Spe - 65631

SPE 65631 Two-Phase Relative Permeability Prediction Using a Linear Regression Model M. N. Mohamad Ibrahim, SPE, University Science of Malaysia L. 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 in Morgantown, West Virginia, 17-19 October 2000.. This paper was selected for presentation by an SPE Program Committee following review of information contained in an abstract submitted by the author(s). Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum Engineers. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgement of 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.

Abstract In the absence of laboratory measured data or in the case when a more general representation of fluid flow in a reservoir is needed, empirical relative permeability correlations become useful. These correlations will also apply to simulation studies which require adjustments to the relative permeability values to account for grid effects. A linear regression model approach is employed to develop prediction equations for water-oil, gas-oil, gas-water, and gas-condensate relative permeability from experimental data. Use of the SPE CD-ROM has allowed a rapid and thorough data retrieval for this study; 416 sets of relative permeability data were obtained from published literature and various industry sources. Improved equations were developed for water-oil and gas-oil systems based on formation type and wettability. Additionally, general equations for gas-condensate and gas-water systems were formulated. Craig’s rule for determining the rock wettability has been modified to cover a wider range of relative permeability data currently available. Available data has increased significantly since the last published work in this area. The prediction equations are compared with previously published correlations where possible.

Introduction Relative permeability, a dimensionless quantity, is the ratio of effective permeability to a base permeability. The effective permeability is a measure of the ability of a single fluid to flow through a rock when the pore spaces of the rock are not completely filled or saturated with the fluid. The base permeability can be absolute air permeability, absolute liquid permeability or effective oil permeability at irreducible water saturation. Relative permeability measurements and concepts become important due to the fact that nearly all hydrocarbon reservoirs contain more than one phase of homogeneous fluid. Relative permeability is a function of pore structure, saturation history and wettability1,2. Laboratory methods for measuring relative permeability were probably introduced to the petroleum industry back in 1944 by Hassler3. Since then various methods of measuring relative permeability have been developed. Some of the more commonly used laboratory methods are Penn-State, Single-Sample Dynamic, Stationary Fluid, Hassler, Hafford, JBN, Capillary Pressure and Centrifuge2. In general, these methods can be categorized into two major groups which consist of steady-state and unsteady-state methods. Laboratory measurement of relative permeability using either steady-state or unsteady-state methods can be expensive and time consuming. Laboratory measurement is considered a micro process because a single measurement is insufficient to represent the entire reservoir. Therefore several core samples from representative facies in the reservoir must be taken and tested. Since results of the relative permeability tests performed on several samples often vary, it is necessary to average the data before a scaling up from core to reservoir scale is performed. An accurate numerical procedure for determining relative 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

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M. N. MOHAMAD IBRAHIM, L. F. KOEDERITZ

which provides a better statistical representation of relative permeability values for the reservoir as a whole. Objectives Realizing that an ample amount of published relative permeability data could be extracted from the Society of Petroleum Engineers' literature (published from 1950 through 1998) plus unpublished data from various oil and gas companies and individuals, an improvement to previously published prediction equations4 for relative permeability is presented and new equations are developed for other systems. Furthermore, the larger amount of data available today will give a better representation of prediction equations since they cover a wider range of domain. In order to create predicting models that represent producing reservoirs, certain criteria in selecting data (relative permeability curves and other pertinent information) were imposed. The data selection criteria used in this study were: 1. The relative permeability curves are generated from either steady-state or unsteady-state experiments. In other words, relative permeability curves obtained from correlations or data obtained from hypothetical simulation studies are excluded in this study; 2. The core used in the experiment must be a naturally formed rock sample. Data obtained from synthetic or man-made cores such as Alundum cores is not considered; 3. Only imbibition data are used for oil-water and gaswater 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 multiple imbibition or drainage processes are presented. The prediction equations for relative permeability of oil-water systems for both sandstones and carbonates, which include limestones and dolomites, are presented for four different types of rock wettability, i.e., strongly water-wet, water-wet, intermediate (or mixed-wet) and oil-wet based on Craig’s rule5; however, many oil-water curves did not strictly follow Craig’s rule. This is not unexpected because Craig’s rule was not based on detailed experimental studies but simply a heuristic rule dating prior to 1971; therefore it will not be true for all cases. Some adjustments to the rule were made by introducing tolerances into it without changing its basic principle in order to categorize data which slightly violated the original rule. Table 1 summarizes the modified Craig’s rule that was used in determining the wettability in this study. While many datasets had no additional wettability indicators, data having an Amott's Index or Modified U.S.B.M. Wettability Test was in agreement with the modified Craig's rule. Additionally, relative permeability equations for gas-oil systems for sandstones and carbonates are also improved. The scope of this study also includes developing predictive relative permeability equations for gas-water and gas-condensate systems.

