Hydraulically optimal porous liner around a porous lens: Strack’s problem revisited

A porous barrier bounded by two confocal ellipses is filled with material of a hydraulic conductivity different from the ambient one in order to control the flow rate inside. Optimization problems are solved by describing the flow through a medium consisting of zones of different properties. For three-component composites, viz. three zones demarcated by two confocal ellipses, an explicit rigorous solution of refraction problems for potential fields of Darcian 2-D saturated flows is used. An aquifer, an elliptical ‘core’ and an elliptical annulus, which serves as a liner of the core, have contrasting hydraulic conductivities (control variables). The magnitude of the velocity in the core, total flow rate, travel time along streamlines, hydraulic gradient in the core are objectives in optimization. A single maximum of the magnitude of velocity in the ‘core’ is found at a certain conductivity of the annulus, similarly to the Strack circular annulus refracting a unidirectional flow. This maximum can be higher or lower than the velocity at infinity, i.e. the liner can amplify or lessen the incident flow. Applications to in situ ore leaching, permeable reactive barriers, porous waste repositories, aquifer permeameters and formation damage due to suffusion in the vicinity of pumping wells are discussed. Abbreviations: ISL: in situ leaching; RAF: reduction–amplification factor.