Analytical Solutions of the Groundwater Flow Equation

Abstract

The differential groundwater flow equation derived in Chap. 2 can be solved analytically in various geometries, provided that certain hypotheses are satisfied. In this chapter, a polar coordinate system with radial geometry, describing the radial groundwater flow towards a well, is considered. The hypotheses underlying the analytical solutions concern the aquifer’s geometry (constant thickness, homogeneity and isotropy, unlimited horizontal extension, initially horizontal potentiometric surface) and the pumping well (fully penetrating, infinitesimal radius, negligible storage, laminar flow and constant pumping rate). Steady state and transient analytical solutions, respectively describing the drawdown as a function of the distance from the well (r), or of r and time, are provided for confined, leaky and unconfined aquifers. Theis’ (and Cooper and Jacob’s approximation) and Thiem’s equations describe, respectively, the transient and steady state solutions of the groundwater flow equation for confined aquifers. Hantush and Jacob, instead, derived the transient analytical solution for leaky aquifers, while De Glee formalized the steady state solution. In the case of unconfined aquifers, the steady state solution formally coincides, except for an adjustment to the drawdown, to Thiem’s solution. The transient solution was, instead, derived by Neuman, under specific simplifying hypotheses, given that a fully rigorous description of flow in unconfined aquifers would entail the use of a nonlinear and nonhomogeneous differential equation due to the inclination of the water table with pumping and the generation of a vertical component of flow velocity.