Content uploaded by Saud M. Al-Fattah
Author content
All content in this area was uploaded by Saud M. Al-Fattah on Apr 17, 2014
Content may be subject to copyright.
Content uploaded by Saud M. Al-Fattah
Author content
All content in this area was uploaded by Saud M. Al-Fattah on Apr 12, 2014
Content may be subject to copyright.
Copyright 2003, Society of Petroleum Engineers, Inc.
This paper was prepared for presentation at the 2003 SPE Annual International Conference
and Exhibition held in Nigeria, Abuja, Aug. 4-6.
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 posi-
tion 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 com-
mercial purposes without the written consent of the Society of Petroleum Engineers is prohib-
ited. 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 acknowledgment 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
Relative permeability data are essential for almost all calcula-
tions of fluid flow in petroleum engineering. Water/oil relative
permeability curves play important roles in characterizing the
simultaneous two-phase flow in porous rocks and predicting
the performance of immiscible displacement processes in oil
reservoirs. This paper presents new empirical equations for
calculating water/oil imbibition relative permeability curves.
The models of relative permeability were developed using
experimental data from 46 displacement core tests from sand-
stone reservoirs of Saudi fields. Three empirical equations are
presented to calculate oil relative permeability, water relative
permeability, and the endpoint of the water relative permeabil-
ity curve. The relative permeability models were derived as a
function of rock and fluid properties using stepwise linear and
nonlinear regression analyses. The new empirical equations
were both evaluated using the data utilized in the development
and validated using published data, which were not used in the
development stage, against previously published equations.
Statistical results show that the new empirical equations de-
veloped in this study are in better agreement with experimen-
tal data than previous empirical equations, for both the data
used in the development and validation stages. The new em-
pirical equations can be used to determine water/oil relative
permeability curves for other fields provided the reservoir data
fall within the range of this study.
Introduction
Relative permeability
1-4
is an important concept in describing
the flow of multiphase systems. It is defined as the ratio of the
effective permeability of a fluid at a given saturation to the
absolute permeability of the rock. Data of relative permeabili-
ties are essential for almost all calculations of fluid flow in
petroleum reservoirs. The data are used in making engineering
estimates of productivity, injectivity, and ultimate recovery.
Some applications of relative permeability data include deter-
mination of free water surface, aid in evaluating drill-stem and
production tests, determination of residual fluid saturations,
fractional flow and frontal advance calculations to determine
the fluid distributions, and making future predictions for all
types of oil reservoir under different operational schemes.
Undoubtedly, these data are considered probably the most
valuable information required in reservoir simulation studies.
The producing gas-oil ratio and the producing water/oil ratio
are two criteria used in history matching, which can be modi-
fied by relative permeability changes. More accurate predic-
tion of relative permeability will reduce the trial and errors
needed to improve the history matching.
Estimates of relative permeabilities are generally obtained
from laboratory experiments with reservoir core samples using
one of the measurement methods: steady state or unsteady
state techniques. The relative permeability data may also be
determined from field data using the production history of a
reservoir and its fluid properties. However, this approach is
not often recommended because it requires complete produc-
tion history data and provides average values influenced by
pressure and saturation gradients, differences in the depletion
stage, and saturation variations in stratified reservoirs. In addi-
tion, the agreement between laboratory determined relative
permeabilities and those calculated from production history is
generally poor. Because the laboratory measurement of rela-
tive permeabilities is rather delicate and time consuming, em-
pirical correlations are usually employed to reproduce experi-
mentally determined relative permeability curves, or to esti-
mate them when experimental data from core samples are not
available.
The purpose of this study is to develop new empirical
equations to predict imbibition water/oil relative permeability
characteristics using experimentally obtained data for sand-
stone reservoir rocks. Multiple linear and nonlinear least-
square regression techniques are applied on the new proposed
models utilizing the experimental rock and fluid saturation
data. The new empirical equations are evaluated against sev-
eral empirical equations published in the literature
4-11
using the
data utilized in the development, and validated using pub-
lished relative permeability data.
