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University of Southern California Dissertations and Theses
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Effective flow and transport properties of deforming porous media and materials: theoretical modeling and comparison with experimental data
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Effective flow and transport properties of deforming porous media and materials: theoretical modeling and comparison with experimental data
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Content
Eective Flow and Transport Properties of Deforming Porous Media and
Materials: Theoretical Modeling and Comparison with Experimental Data
by
Samuel Richesson
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulllment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(CHEMICAL ENGINEERING)
May 2021
Copyright 2021 Samuel Richesson
Acknowledgements
I would like to acknowledge everyone who has played a signicant role in my academic career.
Firstly, my wife, who has since high school been by my side with love, support and encouragement.
It was her passionate pursuit of higher education that inspired mine. I would not be here without
her.
Secondly, my research advisor Professor Muhammad Sahimi. He has been kind and patient
in the process of guiding me in this research. I will be forever grateful for the privilege to learn
from someone as knowledgeable and as passionate as well as with a good sense of humor as him.
Thank you.
Finally, I would like to thank the rest of my committee members. Their encouragement and
curiosity has added greatly to the quality and scope of this work.
ii
Table of Contents
Acknowledgements ii
List of Tables v
List of Figures vi
Abstract ix
Chapter 1: Introduction 1
1.1 Background and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Macroscopic theories based on volume-averaging . . . . . . . . . . . . . . . 3
1.2.2 Mixture theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Scope of this work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Chapter 2: Mean-eld Theory of Deformation and the Eective-Medium Approx-
imation 6
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Hertz-Mindlin theory of contacting grains . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Evolution of the pore-size distribution . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 Eective-medium approximation for permeability . . . . . . . . . . . . . . . . . . . 12
Chapter 3: Predicting Eective Permeability of Deforming Porous Media 14
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2 Estimating the Parameters of the Model . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2.1 The Poisson's Ratio and the Pore-Size Distribution . . . . . . . . . . . . . . 15
3.2.2 The Young's Modulus of the Grains . . . . . . . . . . . . . . . . . . . . . . 17
3.3 Computational Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.4 Comparison with experimental permeability data . . . . . . . . . . . . . . . . . . . 20
3.4.1 Fontainebleau Sandstone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.4.2 Beaver Sandstone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.4.3 Berea Sandstone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.4.4 Boise Sandstone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.4.5 Cambrian Sandstones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.4.6 Fahler Sandstones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.4.7 Indiana Dark Sandstone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.4.8 Massillon Sandstones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.4.9 Miocene Sandstone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.4.10 Pliocene Sandstone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.4.11 Tensleep Sandstone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.4.12 Gulf Coast Sandstone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
iii
3.4.13 Torpedo Sandstone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.4.14 Triassic Sandstones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.4.15 Branford Sandstone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.4.16 Kirkwood Sandstone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.5.1 Limit of the Validity of the EMA . . . . . . . . . . . . . . . . . . . . . . . . 44
3.5.2 Eect of Structural Changes in the Pore Space . . . . . . . . . . . . . . . . 45
3.5.3 The Eect of a Grain-Size Distribution . . . . . . . . . . . . . . . . . . . . 46
3.5.4 Eect of the Deformation Mode . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Chapter 4: Predicting Eective Conductivity of Deforming Porous media 48
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2 Modications for predicting conductivity . . . . . . . . . . . . . . . . . . . . . . . . 49
4.3 Accounting for Surface Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.4 Computational Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.5 The Parameters of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.6 Comparison with experimental conductivity data . . . . . . . . . . . . . . . . . . . 55
4.6.1 Fontainebleau . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.6.2 Beaver Sandstone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.6.3 Berea Sandstones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.6.4 Boise Sandstone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.6.5 Cambrian Sandstones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.6.6 Fahler Sandstones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.6.7 Indiana Dark Sandstone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.6.8 Massillon Sandstones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.6.9 Miocene Sandstone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.6.10 Pliocene Sandstone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.6.11 Tensleep Sandstone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.6.12 Gulf Coast Sandstone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.6.13 Torpedo Sandstone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.6.14 Triassic Sandstones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.6.15 Branford Sandstone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.6.16 Kirkwood Sandstone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Bibliography 79
iv
List of Tables
4.1 Estimates of the Young's modulus G
e
and the Dukhin number of the sandstones. 57
v
List of Figures
2.1 (a) Conguration of two grains and the pore between them for estimating the new
pore radius. (b) The loaded zone for computing the pressure distribution, with
point A being on the contact surface. . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.1 (a) Various PSD distributions generated by Eq. (2.20), with r
min
and r
a
being
the minimum and average pore sizes. (b) The PSD that was utilized in most of the
computations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2 Eect of the PSD on the predicted permeabilities. r
a
is the average pore size. . . 17
3.3 Eect of the Poisson's ratio on the predicted permeabilities. . . . . . . . . . . . 18
3.4 Comparison of the predicted permeabilities with the experimental data for the
Fontainebleau sandstone. The tted value of the Young's modulusG
e
of the grains
is 40 GPa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.5 Comparison of the predicted permeabilities with the experimental data for the
Beaver sandstone. The tted value of the Young's modulus G
e
of the grains is 2.7
GPa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.6 Evolution of the pore-size distribution of the Beaver sandstone as a result of ap-
plying the external hydrostatic pressure P to the sandstone. . . . . . . . . . . . . . 23
3.7 Comparison of the predicted permeabilities with the experimental data for two
Berea sandstones. (a) Berea 100H with tted value of the Young's modulusG
e
25
GPa, and (b) Berea 500 with G
e
13 GPa. . . . . . . . . . . . . . . . . . . . . . . 25
3.8 Comparison of the predicted permeabilities with the experimental data for Boise
sandstone with the tted value of the Young's modulus G
e
38:5 GPa. . . . . . . 26
3.9 Comparison of the predicted permeabilities with the experimental data for three
Cambrian sandstones with the tted value of the Young's modulus being (a) G
e
10 GPa; (b) 12 GPa, and (c) 2 GPa. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.10 Comparison of the predicted permeabilities with the experimental data for six
Fahler sandstones with the tted value of the Young's modulus being (a)G
e
0:12
GPa; (b) 0.15 GPa; (c) 0.32 GPa; (d) 0.11 GPa; (e) 0.17 GPa; (f) 0.27 GPa. . . . . 31
vi
3.11 Comparison of the predicted permeabilities with the experimental data for the
Indiana DH sandstone with the tted value of the Young's modulus being G
e
30
GPa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.12 Comparison of the predicted permeabilities with the experimental data for the
Massillon DH, DV sandstone with the tted value of the Young's modulus being
G
e
9:5 GPa, and 1:7 GPa respectively. . . . . . . . . . . . . . . . . . . . . . . . . 34
3.13 Comparison of the predicted permeabilities with the experimental data for the
Miocene 7 sandstone with the tted value of the Young's modulus being G
e
5
GPa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.14 Comparison of the predicted permeabilities with the experimental data for the
Pliocene 35 sandstone with the tted value of the Young's modulus being G
e
4
GPa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.15 Comparison of the predicted permeabilities with the experimental data for the
Tensleep 35 sandstone with the tted value of the Young's modulus beingG
e
2:6
GPa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.16 Comparison of the predicted permeabilities with the experimental data for the
Tertiary 807 (Gulf Coast) sandstone with the tted value of the Young's modulus
being G
e
6:4 GPa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.17 Comparison of the predicted permeabilities with the experimental data for the
Torpedo sandstone with the tted value of the Young's modulus being G
e
3 GPa. 39
3.18 Comparison of the predicted permeabilities with the experimental data for the
Triassic sandstone with the tted value of the Young's modulusG
e
being (a) 8; (b)
6.5; (c) 83; (d) 40, and (e) 2.8, all in GPa. . . . . . . . . . . . . . . . . . . . . . . . 41
3.19 Comparison of the predicted permeabilities with the experimental data for the
Branford sandstone with the tted value of the Young's modulus being G
e
0:68
GPa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.20 Comparison of the predicted permeabilities with the experimental data for the
Kirkwood sandstone with the tted value of the Young's modulus being G
e
0:14
GPa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.1 Comparison of the predicted conductivities with the experimental data for the
Fontainebleau sandstone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.2 Comparison of the predicted conductivities with the experimental data for the
Beaver sandstone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.3 Comparison of the predicted conductivities with the experimental data for two
Berea sandstones. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.4 Comparison of the predicted conductivities with the experimental data for the Boise
sandstone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
vii
4.5 Comparison of the predicted conductivities with the experimental data for the three
Cambrian sandstones. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.6 Comparison of the predicted conductivities with the experimental data for the four
Fahler sandstones. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.7 Comparison of the predicted conductivities with the experimental data for the
Indiana DH sandstone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.8 Comparison of the predicted conductivities with the experimental data for the
Massillon DH sandstone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.9 Comparison of the predicted conductivities with the experimental data for the
Miocene 7 sandstone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.10 Comparison of the predicted conductivities with the experimental data for the
Pliocene 35 sandstone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.11 Comparison of the predicted conductivities with the experimental data for the
Tensleep 35 sandstone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.12 Comparison of the predicted conductivities with the experimental data for the
Tertiary 807 sandstone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.13 Comparison of the predicted conductivities with the experimental data for the
Torpedo sandstone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.14 Comparison of the predicted conductivities with the experimental data for the
Triassic sandstone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.15 Comparison of the predicted conductivities with the experimental data for the
Branford sandstone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.16 Comparison of the predicted conductivities with the experimental data for the
Kirkwood sandstone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
viii
Abstract
This thesis presents a new theoretical approach for computing the dynamic eective properties
of porous media that are under deformation by a hydrostatic pressure P . Beginning with the ini-
tial pore-size distribution (PSD) of a porous medium before deformation, and given the Young's
modulus and Poisson's ratio of its grains, the model uses an extension of the Hertz{Mindlin theory
of contact between grains to compute the new PSD that results from applying the pressure P to
the medium. It then utilizes the updated PSD in the eective-medium approximation (EMA) to
estimate the eective property. In the case of permeability, this theory directly predicts the ex-
perimental data. For conductivity, we rst account for the contribution from surface conduction,
and then we utilize the theory to update the PSD and, hence, the pore-conductance distribution,
which is then used in the EMA to predict the pressure-dependence of the conductance. In this
work we have used the theory to compute the eective permeability and eective electrical con-
ductivity(when saturated with brine) of twenty-nine dierent sandstones. Comparison between
the predictions and experimental data indicates agreement between the two that ranges from very
good to excellent.
ix
Chapter 1
Introduction
1.1 Background and motivation
The focus of this work is on the eect of deformation in porous media and, in particular, on
the deformation's eect on their
ow properties. This is a phenomenon that is encountered in a
wide variety of contexts and is therefore of fundamental interest to geologists, reservoir engineers,
and groundwater scientists. Examples include consolidating clays (Brown and Brindley, 1980),
geological formations deep underground where the pressure is large (Iliev et al., 2008; Fagbemi et
al., 2018), geothermal, coal-bed methane, oil and gas reservoirs, as well as unconventional energy
resources, such as shale formations. For example, to extract hot water from geothermal reservoirs
at economically attractive rates, their porosity and permeability must be such that the volume
ow rate of water is on the order of tens of m3 per hour or larger (Heiland, 2003). Often though,
such reservoirs have a low permeability and would need stimulation by, for example, hydraulic
fracturing in order to produce hot water economically. Aside from laboratory-scale experiments on
the eect of deformation on the eective permeability that began by Zoback and Byerlee (1975),
who reported on the measurement of the permeability of granite under axial stress over a range of
eective conning pressures, one also needs a predictive model that can provide accurate estimates
for the pressure- or stress-dependence of the eective permeability. Although the problem has
1
been studied experimentally by numerous groups (see, for example, Zhu and Wong, 1997; Keaney
et al., 1998; Ruisten et al., 1999; Ngwenya et al., 2003; Fossein et al., 2007; Baud et al., 2012;
Ballas et al., 2015; Liu et al., 2018; Yang and Hu, 2020), an accurate model for predicting the
pressure-dependence of the permeability is still lacking.
A related phenomenon is deformation of shale formations (Ibanez and Kronenberg, 1993),
an important unconventional energy source, under compression and varying conning pressure.
When shale gas accumulates in rock, it causes very high pressure as a result of the action of
high-pressure gas, coupled with ground stress in the seam, implying that the surrounding rock is
always under high conning stress. Due to their low porosity and permeability, shale formations
will not produce, unless they are subject to hydraulic fracturing, which generates a fracture
network, with the induced fractures intersecting the natural ones (Osborn et al., 2011). After
producing for a while, production of gas decreases, leading to repeated fracturing. Afterwards the
pressures of the shale gas and the fracturing
uid both decrease as the high-pressure fracturing
uid is discharged. Therefore, during the entire process, shale formations are subject to cyclic
loading and water pressure, hence giving rise to stress|permeability coupling (Jiang et al., 2018).
