A higher‐order approximation to effective conductivity in media of anisotropic random structure

Abstract

Properties of the effective conductivity tensor K eff are studied by deriving the second‐order terms in its expansion in the variance σ 2 of normally distributed log conductivity. It is shown that for media of anisotropic structure, the components of the effective conductivity tensor are expressed by a functional of the log conductivity covariance; that is, it depends on the shape of the correlation function and not only on anisotropy ratios, variance σ 2 , and space dimensions. However, the trace of K eff is independent of the log conductivity autocovariance, and for a given mean conductivity depends only on σ 2 . The dependence of the effective conductivity on the correlation structure is illustrated for Gaussian and exponential autocovariances of log conductivity and for two‐ and three‐dimensional flows.