Accurate approximate semi-analytical solutions to the Boussinesq groundwater flow equation for recharging and discharging of horizontal unconfined aquifers

Abstract

The Boussinesq equation is usually used to describe one-dimensional unconfined groundwater movement. Solutions of this equation are important as they provide useful insights regarding the water table response to stream level variations and allow us to quantify the exchange flow between the stream and the aquifer. Due to the nonlinearity of the Boussinesq equation, the solutions are generally obtained using numerical methods. However, for certain classes of initial and boundary conditions there are both exact and approximate analytical solution techniques. This work focuses on the latter approach. A new mathematical technique for approximate solutions of the Boussinesq equation describing flow in horizontal unconfined aquifers induced by sudden change in boundary head is presented. The method applies to the problems of recharging and dewatering of an unconfined aquifer, and approximate solutions to both problems are derived. The solutions were obtained by introducing an empirical function with four parameters which might be obtained using a numerical fitting procedure. Results based on this technique were found to be easily calculated and to be in good agreement with those obtained using numerical calculation based on Runge-Kutta approach. A benchmark between the proposed solutions and five existing approximate analytical solutions shows that the present solutions are the most accurate approximate solutions among those tested. Applications of the solutions are presented in the context of estimating aquifer hydraulic parameters.