Analytical Solution of the Richards Equation under Gravity-Driven Infiltration and Constant Rainfall Intensity

In the field of soil hydrology, the Richards equation is commonly used to model water flow in unsaturated soils. The high nonlinearity of the Richards equation makes it very challenging to solve analytically for situations that are meaningful in practical applications. In this paper, an exact and simple analytical solution of the Richards equation under gravity-driven infiltration and constant rainfall intensity is derived. First, the solution is presented under Torricelli's law, which mimics the soil hydraulic conductivity function and describes the emptying or filling process of a nonlinear water reservoir. Then, following a similar approach, the solution is extended to the Brooks and Corey soil hydraulic conductivity function, which is generally considered to well describe soil hydrological characteristics. The approach followed in this study is a simple hydraulic approach; therefore, the derived solutions are not affected by uncertainty as long as the hypothesis of the gravity-driven infiltration is satisfied for the selected soils. A comparison with the solution numerically derived by the Richards equation for which the gravity-driven assumption is relaxed is performed and discussed. Interestingly, the comparison indicated that the suggested solution delimits the solutions domain of the Richards equation.