Experimental Data
Experimental water/oil imbibition relative permeability data
from different oil fields were gathered for the development of
SPE 85652
Empirical Equations for Water/Oil Relative Permeability in Saudi Sandstone Reservoirs
Saud M. Al-Fattah, SPE, Saudi Aramco
2 S.M. AL-FATTAH SPE 85652
empirical equations. The experimental data are exclusively for
sandstone reservoirs ranging from semi-consolidated to un-
consolidated, fine to very fine-grained, well-sorted rock types.
Steady state and unsteady state techniques of measuring wa-
ter/oil relative permeability were employed on 46 waterflood
core tests in obtaining a total of 827 experimental data points.
Most of the waterflood tests were conducted using the un-
steady state method. A summary of core samples' properties
and test conditions used in the measurement of relative perme-
ability values is given in Table 1. The obtained experimental
data showed a wide range of core length, injection rate of dis-
placing fuid, viscosity and viscosity ratios of fluids, and tem-
perature.
TABLE 1– STATISTICAL DATA DESCRIPTION OF
CORE SAMPLES AND FLUID PROPERTIES
Property Min. Max. Mean
Std
Dev
Core length, cm 4.39 31.55 14.40 10.99
Area of core, cm
2
10.01 11.58 10.96 0.340
Porosity, % 23.07 33.80 28.55 2.680
Pore volume, cm
3
13.80 105.19 43.32 32.05
Displacement rate,
cm
3
/min
1.56 9.02 6.38 2.233
Scaling factor, Lµν,
cm
2
cp/min
1.00 4.98 2.37 0.986
Water viscosity, cp 0.387 1.073 0.62 0.186
Oil viscosity, cp 1.10 16.00 6.35 4.070
Temperature during
flood,
o
F
74.0 165.0 133.3 28.65
The experimental data were checked for the capillary end
effects using the criteria of Rapoport and Leas
12
. The scaling
factor was calculated for each displacement test and the ob-
tained results (Table 1) were within the range of published
experimental values. Relative permeability curves were gener-
ated utilizing the experimental displacement data and core
properties, and using a computer program that implements the
JBN method
13
to calculate relative permeabilities from un-
steady-state two-phase flow in displacement experiment.
Development of Empirical Equations
Non-linear and linear multiple least-square regression analysis
procedures were applied on the experimental data using SAS
software package
14
. Nonlinear and linear least-square regres-
sion Fortran programs
15
were developed and used in this
study. Several model selection techniques including stepwise
regression and R-square were used in selecting the best regres-
sion equations from the specified set of parameters. All these
selection methods showed excellent agreements of regression
results and the stepwise regression technique was selected for
the final results.
Table 2 shows the range of rock and fluid saturation prop-
erties of the 46 sets of imbibition water/oil relative permeabil-
ity curves for sandstone reservoir rocks utilized in the regres-
sion analysis.
TABLE 2– RANGES OF ROCK AND FLUID
SATURATION PROPERTIES
Property Min Max
Water saturation, %
11.728 93.812
Connate water saturation, %
11.728 38.556
Residual oil saturation, %
6.188 39.850
Effective oil permeability at S
w
i
, md
25.30 4790.0
Effective water permeability at S
o
r
,md
4.329 1084.1
Oil Relative Permeability Model. The oil relative permeabil-
ity model was derived as a function of rock and fluid satura-
tion properties.
)
)(
,,,(
wi
S
ro
k
or
S
wi
S
w
Sf
ro
k =
(1)
The best empirical equation for estimating oil relative
permeability curve was found as:
k
r
o
=k
ro(Sw
i
)
−
−
i
w
S
w
S
1
1
3.661763
−−
−−
r
o
S
i
w
S
r
o
S
w
S
1
1
0.7
(2)
The above equation was developed using multiple linear
least-square regression applying the appropriate transforma-
tion, to generate the coefficients of the linearized model. The
physical parameter, k
ro(Sw
i
)
, usually has the value of 1. The
above equation satisfies exactly the requirements that:
(i) at S
w
= S
w
i
, k
r
o
= k
ro(Sw
i
)
, and
(ii) at S
w
= 1- S
o
r
, k
r
o
= 0
Equation (2) was developed with a correlation coefficient
of 0.979 indicating that 98% of the data variation about the
mean is explained by the model. Analysis of the significance
of the independent parameters of the model is presented in
Table 3. All the considered independent variables have very
small probability values indicating very low probability of not
being significant in the model.
