Analytical solutions for radial dispersion with cauchy boundary at injection well

Abstract

In dealing with solute transport from an injection well into an aquifer a macroscopic boundary condition of the Cauchy type (the third type) can be formulated at the well‐aquifer interface if the mass balance principle is invoked. This Cauchy boundary condition includes effects of radial advection and longitudinal dispersion as molecular diffusion is neglected in the problem. Analytical solutions related to a continuous and a pulse injection are determined for the radial dispersion problem concerning such a Cauchy boundary condition. These solutions give resident concentrations in the radial flow system. Results of the continuous injection solution and of that related to a Dirichlet boundary condition (the first type) are different for short injection periods but converged for long injection periods. The deviation caused by using two different boundary conditions is further studied with appropriate asymptotic solutions valid only for short injection periods. It is found that significant concentration gradients may exist across the well‐aquifer interface for short injection periods. Under this condition, neglect of the local dispersion process at the well‐aquifer interface (as assumed in the Dirichlet boundary condition)amounts to forcing more solute mass into the aquifer. As injection is prolonged, the concentration gradient across the interface decreases as the associated local aquifer concentration is increased to the injected concentration level. For relatively long injection periods a local concentration equilibrium at the well‐aquifer interface is reached and thereafter advection dominates the local transport. The Dirichlet boundary condition is a special case neglecting local dispersion at the well bore to the Cauchy boundary condition. For the radial dispersion equation, solutions regarding a Dirichlet boundary condition do not give the flux concentrations due to the nonconstant groundwater velocity. Nevertheless, solutions of flux concentrations for a continuous and a pulse injection are acquired here by applying the flux concentration transformation to appropriate solutions regarding the Cauchy boundary condition.