Explicit solution of a free boundary problem for a nonlinear absorption model of mixed saturated-unsaturated flow

Abstract

In wet soils, zones of saturation naturally develop in the vicinity of impermeable strata, surface ponds and subterranean cavities. Hydrology must be then concerned with transient flow through coexisting unsaturated and saturated zones. The models of advancing saturated zones necessarily involve a nonlinear free boundary problem. A closed-form analytic solution is presented for a nonlinear diffusion model under conditions of ponding at the surface. The soil water diffusivity is restricted to the special functional form D(θ) = a (b − θ) 2 , where θ is the water content field to be determined and a, b are positive constants. The explicit solution depends on a parameter C (determined by the data of the problem), according to two cases: 1 < C < C 1 or C ≥ C 1, where C 1 is a constant which is obtained as the unique solution of an equation. This result complements the study given in P. Broadbridge, Water Resources Research, 1990, 26, 2435–2443, in order to established when the explicit solution is available. The behavior of the bifurcation parameter C 1 as a function of the driving potential is studied with the corresponding limits for small and large values. Moreover, the sorptivity is proven to be continuously differentiable function of the variable C.