Second-order two-dimensional solution for the drainage of recharge based on Picard's iteration technique: A generalized Dupuit-Forchheimer equation

Aquifer recharge is one of the most important problems in hydrology from both theoretical and practical points of view. One of the most widely accepted methods to deal with this problem is the use of the Dupuit-Forchheimer theory. This theory assumes that the water table is almost horizontal, the vertical velocity is zero, and the horizontal velocity is uniform with depth. Surfaces of seepage are not considered. Despite these strong limitations the theory is applied, and success is frequently found in many cases despite its fundamental assumptions being violated. In this work an approximate 2-D solution to the problem is sought on the basis of Picard's iteration technique, from which a second-order differential equation for recharge problems is found. On the basis of this solution, a modified, analytical Dupuit-Forchheimer (DF) ellipse is found which compares favorably with the full 2-D solution of the problem. The analytical developments of this theory provide a generalized DF theory which permits as an outcome the analytical determination of the surface of seepage.