Reservoir Performance During Two-Phase Flow
Authors: Weller W.T. 
Published: December 2007

Abstract
In Part 1, a study of pressure build-up curves calulated for conditions under which both oil and gas flow led to the conclusion that the presence of a dispersed free gas phase in an oil reservoir must be taken into consideration to estimate accurately average reservoir pressure and permeability from build-up curves. Familiar methods based on the assumption of no free gas can be extended to the two-phase case by using total compressibility and mobility in place of single-fluid compressibility and mobility. These methods give correct values for average pressure and permeability when gas saturation is small. Errors become larger as the gas saturation increases. However, for the use to which the results will be put, the methods are satisfactory for reservoir engineering purposes. An improved method of calculating the performance of depletion-type reservoirs is presented in Part 2. Because the mathematical relationships describing simultaneous flow of oil and gas are quite involved, simplifying assumptions are made to provide means of obtaining approximate solutions of reasonable accuracy. One such approximate method now in use is the constant-GOR solution. It involves the assumption that at any instant, the ratio of total gas flow rate (both free and dissolved) to oil flow rate is the same at all points in the reservoir. The approach is not applicable unless the free gas saturation in the reservoir is everywhere greater than the critical gas saturation. This paper presents a modification which, by avoiding the constant-GOR assumption, makes the method applicable to all reservoir conditions, and so far appears to be more accurate than the constant-GOR solution and to be comparable in required computation time.
PART 1-BUILD-UP CURVE ANALYSIS
Introduction
Pressure build-up characteristics of shut-in wells have been used for many years by engineers to evaluate average reservoir pressure, effective permeability thickness of the pay section, effectiveness of well completion (skin effect) and reservoir size. A number of methods of analysis have appeared from time to time. Without exception, these methods are based on the assumption that the reservoir contains but one fluid of constant small compressibility and constant mobility. It has been suggested that these single-fluid methods may be applied to data from reservoirs containing both oil and gas by substitution of some effective total properties of the multiphase system in place of the corresponding single-phase properties. The present investigation was undertaken primarily to evaluate this approach.
METHOD
A number of theoretical build-up curves were calculated for conditions of two-phase flow, under the assumption of certain reservoir and fluid properties, and were analyzed by single-fluid methods with appropriate total compressibility and total mobility values for the corresponding single-fluid properties. Results of the analyses were compared with the assumed conditions. The theoretical build-up curves were completed by procedures similar to those of West, Garvin and Sheldon. Since these calculations require a considerable amount of computer time, an attempt was made to derive an approximate calculation method. The attempt was unsuccessful for calculating build-up curves, but the effort did result in a new approximate method of calculating the performance of solution gas drive reservoirs, which appears to be an improvement over the constant-GOR method-used previously (see Part 2). The West, Garvin and Sheldon calculations involved the following assumptions:
the reservoir is circular and completely bounded, with a completely penetrating well at its center; the porous medium is uniform and isotropic, with a constant water saturation at all points; gravity effects can be neglected; compressibility of rock and water can be neglected; the composition and equilibrium are constant for oil and gas; the same pressure exists in both the oil phase and the gas phase; and no after production occurs, i.e., the well is shut in at the sand face for build-up.
These assumptions make it possible to describe two-phase flow of oil and gas by the partial differential equations:
,......(1)
and
.......(2)
To obtain idealized build-up curves for analysis, these equations were solved numerically with certain hypothetical PVT data, relative permeability data and reservoir parameters. With the use of total compressibility and total mobility, average reservoir pressures were calculated from the curves by the method described by Matthews, Brons and Hazebroek; effective permeability thickness values were calculated from the slopes of the curves by the usual single-fluid method:
..................(3)
JPT
P. 240ˆ