Inversion Calculation of Permeability Coefficient with the Multi-Scale Asymptotic Expansion Method

Abstract

Permeability of porous media is of fundamental importance for many disciplines ranging from earth to material sciences. Homogenization method is a useful tool in the field. The general idea of the homogenization method is the derivation of a macroscopic behavior from the description at the micro-level to the physical phenomenon over a Representative Elementary Volume (REV), or a unit cell. It will make it easier to solve the problem. In this paper, for a given physical phenomenon, the corresponding effective properties of the porous media, such as diffusivity , permeability, can be directly computed via analysis of the unit cell, provided that the microstructure is known. However, complicated techniques to analyze the microstructure are always require. Hence unit cells of simple shapes are chosen. In the framework, Navier-Stokes equation is solved numerically to calculate the permeability coefficient, with the base of homogenization method, and the assumption that the microstructure of porous media is homogeneous and known. And the permeability coefficient can be inversely calculated with the REV which has the circular and square particles. The value of void ratio differs as the length of the unit cell changes. The general process is as bellow: with the value of void ratio and permeability coefficient measured, the program of inversion calculation is used to get the radius of the circular particle and the length of the square particle for sand and clay, respectively. With the characteristic radius or length, the permeability coefficients of different values of void ratio can be calculated. Inversion calculations of sand and clay are computed, from which the computed results and measured results have an agreement.