A Streamline Model for Secondary Recovery

Abstract

A mathematical model particularly suitable for secondary recovery predictions is described. The model is based upon the flow lines generated by the superposition of line sources and sink solutions and is easily adaptable to arbitrary well patterns and fluid displacement mechanisms. Nonunity mobility ratios and reservoir stratification may be modeled. The model may be used with relatively small digital computing equipment. Previous models of this type have used the generated flow lines to outline "flow bundles", thus requiring a prior knowledge of the geometric shape of each of these bundles so that the flow through each could be computed. The modeling method described does not require any such prior knowledge of these flow lines since the volumetric flow in a stream bundle is computed as each chosen streamline is generated. This feature makes the model particularly adaptable to arbitrary well patterns. patterns. Examples are given to show the application of this model to both single and multiphase flow. Introduction Streamline models of secondary recovery projects result from the use of line source and sink solutions to the diffusivity equation to represent injection and production wells. Muskat described these solutions and their application to simplified problems. He used semi-analytic techniques to problems. He used semi-analytic techniques to obtain breakthrough sweep efficiencies for regular, dispersed injection patterns at a mobility ratio of 1. Collins described a finite-difference approximation for determining streamlines for unit mobility ratios in arbitrary well patterns. Both Muskat and Collins described image-well techniques for mathematically bounding the area being studied. Collins' finite-difference method is particularly adaptable to the use of a high-speed computer for obtaining streamlines and travel times along streamlines. Higgins and Leightons have described a technique for approximating the waterflood recovery of oil using streamlines generated by single-fluid how models (such as the method described by Collins). Their technique uses these streamlines to divide the total flow area into "stream channels", which flow in parallel between injection and production wells. Each stream channel is divided into a number of recovery rectilinear flow cells, in series, which closely approximate the shape of the stream channel. Through the use of shape factors determined for each flow cell, a Buckley and Leverett type of frontal displacement in each stream channel is computed. Combining the results from all stream channels gives the waterflood production history by this method. Once the stream channels, the flow cells, and the shape factors have been determined, a single computer program is used to obtain the production history. production history. Hauber has also described a method using stream channels formed by single-fluid streamlines. In this method, the cross-sectional area of each stream channel, as a function of length along a center streamline, is determined mathematically from the stream function, and an integration technique is used to determine the flow through each stream channel. Hauber applied this method to piston-like displacement in a five-spot injection piston-like displacement in a five-spot injection pattern. pattern. The stream-channel concept has been used and elaborated upon in subsequent publications dealing with the displacement of oil by fluids of unequal mobilities. This use of stream channels requires a prior knowledge of the single-fluid streamlines for the well spacing to be studied, so that flow cells and their shape factors, or the area functions of the channels, may be determined. The purpose of this paper is to describe a method which produces equivalent results but requires no prior knowledge of the streamlines or stream channels to be used. It is not necessary to know the eventual destinations of the "stream channels" in order to obtain production histories for arbitrary well patterns and either of the types of displacement patterns and either of the types of displacement mechanism mentioned above. SPEJ P. 7

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