Deformation of coal-bed methane reservoirs represents another example in which the permeability
of the formations varies under uniaxial or triaxial stress conditions. Several groups have developed
theoretical models for the eect of an external load on the porosity and permeability of coals as
they vary with the stress or pressure (see, for example, Liu and Harpalani, 2013; Wu et al.,
2018; Mathias et al., 2019). The cleats in coal formations contribute most to their permeability.
Reducing the pore pressure increases the eective stress, leading to a reduction in the apertures
of the coal cleats and, hence, a reduction in the porosity and permeability.
However, the problem is more general, as there is a wide class of deformable porous media
whose
ow and transport properties, when they are subject to an external load, vary with the
magnitude of that load - pressure or stress. They range anywhere from polymers and hydrogels
(Iritani et al., 2006; Karada, 2010; Sweijen et al., 2017), layers of particles that build up on
2
the surface of lters (Sahimi and Imdakm, 1991; Imdakm and Sahimi, 1987, 1991), and printing
papers (Ghassemzadeh et al., 2001; Ghassemzadeh and Sahimi, 2004; Masoudi and Pillai, 2010),
to diapers (Savoji and Pourjavadi, 2006; Salimi et al., 2010), and foams (Koehler et al., 2000;
Pitois et al., 2009). Another important example of much current interest is deformation and
swelling of geological formations, such as oil and gas reservoirs that no longer produce, as a
result of injecting CO
2
into their pore space. Recent molecular dynamics simulations indicated
(Rahromostaqim and Sahimi, 2018, 2019), in the presence of brine, CO2 causes swelling of pure
and mixed clays that may eventually lead to earthquakes (Maxwell et al., 2008; Tafti et al., 2013;
Rother et al., 2013; Lee et al., 2016).
1.2 Previous Work
1.2.1 Macroscopic theories based on volume-averaging
Physical properties of porous media depend strongly on the properties of their morphol-
ogy, namely, their pore shape, pore-size distribution (PSD), and pore connectivity. Deformation
changes the morphology and, therefore, the macroscopic physical properties. Therefore, a main
goal of studying deformation of porous media is predicting such properties as a function of the
driving force for deformation, such as an external pressure or stress. The foundation for modeling
of
ow and transport in deforming porous medium is provided by the momentum balance and the
equations that govern elastic deformation of solids, together with mass balance and appropriate
constitutive and state equations. In a macroscopic approach to the problem, one averages the
microscopic conservation laws over a suitable volume of a porous formation and supplements the
theory with empirical or semi-empirical constitutive relations for
ow, transport and mechanical
properties. Biot (1941, 1956) pioneered this approach, which has also been extended in order to
study the same phenomena involving nonlinear material behavior (see, for example, Zienkiewicz
3
et al. 1984, 1990; Schre
er et al. 1998; Li et al. 2004; Zhang et al. 2009), as well as when the
pore space is only partially saturated by a
uid in the presence of a second
uid (see, for example,
Khoei and Mohammadnejad, 2011; Pesavento et al., 2017).
1.2.2 Mixture theories
Another macroscopic approach is the so-called mixture theory (see, for example, Atkin and
Craine, 1976a,b; Bowen, 1982; Murad and Cushman, 1996; Huyghe and Janssen, 1997; Cowin
and Cardoso, 2012; Huyghe et al., 2017), which was derived by averaging the microscopic equa-
tions of mass, momentum and energy over a suitable length scale. The entropy inequality that
describes the direction of dissipation of energy due to deformation was not, however, invoked for
deriving the macroscale equations. As a result, the relationships that link macroscopic thermody-
namic variables to the properties of porous media cannot be derived directly. Hassanizadeh and
Gray (1979a,b, 1990) combined the two aforementioned macroscopic approaches together with
the entropy inequality in order to derive a generalization of Darcy's law for the
ow eld in a
deformable porous medium (see also Weinstein et al., 2008; Zhu et al., 2010). Since
ow and
transport in heterogeneous porous media are controlled by their morphology, a few computational
approaches have also been developed that carry out numerical simulations either in the image of
porous media, or in a model of them (see, for example, Zhu and Wong, 1999; Boutt and McPher-
son, 2002; Arns et al. 2001, 2002; Dautriat et al., 2009; Thovert and Adler, 2011; Jasinski et
al., 2015; Bakhshian and Sahimi, 2016; Bakhshian et al., 2018; Fagbemi et al., 2018) in order to
simulate the changes in the morphology of porous media as they undergo deformation.
4
1.3 Scope of this work
In this research we develop a relatively simple, and to our knowledge, new theoretical model
for predicting the eective permeability of porous media that deform under an external hydro-
static pressure P . The model combines a theory of deformation of contacting grains under an
external force with the eective-medium approximation (EMA) in order to predict the eective
permeability as a function of P . Bruggeman (1935) was the rst to develop an EMA for esti-
mating the macroscopic properties of heterogeneous media. The same formulation was developed
independently by Landauer (1952) for computing electrical conductivity of composite solids. The
EMA was used by Koplik et al. (1984) to predict the eective permeability of a porous medium.
In addition, other theories have been suggested for predicting the permeability. For example,
Wadsworth et al. (2016) proposed a universal scaling of the permeability, which is based on the
power laws of percolation theory. But, all the past works, including those that utilized the EMA,
were for predicting physical properties of rigid heterogeneous media (for comprehensive reviews
see Sahimi, 2003, 2011). What we present is, however, the application of a novel combination
of the EMA and a theory of deformation for computing the macroscopic permeability of porous
media that undergo deformation, when a hydrostatic pressure P is applied to them.
5
Chapter 2
Mean-eld Theory of Deformation and the
Eective-Medium Approximation
2.1 Introduction
As mentioned in Chapter 1, we employ the EMA to predict the eective permeability of porous
media that deform as a result of applying an external pressure or stress. The EMA is a sort
of mean-eld approximation (MFA) that replaces a heterogeneous porous medium by an eective
one in which all the pores have the same eective size r
e
. The solution of the
ow problem in the
uniform system is straightforward. Then, one pore in the eective medium is selected at random
and its radius in the original disordered medium is restored, with the rest of the pores still having
the same size r
e
. This generates a perturbation in the solution of the uniform system, whose
magnitude is the dierence between the solution for the uniform medium and one that is uniform
everywhere, but in one pore. The perturbation is calculated, and since the single pore is selected
at random and its size follows a PSD, in order to be able to still represent the disordered medium
with an eective pore radius r
e
and an eective permeability K
e
, one insists that the average of
the perturbation, when the averaging is taken over the PSD, should be zero. In eect, only the
interaction of a single pore with the rest of the eective medium is taken into account, and the
in
uence of the remaining part of the disordered porous medium is represented by the far-eld
6
external pressure gradient. Extensions of the EMA that account for interactions of a pore with
those that are farther than the nearest-neighbor ones were also developed (Sahimi et al., 1983,
1984), but we ignore them in this work.
Thus, in the spirit of the EMA, and for the sake of developing a tractable theoretical approach
that is consistent with the mean-eld nature of the EMA, we consider the interaction between two
grains, the minimum number of grains inan MFA, when an external force F or the corresponding
hydrostatic pressureP is applied to the medium, and determine the deformation that it causes in
the pore between the two grains that changes in size. The eect of the deformation of the rest of a
porous medium is represented by the far-eld applied pressureP or force. The pressureP changes
the PSD of the pore space that, in turn, in
uences its eective
ow and transport properties, and
in particular its eective permeability K
e
. Since one important input to the EMA is the PSD of
the pore space, the rst step toward the goal of predicting the dependence of K
e
on the external
pressure P is to determine the changes in the PSD as P is gradually increased.
2.2 Hertz-Mindlin theory of contacting grains
To determine the interaction between two neighboring grains and the deformation of the pore
space between them, we derive a theory of deformation of contacting grains, rst studied by Hertz
(1892) who assumed no friction between the grains. Goodman (1962) studied the case with friction
between the two grains, while Mindlin (1949) considered the case in which tangential forces and
twisting were coupled at the contact point between two grains. The derivation that we present is
similar to the analysis of Timonshenko and Goodier (1970), but we present the complete details,
simplify the analysis, and our nal results were not actually presented by them. We consider a
typical, or average, grain size R
g
, and normalize all the length scale with respect to it. Consider,
then, Fig. 2.1(a) that shows two grains, 1 and 2, which, in the absence of any applied external
pressure, are in contact at point C along the Z axis. We assume that the grains' surfaces at the
7
Figure 2.1: (a) Conguration of two grains and the pore between them for estimating the new
pore radius. (b) The loaded zone for computing the pressure distribution, with point A being on
the contact surface.
point of contact have radii of curvaturve R
1
andR
2
. If the grains are roughly spherical, then, R
1
andR
2
also represent roughly their radii. Consider the plane tangent atC, and two pointsA and
B that are on the front sections of the surfaces 1 and 2 at a small distancex from theZ axis. The
distancesz
1
andz
2
ofA andB from the tangent plane satisfy the relations,pR
1
z
1
q
2
x
2
R
2
1
,
and pR
2
z
2
q
2
x
2
R
2
2
. Assuming that z
1
and z
2
are small enough that z
2
1
and z
2
2
can be
ignored, we obtain, z
1
x
2
{p2R
1
q and z
2
x
2
{p2R
2
q. Therefore, the distance D z
1
z
2
between A and B is given by
D
R
1
R
2
2R
1
R
2
x
2
: (2.1)
We now apply an external force F to the porous medium to press the grains together, which
causes local deformation near C over a small, roughly circular (spherical) surface, which we refer
to as the contact surface (CS). If the deformation is small, we may assume that both R
1
and
R
2
are much larger than the radius of the CS. Suppose that u
A
and u
B
are, respectively, the
displacements of A and B due to the local deformation along the Z axis. Then, if the tangent
plane at C is held xed, the local compression causes any two points on the surface of the two
grains, which are far from C, to move toward each other by an amount u, implying that the
8
distance between A and B decreases by upu
A
u
B
q and, therefore, the eective pore radius
between the two grains also decreases by u{2. Thus, if the compression caused by applying the
force F brings A and B into the CS, we must have,
upu
A
u
B
qx
2
; with
R
1
R
2
2R
1
R
2
; (2.2)
if we use Eq. (2.1). Thus,
uu
A
u
B
x
2
: (2.3)
Therefore, we must determineu
A
andu
B
in order to computeu, i.e., the decrease in the distance
between non-contacting surfaces of two grains, which leads directly to the change in the size of
the pore between them and, hence, the change in the PSD of the deforming porous medium can
be determined.
Consider a small element of the loaded zone, shown by the shaded area in Fig. 2.1(b) that is
bounded between radii s and sds and angle d where, as shown in Fig. 2.1(a), A is a point on
the CS. If p is the local pressure in the CS, then, the displacement u
A
is given by (Timoshenko
and Goodier 1970),
u
A
1
2
1
G
1
» »
pdsd; (2.4)
which follows directly from the theory of displacement of a spherical grain, where G
1
and
1
are,
respectively, the elastic (Young's) modulus and Poisson's ratio of the grain. A similar equation
also holds for u
B
:
u
B
1
2
2
G
2
» »
pdsd: (2.5)
9
Note that the two equations for u
A
and u
B
are subject to the condition that both A and B are
on the CS. Therefore, by substituting the expressions for u
A
and u
B
in Eq. (2.3), we obtain
1
1
2
1
G
1
1
2
2
G
2
» »
pdsdux
2
: (2.6)
It remains to calculatep, the local pressure distribution over the CS. If a hemisphere of radius
R
c
is constructed on the CS, then building on the Hertz-Mindlin work, we argue that the pressure
distribution is represented by the hemisphere's ordinates. This implies that the pressure p
0
at the
center of the CS (i.e., the maximum pressure in the CS) is simply proportional to R
c
and is given
by,p
0
aR
c
, wherea is a scale factor. As shown in Fig. 2.1(b), the local pressure p varies over a
chord mn, shown by the dashed semicircle. Therefore,
³
pdsp
0
S{R
c
, with S being the area of
the semicircle. Since
S
1
2
pR
2
c
x
2
sin
2
q; (2.7)
then, substituting for
³
pds and S in Eq. (2.6), yields
p
0
R
c
1
2
1
G
1
1
2
2
G
2
»
{2
0
pR
2
c
x
2
sin
2
qdux
2
: (2.8)
Carrying out the integration, we obtain
p
0
4R
c
p2R
2
c
x
2
q
1
2
1
G
1
1
2
2
G
2
ux
2
: (2.9)
Equation (2.9) is an identity in terms of x that must be valid for any of its values. This would be
possible if
R
c
p
0
4
1
2
1
G
1
1
2
2
G
2
; (2.10)
u
1
2
p
0
R
c
1
2
1
G
1
1
2
2
G
2
: (2.11)
10
To relate p
0
to the applied force F , we note that the sum of the pressures in the contact area
multiplied by its surface should be equal to F . Thus,pp
0
{R
c
qp
2
3
R
3
c
qF , or, p
0
3F{p2R
2
c
q,
which, after substituting in Eq. (2.10) and solving for R
c
, yields,
R
c
3F
8
1
2
1
G
1
1
2
2
G
2
1{3
: (2.12)
If we substitute Eq. (2.12) and the result for p
0
in Eq. (2.11), we nd that
u
1
2
3F
1
2
1
G
1
1
2
2
G
2
R
1
R
2
2R
1
R
2
1{2
2{3
: (2.13)
Assuming that the two grains are composed of the same materials, we have,
1
2
and
G
1
G
2
G
e
. We also assume that the two grains have roughly the same radii of curvature,
R
1
R
2
R. With the assumption of local isotropy, the local force is isotropic and homothetic,
i.e., it is a monotonic transformation ofF . Thus,F and the hydrostatic pressureP are related by,
F
?
