Simulation of Three-Dimensional, Two-Phase Flow In Oil and Gas Reservoirs
Authors: Coats K.H. , Nielsen R.L. , Terhune Mary H. , Weber A.G. 
Published: December 2007

COATS, K.H., THE U. OF TEXAS, AUSTIN, TEX. NIELSEN, R.L., ESSO PRODUCTION RESEARCH CO., HOUSTON, TEX. MEMBERS AIME TERHUNE, MARY H., AMERICAN AIRLINES, TULSA, OKLA., WEBER, A.G., ESSO PRODUCTION RESEARCH CO., HOUSTON, TEX. MEMBER AIME

Abstract
Two computer-oriented techniques for simulating the three-dimensional flow behavior of two fluid phases in petroleum reservoirs were developed. Under the first technique the flow equations are solved to model three-dimensional flow in a reservoir. The second technique was developed for modeling flow in three-dimensional media that have sufficiently high permeability in the vertical direction so that vertical flow is not seriously restricted. Since this latter technique is a modified two-dimensional areal analysis, suitably structured three-dimensional reservoirs can be simulated at considerably lower computational expenses than is required using the three-dimensional analysis. A quantitative criterion is provided for determining when vertical communication is good enough to permit use of the modified two-dimensional areal analysis.
The equations solved by both techniques treat both fluids as compressible, and, for gas-oil applications, provide for the evolution of dissolved gas. Accounted for are the effects of relative permeability, capillary pressure and gravity in addition to reservoir geometry and rock heterogeneity. Calculations are compared with laboratory waterflood data to indicate the validity of the analyses. Other results were calculated with both techniques which show the equivalence of the two solutions for reservoirs satisfying the vertical communication criterion.

Introduction
Obtaining the maximum profits from oil and gas reservoirs during all stages of depletion is the fundamental charge to the reservoir engineering profession. In recent years much quantitative assistance in evaluating field development programs has been goaded by computerized techniques for predicting reservoir flow behavior. Because of the spatially distributed and dynamic nature of producing operations, automatic optimization procedures, such as those now in use for planning refining operations, are not now available for planning reservoir development. However, present mathematical simulation techniques do furnish powerful means for making case studies to help in planning primary recovery operations and in selecting and timing supplemental recovery operations.
A number of methods have been reported which simulate the flow of one, two or three fluid phases within porous media of one or two effective spatial dimensions. However, applying computer analyses to actual reservoirs have been limited mostly to two-dimensional areal or cross-sectional flow studies for two immiscible reservoir fluids. To obtain a three-dimensional picture of reservoir performance using such two-dimensional techniques, it has been necessary to interpret the calculations by combining somehow the results from essentially independent areal and cross-sectional studies. To the author's knowledge, the only other three-dimensional computational procedure, in addition to those presented here, was developed by Peaceman and Rachford to simulate the behavior of a laboratory waterflood.
Two computational techniques which may be used to simulate three-dimensional flow of two fluid phases are described in this paper. The first method, called the "three-dimensional analysis", employs a fully three-dimensional mathematical model that treats simultaneously both the areal and cross-sectional aspects of reservoir flow.
SPEJ
P. 377ˆ