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..
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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