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Equations for Water/Oil Relative Permeability in Saudi Arabian Sandstone Reservoirs

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Al-Fattah is a petroleum engineering systems analyst in the Reservoir Engineering Systems Division, Petroleum Engineering Applications Services Dept. (PEASD). He previously worked as a reservoir engineer in the Reservoir Management Department and as a PE systems analyst in the Simulation Systems Division, PEASD. His interests include reservoir engineering, artificial intelligence, operations research, and energy economics and forecasting. In 2000, Al-Fattah completed a PhD degree at Texas A&M University. In 1994, he obtained an MS degree, and in 1992 a BS degree with honors from King Fahd University of Petroleum and Minerals, all in petroleum engineering. Al-Fattah is a member of the Society of Petroleum Engineers and the Dhahran Geoscience Society. ABSTRACT Relative permeability data are essential for almost all calculations 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 sandstone reservoirs of Saudi fields. Three empirical equations are presented to calculate oil relative permeability, water relative permeability and the endpoint of the water relative permeability 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, that were not used in the development stage against previously published equations. Statistical results show that the new empirical equations developed in this study are in better agreement with experimental data than previous empirical equations for the data used in the development and validation stages. The new empirical 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.
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48 SAUDI ARAMCO JOURNAL OF TECHNOLOGY SUMMER 2004
EQUATIONS FOR WATER/OIL
RELATIVE PERMEABILITY IN SAUDI
ARABIAN SANDSTONE RESERVOIRS
Saud Mohammad Al-Fattah
Al-Fattah is a petroleum engineering systems analyst in
the Reservoir Engineering Systems Division, Petroleum
Engineering Applications Services Dept. (PEASD). He
previously worked as a reservoir engineer in the Reservoir
Management Department and as a PE systems analyst in
the Simulation Systems Division, PEASD. His interests
include reservoir engineering, artificial intelligence,
operations research, and energy economics and
forecasting. In 2000, Al-Fattah completed a PhD degree
at Texas A&M University. In 1994, he obtained an MS
degree, and in 1992 a BS degree with honors from King
Fahd University of Petroleum and Minerals, all in
petroleum engineering. Al-Fattah is a member of the
Society of Petroleum Engineers and the Dhahran
Geoscience Society.
ABSTRACT
Relative permeability data are essential for almost all calculations 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 sandstone reservoirs of
Saudi fields. Three empirical equations are presented to calculate oil relative
permeability, water relative permeability and the endpoint of the water relative
permeability 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, that were not used in the
development stage against previously published equations. Statistical results show
that the new empirical equations developed in this study are in better agreement
with experimental data than previous empirical equations for the data used in the
development and validation stages. The new empirical 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.
SAUDI ARAMCO JOURNAL OF TECHNOLOGY SUMMER 2004 49
INTRODUCTION
Relative permeability is an important concept in describing
the flow of multiphase systems (Amyx, J.W., D.M. Bass, Jr.,
and R.L. Whiting, 1960); (Frick, T., 1962); (Heaviside, J.,
C.J.J. Black, J.F. and Berry, 1983); and (Honarpour, M., L.
Koederitz, and A.H. Harvey, 1986). 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
permeabilities is essential for almost all calculations of fluid
flow in petroleum reservoirs. The data is used in making
engineering estimates of productivity, injectivity and ultimate
recovery. Some applications of relative permeability data
include determining the free water surface, aiding in
evaluating drill-stem and production tests, determining the
residual fluid saturations, calculating the fractional flow and
frontal advance to determine the fluid distribution and
making future predictions for all types of oil reservoirs under
different operational schemes. Undoubtedly, this data is
considered the most valuable information required for
reservoir simulation studies. The producing gas/oil ratio and
the producing water/oil ratio are two criteria used in history
matching, which can be modified by relative permeability
changes. More accurate prediction of relative permeability
will reduce the trial and error 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 the complete production history data and
provides average values influenced by pressure and
saturation gradients, differences in the depletion stage and
saturation variations in stratified reservoirs. In addition, the
agreement between laboratory-determined relative
permeabilities and those calculated from production history
is generally poor. Because the laboratory measurement of
relative permeabilities is rather delicate and time
consuming, empirical correlations are usually employed to
reproduce experimentally determined relative permeability
curves or to estimate them when experimental data from
core samples is 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 sandstone 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 several empirical equations published in
the literature (Honarpour, M., L. Koederitz, and A.H.
