Analytical effective coefficient and first-order approximation to linear Darcy's law through block inclusions

Abstract

We present an analytical form for the effective coefficient to linear diffusion equations such that the heterogeneous coefficients are periodic and rapidly oscillating and can be defined as step functions describing inclusions in a main matrix. The new contribution comes from using an analytical approximation for the solution of the well known periodic cell-problem. By defining a correction to the given approximation, the analytical effective coefficient, the zeroth-order approximation in H10(Ømega) and the first-order in L2(Ω) are readily obtained. The known results for effective coefficient are obtained as particular cases, including the geometric average for the checkerboard structure of the medium. We demonstrate numerically that our proposed approximation agrees with the classical theoretical results in homogenization theory. This is done by applying it to problems of interest in flow in porous media, for cases where the contrast ratio between the inclusion and the main matrix are 10:1, 100:1, 1000:1, and 1:10, respectively. © 2008 Springer.