functionals.
Application of Minkowski functionals is in use in understanding pore-scale fluid flow. The pore space
and the phase (oil/water) structures are arbitrary shapes, and their sizes can thus be characterized with the
understanding of Minkowski functionals. Similarly, fluid/fluid interfaces are arbitrary two-dimensional
objects whose sizes can be characterized from Minkowski functionals.
For a three-dimensional object, there will be four Minkowski functionals. The first Minkowski
functional (M0) represents the volume of the element in consideration. This can be used for characterizing
the volume measure for porous media applications such as the porosity and fluid saturations. An example
of the wetting and nonwetting phase in a capillary tube of radius, r, is shown in Figure 49.
170
Figure 49. Schematic of capillary tube filled with wetting and nonwetting phases. A zoomed-in view of the interface across the phases is also shown.
The following equations express the Minkowski functionals of the wetting/nonwetting phases shown in
Figure 49.
( )3
20
2 1 ,
3
ww
rM r L sin
cos
= + − (A.1)
( )3
20
2 1 ,
3
nwnw
rM r L sin
cos
= − − (A.2)
where M0w and M0
nw are the wetting and nonwetting phase first Minkowski functionals for the phases shown
in Figure 49; Lw and Lnw are the lengths of the capillary occupied by the wetting and nonwetting phases in
Figure 49; θ is the contact angle measured through the wetting phase.
The second Minkowski functional (M1) represent the surface area of the element under consideration.
( )1 ,
E
EM ds
= (A.3)
90- r
2
Wetting phase injection
Nonwetting phase recedes
Lnw Lw
L
2 90- rr
171
where M1(E) is the second Minkowski functional of element E; ds and δE are the surface element on and
the entire surface of the object E, respectively. This can be used for characterizing the surface area
properties of importance to porous media applications such as the surface area of the solid (or the specific
surface area of the solid), phase total surface areas of the wetting/nonwetting phases (or the specific fluid
surface areas). The specific surface areas of the fluids can be used for estimating the fluid/fluid interfacial
areas. Examples of second Minkowski functionals for the wetting and nonwetting phase elements in Figure
49 can be estimates as,
1 2 1 ,ww
rM r L sin
cos
= + −
(A.4)
1 2 1 ,nwnw
rM r L sin
cos
= + −
(A.5)
where M1w and M1
nw are the wetting and nonwetting phase second Minkowski functionals for the phases
shown in Figure 49.
The third Minkowski functional (M2) represents the integral of the mean curvature of the element under
consideration and is given as,
( )21 2
1 1,M ds
R R
= +
E
E (A.6)
where R1 and R2 are the principal radii of curvatures of the interfacial element E. It can be used for the
estimation of capillary pressure. For the capillary tube example, with spherical curvature (R1 = R2 = R),
2
1 1 2 2 ,nw cos
MR R R r
= + = =
(A.7)
172
where M2nw is Minkowski functional representing the wetting/nonwetting interphase for the demonstration
in Figure 49; r is the radius of the capillary tube. From Young-Laplace equation, for a spherical interface,
the capillary pressure across the interface can be linked to the third Minkowski functional as,
2 .1 1 2
σ nwc
cosP M
R R r
= + = =
(A.8)
The fourth Minkowski functional (M3) is represented as the integral of Gaussian curvature and is given
as follows,
( )31 2
1,
E E
EM Kds dsR R
= =
(A.9)
where M3(E) is the fourth Minkowski functional of element E; ds and δE are the surface element and the
entire surface, respectively; K=[1/R1R2] is the Gaussian curvature of the element in consideration. Three
examples of different surfaces with different Gaussian curvatures are shown in Figure 50. The special case
of the Gauss-Bonet theorem relates the fourth Minkowski functional (Gaussian curvature) to the Euler
characteristic of the element.
Figure 50. Schematic of different surfaces with different types of Gaussian curvatures.
Positive Gaussian curvature Negative Gaussian curvature Zero Gaussian curvature
E EE
173
The most general form of the Gauss-Bonet theorem states that “The integral of the Gaussian curvature
over a surface E with boundary δE plus the integral of the geodesic curvature (kg) around that boundary
plus all exterior corner angles (θi) is equal to 2π times the Euler characteristic χ(δE) of the surface.”
Mathematically, the Gauss-Bonet theorem is expressed as (Gluck 2012; Watkins),
2 ( ).
E E
Eg iKds k dl
+ + = (A.10)
For a closed element, the integral of the geodesic curvature becomes zero. As such, the following
simplification to the Gauss-Bonet theorem can be made,
( ) 2 .
