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

Authors:

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, which 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 both 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. Introduction Relative permeability1–4 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 permeabilities 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 determination 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 modified by relative permeability changes. More accurate prediction 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 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 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 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 literature4–11 using the data utilized in the development, and validated using published relative permeability data.
Copyright 2003, Society of Petroleum Engineers, Inc.
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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
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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
, 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).
... Therefore, relative permeabilitiy data are essential for almost all calculations of fluid flow in oil and gas reservoirs. In other words, simulation and modeling of reservoirs cannot be performed, unless the relative permeability which represent rock/fluid interaction at reservoir condition be available [5]. ...
... Several empirical and analytical mathematical models have been proposed yet for predicting or estimating the relative permeability of two-phase systems, that each of them has its own characteristics. Generally, most of these models have been developed as a function of wetting phase saturation by using stepwise linear and nonlinear regression analyses [5,9] or some analytical/numerical approach such as capillary models, statistical models and network models which are based on the assumption that a porous media consists of a bundle of capillary tubes [10,11]. ...
Conference Paper
Full-text available
One of the key parameters in porous media is relative permeability especially its application in modeling and simulation of Multiphase flow. The common ways for determination of relative permeability are empirical correlations and analytical mathematical models. Generally, these models are presented as a function of saturation. In this study, ten common and widely used empirical and analytical models, which are presented for estimation of two-phase relative permeability of gas-oil/condensate systems, are evaluated. For evaluation purposes 274 experimental data points related to gas-oil/condensate systems are employed. Adjustment and nonlinear local and global optimization are performed to determine the adjustable parameters of the parametric models. According to the performed adjustment and optimization, Lomeland Ebeltoft Thomas (LET) three parameter analytical correlations showed the best agreement to the experimental values for both gas and oil/condensate relative permeability. 1. Introduction Multiphase flow in porous media is a complex phenomenon. Well understanding of this issue is really essential in reservoir engineering in order to recognize productivity, injectivity, and ultimate recovery. Reservoir simulation and modeling often is employed to predict reservoir performance under different scenarios. For accurate simulation it is necessary to reduce the uncertainties of involving parameters. The main factor which causes the major uncertainty in prediction of reservoir performance is the accurate estimation of reservoir rock/fluid properties as an input data for simulator [1, 2]. The relative permeability is one of the key petrophysical properties, which has a significant effect on the evaluation and forecasting of the multiphase flows in porous medium and reservoir performance [3]. Strictly speaking, the conventional ways of predicting the fluid flow are theoretical empirical models such as Navier-Stokes, Darcy, Brinkman and Darcy-Forchheimer [4]. The mentioned flow models are function of relative permeability in multiphase flow systems. Therefore, relative permeabilitiy data are essential for almost all calculations of fluid flow in oil and gas reservoirs. In other words, simulation and modeling of reservoirs cannot be performed, unless the relative permeability which represent rock/fluid interaction at reservoir condition be available [5]. Generally, relative permeability curves are determined in laboratories using the analysis of multiphase flow in the core. The performed experiments in laboratory commonly divide into two major groups: steady state and unsteady state [6]. For steady state method, the immiscible fluids simultaneously flow in core plugs until saturation and pressure equilibrium is attained, whereas for the unsteady state method a fluid is injected by constant rate or constant pressure to displace in-situ fluid [7]. These methods are usually very sensitive, time consuming and costly [8]. Hence, researchers prefer to obtain these data from other methods that are quick and accurate, such as empirical correlations and analytical mathematical models. In the present study, first ten common and widely used empirical and analytical models of two-phase relative permeability of gas-oil/condensate systems, will be introduced. Then the optimization algorithm, which is utilized to determine parameters of the relative permeability models with adjustable parameters, will be described. The mentioned models are evaluated by implementing 274 experimental data points, which were obtained from open literature. After performing the calculations, the error is calculated for each model separately. It should be noted that in the evaluation of parametric models, adjustment and nonlinear local and global optimization are performed to determine the adjustable parameters of these models.