2R
2
g
P
?
2P , with the second equation being due to normalization of lengths by R
g
(i.e.,
R
g
Ñ R
g
{R
g
1), where
?
2 is due to the geometrical considerations, as shown by Deresiewicz
(1958). Under these conditions, Eqs. (2.12) and (2.13) are simplied to
R
c
pRq
1{3
3
?
2Pp1
2
q
4G
e
1{3
; (2.14)
u
1
R
1{3
3Pp1
2
q
G
e
2{3
: (2.15)
Thus, writing u and R in un-normalized units, Eqs. (2.14) and (2.15) become,
R
c
R
g
R
R
g
1{3
3
?
2Pp1
2
q
4G
e
1{3
; (2.16)
uR
g
R
g
R
1{3
3Pp1
2
q
G
e
2{3
: (2.17)
11
Note that, R
g
R, if the grains are roughly spherical, which we assume to be the case or, at the
minimum, we can dene a radius for an equivalent spherical particle.
2.3 Evolution of the pore-size distribution
Within the framework of the MFA in which the interactions of two neighboring grains with
other grains farther away are ignored, the pore between the two grains does not also interact with
the pores farther away. Thus, as pointed out earlier, to a rst-order approximation, the eective
radius of a pore under an external hydrostatic pressure P decreases by u{2, where u is given by
Eq. (2.17). In other words, the initial PSD distribution f
0
pr
0
q before any pressure is applied
is transformed to a new PSD f
P
pr
P
q at pressure P where, r
P
r
0
u{2. If f
0
pr
0
q is given,
either analytically or numerically, then, since, f
P
pr
P
qf
0
pr
0
u{2q, one either has an analytical
expression for f
P
pr
P
q, or constructs it numerically for any pressure P .
2.4 Eective-medium approximation for permeability
A porous medium consists of pore throats connected together at the pore bodies. The eective
sizes of both the pore throats and pore bodies are distributed according to statistiscal distributions
f
t
pr
t
q and f
b
pr
b
q. It is, however, not straigthforward to measure f
b
pr
b
q and, thus, it is usually
not available. Thus, since the macroscopic permeability is controlled by the pore throats, for
conveniece we refer to the pore throats as pores, and their distribution fprq as the PSD. As
described in Sec. 2.2, in the EMA a heterogeneous pore space is represented by a uniform medium
with the size of all the pores being r
e
. We assume that the pores are cylindrical. Then, for slow
ow the
ow conductance K
f
is given by, K
f
9r
4
(other pore shapes may also be considered).
12
Note that it is possible to consider other pore shapes. The EMA predicts that the macroscopic
permeability K
e
is given by (Doyen, 1988; David et al., 1990)
K
e
C
s
r
4
e
xr
2
b
y
; (2.18)
with being the porosity, is the
ow tortuosity, C
s
is a geometrical factor such that C
s
8 for
cylindrical pores in Hagen-Pasiulle (slow or laminar)
ow, and r
b
is the size of the pore bodies.
Since the distribution f
b
pr
b
q of the size of the pore bodies is typically not available, David et
al. (1990) suggested that one should use, xr
2
b
y xr
2
y
³
r
M
rm
r
2
fprqdr, with r
m
and r
M
being,
respectively, the minimum and maximum pore radii; we do the same in this paper. r
4
e
is computed
by the EMA:
»
r
M
rm
r
4
e
r
4
r
4
pD 1qr
4
e
fprqdr 0; (2.19)
Here, D is the Euclidean dimensionality of the porous medium (D 3 in our calculations). If the
porosity of the porous medium is low enough that the pore space is near its critical porosity
c
or
the percolation threshold, i.e., the porosity at which the sample-spanning cluster of the pores is
barely connected, and the macroscopic permeability and electrical conductivity vanish for¤
c
,
then, as rst derived by Kirkpatrick (1971), one should useZ{2 in Eq. (2.19), instead ofD, where
Z is the mean connectivity of the pore space.
Mukhopadhyay and Sahimi (2000) derived an EMA for predicting the direction-dependent
macroscopic permeabilities of anisotropic porous media; Stroud (1975) presented a continuum
EMA for anisotropic media in which the local conductivity or permeability was a tensor; Ghan-
barian et al. (2016) utilized Eq. (2.19) to predict the relative permeability of water in soil in
the presence of air; Ghanbarian and Javadpour (2017) invoked Eq. (2.19) to estimate the gas
permeability in shales, while saturation-dependent electrical conductivity of partially-saturated
packings of spherical particles was computed by Ghanbarian and Sahimi (2017) using Eq. (2.19).
13
Chapter 3
Predicting Eective Permeability of Deforming Porous
Media
3.1 Introduction
Let us emphasize again that Eqs. (2.16) - (2.19) are not exact, but represent only MFAs to the
problem, which we now utilize to predict the pressure-dependence of the eective permeability
K
e
pPq of a large number of sandstones, and to compare the predictions with the experimental
data. Almost all the experimental data are given by Yale (1984), although as noted below, some of
them were not his measurements, but he had included them in his Doctoral Thesis for comparison
and completeness. Yale (1984) did not provide the sandstones' initial PSD and, therefore, as
mentioned earlier, we used in all the cases described below the PSD presented in Fig. 3.1(b) as
the initial PSD, f
0
pr
0
q, which was reported by Lindquist et al. (2000). The qualitative aspects
of the PSD that we utilize is similar to those for many sandstones, namely, that the distribution
is skewed; it has a maximum close to the smallest pores, and that it also has a relatively long
tail. Clearly, any PSD can be used in the theoretical formulation that we have developed. In the
discussions that follow, all the percentages and fractions that are mentioned are volumetric.
14
3.2 Estimating the Parameters of the Model
Let us rst point out that the model presented in Sec. 2.4 is an MFA. Therefore, similar to any
MFA, the
uctuations in the local properties are ignored, allowing one to analyze the behavior of
the system based on only two grains, the minimum number for a meaningful analysis. Similar to
all the MFAs, the approach has its limitations and strengths, which we will discuss in Sec. 3.5.
For now it suces to mention that since this is a two-grain MFA and, as a result, only an average
grain size is required. We will return to this point in Sec. 3.5.
3.2.1 The Poisson's Ratio and the Pore-Size Distribution
According to Eqs. (2.17) and (2.19), the parameters of the model are the Young's modulus
G
e
, the Poisson's ratio , and the PSD fprq. If experimental data are available for the three
parameters, they can be used directly in the theory. Unfortunately, for the sandstones that
we analyze, the information is not available. Thus, we need to make judicious choice of the
parameters. Our preliminary computations indicated that while the predictions of the model are
sensitive to the value of the Young's modulus, they only change mildly when and the PSD are
varied, which we now demonstrate. Consider, rst, the sensitivity of the predictions to the PSD.
To study this, we used the following theoretical PSD distribution,
fprq
rr
m
pr
a
r
m
q
2
exp
1
2
rr
m
r
a
r
m
2
; (3.1)
where r
a
is the average pore size. We xed the minimum pore size r
m
, and varied r
a
over two
orders of magnitude. Figure 3.1(a) presents the type of the PSD that Eq. (3.1) generates. The
distribution generated by the lowestr
a
in Fig. 3.1 has striking similarities with what was reported
by Fredrich et al. (1991), while those generated by other values of r
a
are qualitatively similar to
those reported by others. We then computed the eective permeability for one of the sandstones
15
0 50 100 150 200 250 300
Pore Size(microns)
0
0.05
0.1
Distribution
r
a
1
10
20
100
0 10 20 30 40 50 60 70 80
Pore Size(microns)
0
0.05
0.1
0.15
0.2
Distribution
(a)
(b)
Figure 3.1: (a) Various PSD distributions generated by Eq. (2.20), with r
min
and r
a
being the
minimum and average pore sizes. (b) The PSD that was utilized in most of the computations.
that we analyze later in this paper, namely, the Tensleep sandstone (see Sec. 3.4.11), xing all
the parameters, but varying the PSD. Figure 3.2 presents the results, where the permeability
is normalized by its value before deformation (see Sec. 3.4). The results do not indicate great
sensitivity to the PSD. Calculations for all the sandstones that we analyzed (see Sec. 3.4) indicated
the same trends. Thus, in the absence of any experimental data for the PSDs of the sandstones,
we used in all the cases described below the distribution presented in Fig. 3.1(b) as the initial
PSD, f
0
pr
0
q, which was reported by Lindquist et al. (2000) for a Fontainebleau sandstone, and
is similar to those for many other sandstones reported by others (see, for example, Cheung et al.,
2012 for Bleurswiller and Boise sandstones). Note that the distribution is also similar to what
Eq. (3.1) generates, and that the pore sizes vary over about two orders of magnitude, a relatively
broad distribution. Next, we studied the sensitivity of the predictions to the value of the Poisson's
ratio, . Once again, all the parameters but were xed, and the model was used to predict the
dependence on the applied pressure of the permeability of the same sandstone as Fig. 3.2. Figure
3.3 presents the results. The Poisson's ratio was varied by a factor of 4, and yet the predictions
vary by at most 2 percent. Calculations with all the other sandstones that we analyzed (see Sec.
16
0 50 100 150 200 250 300 350
Pressure (Bar)
0.7
0.75
0.8
0.85
0.9
0.95
1
Normalized Permeability
r
a
1
10
20
100
Figure 3.2: Eect of the PSD on the predicted permeabilities. r
a
is the average pore size.
3.4) indicated the same trends. Thus, we xed the Poisson's ratio at 0:3, which is in the
middle of the range for sandstones. Note that for pure quartz, 0:2.
3.2.2 The Young's Modulus of the Grains
Let us rst emphasize that the Young's modulus G
e
in Eq. (2.17) is not the modulus of
the porous medium as a whole, which depends on its porosity and for which an EMA has been
developed (Makse et al., 2001), but rather it is that of the grains, or the solid matrix of the porous
medium in the MFA, which should either be measured, estimated theoretically, or be treated as
an adjustable parameter. The grains are, however, hardly pure materials; they usually represent
composites composed of several components. If the composition of the solid matrix or grains
is known, then there are at least two theoretical approaches that can be used to estimate G
e
.
One method of estimating the elastic moduli of the solid matrix is through rigorous upper and
lower bounds. Over the years relatively tight bounds have been derived that provide reasonable
estimates of the elastic moduli of solid composites. These are described in detail by Torquato
17
300 305 310 315 320 325 330 335 340 345 350
Pressure (Bar)
0.75
0.76
0.77
0.78
0.79
0.8
0.81
0.82
0.83
0.84
0.85
Normalized Permeability
0.1
0.2
0.3
0.4
Figure 3.3: Eect of the Poisson's ratio on the predicted permeabilities.