Harvey, 1986); (Wyllie, M.R.J., 1950); (Pirson, S.J., 1958);
(Naar, J. and J.H. Henderson, 1961); (Naar, J. and R.J.
Wygal, 1961); Naar, J., R.J. Wygal and J.H. Henderson,
1962); (Land, C.S., 1968); and (Honarpour, M., L.
Koederitz and A.H. Harvey, 1982) using the data utilized in
the development and validated using published relative
permeability data.
EXPERIMENTAL DATA
Experimental water/oil imbibition relative permeability data
from different oil fields was gathered for the development
of empirical equations. The experimental data is exclusively
for sandstone reservoirs ranging from semi-consolidated to
unconsolidated, from fine to very fine-grained, well-sorted
rock types. Steady state and unsteady state techniques for
measuring water/oil relative permeability were employed on
46 waterflood core tests to obtain a total of 827
experimental data points. Most of the waterflood tests were
conducted using the unsteady state method. A summary of
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, cm210.01 11.58 10.96 0.340
Porosity, % 23.07 33.80 28.55 2.680
Pore volume, cm313.80 105.19 43.32 32.05
Displacement rate, cm3/min 1.56 9.02 6.38 2.233
Scaling factor, L, cm2cp/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, °F 74.0 165.0 133.3 28.65
50 SAUDI ARAMCO JOURNAL OF TECHNOLOGY SUMMER 2004
core samples’ properties and test conditions used in the
measurement of relative permeability values is given in table
1. The obtained experimental data showed a wide range of
core lengths, injection rates of displacing fluid, viscosity and
viscosity ratios of fluids, and temperatures.
The experimental data was checked for the capillary end
effects using the criteria of Rapoport, L.A. and W.J. Leas,
(1953). The scaling factor was calculated for each
displacement test, and the obtained results shown in table 1
were within the range of published experimental values.
Relative permeability curves were generated utilizing the
experimental displacement data and core properties, and
using a computer program that implements the JBN method
(Johnson, E.F., D.P. Bossler and V.O. Naumann, 1959) to
calculate relative permeabilities from unsteady state two-
phase flow in the displacement experiment.
DEVELOPMENT OF EMPIRICAL EQUATIONS
Non-linear and linear multiple least-square regression
analysis procedures were applied on the experimental data
using the SAS software package (SAS/STAT User’s Guide,
1988). Non-linear and linear least-square regression Fortran
programs (Press, W. H., S.A. Teukolsky, W.T. Vetterling and
B.P. Flannery, 1992) were developed and used in this study.
Several model selection techniques including stepwise
regression and R-square were used in selecting the best
regression equations from the specified set of parameters.
All these selection methods showed excellent agreement of
regression results, and the stepwise regression technique was
selected for the final results.
Table 2 shows the range of rock and fluid saturation
properties of the 46 sets of imbibition water/oil relative
permeability curves for the sandstone reservoir rocks
utilized in the regression analysis.
OIL RELATIVE PERMEABILITY MODEL
The oil relative permeability model was derived as a
function of rock and fluid saturation properties.
(1)
The best empirical equation for estimating the oil relative
permeability curve was found as:
(2)
The above equation was developed using multiple linear
least-square regression, applying the appropriate
transformation to generate the coefficients of the linearized
model. The physical parameter, kro(Swi), usually has the
value of 1. The above equation exactly satisfies the
following requirements:
(i) at Sw= Swi, kro = kro(Swi); and
(ii) at Sw= 1- Sor, kro = 0
Equation (2) was developed with a correlation coefficient
of 0.979, indicating that 98 percent 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 a
very low probability of not being significant in the model.
Fig. 1 shows the error distribution histogram constructed
for the deviations frequency versus the residual for this
study’s oil relative permeability empirical equation. Most of
the errors are distributed closely around the mean of zero,
while less than 1 percent of data deviations occur at the
residual extremes.
