An approximation for leaky boundaries in groundwater flow

Abstract

Groundwater flow domains often contain thin layers of low hydraulic conductivity that act as leaky barriers to flow. It is common to include the effects of these leaky layers in the mathematical description of a problem as a mixed boundary condition; we refer to this type of boundary as a leaky boundary. Practical problems containing leaky boundaries have proven difficult to solve analytically. An approximation is presented, for problems of steady groundwater flow, that removes many of the difficulties associated with leaky boundaries. The approximation replaces either a leaky layer or a leaky boundary with a section of aquifer with an equivalent one-dimensional resistance to flow. The leaky boundary is reduced to a combination of impermeable and equipotential boundaries in a transformed domain. The problem may then be dealt with in the transformed domain by classical methods. The nature of the approximation is evaluated in detail. In the case of a leaky layer, the approximation replaces the isotropic layer with an anisotropic one; a small hydraulic conductivity is maintained normal to the layer, but the hydraulic conductivity in the direction parallel to the layer becomes large. Therefore, the approximation is useful where problem boundaries act to constrain the tangential component of flow within the layer. The approximation of a leaky boundary consists of a semi-permeable membrane with no resistance to flow along the membrane. The approximation is analogous to the Dupuit approximation which is used commonly to solve groundwater flow problems. The approximation is applied to investigate a problem of practical interest: flow in the vertical plane to a clogged stream bed. The quality of the solution is judged by comparison with a boundary integral solution to the same problem; the results compare favorably.