SPE 65631

Data Normalization The relative permeability curves used in this study did not originally have the same format, i.e., some of the curves were presented in the classical form while the rest were in normalized form. Even worse are gas-water systems, where the normalization process was not consistent. Some of the curves defined the absolute permeability as the effective permeability of gas at Swc (krgw = 1.0 at Swc where krgw is the relative permeability of gas with respect to water), while others defined the absolute permeability as the 100 percent water saturation permeability (krw = 1.0 at Sg = 0). It is necessary to convert these curves into the same format (either the classical or the normalized) before the regression analysis is performed in order to be consistent. Since less than half of the data collected were in the classical form, the normalized form was chosen to be the standard form throughout this study. Moreover, it is easier to convert the classical data into the normalized form than converting the normalized data into the classical form. This is due to the difficulties in locating the classical relative permeability endpoint absolute permeability values (which most authors did not supply) in the articles reviewed. The classical data form was usually found in much older data, which is another justification to convert all of the data into the more current normalized form. In the case of the gas-water system, the first definition of absolute permeability (krgw = 1.0 at Swc) was chosen to be the standard form since most of the data obtained from the literature were presented in this manner. For the gas-condensate systems, the same normalization procedure as in the gas-oil systems is employed 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 range of saturation values (as far as the abscissa is concerned) due to the fact that some of the curves were longer than others owing to differences in the critical wetting and non-wetting phase saturations, this inconsistency would contribute to high variation in the response (ordinate). Thus, there is a need to find a way to plot each curve in its class on the same horizontal scale in order to reduce this variation so that a better prediction model can be achieved. This can be accomplished by normalizing either the wetting phase saturation or the nonwetting phase saturation which results in the horizontal axis always ranging from zero to one. For oil-water systems, the normalized water saturation is defined as6:

Sw* =

S w − S wi 1 − S wi − S orw

(1)

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

SPE 65631

TWO-PHASE RELATIVE PERMEABILITY PREDICTION USING A LINEAR REGRESSION MODEL

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

Sg* =

S g − S gc 1 − ( S gc+ Slc )

 Sg  Sl * = 1 −    1 − Slc 

(2)

(3)

Regression Analysis In this study, a forward stepwise multiple linear regression technique was employed in developing relative permeability prediction equations. This technique is based on an automatic search procedure concept which develops the best subset of independent variables sequentially, at each regression step adding or deleting one independent variable at a time in an attempt to get the highest possible coefficient of multiple determination, R2 value. The R2 is interpreted as the proportion of observed values of Yi that can be explained by the regression model and is used to measure how well the model fits the data. The higher the value of R2, the more successful the model is in explaining the variation of Y. A value of one indicates that all points lie along the true regression line, whereas a value of zero indicates the absence of a linear relationship between variables. In the latter case, the modeler has to search for an alternative model such as a nonlinear model. Since observational data (data obtained after the experiments were completed) were used in this analysis, an R2 value slightly higher than 60% was considered 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 are introduced into the model. To balance the use of more parameters against the gain in R2, many statisticians use the adjusted R2 value (R2adj)7. Very simply, the R2adj value approaching R2 indicates that excessive terms were not included in the model. Discussion and Comparisons The prediction equations developed in this study are listed in Appendix A and defined in Table 2 which summarizes the characteristics of the equations developed for all four fluid systems. All R2 values well exceed 60% and all R2adj values are within 1.5% of R2 thus indicating a reasonable fit without excessive terms. Figures 1 and 2 graphically illustrate normalized oil-water and gas-oil relative permeability values for both sandstone and carbonate formations. Tables 3 and 4 list the ranges of rock properties and fluid saturations used in developing prediction equations for oil-water and gas-liquid systems, respectively.

