SPE 65631

Two-Phase Relative Permeability Prediction Using a Linear Regression Model

M. N. Mohamad Ibrahim, SPE, University Science of MalaysiaL. F. Koederitz, SPE, University of Missouri-Rolla

Copyright 2000, Society of Petroleum Engineers Inc

This paper was prepared for presentation at the 2000 SPE Eastern Regional Meeting held inMorgantown, West Virginia, 17-19 October 2000..

This paper was selected for presentation by an SPE Program Committee following review ofinformation contained in an abstract submitted by the author(s). Contents of the paper, aspresented, have not been reviewed by the Society of Petroleum Engineers and are subject tocorrection by the author(s). The material, as presented, does not necessarily reflect any positionof the Society of Petroleum Engineers, its officers, or members. Papers presented at SPEmeetings are subject to publication review by Editorial Committees of the Society of PetroleumEngineers. Electronic reproduction, distribution, or storage of any part of this paper forcommercial purposes without the written consent of the Society of Petroleum Engineers isprohibited. Permission to reproduce in print is restricted to an abstract of not more than 300words; illustrations may not be copied. The abstract must contain conspicuous acknowledgementof where and by whom the paper was presented. Write Librarian, SPE, P. O. Box 833836,Richardson, TX 75083-3836, U.S.A., fax 01-972-952-9435.

AbstractIn the absence of laboratory measured data or in the case whena more general representation of fluid flow in a reservoir isneeded, empirical relative permeability correlations becomeuseful. These correlations will also apply to simulation studieswhich require adjustments to the relative permeability values toaccount for grid effects. A linear regression model approach isemployed to develop prediction equations for water-oil, gas-oil,gas-water, and gas-condensate relative permeability fromexperimental data. Use of the SPE CD-ROM has allowed arapid and thorough data retrieval for this study; 416 sets ofrelative permeability data were obtained from publishedliterature and various industry sources. Improved equationswere developed for water-oil and gas-oil systems based onformation type and wettability. Additionally, general equationsfor gas-condensate and gas-water systems were formulated.Craig’s rule for determining the rock wettability has beenmodified to cover a wider range of relative permeability datacurrently available. Available data has increased significantlysince the last published work in this area. The predictionequations are compared with previously published correlationswhere possible.

Introduction Relative permeability, a dimensionless quantity, is the ratio ofeffective permeability to a base permeability. The effectivepermeability is a measure of the ability of a single fluid to flowthrough a rock when the pore spaces of the rock are notcompletely filled or saturated with the fluid. The basepermeability can be absolute air permeability, absolute liquidpermeability or effective oil permeability at irreducible watersaturation. Relative permeability measurements and conceptsbecome important due to the fact that nearly all hydrocarbonreservoirs contain more than one phase of homogeneous fluid.Relative permeability is a function of pore structure, saturationhistory and wettability1,2.

Laboratory methods for measuring relativepermeability were probably introduced to the petroleum industryback in 1944 by Hassler3. Since then various methods ofmeasuring relative permeability have been developed. Some ofthe more commonly used laboratory methods are Penn-State,Single-Sample Dynamic, Stationary Fluid, Hassler, Hafford,JBN, Capillary Pressure and Centrifuge2. In general, thesemethods can be categorized into two major groups whichconsist of steady-state and unsteady-state methods.

Laboratory measurement of relative permeability usingeither steady-state or unsteady-state methods can be expensiveand time consuming. Laboratory measurement is considered amicro process because a single measurement is insufficient torepresent the entire reservoir. Therefore several core samplesfrom representative facies in the reservoir must be taken andtested. Since results of the relative permeability tests performedon several samples often vary, it is necessary to average the databefore a scaling up from core to reservoir scale is performed.

