A derivation of Dupuit solution of steady flow toward wells by matched asymptotic expansions

An approximate solution of the steady and shallow free‐surface flow toward a well in a layer of infinite extent is obtained by expanding the velocity potential in a small parameter power series. This expansion is shown to be valid only in the vicinity of the well and is, therefore, called the inner expansion. An outer expansion, which solves the flow problem at large distance from the well, is derived by using the method of matched asymptotic expansions. The Dupuit approximation coincides with the zero order term of the potential outer expansion. The derivation of a second order outer term makes possible the discussion of the validity of the Dupuit approximation, which tends asymptotically toward the exact solution. In the outer zone, the streamlines are parabolic and are not orthogonal to the equipotentials. The method is illustrated by two numerical examples.