(2002) and Sahimi (2003), to whom the interested reader is referred. The second approach is based
on the so-called self-consistent approximation (SCA) for the eective elastic moduli of a composite
material, rst developed by Budiansky (1965), Hill (1965), and Wu (1966), and developed further
by Berryman (1980), which is the analog of the EMA for the elastic moduli. For example, if we
assume that the solid matrix is composed of two components, say quartz and clay as in many
sandstones, such that the spatial distribution of clay (component 1) with volume fraction
1
is
represented by identical spheres dispersed in the background matrix made of quartz, component
2 with volume fraction
2
, then, according to the SCA the eective bulk modulus B
e
and shear
modulus
e
of the matrix are the solution of the following nonlinear coupled equations:
1
B
e
B
1
4
e
{3B
1
2
B
e
B
2
4
e
{3B
2
0; (3.2)
and
1
e
1
C
e
e
1
2
e
2
C
e
e
2
0; (3.3)
18
whereC
e
p9B
e
8
e
q{p6B
e
12
e
q. The case in which component 1 is spatially distributed in
the background matrix as elliptical particles has also been studied (Berrymnan, 1980). Equation
(3.2) and (3.3) are accurate, provided that
1
!
2
. Suppose, for example, that the matrix
consists of clays with a volume fraction of
1
0:22, for which
1
6:85 GPa and B
1
21
GPa, and quartz with volume fraction of
2
0:78 withB
2
138 GPa and
2
44 GPa. Then,
Eqs. (3.2) and (3.3) yield,
e
30:5 GPa and B
e
33:5 GPa. Thus, since the Poisson's ratio
is given by, pDB
e
2
e
q{rDpD 1qB
e
2
e
s, with D 3 being the dimensionality of the
space, we obtain, 0:15. The eective Young's modulus is then given by,G
e
2
e
p1q 70
GPA. On the other hand, suppose that a sandstone is composed of about 24 percent quartz
(component 1), the same as Fahler 154 analyzed in Sec. 3.4.6, while the rest is made of clay
(or any other compound much softer than quartz) as component 2. Then, Eqs. (3.2) and (3.3)
predict that, B
e
28:7 and
e
10:3, both in GPa, so that 0:28 and G
e
26 GPa, much
smaller than that of pure quartz, G 106 GPa. These are, of course, approximations, but they
do indicate that the Young's modulus of the grains depends strongly on their compositions. The
information on the exact composition of the solid matrix of the deforming sandstones that we
analyze in the present paper is not available. Therefore, in the absence of such information that
we could have used in, for example, Eqs. (3.2) and (3.3) to estimateG
e
for each deforming porous
medium that we analyze, we utilize a single experimental data point for the permeability at a
given pressure P in order to calibrate the model and estimate G
e
. The estimate is then utilized
for predicting K
e
pPq at all other pressures. It is, of course, well-known that the elastic moduli of
composite materials, including porous media's solid matrix, are functions of the applied pressure
P . Consider, however, the elastic moduli of quartz, which is typically the main component of
sandstones. Its elastic moduli do depend on P (Kondo et al., 1981; Wang et al., 2015), but
only at pressures much higher than those considered in the experiments described below. Thus,
ignoring the pressure-dependence ofG
e
is justied. For the predictions that we present below, we
19
used a data point at a pressure in the middle of the range of the pressures at which K
e
had been
measured.
3.3 Computational Procedure
Having developed the necessary theoretical tools for predicting the eective permeability of
deforming porous media, the computational procedure is as follows:
(i) We begin with an initial PSD f
0
pr
0
q of the porous medium to be deformed, and estimate its
initial permeability using Eqs. (2.18) and (2.19).
(ii) For an applied hydrostatic pressure P , we construct the PSD f
P
pr
P
q corresponding toP by
selecting the pore sizes from f
0
pr
0
q, calculating their updated values using Eq. (2.17) and
r
P
r
0
u{2, and repeating it for a large number of pore sizes selected from f
0
pr
0
q, in
order to construct a representative f
P
pr
P
q.
(iii) The resultingf
P
pr
P
q is then utilized to rst update the value ofxr
2
b
pPqyxr
2
P
y and then is
used together withf
P
pr
P
q in Eqs. (2.18) and (2.19) to computeK
e
pPq at the given pressure
P .
As pointed out earlier, if the PSD f
0
pr
0
q can be expressed by an analytical expression, such
as Eq. (3.1), then, so can also f
P
pr
P
q for any P in which case the computations are very fast.
3.4 Comparison with experimental permeability data
3.4.1 Fontainebleau Sandstone
Before presenting the predictions for the sandstones that Yale (1984) experimented on, we
present the results for a Fontainebleau sandstone, since its PSD was reported by Lindquist et
al. (2000), while the data for its pressure-dependent K
e
ppq were reported by Song and Renner
20
(2008). We found the best estimate for the Young's modulus of the sandstone that provides
accurate predictins for the permeability to be,G
e
40 GPa. Figure 3.4 compares the predictions
with the experimental data. Two points are noteworthy about the data. One is that they are
considerably scattered. In particular, the normalized permeability at a pressure of about 180 MPa
(the second data point from the left) is larger than 1. The second noteworthy feature is that the
data vary only in a narrow range (0.85,1) and, therefore, the apparent disagreement between the
predictions and the data is supercial. In fact, the maximum error of the predictions is only 4.4
percent, while the average error is only 2 percent. Thus, it is clear that the theory is capable of
providing accurate predictions for the normalized permeability, when the input data are available.
Let us point out that Fredrich et al. (1991) also reported experimental data for the PSD and
0 200 400 600 800 1000 1200 1400
Pressure (Bar)
0.8
0.85
0.9
0.95
1
1.05
1.1
Normalized Permeability
Fontainebleau FS4
Predicted
Experimental Data
Figure 3.4: Comparison of the predicted permeabilities with the experimental data for the
Fontainebleau sandstone. The tted value of the Young's modulus G
e
of the grains is 40 GPa.
pressure-dependence of the permeability of a Fontainebleau sandstone. Their PSD has striking
similarity with the distribution that Eq. (2.20) produces for r
a
1. However, their data for
21
K
e
pPq have some peculiar features. In particular (see their Fig. 4), K
e
pPq sharply drops when
what is referred to as the \eective pressure" is increased from zero to only 4 MPa. It then stays
essentially constant for 4 MPa P 40 MPa, and then it increases for P ¡ 40 MPa. Fredrich
et al. (1991) remarked that their data has considerable scattering due to inelastic deformations;
that some data may be \artifacts" at the higher pressures due to errors in the opening and closing
valves, and that the core sample that they experimented on split in half when they removed it
from the experimental setup. Thus, we did not try to predict their data. Yale (1984) stated that
in all cases that he experimented on the pore pressure P
p
was constant. Thus, in what follows
the pressure P may be replaced by PP
p
.
3.4.2 Beaver Sandstone
0 50 100 150 200 250 300 350
Pressure (Bar)
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
Normalized Permeability
Beaver
Predicted
Experimental Data
0 200 400
Pressure (Bar)
0.96
0.98
1
/
0
Figure 3.5: Comparison of the predicted permeabilities with the experimental data for the Beaver
sandstone. The tted value of the Young's modulus G
e
of the grains is 2.7 GPa.
22
The Beaver River sandstone (BRS) is a formation on the west side of the Athabasca River near
Mildred Lake and the Beaver River (in Alberta, Canada), and has been identied as as quartzite
(Kristensen et al., 2015) with ne- to medium-size and well-sorted grains, 78 percent of which
was quartz. Its pore space was cemented by 16 percent quartz overgrowth, and also contained
clay. Its initial porosity (before deformation) was
0
0:076. Figure 3.5 compares the predicted
permeabilities, normalized by the initial permeability of the sandstone before deformation (as also
presented by Yale, 1984) as a function of the applied pressure, with the experimental data of Yale
(1984). The agreement between the two is excellent. Because the sandstone is mostly quartz,
its porosity and, thus, PSD do not change much over the range of the applied pressure. This is
conrmed by the inset of Fig. 3.5 that shows that the porosity is reduced by only 5 percent of
its initial value, as well as Fig. 3.6 that presents the PSD of the sandstone at higher pressures,
indicating only small changes.
10 20 30 40 50 60 70 80
0
0.02
0.04
0.06
0.08
0.1
150 bar
10 20 30 40 50 60 70 80
0
0.02
0.04
0.06
0.08
0.1
Distribution
300 bar
10 20 30 40 50 60 70 80
Pore Size (microns)
0
0.02
0.04
0.06
0.08
0.1
450 bar
Figure 3.6: Evolution of the pore-size distribution of the Beaver sandstone as a result of applying
the external hydrostatic pressure P to the sandstone.
23
3.4.3 Berea Sandstone
Yale (1984) presented experimental data for two Berea sandstone. One was Berea 100H (with
H implying that the bedding was horizontal in the experiments) with a sublitharenite environment
- one in which the sandstone is characterized by the presence of less than 15 percent mud matrix -
with 5% - 25% of the grains being rock fragments, more than the feldspar content. The sandstone
consisted of 53% quartz, and had very ne to ne, well-sorted grains with an initial porosity of
0.165. Its cement contained 11% quartz overgrowth, as well as clay. Figure 3.7(a) compares the
predicted permeabilities with the experimental data; the agreement is excellent. The second Berea
sandstone that Yale (1984) experimented on was Berea 500, a quartzenite-type porous medium
composed of up to 90 percent detrital quartz, with limited amounts of other framework grains,
such as feldspar and lithic fragments. Such sandstone can have higher-than-average amounts of
resistant grains, such as chert and minerals. Sixty six percent of Berea 500 was quartz. Its initial
porosity was
0
0:2, while its cement consisted of 5 percent quartz overgrowth, 8 percent Fe
oxide, and 1 percent clay, with the rest being other types of materials. Figure 3.7(b) presents a
comparison of the pressure-dependence of the permeability with the experimental data, indicating
excellent agreement. In both sandstones the porosity was reduced by only 5-6 percent of its initial
value over the entire range of pressure and, therefore, the change in the PSDs was small.
24
Figure 3.7: Comparison of the predicted permeabilities with the experimental data for two Berea sandstones. (a) Berea 100H with tted
value of the Young's modulus G
e
25 GPa, and (b) Berea 500 with G
e
13 GPa.
25
3.4.4 Boise Sandstone
The Boise sandstone used in the experiments by Yale (1984) was of arkosic arenite, or arkose
type, a detrital sedimentary rock that constains at least 25% feldspar, which is why it is sometimes
referred to loosely as feldspathic sandstone. The sandstone was ne to medium grained, very well
sorted, with initial porosity of about 0.26 and minor carbonate-clay cement. Its grains consisted
of 28 percent quartz and 44 percent feldspar. Figure 3.8 compares the predicted permeabilities
with the experimental data. The agreement between the two is excellent.
0 50 100 150 200 250 300 350 400 450
Pressure (Bar)
0.5
0.6
0.7
0.8
0.9
1
1.1
Normalized Permeability
Boise
Predicted
Experimental Data
0 200 400 600
Pressure (Bar)
0.98
0.99
1
/
0
Figure 3.8: Comparison of the predicted permeabilities with the experimental data for Boise
sandstone with the tted value of the Young's modulus G
e
38:5 GPa.
3.4.5 Cambrian Sandstones
Chierici et al. (1967) presented experimental data for the pressure-dependence of the perme-
ability of three Cambrian sandstones. These are low-porosity sandstones from the Cambrian era
26
that consist of sand-size quartz grains held together by quartz cement. The three were referred
to as Cambrian 6, Camnbrian 14 and Cambrian 16 by Yale (1984) and had initial porosities of
0.08, 0.11, and 0.13, respectively. Figures 3.9(a) - 3.9(c) present comparison of the predicted
pressure-dependence of the eective permeability with the experiemental data. In all cases the
agreement between the predictions and the data is good, with the largest dierence being about
12 percent at 450 bar, applied to Cambrian 16. Note that the fact that in the cases of Cambrian 6
and Cambrian 14 the nal porosities of the sandstones at the highest pressure applied were about
95 percent of their initial values indicates the rigidity of their structure. In addition, the lower
value of the Young's modulus G
e
for the Cambrian 16 is consistent with a larger reduction in its
initial porosity.
27
Figure 3.9: Comparison of the predicted permeabilities with the experimental data for three Cambrian sandstones with the tted value of
the Young's modulus being (a) G
e
10 GPa; (b) 12 GPa, and (c) 2 GPa.
28
3.4.6 Fahler Sandstones
Yale (1984) presented experimental data for the pressure-dependence of the eective perme-
abilities of six Fahler sandstones, which he referred to them as Fahler 142, Fahler 154, Fahler 161,
Fahler 162, Fahler 189, and Fahler 192. Of the six, Fahler 142 and 192 were of quartzarenite type,
whereas the other four were sublitharenite sandstones. Moreover, the initial porosity of Fahler
142 was
0
0:08. It was ne grained, very well sorted, with quartz outgrowth, and clay and
carbonate cement. It consisted of 35 percent quartz, 8 percent chert, and 3 percent feldspar, with
the rest being various other types of rock materials. Its cement contained 23 percent carbonate.
