Empirical Equations for Estimating Two-Phase Relative Permeability in Consolidated Rock
Authors: Honarpour Mehdi , Koederitz L.F. , Harvey A. Herbert 
Published: December 2007

Introduction
The permeability of a porous rock to a saturating fluid is determined by the geometry of the rock pore system and not by the physical properties of the fluid. This general statement assumesthe absence of a chemical reaction between the rock and fluid anda single homogeneous fluid phase.
If more than one fluid is present, permeability to any fluid depends not only on the geometry of the permeability to any fluid depends not only on the geometry of the rock pore system but also on the fraction and distribution of each fluid phase, the interfacial tensions, the saturation history, and possibly other factors.
Although direct prediction of relative permeability from theoretical considerations is a worthwhile objective, the most successful techniques for making these predictions are essentially empirical. Rather than predictions are essentially empirical. Rather than attempting a theoretical solution to the problem, we have used an entirely empirical approach. In our study a rather extensive set of relative permeability data was compiled, and conventional stepwise linear regression analysis techniques were used to develop prediction equations from the laboratory data. This procedure is designed to produce a satisfactory fit of the data with a minimum of terms in the equation; it is not intended to provide the best possible data fit.

Development of Empirical Equations
The data used as a basis for the study were derived from oil and gas fields in the continental U.S., Alaska. Canada, Libya, Iran, Argentina, and the United Arab Republic. All the laboratory tests were made at room temperature and atmospheric pressure. We made no attempt to group the data according to laboratory techniques used in measuring relative permeability since this information was not available for many of the data sets. Each set of relative permeability data was classified as either carbonate or noncarbonate mation was insufficient for more detailed lithologic characterization.
In addition to identifying data sets as carbonate or noncarbonate, rough wettability classifications were made according to three arbitrary criteria:The rock was considered water-wet if kro at high oil saturations in an oil/water system greatly exceeded kro in a gas/oil system at the same oil saturations, provided that krg in a gas/oil system greatly exceeded krw in an oil/water system at or near residual oil saturation after waterflooding.The rock was considered oil-wet when kro in the oil/water system was approximately equal to kro in the gas/oil system, provided that krg in the gas/oil system was approximately equal krw in the oil/water system.The rock was considered of intermediate wettability when it did not meet clearly either the water-wet or the oil-wet classification criteria.
After the data sets had been classified according to lithology and wettability, stepwise linear regression analysis was used to develop equations that would approximate the measured relative permeabilities from such factors as fluid saturations, permeability, and porosity. The equations developed in this study are porosity. The equations developed in this study are presented in Appendix A. presented in Appendix A. The "goodness of fit" of each equation was determined according to the statistical concept of R2 (the coefficient of multiple determination), which indicates the amount of variation about the mean that the model accounts for. A low value of R2 indicates an inadequate data fit and suggests that additional variables, higher order terms, or cross-products of the independent variables are needed. An F test was employed to eliminate insignificant variables from the regression equations.
JPT
P. 2905