SIMILARITY APPROXIMATION FOR RADIAL SUBSURFACE FLOW PROBLEM

A useful mathematical technique in the study of radial flow of underground water is the application of similarity transformations, such as Boltzmann's transformation. The transformations lead to approximations that are shown to be valid if certain physical parameters satisfy given limiting conditions. A nonlinear partial differential equation is formulated describing the radial flow of soil moisture. A similarity approximation is used to transform the partial differential equation into a nonlinear ordinary differential equation. Criteria by which to judge the applicability of the similarity approximation are derived. Several monotonicity properties of the transformed moisture as a function of the similarity variable are derived. The properties are used to develop a numeric procedure for the solution of the transformed equation. The procedure is used to study the radial flow of soil moisture to a cylindric sink for diffusivities which vary linearly, quadratically, and exponentially with soil moisture. The solutions show that the moisture front steepens with increasing nonlinearity. Copyright 1969 by the American Geophysical Union.