Stable pumping rates for horizontal wells in bank filtration systems

We consider common bank filtration systems and develop an explicit analytic solution representing steady, two-dimensional, groundwater flow to a horizontal well near a river in an unconfined aquifer. For the boundary-value problem investigated, we find that a unique solution exists for all negative well discharges. For positive discharges, a maximum value exists which corresponds to the formation of a cusp on the free surface. For positive discharges less than the maximum, the solution is not unique, consisting of two alternate configurations of the free surface. One solution includes a stable free surface on a single-valued physical plane, while the alternate solution includes a looped free surface lying on two sheets of a Riemann surface. Imposing a stability condition on the free surface results in a unique solution to the problem. We use the solution to investigate the behavior of the free surface under varying well discharges to identify stable pumping rates and predict well yield. In particular, we examine the well yield and the stability of the free surface when the head in the horizontal well is maintained at the top of the well screen. This condition is shown to produce a stable free surface for a wide range of well radii; the stability is independent of the hydraulic conductivity of the aquifer, the location of the well, or the presence of a skin resistance at the well.