3

The relative permeability equations developed were compared with correlations of Honarpour et al4, Rose8 and Narr et 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-wet systems. Additionally, Rose's and Narr's equations are so general that they do not specify the type of rock. The curves are in close agreement with each other in terms of normalized relative permeability of oil with respect to water as shown in Figure 3 for a water-wet sandstone. The same is also true for a carbonate formation. Figure 4 shows normalized water relative permeability values calculated using the various correlations. The equation developed falls between Honarpour's and Rose's curves. Rose’s plot seems to give unrealistic prediction values for a water-wet system because the endpoint of the krw* curve (krw* at Sw = 1-Sorw) is much higher than expected for a waterwet case. A criticism of Honarpour’s model is that the water relative permeability values appeared low resulting in optimistic recoveries. This criticism lead the present study to separate the strongly water-wet curves from the regular water-wet curves. Figure 5 clearly illustrates this point. For gas-oil systems, the equations presented have eliminated the requirement of an endpoint value for krg. Conclusions Twenty four, two phase relative permeability prediction equations have been developed through extensive trial and error model building processes using linear regression analysis for four different systems which commonly exist in the petroleum industry. In oil-water systems, prediction equations for three types of rock wettability were formed in addition to classification of the equations on the basis of rock type, i.e., sandstone and carbonate. Additionally, completely new correlations for strongly water-wet system for both sandstone and carbonate were developed. As in the oil-water systems, prediction equations according to rock type were successfully developed for gas-oil systems. Completely new correlations based on a linear regression analysis were developed for gaswater and gas-condensate systems. Based on an extensive review of existing data, modifications to wettability determination were developed. Nomenclature Capital Letters R2

=

R2adj

=

Sg Sgc

= =

coefficient of multiple determination adjusted coefficient of multiple determination gas saturation, fraction critical gas saturation

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M. N. MOHAMAD IBRAHIM, L. F. KOEDERITZ

Sl Slc

= =

Sorg

=

Sorw

=

Sw Swc

= =

Swi

=

WW Yi

= =

liquid saturation, fraction total of critical liquid saturations present in the system, fraction residual oil saturation in oilgas system, fraction residual oil saturation in oilwater system, fraction water saturation, fraction critical (connate) water saturation, fraction initial water saturation, fraction water-wet ith observed value where i =1,2,3,.....n

Lowercase Letters ka krcg

= =

krg

=

krgw

=

krlg

=

krog

=

krow

=

krw

=

absolute permeability, md relative permeability of condensate with respect to gas, fraction relative permeability of gas, fraction relative permeability of gas with respect to water, fraction relative permeability of liquid with respect to gas, fraction relative permeability of oil with respect to gas, fraction relative permeability of oil with respect to water, fraction relative permeability of water, fraction

Greek Symbol N

=

porosity, fraction

=

normalized value

Superscript *

SPE 65631

References 1. Unalmiser, S. and Funk, J. J : “Engineering Core Analysis”, Journal of Petroleum Technology, April, (1998). 2. Honarpour, M., Koederitz, L. and Harvey, A. H. : Relative Permeability 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.: "Empirical Equations for Estimating Two Phase Relative Permeability in Consolidated Rock," Trans. AIME, vol. 273, (1982), pp. 2905 ff. 5. Craig, F. F., Jr. : The Reservoir Engineering Aspects of Waterflooding Monograph, Vol. 3, Society of Petroleum Engineers of AIME, Henry L. Doherty Series, Dallas, Texas, (1993), p. 20. 6. Koederitz, L. F., Harvey A. H. and Honarpour M. : Introduction to Petroleum Reservoir Analysis, Gulf Publishing Company, Houston, Texas, (1989). 7. Devore, J. L. : Probability and Statistics for Engineering and the Sciences, Duxbury Press., California, (1995), pp. 474 ff. 8. Rose, W. : “Theoretical Generalizations Leading to the Evaluation of Relative Permeability”, Trans. AIME, vol. 186, (1949), pp. 111 ff. 9. Naar, J. and Henderson, J. H. : “An Imbibition Model, its Applications to Flow Behavior and the Prediction of Oil Recovery”, Trans. AIME, Part II, vol. 222, (1961), p. 61.

