Flow and Transport Properties of Deforming Porous Media. II. Electrical Conductivity
变形多孔介质流动和传输性质。第二部分：电导率

Abstract

In Part I of this series, we presented a new theoretical approach for computing the effective permeability 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 used 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 and utilized the updated PSD in the effective-medium approximation (EMA) to estimate the effective permeability. In the present paper, we extend the theory in order to compute the electrical conductivity of the same porous media that are saturated by brine. We account for the possible contribution of surface conduction, in order to estimate the electrical conductivity of brine-saturated porous media. We then 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 electrical conductivity. Comparison between the predictions and experimental data for twenty-six sandstones indicates agreement between the two that ranges from excellent to good.
在本文的第一部分，我们提出了一种新的理论方法来计算在静水压力 P 作用下变形的多孔介质的有效渗透率。从变形前多孔介质的初始孔隙尺寸分布（PSD）开始，并给出其颗粒的杨氏模量和泊松比，该模型使用 Hertz-Mindlin 理论扩展了颗粒之间的接触来计算由于施加压力 P 到介质而产生的新的 PSD，并利用更新的 PSD 在有效介质近似（EMA）中估计有效渗透率。在本文中，我们扩展了该理论，以便计算被盐水饱和的相同多孔介质的电导率。我们考虑了表面传导的可能贡献，以估计盐水饱和多孔介质的电导率。然后，我们利用该理论更新 PSD，从而更新孔隙电导率分布，然后将其用于 EMA 来预测电导率的压力依赖性。 二十六种砂岩的预测值与实验数据的比较表明，两者之间的一致性从优秀到良好不等。

1 Introduction
1 引言

This paper is the second part in a series devoted to theoretical analysis and computation of flow and transport properties of porous media that undergo deformation as a result of applying an external force or hydrostatic pressure. In the previous paper (Richesson and Sahimi under review), hereafter referred to as Part I, we analyzed the problem of estimating the effective permeability $K_{\text{e}}$ of deforming porous media and presented a new formulation of the problem based on a modified Hertz–Mindlin theory of contacting grain (Hertz 1882; Mindlin 1949) and the effective-medium approximation (EMA) for transport properties of heterogeneous materials. We demonstrated that the new formulation provides accurate estimates of $K_{\text{e}}$ by comparing its predictions with a large set of experimental data for a variety of sandstones. In the present paper, we analyze the problem of predicting the electrical conductivity of brine-saturated porous media and, in particular, in the same sandstones.
本文是该系列论文的第二部分，专注于对因施加外力或静水压力而变形的多孔介质流动和传输特性的理论分析和计算。在前一篇论文（Richesson 和 Sahimi，待审稿），以下简称第一部分，我们分析了估计变形多孔介质有效渗透率 $K_{\text{e}}$ 的问题，并基于改进的赫兹-米林接触颗粒理论（Hertz 1882；Mindlin 1949）和异质材料传输特性的有效介质近似（EMA）提出了新的问题公式。我们通过将其预测与大量不同砂岩的实验数据进行比较，证明了新公式的准确性。在本文中，我们分析了预测盐水饱和多孔介质电导率的问题，特别是在相同的砂岩中。

As is well known, in most porous materials, both natural and synthetic, pores have complex shapes; they are interconnected with their connectivity being stochastically distributed and forming tortuous flow and transport paths. 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 flow 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 ${\text{σ}}_{\text{f}}$. If the effective conductivity of the saturated medium is ${\text{σ}}_{\text{e}}$, then
众所周知，在大多数多孔材料中，无论是天然的还是合成的，孔隙都具有复杂的形状；它们相互连接，其连通性呈随机分布，形成曲折的流动和传输路径。这种多孔材料的表征一直是基本问题中的一个重要问题，并且已经研究了很长时间。获取孔隙空间结构信息的一种方法是将它的流动和传输特性与揭示其形态的量联系起来。特别是，考虑一个被盐水（电导率为 ${\text{σ}}_{\text{f}}$ ）饱和的多孔介质。如果饱和介质的有效电导率为 ${\text{σ}}_{\text{e}}$ ，那么

$$\begin{array}{r}F=\frac{{\text{σ}}_{\text{f}}}{{\text{σ}}_{\text{e}}},\end{array}$$

(1)

where F, the formation factor, is a geometrical characteristic of the pore space. In the limit of long times, the effective diffusion coefficient of a molecule probing the same pore space is given by
其中，F，即形成因子，是孔隙空间的几何特征。在长时间极限下，探测相同孔隙空间的分子的有效扩散系数由下式给出

$$\begin{array}{r}{D}_{\text{e}}=\frac{D_0}{F\varphi },\end{array}$$

(2)

where $\varphi$ is the porosity of the porous medium, and $D_0$ is the free diffusion coefficient outside the pore space. It is, therefore, clear that measurements of $D_{\text{e}}$ and ${\text{σ}}_{\text{e}}$, or the ability to accurately predict them, provide valuable information on the morphology of the pore space (see, for example, Vanderborght et al. 2005; Crook et al. 2008; Ghanbarian and Berg 2017). Thus, in addition to numerous measurements of the diffusivity and electrical conductivity of a wide variety of porous media, various theoretical approaches have also been developed for predicting ${\text{σ}}_{\text{e}}$ and $D_{\text{e}}$, beginning with Archie (1942) and Wyllie and Spangler (1952), and continuing with Woessner (1963); Stejskal and Tanner (1965), and Karger et al. (1981), all the way to the present time (Ghanbarian et al. 2014; Dashtian et al. 2015); for a review, see Cai et al. (2017). Moreover, short-time behavior of $D_{\text{e}}$, before it reaches its asymptotic value given by Eq. (2), has been linked with the ratio of surface $S_{\text{p}}$ of the pores and their volume $V_{\text{p}}$, and has been exploited fruitfully to make progress on the general problem of characterizing a porous medium (Mitra and Sen 1992; Mitra et al. 1992a, 1993).
孔隙介质的孔隙率是 $\varphi$ ，孔隙空间外的自由扩散系数是 $D_0$ 。因此，显然， $D_{\text{e}}$ 和 ${\text{σ}}_{\text{e}}$ 的测量，或者准确预测它们的能力，提供了关于孔隙空间形态的有价值信息（例如，参见 Vanderborght 等人，2005 年；Crook 等人，2008 年；Ghanbarian 和 Berg，2017 年）。因此，除了对各种多孔介质扩散率和电导率的众多测量外，还开发了各种理论方法来预测 ${\text{σ}}_{\text{e}}$ 和 $D_{\text{e}}$ ，从 Archie（1942 年）和 Wyllie 和 Spangler（1952 年）开始，继续到 Woessner（1963 年）；Stejskal 和 Tanner（1965 年），以及 Karger 等人（1981 年），一直到现在（Ghanbarian 等人，2014 年；Dashtian 等人，2015 年）；有关综述，参见 Cai 等人（2017 年）。此外，在达到由方程式给出的渐近值之前， $D_{\text{e}}$ 的短期行为。 (2) 已与孔隙的表面 $S_{\text{p}}$ 比率和它们的体积 $V_{\text{p}}$ 相关联，并已被充分利用以在表征多孔介质的一般问题上取得进展（Mitra 和 Sen 1992；Mitra 等 1992a，1993）。

Almost all the progress that has been made over the past several decades relates to the electrical conductivity and diffusivity of rigid porous media. As discussed in Part I, flow and transport properties of porous media that deform as a result of being subjected to an external driving potential, either pressure or force, are also important and appear in many problems of fundamental scientific importance, as well as practical applications, particularly in characterization of laboratory-scale porous media, as well as gaining information about the structure of field-scale porous media through their resistivity logs.
过去几十年所取得的几乎所有进展都与刚性多孔介质的电导率和扩散率有关。如第一部分所述，由于受到外部驱动势（压力或力）的作用而变形的多孔介质的流动和传输特性也很重要，并出现在许多具有基础科学意义的问题中，以及实际应用中，特别是在表征实验室规模的多孔介质以及通过其电阻率曲线获取关于现场规模多孔介质结构的信息。

