Drainage Areas for Wells in Edge Water-Drive Reservoirs
Authors: Kumar Anil 
Published: December 2007

A computational procedure is presented to determine the drainage areas for wells in an edge water-drive reservoir at steady state or unit mobility, ratio. The drainage areas are found to be less for wells closer to water contact than those away from water contact. A detailed analysis of a two-well system can be used to generate drainage area behavior in large multiwell reservoirs.

Introduction
Many petroleum reservoirs derive a major part of their natural producing energy from water coming from the flanks or edges of the reservoir. This water may come either from an aquifer adjacent to the oil zone, or may be injected from surface - that is, in a peripheral or line flood. A reservoir such as shown in Fig. 1b is called an edge water-drive reservoir and is characterized by (1) a small surface area of contact between the oil and water zone, and (2) a water movement that is essentially parallel to the bedding plane.
The calculation of ultimate recovery from wells in a reservoir with strong edge water drive is of considerable economic importance, particularly in primary production phases of a field and in estimating the effectiveness of peripheral flooding during secondary recovery. The first attempt at quantitative estimation was presented in Ref. 1, which considered a reservoir-well system (Fig. 1) when all wells were producing at the same rate. This example was later reviewed by Craft and Hawkins. More recently, Kumar studied the distribution of drainage areas between two wells in an idealized, rectangular, edge water-drive reservoir (Fig. 2) for five producing rate ratios. Gulati also reported computations for four other rate ratios at steady state. The drainage areas in these studies were obtained by computing the limiting stream line ASA shown in Fig. 2, which separates the flow around a stagnation point S between two wells. The procedure is explained in considerable detail by Ramey.
The assumptions employed by the studies of Kumar, Gulati, and Ramey are as follows:the formation has uniform porosity and permeability,wells are drilled on a uniform spacing,the effect of gravity is negligible,the mobility ratio is unity between encroaching water from the aquifer and swept oil,the well and reservoir system are at steady state, anda constant pressure is maintained at the water contact. These assumptions also hold for the results of this investigation.
In this study, drainage areas for wells in a simple two-well reservoir system (Fig. 2) are considered first. Then multiwell systems (Fig. 1) are discussed, and a matrix method is presented in the Appendix to simplify computations.
It is stressed that the drainage area, a term commonly used for a closed or depletion reservoir, is equivalent to areal sweep for reservoirs under water drive or fluid injection. Areal sweep or coverage, as determined by the method presented here, is applicable only under unit mobility ratio conditions. For mobility ratios other than unity, additional studies need to be conducted.

Two-Well Edge Water-Drive Systems
Consider the rectangular reservoir in Fig. 2, which is twice as long as it wide. Each well is drilled on uniform spacing of area, A. Total area. At, of the rectangular reservoir is, therefore, 2A. Area A2, drained by Well 2 for various producing-rate ratios, can be expressed as the fraction E2 of the total reservoir area.
JPT
P. 1673