SPE 85652 EMPIRICAL EQUATIONS FOR WATER/OIL RELATIVE PERMEABILITY IN SAUDI SANDSTONE RESERVOIRS 3
TABLE 3–T-TEST FOR REGRESSION
COEFFICIENTS OF OIL RELATIVE PERMEABILITY
MODEL
Independent variable T for H0:
Parameter = 0
Prob > |T|
Ln
1 - S
w
1 - S
w
i
24.655 0.0001
Ln
1 - S
w
- S
o
r
1 - S
w
i
- S
o
r
10.896 0.0001
Figure 1 shows the error distribution histogram con-
structed as the deviations frequency versus the residual for this
study's oil relative permeability empirical equation. Most of
the errors distributed closely around the mean of zero while
less than 1% of data deviations occur at the residual extremes.
Water Relative Permeability Model. The water relative
permeability model was developed as a function of fluid satu-
rations and porosity following the work of Chierici
16
for water
relative permeability.
)
)(
,,,(
or
s
rw
k
or
S
wi
S
w
Sf
rw
k = (3)
The following empirical equation was found to reproduce
accurately the experimental water relative permeability data:
−
−−
−
−
⋅=
46163.0
1
05086.1
)(
or
S
w
S
wi
S
w
S
Exp
sor
rw
k
rw
k
(4)
Nonlinear least-square regression was applied on the ex-
perimental data to generate the correlating coefficients of the
model. The first exponential term of equation (4), water rela-
tive permeability at residual oil saturation, represents very
accurately the end point of water relative permeability curve.
A new empirical equation is developed, Eq. (5), to calculate
the relative permeability of water at residual oil saturation. It
was derived as a function of residual oil saturation and poros-
ity and it individually contributes for more than 54% of the
improvement of accuracy of the entire model. Dividing the
water relative permeability data by this term can normalize the
water relative permeability curve. Equation (5) was developed
with a correlation coefficient of 0.919 using regression analy-
sis.
()
[]
or
SExp
or
s
rw
k −−= 141463.5
)(
φ
(5)
The water relative permeability model implicitly satisfies
the initial and end points of water relative permeability curves:
(i) At S
w
= S
w
i
,
r
o
S
w
S
i
w
S
w
S
−−
−
1
= 0, and e
-∞
= 0
implying that k
r
w
= 0
(ii) At S
w
= 1 - S
o
r
,
S
w
- S
w
i
1 - S
w
- S
o
r
= ∞, and e
0
= 1,
implying that k
r
w
= Exp[-5.41463φ(1-S
or
)]
The empirical equation obtained using this model agrees
very closely with experimental data. Equation (4) was devel-
oped with a correlation coefficient of 0.9304 implying that
93% of the data variation around the zero mean is accounted
for by the model. The significance F-test statistic for this
model is 2433.0 with “ Prob > F ” value of 0.0001 indicates
that all the independent variables included in the considered
model contribute significantly to the improvement of the
model. Figure 2 shows the error distribution histogram con-
structed as the deviations frequency vs. the residual for water
relative permeability empirical equation. Most of the errors
distributed closely around the mean of zero while less than 1%
of data deviations occur at the residual extremes.
Behavior of Models
It is important that the dependent variables of regression mod-
els comply with the behavior of their independent correlating
variables. The behaviors of regression models of this study
were examined against their correlating parameters of physical
properties. The average experimental values of irreducible
water saturation, residual oil saturation, and porosity were
used to calculate water and oil relative permeabilities from
regression models for the range of water saturation. Figure 3
depicts a typical behavior of water/oil relative permeability
curves as generated by the models of this study. For the same
case, a semilog plot of water/oil relative permeability curves is
generated in Fig. 4. Figure 5 compares core sample measured
data of relative permeability with empirical equations devel-
oped in this study.
