Steady unsaturated flow in two-dimensional scale-heterogeneous porous media

Unsaturated flow in scale heterogeneous porous media is described by a class of nonlinear partial differential equations involving three general coefficient functions. It is shown that among these, there is a large class of integrable models, being transformable to linear equations even when the spatially varying geometric dilation scale factor is an arbitrary twice differentiable function. Explicit solutions are constructed for basic boundary value problems that relate to field and laboratory measurements. In one of these solutions, the flow pattern is unaffected by the heterogeneity even though the water concentration is modified. From another solution, we predict the bulk conductivity of a heterogeneous soil sample with a constant hydraulic potential difference maintained between the ends.