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

The scaling laws as formulated by Rapport relate dynamically similar flow systems in porous media each involving two immiscible, incompressible fluids. A two-dimensional numerical technique for solving the differential equations describing systems of this type has been employed to assess the practical value of the scaling laws in light of the virtually unscalable nature of relative permeability and capillary pressure curves and boundary conditions.Two hypothetical systems - a gas reservoir subject to water drive and the laboratory scaled model of that reservoir - were investigated with emphasis placed on water coning near a production well. Comparison of the computed behavior of these particular systems shows that water coning in the reservoir would be more severe than one would expect from an experimental study of a laboratory model scaled within practical limits to the reservoir system.This paper also presents modifications of the scaling laws which are available for systems that can be described adequately in two-dimensional Cartesian coordinates. Introduction Present day digital computing equipment and methods of numerical analysis allow realistic and quantitative studies to be carried out for many two-phase flow systems in porous media. Before these tools became available the anticipated behavior of systems of this type could be inferred only from analytical solutions of simplified mathematical models or from experimental studies performed on laboratory models.To reproduce the behavior of a reservoir system on the laboratory scale, certain relationships must be satisfied between physical and geometric properties of the reservoir and laboratory systems. Where the reservoir fluids may be considered as two immiscible and incompressible phases, the necessary relationships have been formulated by Rapoport and others. Rapoport's scaling laws follow from inspectional analysis of the differential equation describing phase saturation distribution in such systems.It will be recalled that these scaling laws presuppose three conditions:the relative permeability curves must be identical for the model and prototype;the capillary pressure curve (function of phase saturation) for the model must be linearly related to that of the prototype; andboundary conditions imposed on the model must duplicate those existing at the boundaries of the prototype. These three requirements seldom if ever can be satisfied in scaling an actual reservoir to the laboratory system because:The laboratory medium normally will be unconsolidated (glass beads or sand) while the reservoir usually is consolidated. Relative permeability and capillary pressure curves are usually quite different for consolidated and unconsolidated porous media.The reservoir usually will be surrounded by a large aquifer which could be simulated in the laboratory only to a limited extent.Wells present in the reservoir would scale to microscopic dimensions in the laboratory if geometric similarity is to be maintained. In view of these considerations, rigorous scaling of even a totally defined reservoir probably would never be possible.The purpose of this paper is to assess the practical value of the scaling laws in the light of the unscalable variables. This has been done by carrying out numerical solutions in two dimensions to the differential equations describing the flow of two immiscible, incompressible fluids in porous media for a field scale reservoir and a laboratory model of that reservoir. While both the reservoir and the laboratory model were purely fictional, each has been made as realistic and representative as possible.The field problem selected as the basis for the investigation was an inhomogeneous, layered gas reservoir initially at capillary gravitational equilibrium and subsequently produced in the presence of water drive. The laboratory model of this reservoir was designed to utilize oil and water in a glass bead pack. SPEJ P. 164^

The paper describes the results of a model test series with the purpose of determining the hydroelastic vibrations of a nearbed pipeline span exposed to flow conditions created by steady current, waves and waves superimposed on steady current. The study has been conducted using a model composed of a spring-mounted rigid pipe segment and a flat plate simulating the sea bed. The hydroelastic cross-flow vibrations of the pipe segment are presented as function of the flow velocity, flow condition (waves and/or steady current) and the relative distance of the pipe to the seabed. A simple approach to analyze the vibrations caused by an irregular wave train is presented.