E
EiKds
+ = (A.11)
A further simplification can be made by consideration of closed surfaces that are smooth. Smooth surfaces
ensure that the sum of the exterior angles goes to zero. This simplifies the Gauss-Bonet theorem as,
( )2 .E
E
Kds
= (A.12)
This simplification helps link the fourth Minkowski functional to the Euler characteristic. From Eqs. (A.9)
and (A.12), we get,
( ) ( )3 2 .E E
E
Kds M
= = (A.13)
174
Further, the Euler characteristic for the surface element ( )( )E and the Euler characteristic for the solid
object ( )( )E are related as follows (Armstrong et al. 2018),
( ) ( )2 .E E = (A.14)
A few examples of 3-D objects and their Euler characteristics are provided in Table 17.
Table 17. Example of different objects and their Euler numbers.
Object M3
Euler number
(Minkowski definition:
A.13 and A.14)
Euler number (Betti
number definition:
χ = β0 – β1 + β2
Hollow sphere
4π 2 1 – 0 + 1 = 2
Solid sphere
4π 1 1 – 0 – 0 = 1
Hollow torus
0 0 1 – 2 + 1 = 0
Solid torus
0 0 1 – 1 + 0 = 0
χ is the Euler characteristic of the object; β0 is the number of objects; β1 is the number of
handles or loops in the objects; β2 is the number of empty cavities in the objects.
175
Appendix B. A consistent approach for χmax determination
The values of χmin and χmax for a porous medium are used in this research to normalize the Euler
characteristic as follows,
.ˆ
max
min max
−=
− (B.1)
The values of χmin and χmax are calculated differently depending on whether experimental data are used, or
pore-network models are used. When using image data, software like Avizo is used for Euler characteristic
estimation of a phase. For example, χmin is estimated for the pore space using the value of V-E+F-O for the
Euler number, where V is the number of vertices; E is the number of edges; F is the number of faces; and
O is the number of objects (voxels). The parameter χmax, however, cannot be directly estimated from image
data. It must be extracted from a pore-network of the dry image of the pore space. The pore-network gives
the information of the np (number of pores) and nt (number of throats). The parameter χmax is then the np
value.
Using the pore-network information, χmin can also be calculated as np – nt from PNM, which we refer
to as χmin-PNM. However, χmin when estimated by Avizo (χmin-image) may not be the same as χmin-PNM. This is
because, the pore space can be different before and after generating the pore-network. A further problem is
that all of the intermediate χ values for different saturations is estimated using the value of V-E+F-O when
image data is used.
Overall, the following methods can be used for consistency.
1. Use the pore-network approach on each individual saturation to find χ for each saturation. Thus,
we use χmax-PNM and χmin-PNM from the pore-network over the pore space. This, however, is not the
usual method for estimating χ at any saturation as is done using imaging data.
176
2. Use the value V-E+F-O for all saturations as well as for the pore space to determine χmin-image. Then,
to find χmax, generate a pore-network of the pore space to find the coordination number (Z = 2nt/np)
instead of directly using χmax-PNM as the number of pores. The value of χmax-image is then determined
from the coordination number as,
.
1 2
mim image
max image Z
−
− =
−
(B.2)
177
Appendix C. Base regression code created for analysis presented in chapter 3
clc
clear all
close all
% % Created by Prakash Purswani
% %----------- Linear Regression with new equality conditions
% %-------4 Equality constraints
% % 1. kr = 1 at S = 1, xhat = 1
% % 2. kr = 0 at S = 0, xhat = 1
% % 3. dkr/dS = 0 at S = 1, xhat = 0
% % 4. dkr/dxhat = 0 at S = 0, xhat = 1
%----------Armstrong et al. (2016): Ca = 10^0
Snw = [0.08784 0.28827 0.30675 0.43963 0.47871 0.47904 0.66023 0.67106 0.76941 0.80114];
krnw = [0.154493616 0.416947583 0.451528114 0.576595911 0.620941523 0.631093263
0.817376274 0.827979284 0.902661087 0.923611531];
Ca = [1.627329193 1.42152634 1.377300437 1.299488153 2.057539683 1.214774557 1.447147458