Parametric analysis was performed to investigate the effect
of other physical parameters, which are not included in the
models, on the characteristics of oil and water relative perme-
ability curves. The physical parameters considered are tem-
perature, porosity, and type of measurement method. Unlike
for oil relative permeability model, the porosity was found a
strong independent variable to be included in the water rela-
tive permeability model particularly in estimating the end-
point. This indicates that the porosity has an influence on the
characteristic of water relative permeability curve. Tempera-
ture and classifying data by type of measurement method were
found to have almost no effect. Reference 21 gives the details.
4 S.M. AL-FATTAH SPE 85652
Evaluation of Published Empirical Equations
The new developed empirical equations were evaluated
against six published empirical equations for checking their
performance and degree of accuracy in estimating the wa-
ter/oil relative permeability curves. These published empirical
equations
4-11
include those developed by Wyllie, Pirson, Naar
et al., Jones, Land, and Honarpour et al. Two error analysis
techniques were used: statistical and graphical error analysis.
Statistical Error Analysis. The statistical parameters used to
compare the degree of accuracy of the water/oil relative per-
meability empirical equations are: average error, absolute av-
erage error, standard deviation, root-mean-square, minimum
and maximum absolute average error, and the F-test statistic.
17
These parameters were computed for the published empirical
equations using the 827 experimentally obtained data points.
Table 4 shows the statistical accuracy of empirical equations
for oil relative permeability. The empirical equation of this
study achieved the lowest errors, standard deviation and root-
mean-square error, with the highest F-test statistic. Empirical
equation of Honarpour et al. stood second in the accuracy of
errors but with lower F-test statistic than Wyllie's empirical
equation. Naar et al.'s empirical equation showed poor accu-
racy, with the highest errors and the lowest F-test statistic.
TABLE 4–STATISTICAL ACCURACY OF OIL
RELATIVE PERMEABILITY EMPIRICAL EQUATIONS
Author Year E
av
E
ab
s
2
rms E
max
F-test
Wyllie 1951
-0.10
0.10 0.02 0.13 0.56 3160
Pirson 1958
-0.13
0.14 0.02 0.15 0.35 676
Naar 1961
-0.22
0.22 0.07 0.26 0.70 450
Jones 1966
-0.15
0.15 0.03 0.18 0.59 644
Land 1968
-0.09
0.09 0.02 0.13 0.59 548
Honar-
pour
1982
-0.06
0.06 0.01 0.09 0.54 1622
Eq. (2) 1994
0.00
0.04 0.00 0.07 0.48 9029
The statistical accuracy of empirical equations for water
relative permeability is given in Table 5. The empirical equa-
tion of this study again achieved the highest accuracy of esti-
mating water relative permeability curves with the lowest er-
rors and highest F-test statistic. Honarpour et al.'s empirical
equation for oil-wet and intermediate wettability stood second
in the accuracy of errors but with unsatisfactory F-test statis-
tic. Naar et al., Jones and Land have all the same empirical
equation and hence have the same unsatisfactory performance
of accuracy. Wyllie's empirical equation showed similar per-
formance of errors to Honarpour et al.'s empirical equation for
oil-wet and intermediate wettability, but with better F-test
statistic. Pirson's empirical equation showed poor performance
of accuracy with the highest errors, and Honarpour et al.'s em-
pirical equation (for water-wet and intermediate) yielded the
lowest F-test statistic.
TABLE 5–STATISTICAL ACCURACY OF WATER
RELATIVE PERMEABILITY EMPIRICAL EQUATIONS
Author Year E
av
E
ab
s
2
rms E
max
F-
test
Wyllie 1951 0.38 0.39 0.19 0.44 0.89 693
Pirson 1958 0.51 0.51 0.33 0.58 1.00 317
Naar * 1961 0.44 0.44 0.24 0.49 0.95 672
Hanarpour,
water-wet
1982 0.45 0.45 0.27 0.52 0.93 128
Honarpour,
oil-wet
1982 0.37 0.37 0.18 0.42 0.83 170
Eq. (4) 1994 0.02 0.07 0.01 0.10 0.46 2433
* Same results also apply to equations of Jones, and Land.
Graphical Error Analysis. The crossplot graphical error
analysis technique is used in the evaluation of performance of
this study and other published empirical equations.
