Partial Integration of Equations of Multiphase Flow
Authors: Martin John C. 
Published: December 2007

Abstract
Equations for three-phase, three-dimensional, compressible flow (including capillarity) are reduced to two-dimensional relations by a partial integration. This reduction allows three-dimensional flow problems to be treated with mathematics for only two spatial dimensions. The results can be used to formulate flow equations for two-dimensional reservoir simulators i-n which the effects of capillarity and fluid segregation in the third dimension are represented. Such reservoir simulators would retain many of the advantages of two-dimensional simulators while simulating three-dimensional effects. The principal restriction of the method is that the thickness of the reservoir should be small, compared to the distance across the reservoir.

Introduction
In recent years, computers have been used to calculate performances of many reservoirs. Most of the detailed calculations, however, are based on finite difference solutions of the flow equations, and present day computers are seldom able to handle a sufficient number of cells to produce entirely satisfactory solutions, even for produce entirely satisfactory solutions, even for reservoirs represented by two-dimensional arrays of cells. The simulation becomes much worse when one wishes to approximate the reservoir by a three-dimensional array.
A great saving in computation or a more detailed solution can be obtained for many reservoirs by using the partial integration of the equations of flow, presented in this paper. The integration reduces the three-dimensional equations to two-dimensional relations; ant for studies of two-dimensional flows in vertical cross-sections, the equations are reduced to one-dimensional relations.
Most reservoir performance calculations currently are based on one- or two-dimensional flow relations. In some cases flow in the third dimension is approximated by assuming a particular type of vertical saturation distribution, such as gravity segregation. The relations developed in this paper approach those for segregated flow as the capillary pressures approach zero, and they approach those for uniformly distributed saturations as the capillary pressures are increased.
For this analysis, the ratio of the reservoir's thickness to the maximum distance across it must be small. The capillary pressures between the oil and water should also be small compared to the maximum pressure difference in the reservoirs. It is requirement is met by most reservoirs. It is assumed that the capillary-pressure curves are well defined, whether or not hysteresis effects are included. Also, the reservoir must have sufficient vertical permeability to allow the fluids to segregate.
The results presented here provide a firm theoretical foundation to Coats' et al. assumption of vertical equilibrium and extend the relations to three-phase flow. Coats' assumption of vertical equilibrium, which he verified by calculations and experiment, is developed here mathematically from basic flow equations.

Discussion

SATURATION AND PRESSURE DISTRIBUTIONS
Appendix A presents a mathematical analysis of fluid flow in reservoirs where the ratio of thickness to maximum distance across the reservoir is small. The results indicate:that the fluids along any line perpendicular to such a reservoir's upper surface are in antic capillary equilibrium (vertical equilibrium);that, to a first approximation, the fluid pressures and properties are functions of only areal position in the reservoir and time; andthat hydrostatic pressure gradients exist along any line perpendicular to the reservoir's upper surface.
The results might be expected after studying several physical considerations. First, no flow is allowed normal to the upper and lower reservoir boundaries, which are relatively close together.
SPEJ
P. 370