Combination Method for Predicting Waterflood Performance for Five-Spot Patterns in Stratified Reservoirs (includes associated discussion and reply)
Authors: Wasson James A. , Schrider Leo A. 
Published: December 2007

The pdf file includes an associated discussion by F.F. Craig Jr., and the authors' reply, published in the February 1969 issue of JPT, pages 233-234.

Abstract
A method of predicting waterflood performance has been developed that combines certain facets of several previously published prediction techniques. The manner of previously published prediction techniques. The manner of combination has required the development and use of some new and some little-known relationships and has eliminated several of the weaker assumptions inherent in the original individual prediction methods. This approach, which was designed for computer solution, has removed the necessity of referring to plotted curves and permits the analytical prediction of waterflood performance for a five-spot well pattern in either homogeneous or stratified reservoirs. The predicted values are expressed in common oilfield units rather than in abstract or dimensionless terms. The calculation procedure has been programmed in FORTRAN IV and the entire program listing is available to potential users. potential users. Introduction
Prior to initiation of secondary recovery of oil by waterflooding, the possibilities of economic success or failure must be evaluated with the greatest practical accuracy. Although many prediction techniques are available, most, if not all, have serious limitations.
In recent years, the Bureau of Mines has attempted to stimulate interest in the secondary recovery of oil from Appalachian pressure-depleted reservoirs. Waterflooding, because of its widespread acceptance and success, is the principal secondary-recovery method considered. During principal secondary-recovery method considered. During the life of the project, several performance prediction methods have been studied and applied. Because these methods, individually, have inherent assumptions and limitations, portions of some of them have been selected and combined in an effort to eliminate such shortcomings wherever possible. We believe that the resulting technique is as accurate as the necessary available data will permit and that it has several distinct advantages over previously published methods. published methods. Development of the Method
The fractional flow equation developed by Buckley and Leverett from Darcy's law is the fundamental mathematical statement describing immiscible fluid-fluid displacement in porous media. A generalized form of this relationship, whose derivation is readily available in the literature, is
......(1)
The formation dip, is usually very small in those petroleum reservoirs being considered for pattern petroleum reservoirs being considered for pattern waterflooding in the Appalachian area. Therefore, in many cases, has an almost negligible value. In addition, the difference between the density of water and the density of oil, is relatively small. For these reasons the gravitational term may often be neglected.
Capillary pressure is a function of saturation, which is in turn a function of distance from the upstream boundary of the system or the injection well.
.........(2)
In this study it is assumed that the frontal zone of the waterflooding (or zone of capillarity) has negligible length and that only those saturations in the water-swept zone at breakthrough and later are of interest. As the distance in a waterflood pattern is comparatively large and the change in water saturation with distance is usually small, the change in capillary pressure with distance is negligible. The simplified form of the fractional flow equation
........(3)
is appropriate therefore, in this application.
Three variables must be considered in determining that fraction of the total fluid flow that consists of water: the viscosity ratio, saturation, and relative permeability ratio.
JPT
P. 1195