Unsteady Groundwater Flow over Sloping Beds: Analytical Quantification of Stream–Aquifer Interaction in Presence of Thin Vertical Clogging Layer

Abstract

In this paper, new analytical solutions are presented to quantify interaction of water between a sloping aquifer and stream of varying heads in presence of a thin vertical sedimentary layer. The initial hydrogeological setting consist of an unconfined sloping aquifer of semi-infinite extent, a fully penetrating stream of varying water level, and a vertical layer of streambank deposits that acts as an interface between stream and aquifer. Unsteady groundwater flow is characterized by a nonlinear Boussinesq equation subject to Robin boundary condition (also referred to as third kind and Cauchy boundary condition). Unlike existing results, which focus only on step changes in stream stage, the current study accounts for gradual rise and decline in stream level. A closed-form analytical solution for water head distribution, discharge rate, and net volumetric exchange of water between stream and aquifer are developed from the solution of a linearized advection-diffusion equation. Performance of the analytical solution is compared with the numerical solution of the corresponding nonlinear Boussinesq equation using L2 and Tchebycheff norms. It is shown that the linearization yields acceptable solutions in many and varied situations where a stream interacts with an aquifer. The analytical solutions are presented in a manner that several other configurations, namely, zero slope, absence of vertical clogging layer, and abrupt changes in stream stage can be deduced as limiting cases of the main results. Combined effects of streambank leakance, bed slope, and stream-stage variation rate on the bank storage characteristic of the aquifer are illustrated with a numerical example.