Effective hydraulic conductivity of a randomly heterogeneous porous medium

Abstract

The effective hydraulic conductivity of a randomly heterogeneous isotropic porous medium has been calculated by means of a new perturbative technique, which differs greatly from that commonly used. In the approach developed, we construct the perturbation series for a seepage velocity and then utilize Darcy's law for calculating pressure. On the basis of the analysis of high-order approximations it is shown that in the general case the effective conductivity in the large-scale limit does depend on the form of the correlation function. Thus the widespread Landau-Lifshitz-Matheron formula, which operates on the conjecture that the first two terms of the perturbation series are those for the Taylor series expansion of an exponential function, proves to be invalid. The calculations were carried out in a space of arbitrary dimension. The dependence of the effective conductivity on the variance has been obtained up to the third-order approximation inclusive in terms of a conductivity logarithm. For convenience and simplification of series analysis the Feynman diagrammatic technique was developed and utilized. In a one-dimensional (1D) case our result in the large-scale limit (when a heterogeneity scale is small enough with respect to the characteristic scale of groundwater flow) gives an exact formula for the effective conductivity.