Numerical Solution of Multiphase, Two-Dimensional Incompressible Flow Using Stream-Tube Relationships
Authors: Martin J.C. , Wegner R.E. 
Published: December 2007

Abstract
A numerical method is presented that overcomes the effects of numerical dispersion. The method applies to two-dimensional, incompressible, two-phase flow, where the effects of gravity can be neglected. The theory presented represents the theoretical foundation for stream-tube methods, including the methods introduced by Higgins and Leighton in 1962. The computer code used to reduce the theory to practice is described, and examples are presented of practice is described, and examples are presented of five-spot, nine-spot, and staggered and direct line drive pattern performances. These results demonstrate the effect of fluid-mobility changes on streamlines. In general, the fixed stream-tube results are within 10% of those obtained using variable stream tubes. Exceptions occur for the direct line drive pattern for mobility ratios of 0.10 and 10. Overall, results indicate that fixed stream-tube methods are satisfactory for many problems and involve much less mathematics and computations than variable-tube methods.

Introduction
The stream-tube approximation was introduced in 1962 by Higgins and Leighton. They presented convincing evidence that the performance of five-spot-pattern waterfloods can be calculated by holding the streamlines constant as the flood progresses and using Buckley-Leverett theory to progresses and using Buckley-Leverett theory to calculate the fluid displacement along streamlines. Extensions, elaborations, and applications of the Higgins-Leighton method are found in Refs. 5 through 11.
Chevron Oil Field Research Co. has used stream-tube computer models for the last 10 years. Most of these models were based on the assumption that the changes in the streamlines could be neglected if the ratios of production and injection between the various wells did not change drastically. Initially, the fluid displacement was determined by Buckley-Leverett theory, and applications were restricted to waterfloods and pressure-maintenance operations by gas injection. Later, the fluid displacement relations were extended, and stream-tube models were used to study CO2 injection, combined CO2 and water injection, water cycling in a geothermal reservoir, and polymer flooding.Results obtained by finite-difference reservoir simulators tend to smooth out or disperse water and oil banks over a number of computing cells. This "numerical-dispersion" effect can result in unrealistic early breakthrough predictions followed by incorrect predictions of the produced water and GOR's. This is a serious problem, particularly in cases where only a few computing cells separate injection and production wells. Stream-tube models can calculate accurately the positions of the fluid banks and, thus, overcome the effects of numerical dispersion. In many cases, these models can be used to calculate reservoir performances at relatively low cost. In addition, computer plots of the streamlines and pressure contours can be used to study overall fluid flow without going to the added trouble and expense of performing the displacement calculations.
SPEJ
P. 313