In the present paper, we study the problem of the pressure dependence of the static electrical conductivity of the brine-saturated porous media. Woodruff et al. (2015) developed an experimental approach for simultaneous measurement of the stress dependence of ultrasonic wave velocities and the components of the elastic stiffness and of the complex conductivity tensors over a wide range of frequency, from 100 mHz to 10 Hz. Their data indicated strong stress dependence of the ultrasonic and complex conductivity measurements, which they attributed to the opening and closing of cracks within the samples. In a more recent paper, Revil et al. (2019) reported extensive experimental data for the effect of the pressure on the complex conductivity of five sandstone core samples from outcrops, as well as a sandstone analog that they had constructed using sintered glass beads. Their data may be analyzed by a method similar to what we describe in the present paper, except that one must use the EMA for frequency-dependent conductivity of heterogeneous materials (Odagaki and Lax 1981; Sahimi et al. 1983); see Sahimi (2003) for a comprehensive discussion).
在本文中，我们研究了盐水饱和多孔介质静态电导率的压力依赖性问题。Woodruff 等人（2015 年）开发了一种实验方法，用于同时测量超声波速度和弹性刚度分量以及复电导率张量在广泛频率范围内的应力依赖性，从 100 mHz 到 10 Hz。他们的数据显示，超声和复电导率测量具有强烈的应力依赖性，他们将此归因于样品内部裂缝的开启和关闭。在最近的一篇论文中，Revil 等人（2019 年）报告了关于压力对五个来自露头的砂岩岩心样品以及他们使用烧结玻璃珠构建的砂岩模拟物的复电导率影响的广泛实验数据。他们的数据可以通过与我们本文中描述的方法类似的方法进行分析，但必须使用 EMA 来测量异质材料的频率相关电导率（Odagaki 和 Lax 1981；Sahimi 等人 1983）；参见 Sahimi（2003 年）以获得全面讨论）。

The rest of this paper is organized as follows: In Sect. 2, we summarize the new theoretical approach to determining the change in the size of the pores when a hydrostatic pressure P is applied to a porous medium. The theory is then used in Sect. 3 to determine the evolution of the pore-conductance distribution (PCD) during deformation. Section 4 describes the EMA for estimating the effective electrical conductivity of brine-saturated porous media. In Sect. 5, we describe how to estimate the contribution of surface conduction to the total conductivity of the pore space. Since the proposed theory involves three parameters, Sect. 6 describes their estimation. The computational procedure is described in Sect. 7, while Sect. 8 presents the theoretical predictions for the electrical conductivity of a wide variety of sandstones and compares them with the experimental data. Section 9 discusses a few aspects of the theoretical approach and its limitations, while the paper is summarized in Sect. 10.
本文的其余部分组织如下：在第 2 节中，我们总结了在将静水压力 P 施加到多孔介质上时确定孔隙尺寸变化的新理论方法。然后在第 3 节中，利用该理论确定孔隙导通率分布（PCD）在变形过程中的演变。第 4 节描述了用于估计盐水饱和多孔介质有效电导率的 EMA。在第 5 节中，我们描述了如何估计表面导通对孔隙空间总电导率的贡献。由于所提出的理论涉及三个参数，第 6 节描述了它们的估计。计算过程在第 7 节中描述，而第 8 节展示了针对各种砂岩的电导率的理论预测，并将它们与实验数据进行了比较。第 9 节讨论了理论方法的一些方面及其局限性，而第 10 节总结了全文。

2 Mean-Field Theory of Deformation of Porous Media under a Hydrostatic Pressure
静水压力下多孔介质变形的平均场理论

As mentioned in Introduction, we employ the EMA to predict the effective electrical conductivity of porous media that deform as a result of applying an external force or hydrostatic pressure. The EMA represents a type of mean-field approximation (MFA) that replaces a heterogeneous porous medium by an uniform one in which all the pores have the same effective size $r_{\text{e}}$. Deriving the solution of the flow or transport problem in the uniform system is straightforward. Then, the radius $r_{\text{e}}$ of a randomly selected pore in the effective medium is replaced by its actual value in the original heterogeneous pore space, with the remaining pores still having the same size $r_{\text{e}}$. The replacement gives rise to perturbation in the solution of the uniform porous medium, whose magnitude is the difference between the solution for the uniform medium and one that is uniform everywhere, except in that one pore. Since the single pore is selected at random and its actual size follows the pore-size distribution (PSD) of the pore space, one sets to zero the average of the perturbation, with the averaging taken over the PSD. In effect, what the EMA does is transforming a multibody system into a one-body problem by accounting for the interaction of only a single pore with the rest of the effective medium. The effect of the remaining part of the disordered porous medium is represented by the far-field external pressure or potential gradient applied to the porous medium for inducing fluid flow or transport.
如引言所述，我们采用 EMA 来预测由施加外部力或静水压力而变形的多孔介质的有效电导率。EMA 代表一种平均场近似（MFA），它将异质多孔介质替换为一个均匀介质，其中所有孔隙具有相同的有效尺寸 $r_{\text{e}}$ 。在均匀系统中推导流动或传输问题的解是直接的。然后，将有效介质中随机选择的孔隙的半径 $r_{\text{e}}$ 替换为其在原始异质孔隙空间中的实际值，其余孔隙仍保持相同的大小 $r_{\text{e}}$ 。这种替换会在均匀多孔介质的解中引起扰动，其大小是均匀介质解与除一个孔隙外处处均匀的解之间的差异。由于单个孔隙是随机选择的，其实际尺寸遵循孔隙空间孔隙尺寸分布（PSD），因此将扰动的平均值设为零，平均是在 PSD 上进行的。 实际上，EMA 所做的是通过只考虑单个孔隙与有效介质其余部分的相互作用，将多体系统转化为单体问题。无序多孔介质剩余部分的影响由施加于多孔介质上的远场外部压力或势梯度来表示，以诱导流体流动或传输。

In the spirit of the EMA, and in order to develop a tractable theoretical approach consistent with the mean-field nature of the EMA, we considered in Part I the interaction between only two grains and its influence on the pore between the two, subject to an external force F or the corresponding hydrostatic pressure P applied to the porous medium. We then derived a formulation for determining how the deformation of the pore space changes the size of pore between the two grains. Similar to the EMA, the effect of the deformation of the rest of the pore space was represented by the far-field applied pressure P or force F. The applied pressure P changes the PSD of the pore space, which in turn alters its effective permeability and electrical conductivity. The input to the EMA is the PSD or (PCD) of the pore space. Therefore, if the change in the PSD or PCD as a result of applying a pressure or force to the pore space is determined, the EMA may be used to estimate the effective permeability $K_{\text{e}}$ and electrical conductivity ${\text{σ}}_{\text{e}}$ of the porous medium at that pressure. In Part I, the idea was developed for computing $K_{\text{e}}$. In the present paper, we further extend the theory in order to compute ${\text{σ}}_{\text{e}}$.
在 EMA 的精神指导下，为了发展一个与 EMA 的平均场性质一致的可处理理论方法，我们在第一部分考虑了只有两个颗粒之间的相互作用及其对孔隙的影响，该孔隙受到外部力 F 或施加在多孔介质上的相应静水压力 P。然后，我们推导出一个公式，以确定孔隙空间的变形如何改变两个颗粒之间的孔隙大小。与 EMA 类似，其余孔隙空间的变形效应由远场施加的压力 P 或力 F 表示。施加的压力 P 改变了孔隙空间的 PSD，进而改变其有效渗透率和电导率。EMA 的输入是孔隙空间的 PSD 或（PCD）。因此，如果确定了施加压力或力到孔隙空间后 PSD 或 PCD 的变化，可以使用 EMA 来估计在该压力下多孔介质的有效渗透率 $K_{\text{e}}$ 和电导率 ${\text{σ}}_{\text{e}}$ 。在第一部分，开发了计算 $K_{\text{e}}$ 的想法。 在本文中，我们进一步扩展了理论以便计算 ${\text{σ}}_{\text{e}}$ 。