An accurate numerical procedure for determiningrelative permeability values provides an alternative technique,and at the same time it can overcome the previous shortcomings.In contrast to laboratory measurement, this is a macro process

2 M. N. MOHAMAD IBRAHIM, L. F. KOEDERITZ SPE 65631

SS S

S Sww wi

wi orw

* ( )=−

− −11

which provides a better statistical representation of relativepermeability values for the reservoir as a whole.

ObjectivesRealizing that an ample amount of published relativepermeability data could be extracted from the Society ofPetroleum Engineers' literature (published from 1950 through1998) plus unpublished data from various oil and gas companiesand individuals, an improvement to previously publishedprediction equations4 for relative permeability is presented andnew equations are developed for other systems. Furthermore,the larger amount of data available today will give a betterrepresentation of prediction equations since they cover a widerrange of domain.

In order to create predicting models that representproducing reservoirs, certain criteria in selecting data (relativepermeability curves and other pertinent information) wereimposed. The data selection criteria used in this study were:

1. The relative permeability curves are generated fromeither steady-state or unsteady-state experiments. In otherwords, relative permeability curves obtained from correlationsor data obtained from hypothetical simulation studies areexcluded in this study;

2. The core used in the experiment must be a naturallyformed rock sample. Data obtained from synthetic or man-madecores such as Alundum cores is not considered;

3. Only imbibition data are used for oil-water and gas-water systems whereas for gas-oil and gas-condensate systems,only drainage data are used in the analysis; and

4. Only the primary data is selected when multipleimbibition or drainage processes are presented.

The prediction equations for relative permeability ofoil-water systems for both sandstones and carbonates, whichinclude limestones and dolomites, are presented for fourdifferent types of rock wettability, i.e., strongly water-wet,water-wet, intermediate (or mixed-wet) and oil-wet based onCraig’s rule5; however, many oil-water curves did not strictlyfollow Craig’s rule. This is not unexpected because Craig’s rulewas not based on detailed experimental studies but simply aheuristic rule dating prior to 1971; therefore it will not be truefor all cases. Some adjustments to the rule were made byintroducing tolerances into it without changing its basicprinciple in order to categorize data which slightly violated theoriginal rule. Table 1 summarizes the modified Craig’s rule thatwas used in determining the wettability in this study. Whilemany datasets had no additional wettability indicators, datahaving an Amott's Index or Modified U.S.B.M. Wettability Testwas in agreement with the modified Craig's rule.

Additionally, relative permeability equations for gas-oilsystems for sandstones and carbonates are also improved. Thescope of this study also includes developing predictive relativepermeability equations for gas-water and gas-condensatesystems.

Data NormalizationThe relative permeability curves used in this study did notoriginally have the same format, i.e., some of the curves werepresented in the classical form while the rest were innormalized form. Even worse are gas-water systems, where thenormalization process was not consistent. Some of the curvesdefined the absolute permeability as the effective permeabilityof gas at Swc (krgw = 1.0 at Swc where krgw is the relativepermeability of gas with respect to water), while others definedthe absolute permeability as the 100 percent water saturationpermeability (krw = 1.0 at Sg = 0).

It is necessary to convert these curves into the sameformat (either the classical or the normalized) before theregression analysis is performed in order to be consistent. Sinceless than half of the data collected were in the classical form,the normalized form was chosen to be the standard formthroughout this study. Moreover, it is easier to convert theclassical data into the normalized form than converting thenormalized data into the classical form. This is due to thedifficulties in locating the classical relative permeability end-point absolute permeability values (which most authors did notsupply) in the articles reviewed. The classical data form wasusually found in much older data, which is another justificationto convert all of the data into the more current normalized form.In the case of the gas-water system, the first definition ofabsolute permeability (krgw = 1.0 at Swc) was chosen to be thestandard form since most of the data obtained from the literaturewere presented in this manner. For the gas-condensate systems,the same normalization procedure as in the gas-oil systems isemployed where the effective permeability of liquid(condensate) at Sg = 0 is defined as the absolute permeability.

