Non-linear radiation in the Boussinesq equation of the agricultural drainage

Abstract

Water flow to subterranean drains is described by using the Boussinesq equation subjected to radiation type boundary conditions; these conditions establish a relationship between drainage flux and water head at the drain. Analyzed radiation conditions are a quadratic polynomial equivalent to the Hooghoudt equation and a power function; both conditions contain linear and quadratic radiations. Evidence is given that these last radiations correspond to the extreme probabilistic models of water flow proposed, respectively, by Purcell [Purcell, W.R., 1949. Capillary pressures their measurement using mercury and the calculation of permeability thereform. Petr. Trans., Amer. Inst. Mining Metall. Eng. 186, 39–48] and Childs and Collis-George [Childs, E.C., Collis-George, N., 1950. The permeability of porous materials. Proc. R. Soc., Ser. A 201, 392–405]. Hooghoudt type radiation results from a convex combination of extreme radiations using a reference flux and an interpolation factor as parameters. Power radiation is established from both fractal geometry concepts and partially correlated water flow by soil structure. This fractal radiation contains a reference flux and an exponent equal to double of the surface soil-particles fractal dimension with regards to Euclidean soil dimension. Considering that convex radiation is an approximation to fractal radiation, the least square method allows us to establish a relationship between the interpolation factor and the exponent of the fractal radiation. We use a numerical solution to the Boussinesq equation subjected to radiation conditions to describe a drainage experiment performed in the laboratory. The results shows that the cumulative drained depth is better represented by fractal and convex radiations rather than by extreme ones.