1.400074764 1.041695422 0.897429262];
X = [2933 -2286 -2353 -5011 -7221 -5957 -10300 -10544 -10682 -10594];
%---------Planar equation fitting AS+BX+C = kr (lower end)
syms A1 B1 C1
for i = 1:3
eqn1(i) = A1*Snw(i) + B1*X(i) + C1 == krnw(i);
end
%-------Linear equation fitting (aS+b = X)
J1 = [Snw(1:3)' ones(3,1)];
178
j1 = regress(X(1:3)',J1);
%---------Planar equation fitting AS+BX+C = kr (upper end)
syms A2 B2 C2
for i = 1:3
eqn2(i) = A2*Snw(length(Snw)-3+i) + B2*X(length(Snw)-3+i) + C2 == krnw(length(Snw)-3+i);
end
%-------Linear equation fitting (aS+b = X)
J2 = [Snw(end-2:end)' ones(3,1)];
j2 = regress(X(end-2:end)',J2);
Sol1 = solve([eqn1(1), eqn1(2), eqn1(3)], [A1, B1, C1]);
ansA1 = double(Sol1.A1);
ansB1 = double(Sol1.B1);
ansC1 = double(Sol1.C1);
Sol2 = solve([eqn2(1), eqn2(2), eqn2(3)], [A2, B2, C2]);
ansA2 = double(Sol2.A2);
ansB2 = double(Sol2.B2);
ansC2 = double(Sol2.C2);
%---------For Xmax, kr = 0
syms m n
eq1 = ansA1*m + ansB1*n + ansC1 == 0;
eq2 = j1(1)*m - n + j1(2) == 0;
Sol3 = solve([eq1, eq2], [m, n]);
ansm = double(Sol3.m);
ansn = double(Sol3.n);
179
%---------For Xmin, kr = 1
syms f g
eq11 = ansA2*f + ansB2*g + ansC2 == 1;
eq22 = j2(1)*f - g + j2(2) == 0;
Sol4 = solve([eq11, eq22], [f, g]);
ansf = double(Sol4.f);
ansg = double(Sol4.g);
%------------ Pore structure Euler connectivity values identified
Xmax = ansn;
Xmin = ansg;
%---------------Tuning against Ca = 10^-4
%----------Armstrong et al. (2016): Ca = 10^-4
Snw = [0.09204 0.29338 0.31264 0.44507 0.48577 0.66348 0.67386 0.77207 0.80465 0.88334
0.900609];
krnw = [0.086075394 0.197302896 0.298669867 0.357956049 0.451574991 0.626190811
0.614427609 0.763573275 0.789863744 0.908177118 0.924545659];
Ca = [0.000492236 0.000400132 0.000380349 0.000347567 0.000371549 0.000974638 0.000835806
0.000555354 0.000523234 0.000336238 0.000286985];
X = [158 -400 -661 -1883 -2449 -3341 -3374 -4553 -5306 -7054 -7539];
xhat = (X-Xmax)/(Xmin-Xmax); % normalized Euler connectivity
%----------Linear Regression
X = [(ones(size(Snw)) - 2*xhat + xhat.^2)' (-2*Snw + Snw.^2 + xhat.*Snw)'];
xx = regress((krnw-xhat.*Snw)',X)
a = xx(1);
b = xx(2);
180
% --------------- Check the goodness of fit ---------------
vtest = a*(1 - 2*xhat + xhat.^2) + b*(-2*Snw + Snw.^2 + xhat.*Snw) + xhat.*Snw;
for i = 1:length(vtest)
if vtest(i) < 0
vtest(i) = 0;
end
end
Residual = vtest - krnw;
MSE = (sum((vtest - krnw).^2));
M = mean(vtest);
SStotal = (sum((vtest - M).^2));
R2_10_4 = 1 - MSE/SStotal;
Mean_Residual = mean(Residual);
count = 1;
for i = 0:0.1:1
count_i(count) = i;
mean_R(count) = Mean_Residual;
mean_for_plot(count) = 0;
count = count + 1;
end
% --------------- Plotting Surface ---------------
syms x y
h1 = ezsurf(a*(1 - 2*y + y.^2) + b*(-2*x + x.^2 + y.*x) + x.*y, [0, 1, 0, 1]);
h1 = findall(h1,'Type','Surface');
X1 = get(h1,'XData');
Y1 = get(h1,'YData');
Z1 = get(h1,'ZData');
181
%---------Setting negative data for kr in the surface to zero
for i = 1:length(Z1)
for j = 1:length(Z1)
if Z1(i,j) < 0
Z1(i,j) = 0;
end
end
end
figure
plot3(Snw,xhat,krnw,'k.','markersize',30,'linewidth',2)
hold on
contourf(X1,Y1,Z1)
alpha 0.9
shading interp
colorbar
xlabel('Phase saturation','fontsize',20)
ylabel('Normalized Euler characteristic','fontsize',20)
zlabel('Phase relative permeability','fontsize',20)
%title('Fitting Response Surface-Data-Armstrong et al 2017')
%legend('Experimental data')
set(gca,'fontsize',25,'linewidth',2)
% set(get(gca,'YLabel'),'Rotation',10);
% set(get(gca,'XLabel'),'Rotation',-6);
%zlim([0 1])
view(0,90)
grid on
%------------ Plot Error for each data point
figure
182
plot(Snw,Residual,'ko','markersize',10,'linewidth',2)
hold on
plot(count_i,mean_for_plot,'r--','markersize',10,'linewidth',2)
xlabel('Phase saturation','fontsize',20)
ylabel('Residual','fontsize',20)
set(gca,'fontsize',25,'linewidth',2)
% End of code %
183
Appendix D. Data used in chapter 3 (adapted from Armstrong et al. 2016)
Table 18. Data used in chapter 3 for nonwetting phase saturation, connectivity, and relative permeability (adapted from Armstrong et al. 2016).
