Bi-Radial Model for the Study of Displacement Processes in Pattern-Developed Reservoirs
Authors: Fraser John W. , Crawford Paul B. 
Published: December 2007

This paper was prepared for the 49th Annual Fall Meeting of the Society of Petroleum Engineers of AIME, to be held in Houston, Texas, Oct. 6–9, 1974. Permission to copy is restricted to an abstract of not more than 300 words. Illustrations may not be copied. The abstract should contain conspicuous acknowledgement of where and by whom the paper is presented. Publication elsewhere after publication in the JOURNAL OF PETROLEUM TECHNOLOGY or the SOCIETY OF PETROLEUM ENGINEERS JOURNAL is PETROLEUM TECHNOLOGY or the SOCIETY OF PETROLEUM ENGINEERS JOURNAL is usually granted upon request to the Editor of the appropriate journal provided agreement to give proper credit is made. provided agreement to give proper credit is made. Discussion of this paper is invited. Three copies of any discussion should be sent to the Society of Petroleum Engineers office. Such discussion may be presented at the above meeting and, with the paper, may be considered for publication in one of the two SPE magazines.

Abstract
The behavior of displacement mechanisms through pattern developed reservoirs has been studied extensively utilizing a great variety of methods. Basically, the methods of study have involved observation of physical laboratory models, analytical solutions, and numerical solutions. All three methods have their own unique disadvantages. Physical models are usually small, simple, and very time consuming to build and operate. Analytical solutions can only be treated as approximations when heterogeneities are introduced. Numerical solutions become quite costly when a great deal of detail is involved in the study because of the large number of cells required for proper simulation.
A streamline approach has been applied to analytical methods to provide a greater degree of accuracy in the study of displacement processes. This approach has been an improvement processes. This approach has been an improvement over past methods; however, it is still not conducive for studying heterogeneities in the reservoir, and no provision is made for either horizontal or vertical crossflow components during the displacement program.
This paper proposes the study and application whereby the streamlines closely approach the numerical grid of the digital computer. The five-spot pattern has been treated by aligning a numerical simulator into a bi-radial divergent-convergent grid arrangement. Such representations correspond closely to flow from input to output wells. This grid alignment has made critical observation possible in any region desired without excessive cells in unnecessary regions of the system in order to satisfy the boundary or symmetry requirement of the numeric model. The divergent-convergent radial grid arrangement allows solutions through the five-spot octant at considerably less expense than with the conventional rectangular grid arrangement. In addition, well pressures may be computed in keeping with the numerical solution, heterogeneous reservoirs can be simulated, and crossflow components are provided.

Introduction
The success of a displacement process in a reservoir is primarily influenced by the volumetric sweep efficiency. Volumetric sweep efficiency is defined as the displaceable reservoir volume divided by the total reservoir volume. Sweep efficiencies are known to vary considerably as a function of the injection pattern and the mobility ratio between the pattern and the mobility ratio between the injected and displaced fluids.
Sweep behavior has been the subject of extensive research through analytical and laboratory studies.