Clean two‐ and three‐dimensional analytical solutions of Richards' equation for testing numerical solvers

This technical note derives clean analytical solutions of Richards' equation for three‐dimensional unsaturated groundwater flow. Clean means that the boundary conditions and steady state solutions are closed form expressions and the transient solutions have relatively simple additional Fourier series terms. Two‐dimensional versions of these solutions are also given. The primary purpose for the solutions is to test linear and nonlinear solvers in finite difference/volume/element computer programs for accuracy and scalability using architectures ranging from PCs to parallel high‐performance computers. This derivation starts from the quasi‐linear assumption of relative hydraulic conductivity varying exponentially with pressure head and the separate approximation that relative hydraulic conductivity varies linearly with moisture content. This allows a transformation to be used to create a linear partial differential equation. Separation of variables and Fourier series are then used to obtain the final solution. Physically reasonable material properties are also used. A total of four solutions are given in this technical note (steady state and transient solutions for two different boundary conditions of the sample problem).