-(log(NCa)) Snw Sw krnw χnw nw
0
0.09 0.91 0.15 2933 0.17
0.29 0.71 0.42 -2286 0.49
0.31 0.69 0.45 -2353 0.49
0.44 0.56 0.58 -5011 0.65
0.48 0.52 0.62 -7221 0.79
0.48 0.52 0.63 -5957 0.71
0.66 0.34 0.82 -10300 0.98
0.67 0.33 0.83 -10544 0.99
0.77 0.23 0.90 -10682 1.00
0.80 0.20 0.92 -10594 0.99
1
0.09 0.91 0.16 1116 0.28
0.29 0.71 0.41 -1140 0.42
0.31 0.69 0.45 -1426 0.44
0.44 0.56 0.57 -3458 0.56
0.48 0.52 0.61 -4190 0.60
0.67 0.33 0.84 -7084 0.78
0.77 0.23 0.92 -8206 0.85
0.80 0.20 0.93 -8583 0.87
0.88 0.12 0.97 -9461 0.92
0.90 0.10 0.97 -10484 0.99
2
0.09 0.91 0.15 591 0.32
0.29 0.71 0.40 -508 0.38
0.31 0.69 0.45 -1426 0.44
0.31 0.69 0.44 -1371 0.43
0.31 0.69 0.43 -1112 0.42
0.44 0.56 0.56 -2467 0.50
0.48 0.52 0.60 -3176 0.54
0.66 0.34 0.81 -5266 0.67
0.67 0.33 0.82 -5390 0.68
0.77 0.23 0.90 -6716 0.76
0.80 0.20 0.92 -7204 0.79
0.90 0.10 0.98 -9783 0.94
184
3
0.09 0.91 0.09 142 0.34
0.29 0.71 0.22 -399 0.38
0.31 0.69 0.32 -677 0.39
0.44 0.56 0.38 -1889 0.47
0.49 0.51 0.45 -2443 0.50
0.66 0.34 0.72 -3669 0.57
0.67 0.33 0.71 -3597 0.57
0.77 0.23 0.82 -4880 0.65
0.80 0.20 0.85 -5624 0.69
0.88 0.12 0.93 -7322 0.79
0.90 0.10 0.94 -7752 0.82
4
0.09 0.91 0.09 158 0.34
0.29 0.71 0.20 -400 0.38
0.31 0.69 0.30 -661 0.39
0.45 0.55 0.36 -1883 0.47
0.49 0.51 0.45 -2449 0.50
0.66 0.34 0.63 -3341 0.55
0.67 0.33 0.61 -3374 0.56
0.77 0.23 0.76 -4553 0.63
0.80 0.20 0.79 -5306 0.67
0.88 0.12 0.91 -7054 0.78
0.90 0.10 0.92 -7539 0.81
5
0.09 0.91 0.02 167 0.34
0.29 0.71 0.13 -62 0.35
0.31 0.69 0.23 -498 0.38
0.44 0.56 0.31 -1481 0.44
0.48 0.52 0.41 -2071 0.48
0.66 0.34 0.53 -2910 0.53
0.77 0.23 0.77 -4535 0.63
0.80 0.20 0.78 -5265 0.67
0.88 0.12 0.91 -7029 0.78
0.90 0.10 0.93 -7518 0.81
6
0.29 0.71 0.22 -62 0.35
0.44 0.56 0.31 -1288 0.43
0.48 0.52 0.27 -1839 0.46
0.66 0.34 0.56 -2905 0.53
0.67 0.33 0.78 -2966 0.53
0.77 0.23 0.68 -4097 0.60
0.81 0.19 0.71 -4797 0.64
0.88 0.12 0.87 -6509 0.75
0.90 0.10 0.89 -6970 0.77
Table 16. (Continued)
185
Appendix E. Procedure for developing iso-quality curves discussed in chapter 4
Figure 51. Schematic showing the procedure for developing the iso-quality curves discussed in chapter 4 (Figure 27).
