The purpose of this study is to develop exact analytic solutions for the pressure response of a finite conductivity fracture. This model should be able to verify the existence of various flow regimes found in earlier studies. It is hoped that this solution could be modified to give simplified expressions for well pressures for all times and all fracture conductivity ranges.

The present work poses and solves the problem of a vertical finite conductivity fracture of elliptical cross-section. The flow within the fracture is assumed to be incompressible and the reservoir is assumed to be infinite. The elliptical fracture geometry was chosen to facilitate the expression of fracture and reservoir pressures as eigenfunction expansions.

The solution is obtained by expressing the reservoir pressure as a series of Mathieu functions, and the fracture pressure as a series of cosines. The coefficients in these series satisfy an infinite set of linear relations, termed Fredholm sum equations. Exact solutions to these sum equations are obtained in forms which resemble continued fractions of summations, or equivalently, which require iteration of rational forms. A great deal of effort has been expended to speed the calculation of the solutions, however, only partial success has been achieved.

The solutions become increasingly difficult to compute as time decreases. So, approximate solutions for well pressures are given for extremely low values of time. These solutions indicate that behavior of an elliptical fracture is essentially the same as that of a rectangular fracture. Indeed, the well pressures calculated in this work are quite close to those for a rectangular fracture. Generally applicable simplified well solutions have not been found.

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