Using the stream function for flow governed by Poisson's equation

Abstract

For two-dimensional groundwater flow governed by Laplace's equation there exists both a discharge potential function and a stream function, which are each other's conjugate harmonics. The stream function is constant along a streamline, and the difference in its value for two streamlines represents the total flow between these streamlines. Most regional groundwater flow problems, however, exhibit areal recharge and leakage from adjacent aquifers. These problems are governed by Poisson's equation; no stream function exists for that case. In the context of the analytic element method (AEM), a technique has been developed which allows the stream function to be used in models which include areal recharge or leakage. The AEM uses superposition of analytic functions; the majority satisfy Laplace's equation, whereas only a few functions satisfy Poisson's equation. By treating these solutions to Poisson's equation separately, the stream function can be utilized in a manner similar to that for flow without areal recharge or leakage. The approach is applied to streamline tracing and, more importantly, to the modeling of leaky slurry walls and no-flow boundaries, both open and closed.