Since the collected curves did not have the same rangeof saturation values (as far as the abscissa is concerned) due tothe fact that some of the curves were longer than others owingto differences in the critical wetting and non-wetting phasesaturations, this inconsistency would contribute to highvariation in the response (ordinate). Thus, there is a need tofind a way to plot each curve in its class on the same horizontalscale in order to reduce this variation so that a better predictionmodel can be achieved. This can be accomplished bynormalizing either the wetting phase saturation or the non-wetting phase saturation which results in the horizontal axisalways ranging from zero to one.

For oil-water systems, the normalized water saturationis defined as6:

Except for the oil-water system, the rest of the systemshave the relative permeability to liquid with respect to gas (krlg)curves which are almost always longer than the relativepermeability to gas (krg) curves due to the presence of the critical

SPE 65631 TWO-PHASE RELATIVE PERMEABILITY PREDICTION USING A LINEAR REGRESSION MODEL 3

SS S

S Sgg gc

gc lc

*( )

( )=−

− +12

SS

Slg

lc

* ( )= −−

1

13

gas saturations (Sgc). Therefore, separate saturationnormalization equations must be used for each curve as follows:

Regression AnalysisIn this study, a forward stepwise multiple linear regressiontechnique was employed in developing relative permeabilityprediction equations. This technique is based on an automaticsearch procedure concept which develops the best subset ofindependent variables sequentially, at each regression stepadding or deleting one independent variable at a time in anattempt to get the highest possible coefficient of multipledetermination, R2 value. The R2 is interpreted as the proportionof observed values of Yi that can be explained by the regressionmodel and is used to measure how well the model fits the data.The higher the value of R2, the more successful the model is inexplaining the variation of Y. A value of one indicates that allpoints lie along the true regression line, whereas a value of zeroindicates the absence of a linear relationship between variables.In the latter case, the modeler has to search for an alternativemodel such as a nonlinear model. Since observational data(data obtained after the experiments were completed) were usedin this analysis, an R2 value slightly higher than 60% wasconsidered highly satisfactory6. One criticism of using the R2

criteria as the only indication of goodness of fit is that the R2

value will keep increasing if more independent variables areintroduced into the model. To balance the use of moreparameters against the gain in R2, many statisticians use theadjusted R2 value (R2

adj)7. Very simply, the R2

adj valueapproaching R2 indicates that excessive terms were not includedin the model.

Discussion and ComparisonsThe prediction equations developed in this study are listed inAppendix A and defined in Table 2 which summarizes thecharacteristics of the equations developed for all four fluidsystems. All R2 values well exceed 60% and all R2

adj values arewithin 1.5% of R2 thus indicating a reasonable fit withoutexcessive terms. Figures 1 and 2 graphically illustratenormalized oil-water and gas-oil relative permeability values forboth sandstone and carbonate formations. Tables 3 and 4 list theranges of rock properties and fluid saturations used indeveloping prediction equations for oil-water and gas-liquidsystems, respectively.

The relative permeability equations developed werecompared with correlations of Honarpour et al4, Rose8 and Narret al9. These works either did not employ wettability preferences,or did not distinguish between oil-wet and intermediate (mixed)wettability, and between water-wet and strongly water-wetsystems. Additionally, Rose's and Narr's equations are so generalthat they do not specify the type of rock. The curves are inclose agreement with each other in terms of normalized relativepermeability of oil with respect to water as shown in Figure 3 fora water-wet sandstone. The same is also true for a carbonateformation. Figure 4 shows normalized water relativepermeability values calculated using the various correlations.The equation developed falls between Honarpour's and Rose'scurves. Rose’s plot seems to give unrealistic prediction valuesfor a water-wet system because the endpoint of the krw* curve(krw* at Sw = 1-Sorw) is much higher than expected for a water-wet case. A criticism of Honarpour’s model is that the waterrelative permeability values appeared low resulting in optimisticrecoveries. This criticism lead the present study to separate thestrongly water-wet curves from the regular water-wet curves.Figure 5 clearly illustrates this point. For gas-oil systems, theequations presented have eliminated the requirement of anendpoint value for krg.

