Graphical Determination of the Constants in the Corey Equation for Gas-Oil Relative Permeability Ratio

Abstract

The Corey equations for relative gas and oil permeability and the gas-oil relative permeability ratio are: (1) SL − SLR 2 SL − SLR 2krg = 1 − 1 − Sm − SLR 1 − SLR (2) SL − SLR 4kro = 1 − SLR (3) SL − SLR 2 SL − SLR 21 − 1 − krg Sm − SLR 1 − SLR=kro SL − SLR 4 1 − SLR where SL is the total liquid saturation and equal to 1 minus the gas saturation (1 - Sg), and where Sm and SLR are constants. Eq. 3 for the relative permeability ratio has been found to give a fairly accurate representation of actual relative permeability ratios in many cases. It is sometimes used to test, extrapolate and smooth experimental data. It can also be useful when a convenient analytical expression for relative permeability ratio is required in computer programming of reservoir engineering problems. The shapes of Corey relative permeability ratio curves for some selected values of the constants Sm and SLR are shown in Fig. 1. Since Eq. 3 contains only two constants, knowledge of two points on the curve of krg/kro vs Sg is sufficient to evaluate both constants. The algebraic solution of the two simultaneous equations is tedious, however; and the graphical method developed by Corey in his original papers appears to be incorrect and is at best a method of successive approximations that requires repetitive graphical constructions and numerical computations as well. This inconvenience has probably prevented the Corey equation from receiving the attention it deserves from reservoir engineers. The first objective of this paper is to present a simple graphical method for determining the constants of the Corey equation from two known points. The second is to present a graphical method, eliminating numerical computations, for determining enough additional points to permit construction of a reasonably accurate plot of relative permeability ratio vs gas saturation over the region of interest in reservoir engineering calculations. The method makes use of three figures (Figs. 2, 3, and 4). Each intersection of the lines plotted on these figures corresponds to one value of SLR and one value of Sm as indicated by the labels. The values plotted on the vertical and horizontal scales are the gas saturations at two different values of the gas-oil relative permeability ratio, krg/kro. The location of each intersection is found by substituting the chosen values of SLR, Sm, and krg/kro into Eq. 3 and solving for SL. Since (1 - SL) is Sg this gives the gas saturation corresponding to the given values of SLR, Sm and krg/kro. This was done for each combination of SLR and Sm at the same two chosen values for krg/kro, yielding two values of Sg for each combination of SLR and Sm. These two values of Sg were then plotted, one vertically and the other horizontally, to mark the intersections, and then lines were drawn connecting points of constant SLR and constant Sm. Each of the three figures was constructed in the same way, but for a different pair of krg/kro values. Once each figure is constructed, it provides a compact source of information on the relationships between the two gas saturations at the two different values of krg/kro and the Corey equation constants, SLR and Sm. P. 1111ˆ