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 are 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.
Due to the large disparity in length scale between wellbore diameter and reservoir simulation grid, a well is generally treated as a point source or sink in finite difference reservoir simulation. A productivity index, PI, is used to relate the pressure difference between well-block and wellbore to the production rate of the well. The productivity index of a vertical or horizontal well in a homogeneous formation has been derived for two-dimensional flow (e.g., Peaceman, Babu et al.). However, these formulas are limited in their application. A more general description that admits three-dimensional, layered, anisotropic formations and arbitrary well orientation is lacking.
The Darcy flow around a well can be calculated numerically using finite difference or finite element methods. In addition, Lee has developed a boundary element method that efficiently calculates the flow in anisotropic formations between two impermeable boundaries. However, these numerical methods are complex and computationally expensive, especially for a formation with layered boundaries. An analytical solution would be quite valuable in the accurate calculation of the flow. Slender body theory and the method of reflections provides the means to obtain such a solution.
The analytical solution for flow to a well in a two-dimensional porous media is the well-known logrithmic solution (Muscat). While this solution can account for anisotropies in the permeability tensor, it does not presently account for layered boundaries or even impermeable barriers in a formation. By combining this result with a technique known as the method of reflections, we can eliminate this limitation. The method of reflections has enjoyed widespread use in the solution of problems in suspension mechanics (see, for example, Happel and Brenner, Lee, Chadwick, and Leal).