Since the equation that describes the change in the PSD of a porous medium as a result of applying a hydrostatic pressure P was derived in Part I, we only present the final result here and refer the interested reader to Part I for complete details. In Part I, we showed that by applying a hydrostatic pressure to a porous medium, the closest distance between the surfaces of the aforementioned two grains decreases by u, given by
由于在第一部分中已经推导出描述孔隙介质 PSD 随静水压力 P 变化规律的方程，这里我们只给出最终结果，并请感兴趣的读者参阅第一部分以获取完整细节。在第一部分中，我们展示了通过对孔隙介质施加静水压力，上述两种颗粒之间的最近距离减少 u，其中 u 由以下公式给出：

$$\begin{array}{r}u=R_{\text{g}}{\left(\frac{R_{\text{g}}}{R}\right)}^{1/3}{\left[\frac{3P(1-{\nu }^2)}{E_{\text{e}}}\right]}^{2/3},\end{array}$$

(3)

where $E_{\text{e}}$, $R_{\text{g}}$, R, and $\nu$ are, respectively, the effective Young’s modulus, radius, radius of curvature, and the Poisson’s ratio of the grains. If the grains are roughly spherical, then $R_{\text{g}}\approx R$, which we assume to be the case or that, at the minimum, we can define an effective radius for an equivalent spherical particle.
其中， $E_{\text{e}}$ 、 $R_{\text{g}}$ 、R 和 $\nu$ 分别代表颗粒的有效杨氏模量、半径、曲率半径和泊松比。如果颗粒大致呈球形，那么 $R_{\text{g}}\approx R$ ，我们假设这种情况成立，或者至少可以定义一个等效球形颗粒的有效半径。

3 Evolution of Pore-Size and Pore-Conductance Distributions as Functions of the Hydrostatic Pressure
孔隙尺寸和孔隙电导率分布随静水压力的变化演化

Since the MFA neglects the interactions of two neighboring grains with other grains, the pore between them does not also interact with the pores farther away. Therefore, to a first-order approximation, the effective radius of the pore between two grains in a porous medium under an external hydrostatic pressure P decreases by u/2, where u is given by Eq. (3). In other words, the initial PSD distribution $f_0(r_0)$ before any pressure is applied is transformed to a new PSD $f_{\text{P}}(r_{\text{P}})$ at pressure P where $r_{\text{P}}=r_0-u/2$. If $f_0(r_0)$ is given, either analytically or numerically, then, since, $f_{\text{P}}(r_{\text{P}})=f_0(r_0-u/2)$, one either has an analytical expression for $f_{\text{P}}(r_{\text{P}})$, or constructs it numerically for any pressure P.
由于 MFA 忽略了两个相邻颗粒与其他颗粒的相互作用，因此它们之间的孔隙也不会与更远处的孔隙相互作用。因此，在施加外部静水压力 P 的情况下，多孔介质中两个颗粒之间的孔隙有效半径按一阶近似减少 u/2，其中 u 由公式(3)给出。换句话说，在施加任何压力之前，初始 PSD 分布 $f_0(r_0)$ 转变为压力 P 下的新 PSD 分布 $f_{\text{P}}(r_{\text{P}})$ ，其中 $r_{\text{P}}=r_0-u/2$ 。如果 $f_0(r_0)$ 给出，无论是解析还是数值，那么，由于 $f_{\text{P}}(r_{\text{P}})=f_0(r_0-u/2)$ ，对于任何压力 P，要么有 $f_{\text{P}}(r_{\text{P}})$ 的解析表达式，要么数值构造它。

Suppose that ${\text{σ}}_{\text{P}}$ is the conductance of a pore when the pore space is under the pressure P. Then, since ${\text{σ}}_{\text{P}}\propto r_{\text{P}}^2$, given a PSD $f_{\text{P}}(r_{\text{P}})$, one obtains the PCD by $h_{\text{P}}({\text{σ}}_{\text{P}})d{\text{σ}}_{\text{P}}=f_{\text{P}}(r_{\text{P}})d{r}_{\text{P}}$. That is, for any pressure P, the PSD $f_{\text{P}}(r_{\text{P}})$, updated by using Eq. (3), yields the corresponding PCD $h_{\text{P}}({\text{σ}}_{\text{P}})$, which is then used in the EMA in order to determine the effective electrical conductivity ${\text{σ}}_{\text{e}}$ of the porous medium.
假设当孔隙空间处于压力 P 下时， ${\text{σ}}_{\text{P}}$ 是孔隙的电导率。那么，由于 ${\text{σ}}_{\text{P}}\propto r_{\text{P}}^2$ ，给定一个 PSD $f_{\text{P}}(r_{\text{P}})$ ，通过 $h_{\text{P}}({\text{σ}}_{\text{P}})d{\text{σ}}_{\text{P}}=f_{\text{P}}(r_{\text{P}})d{r}_{\text{P}}$ 得到 PCD。也就是说，对于任何压力 P，PSD $f_{\text{P}}(r_{\text{P}})$ ，通过使用公式(3)更新，得到相应的 PCD $h_{\text{P}}({\text{σ}}_{\text{P}})$ ，然后用于 EMA 以确定多孔介质的有效电导率 ${\text{σ}}_{\text{e}}$ 。

4 Effective-Medium Approximation for the Effective Conductivity
有效电导率的有效介质近似

The morphology of any porous medium consists of pore throats that are connected together via the pore bodies. The effective sizes of both pore throats and pore bodies are distributed according to statistical distributions $f_{\text{t}}(r_{\text{t}})$ and $f_{\text{b}}(r_{\text{b}})$. But, whereas experimental measurement of $f_{\text{t}}(r_{\text{t}})$ is straightforward (see, for example, Sahimi 2011; Blunt 2017 for discussions of various measurement methods), measuring $f_{\text{b}}(r_{\text{b}})$ is not straightforward. On the other hand, both macroscopic permeability and electrical conductivity of porous media are controlled by the pore throats, which, for convenience, are referred to as pores, and their distribution f(r) 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 $r_{\text{e}}$, which we assume to be cylindrical, then its electrical conductivity is, ${\text{σ}}_{\text{P}}\propto r_{\text{P}}^2$. One can, of course, consider other pore shapes. The EMA predicts that the macroscopic electrical conductivity ${\text{σ}}_{\text{e}}$ is given by Doyen (1988); David et al. (1990)
任何多孔介质的形态由相互连接的孔隙通道组成，这些孔隙通道通过孔隙体连接。孔隙通道和孔隙体的有效尺寸都按照统计分布 $f_{\text{t}}(r_{\text{t}})$ 和 $f_{\text{b}}(r_{\text{b}})$ 分布。然而，虽然 $f_{\text{t}}(r_{\text{t}})$ 的实验测量是直接的（例如，参见 Sahimi 2011；Blunt 2017 关于各种测量方法的讨论），但测量 $f_{\text{b}}(r_{\text{b}})$ 并不直接。另一方面，多孔介质的宏观渗透率和电导率都受孔隙通道的控制，为了方便起见，孔隙通道被称为孔隙，其分布 f(r)称为 PSD。由于在 EMA 中，非均匀孔隙空间被表示为所有孔隙尺寸为 $r_{\text{e}}$ 的均匀介质，我们假设其为圆柱形，因此其电导率为 ${\text{σ}}_{\text{P}}\propto r_{\text{P}}^2$ 。当然，也可以考虑其他孔隙形状。EMA 预测宏观电导率 ${\text{σ}}_{\text{e}}$ 由 Doyen（1988）；David 等人（1990）给出。

$$\begin{array}{r}{\text{σ}}_{\text{e}}=\left(\frac{\varphi }{\tau }\frac{r_{\text{e}}^2}{⟨r_{\text{b}}^2⟩}\right){\text{σ}}_{\text{f}},\end{array}$$

