Approximate solution to the generalized radial Boussinesq equation

Abstract

This paper presents a method for constructing approximate semi-analytical solutions for the radially symmetric generalized Boussinesq equation, also known as the porous medium equation (PME). This work is a further extension of analysis done by Telyakovskiy et al. (2016), that studied water injection governed by the Boussinesq equation into an unconfined initially dry aquifer. The newly derived approximate solution also has two parts; one to approximate singular behavior near the well and a polynomial part to model far-field behavior. With the prescribed initial and boundary conditions, the analyzed problem can be rewritten in terms of similarity variables resulting in a boundary value problem for a nonlinear ordinary differential equation. Using higher-order physically significant moments of the solution, a closed-form expression for the position of the wetting front and the shape of the phreatic surface is obtained. The introduced solution is valid over a wide range of injection regimes: the time-independent, the power-law injection function with positive exponents, and exponential-law of injection. A highly accurate numerical solver similar to Telyakovskiy et al. (2016) is used to validate the presented approximate solutions. Plus, comparison is made with the experimental data on prediction of the wetting front position for the special case of the Boussinesq equation.