Figures 6 through 12 show the crossplots of estimated vs.
measured values for oil relative permeability empirical equa-
tions. Most of the plotted points of this study's empirical equa-
tion fall very close to the perfect 45º straight line. All other
published empirical equations overestimate the experimental
data of oil relative permeabilities with high deviations from
the perfect 45º line.
Crossplots for water relative permeability empirical equa-
tions are shown in Figs. 13 through 18. The closeness of the
data points to the perfect 45º line for this study's empirical
equation is obvious. All other published empirical equations
reveal their underestimation of the experimental data of water
relative permeability.
Validation of New Empirical Equations
The new developed empirical equations for water/oil imbibi-
tion relative permeability calculations were validated using
published relative permeability data in the literature that were
not utilized in the models developed in this study. This proce-
dure was undertaken to examine the applicability of the new
empirical equations and evaluate their performance of accu-
racy against previously published empirical equations. A total
of 45 data points from four published sets
18-21
of water/oil rela-
tive permeability were used for validation of empirical equa-
tions. These data are summarized in Table 6.
SPE 85652 EMPIRICAL EQUATIONS FOR WATER/OIL RELATIVE PERMEABILITY IN SAUDI SANDSTONE RESERVOIRS 5
TABLE 6–SUMMARY OF PUBLISHED DATA USED
FOR VALIDATION OF EMPIRICAL EQUATIONS
Source Field Data
Points
Data
Sets
Willhite Chesney
MP-4
28 2
Braun & Blackwell Berea 8 1
Jones & Roszelle - 9 1
The new empirical equations were compared with pub-
lished empirical equations of Wyllie, Pirson, Naar et al.,
Jones, Land, and Honarpour et al. Each set of published data
was used for comparison of published empirical equations
with those developed in this study. The results showed that the
new empirical equations reproduced very accurately experi-
mental relative permeability data than other published equa-
tions. Table 7 shows results of statistical accuracy for oil rela-
tive permeability models using the compiled published data
sets. Empirical equations of oil relative permeability were
compared in terms of average error, average absolute error,
standard deviation, maximum absolute error, and coefficient
of determination. The oil relative permeability model of this
study yields better results of accuracy than other published
models for oil relative permeability by showing lower values
of errors and higher coefficient of determination. Empirical
equations of Pirson and Honarpour et al. show similar per-
formance of accuracy; however, Pirson’s equation has a higher
coefficient of determination than Honarpour’s et al. does. Em-
pirical equation of Naar et al. performs poorly compared to
other empirical equations.
TABLE 7–STATISTICAL ACCURACY OF OIL
RELATIVE PERMEABILITY EMPIRICAL EQUATIONS
Author Year E
av
E
ab
s
2
Rms
E
max
R
2
Wyllie 1951
-0.14
0.14 0.03 0.16 0.30 0.70
Pirson 1958
-0.10
0.13 0.02 0.15 0.30 0.91
Naar 1961
-0.28
0.28 0.10 0.31 0.51 0.12
Jones 1966
-0.19
0.19 0.05 0.22 0.38 0.44
Land 1968
-0.09
0.09 0.02 0.13 0.33 0.79
Honar-
pour
1982
0.01
0.05 0.01 0.09 0.39 0.91
Eq. (2) 1994
-0.01
0.04 0.01 0.07 0.23 0.94
Table 8 gives statistical accuracy of results for water rela-
tive permeability models using compiled sets of published
data. Empirical equations of water relative permeability were
compared in terms of average error, average absolute error,
standard deviation, maximum absolute error, and F-test statis-
tic. Inclusion of r
2
parameter in this error analysis was imprac-
tical because small values of F-test given by empirical equa-
tions of Honarpour et al. and Pirson resulted in negative r
2
values. It should be noted that the F-test statistic has a strong
direct relation with the coefficient of determination. The high-
est performance of accuracy is achieved by this study which
gives lower errors and higher F-test statistic than the results
obtained for other studies.