... e reservoir physical properties are essential for the calculation of fluid flow in oil and gas reservoirs. In other words, reservoir simulation and modeling cannot be performed unless the reservoir physical properties are available [17]. Reservoir simulation is often used to predict reservoir performance under different scenarios. ...
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When the reservoir physical properties are distributed very dispersedly, the matching precision of these reservoir parameters is not good. We propose a novel method for matching the reservoir physical properties based on particle swarm optimization (PSO) and support vector machine (SVM) algorithm. First, the data structure characteristics of the reservoir physical properties are analyzed. Then, the particle swarm differential perturbation evolution algorithm is used to cluster and characterize the reservoir physical properties. Finally, by using the SVM algorithm for feature reorganization and the least squares matching of the extracted reservoir physical properties, the feature quantity of the reservoir physical properties can be accurately mined and the pressure matching precision is improved. The experimental results show that employing the proposed method to analyze and sample the data characteristics of the physical properties of the reservoir is better. The extracted parameters can effectively reflect the physical characteristics of oil reservoirs. The proposed method has potential applications in guiding the exploration and development of oil reservoirs.
... Relative permeability data are essential for almost all calculations of fluid flow in oil and gas reservoirs. In other words, simulation and modeling of reservoirs cannot be performed unless the relative permeability at reservoir condition is available [7]. Reservoir simulation is often employed to predict reservoir performance under different scenarios. ...
... Relative permeability data are essential for almost all calculations of fluid flow in oil and gas reservoirs. In other words, simulation and modeling of reservoirs cannot be performed unless the relative permeability at reservoir condition is available [7]. Reservoir simulation is often employed to predict reservoir performance under different scenarios. ...
Article
As the ages of most oil fields fall in the second half of their lives, many attempts have been made to enhance oil recovery in an efficient way. Gas injection into oil reservoirs for enhanced oil recovery (EOR) purposes requires relative permeability as a crucial issue in reservoir engineering. In this study, a new method is applied to predict relative permeabilities of gas/oil system related to various rock and fluid types. For this reason, a soft computing technique- Multi-gene genetic programming (MGGP) is employed to develop tools for prediction of relative permeability. The new methods are evaluated by experimental data extracted from open literature and are validated by extensive error analysis. The generated smart mathematical equations are able to predict relative permeabilities of gas/oil system with high accuracy and are applicable for various types of rock and fluid as well. In contrary to other reported correlations, the new novel equations require oil API and gas molecular weight as extra input variables to improve their estimating ability for every type of rock and fluid. The proposed technique is promising and encouraging for petroleum and reservoir engineers to be implemented for other gas/oil petro-physical properties.
... Several empirical formulations are available to characterize observed water-oil relative permeability curves (Al-Fattah, 2003;Chierici, 1984;Corey, 1954;Honarpour et al., 1982;Lomeland et al., 2005). The structure of these models is typically driven by experimental observations, theoretical arguments and/or heuristic concepts. ...
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We illustrate the way formal model identification criteria can be employed to rank and evaluate a set of alternative models in the context of the interpretation of laboratory scale experiments yielding two-phase relative permeability curves. We consider a set of empirical two-phase relative permeability models (i.e., Corey, Chierici and LET) which are typically employed in industrial applications requiring water/oil relative permeability quantifications. Model uncertainty is quantified through the use of a set of model weights which are rendered by model posterior probabilities conditional on observations. These weights are then employed to (a) rank the models according to their relative skill to interpret the observations and (b) obtain model averaged results which allow accommodating within a unified theoretical framework uncertainties arising from differences amongst model structures. As a test bed for our study, we employ high quality two-phase relative permeability estimates resulting from steady-state imbibition experiments on two diverse porous media, a quartz Sand-pack and a Berea sandstone core, together with additional published datasets. The parameters of each model are estimated within a Maximum Likelihood framework. Our results highlight that in most cases the complexity of the problem appears to justify favoring a model with a high number of uncertain parameters over a simpler model structure. Posterior probabilities reveal that in several cases, most notably for the assessment of oil relative permeabilities, the weights associated with the simplest models is not negligible. This suggests that in these cases uncertainty quantification might benefit from a multi-model analysis, including both low- and high-complexity models. In most of the cases analyzed we find that model averaging leads to interpretations of the available data which are characterized by a higher degree of fidelity than that provided by the most skillful model.