ConclusionsTwenty four, two phase relative permeability predictionequations have been developed through extensive trial and errormodel building processes using linear regression analysis forfour different systems which commonly exist in the petroleumindustry. In oil-water systems, prediction equations for threetypes of rock wettability were formed in addition toclassification of the equations on the basis of rock type, i.e.,sandstone and carbonate. Additionally, completely newcorrelations for strongly water-wet system for both sandstoneand carbonate were developed. As in the oil-water systems,prediction equations according to rock type were successfullydeveloped for gas-oil systems. Completely new correlationsbased on a linear regression analysis were developed for gas-water and gas-condensate systems. Based on an extensivereview of existing data, modifications to wettabilitydetermination were developed.

NomenclatureCapital Letters

R2 = coefficient of multiple

determinationR2

adj = adjusted coefficient of multiple determination

Sg = gas saturation, fractionSgc = critical gas saturation

4 M. N. MOHAMAD IBRAHIM, L. F. KOEDERITZ SPE 65631

Sl = liquid saturation, fractionSlc = total of critical liquid saturations present in the system, fraction

Sorg = residual oil saturation in oil- gas system, fraction

Sorw = residual oil saturation in oil- water system, fraction

Sw = water saturation, fractionSwc = critical (connate) water

saturation, fractionSwi = initial water saturation,

fraction WW = water-wet

Yi = ith observed value where i =1,2,3,.....n

Lowercase Letters

ka = absolute permeability, mdkrcg = relative permeability of

condensate with respect to gas, fraction

krg = relative permeability of gas, fraction krgw = relative permeability of gas

with respect to water, fractionkrlg = relative permeability of liquid

with respect to gas, fraction krog = relative permeability of oil

with respect to gas, fractionkrow = relative permeability of oil

with respect to water, fractionkrw = relative permeability of

water, fraction

Greek Symbol

N = porosity, fraction

Superscript

* = normalized value

References1. Unalmiser, S. and Funk, J. J : “Engineering Core Analysis”,Journal of Petroleum Technology, April, (1998).

2. Honarpour, M., Koederitz, L. and Harvey, A. H. : RelativePermeability of Petroleum Reservoirs, CRC Press. Inc., Florida, (1986).

3. Hassler, G. L., U.S. Patent 2,345,935, (1944).

4. Honarpour, M., Koederitz, L. F. and Harvey, A. H.: "EmpiricalEquations for Estimating Two Phase Relative Permeability inConsolidated Rock," Trans. AIME, vol. 273, (1982), pp. 2905 ff.

5. Craig, F. F., Jr. : The Reservoir Engineering Aspects ofWaterflooding Monograph, Vol. 3, Society of Petroleum Engineersof AIME, Henry L. Doherty Series, Dallas, Texas, (1993), p. 20.

6. Koederitz, L. F., Harvey A. H. and Honarpour M. : Introduction toPetroleum Reservoir Analysis, Gulf Publishing Company, Houston,Texas, (1989).

7. Devore, J. L. : Probability and Statistics for Engineering and theSciences, Duxbury Press., California, (1995), pp. 474 ff.

8. Rose, W. : “Theoretical Generalizations Leading to the Evaluationof Relative Permeability”, Trans. AIME, vol. 186, (1949), pp. 111 ff.

9. Naar, J. and Henderson, J. H. : “An Imbibition Model, itsApplications to Flow Behavior and the Prediction of OilRecovery”, Trans. AIME, Part II, vol. 222, (1961), p. 61.

