Spreading of the liquid over the surface with penetration into the soil

A semi-empirical theory of liquid spreading induced by the gravity force and accompanied by penetration into the soil is constructed in a quasi-one-dimensional approximation. Some specific features of spreading with allowance for the vegetation type are considered. Under the assumption that the dependence of the resistance force on the spreading velocity is linear or quadratic, the system of equations of liquid motion on the surface with dense and scarce vegetation is reduced to one nonlinear equation. Approximate analytical solutions for constant-power sources are obtained. A situation with no plants on the surface (i.e., the hydraulic resistance of the surface is determined by specific features of the soil) is analyzed.