Modelling unsaturated flow by using Mathematica

The built-in Mathematica function NDSolve is extended to solve nonlinear boundary-value problems and a second-order partial differential equation subjected to inconsistent boundary and initial conditions. This communication demonstrates the ability of the extended NDSolve to find steady-state and transient solutions of unsaturated flow described by the one-dimensional Richards equation. Only first kind boundary conditions are considered, but the extended NDSolve is applicable for first, second, and third kind boundary conditions. To verify the steady-state numerical results, an original analytical solution is derived. The results given by numerical and analytical steady-state solutions coincide. The residuum of transient solution is plotted to show that the numerical results satisfy the transient problem except in a singular point, where the boundary and initial conditions are inconsistent. The transient pressure head and soil moisture content are plotted for typical soil properties, starting from different initial steady-state distribution up to the time needed to reach the final steady-state distribution. Copyright © 2007 John Wiley & Sons, Ltd.