Computer Prediction of Water Drive of Oil and Gas Mixtures Through Irregularly Bounded Porous Media Three-Phase Flow
Authors: Higgins R.V. , Leighton A.J. 
Published: December 2007

Abstract
Interest by petroleum engineers in the flow of three phases oil, gas and water in irregularly bounded porous media lies mostly in the performance calculation of water floods of reservoirs that have been partially depleted as the result of expansion of much of the originally dissolved gas. The authors present a method to forecast three-phase flow in complex geometry and explain the details by the use of a specific example of a five-spot water flood of a partially depleted stratified reservoir. In this example, the fluid and rock properties of a field given by Prats, et al, were used. The computed results fit the field performance more accurately than Prats, et al, and Slider. The time required for the high-speed digital computer to make the calculations, including the contributions from the different layered zones, is about one minute.

Introduction
The declining rate of discoveries of new oil fields in the United States makes the recovery of more oil from known reservoirs more attractive than previously. Water flooding has an excellent proved background for recovering additional oil economically. Accordingly, the main interest of this paper is in this type of recovery. In petroleum engineering studies, commercial interest in three-phase flow is mostly in the water flooding of reservoirs in which the oil has been partially produced by the expansion of dissolved gas. When water is pumped into these reservoirs, three-phase flow takes place. Although this paper is concerned with these conditions, the principles involved could be used for other conditions when and where they come to the fore. Many investigators have made contributions to non-empirical forecast methods using basic scientific engineering principles. Several of these used the oil in place at the start of the water flood and the oil remaining after a large quantity of water has passed through a core for key values in their calculations. Recently, Prats, et al, and Slider have reduced assumptions by adding the third phase-gas-in their forecasting methods. Slider uses the mobilities in the immediate vicinity of inlet and outlet wells as an aid to simulate the resistance effect in the five-spot pattern of the flow of oil, gas and water. Prats, et al, minimize assumptions by using previously determined laboratory data for determining sweep efficiency in the five-spot pattern. In neither of these papers is the saturation profile continuously affected by permeability-saturation curves. Sheldon and Dougherty recently described a method that employs continuously changing saturation profiles using permeability curves, has a minimum of assumptions and needs no prior sweep efficiency. The Higgins-Leighton method, described in this paper, has all of these features; however, many of the techniques are different from those of Sheldon and Dougherty. The Higgins-Leighton method, tested in May, 1961, is direct and easy to apply and requires very little computer time to calculate a forecast. The short computer time is especially helpful in the study of a reservoir containing many layers of different relative permeabilities. In the Higgins-Leighton method, the individual pressures do not have to be calculated, as the resistance to flow in each cell in the flow pattern is readily determined without the use of any iterative techniques. The saturation and permeability distributions are readily determined. These data and the shape factor, which is measured only once from the potentiometric model when mobility ratio is one, determine the resistance to flow in each cell.

THEORY
The authors showed in a previous paper that, as an aid to calculating performance, the reservoir can be divided into channels using the streamlines of a potentiometric model as a guide. See Fig. 1. This procedure also was used in this paper. The authors also showed that, by treating conduits as approximately one-dimensional and neglecting pressure gradients transverse to the main flow, the Buckley-Leverett equation may be expressed as
..............(1)
The principles expressed by this equation are employed extensively in the three-phase flow, as they were in two-phase flow.
JPT
P. 1048^