TABLE 8–STATISTICAL ACCURACY OF WATER
RELATIVE PERMEABILITY EMPIRICAL EQUATIONS
Author Year E
av
E
ab
s
2
rms E
max
F-
test
Wyllie 1951 0.05 0.05 0.01 0.08 0.21 72
Pirson 1958 0.12 0.12 0.03 0.16 0.45 11
Naar * 1961 0.08 0.08 0.01 0.11 0.30 34
Hanarpour,
water-wet
1982 0.08 0.09 0.02 0.12 0.36 4
Honarpour,
oil-wet
1982 0.00 0.05 0.01 0.07 0.18 19
Eq. (4) 1994 0.00 0.02 0.00 0.03 0.07 368
* Same results also apply to equations of Jones, and Land.
Conclusions
Multiple linear and nonlinear least-square regression analyses
were applied on collected laboratory experimental data for
development of empirical equations for water/oil relative per-
meability. The following conclusions were reached as a result
of this study:
1. New empirical models are presented for calculating im-
bibition water/oil relative permeability curves of sandstone
reservoir rocks. These empirical equations can predict relative
permeability in sandstones with properties falling within the
range of data used in this development.
2. The newly developed empirical equations satisfy exactly
the initial and end points requirements of water/oil relative
permeability curves. The endpoint water relative permeability
empirical equation developed in this study was found to re-
produce accurately experimental values.
3. Comparative evaluation of existing empirical equations
was made and the new empirical equations give better results
of accuracy than previously published equations for the data
used in this study.
4. The new empirical equations were validated using pub-
lished relative permeability data. They showed better results of
accuracy in estimating measured relative permeability data
than other published equations.
5. Parametric analysis showed that porosity has an influ-
ence on the characteristic of water relative permeability curve,
and therefore it should be included in the model for better pre-
diction of the endpoint in particular. Temperature and the type
of measurement method of relative permeability data showed
insignificant effects on water and oil relative permeability
curves.
6 S.M. AL-FATTAH SPE 85652
Nomenclature
E
= error or residual
Exp = exponential function (Exp x = e
x
)
E
ab
= average absolute error
E
av
= average error
E
max
= maximum absolute error
E
min
= minimum absolute error
F = F-test statistic
k
ro
= relative permeability to oil, fraction
k
rw
= relative permeability to water, fraction
k
ro(Swi)
= relative permeability to oil at irreducible water
saturation, frac.
k
rw(Sor)
= relative permeability to water at residual oil
saturation, frac.
n = number of data points
r = correlation coefficient
R
2
= coefficient of determination
Rms = root mean square
S
o
= oil saturation, fraction
S
or
= residual oil saturation, fraction
S
w
= water saturation, fraction
S
wf
= water saturation at breakthrough, fraction
S
wi
= irreducible water saturation, fraction
s = standard deviation
t = T-test statistic
φ
= porosity, fraction
Subscripts
est = estimated value
exp = experimental value
o = oil phase
w = water phase
Acknowledgments
Thanks to Saudi Aramco management for their permission to
publish this paper. I am grateful to M.A. Al-Marhoun, King
Fahd University of Petroleum and Minerals, for his valuable
comments and review of the manuscript. I also would like to
thank K. Al-Fossail and H. Al-Yousef, King Fahd University
of Petroleum and Minerals, for their comments.
References
1. Amyx, J.W., Bass, D.M. Jr., and Whiting, R.L.: Petroleum Reser-
voir Engineering, McGraw-Hill Book Co., New York (1960).
2. Frick, T.: Petroleum Production Handbook, SPE of AIME, Dal-
las, TX. (1962).
3. Heaviside, J., Black, C.J.J. and Berry, J.F.: “Fundamentals of
Relative Permeability: Experimental and Theoretical Considera-
tions,” paper SPE 12173 presented at the 1983 SPE Annual
Technical Conference and Exhibition, San Francisco, CA, Oct. 5-
8.
4. Honarpour, M., Koederitz, L., and Harvey, A.H.: Relative Perme-
ability of Petroleum Reservoirs, CRC Press Inc., Boca Raton
(1986).
5. Wyllie, M.R.J.: “Interrelationship between Wetting and Nonwet-
ting Phase Relative Permeability,” Trans., AIME (1950) 192,
381-82.
6. Pirson, S.J.: Oil Reservoir Engineering, McGraw-Hill Book Co.
Inc., New York City (1958).
