Improving Numerical Modeling of Two‐Dimensional Water Flow in Variably Saturated, Heterogenous Porous Media

Previous work has shown that a transformation of the pressure head greatly improved the efficiency of the algorithms for simulating one‐dimensional infiltration into very dry heterogenous soils. We expanded the approach to solve two‐dimensional problems and introduced an adaptively optimum relaxation procedure to improve the efficiency and the robustness of the numerical solutions. A theoretical framework was developed for analysis of convergence problems in solving Richards' equation. Based on this framework, several simple calculation methods are proposed to estimate the relative convergence of different transformation methods. They can also be used to select “good” transformation parameters and a “good” time steps. Two transformation‐based methods, Kirkland's ϕ‐based method and Ross' p‐based method, as well as Celia's h‐based method are used as comparisons. Test cases are infiltrations into very dry heterogenous porous media. The results show that the Pt‐based method is the most effective of the four methods and the simplest of the three transformation methods. It is several orders of magnitude faster than the h‐based method and also reduces the truncation error using the same grid. The proposed adaptively optimum relaxation procedure enhances the efficiency and robustness of all four methods to different degrees. Combination of the Pt‐based method with the adaptively optimum relaxation procedure results in a very efficient and robust algorithm. Using a coarser grid in the horizontal direction can further improve the efficiency in both computer time and memory requirements of the two‐dimensional model without loss of accuracy.