Exact theory for the shape of the free water surface about a well in a semiconfined aquifer

An exact theory is developed for the height of the water table about a well penetrating to a barrier of a semiconfined aquifer. The flow is steady state, and the free surface is a streamline. The problem is a classic one for which the Dupuit-Forchheimer theory happens to give an exact formula for the well discharge but fails to give the proper shape of the free surface. The height of the free surface, at any point, is obtained through an equation in which the height occurs both linearly and in terms of an infinite series of transcendental functions. The equation is solvable by iteration. The theory starts with the development of a potential function. Any capillary fringe is neglected. The potential function is obtained by assuming that a fictitious flow medium occupies the space of the cone of depression. The lower boundary of the fictitious flow medium has the same boundary conditions as the upper boundary of the actual flow medium. The problem is important in well pollution. The theoretical methods may also be applied to the free surface problem when the free surface is not a streamline but receives steady flux.