General Reference Mohamad Ibrahim, M. N. :"Two-Phase Relative Permeability Prediction Using A Linear Regression Model", Ph.D. Dissertation, University of Missouri, Rolla, (1999)

SPE 65631

TWO-PHASE RELATIVE PERMEABILITY PREDICTION USING A LINEAR REGRESSION MODEL

5

Appendix A 2

3

krow * = 1 − 21529 . S w * + 0.6389135 S w * + 14345325 . S w * −0.919704 S w *

4

( A1)

. krw * = 0.09101641 S w *1.5 −01841405 φ 0.5 S w *1.4 −0.0001629 S w * ka S wc 2.6 S orw 5 S w *2.5 . . − 11810931 S orw S w * +0.64933067 + 212270704 φ 6Sw * φ + 0.01375097 (ln ka )S w * 3

1.5

( A2 )

krow * = 1 − 0.7233267 S w * −17720584 . S w *2 + 150407908 . S w *3

( A3)

krw * = 0.28483482 S orw S w * − 0.0324527 S w * +0.07113168 S w *1.5 − 2.2461759 S wc Sorw1.7 S w *2

( A4)

krow * = 1 − 3090996 . S w * +2.8670229 S w *1.6 − 0.768952 S w *2

krw * = 0.22120304 S w *1.6 +0.24933592 S orw 2

( A5)

S w *3 . + 21370925 S w *2 S orw 5 φ

. (φ 2 S wc S orw S w *) 2 + 83.491972 φ 4 S w *5 S orw1.5 − 0.4562939 S wc 3S w *4 +116107198 − 8.7866012 S orw 3S w *2.3 +0.00000578 S w *3 ( S wc ln ka )10 (1 − S orw )0.4 − 12.841061 S w *2 ((ln ka )S wc )3 φ 6 krow * = 1 − 2.65253 S w * +2.4720911 S w *2 − 0.814367 S w *3

( A6 ) ( A7)

krw * = 01163954 S w *4 +2.66958338 S w *0.8 ( S wc S orw )2 + 0.47536676 S w *( S wc ln ka )2 . − 0.3912824 S w *( S wc ln ka )3 + 752.014909 ( S wc S w *)2 φ 3 − 398.40214 φ 2.5 S wc 2 S w *2.2 − 152.43629 (φ S orw )3 S w *2.7 +0.22964285 S orw 0.5 S w *

( A8 )

krow * = 1 − 2.985766 S w * + 31548084 . S w *2 − 1171486 . S w *3

( A9)

krw * = 0.2441795 S w * − 0.355058 S w *2 + 0.5117625 S w *3

( A10)

krow * = 1 − 4.985409 Sw * + 21322192 . Sw *2 − 29.04644 Sw *2.5 + 11723526 . Sw *3 ( A11)

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M. N. MOHAMAD IBRAHIM, L. F. KOEDERITZ

SPE 65631

krw * = 0.46689293 (ln ka )S wc S w * −0.0589939 S w * −01938748 . S w * φ 1.5 ( S orw ln ka )2 + 0.24253563 S w *3 −0195414 . S w *0.9 ( S wc ln ka )2 + 66.6228413 φ S orw S w * S wc 2 − 11126159 φ 1.5 ( S wc S w *)2.5 + 125.291504 S w * S orw 2.5φ 3 .

( A12 )

krow * = 1 − 3254725 . S w * +38176666 . S w *2 − 1563216 . S w *3

( A13)

krw * = 0.3643225 S w * − 0.7458182 S w *2 +106090802 . S w *3

( A14)

krow * = 1 − 8.6102768 S w * +87.9417721 S w *2 − 207.03656 S w *2.5 +187.099163 S w *3 − 60.388661 S w *3.5

( A15)

krw * = 0.2178721 S w *0.4 +0.00536612 (ln ka )2 S w *0.5 − 9.7494266 S w *0.7 ( S orw S wc )2 − 51295364 . S w * φ S orw 2 − 4.5717726 φ S w *0.5 S wc 2 + 0.57604803 S w *1.5

( A16)

krog * = 01599039 . S l * − 1045545 . Sl *2 +4.0843698 S l *3 −5.414161 S l *4 + 3.2103149 Sl *5

( A17)

krg * = 0.9396949 S g *2 −0.774167 S org S g *2 − 1216298 . S wc S g *2 +11628119 . φ S g *2 − 1248192 . S gc S g *2

( A18)

krog * = 4.465936 S l *2 − 0.252752 Sl * −22.93637 S l *3 + 53000956 . S l *4 −5519912 . S l *5 + 21917911 . Sl *6

( A19)

krg * = 0.3296593 S g *2 −0.001723 ka S g *2 +2.0568057 S wc S g * +12314265 . φS g *2 ( A20 )

krgw * = 13046802 . S g * −8159598 . S g *2 + 2550978 . S g *3 −3153754 . S g *4 + 13883828 . S g *5