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Predicting the incremental recovery factor with an enhanced oil recovery (EOR) technique is a very crucial task. It requires a significant investment and expert knowledge to evaluate the EOR incremental recovery factor, design a pilot, and upscale pilot result. Water-alternating-gas (WAG) injection is one of the proven EOR technologies, with an incremental recovery factor typically ranging from 5 to 10%. The current approach of evaluating the WAG process, using reservoir modeling, is a very time-consuming and costly task. The objective of this research is to develop a fast and cost-effective mathematical model for evaluating hydrocarbon-immiscible WAG (HC-IWAG) incremental recovery factor for medium-to-light oil in undersaturated reservoirs, designing WAG pilots, and upscaling pilot results. This integrated research involved WAG literature review, WAG modeling, and selected machine learning techniques. The selected machine learning techniques are stepwise regression and group method of data handling. First, the important parameters for the prediction of the WAG incremental recovery factor were selected. This includes reservoir properties, rock and fluid properties, and WAG injection scheme. Second, an extensive WAG and waterflood modeling was carried out involving more than a thousand reservoir models. Third, WAG incremental recovery factor mathematical predictive models were developed and tested, using the group method of data handling and stepwise regression techniques. HC-IWAG incremental recovery factor mathematical models were developed with a coefficient of determination of about 0.75, using 13 predictors. The developed WAG predictive models are interpretable and user-friendly mathematical formulas. These developed models will help the subsurface teams in a variety of ways. They can be used to identify the best candidates for WAG injection, evaluate and optimize the WAG process, help design successful WAG pilots, and facilitate the upscaling of WAG pilot results to full-field scale. All this can be accomplished in a short time at a low cost and with reasonable accuracy.
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Full-text available
Determination of relative permeability data is required for almost all calculations of fluid flow in petroleum reservoirs. Water-oil relative permeability data play important roles in characterizing the simultaneous two-phase flow in porous rocks and predicting the performance of immiscible displacement processes in oil reservoirs. They are used, among other applications, for determining fluid distributions and residual saturations, predicting future reservoir performance, and estimating ultimate recovery. Undoubtedly, these data are considered probably the most valuable information required in reservoir simulation studies. Estimates of relative permeability are generally obtained from laboratory experiments with reservoir core samples. Because the laboratory measurement of relative permeability is rather delicate, expensive and time consuming, empirical correlations are usually used to predict relative permeability data, or to estimate them in the absence of experimental data. However, developing empirical correlations for obtaining accurate estimates of relative permeability data showed limited success and proved difficult especially for carbonate reservoir rocks. Artificial neural network (ANN) technology has proved successful and useful in solving complex structured and nonlinear problems. This paper presents a new modeling technology to predict accurately water-oil relative permeability using ANN. The ANN models of relative permeability were developed using experimental data from waterflood core tests samples collected from carbonate reservoirs of giant Saudi Arabian oil fields. Three groups of data sets were used for training, verification, and testing the ANN models. Analysis of results of the testing data set show excellent agreement with the experimental data of relative permeability. In addition, error analyses show that the ANN models developed in this study outperform all published correlations. The benefits of this work include meeting the increased demand for conducting special core analysis, optimizing the number of laboratory measurements, integrating into reservoir simulation and reservoir management studies, and providing significant cost savingson extensive lab work and substantial required time. Introduction Artificial neural networks have seen an explosion of interest over the past few years. They are powerful and useful tools for solving practical problems in the petroleum industry (Mohaghegh 2005; Al-Fattah and Startzman 2003). Advantages of neural network techniques (Bishop 1995; Fausett 1994; Haykin 1994; Patterson 1996) over conventional techniques include the ability to address highly nonlinear relationships, independence from assumptions about the distribution of input or output variables, and the ability to address either continuous or categorical data as either inputs or outputs. In addition, neural networks are intuitively appealing as they are based on crude low-level models of biological systems. Neural networks, as in biological systems, simply learn by examples. The neural network user provides representative data and trains the neural networks to learn the structure of the data.
