Innovative analytical solutions to 1, 2, and 3D water infiltration into unsaturated soils for initial-boundary value problems

Fluid infiltration into unsaturated soil is of vital significance from many perspectives. Mathematically, such transient infiltrations are described by Richards equation: a nonlinear parabolic Partial Differential Equation (PDE) with limited analytical solutions in the literature. The current study uses separation of variables and Fourier series expansion techniques and presents new analytical solutions to the equation in one, two, and three dimensions subject to various boundary and initial conditions. Solutions for ID horizontal and vertical water infiltration are derived and compared to numerical finite-difference method solutions, whereby both solutions are shown to coincide well with one another. Solutions to 2 and 3D vertical water infiltration are derived for constant, no-flow, and sinusoidal boundary and initial conditions. The presented analytical solutions are such that both steady and unsteady solutions may be obtained from a single closed form solution. The solutions may be utilized to test numerical models that use different computational techniques.