Addressing nonlinear transient diffusion in porous media through transformations

The nonlinear differential equation describing flow of a constant compressibility liquid in a porous medium is examined in terms of the Kirchhoff and Cole-Hopf transformations. A quantitative measure of the applicability of representing flow by a slightly compressible liquid – which leads to a linear differential equation, the Theis equation – is identified. The classical Theis problem and the finite-well-radius problem in a system that is infinite in its areal extent are used as prototypes to address concepts discussed. This choice is dictated by the ubiquity of solutions that depend on these archetypal examples for examining transient diffusion. Notwithstanding that the Kirchhoff and Cole-Hopf transformations arrive at a linear differential equation, for the specific purposes of this work – the estimation of the hydraulic properties of rocks, the Kirchhoff transformation is much more advantageous in a number of ways; these are documented. Insights into the structure of the nonlinear solution are provided. The results of this work should prove useful in many contexts of mathematical physics though developed in the framework of applications pertaining to the earth sciences.