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Adaptive neuro-fuzzy inference system (ANFIS) is a powerful nonlinear, multivariable regression technique. Here ANFIS was used to identify complex relation between water-oil relative permeability key points and rock and fluid properties. Some 260 relative permeability curves from Iranian carbonate and sandstone reservoirs were used in this study. For each curve six key points (i.e., end points and the crossover points) were considered. ANFIS was then used to predict these key points from different rock and fluid properties. The results showed that very high correlation coefficients in the range of 0.8–0.92 are achievable for Kr key points. ANFIS is a very suitable tool therefore to obtain un-normalized water-oil relative permeability curves with high accuracy when the required core and fluid properties are available.
Article
Adaptive neuro-fuzzy inference system is an intelligent nonlinear, multivariable technique. In this study, an adaptive neuro-fuzzy inference system was used to identify complex relations between water-oil relative permeability curves and rock and fluid properties. Sixty-seven relative permeability curves from Iranian carbonate and sandstone reservoirs were used in this study. An adaptive neuro-fuzzy inference system was then used to predict un-normalized relative permeability full curves as a function of different rock and fluid properties. High correlation coefficients (R 2) of 0.88 and 0.94 were obtained for oil and water relative permeability curves, respectively. Water relative permeability curves also show no dependency to wettability index of the core and fluid system while oil curves show weak dependency. Adaptive neuro-fuzzy inference system models showed very promising to obtain real water-oil relative permeability full curves when the required core and fluid properties are available. Finally, the effect of porosity, initial water saturation, and wettability on relative permeability curves was investigated by constructed models. Relative permeability of both phases increased with porosity and showed decreasing with initial water saturation. Wettability has no considerable effect on relative permeability curves.
Article
Published in Petroleum Transactions, AIME, Vol. 216, 1959, pages 370–372. A method is presented for calculating individual gas and oil or water and oil relative permeabilities from data obtained during a gas drive or a waterflood experiment performed on a linear porous body. The method has been tested and found both rapid and reliable for normal-sized core samples. Introduction Individual oil and gas or oil and water relative permeabilities are required for a number of reservoir engineering applications. Chief among these is the evaluation of oil displacement under conditions where gravitational effects are significant, such as a water drive or crestal gas injection in a steeply dipping oil reservoir. Numerous proposed methods of obtaining relative permeability data on reservoir core samples have been too tedious and time consuming for practical use, or have yielded questionable and sometimes inconsistent results. A method bas been developed by which the individual relative permeability curves can be calculated from data collected during a displacement test. The method is based on sound. Using this method, with a properly designed experimental procedure, relative permeability curves can be obtained using core samples of normal size (i.e., 2 to 3 in. in length and 1 to 2 in. in diameter) within a few days after receipt of the core. In a recent publication D. A. Efros describes an approach to the calculation of individual relative permeabilities that is based on the same theoretical considerations. We believe the approach described in the present paper is more adaptable to practical application than the method implied by Efros. In addition, comparisons with independently determined relative permeabilities are furnished to substantiate the reliability of the new method.
Article
An entirely empirical approach was applied to predict the permeability of a porous rock to a saturating fluid. In this study, a rather extensive set of relative permeability data was compiled, and conventional stepwise linear regression analysis techniques were used to develop prediction equations from the laboratory data. This procedure is designed to produce a satisfactory fit of the data with a minimum of terms in the equation; it is not intended to provide the best possible data fit.
