Stochastic Analysis of Unsaturated Flow in Heterogeneous Soils: 1. Statistically Isotropic Media

TLDR

Hydraulic conductivity in unsaturated soils follows K = Ks exp(−αψ), linking conductivity to capillary pressure head. The study presents a general formulation for modeling Ks and α as statistically homogeneous spatial random fields. Steady unsaturated flow with vertical mean infiltration is analyzed via a perturbation approximation of the stochastic flow equation, solved by spectral representation, with solutions developed assuming constant α and representing Ks variability by one‑ and three‑dimensional isotropic random fields. The analysis yields head variances, covariance functions, effective hydraulic conductivities, and flux variances; larger α values reduce head variance and drive the system toward gravitationally dominated vertical flow, while effective conductivity depends on the correlation scale of ln Ks and the mean hydraulic gradient.

Abstract

Steady unsaturated flow with vertical mean infiltration through unbounded heterogeneous porous media is analyzed using a perturbation approximation of the stochastic flow equation which is solved by spectral representation techniques. The hydraulic conductivity K is related to the capillary pressure head ψ by K = K s exp (−αψ), where K s is the saturated conductivity, and α is a soil parameter. A general formulation is presented for the case with K s and α represented as statistically homogeneous spatial random fields. In part 1, solutions are developed assuming α is constant and representing K s variability by one‐dimensional and three‐dimensional isotropic random fields. Results are obtained for head variances and covariance functions, effective hydraulic conductivities, variances of the unsaturated hydraulic conductivity, flux variances, and variance of pressure gradient. When the parameter α is relatively large, corresponding to coarse textured soils, the head variance decreases and all of the results demonstrate a trend toward gravitationally dominated one‐dimensional vertical flow. The effective conductivity is dependent on the correlation scale of ln K s and the mean hydraulic gradient.