Iso-connectivityIso-saturation
2-D interpolation to find kr at gridded pointsIso-quality curves
Gridded data at equal -S spacingOriginal PNM data
186
Appendix F. Residual curves for different stopping criteria and wettability alteration in PNM
Figure 52 shows the initial-residual (IR) saturation curves generated using pore-network modeling for four
different contact angles. The contact angles here are reported through phase1 (receding phase during the
secondary process). Four different stopping criteria are shown for obtaining the residual saturations to
compare residual saturations that may be obtained from typical core flooding experiments where the
endpoint of the experiments is judged based on a very high fractional flow near 0.99. These include, three
kr stopping criteria of 10-2,10-3, and 10-5, while the fourth criterion corresponds to the high capillary pressure
of -105 Pa.
Figure 52. Initial-residual saturation curves generated from pore-network modeling for different contact angle cases measured through the receding phase. The residual saturations are obtained for different stopping criteria and are shown by different colors.
0
0.1
0.2
0.3
0.4
0.5
0.6
0 0.2 0.4 0.6 0.8 1
S r
Si
Stop at kr = 10^-2
Stop at kr = 10^-3
Stop at kr = 10^-5
Stop at Pc = -10^5 Pa
0
0.1
0.2
0.3
0.4
0.5
0.6
0 0.2 0.4 0.6 0.8 1
S r
Si
0
0.1
0.2
0.3
0.4
0.5
0.6
0 0.2 0.4 0.6 0.8 1
S r
Si
0
0.1
0.2
0.3
0.4
0.5
0.6
0 0.2 0.4 0.6 0.8 1
S r
Si
θ1 = 180o θ1 = 120o
θ1 = 60o θ1 = 0o
187
Figure 52 shows that residual curves depend on the criteria adopted for marking the residual state. The
capillary pressure-based criterion shows the lowest values of the residual saturation for all contact angles
and for all initial oil saturations, suggesting the ultimate residual saturations at those respective conditions.
Further, the difference in the residual saturations is significant for lower receding phase contact angles of
60° and 0°. This is because at these contact angles pore-scale flow regimes of layer flow and piston-like
advance by the advancing phase play a crucial role in driving low saturations. All four contact angle cases
in Figure F1 show a deviation (around Si ~ 0.35) in the residual saturation curves for the kr stopping criterion
cases. This is because for such low initial oil saturation the phase is not able to flow and as such the relative
permeability remains negligible.
Figure 53 shows capillary pressure scanning curves for two different contact angles generated using
pore-network modeling. For both sets, the primary phase1 injection was conducted at a receding contact
angle of 180°, while during the water injection step, the receding contact angles were 180° (no alteration)
and 30° (after wettability alteration). The capillary pressure for a no wettability alteration remains positive
while a sudden shift in the capillary pressure for the 30° case is observed after wettability alteration.
Figure 53. Primary drainage and water injection capillary pressure scanning curves generated using pore-network modeling. The different colors (and the arrows mark) give different cycles of injection. These scanning curves are generated with the endpoint of primary drainage (or starting point of water injection) at Si = 0.9.