General ReferenceMohamad Ibrahim, M. N. :"Two-Phase Relative Permeability PredictionUsing A Linear Regression Model", Ph.D. Dissertation, University ofMissouri, Rolla, (1999)

SPE 65631 TWO-PHASE RELATIVE PERMEABILITY PREDICTION USING A LINEAR REGRESSION MODEL 5

k S S S S

S S S A

rw orw w w w

wc orw w

* . * . * . *

. * ( )

.

.

= − +

−

0 28483482 0 0324527 0 07113168

2 2461759 4

15

1 7 2

k S S S Arow w w w* . * . * . * ( ).= − + −1 3090996 2 8670229 0 768952 51 6 2

k S SS

S S

S S S S S S S

S S S S k S

S k S A

rw w orww

w orw

w orw wc w wc orw w

orw w w wc a orw

w a wc

* . * .*

. *

. * . * . ( *)

. * . * ( ln ) ( )

. * ((ln ) ) ( )

.

.

. .

= + +

+ − +

− + −

−

0 22120304 0 24933592 21370925

83491972 0 4562939 116107198

8 7866012 0 00000578 1

12 841061 6

1 6 23

2 5

4 5 15 3 4 2 2

3 2 3 3 10 0 4

2 3 6

φ

φ φ

φ

k S S S Arow w w w* . * . * . * ( )= − + −1 2 65253 2 4720911 0814367 72 3

k S S S S S S k

S S k S S S S

S S S S A

rw w w wc orw w wc a

w wc a wc w wc w

orw w orw w

* . * . * ( ) . *( ln )

. *( ln ) . ( *) . *

. ( ) * . * ( )

.

. .

. .

= + +

− + −

− +

01163954 2 66958338 0 47536676

0 3912824 752 014909 398 40214

152 43629 0 22964285 8

4 0 8 2 2

3 2 3 2 5 2 2 2

3 2 7 0 5

φ φ

φ

k S S S Arow w w w* . * . * . * ( )= − + −1 2 985766 31548084 1171486 92 3

k S S S Arw w w w* . * . * . * ( )= − +0 2441795 0 355058 0 5117625 102 3

k S S S S Arow w w w w* . * . * . * . * ( ).= − + − +1 4 985409 21322192 29 04644 11723526 112 2 5 3

k S S S S Arow w w w w* . * . * . * . * ( )= − + + −1 21529 0 6389135 14345325 0 919704 12 3 4

k S S S Arow w w w* . * . * . * ( )= − − +1 0 7233267 17720584 150407908 32 3

k S S S k S

S SS S

S

k S A

rw w w w a wc

orw worw w

w

a w

* . * . * . *

. * .*

. *

. (ln ) * ( )

. . . .

..

= − −

− + +

+

0 09101641 01841405 0 0001629

11810931 0 64933067 212270704

0 01375097 2

1 5 0 5 1 4 2 6

3 1 55 2 5

6

φ

φφ

Appendix A

6 M. N. MOHAMAD IBRAHIM, L. F. KOEDERITZ SPE 65631

k k S S S S S k

S S S k S S S

S S S S A

rw a wc w w w orw a

w w wc a orw w wc

wc w w orw

* . (ln ) * . * . * ( ln )

. * . * ( ln ) . *

. ( *) . * ( )

.

.

. . .

= − −

+ − +

− +

0 46689293 0 0589939 01938748

0 24253563 0195414 66 6228413

11126159 125 291504 12

1 5 2

3 0 9 2 2

1 5 2 5 2 5 3

φ

φ

φ φ

k S S S Arow w w w* . * . * . * ( )= − + −1 3254725 38176666 1563216 132 3

k S S S Arw w w w* . * . * . * ( )= − +0 3643225 0 7458182 106090802 142 3

k S S S S

S A

row w w w w

w

* . * . * . * . *

. * ( )

.

.

= − + − +

−

1 8 6102768 87 9417721 207 03656 187 099163

60 388661 15

2 2 5 3

3 5

k S k S S S S

S S S S S A

rw w a w w orw wc

w orw w wc w

* . * . (ln ) * . * ( )

. * . * . * ( )

. . .

