Prediction of Waterflood Performance for Arbitrary Well Patterns and Mobility Ratios
Authors: Hauber W.C. 
Published: December 2007

Abstract
Techniques previously published to predict the production performance of a water flood when the mobility ratio is not unity have been primarily restricted to a five-spot well pattern. When other types of well pattern are considered, the prediction of performance has required either a model study or a complex numerical solution of a multiple-fluid system with a high-speed digital computer. Simple methods are developed in this report for prediction the areal sweep efficiency and injectivity history of a water flood for arbitrary well patterns and mobility ratios. The effect of an initial gas saturation can be incorporated into this method of prediction. With the layer superposition concepts published by Prats, et al., these methods can be used to predict the production performance of a water flood in a reservoir having a wide range in permeability. A reasonably good agreement is shown by comparisons of the results calculated by this method with those obtained either by model studies or by complex numerical solution of multiple-fluid systems. Therefore, it has been concluded that a reasonable accurate method has been developed for predicting the production performance of a water flood employing an arbitrary type of will pattern for any given mobility ratio. A medium-speed digital computer can be used in applying this method.
Introduction
Production performance of multi-fluid five-spot water floods can be predicted by the techniques published by Prats, et al. This technique takes into account the effects of initial gas saturation, mobility ratio and vertical variations in horizontal permeability. A basic assumption in applying this technique is that unit displacement efficiency does not vary with time after the passage of the flood front. Prats, et al. demonstrated that this assumption is valid for normal waterflood applications with small mobility ratios. To extend Prats' technique to other types of well patterns has required either an experimental model study or a numerical solution of a multiple-fluid system to describe the performance of an individual homogeneous layer. Sheldon and Dougherty presented an exact numerical solution employing any arbitrary well pattern. This technique is complex to apply, requiring
numerical solution of the single-fluid system, conformal mapping of the streamlines and isopotential lines to form a new grid network, and a numerical solution of the multi-fluid system on the new grid network.
A method was developed in this paper to approximate areal sweep efficiency and injectivity as functions of time for an individual homogeneous layer for arbitrary well patterns and mobility ratios. Analytic expressions were derived to directly apply this method to five-spot and direct line drive well patterns. For other types of well patterns, this method is similar to the approach of Higgins and Leighton, in that it utilizes the stream tubes and isopotential lines obtained by the solution of differential equations when the mobility ratio is unity. However, this method is easier to apply than the Higgins-Leighton method since it does not require knowledge of the variation between oil and water relative permeabilities and saturation. Using the relationship between areal sweep efficiency, injectivity and the volume of water injected into an individual homogeneous layer as obtained by the method presented in this paper, and Prats' technique for the superposition of individual layers, one can predict the production performance of any water flood.
ASSUMPTIONS
GENERAL
Basic assumptions made in this report are that fluids are incompressible and the layers are horizontal, homogeneous and uniformly thick. It is assumed that as water is injected into the layer, three distinct regions are formed--a water bank, an oil bank, and an unflooded region. Each region is assumed to displace other regions in a pistonlike manner, with only one fluid flowing in each region, so that there is no change in unit displacement efficiency with time after the flood front has passed in the reservoir. Therefore, the residual oil saturation in the water bank and the oil saturations in the unflooded region are constant throughout the layer. The ratio (F) of the bulk volume of a layer occupied by the oil and water banks at any time before oil breakthrough, to the corresponding bulk volume of a layer occupied by the water bank, is given by the equation.
JPT
P. 95ˆ