Primary phase1 injection θ1 = 180o
Primary phase2 injection θ1 = 180o
Primary phase2 injection θ1 = 30o
Si = 0.9
(Pa)
188
Appendix G. Base code for running PNM for generating numerical data sets for chapters 4 and 5
clc
clear all
close all
fclose('all');
% Created by Prakash Purswani
cd C:\Users\pxp5185 % Location for input files of extracted pore network
% Extract pore network’s Euler characteristic (Xmax and Xmin)
[np, nt] = getNpNt('Bentheimer_Lin/Bentheimer'); % get these from _node1.dat and _link1.dat files
Z = 2*nt/np;
Xmax = np;
Xmin = np - nt;
cd C:\Users\pxp5185\S_01_02_remaining_Copy_2_PD_0 % Current location
% Subroutine for running PNM code (User can keep/remove secondary drainage as per requirement)
counter = 1;
for ii = 0:0.1:0.9
filecontent = fileread('input_pnflow.dat');
newcontent = regexprep(filecontent,'0.000', sprintf('%.3f',ii));
fid = fopen('input_pnflow_run.dat', 'w');
fwrite(fid, newcontent);
fclose(fid);
system('"pnflow_tom.exe" input_pnflow_run.dat');
movefile ('C:\Users\pxp5185\S_01_02_remaining_Copy_2_PD_0\*.m', 'Theta_0_S_0.1_0.2')
movefile ('C:\Users\pxp5185\S_01_02_remaining_Copy_2_PD_0\Bentheimer_out.prt',
'Theta_0_S_0.1_0.2')
movefile ('C:\Users\pxp5185\S_01_02_remaining_Copy_2_PD_0\*.txt', 'Theta_0_S_0.1_0.2')
189
movefile ('C:\Users\pxp5185\S_01_02_remaining_Copy_2_PD_0\Bentheimer_res\*.vtu',
'Theta_0_S_0.1_0.2')
sub_folder = sprintf('S=%.3f',ii);
mkdir(['C:\Users\pxp5185\S_01_02_remaining_Copy_2_PD_0\Theta_0_S_0.1_0.2\',sub_folder]);
cd Theta_0_S_0.1_0.2
movefile ('C:\Users\pxp5185\S_01_02_remaining_Copy_2_PD_0\Theta_0_S_0.1_0.2\*.vtu', sub_folder)
movefile ('C:\Users\pxp5185\S_01_02_remaining_Copy_2_PD_0\Theta_0_S_0.1_0.2\*.txt', sub_folder)
movefile ('C:\Users\pxp5185\S_01_02_remaining_Copy_2_PD_0\Theta_0_S_0.1_0.2\*.m', sub_folder)
movefile ('C:\Users\pxp5185\S_01_02_remaining_Copy_2_PD_0\Theta_0_S_0.1_0.2\*.prt', sub_folder)
files = dir;
directoryNames = {files([files.isdir]).name};
for k = 3:length(directoryNames)
currD = files(k).name;
cd(currD)
% kr and S measurements
Bentheimer_cycle1_drain;
Bentheimer_cycle2_imb;
Bentheimer_cycle3_drain;
[row_cycle1,cols_cycle_1] = size(Res_draincycle_1);
for i = 1:row_cycle1
Sw_PD(counter,i) = Res_draincycle_1(i,1);
Pc_PD(counter,i) = Res_draincycle_1(i,2);
krw_PD(counter,i) = Res_draincycle_1(i,3);
kro_PD(counter,i) = Res_draincycle_1(i,4);
end
[row_cycle2,cols_cycle_2] = size(Res_imb_2);
for i = 1:row_cycle2
190
Sw_I(counter,i) = Res_imb_2(i,1);
Pc_I(counter,i) = Res_imb_2(i,2);
krw_I(counter,i) = Res_imb_2(i,3);
kro_I(counter,i) = Res_imb_2(i,4);
end
[row_cycle3,cols_cycle_3] = size(Res_draincycle_3);
for i = 1:row_cycle3
Sw_SD(counter,i) = Res_draincycle_3(i,1);
Pc_SD(counter,i) = Res_draincycle_3(i,2);
krw_SD(counter,i) = Res_draincycle_3(i,3);
kro_SD(counter,i) = Res_draincycle_3(i,4);
end
% Subroutine for extracting Euler number information
count = 1;
EulerFiles_PD = dir('*1_OInj*.vtu'); % For Primary Drainage cycle
for n = 1:length(EulerFiles_PD)
filename = EulerFiles_PD(n).name;
filetext = fileread(filename);
addpath('C:\Users\pxp5185')
ffaz_data = int8(getVtuData( filetext, 'PointData', 'ffaz'))-1;
connectivity_data = getVtuData( filetext, 'Cells', 'connectivity');
