Optimal placement of a wellbore seal impeding seepage from a tilted fracture

Steady, one-phase, 2D Darcian seepage in a homogeneous pay from one horizontal isobar to another with a tilted fracture intersecting the two isobars is studied by the methods of complex analysis. The shortcut through the fracture is foiled by an impermeable seal of a constant length which is placed at the fracture-isobar intersection with an objective to minimise either the total flow rate from the fracture or the travel time of marked particles moving along the seal. Conformal mappings of the slanted-face half-strip in the physical domain onto a half-strip in the complex potential domain give explicit expressions of the seepage rate and travel time through hypergeometric functions. The optimal seal position is found by computer algebra routines for any given distance between isobaric horizons, tilt angle, pressure drop between isobars and hydraulic conductivity of the rock.     

A Well in a ??Target?? Stratum of a Two-Layered Formation: The Muskat?CRiesenkampf Solution Revisited

Explicit analytical solutions are obtained in terms of hydraulic head (pressure) and Darcian velocity for a steady Darcian flow to a point/line sink and array of sinks with refraction of streamlines on a horizontal interface between two layers of constant hydraulic conductivities. The sinks are placed in a ??target?? layer between a constant potential plane and interface. An equipotential surface, encompassing the sink represents a horizontal or vertical well, is reconstructed as a quasi-cylinder or quasi-sphere. The method of electrostatic images and theory of holomorphic functions are employed for obtaining series expansion solutions of two conjugated Laplace equations. If the conductivity of the ??target?? layer is less than that of the super/sub-stratum, then there is a minimum of the flow rate into the well of a given size. Applications to agricultural drainage and surface DC-electrical resistivity surveying are discussed.