Three-Phase Fluid Flow Including Gravitational, Viscous and Capillary Forces
Authors: Sheffield M. 
Published: December 2007

Abstract
This paper presents a technique for predicting the flow, of oil, gas and water through a petroleum reservoir. Gravitational, viscous and capillary forces are considered, and all fluids are considered to be slightly compressible. Some theoretical work concerning the fluid flow in one-, two- and three-space dimensions is given along with example performance predictions in one- and two-space dimensions. predictions in one- and two-space dimensions

Introduction
Since the introduction of high speed computing equipment one of the goals of reservoir engineering research has been to develop more accurate methods of describing fluid movement through underground reservoirs. Various mathematical methods have been developed or used by reservoir engineers to predict reservoir performance. The work reported in this paper extends previously published work on three-phase fluid flow (1) by including a rigorous treatment of capillary forces and (2) by showing certain theoretical mathematical results proving that these equations can be approximated by certain numerical techniques and that a unique solution exists.

Discussion
The method of predicting three-phase compressible fluid flow in a reservoir can be summarized briefly by the following steps. 1.The reservoir, or a section of a reservoir, is characterized by a series of mesh points with varying rock and fluid properties simulated at each mesh point. point. 2.Three partial differential equations are written to describe the movement at any point in the reservoir of each of the three compressible fluids. All forces influencing movement are considered in the equations. 3. At each mesh point, the partial differential equations are replaced by a system of analogous difference equations. 4. A numerical technique is used to solve the resulting system of difference equations.
Capillary forces have been included in two-phase flow calculations. The literature, however, does not contain examples of prediction techniques for three-phase flow that include capillary forces. Where capillary Races are considered, each of the three partial differential equations previously discussed has a different dependent variable, namely pressure in one of the three fluid phases. Therefore, pressure in one of the three fluid phases. Therefore, three difference equations must be solved at each point in the reservoir. Where large systems of point in the reservoir. Where large systems of equations must be solved simultaneously, an engineer might question whether a unique solution to this system of equations actually exists and, if so, what numerical techniques may be used to obtain a good approximation to the solution. It is shown in the Appendix that a unique solution to the three-phase flow problem, as formulated, always exists. It is also shown that several methods may be used to obtain a good approximation to the solution. The partial differential equations and difference equations partial differential equations and difference equations used are shown in the Appendix. Matrix notation has been used in developing the mathematical results.
Two sample problems were solved on a CDC 1604. They illustrate the type of problems that can be solved using a three-phase prediction technique.

SAMPLE ONE-DIMENSIONAL RESERVOIR PERFORMANCE PREDICTION
A hypothetical reservoir was studied to provide an example of a one-dimensional problem that can be solved. Of the several techniques available, the direct method A solution as shown in the Appendix was used.
The reservoir section studied was a truncated, wedge-shaped section, 2,400 ft long, with a 6' dip. (A schematic is shown in Fig. 1.) This section was represented by 49 mesh points, uniformly spaced at 50-ft intervals. The upper end of the wedge was 2 ft wide, and the lower end was 6 ft wide.
SPEJ
P. 255