Effective Hydraulic Conductivity of Unsaturated Isotropic Soils with Multidimensional Heterogeneity

Accurate simulation and prediction of flow and transport of solutes in a heterogeneous vadose zone requires the appropriate hydraulic properties corresponding to the spatial scale of interest. Upscaling techniques are needed to provide effective properties for describing the vadose zone system's behavior with information collected at a much smaller scale. Numerical experiments were performed to investigate the effective unsaturated hydraulic conductivity of soils with different degrees and dimensionalities of heterogeneity. Researchers have extended Matheron's method for determining the hydraulic conductivity of soils with one-dimensional heterogeneity under a saturated condition to unsaturated conditions. In this work, Matheron's method was further extended to the unsaturated soils with two- and three-dimensional heterogeneity. It was found that the first-order approximation of the extended formula is similar to those based on the small-perturbation approach. The extended Matheron's method was verified using multistep numerical experiments of gravity-induced flow into synthetic soils with different degrees of heterogeneity. Results showed that the dimensionality of soil heterogeneity has a significant impact on the effective unsaturated hydraulic conductivity, and the extended Matheron's method can well estimate the effective conductivity of the soils with multidimensional heterogeneity.