Numerical solution of the Boussinesq equation: Application to the agricultural drainage

Abstract

Water flow to subterranean drains is described by the Boussinesq equation. To solve this equation, analytical solutions comprising constants, such as the transmissivity and drainable porosity have been developed; however, these solutions assume that free surface of the water falls instantly over the drains. The aim of this investigation is to present a finite difference solution of the differential equation using a drainable porosity variable and a fractal radiation condition. Here, two schemes are presented: the first one, with an explicit head and drainable porosity, both joined by a functional relationship, called mixed formulation; and the second one, called head formulation, with only the head. By using a lineal analytical solution, both methods have been validated and the nonlinear part was stable and brief. The proposed numerical solution is useful for the hydraulic characterization of soils with inverse modelation and to improve the designs of agricultural drainage systems, when taking into consideration that the assumptions of the classical solution have been eliminated. To evaluate the descriptive capacity of the numerical solution, these solutions were used to describe a drainage experiment performed in the laboratory. The results show that the cumulative drained depth is well represented by these solutions with the fractal radiation and the variable drainable porosity. Key words: Fractal radiation condition, variable drainable porosity, analytical solution, mixed formulation, head formulation.