Fahler 154 with an initial porosity of 0.044 was very ne to ne grained and very well sorted,
24 percent of which was quartz, 8 percent chert, 4 percent lithics, and 2 percent feldspar, with
the rest being other rock materials. Its cement contained chalcedony (a cryptocrystalline form of
silica), Fe oxide and carbonate. Fahler 161 had an initial porosity of 0.023 and medium grained.
It was well sorted with quartzovergrowth consisting of 32% chert, 25% quartz, and 11% lithics.
Likewise, Fahler 162 was a sandstone with an initial porosity of 0.03, ne to medium grained,
consisting of 46 percent quartz, 8 percent various lithics, and 6 percent chert, with the rest being
other types of rock materials. Its cement consisted of 25 percent quartz overgrowth, 8 percent
Fe oxide, and 8 percent clay. Fahler 189 had an initial porosity of 0.02, medium grained, and
contained 27 percent quartz, 27 percent chert, 11 percent various lithics, and 3 percent feldspar.
Its cement consisted of 13 percent quartz overgrowth, 9 percent carbonate, 4 percent chalcedony,
and 2 percent clay. Finally, Fahler 192 had an initial porosity of 0.0458 with ne well sorted
grains that consisted of 28% quartz, 3% chert, and 2% lithics. The rest consisted of other rock
materials. Its cement matrix consisted of 22% quartz overgrowth, 11% interparticle cement, 9%
intergrowth with clay, 4% framboidal-pyrite, 2% brous clay, and 19% carbonate. Figures 3.10(a)
- 3.10(f) present the predictions and compare them with the experimental data. We rst note the
consistency between the estimated values ofG
e
for the six sandstones, which indicates the internal
29
consistency of the theoretical model. As Figures 3.10 indicate, in all cases the agreement between
the predictions and the data is excellent for pressures as high as 250 bars, but they deviate from
the data at higher pressures. We will come back to this point in Sec 3.3.
30
Figure 3.10: Comparison of the predicted permeabilities with the experimental data for six Fahler sandstones with the tted value of the
Young's modulus being (a)G
e
0:12 GPa; (b) 0.15 GPa; (c) 0.32 GPa; (d) 0.11 GPa; (e) 0.17 GPa; (f) 0.27 GPa.
31
3.4.7 Indiana Dark Sandstone
The next sandstone whose pressure-dependence permeability we predicted is the Indiana Dark
sandstone, referred to as the Indiana DH, with DH indicating that the sample was taken (after
drilling) horizontally. The sandstone was of subarkose type, one in which feldspar sand grains
exceed rock fragments, but make up 5 to 15 percent of the rock. Its initial porosity was relatively
high, 0.27, with its major components being 57 percent quartz and 7 percent feldspar. Its cement
contained clay, as well as 22 percent hematite, Fe
2
O
3
. Figure 3.11 compares the predictions with
the experimental data of Yale (1984); the agreement is excellent. Note that since the initial
porosity of the sandstone was high and it reduced by only 4 percent even at the highest pressure,
then, consistent with our arguments about the Fahler sandstone, the porous medium remained
well connected, precisely in the range of porosity in which the EMA is highly accurate.
0 50 100 150 200 250 300 350 400 450
Pressure (Bar)
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
1.05
1.1
Normalized Permeability
Indiana DH
Predicted
Experimental Data
0 200 400
Pressure (Bar)
0.96
0.97
0.98
0.99
/
0
Figure 3.11: Comparison of the predicted permeabilities with the experimental data for the
Indiana DH sandstone with the tted value of the Young's modulus being G
e
30 GPa.
32
3.4.8 Massillon Sandstones
The sandstone, referred to as Massillon (from Massillon, Stark County, Ohio) DH or DV (for
horizontal or vertical compared with the bedding planes) by Yale (1984), was of quatzarenite type
with an initial porosity of 0.161, and medium-size and well-sorted grains. Sixty one percent of it
was quartz, with feldspar, chert, and lithics each contributing 1 percent, and the rest being other
types of rock material. Its cement contained Fe-oxide at 15 percent and clay at 5 percent. Figure
3.12 compares the predictions with the experimental data of Yale (1984). The largest dierence
between the two sets is about 6 percent at a pressure of 400 bars. This example shows the ecacy
of MFAs in anisotropic media.
33
Figure 3.12: Comparison of the predicted permeabilities with the experimental data for the Massillon DH, DV sandstone with the tted value
of the Young's modulus being G
e
9:5 GPa, and 1:7 GPa respectively.
34
3.4.9 Miocene Sandstone
Chierici et al. (1967) reported experimental data for the pressure-dependence of Miocene
sandstone, a low-porosity rock that is of the feldspathic arenite type. Its initial porosity was
0.083. The roundness and sorting of the grains of such sandstones are typically high, implying the
existence of long
ow and transport distances (Saitoh and Masuda, 2004). Figure 3.13 presents the
comparison between the predictions for the pressure-dependent permeability and the experimental
data. The agreement is excellent. The existence of well-connected and long transport and
ow
paths practically guarantees that the predictions would be accurate, because it is precisely under
such conditions that the EMA is accurate.
0 50 100 150 200 250 300 350 400 450
Pressure (Bar)
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
1.05
1.1
Normalized Permeability
Miocene 7
Predicted
Experimental Data
0 200 400
Pressure (Bar)
0.92
0.94
0.96
/
0
Figure 3.13: Comparison of the predicted permeabilities with the experimental data for the
Miocene 7 sandstone with the tted value of the Young's modulus being G
e
5 GPa.
35
3.4.10 Pliocene Sandstone
Chierici et al. (1967) also reported experimental data for the pressure-dependence of the
permeability of a Pliocene sandstone, which Yale (1984) referred to it as Pliocene 35. Similar
to Miocene sandstones, Pliocene sandstones also have round grains. The initial porosity of the
sample was 0.2. Figure 3.14 compares the predictions with the data. The agreement between the
two is excellent.
0 50 100 150 200 250 300 350 400 450
Pressure (Bar)
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
1.05
1.1
Normalized Permeability
Pliocene 35
Predicted
Experimental Data
0 200 400
Pressure (Bar)
0.94
0.96
0.98
/
0
Figure 3.14: Comparison of the predicted permeabilities with the experimental data for the
Pliocene 35 sandstone with the tted value of the Young's modulus being G
e
4 GPa.
3.4.11 Tensleep Sandstone
Fatt (1957) reported on his measurements of the pressure-dependence of the permeability of
Tensleep sandstone. The porous medium represents a geological formation from the entire Penn-
sylvanian sequence in central and northern Wyoming in the very early Permian age (Branson and
36
Branson, 1941). Such rocks are predominantly cross-bedded sandstone and have thin limestone
and dolomite beds (Kerr et al., 1986). The initial porosity of the sample was 0.146. In Fig.
3.15 we compare the predictions with the experimental data of Fatt (1957), also reported by Yale
(1984). The agreement is excellent over much of the range of the applied pressure. The largest
dierence, at the highest pressure, is about 10 percent.
0 50 100 150 200 250 300 350
Pressure (Bar)
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
Normalized Permeability
Tensleep
Predicted
Experimental Data
0 200 400
Pressure (Bar)
0.988
0.99
0.992
0.994
0.996
/
0
Figure 3.15: Comparison of the predicted permeabilities with the experimental data for the
Tensleep 35 sandstone with the tted value of the Young's modulus being G
e
2:6 GPa.
3.4.12 Gulf Coast Sandstone
The experimental data for this sandstone were reported by Yale (1984), who referred to the rock
as Tertiary 807. In general, tertiary rocks are those that were formed during part of the Cenozoic
era, covering the Paleogene and Neogene periods. The sample with which the experiments were
carried out was of the subarkose type, with ne, well-sorted grains and high intergranular porosity.
It contained 52 percent quartz, 9 percent feldspar, 5 percent chert, and 3 percent lithics with an
37
initial porosity of 0.22. Figure 3.16 compares the theoretical predictions with the experimental
data. Once again, over much of the pressure range the agreement between the predictions and the
data is excellent. The theory predicts slightly higher permeabilities at the two highest pressures,
with the largest deviation from the data being, however, about 5 percent at 400 bars.
0 50 100 150 200 250 300 350 400 450
Pressure (Bar)
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
1.05
1.1
Normalized Permeability
Gulf Coast
Predicted
Experimental Data
0 200 400 600
Pressure (Bar)
0.94
0.96
0.98
/
0
Figure 3.16: Comparison of the predicted permeabilities with the experimental data for the
Tertiary 807 (Gulf Coast) sandstone with the tted value of the Young's modulus being G
e
6:4
GPa.
3.4.13 Torpedo Sandstone
Dobrynin (1962) reported permeability data for the Torpedo sandstone from Kansas. The
initial porosity of the sample was 0.202, and it contained about 5 percent clay minerals that
consisted mostly of kaolinite and chlorite, distributed evenly throughout the sample. As Figure
3.17 indicates, except at 450 bars where the predicted permeability is larger than the measured
38
value by 5 percent, the agreement between the predictions and the experimental data is excellent.
0 50 100 150 200 250 300 350 400 450
Pressure (Bar)
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
1.05
1.1
Normalized Permeability
Torpedo
Predicted
Experimental Data
0 200 400
Pressure (Bar)
0.97
0.98
0.99
1
/
0
Figure 3.17: Comparison of the predicted permeabilities with the experimental data for the
Torpedo sandstone with the tted value of the Young's modulus being G
e
3 GPa.
3.4.14 Triassic Sandstones
Yale (1984) reported measurements of the pressure-dependence of the permeability of ve
Triassic sandstones. Such rocks were formed in the Triassic period, between 200 and 251 million
years ago. The morphology of such sandstones varies greatly, from very ne- to very coarse-
grained. They represent porous formations with low- or ultra-low permeability, but they often
contain both tectonic and diagenetic fractures that provide
ow paths. The ve sandstones studied
by Yale (1984) were referred to as Triassic 26, 27, 34, 38, and 41 with initial porosities that were,
respectively, 0.18, 0.18, 0.2, 0.2, and 0.21. Figures 3.18(a) - 3.18(e) compare the predictions of
the permeabilities with the experimental data. Except for Triassic 27 sandstone, the agreement
39
between the predictions and the data is uniformly excellent. Even in the case of Triassic 27,
the maximum dierence between the predictions and the data at high pressures is only about 5
percent.
40
Figure 3.18: Comparison of the predicted permeabilities with the experimental data for the Triassic sandstone with the tted value of the
Young's modulus G
e
being (a) 8; (b) 6.5; (c) 83; (d) 40, and (e) 2.8, all in GPa.
41
3.4.15 Branford Sandstone
Wyble (1958) reported permeability data for this sandstone, referred to as Branford sandstone
in Yale (1984), from Connecticut. The initial porosity of the sublitharenite type sandstone was
0.11. The grains were medium sized with well sorted grains that consisted mostly of quartz,
carbonate, and clay cement. We again see excellent agreement between the predictions and the
experimental data in Figure 3.19. The largest deviation exists at the highest pressures at about
ve percent.
0 50 100 150 200 250 300 350
Pressure (Bar)
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
Normalized Permeability
Branford
Predicted
Experimental Data
0 200 400
Pressure (Bar)
0.8
0.9
1
/
0
Figure 3.19: Comparison of the predicted permeabilities with the experimental data for the
Branford sandstone with the tted value of the Young's modulus being G
e
0:68 GPa.
3.4.16 Kirkwood Sandstone
Finally, Figure 3.20 shows the predictions of the Kirkwood sandstone against normalized
permeability data, originally presented by Wyble (1958), with initial porosity of 0.152. The
sandstone was described as fairly clean orthoquartzite. The predictions again show excellent
42
agreement over the lower pressure range. And even at the highest pressures, we see a maximum
dierence of about 10 percent. This error could be attributed to signicant morphology changes
that our model fails to consider(see Sec. 3.3).
0 50 100 150 200 250 300 350
Pressure (Bar)
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Normalized Permeability
Kirkwood
Predicted
Experimental Data
0 200 400
Pressure (Bar)
0.92
0.94
0.96
0.98
1
/
0
Figure 3.20: Comparison of the predicted permeabilities with the experimental data for the
Kirkwood sandstone with the tted value of the Young's modulus being G
e
0:14 GPa.
