Stochastic analysis of steady-state unsaturated flow in heterogeneous media: Comparison of the Brooks-Corey and Gardner-Russo models

Existing stochastic models of unsaturated flow and transport are usually developed using the simple Gardner-Russo constitutive relationship though it is generally accepted that the more complex van Genuchten and Brooks-Corey relationships may perform better in describing experimental data. In this paper, we develop first-order stochastic models for gravity-dominated flow in second-order stationary media with both the Brooks-Corey and the Gardner-Russo constitutive relationships. These models also account for the spatial variability in effective water content, while the spatial variability is generally neglected in most existing stochastic models. Analytical solutions are obtained for the case of one-dimensional gravity-dominated flow. On the basis of the solutions, we illustrate the differences between results from these two constitutive models through some one-dimensional examples. It is found that the impacts of the constitutive models on the statistical moments of suction head, effective water content, unsaturated hydraulic conductivity, and velocity depend on the saturation ranges. For example, the mean head and the mean effective water content for the Brooks-Corey model differ in a great manner with their counterparts for the Gardner-Russo model near the dry and wet limits while the differences are small at the intermediate range of saturation. This finding is confirmed with some two-dimensional examples. It is also found that the Brook-Corey model has certain advantages over the Gardner-Russo model in analyzing unsaturated flow in heterogeneous media. For example, the stochastic model developed based on the Brooks-Corey function requires the coefficient of variation of head and soil parameter 'α(Bc)' to be small (<<1), whereas that based on the Gardner-Russo function assumes the one-point cross covariance of head and α(GR) to be small (<<1). Illustrative examples reveal that the latter condition may be violated because the one-point covariance is found to increase rapidly to beyond unity as the soil becomes dry, whereas the former may be readily satisfied.