Innovative Analytical Solutions to 1, 2 and 3D Water Infiltration into Unsaturated Soils for Initial-Boundary Value Problems

Abstract

Abstract. Fluid in ltration into unsaturated soil is of vital signi cance from many perspectives. Mathematically, such transient in ltrations are described by Richards' equation: a nonlinear parabolic Partial Di erential 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 1D horizontal and vertical water in ltration are derived and compared to numerical nite-di erence method solutions, whereby both solutions are shown to coincide well with one another. Solutions to 2 and 3D vertical water in ltration are derived for constant, no-ow, 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 di erent computational techniques.