Solution of Potential Flow Problems in Porous Media within Complex Geometrical Domains
Authors: Rampersad P.R. , Hareland G. , Ochas D. 
Published: December 2007

Abstract
This paper presents a new method for determining the effect of well location within any reservoir boundary on well performance. There are several types of analytical as well as numerical methods used to solve potential flow problems in bounded systems. However, these are limited in their treatment of the reservoirs geometrical shape. For two dimensional problems a powerful tool used is conformal mapping. In conformal mapping, a problem is transferred from a geometrical domain in which the solution is sought, to a domain in which the solution is known.
The transformation then provides the solution in the original domain. The practical limitation of conformal maps has always been that they must be computed numerically, except for simple domains where the exact conformal map is known. With improvements in computer technology the method can be used for fast, accurate and flexible computation of solutions to these problems. Traditionally, Dietz shape factors have been used to account for wellbore location within the drainage area. These have been presented for certain well locations in specific geometric domains. However, the technique described in this paper has been shown to be useful in determining solutions to flow problems in complex geometrical shapes, such as flow in fractured horizontal wells, under steady state flow conditions. In this study the basic techniques and their application to simple reservoir geometries will be presented. The results obtained compare closely with results obtained using the Dietz shape factors for certain limited wellbore and drainage area configuration. The application is presently limited to the steady state solution.

Introduction
For several decades the majority of potential flow problems were solved using several simplifying assumptions and relatively simple geometrical domains. In recent times we have used numerical computing to solve these problems in virtually any domain as well as in three dimensional space. This computing power and expertise has not been wide spread and extensively harnessed. This is in part due to the fact, that most comprehensive reservoir simulators are not simple to use and the cost of such a resource can be prohibitively high, for small operators. Thus, the industry has continued to solve flow problems analytically by continued use of simplified flow domains, modified to account for non conformance from those domains. The objective of this paper is to highlight a technique used extensively in the past, with specific application to solving flow problems in reservoirs, with polygonal boundaries. The method provides some useful insights as wells as ease in solving flow problems for a variety of reservoir conditions, geometries and wellbore reservoir configurations. The method can be described as semi-analytical, since it requires both analytical as well as numerical computations.

Well Inflow Performance Model
The inflow performance model assumes that flow in the two dimensional plane can be extended into three dimensions by combining the two dimensional flow with the effects of flow in the vertical plane using an average reservoir thickness derived from structural contour maps or seismic surveys. The two dimensional flow is solved by use of conformal mapping theory, which guaranties the transformation of all boundary conditions in the physical plane to the mapped plane. A simple flow solution can then be applied to the mapped plane, to determine the flow in the physical plane. This theory has been shown to be accurate in the solving potential flow problems in other engineering disciplines. One of the key uses of this technique is that it is not necessary to have any geometric limitations on the reservoir and wellbore location.
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