( A21)

krw * = 0.94555376 S l * − 12967293 . Sl *1.7 + 169592185 . S l *3 −0.0424518 S gc (ln ka )3 S l *5 − 14583028 . S wc1.5 (φ S l *)2 + 0.02764389 S wcφ ( ka S gc )2 Sl *4

( A22)

SPE 65631

TWO-PHASE RELATIVE PERMEABILITY PREDICTION USING A LINEAR REGRESSION MODEL

krcg * = 01194373 . S l * −0.089246 S l *2 +0.9606793 S l *3

( A23)

krg * = 333929676 . S g *1.2 + 6.75670631 S g *0.9 S lc1.2 − 20.926791 S g * Slc 2 S gc + 101654474 . S g * S gc 4 − 7.3835856 Slc 0.5 S g *

( A24)

Table 1. Modified Craig's rule Swc

Rock Wettability Strongly

Sw at which krw * and krow* are equal

Water-Wet:

≥

15%

≥ 45%

Water-Wet:

≥

10%

≥

45%

Oil-Wet:

≤

15%

≤

55%

≥

10%

Intermediate: (Mixed-Wet)

≤ 15%

45%

≤ Sw ≤

krw* at Sw = 100-Sorw (fraction)

≤

0.07

0.07 < krw*

≥

≤ 0.3

0.5

55%

> 0.3

55%

< 0.5

OR

45%

≤ Sw ≤

7

8

M. N. MOHAMAD IBRAHIM, L. F. KOEDERITZ

SPE 65631

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

System

Wettability

Strongly Water-Wet

Lithology

Equation

Number of Data Sets

Number of Data Points

R2

R2adj

Sandstone

A1

16

127

87

86

A2

16

127

82

81

A3

6

49

99

99

A4

6

49

89

88

A5

102

870

93

93

A6

102

870

75

75

A7

28

317

94

94

A8

28

317

80

80

A9

43

396

93

93

A10

43

396

90

90

A11

29

278

93

93

A12

29

278

85

85

A13

31

245

95

95

A14

31

245

89

88

A15

19

184

86

86

A16

19

184

86

85

A17

98

962

95

95

A18

92

799

90

90

A19

14

133

94

94

A20

14

119

89

89

A21

19

144

89

89

A22

19

166

88

87

A23

17

123

95

95

A24

17

115

85

85

Carbonate

Sandstone Water-Wet Carbonate Oil-Water Sandstone Intermediate Wettability (Mixed-Wet)

Carbonate

Sandstone Oil-Wet Carbonate

Sandstone Gas-Oil Carbonate

Gas-Water

Gas-Condensate

All

All

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

Figure 1

Figure 2

Oil-Water Relative Permeability

Gas-Oil Relative Permeability

Water-Wet Rocks

Sandstone & Carbonate

1

1

φ = 15% ka = 100 0.8

0.8

Sandstone

md Sandstone

Swc = 15%

0.6

Carbonate

Sorg =15%

0.6

kr *

Carbonate

kr *

φ = 15% ka =100 md Swc =20% Sorw =20%

0.4

0.4

0.2 0.2

0 0

0.2

0.4 0.6 Sw (fraction)

0.8

1 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Sg (fraction)

Figure 3

Figure 4

Water-wet Sandstone

Water-Wet Sandstone

Oil Relative Permeability to Water

Water Relative Permeability

1 0.8 Equation A6

0.8

Honarpour's

0.6

0.4

φ = 15% ka = 100 md Swc = 20% Sorw = 20%

Honarpour's

Rose's

krw *

krow *

0.6 φ = 15% ka = 100 md Swc = 20% Sorw = 20%

Equation A5

0.4 Rose's

Narr's

0.2

Narr's

0.2

0 0

0.2

0.4

0.6

Sw (fraction)

0.8

1

0 0

0.2

0.4

0.6

Sw (fraction)

0.8

1

TWO-PHASE RELATIVE PERMEABILITY PREDICTION USING A LINEAR REGRESSION MODEL

Figure 5

Water-Wet System Carbonate 0.6

0.5

φ = 10% ka = 100 md Swc = 20% Sorw=20%

0.4

kr *

SPE 65631

0.3

WW

Strongly WW

Honarpour's WW

0.2

0.1

0 0

0.2

0.4

0.6

Sw (fraction)

0.8

1

11