Advances in Materials Science and Engineering

Volume 2016 (2016), Article ID 5172148, 8 pages

http://dx.doi.org/10.1155/2016/5172148

## Calculation of Clay Permeability Using a Rectangular Particle-Water Film Model by the Double-Scale Asymptotic Expansion Method

^{1}Research Center of Coastal and Urban Geotechnical Engineering, Zhejiang University, Hangzhou 310058, China^{2}Huadian Electric Power Research Institute, Hangzhou 310030, China

Received 19 June 2016; Revised 20 September 2016; Accepted 3 October 2016

Academic Editor: Giorgio Pia

Copyright © 2016 Xiaowu Tang 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

Permeability of soil plays an important role in geotechnical engineering and is commonly determined by methods combining measurements with theory. Using the double-scale asymptotic expansion method, the Navier-Stokes equation is numerically solved to calculate the permeability, based on the homogenization method and the assumption that the homogeneous microstructure of the relevant porous media is represented accurately as the Representative Elemental Volume (REV). In this study, the commonly used square model is tested in the calculation of sea clay permeability. The results show large deviations. It is suspected that the square model could not represent the flattened shape of the clay particles and the bound water film wrapping around them. Hence, the Rectangle Particle-Water Film Model (i.e., the R-W model) is proposed. After determining the horizontal and vertical characteristic length of the unit cell using two pairs of initial data, the permeabilities of other different void ratios could be inversely calculated. The results of three types of clay obtained using the R-W model agree well with the experimental data. This shows the efficient feasibility and accuracy of the R-W model by providing a good representation of the clay particles when using the double-scale asymptotic expansion method to calculate clay permeability.

#### 1. Introduction

Permeability of soil is of fundamental importance in geotechnical engineering. Experimental methods, such as the classical Darcy law, are commonly used to obtain the permeability of soil. These methods are convenient to be conducted but always require much time and effort. Moreover, the experimental methods focus only on the external phenomenon from the macroscopic view and ignore the seepage processes inside the porous materials at the microscopic scale. Actually, as a type of porous material, soils can be distinguished at three different scales: the microscopic scale, the Darcy scale, and the macroscopic scale [1]. To produce a more comprehensive and thorough analysis, the seepage process at both the macroscopic and the microscopic scale should be studied.

The homogenization method [2], which is also called the multiscale asymptotic expansion method, is a powerful tool to investigate the characteristics of nonhomogeneous materials. Starting from the physical phenomenon at the microscopic scale, the homogenization method can obtain the characteristic descriptions of the material at the macroscopic scale using asymptotic expansion of the governing equations. The homogenization method has been widely used in the researches of characteristics of composite materials, which generally have a uniform and regular structure. Andreassen and Andreasen [3] used short and self-contained Matlab implementation to provide the calculation method of the elasticity tensor and fluid permeability of composite materials by homogenization. Keip et al. [4] presented a two-scale computational homogenization framework for the simulation of electroactive solids at finite strains. The results of these researches showed that the homogenization method was typically well suited to study composite materials.

In geotechnical engineering, Wang et al. [5] proposed a simplified homogenization algorithm for composite soils to simulate the characteristics of soils via numerical calculations which assumed that the pressure in the microscopic scale was homogeneous. Specific to the seepage in soils, Tang et al. [6] proposed a multiscale method to calculate the permeability coefficient of soils. Based on the homogenization method and the assumption that the homogeneous microstructure of the soils is well represented by the Representative Elemental Volume (REV), the Navier-Stokes equation was numerically solved. Sun et al. [7] used the double-scale asymptotic expansion method to inversely calculate the permeability of clay using a square particle unit cell as the REV. The results were within the allowance but presented significant errors. Considering the calculation of permeability for kaolin clay as an example, the calculated permeability was m^{2} at the largest void ratio , while the measured permeability m^{2}; also, the calculated value was m^{2} when the smallest void ratio was determined to be , while the measured value m^{2}.

In this paper, the existing square model, which was used in the inversed calculation by the double-scale asymptotic expansion method, was tested to find the defects of the model. Additionally, a new model would be proposed based on the real features of clay particles aiming to improve the calculated accuracy of clay permeability.

#### 2. Calculation of Clay Permeability by Multiscale Method

##### 2.1. Multiscale Expansion of Navier-Stokes Equations

Previous studies have shown that, for a given type of statistically homogeneous soil, an equivalent REV always exists to represent the structure of the soil. A similar description could be used for nonhomogeneous porous materials [8, 9]. The basic assumption of the calculation performed in this study was that the porous media are homogeneous, and the arrangement of their particles was periodic; thus, an REV should be chosen to represent the porous media, as shown in Figure 1. The pores are fully saturated by an incompressible Newtonian fluid with a small Reynolds number; denotes the solid part of the particle, the space of the fluid in the pore volume, and Γ the interface between the solid and the fluid.