WATER RELATIVE PERMEABILITY MODEL
The water relative permeability model was developed as a
function of fluid saturations and porosity following the
work of Chierici, G.L., (1984) for water relative
permeability.
(3)
The following empirical equation was found to
accurately reproduce the experimental water relative
permeability data:
Table 2. Ranges of rock and fluid saturation properties.
Table 3. T-Test for regression coefficients of the oil relative permeability model.
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 Swi, md 25.30 4790.0
Effective water permeability at Sor, md 4.329 1084.1
Independent variable T for H0:
Parameter = 0 Prob > |T|
Ln 1 - Sw
1 - Swi
24.655 0.0001
Ln 1 - Sw - Sor
1 - Swi - Sor
10.896 0.0001
SAUDI ARAMCO JOURNAL OF TECHNOLOGY SUMMER 2004 51
(4)
Nonlinear least-square regression was applied on the
experimental data to generate the correlating coefficients of
the model. The first exponential term of equation (4), water
relative permeability at residual oil saturation, represents
very accurately the end point of the water relative
permeability curve. A new empirical equation is developed,
equation (5), to calculate the relative permeability of water
at residual oil saturation. It was derived as a function of
residual oil saturation and porosity, and it individually
contributes more than 54 percent of the improvement in
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 analysis.
(5)
The water relative permeability model implicitly satisfies
the initial and end points of water relative permeability
curves:
The empirical equation obtained using this model agrees
very closely with experimental data. Equation (4) was
developed with a correlation coefficient of 0.9304, implying
that 93 percent 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 the “ Prob > F ”
value of 0.0001, indicating that all the independent
variables included in the considered model contribute
significantly to the improvement of the model. Fig. 2 shows
the error distribution histogram constructed for the
deviation frequency vs. the residual for water relative
permeability empirical equation. Most of the errors were
distributed closely around the mean of zero, while less than
1 percent of data deviations occur at the residual extremes.
BEHAVIOR OF MODELS
It is important that the dependent variables of regression
models 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. Fig. 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.
Fig. 5 compares the core sample measured data of
relative permeability with empirical equations developed
in this study.
Parametric analysis was performed to investigate the
effect of other physical parameters, which are not included
Fig. 1. Error distribution plot for oil relative permeability correlation
(this study).
-0.475
-0.425
-0.375
-0.325
-0.275
-0.225
-0.175
-0.125
-0.075
-0.025
0.025
0.075
0.125
0.175
0.225
0.275
0.325
0.375
0.425
0.475
Residual, fraction
0
50
100
150
200
250
300
350
Frequency
Fig. 2. Error distribution plot for water relative permeability correlation
(this study).
-0.475
-0.425
-0.375
-0.325
-0.275
-0.225
-0.175
-0.125
-0.075
-0.025
0.025
0.075
0.125
0.175
0.225
0.275
0.325
0.375
0.425
0.475
Residual, fraction
0
50
100
150
200
Frequency
52 SAUDI ARAMCO JOURNAL OF TECHNOLOGY SUMMER 2004
in the models, on the characteristics of oil and water relative
permeability curves. The physical parameters considered are
temperature, porosity and measurement method. Unlike the
oil relative permeability model, the porosity was found a
strong independent variable to be included in the water
relative permeability model, particularly in estimating the
endpoint. This indicates that the porosity has an influence
on the characteristic of the water relative permeability curve.
Measuring temperature and classifying data by type of
measurement method were found to have almost no effect,
and Al-Fattah, S.M., (1994), gives the details.
EVALUATION OF PUBLISHED EMPIRICAL
EQUATIONS
The newly developed empirical equations were evaluated
against six published empirical equations for checking their
performance and degree of accuracy in estimating the
water/oil relative permeability curves. These published
empirical equations (Honarpour, M., L. Koederitz and A.H.
Harvey, 1986); (Wyllie, M.R.J., 1950); (Pirson, S.J., 1958);
(Naar, J. and J.H. Henderson, 1961); (Naar, J. and R.J.
Wygal, 1961); (Naar, J., R.J. Wygal and J.H. Henderson,
1962); (Land, C.S., 1968); and (Honarpour, M., L.
Table 4. Statistical accuracy of oil relative permeability empirical equations.
