Nonlocal properties of nonuniform averaged flows in heterogeneous media

Abstract

The properties of nonuniform average potential flows in media of stationary random conductivity are studied. The mathematical model of average flow is derived as a system of governing equations to be satisfied by mean velocity and mean head. The averaged Darcy's law determines the effective conductivity as an integral operator of the convolution type, relating the mean velocity to the mean head gradient in a nonlocal way. In Fourier domain the mean velocity is proportional to the mean head gradient. The coefficient of proportionality is referred to as the effective conductivity tensor and is derived by perturbation methods in terms of functionals of the conductivity spatial moments. It is shown that the effective conductivity cannot be defined uniquely for potential flows. However, this nonuniqueness does not affect the spatial distribution of the mean head and velocity. Analytical expressions of the effective conductivity tensor are derived for two‐ and three‐dimensional flows and for exponential and Gaussian correlations of isotropic conductivity. The fundamental solution of the governing equations in effective media (mean Green function) is calculated for the same cases. Two new asymptotic models of the averaged Darcy's law are developed to be applicable to large and small scales of heterogeneity. Several asymptotic expansions of tue two‐dimensional and three‐dimensional mean Green functions are derived for exponential and Gaussian correlations.