Stream filtration induced by pumping in a confined, unconfined or leaky aquifer bounded by two parallel streams or by a stream and an impervious stratum

A mathematical model is developed for describing three-dimensional groundwater flow induced by a fully-penetrating vertical well in aquifers between two parallel streams. A general equation is adopted to represent the top boundary condition which is applicable to either a confined, unconfined or leaky aquifer. The Robin (third-type) boundary condition is employed to represent the low-permeability streambeds. The Laplace-domain head solution of the model is derived by the double-integral and Laplace transforms. The Laplace-domain solution for a stream depletion rate (SDR) describing filtration from the streams is developed based on Darcy's law and the head solution and inverted to the time-domain result by the Crump method. In addition, the time-domain solution of SDR for the confined aquifer is developed analytically after taking the inverse Laplace transform and the time-domain solutions of SDR for the leaky and unconfined aquifers are developed using the Padé approximation. Both approximate solutions of SDR are expressed in terms of simple series and give fairly good match with the Laplace-domain SDR solution and measured data from a field experiment in New Zealand. The uncertainties in SDR predictions for the aquifers are assessed by performing the sensitivity analysis and Monte Carlo simulation. With the aid of the time-domain solutions, we have found that the effect of the vertical groundwater flow on the temporal SDR for a leaky aquifer is dominated by two lumped parameters: κ=Kvx02/(KhD2) and κ'=K'D/(B'K_v) where D is the aquifer thickness, x₀ is a distance between the well and nearer stream, K_h and K_v are the aquifer horizontal and vertical hydraulic conductivities, respectively, and K' and B' are the aquitard hydraulic conductivity and thickness, respectively. When κ<10, neglecting the vertical flow underestimates the SDR. When κ ≥ 10, the effect of vertical flow is negligible. When ^{κ '} ≤ ^{10 - 4}, the aquitard can be regarded as impermeable, and the leaky aquifer behaves as a confined one.