Exact solutions for water infiltration with an arbitrary surface flux or nonlinear solute adsorption

Several authors have now used the Backlund transformation to linearize Richard's equation and so obtain analytical solutions for infiltration and redistribution processes for the linearizable class of diffusivities and hydraulic conductivities. After transformation a convection- diffusion equation is obtained. To date, all the analytical results making use of the Backlund transform have been based on a specified, constant flux that is assumed to exist at the surface boundary. We generalize the previous work by removing this restriction and obtain a general solution for an arbitrary temporal variation in surface flux. The solution contains a function that is specified by a Volterra integral equation. In practice, the Volterra equation must be evaluated numerically. This task is accomplished using a relatively straightforward algorithm. The analytical results for infiltration can be applied to the process of solute transport subject to a nonlinear adsorption isotherm since these two phenomena are equivalent under another mapping. Our new analytical results, therefore, also apply to this class of solute transport processes. The general form of the adsorption isotherm for which exact results can be obtained is derived. This isotherm appears capable of representing a relatively large family of physically relevant cases, although this topic remains open for future investigation