Approximation of surface-groundwater interaction mediated by vertical streambank in sloping terrains

New analytical solutions are derived to estimate the interaction of surface and groundwater in a stream-aquifer system. The analytical model consists of an unconfined sloping aquifer of semi-infinite extant, interacting with a stream of varying water level in the presence of a thin vertical sedimentary layer of lesser hydraulic conductivity. Flow of subsurface seepage is characterized by a nonlinear Boussinesq equation subjected to mixed boundary conditions, including a nonlinear Cauchy boundary condition to approximate the flow through the sedimentary layer. Closed form analytical expressions for water head, discharge rate and volumetric exchange are derived by solving the linearized Boussinesq equation using Laplace transform technique. Asymptotic cases such as zero slope, absence of vertical clogging layer and abrupt change in stream-stage can be derived from the main results by taming one or more parameters. Analytical solutions of the linearized Boussinesq equation are compared with numerical solution of corresponding nonlinear equation to assess the validity of the linearization. Advantages of using a nonlinear Robin boundary condition, and combined effects of aquifer parameters on the bank storage characteristic of the aquifer are illustrated with a numerical example.