Author Year Eav Eab s2rms Emax 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
Honarpour 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
Fig. 3. Behavior of water/oil relative permeability models against their physical
correlating properties, sandstone Saudi field.
00.1 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
Normalized
Eq.(4)
Eq.(2)
Fig. 4. Semilog plot of water/oil relative permeability from empirical equations,
Saudi sandstone reservoir.
00.1 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)
Fig. 5. Water/oil relative permeability measured data from core sample
compared to empirical equations of this study.
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
SAUDI ARAMCO JOURNAL OF TECHNOLOGY SUMMER 2004 53
Koederitz and A.H. Harvey, 1982) include those developed
by Wyllie, Pirson, Naar, et al., (1986) and Jones, Land and
Honarpour, et al., (1982). 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 permeability empirical
equations are: average error, absolute average error,
standard deviation, root-mean-square, minimum and
maximum absolute average error, and the F-test statistic
(Walpore, R.E. and R.H. Myers, 1993). 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. The 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 accuracy, with the highest errors and the lowest
F-test statistic.
The statistical accuracy of empirical equations for water
relative permeability is given in table 5. The empirical
equation of this study again achieved the highest accuracy
in estimating water relative permeability curves with the
lowest errors 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 statistic. Naar, et al. (1961/1962);
Jones, S.C., (1978); and Land, C.S., (1968) have all the
same empirical equations and hence have the same
unsatisfactory accuracy. Wyllie’s empirical equation showed
a similar performance of errors to Honarpour, et al.’s
empirical equation for oil-wet and intermediate wettability,
but with a better F-test statistic. Pirson’s empirical equation
showed poor accuracy, with the highest errors, and
Honarpour, et al.’s empirical equation (for water-wet and
intermediate) yielded the lowest F-test statistic.
GRAPHICAL ERROR ANALYSIS
The crossplot graphical error analysis technique is used in
the evaluation of the performance of this study and other
published empirical equations.
Table 5. Statistical accuracy of water relative permeability empirical equations.
Author Year Eav Eab s2rms Emax 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
Honarpour, 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.
Fig. 6. Crossplot for oil relative permeability empirical equations (Wyllie).
Fig. 7. Crossplot for oil relative permeability empirical equations (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 permeability
54 SAUDI ARAMCO JOURNAL OF TECHNOLOGY SUMMER 2004
Figs. 6 to 12 show the crossplots of estimated vs.
measured values for the oil relative permeability empirical
equations. Most of the plotted points of this study’s
empirical equation 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
equations are shown in figs. 13 to 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 newly-developed empirical equations for water/oil
imbibition 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 procedure was undertaken to examine the applicability
of the new empirical equations and to evaluate their
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
Fig. 8. Crossplot for oil relative permeability empirical equations (Naar, 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 permeability
Fig. 9. Crossplot for oil relative permeability empirical equations (Jones).
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 permeability
Fig. 10. Crossplot for oil relative permeability empirical equations (Land).
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 permeability
Fig. 11. Crossplot for oil relative permeability empirical equations (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 permeability
Fig. 12. Crossplot for oil relative permeability empirical equations (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 permeability
SAUDI ARAMCO JOURNAL OF TECHNOLOGY SUMMER 2004 55
accuracy against previously published empirical equations. A
total of 45 data points from four published sets (Willhite,
G.P., 1986); (Braun, E.M. and R.J. Blackwell, 1991); (Jones,
S.C. and W.O. Roszelle, 1978); and (Al-Fattah, S.M., 1994)
of water/oil relative permeability used for validation of the
empirical equations. This data is summarized in table 6.
The new empirical equations were compared with
published empirical equations of Wyllie, G.P., (1986);
Pirson, S.J. (1958); Naar, J., et al., (1961/1962); Jones, S.C.,
(1978); Land, C.S., (1968); and Honarpour, M.L., et al.,
(1986/1982). Each set of published data was used for the
comparison of published empirical equations with those
developed in this study. The results showed that the new
empirical equations more accurately reproduced
Fig. 13. Crossplot for oil relative permeability empirical equations (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 permeability
Fig. 14. Crossplot for oil relative permeability empirical equations (Jones).