3.5 Discussion
Let us rst point out that in a previous paper (Richesson and Sahimi, 2019) we used the
radius of contactR
c
, given by Eq. (2.16) (originally presented by Yale, 1984, without derivation)
in order to develop an expression for the PSD of a deforming porous medium. Within the MFA
that we have developed, `
p
pPq, the pore length at pressure P , is given by (Yale, 1984)
`
p
pPq`
0
R
c
pPq`
0
1
R
c
pPq
`
0
; (3.4)
43
where`
0
is the length of the pore before deformation. If r
0
{`
0
is the aspect ratio of a pore before
it is deformed, then, within the MFA the pore size at pressure P is given by
r
P
pPq`
p
pPq
r
0
`
0
: (3.5)
Thus, if a hydrostatic pressure P is applied to a porous medium, the change in any pore size can
be computed by Eq. (3.2), implying that the initial PSD can be updated. However, although
as we demonstrated previously (Richesson and Sahimi, 2019), we obtained excellent agreement
between the predicted permeabilities and the experimental data for ve sandstones, when we
used Eq. (3.2) for updating the PSDs of the sandstones considered in this paper, good agreement
between the theoretical predictions and the data could often be obtained only when the Young's
modulusG
e
of the grains or the solid matrix was nonphysically very high, ranging from hundreds
of GPa to even thousands. Thus, we believe that only when the change in the PSD is determined
through the quantityu, given by Eq. (2.17), can one expect physically acceptable tted values of
G
e
and good agreement between the predictions and the experimental data. Several other aspects
of our proposed model deserve discussions, which we now present.
3.5.1 Limit of the Validity of the EMA
An important question is the range of the validity of the EMA, as well as the MFA that we
developed for the deformation. As the extensive comparison between the theoretical predictions
and the experimental data demonstrated, if the porosity of a porous medium is not extremely
low, to the extent that it is barely connected, the MFA developed in this paper together with the
EMA is accurate for predicting the eective permeability of porous media. Koplik (1981) and
Adler and Berkowitz (2000) studied the limit of the accuracy of the EMA. Generally speaking,
the EMA is (i) more accurate for two-dimensional (2D) media than for 3D, and (ii) not very
accurate near the critical porosity or the percolation threshold, a region that is called the critical
44
region. In random media the critical region is dened roughly by (Sahimi, 1994),
c
¤ 1{Z,
where Z is the mean pore connectivity, and
c
is the critical porosity. (iii) If there are extended
correlations between the pores' sizes, then the EMA is less accurate than in completely random
porous media, although Mukhopadhyay and Sahimi (2000) suggested ways of taking into account
the eect of such correlations. Comprehensive discussions of the strengths and shortcomings of
the EMA are provided by Sahimi (2003) and Hunt and Sahimi (2017).
3.5.2 Eect of Structural Changes in the Pore Space
As the discussions of the last section indicated, in some cases, such as the Cambrian and
Fahler sandstones, the eective permeability at high pressures decreases a bit more slowly than
what the model predicts. One possible reason for this is that the morphology of the porous media
undergoes fundamental changes at high pressures, such as opening up new cracks that provide new
ow paths for the
uid and, hence, arrest to some extent the decline in the permeability at high
pressures. We already mentioned a good example of this type of change in the morphology, namely,
a Fontainebleau sandstone that Fredrich et al. (1991) had experimented with. If such changes
do occur, the present approach may be modied in order to provide more accurate predictions.
For example, an EMA was developed by Hughes and Sahimi (1993a,b) for
ow and transport
in porous media with two distinct types of
ow and transport paths, such as pores and cracks,
which may be used in this context, but one would need not only the PSD distribution, but also
the distribution of the cracks' apertures. Alternatively, one may use the present EMA but with
a bimodal PSD, so that by viewing the cracks as large pores, their presence can be taken into
account.
45
3.5.3 The Eect of a Grain-Size Distribution
As described earlier, the theory of grain deformation, Eqs. (2.16) and (2.17), requires an
average grain sizeR
g
. This is due to the fact that, as emphasized earlier, the theory deformation
is a MFA and, as such, it considers the deformation of only two neighboring grains and the
change in the eective radius of the pore between them. Thus, if a grain-size distribution (GSD)
is available (see, for example, Cheung et al., 2012), one can compute R
g
. Beyond that, and in
order to take into account the eect of a GSD of a collection of grains, the MFA must be rened,
and the eect of the interactions between a collection of grains must be taken into account. While
numerical simulations in this direction have been made in the past (see, for example, Bakhshian
and Sahimi, 2016; Das and Singh, 2017), to our knowledge none has produced a tractable theory
for the evolution of the PSD. This is an issue that we are currently studying.
3.5.4 Eect of the Deformation Mode
Deformation of porous geological formations, such as depleted oil reservoirs, is often caused
by uniaxial stress. What we have studied is the case in which the conning pressure is applied
hydrostatically. It is clear that the deformations that result from the two types of the bound-
ary conditions are dierent, because the spatial distributions of pressure in the two systems are
dierent. But, when, for example, the overburden pressure exerts itself uniaxially in an oil reser-
voir, the surrounding rock limits the resulting lateral deformation. This implies that one obtains
mostly vertical compaction, which represents smaller changes in the pore sizes than what is caused
by hydrostatic pressure. Section 3.2 demonstrated that our theory provides accurate predictions
for the macroscopic permeability as a function of the hydrostatic pressure, and such a pressure
deforms the pore space much more extensively than a uniaxial stress would. We therefore believe
that a slightly modied theory would be at least equally accurate for the case in which a uniaxial
stress is exerted on a porous medium.
46
3.6 Summary
Predicting physical properties of deforming porous media, and in particular their permeability
and transport characteristics, such as their electrical conductivity when they are saturated with
brine, is important to many physical processes. They include shale formations undergoing fracking,
and oil, gas, and coal-bed reservoirs, as well as composite materials that are used in everyday life.
We presented a new theoretical model for predicting the permeability K
e
of such porous media
and materials. The theory, a mean-eld approximation, determines the change in the size of a
pore between two grains that deform when a hydrostatic pressure is applied to them. Then, given
the initial PSD of a porous medium before deformation and the Young's modulus of the grains as
the input, the theory determines the PSD of a porous medium that is deformed by applying an
external pressure P , which is then used with the eective-medium approximation to predict the
eective permeability of the porous medium at the same pressure. Extensive comparison between
the theoretical predictions and experimental data for the pressure-dependence of the eective
permeabilities of twenty-nine sandstones indicated agreement between the two in almost all cases,
ranging from very good to excellent.
47
Chapter 4
Predicting Eective Conductivity of Deforming Porous
media
4.1 Introduction
In most porous materials, both natural and synthetic, pores have complex shapes; they are
interconnected with the connectivity being stochastically distributed, and they form
ow and
transport paths that are tortuous. Characterization of such porous media has always been a
problem of fundamental importance, and has been studied for a long time. One way of gaining
information on the structure of a pore space is by relating its
ow and transport properties to
the quantities that shed light on its morphology. In particular, consider a porous medium that is
saturated by brine with electrical conductivity
w
. If the eective conductivity of the saturated
medium is
e
, then,
e
w
F
; (4.1)
whereF is a geometrical characteristic of the pore space, called the formation factor. In the limit
of long times, the eective diusion coecient of a molecule probing the same pore space is given
by
D
e
D
0
F
; (4.2)
48
where is the porosity of the porous medium, and D
0
is the free diusion coecient outside
the pore space. It is, therefore, that measurements of D
e
and
e
, or the ability to predict them
accurately, provide valuable information on the morphology of the pore space. Thus, in addition
to numerous measurements for a wide variety of porous media, various theoretical approaches have
also been developed for predicting
e
and D
e
[1,2]. Moreover, short-time behavior of D
e
, before
it reaches its asymptotic value given by Eq. (4.1) has been linked with the ratio of surface S
p
of
the pores and their volume V
p
, and has been exploited fruitfully to make progress on the general
problem of characterizing a porous medium[3-6]. Almost all the progress that has been made over
the past several decades relates to the conductivity and diusivity of rigid porous media. But,
what if the porous medium is deformable? As discussed in Chapters 2 and 3,
ow and transport
properties of porous media that deform as a result of being subjected to an external driving force,
either pressure or stress, and are also important, and appear in many problems of fundamental
scientic importance, as well as practical applications. In this chapter, we use the same type of
theoretical model that we developed in Chapters 2 and 3 for the eective permeability in an eort
to predict the eective electrical conductivity of a deforming porous medium, saturated by an
electrically-conducting
uid, such as brine.
4.2 Modications for predicting conductivity
Note that we use the same theory of deformation that we developed in Chapter 2. Recall that
in the mean-eld theory of Chapter 2, two neighboring pores are embedded in a uniform porous
medium, and an external hydrostatic pressure P is applied to the system. Then, the eective
radius between the two pores is reduced by a quantity u, given previously by
uR
g
R
g
R
1{3
3Pp1
2
q
G
e
2{3
: (4.3)
49
As described in Chapters 2 and 3, the morphology of any porous medium consists of pore
throats that are connected together via the pore bodies. The eective sizes of both pore throats
and pore bodies are distributed according to statistiscal distributions f
t
pr
t
q and f
b
pr
b
q. But,
whereas experimental measurement of f
t
pr
t
q is straightforward (see, for example, Sahimi 2011;
Blunt 2017 for discussions of various measurement methods), measuring f
b
pr
b
q is not straigthfor-
ward. On the other hand, both macroscopic permeability and electrical conductivity of porous
media are controlled by the pore throats, which, for conveniece, are referred to as pores, and their
distribution fprq as the PSD. Since in the EMA a heterogeneous pore space is represented by a
uniform medium in which the size of all the pores is r
e
, which we assume to be cylindrical, then,
its electrical conductantivity is,
P
9r
2
P
. One can, of course, consider other pore shapes. The
EMA predicts that the macroscopic eletrical conductivity
e
is given by (Doyen 1988; David et
al. 1990)
e
r
2
e
xr
2
b
y
f
; (4.4)
where is the porosity, and is the tortuosity for which various theories, as well as empirical and
semi-empirical relations, have been developed (for a review see Ghanbarian-Alavijeh et al. 2013).
Since the distribution f
b
pr
b
q of the size of the pore bodies is typically not available, David et al.
(1990) suggested that one should use,
xr
2
b
yxr
2
y
»
r
M
rm
r
2
fprqdr; (4.5)
withr
m
andr
M
being, respectively, the minimum and maximum pore radii. We assume the same
in this paper. In the EMA an eective conductance g
e
is computed by
»
g
M
gm
g
e
g
gpD 1qg
e
hpgqdg 0: (4.6)
50
Here,g
e
andg
M
are, respectively, the minimum and maximum conductances,D is the dimension-
ality of space (D 3), and hpgq is the pore-conductance distribution. Since g
e
9r
2
e
, g9r
2
, and
hpgqdgfprqdr, so that hpgqfpr
2
qdr{dgfpr
2
qp1{2rq, we obtain an alternative formulation
directly in terms of fprq:
»
r
2
M
r
2
m
r
2
e
r
2
r
2
pD 1qr
2
e
fpr
2
q
2r
dr 0: (4.7)
As for the tortuosity factor , we used
1m
. The review by Ghabbarian et al. (2013)
indicates that 1:2 ¤ m ¤ 4:4 for a wide variety of porous media. We set m 4, which is for
tortuous porous media.
Mukhopadhyay and Sahimi (2000) derived an EMA for predicting direction-dependent macro-
scopic conductivity of anisotropic porous media, while Stroud (1975) presented a continuum EMA
for anisotropic media in which the local conductivity was a tensor. Other applications of the EMA
were pointed out in Part I, to which the interested reader is referred.
4.3 Accounting for Surface Conduction
The experimental data for the electrical conductvity of porous media that we compare with the
predictions of the model are in terms of the formation resistivity factorF , reported by Yale (1984).
Equation (1) that denes F is based on the eective conductivity of the saturated pore space,
and does not include possible contribution by conduction across the pores' surface. Depending
on the chemical composition of a porous medium, however, particularly in a clay-bearing one,
surface conduction may contribute signicantly to the overall measured conductivity, since the
clay grains' surface allows for a layer of counterions that facilitates development of a signicant
negative surface charge (Waxman and Smits 1968; Clavier et al. 1984). The data reported by
Yale (1984) are for a broad variety of sandstones, which do have signicant clay content (see
also below). Therefore, it is imperative to account for surface conduction, before comparing the
51
predictions of the theory, which does not take into acount the eect of surface conduction, with
the data.
Revil et al. (1998) derived the following equation for the total conductivity
t
of a pore
space, including the conctribution by surface conductivity, which is saturated by a
uid with a
pH between 5 and 8:
t
f
F
$
&
%
1t
f
pq
F
1
2
t
f
pq
1
t
f
pq
g
f
f
e
1
t
f
pq
2
4F
t
f
pq
,
.