(4)

where $\varphi$ is the porosity, and $\tau$ 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 $f_{\text{b}}(r_{\text{b}})$ of the size of the pore bodies is typically not available, David et al. (1990) suggested that one should use,
孔隙率 和曲折率 $\varphi$ ，对于各种理论以及经验性和半经验性关系都已有研究（参见 Ghanbarian-Alavijeh 等人 2013 年的综述）。由于孔隙体尺寸的分布 $\tau$ 通常不可用，David 等人（1990 年）建议应使用，

$$\begin{array}{r}⟨r_{\text{b}}^2⟩\simeq ⟨r^2⟩=\underset{r_{\text{m}}}{\overset{r_{\text{M}}}{\int }}{r}^{2}f(r)dr,\end{array}$$

(5)

with $r_{\text{m}}$ and $r_{\text{M}}$ being, respectively, the minimum and maximum pore radii. We assume the same in this paper. In the EMA, an effective conductance $g_{\text{e}}$ is computed by
与 $r_{\text{m}}$ 和 $r_{\text{M}}$ 分别为最小和最大孔隙半径相对应。本文假设相同。在 EMA 中，通过计算 $g_{\text{e}}$ 的有效电导率。

$$\begin{array}{r}\underset{g_{\text{m}}}{\overset{g_{\text{M}}}{\int }}\frac{g_{\text{e}}-g}{g+(D-1)g_{\text{e}}}h(g)dg=0.\end{array}$$

(6)

Here, $g_{\text{e}}$ and $g_{\text{M}}$ are, respectively, the minimum and maximum conductances, D is the dimensionality of space ($D=3$), and h(g) is the pore-conductance distribution. Since $g_{\text{e}}\propto r_{\text{e}}^2$, $g\propto r^2$, and $h(g)dg=f(r)dr$, so that $h(g)=f(r^2)dr/dg=f(r^2)(1/2r)$, we obtain an alternative formulation directly in terms of f(r):
这里， $g_{\text{e}}$ 和 $g_{\text{M}}$ 分别是最小和最大电导率，D 是空间维度（ $D=3$ ），h(g) 是孔隙电导率分布。由于 $g_{\text{e}}\propto r_{\text{e}}^2$ ， $g\propto r^2$ ，和 $h(g)dg=f(r)dr$ ，因此 $h(g)=f(r^2)dr/dg=f(r^2)(1/2r)$ ，我们直接以 f(r) 为基础得到一个替代公式。

$$\begin{array}{r}\underset{r_{\text{m}}^2}{\overset{r_{\text{M}}^2}{\int }}\frac{r_{\text{e}}^2-r^2}{r^2+(D-1)r_{\text{e}}^2}\frac{f(r^2)}{2r}dr=0.\end{array}$$

(7)

As for the tortuosity factor $\tau$, we used $\tau ={\varphi }^{1-m}$. The review by Ghabbarian et al. (2013) indicates that $1.2\le m\le 4.4$ for a wide variety of porous media. After some preliminary simulations by which we tested the accuracy of various values of m, we set $m=4$, which is for tortuous porous media.
关于曲折因子 $\tau$ ，我们使用了 $\tau ={\varphi }^{1-m}$ 。Ghabbarian 等人（2013）的综述表明， $1.2\le m\le 4.4$ 适用于各种多孔介质。在进行了初步模拟以测试 m 值各种值的准确性之后，我们设定了 $m=4$ ，这是针对曲折多孔介质的。

Mukhopadhyay and Sahimi (2000) derived an EMA for predicting direction-dependent macroscopic 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.
Mukhopadhyay 和 Sahimi（2000）推导出了一种 EMA 来预测各向异性多孔介质方向相关的宏观电导率，而 Stroud（1975）提出了一种连续 EMA，用于各向异性介质，其中局部电导率是一个张量。EMA 的其他应用在第 I 部分中已指出，感兴趣的读者可参考。

5 Accounting for Surface Conduction
考虑表面传导

The experimental data for the electrical conductivity of porous media that we compare with the predictions of the model are in terms of the formation resistivity factor F, reported by Yale (1984). Equation (1) that defines F is based on the effective 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 significantly to the overall measured conductivity, since the clay grains’ surface allows for a layer of counterions that facilitates the development of a significant negative surface charge (Waxman and Smits 1968; Clavier et al. 1984). In fact, Revil et al. (2019) reported that the conductivity of the sandstones that they measured was dominated by their surface conductivity, since pore water salinity was low in their experiments. They interpreted their data using the Stern layer polarization model (Stern 1924). The Stern layer represents the inner part of the electrical double layer that coats the grains’ surface. The data reported by Yale (1984) are for a broad variety of sandstones, which do have significant clay content (see also below). Therefore, it is imperative to account for surface conduction, before comparing the predictions of the theory, which does not take into account the effect of surface conduction, with the data.
实验数据中，我们比较模型预测的孔隙介质电导率，是以耶鲁（1984 年）报道的孔隙形成电阻率因子 F 来表示的。定义 F 的方程（1）基于饱和孔隙空间的有效电导率，不包括孔隙表面传导的可能贡献。然而，根据孔隙介质的化学成分，尤其是在含粘土的介质中，表面传导可能对总测量的电导率有显著贡献，因为粘土颗粒的表面允许形成一层反离子层，从而促进显著负表面电荷的发展（Waxman 和 Smits 1968；Clavier 等 1984）。事实上，Revil 等人（2019 年）报告说，他们测量的砂岩的电导率主要由其表面电导率决定，因为他们的实验中孔隙水盐度低。他们使用 Stern 层极化模型（Stern 1924）来解释他们的数据。Stern 层代表覆盖颗粒表面的电双层内层。 报告的数据来自耶鲁（1984 年），涉及多种砂岩，这些砂岩具有显著的粘土含量（参见下文）。因此，在将理论的预测与数据比较之前，必须考虑表面传导，因为该理论没有考虑表面传导的影响。

Revil et al. (1998) derived the following equation for the total conductivity ${\text{σ}}_{\text{t}}$ of a pore space, including the contribution by surface conductivity, which is saturated by a fluid with a pH between 5 and 8:
Revil 等人（1998 年）推导出以下孔隙空间总电导率 ${\text{σ}}_{\text{t}}$ 的方程，包括表面电导率的贡献，该表面电导率被 pH 值在 5 到 8 之间的流体所饱和：

$$\begin{array}{r}{\text{σ}}_{\text{t}}=\left(\frac{{\text{σ}}_{\text{f}}}{F}\right)\{1-t_{(+)}^{\mathrm{f}}+F\xi +\frac{1}{2}(t_{(+)}^{\mathrm{f}}-\xi )[1-\frac{\xi }{t_{(+)}^{\mathrm{f}}}+\sqrt{{(1-\frac{\xi }{t_{(+)}^{\mathrm{f}}})}^2+\frac{4F\xi }{t_{(+)}^{\mathrm{f}}}}]\},\end{array}$$

(8)

where $t_{(+)}^{\mathrm{f}}$ is the Hittorf transport number of cations in the free electrolyte (brine in pores), and $\xi$ is known as the Dukhin number (Lyklema 1993), which is the ratio of surface and fluid conductivities. Equation (8) reduces to Eq. (1) in the limit $\xi \to 0$. The brine used in the experiments of Yale (1984) was NaCl, for which $t_{(+)}^{\mathrm{f}}\approx 0.38$. Thus, if the second parameter of Eq. (8), namely $\xi$, is also known, then, for every measured ${\text{σ}}_{\text{t}}$ one can use it to compute the corresponding formation factor F.
式中， $t_{(+)}^{\mathrm{f}}$ 是自由电解质（孔隙中的卤水）中阳离子的 Hittorf 迁移数， $\xi$ 被称为杜金数（Lyklema 1993），它是表面电导率和流体电导率的比值。当 $\xi \to 0$ 的极限时，方程（8）简化为方程（1）。耶鲁（1984）实验中使用的卤水是 NaCl，对于它， $t_{(+)}^{\mathrm{f}}\approx 0.38$ 。因此，如果方程（8）的第二个参数，即 $\xi$ 也为人所知，那么对于每个测量的 ${\text{σ}}_{\text{t}}$ ，都可以用它来计算相应的形成因子 F。