. .

= + −

− − +

0 2178721 0 00536612 9 7494266

51295364 4 5717726 0 57604803 16

0 4 2 0 5 0 7 2

2 0 5 2 15φ φ

k S S S S

S A

rog l l l l

l

* . * . * . * . *

. * ( )

= − + −

+

01599039 1045545 4 0843698 5 414161

32103149 17

2 3 4

5

k S S S S S S

S S A

rg g org g wc g g

gc g

* . * . * . * . *

. * ( )

= − − +

−

0 9396949 0 774167 1216298 11628119

1248192 18

2 2 2 2

2

φ

k S S S S S

S A

rog l l l l l

l

* . * . * . * . * . *

. * ( )

= − − + −

+

4 465936 0 252752 22 93637 53000956 5519912

21917911 19

2 3 4 5

6

k S k S S S S Arg g a g wc g g* . * . * . * . * ( )= − + +0 3296593 0 001723 2 0568057 12314265 202 2 2φ

k S S S S

S A

rgw g g g g

g

* . * . * . * . *

. * ( )

= − + −

+

13046802 8159598 2550978 3153754

13883828 21

2 3 4

5

k S S S S k S

S S S k S S A

rw l l l gc a l

wc l wc a gc l

* . * . * . * . (ln ) *

. ( *) . ( ) * ( )

.

.

= − + −

− +

0 94555376 12967293 169592185 0 0424518

14583028 0 02764389 22

1 7 3 3 5

15 2 2 4φ φ

SPE 65631 TWO-PHASE RELATIVE PERMEABILITY PREDICTION USING A LINEAR REGRESSION MODEL 7

k S S S Arcg l l l* . * . * . * ( )= − +01194373 0 089246 0 9606793 232 3

k S S S S S S

S S S S A

rg g g lc g lc gc

g gc lc g

* . * . * . *

. * . * ( )

. . .

.

= + −

+ −

333929676 6 75670631 20 926791

101654474 7 3835856 24

1 2 0 9 1 2 2

4 0 5

Table 1. Modified Craig's rule

Rock Swc Sw at which krw * krw* at Sw = 100-Sorw

Wettability and krow* are equal (fraction)

Strongly

Water-Wet: 15% 45% 0.07≥ ≥ ≤

Water-Wet: 10% 45% 0.07 < krw* 0.3 ≥ ≥ ≤

Oil-Wet: 15% 55% 0.5 ≤ ≤ ≥

Intermediate: 10% 45% Sw 55% > 0.3 ≥ ≤ ≤(Mixed-Wet) OR

15% 45% Sw 55% < 0.5≤ ≤ ≤

8 M. N. MOHAMAD IBRAHIM, L. F. KOEDERITZ SPE 65631

Table 2. Summary of characteristics of equations developed in this study

System Wettability Lithology Equation Number of Data Sets

Number ofData Points

R2 R2adj

Oil-Water

StronglyWater-Wet

Sandstone A1 16 127 87 86

A2 16 127 82 81

Carbonate A3 6 49 99 99

A4 6 49 89 88

Water-Wet

Sandstone A5 102 870 93 93

A6 102 870 75 75

Carbonate A7 28 317 94 94

A8 28 317 80 80

IntermediateWettability

(Mixed-Wet)

Sandstone A9 43 396 93 93

A10 43 396 90 90

Carbonate A11 29 278 93 93

A12 29 278 85 85

Oil-Wet

Sandstone A13 31 245 95 95

A14 31 245 89 88

Carbonate A15 19 184 86 86

A16 19 184 86 85

Gas-Oil

Sandstone A17 98 962 95 95

A18 92 799 90 90

Carbonate A19 14 133 94 94

A20 14 119 89 89

Gas-Water All A21 19 144 89 89

A22 19 166 88 87

Gas-Condensate All A23 17 123 95 95

A24 17 115 85 85

SPE 65631 TWO-PHASE RELATIVE PERMEABILITY PREDICTION USING A LINEAR REGRESSION MODEL 9

Table 3. Ranges of rock properties and fluid saturations used in developing oil-water relative permeability equations

