On the use of analytical and numerical advection‐dispersion solvers for solving Richardson–Richards equation

Water flow in unsaturated soils is modeled using the Richardson–Richards (RR) equation, a nonlinear partial differential equation. Contaminant transport processes, on the other hand, are modeled using the advection‐dispersion equation (ADE) that combines a hyperbolic term and parabolic dispersion term to simulate the advection and dispersion processes. In this paper, we examine various mathematical transformations used to transform both pressure‐head‐ and water‐content‐based RR equations to a form similar to the ADE. The transformed RR equations are solved using both analytical and numerical solution methods. We evaluated the efficiency of various discretization strategies for numerically solving an infiltration test problem using this transformation approach. In addition, we also derived a new type of analytical solution for the pressure‐head form of the transformed RR equation using this approach. To test the performance of different types of ADE‐based RR solutions, we employed a standard implicit solver as a benchmark to evaluate these solutions. Results of our comparison study, assessed in terms of computational performance and accuracy, demonstrated that ADE‐based solutions produce results comparable to those produced with standard solvers. In terms of computing efficiency, the dual time step explicit–implicit solution is the most promising strategy for solving the ADE‐form of the RR equation used to simulate the infiltration problem considered in this work. However, the other solution strategies also yielded comparable results. Our findings demonstrate that the use of ADE‐based solvers, which are primarily used in the contaminant transport literature, offers a promising alternative approach for solving the RR equation.