7. Naar, J. and Henderson, J.H.: “An Imbibition Model - Its Applica-
tion to Flow Behavior and the Prediction of Oil Recovery,” SPEJ
(June 1961) 61-70; Trans., AIME, 222.
8. Naar, J. and Wygal, R.J.: “Three-Phase Imbibition Relative Per-
meability,” SPEJ (Dec. 1961) 254-58; Trans., AIME, 222.
9. Naar, J., Wygal, R.J. and Henderson, J.H.: “Imbibition Relative
Permeability in Unconsolidated Porous Media,” SPEJ (March
1962) 254-58; Trans., AIME, 223.
10. Land, C.S.: “Calculation of Imbibition Relative Permeability for
Two- and Three-Phase Flow from Rock Properties” SPEJ (June
1968) 149-56.
11. Honarpour, M., Koederitz, L., and Harvey, A.H.: “Empirical
Equations for Estimating Two-Phase Relative Permeability in
Consolidated Rock,” JPT (Dec. 1982) 2905-08.
12. Rapoport, L.A. and Leas, W.J.: “Properties of Linear Water-
floods,” JPT (May 1953) 139-48; Trans., AIME, 198.
13. Johnson, E.F., Bossler, D.P., and Naumann, V.O.: “Calculation of
Relative Permeability from Displacement Experiments,” Trans.,
AIME (1959) 216, 370-72.
14. SAS/STAT User's Guide, Release 6.03 ed., SAS Institute Inc.,
Cary, NC (1988).
15. Press, W. H., Teukolsky, S.A., Vetterling, W.T. and Flannery,
B.P.: Numerical Recipes in Fortran, Second ed., Cambridge
Univ. Press (1992).
16. Chierici, G.L.: “Novel Relations for Drainage and Imbibition
Relative Permeabilities,” SPEJ (June 1984) 275-76.
17. Walpore, R.E. and Myers, R.H.: Probability and Statistics for
Engineers and Scientists, 5
th
ed., Macmillan Pub. Co., New York
(1993).
18. Willhite, G.P.: Waterflooding, Textbook Series, SPE, Richardson,
TX (1986) 3.
19. Braun, E.M., and Blackwell, R.J.: “A Steady-State Technique for
Measuring Oil-Water Relative Permeability Curves at Reservoir
Conditions,” paper SPE 10155 presented at the 1981 SPE 56
th
Annual Technical Conference and Exhibition, San Antonio, TX,
Oct. 5-7.
20. Jones, S.C., and Roszelle, W.O.: “Graphical Techniques for De-
termining Relative permeability from Displacement Experi-
ments,” JPT (May 1978) 807-817.
21. Al-Fattah, S.M.: “Development of Empirical Equations for Wa-
ter/Oil Relative Permeability,” MS Thesis, King Fahd University
of Petroleum and Minerals, Dhahran, Saudi Arabia (Dec. 1994).
SPE 85652 EMPIRICAL EQUATIONS FOR WATER/OIL RELATIVE PERMEABILITY IN SAUDI SANDSTONE RESERVOIRS 7
-
0.
47
5
-
0.4
2
5
-
0.3
7
5
-
0
.32
5
-
0.
27
5
-
0
.225
-
0
.175
-
0
.12
5
-
0.0
7
5
-
0
.025
0
.
02
5
0
.
07
5
0
.
125
0.
1
75
0.
22
5
0
.
27
5
0.
3
25
0.
3
75
0
.
425
0.475
Residual, fraction
0
50
100
150
200
250
300
350
400
Frequency
Fig. 1- Error distribution plot for oil relative permeability correla-
tion (This study).
-0.475
-0.425
-0.375
-0.3
2
5
-
0
.2
7
5
-
0
.2
25
-
0
.1
75
-0.
1
25
-0.075
-0.025
0
.0
25
0
.
075
0
.1
25
0
.1
75
0
.2
25
0
.27
5
0.32
5
0.375
0
.
425
0
.
475
Residual, fraction
0
50
100
150
200
250
Frequency
Fig. 2- Error distribution plot for water relative permeability corre-
lation (This study).
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Water saturation
,
fraction
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Relative permeability, fraction
Normalize
d
E
q
.