-
; (4.8)
where t
f
pq
is the Hittorf transport number of cations in the free electrolyte (brine in pores),
and is known as the Dukhin number (Lyklema 1993), which is the ratio of surface and
uid
conductivities. Equation (8) reduces to Eq. (1) in the limit Ñ 0. The brine used in the
experiments of Yale (1984) was NaCl, for which t
f
pq
0:38. Thus, if the second parameter of
Eq. (8), namely, , is also known, then, for every measured
t
one can use it to compute the
corresponding formation factor F .
Revil et al. (1998) derived the following equation for the Dukhin number,
2
g
C
3
f
: (4.9)
Here,
g
is the density of the solid matrix,C is the cation exchange capacity, and is the equivalent
mobility for surface conduction. Equation (9) provided accurate predictions for clay-rich sediments
(Diagle et al. 2015). Because the grains' density and the exact chemical compositions of the
sandstones that we consider were not given by Yale (1984), we cannot determine the Dukhin
number for them. Thus, since is independent of the hydrostatic pressure, we use one point
for every sandstone that we consider in order to estimate the Dukhin number using the above
equation. This is explained in the next section.
52
4.4 Computational Procedure
Given the theoretical formulation for predicting the eective electrical conductivity
e
of de-
forming porous media, the following computational procedure was used to calculate
e
.
(i) Given the initial PSD f
0
pr
0
q and, therefore, an initial conductance distribution h
0
pg
0
q of
an undeformed porous medium, its electrical conductivity was computed using Eqs. (4.4),
(4.5), and (4.7).
(ii) For a given hydrostatic pressure P , the corresponding PSD f
P
pr
P
q was constructed by
selecting the pore sizes from f
0
pr
0
q, calculating their updated values using Eq. (4.1) and
r
P
r
0
u{2, and repeating it for a large number of pore sizes selected fromf
0
pr
0
q, so that
an accurate f
P
pr
P
q was obtained.
(iii) The PSD f
P
pr
P
q and Eq. (4.5) were then utilized to determine xr
2
b
pPqyxr
2
P
y. The
result was used together with f
P
pr
P
q in Eqs. (4.4) and (4.7) to compute
e
pPq at pressure
P .
(iv) To compare the predictions in (iii) with the experimental data for a given porous
medium, we utilized the experimental value of the total conductivity
t
on the left side of
Eq. (4.8) at a single pressure and determined the parameter, using as the formation factor
F its theoretical prediction. In eect, for every sandstone we used a single experimental
point to estimate . Since is independent of P , we used the same estimate in Eq. (4.8)
and solved for the formation factor F at various pressures P , given that the left side of Eq.
(4.8) is the experimental value of the total conductivity.
(v) The resulting values ofF at various pressuresP represent the true values of the formation
factor over the pressure range for a given porous medium, which are then compared with
the theoretical predictions.
53
Note that if all the parameters of Eq. (4.9) are known, it can be utilized to estimate directly,
without any need for step (iv). In that case, the estimate of and the experimental data for
t
for
a given porous medium are used drectly in Eq. (4.8) and the resulting nonlinear equation is solved
numerically for the true formation factors F . Note also that if the PSD f
0
pr
0
q is expressed by an
analytical expression,f
P
pr
P
q will also be determined analytically, in which case the computations
will be very fast.
4.5 The Parameters of the Model
The parameters of the model are the Poisson's ratio and the Young's modulus G
e
of the
grains (not the porous medium) that appear in Eq. (4.3), the PSD, and the Dukhin number . If
experimental data for the parameters are available, they can be used directly in the theory, but
they are not available for the sandstones that we analyze.
We already described how we estimate . As discussed in Chapter 3, the predictions of the
model are sensitive to the value of the Young's modulus. In Chapter 3 we explained how we
estimate G
e
. Since the experimental data that we compare with the theoretical predictions are
for the same sandstones as those in Chapter 3, we use the same values of G
e
.
As for the Poisson's ratio, in Chapter 3 we demonstrated that if all the parameters but are
xed, the Poisson's ratio is varied by a factor of 4, and the model is used to predict the dependence
on the applied pressure of the permeability of the sandstones, the predictions vary by at most 2
percent. Thus, we x the Poisson's ratio at 0:3, the same value that we used in Chapter 3,
which is in the middle of the range for sandstones; for pure quartz, for example, 0:2.
The sensitivity of the predictions to the PSD was also studied in Chapter 3. To do so, we used
the following theoretical PSD distribution,
fprq
rr
m
pr
a
r
m
q
2
exp
1
2
rr
m
r
a
r
m
2
; (4.10)
54
where r
a
is the average pore size. The minimum pore size r
m
was xed, and r
a
was varied
over two orders of magnitude. Figure 3.1(a) presents the PSD that Eq. (4.10) generates. The
distribution corresponding to the lowestr
a
in Fig. 3.1(a) is completely similar to the PSD reported
by Fredrich et al. (1993) for a Fontainebleau sandstone, while those generated by other values
of r
a
are qualitatively similar to those reported by others for other types of sandstone. We
then demonstrated in Chapter 3 that predicted eective permeability for the sandstones that we
analyze later in this paper are not greatly sensitive to the PSD. Therefore, similar to Chapter 3,
in the absence of any experimental data for the PSDs of the sandstones that we analyze below,
we used in all the cases described below the distribution presented in Fig. 3.1(b) as the initial
PSD, f
0
pr
0
q, which was reported by Lindquist et al. (2000) for a Fontainebleau sandstone, and
is similar to those for many other sandstones reported by others (see, for example, Cheung et al.
2012 for Bleurswiller and Boise sandstones).
Table 1 summarizes the parameters G
e
and for all the sandstone that we analyze in the
following section.
4.6 Comparison with experimental conductivity data
Let us rst emphasize that the model presented is a MFA, which, similar to any MFA, neglects
the
uctuations in the local properties, hence making it possible to analyze the behavior of a
heterogeneous porous medium based on only two grains, the minimum number for a meaningful
analysis. This also implies that only an average grain size is required. As such, similar to all the
MFAs, the approach has its limitations and strengths.
We used the theory to predict the pressure-dependence of the eective electrical conductivity
e
pPq of a large number of sandstones, and to compare the predictions with the experimental
data, almost all of which are given by Yale (1984). He did not provide the sandstones' initial PSD
and, therefore, as mentioned earlier, we used in all the cases described below the PSD presented
55
in Sec.2.5.1. In Chapter 3 we described the geological characteristics of each sandstone and,
therefore, they will not be repeated here.
Yale (1984) stated that in all the cases that he experimented on, the pore pressure P
p
was
constant. Thus, in what follows the pressure P may be replaced by PP
p
.
56
Table 4.1: Estimates of the Young's modulus G
e
and the Dukhin number of the sandstones.
Sandstone G
e
(GPa) 10
3
Sandstone G
e
(GPa) 10
3
Fontainebleau 40 0.0 Beaver 2.7 3.3
Berea 100H 25 18.9 Berea 500 13 2.6
Boise 38.5 16.2 Cambrian 6 10 2.3
Cambrian 14 12 6.1 Cambrian 16 2 1.3
Fahler 142 0.12 4 Fahler 154 0.15 3.4
Fahler 161 0.32 0 Fahler 162 0.11 1.3
Fahler 189 0.17 0 Fahler 192 0.27 2
Massillon DH 9.5 17.3 Massillon DV 0.54 15.4
Miocene 7 5 1.2 Pliocene 35 4 4.1
Tensleep 2.6 29.6 Gulf Coast 6.4 22.7
Torpedo 3 8.6 Triassic 26 8 31.6
Triassic 27 6.5 29.9 Triassic 34 83 37.1
Triassic 38 40 33.1 Triassic 41 2.8 3.3
Branford 0.7 4.2 Kirkwood 0.5 12.7
Indiana DH 30 49.5
4.6.1 Fontainebleau
We rst present the predictions for a Fontainebleau sandstone, for which Farid et al. (2016)
reported measurements of the pressure-dependence of its conductivity. The initial porosity
0
of the sandstone, before deformation, was 0.052. Farid et al. (2016) also stated that the clay
content of the sandstone was negligible. Therefore, we took the Dukhin number 0. Figure
4.1 compares the predictions, normalized by the initial conductivity, with the experimental data.
57
0 50 100 150 200 250 300 350 400 450 500
Pressure (Bar)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
Normalized Conductivity
Fontainebleau FB11
Predicted
Experimental Data
0 200 400 600
Pressure (Bar)
0.94
0.96
0.98
1
/
0
Figure 4.1: Comparison of the predicted conductivities with the experimental data for the
Fontainebleau sandstone.
Note that since 0, no tting parameter was used. The agreement between the two sets is very
good, with the maximum dierence being no more than 7 percent.
4.6.2 Beaver Sandstone
The Beaver River sandstone is a formation on the west side of the Athabasca River near Mildred
Lake and the Beaver River (in Alberta, Canada), with an initial porosity (before deformation) of
0
0:076. Figure 4.2 compares the predicted pressure-dependence of the sandstone's electrical
conductivity, normalized by its value before deformation, with the experimental data of Yale
(1984) who also presented all of his data in normalized fashion. The agreement between the two
is excellent, with the largest dierence between the predictions and the data being about 8 percent
at the highest pressure.
58
0 50 100 150 200 250 300 350 400 450
Pressure (Bar)
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
1.05
1.1
Normalized Conductivity
Beaver
Predicted
Experimental Data
0 200 400 600
Pressure (Bar)
0.96
0.98
1
/
0
Figure 4.2: Comparison of the predicted conductivities with the experimental data for the Beaver
sandstone.
4.6.3 Berea Sandstones
The electrical conductivities of two Berea sandstones were reported by Yale (1984). One was
Berea 100H, a sandstone whose bedding was horizontal, with an initial porosity of 0.165. Figure
4.3 compares the predictions with the experimental data. The theory predicts continuous decline
of the conductivity with increasing pressure. The data indicate that at the highest pressure the
conductivity levels o, hence indicating that the morphology of the sandstone did not change at
the highest pressure. Despite this, the maximum dierence between the predictions and the data
is about 2.5 percent.
Berea 500, with an initial porosity of 0.2, was the second sandstone whose conductivity was
measured by Yale (1984). Figure 4.3 also compares the predictions with the experimental data
with the same level of accuracy as the predictions for Berea 100H, except that in this case the
measured electrical conductivity decays a bit faster than the predictions at the highest pressures.
We shall come back to this point shortly.
59
Figure 4.3: Comparison of the predicted conductivities with the experimental data for two Berea sandstones.
60
4.6.4 Boise Sandstone
The Boise sandstone had an initial porosity of 0.26. Figure 4.4 compares the predictions with
the experimental data. At lower pressures the measured electrical conductivity seems to decay a
bit faster than the predictions, while the opposite trends develop at the highest pressures. Note,
however, that the precentage dierence between the two sets is no more than 4 percent, well
within the measurements' uncertainties.
0 50 100 150 200 250 300 350 400
Pressure (Bar)
0.85
0.9
0.95
1
1.05
Normalized Conductivity
Boise
Predicted
Experimental Data
0 200 400 600
Pressure (Bar)
0.98
0.99
1
/
0
Figure 4.4: Comparison of the predicted conductivities with the experimental data for the Boise
sandstone.
4.6.5 Cambrian Sandstones
Cambrian sandstones are low-porosity formations from the Cambrian era. Yale reported their
electrical conductivity for three samples, referred to as Cambrian 6, 14 and 16 with initial porosi-
ties of 0.08, 0.11, and 0.13, respectively. Figures 4.5 compare the predicted pressure-dependence
of the eective conductivity with the experiemental data. In all cases the agreement between the
predictions and the data is excellent.
61
Figure 4.5: Comparison of the predicted conductivities with the experimental data for the three Cambrian sandstones.
62
4.6.6 Fahler Sandstones
Pressure-dependence of the electrical conductivities of four samples of Fahler sandstones, from
Fahler strata in Spirit River formation (in Grande Cache in Alberta, Canada) were reported by
Yale (1984). They were referred to as Fahler 142, 154, 162, and 189, with their geological char-
acteristics described in Part I. Their initial porosities were, respectively,
0
0:08; 0:044; 0:03;
and 0.02. Figure 4.6 compares the predictions with the experimental data, with the agreement
between the two being generally excellent for three of the sandstones, namely, Fahler 142, 154,
and 189.
In the case of Fahler 162 the agreement between the data and the predictions is good at higher
pressures, but the dierence between the two sets for the three smallest pressures is relatively
large, although their trends are completely similar. Note that, among the four Fahler sandstones,
the largest deformation-induced decline in the porosity belongs to Fahler 162, whose porosity was
reduced by about 20 percent over the intermediate values of the applied pressure. Given that
the initial porosity of the sandstone was only 0.03, its corresponding value at such pressures is
about 0.022, which is very low. The EMA does not usually provide accurate predictions for such
low-porosity materials. But the decline in the porosity is only one aspect of the problem. The
other, more important, aspect is how the porosity is distributed in the porous medium, as well
as the grains' chemical composition. This point is well demonstrated by the data for Fahler 189.