the recovery,of gas from water drive ' reservoirs and in understanding the This paper examines the knowledge results of water movement in gas storage available on predicting the residual gas operations. Geff'en,et al. in lgbp made left behind an advancing water front"in an experiment+ study of gas displacea water-drive producing gas reservoir or" ment by wqt~r,The research of Gorring2 gas storage reservoir. It is concluded and Nielsen was directed toward a furthat "measurements of 'residualgas should "th$runderstanding of this problem, This be made in the laboratory on cores by paper summarizes the current knowledge water flooding since only ,ageneral from the literature.and presents the relati(~.nship between residual gas and recent studies on the process by which porosj,tywas found. Calculations of gas is entrapped behind m encroaching" saturation distribution can be made for water front,' / water flooding of reservoir rock but such calculations require a knowledge of the Consider the p'rdblemsfaced by the residual gas saturation by imbibition engineer dealing with a gas reservoir and the saturation at zero gas permea-subject to water drive,,oraquifer gas bilityalternate ways of noting the storage reservoir. Nhat is the gas-wat&r residual gas content.saturation distribution at the gas-water interface and what will be the effective-Imbibition rate measurements are , ness of the water in displacing gas when shown to provide a non-destructive gas is produced? What information should method for scaq.hi,ng cores for inhomo-be procured in order that"the best possigeneities.

The purpose of this paper is to further the understanding of reservoir response to hot-water injection by describing a two-dimensional, mathematical model of the process. Key assumptions are that no gas phase is present, and that the injected fluid reaches thermal equilibrium instantaneously with the reservoir fluids and sand. The resulting system of three partial differential equations is solved simultaneously through the use of a "leap-frog" application of standard alternating direction implicit methods for the solution of the mass-balance equations and the method of characteristics for solution of the energy-balance equation. The utility of the mathematical model is demonstrated by comparing numerical and analytic temperature distributions for hot-water bank injection and by comparing calculated with observed field behavior. Additional calculations show that hot waterflooding can recover significantly more oil than cold waterflooding, and that a hot-water bank recovers, with less energy input, nearly as much oil as continuous injection. Introduction Consistent field success with hot fluid injection processes requires good reservoir description and a thorough understanding of recovery mechanisms. The latter is fostered by a combination of well designed experimental studies and physically sound mathematical modeling. The purpose of this paper is to further the understanding of reservoir response to hot-water injection by describing a two- dimensional mathematical technique, indicating its validity, and demonstrating its utility for studying the effects of reservoir and operating parameters. Published experimental studies of hot fluid injection processes are few. Even if extensive data were available two considerations discourage exclusive reliance on laboratory or field work. First, because of the severity of scaling requirements, laboratory results must be interpreted with care. Second, because of the one-shot nature of field experiments - injectivity/productivity tests and pilot floods - results seldom are available over the desired range of operating conditions. These factors emphasize the need for devoting attention to mathematical modeling to complement laboratory and field work. Through the use of mathematical models, scaling uncertainties can be bridged and response can be predicted for unique combinations of reservoir and operating conditions. Numerous mathematical models developed during recent years enable us to calculate temperature distributions, thermal efficiency (or conversely, fraction of heat lost), and oil recovery behavior for hot fluid injection. Ramey and Spillette recently have provided reviews of these methods. Most models have concentrated on predicting temperature distributions and thermal efficiency; few have been directed at predicting oil recovery. Although these techniques are useful for estimating effects of reservoir parameters and operating conditions on process performance, results are limited by the assumptions made and by the methods of coupling independently solved fluid-flow and energy-balance equations. A computer-based method is presented for predicting total reservoir response to hot-water injection that obviates most simplifying assumptions. The model simulates fluid flow and heat transfer in two dimensions within a vertical cross-section spanning the oil sand and adjacent unproductive strata. Known numerical procedures are used to solve the governing partial differential equations. The mathematical model handles the effects of reservoir heterogeneity, gravity, capillarity, relative permeability and temperature-dependent fluid properties. In addition, a wide range of operating conditions can be modeled, including hot-water followed by cold-water injection. Mathematical Description of Hot-Water Injection Key assumptions in the mathematical model for hot- water injection are thatno gas phase is present andthe injected fluid reaches thermal equilibrium instantaneously with the reservoir fluids and sand. Relative permeability and capillary pressure are assumed independent of temperature. JPT P. 627ˆ