Numerical homogenization of the Richards equation for unsaturated water flow through heterogeneous soils

Homogenized equations and the corresponding effective constitutive relations are generally necessary for numerically modeling large‐scale unsaturated flow processes in soils. Recently, based on the Kirchhoff transformation and the two‐scale convergence theory, a homogenization method for the Richards equation with the Mualem‐van Genuchten model has been proposed, with a constant model parameter α relating to the inverse of the air‐entry pressure and the soil pore size distribution. The homogenized model is computationally efficient and convenient to use because of its explicit expression. In this study, we generalize this method, allowing α to be a spatially distributed random field and proposing a homogenized Richards equation in the mixed form ( θ / h ) under the condition that the effective hydraulic conductivity tensor is diagonal. This generalization eliminates the limitation of a constant α in practical applications; the proposed homogenized model is meaningful in most situations because the flow problems are influenced mainly by the diagonal terms of conductivity and the off‐diagonal terms are often neglected. Two‐dimensional numerical tests are conducted in soil profiles with different degrees of spatial heterogeneity structure to illustrate that the homogenized model can capture the fine‐scale flow behaviors on coarse grids effectively. Homogenization for the Richards equation with other two commonly used constitutive relations—the Brooks‐Corey model and the Gardner‐Russo model—is also illustrated in this study.