In finite difference reservoir simulation, a well is generally treated as a point source or sink. As a result, a well productivity index must be specified to relate the difference between wellblock pressure and wellbore pressure to the production rate. However, analytical formulas for productivity indices are known only for a vertical or horizontal well in a homogeneous formation. In this paper, the method of reflections and slender body theory are shown to provide a means to obtain accurate measures of the productivity index in complex reservoir formations.

Using slender body theory, we have derived an asymptotic solution of single-phase flow around an inclined well that is applied here to the determination of the productivity index. The method of reflections can account for the presence of layer boundaries through the application of image solutions. In combination, these techniques permit the calculation of the productivity index in anisotropic formations with layer boundaries. A comparison of the results with numerical solutions verifies their accuracy. The previous formulas sire shown to be recovered as a special case.

Several examples are presented. The effect of layer boundaries on the productivity index is shown in both two and three dimensions. We consider both permeable and impermeable layers. We also investigate the productivity index of an inclined well in a layered anisotropic formation.

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