G2. Snippets from Infiltration: Where Approximate Becomes Exact

Abstract

The Darcy-Buckingham macroscopic approach to soil-water modelling, leading to a nonlinear Richards’ diffusion-convection equation, has been very useful for many decades. Some sharp results of the 1970s by W. Brutsaert and J.-Y. Parlange have been an influence on many, including myself. Since the 1980s, several groups have used an integrable one-dimensional version of Richards’ equation, with realistic nonlinear transport coefficients, to predict experimentally verifiable quantities. Neat expressions have been derived for time to incipient ponding, for the dependence of sorptivity on pond depth and for the second and higher infiltration coefficients. These exact results are at odds with those of the traditional Green-Ampt model. In the limit of delta-function diffusivity, the water content profile approaches a step function, so the water content is everywhere close to either the boundary value or the initial value. As explained by Barry et al (1995), far from there being a unique “Green-Ampt limit”, practical predictions in the limit of a delta function diffusivity depend subtly on the relationship between diffusivity and conductivity at intermediate values of water content. In fact, the traditional Green-Ampt predictions, with a constant potential at the wet front, may be recovered from a linear, rather than step-function behaviour of conductivity vs water content. A number of practical predictions of the integrable model agree exactly with those of the approximate analytic method originated earlier by Parlange, involving approximations within an integrand after expressing the water conservation equation in integral form. The exactly solvable model refutes the traditional Green-Ampt model and validates the quasi-analytic integral formulation.