Analytical solution of transient radial air flow to an extraction well

Abstract

The non-linear partial differential equation describing the transient radial flow of air towards an extraction well has attracted extensive attention. Several linearized analytical solutions have been derived by adapting analytical solutions for groundwater flow. However, the validity of these linearized solutions remains unjustified because the exact non-linear solution for transient radial air flow to a well has not been available. In this study, an exact analytical solution is obtained using the Boltzmann transformation, and the pressure is calculated using the Newton–Raphson method. The solution provides a means for evaluating linearized analytical solutions and verifying numerical models. Examples are presented to test linearized analytical solutions for air flow towards an extraction well. Tests indicate that errors in pressure calculation using linearized analytical solutions are relatively small when the governing equation and boundary conditions are linearized with respect to pressure-squared. However, the errors owing to linearization are unacceptably large when the governing equation and boundary conditions are linearized with respect to pressure. Linearization with respect to pressure-squared is therefore recommended in adapting groundwater flow solutions. The new technique developed in this paper results in a solution that is simple to implement and requires very little computational effort. The technique can be applied to solve many other nonlinear diffusion problems.