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 permeability
Fig. 15. Crossplot for oil relative permeability empirical equations (Land).
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 permeability
Fig. 16. Crossplot for oil relative permeability empirical equations (Honarpour,
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 permeability
Fig. 17. Crossplot for oil relative permeability empirical equations (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 permeability
Fig. 18. Crossplot for oil relative permeability empirical equations (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 permeability
56 SAUDI ARAMCO JOURNAL OF TECHNOLOGY SUMMER 2004
experimental relative permeability data than other
published equations. Table 7 shows the results of statistical
accuracy for oil relative 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
accuracy than other published models for oil relative
permeability by showing lower values of error and a higher
coefficient of determination. Empirical equations of Pirson,
S.J., (1958) and Honarpour, M., et al., (1986/1982) show
similar accuracy; however, Pirson’s equation has a higher
coefficient of determination than Honarpour, M., et al.,
(1986/1982) does. The empirical equation of Naar, et al.
performs poorly compared to other empirical equations.
Table 8 gives the statistical accuracy of the results for
water relative 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 statistic. Inclusion of the r2
parameter in this error analysis was impractical because
small values of the F-test given by empirical equations of
Honarpour, et al. and Pirson resulted in negative r2values.
It should be noted that the F-test statistic has a strong direct
relation with the coefficient of determination. The highest
accuracy is achieved by this study, which gives lower errors
and higher F-test statistics than the results obtained by
other studies.
CONCLUSIONS
Multiple linear and non-linear least-square regression
analyses were applied on collected laboratory experimental
data for the development of empirical equations for
water/oil relative permeability. The following conclusions
were reached as a result of this study:
1. New empirical models are presented for calculating
imbibition water/oil relative permeability curves of
sandstone reservoir rocks. These empirical equations
can predict the relative permeability in sandstone, with
properties falling within the range of data used in this
development.
2. The newly-developed empirical equations exactly
satisfy the initial and endpoint requirements of
water/oil relative permeability curves. The endpoint
water relative permeability empirical equation
developed in this study was found to accurately
reproduce experimental values.
3. Comparative evaluation of existing empirical
equations was made, and the new empirical equations
give better accuracy than previously published
Table 8. Statistical accuracy of water relative permeability empirical equations.
Author Year Eav Eab s2Rms Emax 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
Table 7. Statistical accuracy of oil relative permeability empirical equations.
Author Year Eav Eab s2Rms Emax R2
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
Honarpour 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
SAUDI ARAMCO JOURNAL OF TECHNOLOGY SUMMER 2004 57
equations for the data used in this study.
4. The new empirical equations were validated using
published relative permeability data. They showed
better accuracy in estimating measured relative
permeability data than other published equations.
5. Parametric analysis showed that porosity has an
influence on the characteristic of the water relative
permeability curve, and therefore, it should be
included in the model for better prediction of the
endpoint in particular. The temperature and the type
of measurement method of relative permeability data
showed insignificant effects on the water and oil
relative permeability curves.
NOMENCLATURE
Error or residual E
Exponential function (Exp x = ex)
Average absolute error Eab
Average error Eav
Maximum absolute error Emax
Minimum absolute error Emin
F-test statistic F
Relative permeability to oil, fraction kro
Relative permeability to water, fraction krw
Relative permeability to oil at irreducible
water saturation, fraction kro(Swi)
Relative permeability to water at residual
oil saturation, fraction krw(Sor)
Number of data points n
Correlation coefficient r
Coefficient of determination R2
Root mean square Rms
Oil saturation, fraction So
Residual oil saturation, fraction Sor
Water saturation, fraction Sw
Water saturation at breakthrough, fraction Swf
Irreducible water saturation, fraction Swi
Standard deviation s
T-test statistic t
Porosity, fraction
φ
SUBSCRIPTS
Estimated value est
Experimental value exp
Oil phase o
Water phase w
ACKNOWLEDGMENTS
The author would like to thank M.A. Al-Marhoun, King
Fahd University of Petroleum and Minerals, for his valuable
comments and review of the manuscript. He would like to
thank K. Al-Fossail and H. Al-Yousef, King Fahd University
of Petroleum and Minerals, for their comments.
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