sumNhtO = 0;
sumNhtO_NW = 0;
npnt2=np+nt+2;
count2 = 1;
for i = 0:nt-1
iPt(count2) = connectivity_data(i*3+3)+1;
iP1(count2) = connectivity_data(i*3+1)+1;
iP2(count2) = connectivity_data(i*3+2)+1;
191
ffT(count2) = ffaz_data(iPt(count2));
ffP1(count2) = ffaz_data(iP1(count2));
ffP2(count2) = ffaz_data(iP2(count2));
if(ffT(count2) == 0)
sumNhtO = sumNhtO +((ffP1(count2)==0) && (iP1(count2) <= npnt2)) + ((ffP2(count2)==0) &&
(iP2(count2) <= npnt2));
end
if(ffT(count2) == 1)
sumNhtO_NW = sumNhtO_NW +((ffP1(count2)==1) && (iP1(count2) <= npnt2)) +
((ffP2(count2)==1) && (iP2(count2) <= npnt2));
end
count2 = count2 + 1;
end
poresffaz=ffaz_data(3:np+2);
throatffaz=ffaz_data(np+3:np+nt+2);
Euler_NW_PD(counter,count) = sum(poresffaz == 1) + sum(throatffaz == 1) - sumNhtO_NW;
Euler_W_PD(counter,count) = sum(poresffaz == 0) + sum(throatffaz == 0) - sumNhtO;
count = count + 1;
end
count = 1;
EulerFiles_I = dir('*2_WInj*.vtu'); % For imbibition cycle
for n = 1:length(EulerFiles_I)
filename = EulerFiles_I(n).name;
filetext = fileread(filename);
addpath('C:\Users\pxp5185')
ffaz_data = int8(getVtuData( filetext, 'PointData', 'ffaz'))-1;
connectivity_data = getVtuData( filetext, 'Cells', 'connectivity');
sumNhtO = 0;
sumNhtO_NW = 0;
npnt2 = np+nt+2;
count2 = 1;
192
for i = 0:nt-1
iPt(count2) = connectivity_data(i*3+3)+1;
iP1(count2) = connectivity_data(i*3+1)+1;
iP2(count2) = connectivity_data(i*3+2)+1;
ffT(count2) = ffaz_data(iPt(count2));
ffP1(count2) = ffaz_data(iP1(count2));
ffP2(count2) = ffaz_data(iP2(count2));
if(ffT(count2) == 0)
sumNhtO = sumNhtO +((ffP1(count2)==0) && (iP1(count2) <= npnt2)) + ((ffP2(count2)==0) &&
(iP2(count2) <= npnt2));
end
if(ffT(count2) == 1)
sumNhtO_NW = sumNhtO_NW +((ffP1(count2)==1) && (iP1(count2) <= npnt2)) +
((ffP2(count2)==1) && (iP2(count2) <= npnt2));
end
count2 = count2 + 1;
end
poresffaz = ffaz_data(3:np+2);
throatffaz = ffaz_data(np+3:np+nt+2);
Euler_NW_I(counter,count) = sum(poresffaz == 1) + sum(throatffaz == 1) - sumNhtO_NW;
Euler_W_I(counter,count) = sum(poresffaz == 0) + sum(throatffaz == 0) - sumNhtO;
count = count + 1;
end
count = 1;
EulerFiles_SD = dir('*3_OInj*.vtu'); % For Secondary Drainage cycle
for n = 1:length(EulerFiles_SD)
filename = EulerFiles_SD(n).name;
filetext = fileread(filename);
addpath('C:\Users\pxp5185')
ffaz_data = int8(getVtuData( filetext, 'PointData', 'ffaz'))-1;
connectivity_data = getVtuData( filetext, 'Cells', 'connectivity');
sumNhtO = 0;
193
sumNhtO_NW = 0;
npnt2=np+nt+2;
count2 = 1;
for i = 0:nt-1
iPt(count2) = connectivity_data(i*3+3)+1;
iP1(count2) = connectivity_data(i*3+1)+1;
iP2(count2) = connectivity_data(i*3+2)+1;
ffT(count2) = ffaz_data(iPt(count2));
ffP1(count2) = ffaz_data(iP1(count2));
ffP2(count2) = ffaz_data(iP2(count2));
if(ffT(count2) == 0)
sumNhtO = sumNhtO +((ffP1(count2)==0) && (iP1(count2) <= npnt2)) + ((ffP2(count2)==0) &&
(iP2(count2) <= npnt2));
end
if(ffT(count2) == 1)
sumNhtO_NW = sumNhtO_NW +((ffP1(count2)==1) && (iP1(count2) <= npnt2)) +
((ffP2(count2)==1) && (iP2(count2) <= npnt2));
end
count2 = count2 + 1;
end
poresffaz=ffaz_data(3:np+2);
throatffaz=ffaz_data(np+3:np+nt+2);
Euler_NW_SD(counter,count) = sum(poresffaz == 1) + sum(throatffaz == 1) - sumNhtO_NW;