Revil et al. (1998) derived the following equation for the Dukhin number,
Revil 等人（1998 年）推导出以下关于杜金数的方程：

$$\begin{array}{r}\xi =\frac{2{\rho }_{\text{g}}C\beta }{3{\text{σ}}_{\text{f}}}.\end{array}$$

(9)

Here, ${\rho }_{\text{g}}$ is the density of the solid matrix, C is the cation exchange capacity, and $\beta$ is the equivalent mobility for surface conduction. Equation (9) provided accurate predictions for clay-rich sediments (Daigle 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 $\xi$ is independent of the hydrostatic pressure, we use one data point for the conductivity of every sandstone that we consider in order to estimate the Dukhin number using the above equation. This point was selected in the middle of the experimental pressure range.
这里， ${\rho }_{\text{g}}$ 是固体基质的密度，C 是阳离子交换容量， $\beta$ 是表面传导的等效迁移率。方程（9）为富含粘土的沉积物提供了准确的预测（Daigle 等人，2015 年）。由于耶鲁（1984 年）没有给出我们考虑的砂岩的颗粒密度和确切的化学成分，我们无法确定它们的杜金数。因此，由于 $\xi$ 与静水压力无关，我们使用每个砂岩的一个数据点来估计杜金数，使用上述方程。这个点是在实验压力范围的中间选择的。

6 Computational Procedure
6 计算过程

Given the theoretical formulation for predicting the effective electrical conductivity ${\text{σ}}_{\text{e}}$ of deforming porous media, the following computational procedure was used to calculate ${\text{σ}}_{\text{e}}$.
给出了预测变形多孔介质有效电导率的理论公式，随后采用了以下计算步骤进行计算。

• 1. Given the initial PSD $f_0(r_0)$ and, therefore, an initial conductance distribution $h_0(g_0)$ of an undeformed porous medium, its electrical conductivity was computed using Eqs. (4), (5), and (7).
  给定初始的 PSD 和因此未变形多孔介质的初始电导率分布，其电导率通过公式（4）、（5）和（7）进行计算。

• 2. For a given hydrostatic pressure P, the corresponding PSD $f_{\text{P}}(r_{\text{P}})$ was constructed by selecting the pore sizes from $f_0(r_0)$, calculating their updated values using Eq. (1) and $r_{\text{P}}=r_0-u/2$, and repeating it for a large number of pore sizes selected from $f_0(r_0)$, so that an accurate $f_{\text{P}}(r_{\text{P}})$ was obtained.
  对于给定的静水压力 P，通过从给定的孔径中选择孔径，使用公式（1）和（4）计算它们的更新值，并重复这一过程，对从给定的孔径中选择的大量孔径进行计算，从而获得准确的结果。

• 3. The PSD $f_{\text{P}}(r_{\text{P}})$ and Eq. (5) were then utilized to determine $⟨r_{\text{b}}^2(P)⟩=⟨r_{\text{P}}^2⟩$. The result was used together with $f_{\text{P}}(r_{\text{P}})$ in Eqs. (4) and (7) to compute ${\text{σ}}_{\text{e}}(P)$ at pressure P.
  PSD $f_{\text{P}}(r_{\text{P}})$ 和方程（5）随后被用来确定 $⟨r_{\text{b}}^2(P)⟩=⟨r_{\text{P}}^2⟩$ 。结果与 $f_{\text{P}}(r_{\text{P}})$ 一起用于方程（4）和（7）中，以计算压力 P 下的 ${\text{σ}}_{\text{e}}(P)$ 。

• 4. To compare the predictions in (3) with the experimental data for a given porous medium, we utilized the experimental value of the total conductivity ${\text{σ}}_{\text{t}}$ on the left side of Eq. (8) at a single pressure and determined the parameter $\xi$, using as the formation factor F its theoretical prediction. In effect, for every sandstone we used a single experimental point to estimate $\xi$. Since $\xi$ is independent of P, we used the same estimate in Eq. (8) and solved for the formation factor F at various pressures P, given that the left side of Eq. (8) is the experimental value of the total conductivity.
  为了将(3)中的预测与给定多孔介质的实验数据进行比较，我们利用了在单个压力下方程(8)左侧的总电导率 ${\text{σ}}_{\text{t}}$ 的实验值，并使用其理论预测作为形成因子 F 来确定参数 $\xi$ 。实际上，对于每种砂岩，我们使用一个实验点来估计 $\xi$ 。由于 $\xi$ 与 P 无关，我们在方程(8)中使用了相同的估计值，并在各种压力 P 下求解形成因子 F，前提是方程(8)的左侧是总电导率的实验值。

• 5. The resulting values of F at various pressures P 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.
  在各种压力 P 下 F 的结果值代表了给定多孔介质在压力范围内的真实地层因子，然后与理论预测进行比较。

Note that if all the parameters of Eq. (9) are known, it can be utilized to estimate $\xi$ directly, without any need for step (4). In that case, the estimate of $\xi$ and the experimental data for ${\text{σ}}_{\text{t}}$ for a given porous medium are used directly in Eq. (8) and the resulting nonlinear equation is solved numerically for the true formation factors F. Note also that if the PSD $f_0(r_0)$ is expressed by an analytical expression, $f_{\text{P}}(r_{\text{P}})$ will also be determined analytically, in which case the computations will be very fast.
注意，如果方程（9）的所有参数都已知，则可以直接利用它来估计 $\xi$ ，无需进行步骤（4）。在这种情况下，将直接使用给定多孔介质的 $\xi$ 的估计值和实验数据 ${\text{σ}}_{\text{t}}$ 代入方程（8），然后对得到的非线性方程进行数值求解，以得到真实的地层因子 F。此外，如果 PSD $f_0(r_0)$ 用解析表达式表示， $f_{\text{P}}(r_{\text{P}})$ 也将被解析确定，在这种情况下，计算将非常快速。

7 The Parameters of the Model
7 模型的参数

The parameters of the model are the Poisson’s ratio $\nu$ and the Young’s modulus $E_{\text{e}}$ of the grains (not the porous medium) that appear in Eq. (3), the PSD, and the Dukhin number $\xi$. 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.
模型参数为式(3)、PSD 和杜金数中的颗粒（而非多孔介质）的泊松比 $\nu$ 和杨氏模量 $E_{\text{e}}$ 。如果可获得参数的实验数据，它们可以直接用于理论中，但我们所分析砂岩的数据不可用。

We already described how we estimate $\xi$. As discussed in Part I, the predictions of the model are sensitive to the value of the Young’s modulus. In Part I, we explained how we estimate $E_{\text{e}}$. Since the experimental data that we compare with the theoretical predictions are for the same sandstones as those in Part I, we use the same values of $E_{\text{e}}$.
我们已经描述了如何估计 $\xi$ 。正如第一部分所讨论的，该模型的预测对杨氏模量的值很敏感。在第一部分中，我们解释了如何估计 $E_{\text{e}}$ 。由于与我们比较的理论预测的实验数据与第一部分中的相同砂岩，我们使用相同的 $E_{\text{e}}$ 值。

As for the Poisson’s ratio, in Part I we demonstrated that if all the parameters but $\nu$ are fixed, 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 fix the Poisson’s ratio at $\nu \approx 0.3$, the same value that we used in Part I, which is in the middle of the range reported for sandstones.
关于泊松比，在第一部分中，我们证明了如果除了 $\nu$ 之外的所有参数都固定，泊松比会变化 4 倍，该模型用于预测砂岩的渗透率对施加压力的依赖性，预测值的变化不超过 2%。因此，我们将泊松比固定在 $\nu \approx 0.3$ ，与我们在第一部分中使用的相同值，这个值位于报道的砂岩范围中间。

