On the effective hydraulic conductivity in mean vertical unsaturated steady flows

Water flow in partially saturated heterogeneous porous formations is modelled by regarding the hydraulic parameters as stationary random space functions (RSFs). As a consequence, the flow variables are also RSFs, and we aim to develop a procedure to derive the effective hydraulic conductivity (EHC). The methodology relies on a perturbation approach which regards the variances of the hydraulic parameters as small quantities. By using the Gardner’s [Gardner WR. Some steady state solutions of unsaturated moisture flow equations with application to evaporation from a water table. Soil Sci 1958;85:228–32] two-parameters (Ks, α) model for the local unsaturated conductivity, we obtain the EHC for any dimensionality d of the flow domain, and arbitrary correlation functions of the input RSFs. Unlike previous studies [e.g. Yeh T-CJ, Gelhar J, Gutjahr A. Stochastic analysis of unsaturated flow in heterogenous soils. 1. Statistically isotropic media. Water Resour Res 1985;21;447–56, Yeh T-CJ, Gelhar J, Gutjahr A. Stochastic analysis of unsaturated flow in heterogenous soils. 2. Statistically anisotropic media with variable α. Water Resour Res 1985:21:457–64], the EHC is represented here as product between the local scale conductivity valid for a domain of mean parameters, and a correction function κ∗ which depends on the medium heterogeneity structure and the mean pressure head. Generally, the correction function κ∗ is expressed by d-fold quadrature. These quadratures are further reduced after adopting specific (i.e. exponential and Gaussian) structure for the (cross) correlation functions involved in the computation of κ∗. We have also focused on some particular formation structures which are relevant for the applications, and permit simplification of the computational aspect, as well. We investigate effects of the heterogeneity formation properties as well as the mean head on the structure of κ∗. Overall, results suggest that, given the formation statistics, the impact of the heterogeneity upon κ∗ is enhanced as the medium becomes drier. This is particularly so when the variability of the fluctuation of Y = ln Ks is small compared with that of ζ = ln α. Conversely, when the heterogeneity of Y is prevalent upon that of ζ, κ∗ is influenced solely by the anisotropic structure of the formation unless the horizontal correlation scales are much greater than the vertical ones.