A simple and accurate closed-form analytical solution to the Boussinesq equation for horizontal flow

Abstract

The classical problem used to model the response of an unconfined aquifer to a sudden change in boundary head is considered in this paper. This problem is usually modeled using the nonlinear groundwater Boussinesq equation. Due to the nonlinearity of this problem, no exact solutions exist. Solutions to the Boussinesq equation are therefore obtained using numerical techniques or approximate analytical methods. In this paper, we present a novel closed-form approximate analytical solution to this problem. The Boussinesq equation is converted to a first-order ordinary differential equation (ODE) by means of the Boltzmann transformation and by introducing a new variable related to the water flow. The first-order ODE is then solved analytically after introducing an intermediate approximation involving two fitting parameters. To avoid any numerical treatment, closed-form polynomial expressions of the fitting parameters are proposed. The final-form solution is simple to use and is obtained in terms of the incomplete gamma function, which is valid for both recharge and discharge. The derived solution is tested and compared to efficient numerical solutions, as well as to two types of analytical solutions: an accurate series expansion solution and an equivalent closed-form solution. The results show excellent agreement between the proposed solution and the numerical and series solutions. The proposed solution offers a key advantage in terms of both accuracy and simplicity; notably, it can be implemented using a simple spreadsheet.