(
4
)
E
q
.
(
2
)
Fig. 3- Behavior of water/oil relative permeability models against
their physical correlating properties, sandstone Saudi field.
Fig. 4- Semilog plot of water/oil relative permeability from empiri-
cal equations, Saudi sandstone reservoir.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Sw, fraction
0.001
0.01
0.1
1
kr
, fraction
Normalized
Eq.(4)
Eq.(2)
8 S.M. AL-FATTAH SPE 85652
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Sw, fraction
kr, fraction
Measuerd data
This study
Fig. 5- Water/oil relative permeability measured data from core
sample compared to empirical equations of this study.
0.0001
0.001
0.01
0.1
1
0.0001 0.001 0.01 0.1 1
Measured oil relative permeability
Estimated oil relative permeabili
t
Fig. 6- Crossplot for oil relative permeability empirical equation
(Wyllie).
0.0001
0.001
0.01
0.1
1
0.0001 0.001 0.01 0.1 1
Measured oil relative permeability
Estimated oil relative permeabili
t
Fig. 7- Crossplot for oil relative permeability empirical equation
(Pirson).
0.0001
0.001
0.01
0.1
1
0.0001 0.001 0.01 0.1 1
Measured oil relative permeability
Estimated oil relative permeabili
t
Fig. 8- Crossplot for oil relative permeability empirical equation
(Naar et al.).
SPE 85652 EMPIRICAL EQUATIONS FOR WATER/OIL RELATIVE PERMEABILITY IN SAUDI SANDSTONE RESERVOIRS 9
0.0001
0.001
0.01
0.1
1
0.0001 0.001 0.01 0.1 1
Meas ured oil relative permeability
Estimated oil relative permeabili
t
Fig. 9- Crossplot for oil relative permeability empirical equation
(Jones).
0.0001
0.001
0.01
0.1
1
0.0001 0.001 0.01 0.1 1
Meas ured oil relative permeability
Estimated oil relative permeabili
t
Fig. 10- Crossplot for oil relative permeability empirical equation
(Land).
0.0001
0.001
0.01
0.1
1
0.0001 0.001 0.01 0.1 1
Meas ured oil relative permeability
Estimated oil relative permeabili
t
Fig. 11- Crossplot for oil relative permeability empirical equation
(Honarpour et al.).
0.0001
0.001
0.01
0.1
1
0.0001 0.001 0.01 0.1 1
Measured oil relative permeability
Estimated oil relative permeabili
t
Fig. 12- Crossplot for oil relative permeability empirical equation
(This study).
0.0001
0.001
0.01
0.1
1
0.0001 0.001 0.01 0.1 1
Measured water relative permeability
Estimated water relative permeabilit
y
Fig. 13- Crossplot for water relative permeability empirical equa-
tion (Wyllie).
0.0001
0.001
0.01
0.1
1
0.0001 0.001 0.01 0.1 1
Meas ured water relative permeability
Estimated water relative permeabilit
y
Fig. 14- Crossplot for water relative permeability empirical equa-
tion (Pirson).
10 S.M. AL-FATTAH SPE 85652
0.0001
0.001
0.01
0.1
1
0.0001 0.001 0.01 0.1 1
Measured water relative permeability
Estimated water relative permeabilit
y
Fig. 15- Crossplot for water relative permeability empirical equa-
tion (Naar et al.).
0.0001
0.001
0.01
0.1
1
0.0001 0.001 0.01 0.1 1
Measured water relative permeability
Estimated water relative permeabilit
y
Fig. 16- Crossplot for water relative permeability empirical equa-
tion (Honarpour et al., water-wet).
0.0001
0.001
0.01
0.1
1
0.0001 0.001 0.01 0.1 1
Measured water relative permeability
Estimated water relative permeabilit
y
Fig. 17- Crossplot for water relative permeability empirical equa-
tion (Honarpour et al., oil-wet).
0.0001
0.001
0.01
0.1
1
0.0001 0.001 0.01 0.1 1
Measured water relative permeability
Estimated water relative permeabilit
y
Fig. 18- Crossplot for water relative permeability empirical equa-
tion (This study).