Even though its porosity is very low, the predictions are still accurate.
Fahler 162 is a ne-to-medium grained sandstone that consists of 46 percent quartz, 8 percent
various lithics, and 6 percent chert, with the rest being other types of rock materials. Its cement
contains 25 percent quartz overgrowth, 8 percent Fe oxide, and 8 percent clay. On the other hand,
Fahler 189 is a medium grained sandstone with 27 percent quartz, 27 percent chert, 11 percent
various lithics, and 3 percent feldspar, with its cement consisting of 13 percent quartz overgrowth,
9 percent carbonate, 4 percent chalcedony, and 2 percent clay. Thus, Fahler 162 contains far more
63
quartz, an extremely hard material, than Fahler 189, implying that the mechanism of porosity
reduction in the two sandstones may be dierent.
64
Figure 4.6: Comparison of the predicted conductivities with the experimental data for the four Fahler sandstones.
65
4.6.7 Indiana Dark Sandstone
Indiana dark DH sandstone, with DH indicating that the sample was taken horizontally after
drilling (parallel to bedding), had a relatively high initial porosity of 0.27. Figure 4.7 compares
the predictions with the experimental data of Yale (1984). The agreement is excellent, with the
largest dierence between the prediction and data being about 5 percent at the highest pressure.
0 50 100 150 200 250 300 350 400 450
Pressure (Bar)
0.7
0.75
0.8
0.85
0.9
0.95
1
1.05
1.1
Normalized Conductivity
Indiana DH
Predicted
Experimental Data
0 200 400 600
Pressure (Bar)
0.96
0.98
1
/
0
Figure 4.7: Comparison of the predicted conductivities with the experimental data for the Indiana
DH sandstone.
4.6.8 Massillon Sandstones
Massillon DH sandstone (from Massillon, Stark County, Ohio) is of quatzarenite type with
an initial porosity of 0.161, and medium-size and well-sorted grains. Figure 4.8 compares the
predictions with the experimental data of Yale (1984). Although the largest dierence between
66
the two sets is only about 3.5 percent at a pressure of 175 bars, the trends in the two sets are
somewhat dierent. The data indicate that the porosity at lower pressures reduced more strongly
and, then, it leveled o, because 60 percent of the sandstone is quartz, which means it is dicult
to reduce the porosity further even at 500 bars. On the other hand, the EMA does not take into
account the deformation mechanism, and assumes simply that the porosity is reduced randomly.
67
Figure 4.8: Comparison of the predicted conductivities with the experimental data for the Massillon DH sandstone.
68
4.6.9 Miocene Sandstone
The Miocene formation is a low-porosity sandstone of the feldspathic arenite type with an
initial porosity of 0.083. It is known that due to high roundness and sorting of its grains, the
sandstone contains long
ow and transport paths over large distances (Saitoh and Masuda 2004).
Figure 4.9 presents the comparison between the predictions with the pressure-dependence of the
electrical conductivity data of Yale (1984). The agreement is excellent. The well-connected and
long transport and
ow paths of the sandstone practically guarantee accurate predictions, because
it is precisely under such conditions that the EMA is accurate.
0 50 100 150 200 250 300 350 400 450
Pressure (Bar)
0.5
0.6
0.7
0.8
0.9
1
1.1
Normalized Conductivity
Miocene 7
Predicted
Experimental Data
0 200 400 600
Pressure (Bar)
0.9
0.95
1
/
0
Figure 4.9: Comparison of the predicted conductivities with the experimental data for the Miocene
7 sandstone.
69
4.6.10 Pliocene Sandstone
Pliocene is the second and terminal epoch of Neogene period. The Pliocene formations are
found in both marine form found in the Indian Ocean and western part of Yemen, and in the
form of nonmarine sedimentary rock in the continental United States in, for example, Washington
State (Walsh et al. 1987) and Oklahoma (Heran et al. 2003). The initial porosity of the sample,
referred to as Pliocene 35 by Yale (1984), was 0.2. Figure 4.10 compares the predictions with the
data. The experimental data are somewhat scattered, but the largest dierence between the data
and predictions is about 9 percent at 250 bars.
70
0 50 100 150 200 250 300 350 400 450
Pressure (Bar)
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
1.05
1.1
Normalized Conductivity
Pliocene 35
Predicted
Experimental Data
0 200 400 600
Pressure (Bar)
0.94
0.96
0.98
1
/
0
Figure 4.10: Comparison of the predicted conductivities with the experimental data for the
Pliocene 35 sandstone.
4.6.11 Tensleep Sandstone
Tensleep sandstone is from a geological formation in the entire Pennsylvanian sequence in
central and northern Wyoming in the very early Permian age (Branson and Branson, 1941), and
represents crossbedded sandstone with thin limestone and dolomite beds (Kerr et al. 1986). The
initial porosity of the sandstone was 0.146. In Figure 4.11 we compare the predictions with the
experimental data reported by Yale (1984). The agreement is very good over much of the range
of the applied pressure. Note that after initially declining, the porosity remains unchange over
a range of pressure, and then declines again, which explains the slower decline of the measured
conductivity than the predictions at the highest pressures.
71
0 50 100 150 200 250 300 350
Pressure (Bar)
0.7
0.75
0.8
0.85
0.9
0.95
1
1.05
Normalized Conductivity
Tensleep
Predicted
Experimental Data
0 200 400
Pressure (Bar)
0.985
0.99
0.995
1
/
0
Figure 4.11: Comparison of the predicted conductivities with the experimental data for the
Tensleep 35 sandstone.
4.6.12 Gulf Coast Sandstone
Yale (1984) referred to the sandstone as Tertiary 807. Tertiary rocks were formed during
part of the Cenozoic era. The initial porosity of the sample was 0.22. Figure 4.12 compares the
theoretical predictions with the experimental data. The agreement between the predictions and
the data is excellent.
72
0 50 100 150 200 250 300 350 400
Pressure (Bar)
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
1.05
1.1
Normalized Conductivity
Gulf Coast
Predicted
Experimental Data
0 200 400 600
Pressure (Bar)
0.94
0.96
0.98
1
/
0
Figure 4.12: Comparison of the predicted conductivities with the experimental data for the Ter-
tiary 807 sandstone.
4.6.13 Torpedo Sandstone
The Torpedo sandstone was from Kansas with an initial porosity of 0.202. As Figure 4.13
indicates, the predicted electrical conductivity closely matches the measured data.
73
0 50 100 150 200 250 300 350 400 450
Pressure (Bar)
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
1.05
1.1
Normalized Conductivity
Torpedo
Predicted
Experimental Data
0 200 400 600
Pressure (Bar)
0.97
0.98
0.99
1
/
0
Figure 4.13: Comparison of the predicted conductivities with the experimental data for the Tor-
pedo sandstone.
4.6.14 Triassic Sandstones
Triassic rocks were formed in the Triassic period, between 200 and 251 million years ago,
with the morphology of such sandstones varying greatly, from very ne- to very coarse-grained.
Although, generally speaking, they are porous formations with low- or ultra-low
ow properties,
they often have both tectonic and diagenetic fractures that provide
ow paths. Yale (1984)
reported the data for ve samples of such sandstones, referred to as Triassic 26, 27, 34, 38,
and 41, with their initial porosities being, respectively, 0.18, 0.18, 0.2, 0.2, and 0.21. Figures
4.14 compare the predictions for the conductivities with the experimental data. The agreement
between the predictions and the data is uniformly very good for all samples.
74
Figure 4.14: Comparison of the predicted conductivities with the experimental data for the Triassic sandstone.
75
4.6.15 Branford Sandstone
Figure 4.15 compares the predicted pressure-dependence of the electrical conductivity of Bran-
ford (Connecticut) sandstone with the experimental data of Yale (1984). The initial porosity
0
of the sandstone, which is of sublitharenite type with medium-size and well-sorted grains made
mostly of quartz, carbonate, and clay cement is 0.11 (Bernab e 1989). Execept for pressures
P ¡ 300 bars the agreement between the two is excellent. We shall return to the issue of the tail
of the curve in Section 9.
0 50 100 150 200 250 300 350
Pressure (Bar)
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
Normalized Conductivity
Branford
Predicted
Experimental Data
0 200 400 600
Pressure (Bar)
0.9
0.95
1
/
0
Figure 4.15: Comparison of the predicted conductivities with the experimental data for the Bran-
ford sandstone.
4.6.16 Kirkwood Sandstone
Finally, we show in Figure 4.16 the predictions for the conductivity of the Kirkwood sandstone
and compare them with the data. The sandstone is a fairly clean orthoquartzite (Wyble 1958)
76
with a porosity varying between 0.13 and 0.19. Once again, except for the tail of the curve
(to which we shall return in the next section), the agreement between the predictions and the
data is essentially perfect, hence demonstrating the accuracy of the proposed model. Note that
we showed in a previous paper (Richesson and Sahimi 2019) that the theory provides highly
accurate predictions for the pressure-dependence of the eective permeability of both Branford
and Kirkwood sandstone.
0 50 100 150 200 250 300 350
Pressure (Bar)
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Normalized Conductivity
Kirkwood
Predicted
Experimental Data
0 200 400 600
Pressure (Bar)
0.94
0.96
0.98
1
/
0
Figure 4.16: Comparison of the predicted conductivities with the experimental data for the Kirk-
wood sandstone.
4.7 Discussion
Overall, the MFA and EMA for conductivity agrees well with the experimental data. The
largest deviations are under 10% and they are usually near the largest pressure where our model
under predicts the eective conductivity. This is not dissimilar to what was seen in Chapter 3,
and we conclude a similar reasoning for this. See section 3.5 for an applicable discussion of the
77
MFA, EMA, morphology, and deformation mode. Specically for the conductivity predictions, we
want to point out that all the conductivity predictions are consistent with the Young's modulus
used for the permeability predictions. This at the very least creates a self-consistency for the
model.
4.8 Summary
The introduction of this Chapter argued that the electrical conductivity of a porous medium
lled with a conducting
uid such as brine gives light to the complex morphology of that medium.
Additionally, Chapter 1 argued that it is this complex morphology that governs the important
industrial transport properties of that medium, and that predicting these properties can be nan-
cially advantageous for each industry. The excellent agreement between most of our predictions
and the experimental data show that this predictive and dynamic model, while relatively simple,
can be an important step towards simplifying this dicult problem. It is also clear that we have
kept the model as simple as possible. Adding complexities to the system, such as two-phase
ow,
allowing for non-spherical pores, or including anisotropic features could add more robustness to
its predictive power. Work in this direction is in progress.
78
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Abstract (if available)
Abstract
This thesis presents a new theoretical approach for computing the dynamic effective properties of porous media that are under deformation by a hydrostatic pressure P. Beginning with the initial pore-size distribution (PSD) of a porous medium before deformation, and given the Young’s modulus and Poisson’s ratio of its grains, the model uses an extension of the Hertz–Mindlin theory of contact between grains to compute the new PSD that results from applying the pressure P to the medium. It then utilizes the updated PSD in the effective-medium approximation (EMA) to estimate the effective property. In the case of permeability, this theory directly predicts the experimental data. For conductivity, we first account for the contribution from surface conduction, and then we utilize the theory to update the PSD and, hence, the pore-conductance distribution, which is then used in the EMA to predict the pressure-dependence of the conductance. In this work we have used the theory to compute the effective permeability and effective electrical conductivity (when saturated with brine) of twenty-nine different sandstones. Comparison between the predictions and experimental data indicates agreement between the two that ranges from very good to excellent.
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Asset Metadata
Creator
Richesson, Samuel Wood
(author)
Core Title
Effective flow and transport properties of deforming porous media and materials: theoretical modeling and comparison with experimental data
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Chemical Engineering
Publication Date
04/14/2021
Defense Date
03/15/2021
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
conductivity,deformation,effective-medium approximation,formation factor,OAI-PMH Harvest,permeability,porous media,sandstones
Language
English
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Electronically uploaded by the author
(provenance)
Advisor
Sahimi, Muhammad (
committee chair
), de Barros, Felipe (
committee member
), Jessen, Kristian (
committee member
), Jha, Birendra (
committee member
), Kalia, Rajiv (
committee member
)
Creator Email
srich3sson@gmail.com,srichess@usc.edu
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https://doi.org/10.25549/usctheses-c89-442317
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Richesson, Samuel Wood
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Tags
conductivity
deformation
effective-medium approximation
formation factor
permeability
porous media
sandstones