Finite difference solution of the agricultural drainage Boussinesq equation with variable drainable porosity subject to a fractal radiation boundary condition

Abstract

Subsurface drainage systems are used to control the depth of the water table and to reduce or prevent soil salinity. Generally, the flow of the groundwater is studied with the Bousssinesq equation, whose analytical solutions are obtained assuming that aquifer transmissivity and drainable porosity are constant. These solutions assume as well that the free surface of the water falls instantly over the drains. The general solution requires numerical methods. Some authors have demonstrated that the drain boundary condition is a fractal radiation condition and that the drainable porosity is a variable which is related to the soil retention curve. This solution has been obtained with a finite element method, which in one-dimensional form is equivalent to a finite difference method. Here, we propose a finite difference solution of the differential equation with variable drainable porosity and a fractal radiation condition. The proposed finite differences method has two formulations: the first one, with an explicit head and drainable porosity, both joined with a functional relationship, which we call mixed formulation; and the second one, which we call head formulation, with only the head. Both methods have been validated with a lineal analytical solution, and the nonlinear part is stable and brief. The proposed numerical solution is useful for the hydraulic characterization of soils with inverse modeling and for improving the design of agricultural drainage systems, considering that the assumptions of the classical solution have been eliminated.