Three-dimensional analytical solutions of Richards’ equation for a box-shaped soil sample with piecewise-constant head boundary conditions on the top

Summary In a recent paper by this author, analytical solutions of Richards’ equation for three-dimensional (3-D) unsaturated flow were derived for a box-shaped soil sample. The same concepts are used in this paper, except that a complicated specified head boundary condition on the top of the soil sample used in the previous paper is replaced by a simple piecewise-constant specified head boundary condition. This author has concluded that both solutions are very valuable in testing numerical models using the finite element/volume/difference computational techniques. These analytical solutions were originally derived to test parallel high performance computing groundwater programs, but it has been found that two-dimensional versions of these solutions are also good for testing on PCs. In particular, the efficiency and accuracy of the linear and nonlinear solvers can be scrutinized effectively using these solutions. Analytical solutions for Richards’ equation are difficult to derive because of the highly nonlinear aspect of this partial differential equation. However, the quasi-linear approximation of the log of relative hydraulic conductivity varying linearly with pressure head coupled with relative hydraulic conductivity varying linearly with moisture content allows Richards’ equation to be transformed into a linear partial differential equation. Physically reasonable material properties are also achieved in this approximation, although modeling of real-world problems is limited by this assumption. Despite these limitations, the resulting analytical solutions have proven to be extremely valuable in testing groundwater models. As in the previous paper, a transformation based on the assumptions stated above, separation of variables, and Fourier series are used to obtain the final solution. Steady-state and transient solutions for two different boundary conditions of the test problem are provided in this paper.