The sensitivity of the predictions to the PSD was also studied in Part I. To do so, we used the following theoretical PSD distribution,
第一部分也研究了预测对 PSD 的敏感性。为了做到这一点，我们使用了以下理论 PSD 分布，

$$\begin{array}{r}f(r)=\left[\frac{r-r_{\text{m}}}{(r_0-r_{\text{m}}{)}^2}\right]\exp [-\frac{1}{2}{\left(\frac{r-r_{\text{m}}}{r_0-r_{\text{m}}}\right)}^2],\end{array}$$

(10)

where $r_0$ is a parameter related to the average pore size $r_{\text{a}}$ through $r_{\text{a}}=(r_0{r}_{\text{m}})\sqrt{pi/2}+r_{\text{m}}$. We held the minimum pore size $r_{\text{m}}$ constant at $0.18\mu$m, the lowest pore sizes that we identified in the published PSDs for sandstones and varied $r_0$ over about two orders of magnitude. Figure 1a presents the PSD that Eq. (10) generates. The distribution corresponding to the lowest $r_0$ in Fig. 1a is completely similar to the PSD reported by Fredrich et al. (1993) for a Fontainebleau sandstone, while those generated by other values of $r_0$ are qualitatively similar to those reported by others for other types of sandstone. As we demonstrated in Part I, the predicted effective pressure-dependent permeabilities of the sandstones that we analyzed in Part I were not greatly sensitive to the PSD. Therefore, similar to Part I, in the absence of any experimental data for each of the PSDs of the sandstones that we analyze below, we used in all cases described below the distribution presented in Fig. 1b as the initial PSD, $f_0(r_0)$, which was reported by Lindquist et al. (2000) for a Fontainebleau sandstone, which is also similar qualitatively to those for many other sandstones reported by others (see, for example, Cheung et al. 2012 for the Bleurswiller and Boise sandstones).
$r_0$ 是与平均孔径 $r_{\text{a}}$ 通过 $r_{\text{a}}=(r_0{r}_{\text{m}})\sqrt{pi/2}+r_{\text{m}}$ 相关的参数。我们将最小孔径 $r_{\text{m}}$ 保持为 $0.18\mu$ m，这是我们在发表的沙岩 PSD 中识别出的最低孔径，并在大约两个数量级内变化 $r_0$ 。图 1a 展示了由方程(10)生成的 PSD。图 1a 中最低 $r_0$ 对应的分布与 Fredrich 等人(1993)报道的 Fontainebleau 沙岩的 PSD 完全相似，而由其他 $r_0$ 值生成的分布与其他人报道的其他类型沙岩的分布定性相似。正如我们在第一部分所证明的，我们在第一部分分析的沙岩的有效渗透率对压力的预测并不对 PSD 非常敏感。因此，与第一部分类似，在没有我们下面分析的沙岩每个 PSD 的实验数据的情况下，我们在下面所有情况下都使用了图 1b 中展示的分布作为初始 PSD， $f_0(r_0)$ ，这是 Lindquist 等人报道的。 （2000）对于一种枫丹白露砂岩，其性质也与许多其他由他人报道的砂岩相似（例如，参见 Cheung 等人 2012 年关于 Bleurswiller 和 Boise 砂岩的报道）。

Fig. 1
图 1

a The PSDs generated by Eq. (10) for various average pore sizes $r_{\text{a}}$. b The PSD used in all the calculations
a. 由方程（10）生成的各种平均孔径的 PSDs $r_{\text{a}}$ 。b. 所有计算中使用的 PSD

Table 1 summarizes the parameters $E_{\text{e}}$ and $\xi$, as well as the initial porosity ${\varphi }_0$ of all the sandstone that we analyze in the following section.
表 1 总结了以下参数 $E_{\text{e}}$ 和 $\xi$ ，以及我们在下一节中分析的所有砂岩的初始孔隙率 ${\varphi }_0$ 。

8 Theoretical Predictions and Comparison with Experimental Data
8 理论预测与实验数据的比较

Before we present the predictions of the theory and compare them with the experimental data; three points are worth mentioning.
在我们展示理论预测并与实验数据比较之前，有三点值得提及。

• 1. First, we emphasize that the model presented in Sects. 2–5 is a MFA that, similar to any MFA, neglects the fluctuations in the local properties, hence making it possible to analyze the behavior of a heterogeneous porous medium based on only a single pore between 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.
  首先，我们强调第 2-5 节中提出的模型是一种 MFA，类似于任何 MFA，它忽略了局部性质的波动，因此可以基于两个颗粒之间的单个孔隙来分析非均质多孔介质的行为，这是有意义的分析所需的最小数量。这也意味着只需要平均颗粒尺寸。因此，类似于所有 MFA，这种方法有其局限性和优点。

• 2. Second, for at least some of the sandstones that we analyze below, if we plot ${\text{σ}}_{\text{e}}(P)/{\text{σ}}_{\text{e}}(0)$ versus a suitably selected rescaled P, we obtain a more or less single universal master curve. The master curve represents essentially the dimensionless response of the porous media under study to the changes in the pressure, since the initial PSD, the Poisson’s ratio, and the exponent m are all set to constant values. A clue to the proper rescaling of P is provided by Eq. (3), since it implies that the quantity $u/R_{\text{g}}$ is a function only of $P/E_{\text{e}}$.
  其次，对于我们下面分析的一些砂岩，如果我们绘制 ${\text{σ}}_{\text{e}}(P)/{\text{σ}}_{\text{e}}(0)$ 与适当选择的归一化 P 的关系图，我们得到一条或多或少的单一直观主曲线。主曲线本质上代表了所研究多孔介质对压力变化的无量纲响应，因为初始 PSD、泊松比和指数 m 都被设定为常数。方程（3）提供了对 P 正确归一化的线索，因为它暗示了 $u/R_{\text{g}}$ 这个量仅是 $P/E_{\text{e}}$ 的函数。

  The existence of such a master curve implies that it can predict the measurements by rescaling the pressure axis to generate a dimensionless form based on the input data—the initial porosity and an experimental data point that provides an estimate for the effective Young’s modulus $E_{\text{e}}$. Other mechanical models than the HM theory of point contact may provide a different master curve. A master curve for another important property of large-scale porous media was proposed by Rassamdana et al. (1996).
  这样的主曲线的存在意味着它可以通过调整压力轴的缩放来预测测量值，生成基于输入数据（初始孔隙率和提供有效杨氏模量估计的实验数据点）的无量纲形式——HM 理论点接触以外的其他力学模型可能提供不同的主曲线。Rassamdana 等人（1996 年）提出了另一个重要的大规模多孔介质特性的主曲线。

• 3. We point out that, in principle, there is no rigorous theoretical relation between the permeability—essentially the square of a relevant length scale of a pore space and, hence, a static property—and electrical conductivity, which is a dynamic property. Therefore, aside from their sharing the same flow and transport paths, one cannot necessarily draw any conclusion for one based on the behavior of the other one. This is particularly true for porous media, such as the almost all the sandstones that we analyze, which contain a significant volume fraction of clays, and therefore, surface conduction may play a significant role in electrical conduction.
  我们指出，从原则上讲，渗透率——本质上是一个孔隙空间相关长度尺度的平方，因此是一个静态属性——与电导率，这是一个动态属性，之间没有严格的理论关系。因此，除了它们共享相同的流动和传输路径外，不能必然从另一个的行为中得出任何结论。这对于多孔介质尤其如此，例如我们分析的几乎所有砂岩，它们含有大量的粘土，因此，表面传导可能在电导中发挥重要作用。

We used the theory to predict the pressure dependence of the effective electrical conductivity ${\text{σ}}_{\text{e}}(P)$ 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 in Fig. 1b. In Part I, we described the geological characteristics of each sandstone, and therefore, they will not be repeated here.
我们使用该理论预测了大量砂岩的有效电导率 ${\text{σ}}_{\text{e}}(P)$ 的压力依赖性，并将预测结果与实验数据进行比较，其中几乎所有数据都由耶鲁（1984 年）提供。他没有提供砂岩的初始粒度分布，因此，如前所述，我们在下面描述的所有情况下都使用了图 1b 中所示的粒度分布。在第一部分中，我们描述了每种砂岩的地质特征，因此，这里将不再重复。

