A semi-analytical solution of Richards Equation for two-layered one-dimensional soil

A semi-analytical solution of the Richards Equation posed on a two-layered one-dimensional soil supplied with various boundary conditions is derived under a constraint that the constitutive relations are exponentially dependent on the pressure head. It allows for a transformation of the Richards Equation into a linear parabolic partial differential equation that governs a spatial-temporal function that represents the hydraulic conductivity. The procedure is proceeded with expressing this function as a linear combination of a set of eigenfunctions associated with a novel Sturm-Liouville problem that reflects the layer system and an auxiliary function that depends only on the spatial variable and the pressure head at the interface at the time of interest. All the relevant coefficients in the representation satisfy a nonlinear differential-algebraic system gathered from imposing the continuity of the pressure head and its flux at the interface. Two different approximations of the derivatives yield algebraic systems to be solved by the Newton method of iteration. Several pertinent numerical experiments demonstrating the approach are discussed and compared with the standard finite volume method.