Exact closed‐form solution of the two‐dimensional Laplace Equation for Steady Groundwater flow with nonlinearized free‐surface boundary condition

Abstract

Steady groundwater flow with steep gradients of the phreatic surface can be described by the Laplace equation. Without employing either the Dupuit assumption nor any other linearization of the nonlinear free‐surface boundary condition, we develop for an aquifer of infinite horizontal extension the exact closed‐form analytical solution of the two‐dimensional Laplace equation with nonsymmetric arbitrary recharge and/or drainage. Conformal mapping, transformation procedures, and inversion of the resulting integral equation represent the essential steps in the development of the solution, providing the location of the phreatic surface throughout the considered domain. The solution is expressed exclusively in algebraic terms without the need for iterative procedures thus offering a new insight and transparency into the solution's structure. The closed‐form solution relates straightforwardly to real‐world problems and can also be used for the evaluation of recharge or drainage in a river‐aquifer system through a relatively simple solution of the inverse problem without numerical inconvenience. The closed‐form solution creates a new standard for investigating both the behavior of numerical solutions and the domain of validity for simplified approaches. The computer code SURF can be downloaded from http://www.tu‐dresden.de/fghhihm/hydrologie.html.