Yale (1984) stated that in all the cases that he experimented on, the pore pressure $P_{\text{P}}$ was constant. Thus, in what follows the pressure P may be replaced by $P-P_{\text{P}}$.
耶鲁（1984 年）指出，在他所进行的所有实验中，孔隙压力 $P_{\text{P}}$ 均保持恒定。因此，在以下内容中，压力 P 可以被 $P-P_{\text{P}}$ 替换。

Table 1 Estimates of the Young’s modulus $E_{\text{e}}$ (in GPa), the initial porosity ${\varphi }_0$, and the Dukhin number $\xi$ of the sandstones
表 1 沙岩的杨氏模量 $E_{\text{e}}$ （单位：GPa）、初始孔隙率 ${\varphi }_0$ 和杜金数 $\xi$ 的估计值

8.1 Fontainebleau Sandstone
8.1 丰坦布勒沙岩

We first present the predictions for a Fontainebleau sandstone, for which Farid (2016) reported measurements of the pressure dependence of its conductivity. The initial porosity ${\varphi }_0$ of the sandstone, before deformation, was 0.052. Farid (2016) also stated that the clay content of the sandstone was negligible. Therefore, we took the Dukhin number $\xi =0$. Figure 2 compares the predictions, normalized by the initial conductivity, with the experimental data. Note that since $\xi =0$, no fitting parameter was used. The agreement between the two sets is very good, with the maximum difference being no more than 7 percent.
我们首先展示了对于丰坦布勒沙岩的预测，Farid（2016 年）报道了其导电率压力依赖性的测量结果。在变形之前，沙岩的初始孔隙率 ${\varphi }_0$ 为 0.052。Farid（2016 年）还指出，沙岩中的粘土含量可以忽略不计。因此，我们采用了杜金数 $\xi =0$ 。图 2 比较了通过初始导电率归一化的预测值与实验数据。请注意，由于 $\xi =0$ ，没有使用拟合参数。两组数据之间的吻合度非常好，最大差异不超过 7%。

Fig. 2
图 2

Comparison of the predicted conductivities with the experimental data for the Fontainebleau sandstone
与 Fontainebleau 砂岩的实验数据预测电导率的比较

8.2 Beaver River Sandstone
8.2 比弗河砂岩

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 ${\varphi }_0\approx 0.076$. Figure 3 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 difference between the predictions and the data being about 8 percent at the highest pressure.
比弗河砂岩位于加拿大艾伯塔省的米尔德雷德湖和比弗河（阿萨巴斯卡河西岸）附近，其初始孔隙率（变形前）为 ${\varphi }_0\approx 0.076$ 。图 3 比较了砂岩电导率的预测压力依赖性，以变形前的值归一化，与耶鲁（1984 年）的实验数据，他也将所有数据以归一化形式呈现。两者之间的一致性非常好，预测值与数据之间的最大差异在最高压力下约为 8%。

Fig. 3
图 3

Comparison of the predicted conductivities with the experimental data for the Beaver sandstone
与实验数据相比，预测的电导率与比埃弗尔砂岩

We note that, at high pressures, there is a qualitative difference between the dependence of the conductivity of the Beaver sandstone and its permeability, shown in Fig. 6 of Part I. The theory predicts faster decay of the conductivity than the experimental data, whereas its predictions for the permeability are highly accurate at all pressures. This may be due to the inadequacy of Eq. (8) for accounting for the effect of surface conduction; the inadequacy of the way we accounted for it by fitting the parameter $\xi$ to a single data point, or both, since slower decay of the conductivity, and the absence of the same trends in the permeability clearly point toward the significance of surface conduction.
我们注意到，在高压力下，比埃沙岩的电导率与其渗透率之间的依赖关系存在定性差异，如图 1 部分第 6 图所示。理论预测电导率的衰减速度比实验数据快，而其对渗透率的预测在所有压力下都非常准确。这可能是由于方程（8）不足以解释表面传导效应；我们通过将参数 $\xi$ 拟合到单个数据点的方式来解释它的不足，或者两者都有，因为电导率的衰减较慢，渗透率中缺乏相同的趋势明显指向表面传导的重要性。

8.3 Berea Sandstones
8.3 贝雷砂岩

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 compares the predictions with the experimental data. Although the maximum difference between the predictions and the data is about 2.5 percent, the theory predicts continuous decline of the conductivity with increasing pressure, whereas the experimental data indicate that at the highest pressures the conductivity levels off, hence indicating that either the morphology of the sandstone did not change at the highest pressure, or we did not account for the effect of surface conduction properly.
两种贝雷砂岩的电导率由耶鲁（1984 年）报道。其中一种是贝雷 100H，一种水平层理的砂岩，初始孔隙率为 0.165。图 4 比较了预测值与实验数据。尽管预测值与数据之间的最大差异约为 2.5%，但理论预测电导率随压力增加而连续下降，而实验数据表明，在最高压力下，电导率趋于平稳，这表明砂岩的形态在最高压力下没有发生变化，或者我们没有正确考虑表面传导的影响。

Fig. 4
图 4

Comparison of the predicted conductivities with the experimental data for two Berea sandstones
比较预测的电导率与两种贝雷亚砂岩的实验数据

Berea 500, with an initial porosity of 0.2, was the second sandstone whose conductivity was reported by Yale (1984). Figure 4 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.
Berea 500，初始孔隙率为 0.2，是第二个由耶鲁（1984 年）报道导电性的砂岩。图 4 还比较了预测值与实验数据，其准确度与 Berea 100H 的预测值相同，只是在最高压力下，测量的电导率衰减速度略快于预测值。

A clue to this is obtained from Fig. 8 of Part I in which we compare the theoretical predictions with the experimental data for the permeability of the same sandstones. That figure indicates excellent agreement between the predictions and the data at all the pressures. Since the porous media in both cases are the same, it may imply that the model may underestimate the contribution of surface conduction in case of Berea 100H, but overestimates it in the case of Berea 500.
在第一部分的图 8 中，我们比较了相同砂岩的渗透率的理论预测值与实验数据。该图表明，在所有压力下，预测值与数据之间的一致性非常好。由于两种情况下的多孔介质都相同，这可能会暗示模型可能低估了 Berea 100H 情况下表面导电的贡献，但在 Berea 500 的情况下可能高估了它。

8.4 Boise Sandstone
8.4 博伊西砂岩

The Boise sandstone had an initial porosity of 0.26. Figure 5 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 percentage difference between the two sets is no more than 4 percent, well within the measurements’ uncertainties.

Fig. 5

Comparison of the predicted conductivities with the experimental data for the Boise sandstone

8.5 Cambrian Sandstone

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 porosities of 0.08, 0.11, and 0.13, respectively. Figure 6 compares the predicted pressure dependence of the effective conductivity with the experimental data. In all cases, the agreement between the predictions and the data is excellent.

Fig. 6

Comparison of the predicted conductivities with the experimental data for the three Cambrian sandstones

8.6 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 characteristics described in Part I. Their initial porosities were, respectively, ${\varphi }_0\approx 0.08,0.044,0.03,$ and 0.02. Figure 7 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.

Fig. 7

Comparison of the predicted conductivities with the experimental data for the four Fahler sandstones

In the case of Fahler 162, the agreement between the data and the predictions is good at higher pressures, but the difference 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 fine-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 quartz, an extremely hard material, than Fahler 189, implying that the mechanism of porosity reduction in the two sandstones may be different.

8.7 Indiana Dark Sandstone

Indiana dark DH sandstone, with DH indicating that the sample was taken after drilling horizontally (parallel to bedding), had a relatively high initial porosity of 0.27. Figure 8 compares the predictions with the experimental data of Yale (1984). The agreement is excellent over much of the pressure range, with the largest difference between the prediction and data being about 5 percent at the highest pressure.

