# Empirical Equations for Water/Oil Relative Permeability in Saudi Sandstone Reservoirs

## Figures

Figures - uploaded by Saud M. Al-Fattah

Author content

All figure content in this area was uploaded by Saud M. Al-Fattah

Content may be subject to copyright.

... Therefore, Al-Fattah aroused an empirical equation for calculating water/oil imbibition relative permeability curves. The derived relative permeability model is dependent to rock and fluid properties and it is gained from different mathematical analyses [87]. Using the most effective and compatible model of multiphase relative permeability strongly affects the final recovery efficiency for a special reservoir and formation. ...

As the ages of most oil fields fall in the second half of their lives, many attempts have been made to enhance oil recovery in an efficient way. Gas injection into oil reservoirs for enhanced oil recovery (EOR) purposes requires relative permeability as a crucial issue in reservoir engineering. In this study, a new method is applied to predict relative permeabilities of gas/oil system related to various rock and fluid types. For this reason, a soft computing technique- Multi-gene genetic programming (MGGP) is employed to develop tools for prediction of relative permeability. The new methods are evaluated by experimental data extracted from open literature and are validated by extensive error analysis. The generated smart mathematical equations are able to predict relative permeabilities of gas/oil system with high accuracy and are applicable for various types of rock and fluid as well. In contrary to other reported correlations, the new novel equations require oil API and gas molecular weight as extra input variables to improve their estimating ability for every type of rock and fluid. The proposed technique is promising and encouraging for petroleum and reservoir engineers to be implemented for other gas/oil petro-physical properties.

Published in Petroleum Transactions, AIME, Vol. 216, 1959, pages 370–372.
A method is presented for calculating individual gas and oil or water and oil relative permeabilities from data obtained during a gas drive or a waterflood experiment performed on a linear porous body. The method has been tested and found both rapid and reliable for normal-sized core samples.
Introduction
Individual oil and gas or oil and water relative permeabilities are required for a number of reservoir engineering applications. Chief among these is the evaluation of oil displacement under conditions where gravitational effects are significant, such as a water drive or crestal gas injection in a steeply dipping oil reservoir. Numerous proposed methods of obtaining relative permeability data on reservoir core samples have been too tedious and time consuming for practical use, or have yielded questionable and sometimes inconsistent results.
A method bas been developed by which the individual relative permeability curves can be calculated from data collected during a displacement test. The method is based on sound. Using this method, with a properly designed experimental procedure, relative permeability curves can be obtained using core samples of normal size (i.e., 2 to 3 in. in length and 1 to 2 in. in diameter) within a few days after receipt of the core.
In a recent publication D. A. Efros describes an approach to the calculation of individual relative permeabilities that is based on the same theoretical considerations. We believe the approach described in the present paper is more adaptable to practical application than the method implied by Efros. In addition, comparisons with independently determined relative permeabilities are furnished to substantiate the reliability of the new method.

An entirely empirical approach was applied to predict the permeability of a porous rock to a saturating fluid. In this 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.

In a recent publication wetting phase relative permeability was expressedas:
(Equation 1)
and it was stated that a similar expression applied, mutatis mutandis, tonon-wetting phase relative permeability; i.e.,
(Equation 2)
In Equations (1) and (2), Sw and So are respectively the wetting andnon-wetting phase saturations as a fraction of the pore volume; Pc is thecapillary pressure; I, the wetting phase electrical resistivity index; IN, theanalogous non-wetting phase electrical resistivity index and Krw and Krnw, thewetting and non-wetting phase relative permeabilities.
Equation (3) is interesting because it expresses the nonwetting phaserelative permeability of a porous medium at any saturation as a function of thewetting phase relative permeability at that saturation, the saturation itselfand parameters which bear directly on the distribution within the pores of thenon-wetting and wetting phase fluid networks. It is thus inherentlyplausible.
Equation (3) is not readily checked against experimental data since fewreliable relative permeability figures have appeared in the literature; also nopublished information exists on the probable relationship between IN and So.However, if the Krw and Krnw figures of Leverett for unconsolidated sands areused in conjunction with the relationship I = Sw-2 for sands of this type, itis possible for Equation (3) to compute IN as a function of So. The results areshown in the accompanying figure and it will be seen that IN = So-1.72. Asynthetic sandstone for which Muskat quotes relative permeability data, alsogives a straight line plot between IN and So if I = Sw-1.8 is assumed. (Aresistivity index exponent of 1.8 appears to be the best average forconsolidated porous media.)
T.N. 93

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

This paper presents graphical constructions that simplify the calculation of relative permeability from displacement data. These constructions convert raw data to relative permeability in a less tedious, more accurate manner than the usual computations. Fractional-flow saturation curves derived from waterflood displacements are always concave downward and never yield multiple-value saturations.
Introduction
To find oil and water relative permeabilities by the displacement or unsteady-state method, a small linear core usually is saturated with water, then oilflooded to irreducible water saturation. Subsequently, the core is waterflooded, and during the process, pressure drop (either constant or variable) across the entire core and water injection rate (constant or variable) are determined. Effluent fractions are collected and the amount of water and oil in each is measured. Augmented by the absolute permeability and pore volume of the core and by oil and permeability and pore volume of the core and by oil and water viscosities, these data are sufficient to develop relative permeability curves.
The average saturation in the core at any time in the flood can be found from an over-all material balance. However, to calculate relative permeability, the saturation history at some point in the core must be determined, not the average saturation history. The Welge equation yields saturations at the effluent end of the core when the average saturation history is known.
Similarly, to compute relative permeability, the point pressure gradient per unit injection rate is needed, not the pressure gradient per unit injection rate is needed, not the average. The equation developed by Johnson et al. converts average relative injectivity to a point value, accomplishing the required task.
While the equations of Welge and Johnson et al. have been used successfully for years, they require tedious computation and are subject to error because of the evaluation of derivatives. The graphical techniques presented in this study are equivalent to these equations, but are easier to use and can give a more accurate evaluation of relative permeability.
Lefebvre du Prey has presented graphical constructions based on curves of volume of oil produced vs time and pressure drop vs time to develop the required point functions. These constructions are limited to constant rate displacements. The constructions presented here are general and apply to constant rate, constant pressure, or variable rate-pressure displacements. Constant-rate and constant-pressure examples are given to help clarify the methods.
The graphical techniques make it easy to see that double or triple saturation values, so extensively discussed in the past simply do not result from the fractional flow curve generated by a single displacement, such as a waterflood or an oilflood.
Theory
Ignoring gravity effects and capillary pressure, water and oil relative permeabilities (expressed as functions of saturation) are
(1)
(2)
To use these equations, the fractional flow of water or oil and effective viscosity, lambda-1, must be determined as functions of saturation.
JPT
P. 807