Euler_W_SD(counter,count) = sum(poresffaz == 0) + sum(throatffaz == 0) - sumNhtO;
count = count + 1;
end
cd C:\Users\pxp5185\S_01_02_remaining_Copy_2_PD_0\Theta_0_S_0.1_0.2
rmdir(['C:\Users\pxp5185\S_01_02_remaining_Copy_2_PD_0\Theta_0_S_0.1_0.2\',sub_folder],'s');
end
counter = counter + 1;
cd C:\Users\pxp5185\S_01_02_remaining_Copy_2_PD_0
end
% Calculating final outputs
194
Snw_PD = 1- Sw_PD;
Xhat_NW_PD = (Euler_NW_PD-Xmax)/(Xmin-Xmax);
Xhat_W_PD = (Euler_W_PD-Xmax)/(Xmin-Xmax);
Snw_I = 1- Sw_I;
Xhat_NW_I = (Euler_NW_I-Xmax)/(Xmin-Xmax);
Xhat_W_I = (Euler_W_I-Xmax)/(Xmin-Xmax);
Snw_SD = 1- Sw_SD;
Xhat_NW_SD = (Euler_NW_SD-Xmax)/(Xmin-Xmax);
Xhat_W_SD = (Euler_W_SD-Xmax)/(Xmin-Xmax);
% For making matrices vertical
Snw_PD = Snw_PD';
Snw_I = Snw_I';
Snw_SD = Snw_SD';
Sw_PD = Sw_PD';
Sw_I = Sw_I';
Sw_SD = Sw_SD';
Euler_NW_PD = Euler_NW_PD';
Euler_NW_I = Euler_NW_I';
Euler_NW_SD = Euler_NW_SD';
Euler_W_PD = Euler_W_PD';
Euler_W_I = Euler_W_I';
Euler_W_SD = Euler_W_SD';
Xhat_NW_PD = Xhat_NW_PD';
Xhat_NW_I = Xhat_NW_I';
Xhat_NW_SD = Xhat_NW_SD';
Xhat_W_PD = Xhat_W_PD';
Xhat_W_I = Xhat_W_I';
Xhat_W_SD = Xhat_W_SD';
kro_PD = kro_PD';
kro_I = kro_I';
kro_SD = kro_SD';
195
krw_PD = krw_PD';
krw_I = krw_I';
krw_SD = krw_SD';
Pc_PD = Pc_PD';
Pc_I = Pc_I';
Pc_SD = Pc_SD';
% Saving excel files in the same folder
writematrix(Snw_PD,'Snw_PD.csv')
writematrix(Snw_I,'Snw_I.csv')
writematrix(Snw_SD,'Snw_SD.csv')
writematrix(Sw_PD,'Sw_PD.csv')
writematrix(Sw_I,'Sw_I.csv')
writematrix(Sw_SD,'Sw_SD.csv')
writematrix(Xhat_NW_PD,'Xhat_NW_PD.csv')
writematrix(Xhat_NW_I,'Xhat_NW_I.csv')
writematrix(Xhat_NW_SD,'Xhat_NW_SD.csv')
writematrix(Xhat_W_PD,'Xhat_W_PD.csv')
writematrix(Xhat_W_I,'Xhat_W_I.csv')
writematrix(Xhat_W_SD,'Xhat_W_SD.csv')
writematrix(Pc_PD,'Pc_PD.csv')
writematrix(Pc_I,'Pc_I.csv')
writematrix(Pc_SD,'Pc_SD.csv')
writematrix(kro_PD,'kro_PD.csv')
writematrix(kro_I,'kro_I.csv')
writematrix(kro_SD,'kro_SD.csv')
writematrix(krw_PD,'krw_PD.csv')
writematrix(krw_I,'krw_I.csv')
writematrix(krw_SD,'krw_SD.csv')
% End of code %
196
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Curriculum Vita
Prakash Purswani
Education
Ph.D., Energy and Mineral Engineering: Petroleum & Natural Gas Engineering, 2021
Department of Energy and Mineral Engineering, The Pennsylvania State University, University Park, USA
M.S., Energy and Mineral Engineering: Petroleum & Natural Gas Engineering, 2017
Department of Energy and Mineral Engineering, The Pennsylvania State University, University Park, USA
B.E., Chemical Engineering, 2015
Department of Chemical Engineering, BITS-Pilani, Hyderabad, India
Work experience
- Petroleum Engineering Intern, Chevron Corporation (Summer, 2020)
- Course Instructor (PNGE 405), The Pennsylvania State University (Fall, 2019)
- Petroleum Engineering Intern, Chevron Corporation (Summer, 2019)
- Study Abroad Research Scholar, Imperial College London (Spring, 2019)
Awards
- Graduate Student Award, EME Department, Penn State (April, 2019)
- Holleran and Bowman Academic Excellence Award, EME Department, Penn State (February, 2019)
Contact: [email protected]
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