Fig. 8

Comparison of the predicted conductivities with the experimental data for the Indiana DH sandstone

Note, however, that, similar to the Beaver and Berea sandstones, there is a qualitative difference between the behavior of the conductivity of Indiana DH sandstone and its permeability counterpart, shown in Fig. 12 of Part I. In the case of the permeability, the agreement between the theoretical predictions and the data is excellent at all the pressures. Since the only difference between the two cases appears to be the contribution of the surface conductivity, the fact that the theory predicts a faster decay of the conductivity at the highest pressures may imply that the model does not account adequately for the effect of surface conduction.

8.8 Massillon Dark Sandstone

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 9 compares the predictions with the experimental data of Yale (1984). Although the largest difference between the two sets is only about 3.5 percent at a pressure of 17.5 MPa, the trends in the two sets are somewhat different. The data indicate that the porosity at lower pressures reduced more strongly and, then, it leveled off, because 60 percent of the sandstone is quartz, which means it is difficult to reduce the porosity further even at 50 MPa. On the other hand, the EMA does not take into account the deformation mechanism and assumes simply that the porosity is reduced randomly.

Fig. 9

Comparison of the predicted conductivities with the experimental data for the Massillon DH sandstone

8.9 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 flow and transport paths over large distances (Saitoh and Masuda 2004). Figure 10 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 flow paths of the sandstone practically guarantee accurate predictions, because it is precisely under such conditions that the EMA is accurate.

Fig. 10

Comparison of the predicted conductivities with the experimental data for the Miocene 7 sandstone

8.10 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 and et al. 2003). The initial porosity of the sample, referred to as Pliocene 35 by Yale (1984), was 0.2. Figure 11 compares the predictions with the data. The experimental data are somewhat scattered, but the largest difference between the data and predictions is about 9 percent at 25 MPa.

Fig. 11

Comparison of the predicted conductivities with the experimental data for the Pliocene 35 sandstone

8.11 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 Fig. 12, 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 unchanged 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.

Fig. 12

Comparison of the predicted conductivities with the experimental data for the Tensleep 35 sandstone

8.12 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 13 compares the theoretical predictions with the experimental data. The agreement between the predictions and the data is excellent.

Fig. 13

Comparison of the predicted conductivities with the experimental data for the Tertiary 807 sandstone

8.13 Torpedo Sandstone

The Torpedo sandstone was from Kansas with an initial porosity of 0.202. As Fig. 14 indicates, the predicted electrical conductivity closely matches the measured data.

Fig. 14

Comparison of the predicted conductivities with the experimental data for the Torpedo sandstone

8.14 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 fine- to very coarse-grained. Although, generally speaking, they are porous formations with low- or ultra-low flow properties, they often have both tectonic and diagenetic fractures that provide flow paths. Yale (1984) reported the data for five 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. Figure 15 compares the predictions for the conductivities with the experimental data. The agreement between the predictions and the data is uniformly very good for all samples.

Fig. 15

Comparison of the predicted conductivities with the experimental data for the Triassic sandstone

8.15 Branford Sandstone

Figure 16 compares the predicted pressure dependence of the electrical conductivity of Branford (Connecticut) sandstone with the experimental data of Yale (1984). The initial porosity ${\varphi }_0$ 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 (Bernabé 1989). Except for pressures $P>30$ MPa, the agreement between the two is excellent. We shall return to the issue of the tail of the curve in Sect. 9.

Fig. 16

Comparison of the predicted conductivities with the experimental data for the Branford sandstone

8.16 Kirkwood Sandstone

Finally, we show in Fig. 17 the predictions for the conductivity of the Kirkwood sandstone and compare them with the data. The sandstone is a fairly clean orthoquartzite (Wyllie and Spangler 1952) 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 effective permeability of both Branford and Kirkwood sandstone.

Fig. 17

Comparison of the predicted conductivities with the experimental data for the Kirkwood sandstone

9 Discussion

Several aspects of the predictions deserve to be considered in more details. In what follows, we discuss each aspect.

9.1 Limits of Accuracy of the Effective-Medium Approximation

The range of the validity of the EMA, as well as the MFA that we have developed for the deformation, are important issues. Sahimi (2003) and Hunt and Sahimi (2017) provide comprehensive discussions of the strengths and shortcomings of the EMA. As is well known, the EMA is (1) more accurate for two-dimensional (2D) media than for 3D, and (2) not very accurate in the critical region, i.e., the region near the critical porosity or the percolation threshold. In random media, the critical region is defined roughly by Sahimi (1994), $\varphi -{\varphi }_{\text{c}}\le 1/Z$, where Z is the mean pore connectivity, and $\varphi \text{c}$ is the critical porosity. (3) 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 effect of such correlations.

9.2 The Reduction of the Conductivity at High Pressures

As described earlier, in some of the sandstones that we analyzed, the measured effective conductivity at high pressures decreases a bit more slowly than the theoretical predictions. One possible reason for this is that high pressures change the morphology of the porous media by mechanism(s) that the MFA and the EMA do not account for, such as opening up new cracks that provide new transport paths for the fluid and, hence, arrest to some extent the decline in the conductivity. Accounting for such effects would require modifications of the deformation theory that is beyond the scope of the present work. As discussed earlier, another possible reason for this behavior could be the inadequacy of the way we account for the effect of surface conduction.

9.3 Effect of the Boundary Conditions

We recall that deformation of geological formations is often caused by uniaxial stress, whereas in the experimental system and the present theoretical modeling the confining pressure is applied hydrostatically. Needless to say, the deformations that result from the two types of the boundary conditions are different, because the spatial distributions of the applied pressure in the two systems are different. But, when, for example, the overburden pressure exerts itself uniaxially in an oil reservoir, 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. We already demonstrated that the theory provides accurate predictions for the macroscopic conductivity (and the permeability, as reported in Part I) as a function of the hydrostatic pressure that deforms the pore space much more extensively than a uniaxial stress would. Therefore, a slightly modified theory would be at least equally accurate for the case in which a uniaxial stress is exerted on a porous medium.

9.4 Effect of the Contact Law for the Grains

The theory that was developed in Part I and utilized in the present paper for determining the change in the size of a pore as a result of deforming a porous medium was based on the Hertz–Mindlin theory of contacting grains in unconsolidated porous media. The experimental data that were compared with the theoretical predictions were, however, for mostly consolidated sandstones that have been cemented. As shown by Dvorkin and Yin (1995), the cementation influences strongly the contact laws. This could provide an explanation as to why the fitted Young’s moduli of various sandstones did not agree with what one might expect for such porous media, which contain a significant amount of quartz.

One way of addressing the theoretical shortcoming is to use the Hertz–Mindlin theory for cemented sandstones, rederive the expression for the change in the effective size of the pores, and recompute everything. An alternative, and perhaps simpler, approach would be to determine the change in the radius of a hollow cylinder (a pore throat), embedded in a solid material of a given Young’s modulus, as a result of exposing the entire system to a hydrostatic pressure, which would indeed represent a mean-field approximation. The result can then be used to update the PSD for a given pressure and, hence, the effective conductivity. We will report on both approaches in a future paper.

10 Summary

This paper, the second a in series, presented a new theoretical approach for predicting the electrical conductivity ${\text{σ}}_{\text{e}}$ of brine-saturated porous media that deform as a result of a hydrostatic pressure applied to them. The theory, a mean-field approximation, determines the change in the size of a pore between two grains that deform when the hydrostatic pressure is applied to them. Given the initial PSD of a porous medium before deformation, the Young’s modulus, and the Poisson’s ratio of the grains as the input, the theory determines the PSD of the pore space at pressure P. The updated PSD is used to determine the pore-conductance distribution, which is then used in the effective-medium approximation to predict the effective conductivity of the porous medium at the same pressure. Extensive comparison between the theoretical predictions and experimental data for the pressure dependence of ${\text{σ}}_{\text{e}}$ of twenty-six sandstones indicated agreement between the two in almost all cases, ranging from very good to excellent.

The same type of approach can be used to estimate the effect of deformation on the dispersion coefficients in flow through deforming porous media. Work in this direction is in progress and will be reported in the future.