Article
This paper presents graphical constructions that simplify the calculation of relative permeability from displacement data. These constructions convert raw data to relative permeability in a less tedious, more accurate manner than the usual computations. Fractional-flow saturation curves derived from waterflood displacements are always concave downward and never yield multiple-value saturations. Introduction To find oil and water relative permeabilities by the displacement or unsteady-state method, a small linear core usually is saturated with water, then oilflooded to irreducible water saturation. Subsequently, the core is waterflooded, and during the process, pressure drop (either constant or variable) across the entire core and water injection rate (constant or variable) are determined. Effluent fractions are collected and the amount of water and oil in each is measured. Augmented by the absolute permeability and pore volume of the core and by oil and permeability and pore volume of the core and by oil and water viscosities, these data are sufficient to develop relative permeability curves. The average saturation in the core at any time in the flood can be found from an over-all material balance. However, to calculate relative permeability, the saturation history at some point in the core must be determined, not the average saturation history. The Welge equation yields saturations at the effluent end of the core when the average saturation history is known. Similarly, to compute relative permeability, the point pressure gradient per unit injection rate is needed, not the pressure gradient per unit injection rate is needed, not the average. The equation developed by Johnson et al. converts average relative injectivity to a point value, accomplishing the required task. While the equations of Welge and Johnson et al. have been used successfully for years, they require tedious computation and are subject to error because of the evaluation of derivatives. The graphical techniques presented in this study are equivalent to these equations, but are easier to use and can give a more accurate evaluation of relative permeability. Lefebvre du Prey has presented graphical constructions based on curves of volume of oil produced vs time and pressure drop vs time to develop the required point functions. These constructions are limited to constant rate displacements. The constructions presented here are general and apply to constant rate, constant pressure, or variable rate-pressure displacements. Constant-rate and constant-pressure examples are given to help clarify the methods. The graphical techniques make it easy to see that double or triple saturation values, so extensively discussed in the past simply do not result from the fractional flow curve generated by a single displacement, such as a waterflood or an oilflood. Theory Ignoring gravity effects and capillary pressure, water and oil relative permeabilities (expressed as functions of saturation) are (1) (2) To use these equations, the fractional flow of water or oil and effective viscosity, lambda-1, must be determined as functions of saturation. JPT P. 807
Article
This book enables petroleum reservoir engineers to predict the flow of fluids within a hydrocarbon deposit. Laboratory techniques are described for both steady-state and unsteady state measurements, and the calculation of relative permeability from field data is illustrated. A discussion of techniques for determing wettability is included, along with theoretical and empirical methods for the calculation of relative permeability, and prediction techniques. Contents include: Measurement of Rock Relative Permeability; Two-Phase Relative Permeability; Factors Affecting Two-Phase Relative Permeability; Three-Phase Relative Permeability; and Index.
Article
The original Buckley-Leverett theory has been extended and a more detailed formulation of the waterflood behavior in linear horizontal systems is presented. Particular consideration has been given to the evaluation of capillary pressure effects and differential equations permitting an explicit evaluation of these effects have been derived. On the basis of the developed theory it is recognized that the flooding behavior is dependent upon the length of the system and the rate of injection. At the same time it has been determined that systems of different lengths yield the same flooding behavior if the injection rates and/or the fluid viscosities are properly adjusted or "scaled." It has also been found that the sensitivity of the flooding behavior with respect to rate and length decreases as any one of these factors increases in value and that for sufficiently long systems and high rates of water injection the flooding behavior becomes independent of rate and length, or "stabilized." To such stabilized conditions the theory formulated by Buckley and Leverett is applicable. A number of laboratory flooding tests have been made and good agreement has been found between theory and experimental observations. The experimental results are discussed and it is shown that under field conditions the flooding behavior is usually stabilized. As a result of these findings a procedure is indicated for evaluating field performances either on the basis of tests performed with commonly available core samples or by means of calculations using relative permeability data.