Flow in Heterogeneous Media Displaying a Linear Trend in the Log Conductivity

Abstract

The purpose of this study is to derive a general solution for the problem of flow in nonstationary geological formations where the nonstationarity manifests itself in the form of a spatial trend in the mean log conductivity. A stochastic frame of reference is adopted to account for the spatial variability of the hydraulic conductivity. For a complete stochastic description we derive the expected values and spatial covariances of the hydraulic head and the fluid flux vector, as well as a relation between the expected values of the head and the fluxes. These expressions are obtained using a perturbation expansion of the log conductivity about its nonstationary mean, and they are correct to the first order in the variance of the log conductivity. The expressions we derive are applicable for any space dimensionality and for arbitrary orientation of the trend in space. A general methodology is outlined for derivation of these expressions for any type of spatial covariance; and for demonstration, explicit results are obtained for a Gaussian isotropic covariance.