Stochastic Analysis of Unsaturated Flow in Heterogeneous Soils: 2. Statistically Anisotropic Media With Variable α

Abstract

Steady unsaturated flow in heterogeneous soil with an arbitrarily oriented mean hydraulic gradient is analyzed using spectral solutions of the stochastic perturbation equation which describes the capillary pressure head ψ . The unsaturated hydraulic conductivity is related to ψ by K = K s exp (−α ψ ), where K s is the saturated hydraulic conductivity and α is a soil parameter, and both K s and α are treated as three‐dimensional statistically homogeneous, anisotropic random fields. Analytical results are obtained for the capillary pressure head variance and the effective (mean) unsaturated hydraulic conductivity. The head variance depends upon the degree of anisotropy of the ln K s covariance; when α is random, the head variance increases with mean capillary pressure head. The effective hydraulic conductivity for arbitrary orientation of the mean hydraulic gradient J is shown to have tensorial properties, but its components depend on the magnitude and direction of J and the orientation of the stratification in the soil. When α is random, the degree of anisotropy of the effective conductivity depends strongly on mean capillary pressure.