On transport through heterogeneous media: application of conjugated reciprocal transformations

Abstract

Conjugation of reciprocal transformations is used to solve a class of boundary value problems involving a source term relevant to water transport through a heterogeneous medium with a volumetric extraction mechanism. The main emphasis is the solution method that involves conjugation of reciprocal transformations, as well as other changes of variable, applied to a newly identified integrable model. The transport equation is a version of the nonlinear Richards equation, based on Buckingham’s extension of Darcy’s law to unsaturated media, now allowing for heterogeneous transport coefficients. In addition, there is a plant-root water extraction term that depends on both water content and position. The nonlinear boundary conditions have prescribed flux at one boundary and zero flux at the other barrier boundary of a one-dimensional heterogeneous medium.