Geofluids

Volume 2019, Article ID 9240203, 10 pages

https://doi.org/10.1155/2019/9240203

## Permeability Estimation Based on the Geometry of Pore Space via Random Walk on Grids

Department of Hydrosciences, School of Earth Sciences and Engineering, Nanjing University, Nanjing 210093, China

Correspondence should be addressed to Tongchao Nan; nc.ude.ujn@nanct and Jichun Wu; nc.ude.ujn@uwcj

Received 25 May 2018; Revised 13 September 2018; Accepted 1 October 2018; Published 8 January 2019

Guest Editor: Joaquín Jiménez-Martínez

Copyright © 2019 Tongchao Nan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In the literature, the mean penetration depth (MPD) calculated by “walk on spheres” or “walk on cubes” was used to quickly estimate the intrinsic permeability of digitized porous media. However, these two methods encounter difficulties such as irregular boundaries and the determination of arrivals at a boundary. In this study, an MPD method that is based on a more flexible “random walk on grid” (WOG) is explored. Moreover, the accurate MPDs for the pores of simple shapes are derived with Green’s functions to validate the WOG-based MPD. The results suggested that MPDs based on Green’s functions and WOG are consistent with each other; the factor in the permeability expression is slightly dependent on roundness of the cross sections and is approximately 1.125 on average, according to analytical and numerical results. In a synthetic complex pore, the permeability estimated by WOG is comparable to, but greater than, the estimate based on the pore-scale dynamics simulation in COMSOL.

#### 1. Introduction

Permeability, which is a measure of the ability of a porous material to allow fluid to pass through it, is one of the most important properties of geologic formations to hydrogeologists, petrophysicists, and other geologic fluid researchers. Since permeability (or hydraulic conductivity) greatly influences the fluid movements and flow fields of groundwater, it is generally the fundamental factor in all groundwater-related problems, such as groundwater exploitation, surface-groundwater interaction, contaminant transport, agricultural irrigation, and site selection for radioactive waste disposal [1, 2].

As a macroscopic parameter of formations, permeability is closely related to pore microstructures. In addition to the direct measurement of samples, indirect measurement is also important for different environmental and technological applications. Some studies have been devoted to establishing the relationship between microscopic geometry and macroscopic permeability by theoretical models and numerical calculations [3–8], which are becoming increasingly valuable for the growing availability of microscopic tomographic data, such as nuclear magnetic resonance, X-ray microtomography, and scanning-electron-microscopic imaging ([9–12]; among many others).

For pores of some of the simplest geometries, the permeability can be calculated by the exact solution to the Poiseuille flow equation. For example, in a straight tube of circle-shaped cross sections, one obtains by combining three equations: (Hagen-Poiseuille law), (Darcy’s law), and (relation between permeability and conductivity) (see, e.g., [13]). For tubes of elliptical cross sections, , where and are the semiaxes [14]; similar results are available for several other shapes [15, 16]. For media of higher complexity, for example, in a packed bed of solids, where is the mean grain diameter, is the porosity, and is a parameter that is dependent on cross-sectional geometry [6, 17, 18].

However, more powerful tools are required in media of general microstructure. By analyzing diffusion processes in porous media, Torquato [8] linked the permeability tensor of the media to a so-called trapping constant, which is not easy to evaluate and thus is not practical. However, the idea of using diffusion or Brownian processes to reflect the properties of porous media is useful. Based on the Brownian motion model and the Debijf-Brinkman equation (see, e.g., [19]), Hwang et al. [5] found that the mean penetration depth (MPD, denoted by ), which is the expected depth that a Brownian motion can enter before reaching solid surface for the first time, is a good measure of the permeability of the medium and suggested that . Simonov and Mascagni [7] took porosity into account and suggested that , where is the porosity and is a factor. They estimated by using the Poiseuille law-based permeability in an ideal, straight pore of the square cross section with a side length that is equal to , and . It is still not clear whether is also valid for pores of other geometries. Sabelfeld [20] used a spectral projection method to avoid the numerical calculation on overlapping discs and spheres which, however, applies to simplified pore surfaces and is unsuitable for real media with rough surfaces.

The equations above are all in a quadratic form; i.e., , where is some type of characteristic length (e.g., hydraulic radius, mean pore size, mean grain diameter, or MPD), and the coefficient may depend on the cross section and/or porosity ([13]; and references therein).

If one uses MPD as the characteristic length, the coefficient may depend on both cross-section shape and porosity. The influence of porosity on was also investigated by Simonov and Mascagni [7] using “walk on cubes” and “walk on spheres.” However, “walk on cubes” is only suitable for voids of polyhedron and is inefficient for curly boundaries [21]. “Walk on spheres” has difficulty in determining exactly when and where the random walk ends at a boundary. The difficulty leads to significant error and generates some bias (Milstein and Tretyakov 1999; Deaconu and Lejay 2006). To avoid these problems, in this study, we extend the MPD method by using a more flexible “random walk on grid” (WOG), which is able to easily determine the arrival at boundaries and handle irregular surfaces, including cambering, narrow gaps, and wedge outs.

To validate the effectiveness of the WOG-based MPD method and to investigate the extent of ’s dependence on the roundness of pore cross sections, we derived the theoretical values of MPD and using Green’s function in the pores of several basic geometries and compared them to their WOG-based counterparts. Simonov and Mascagni [7] realized the possible weak dependence of on the cross-sectional geometry of void space, but they did not investigate the problem further. Here, the dependence is investigated through derived by Green’s functions, as well as the estimated by WOG.

To date, the permeability calculated by the MPD method has not been verified in a porous medium against dynamic simulation or experimental results in the literature. A synthetic porous sample is constructed for permeability estimation using COMSOL simulation and the WOG-based MPD method.

The paper is organized as follows: In Section 2, the MPD method based on WOG and Green’s functions, as well as COMSOL simulation, is introduced. In Section 3, the results of the MPD estimation in simple pores of different geometries are reported, and the factor is calibrated for each geometry. Furthermore, the average factor is used to estimate the permeability of a synthetic pore, and the result is verified by a COMSOL model. In Section 4, the results are discussed and conclusions are drawn.

#### 2. Methodology

##### 2.1. Mean Penetration Depth (MPD) and MPD Calculation via Green’s Function

Given a porous medium, let a Brownian motion initiate at (denoted by ) on its outer boundary plane (say, ; Figure 1) and enter the medium until the walker hits a solid surface (i.e., the inner boundary, represented by blue surfaces in Figure 1) at a random position, . The depth of the hit point relative to the outer boundary plane (i.e., ; is the ensemble index) is called the penetration depth of the Brownian motion, denoted by . The mean penetration depth (MPD) is the ensemble mean of the penetration depths in Monte Carlo experiments, starting from an area on the outer boundary (termed “departure region” , i.e., the green area in Figure 1), i.e., , where stands for expectation with respect to the ensemble index , denotes the MPD estimated by the Brownian motion (or random walk), and is the area of .