General analytical solutions of the linearized Richards equation for a half-space and a limited-layer domains

This paper aims to describe a compilation of solutions of the linearized one-dimensional Richards equation, solved both in a half-space and in a limited-layer domain. The solution, that is the soil water content at any required time and depth, is the sum of two components, one related to the initial condition and to null boundary conditions and the other related to the boundary conditions and to a null initial condition. The sum of the two quoted components is the general solution of the Richards equation in integral form; the analytical expression of the soil water content distribution is therefore obtained if the integrals in the solution can be solved. Besides the integral form solution, another solution holding for any initial and boundary conditions represented with step functions is described in the paper. The initial condition is always the soil water content profile (e.g. the one experimentally measured) while the boundary conditions are different for the two domains. For the half-space domain, the boundary condition can be the soil water content at the surface or the surface flux (e.g. the measured precipitation or evaporation). For this domain, the solution, with the supplementary conditions expressed as step functions, is obtained using a procedure, which accounts for the effects of the hydrological conditions of the soil on the flux at the surface. Therefore, this procedure is able to switch between successive atmosphere-controlled and soil-controlled phases of infiltration or evaporation, as required by the given boundary condition. The procedure provides the ponding time, the desiccation time and the surface water flux during the soil-controlled phases. For the limited-layer domain, the top and bottom boundary conditions are given as time dependent soil water content trends. Also for the limited-layer, a solution, obtained approximating the supplementary conditions with step functions, is derived using the basic solution, which is here presented. It provides the soil water content profile evolution, the top and bottom instantaneous and cumulative fluxes and the water gained by the soil layer in a specified time interval. Lastly, a comparison between the procedure results and an exact analytical solution is discussed.