Uniform steady free-surface flow in heterogeneous porous formations

Abstract

The effect of spatial variability of the hydraulic conductivity upon free-surface flow is investigated in a stochastic framework. We examine the three-dimensional free-surface gravitational flow problem for a sloped mean uniform flow in a randomly heterogeneous porous medium. The model also describes the interface between two fluids of differing densities, e.g. freshwater/saltwater and water/oil with the denser fluid at rest. We develop analytic solutions for the variance and integral scale of free-surface fluctuations and of specific discharge on the free surface. Additionally, we obtain semi-analytic solutions for the statistical moments of the head and the specific discharge beneath the free surface. Statistical moments are derived using a first-order approximation and then compared with their counterpart in an unbounded medium. The effect of anisotropy and angle of mean uniform flow on the statistical moments is analysed. The solutions can be used for solving more complex flows, slowly varying in the mean.