Equations N (%) ka (md) Swc (%) Sorw (%)

A1 & A2 9.9 - 63.2 2.23 - 3,070 15.3 - 50.0 15.0 - 51.1

A3 & A4 11.7 - 18.0 0.84 - 7.2 30.0 - 46.2 7.0 - 32.2

A5 & A6 8.4 - 37.1 0.52 - 8,440 3.6 - 67.5 6.6 - 47.3

A7 & A8 6.2 - 33.0 0.27 - 3,100 6.0 - 43.5 13.0 - 50.6

A9 & A10 8.0 - 32.6 3.4 - 10,500 5.0 - 38.9 11.09 - 44.4

A11 & A12 5.9 - 38.3 1.08 - 4,018.7 7.0 - 41.0 13.9 - 50.0

A13 & A14 9.1 - 33.0 1 - 5,010 4.7 - 44.0 7.67 - 55.0

A15 & A16 9.8 - 35.0 1.3 - 1,420 8.0 - 53.6 9.8 - 57.0

Table 4. Ranges of rock properties and fluid saturations used in developing relative permeability equations for gas-liquid systems

Equation N (%) ka (md) Sgc (%) Swc (%) Sorg (%)

A17 6.3 - 39.0 1.48 - 5580 0.6 - 25.0 3.28 - 50.0 3.5 - 48.0

A18 6.3 - 39.0 1.48 - 3650 0.6 - 25.0 3.28 - 50.0 5.0 - 48.0

A19 & A20 9.0 - 34.9 4.3 - 731 0.01 - 13.52 6.0 - 51.1 5.0 - 38.6

A21 & A22 5.0 - 25.0 0.1 - 345 3.0 - 47.9 10.0 - 61.2 Not Applicable

A23 & A24 6.0 - 26.6 Not Available 2.0 - 30.0 9.0 - 60.0

10 M. N. MOHAMAD IBRAHIM, L. F. KOEDERITZ SPE 65631

0

0.2

0.4

0.6

0.8

1

kr *

0 0.2 0.4 0.6 0.8 1 Sw (fraction)

Sandstone

Carbonate

Oil-Water Relative PermeabilityWater-Wet Rocks

φ=15%ka =100 mdSwc =20%Sorw =20%

Figure 1

0

0.2

0.4

0.6

0.8

1

kr *

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Sg (fraction)

Sandstone

Carbonate

Gas-Oil Relative PermeabilitySandstone & Carbonate

φ = 15%

ka = 100

md

Swc = 15%

Sorg =15%

Figure 2

0

0.2

0.4

0.6

0.8

1

krow

*

0 0.2 0.4 0.6 0.8 1 Sw (fraction)

Equation A6

Honarpour's

Rose's

Narr's

Water-wet SandstoneOil Relative Permeability to Water

φ = 15%ka = 100 mdSwc = 20%Sorw = 20%

Figure 3

0

0.2

0.4

0.6

0.8

krw

*

0 0.2 0.4 0.6 0.8 1

Sw (fraction)

Equation A5

Honarpour's

Rose's

Narr's

Water-Wet SandstoneWater Relative Permeability

φ = 15%

ka = 100 md

Swc = 20%

Sorw = 20%

Figure 4

SPE 65631 TWO-PHASE RELATIVE PERMEABILITY PREDICTION USING A LINEAR REGRESSION MODEL 11

0

0.1

0.2

0.3

0.4

0.5

0.6

kr *

0 0.2 0.4 0.6 0.8 1

Sw (fraction)

WW

Strongly WW

Honarpour's WW

Water-Wet SystemCarbonate

φ = 10%ka = 100 mdSwc = 20%Sorw=20%

Figure 5