A vector potential for a partly penetrating well and flux in an approximate method of images

Abstract

A vector potential is formulated for three-dimensional steady groundwater flow within a framework that is consistent with the Lagrange stream function for twodimensional flow. Analytical expressions are developed for vector potentials of a point–sink and a line–sink (of both finite and infinite length). A previously unpublished vector potential for a partly penetrating well in a horizontal confined aquifer is obtained using the method of images, where an infinite series of image line–sinks is approximated using a finite number of images and two semi–infinite elements. Singularities and discontinuities within these vector potentials are interpreted as producing a virtual discharge connecting vortex lines into continuous loops over infinity. The vector potential is used to accurately and efficiently compute the net flux through surfaces containing singularities in the specific discharge. This net flux quantifies errors in the elevation of streamlines associated with a finite number of images, and provides a measure to ensure a relatively seamless link between local three–dimensional flow and farfield two–dimensional flow.