Analytical solution for a phreatic groundwater fall: the Riesenkampf and Numerov solutions revisited

Abstract

Steady, two-dimensional Darcian flow in a homogeneous isotropic unconfined aquifer, bounded from below by a rectangular wedge representing bedrock, is studied by the theory of holomorphic functions. A triangle of the complex potential domain is mapped onto a circular triangle in the hodograph plane with the help of an auxiliary variable. A full potential theory results in closed-form integral representations for the complex potential and complex velocity, from which the flow rate and free surface are calculated using computer algebra built-in functions. This solution, uniformly valid in the whole flow domain, is compared with simpler approximate ones, retrieved from an analytical archive. Two flow zones are distinguished: a tranquil subdomain where the Dupuit-Forchheimer approximation is suitable and a nappe (a subdomain with a rapidly changing Darcian velocity and steep slope of the phreatic surface) where the Numerov or Polubarinova-Kochina solutions, in terms of the full potential model, are available. Approximations in the two zones are conjugated by matching the positions of the water table and the flow rates, which eventually agree well with the obtained comprehensive solution.