Semianalytical Solution to Richards' Equation for Layered Porous Media

Abstract

Traditional finite-difference and finite-element solutions to Richards' equation can exhibit stability problems and mass balance errors when sharp saturation fronts migrate across material interfaces. Several semianalytical solutions to Richards' equation have been developed for layered media, but none allow for arbitrary constitutive relationships between capillary pressure, water saturation, and relative water permeability. This paper develops a more general semianalytical solution to Richards' equation that can simulate unsaturated flow in layered media and utilize arbitrary constitutive relationships. Based on a combination of the Runge-Kutta and shooting methods, initial-boundary-value problems are solved for layered systems without experiencing the types of stability problems commonly associated with numerical models. The proposed numerical scheme is computationally efficient and numerically stable, and it compares favorably with other analytical solutions. Results of two example simulations demonstrate the versatility of the method for solving both horizontal and vertical unsaturated flow problems. The primary disadvantages of the numerical approach are that it is limited to one-dimensional flow and that it requires the development of specific iteration schemes for different categories of boundary-value problems.