ARTICLE pubs.acs.org/EF
Quantitative Prediction Model for the WaterOil Relative Permeability Curve and Its Application in Reservoir Numerical Simulation. Part 1: Modeling Jian Hou,*,† Fuquan Luo,†,‡ Chuanfei Wang,§ Yanhui Zhang,† Kang Zhou,† and Guangming Pan† †
College of Petroleum Engineering, China University of Petroleum, Dongying, Shandong 257061, People’s Republic of China Downhole Operation Company, Jidong Oilfield Company, Tangshan, Hebei 063000, People’s Republic of China § Geoscience Research Institute, Shengli Oilfield Company, Dongying, Shandong 257015, People’s Republic of China ‡
ABSTRACT: The wateroil relative permeability curve has a great effect on the rules of water cut increase and production variation. It is one of the most important data in reservoir development. With regard to a reservoir with a high degree of heterogeneity, the flow properties are various in different positions of the reservoir. Therefore, neither a single average relative permeability curve for the whole reservoir nor different curves for different sedimentary facies can precisely describe the reservoir flow characteristics, which will cause great difficulties for the remaining oil prediction and potential tapping. Therefore, it is of great importance to build a prediction model for the wateroil relative permeability curve, which can provide a calculation theory of the relative permeability curve for reservoir simulation using different relative permeability curves in different grid cells. The existing prediction models for the relative permeability curve have established the correlations between petrophysical properties and endpoint values of the relative permeability curve and between end-point values of the relative permeability curve and the relative permeability curve independently. However, the relationship between petrophysical properties and the relative permeability curve has not been developed. For this reason, it is impossible to achieve the spatial distribution of the relative permeability curve according to petrophysical properties. Furthermore, the end-point values of the relative permeability curve are usually calculated directly by petrophysical properties, which may result in all predicted relative permeability curves shifting to the left or right unrealistically compared to the reservoir average relative permeability curve. As a result of the above-mentioned problems, taking into consideration the correction effect of the average relative permeability curve on predicting relative permeability curves in different positions of the reservoir, this paper obtains a correlation between petrophysical properties and the relative permeability curve on the basis of the statistical analysis technique and normalization method of the reservoir average relative permeability curve. Finally, a prediction model for the wateroil relative permeability curve is established. A test based on the basic data of the relative permeability curve is performed to verify the effect of this model. The test result shows that the prediction procedure of this model is quick, easy, and reliable. It also indicates that the spatial distribution of the relative permeability curve satisfying the migration rule can be generated from petrophysical properties, which provides a basic calculation theory of the wateroil relative permeability curve for reservoir simulation using different relative permeability curves in different grid cells.
1. INTRODUCTION It has been proven in the reservoir exploration and development that the key to success is up to whether the reservoir recognition is in accordance with the objective reality. The reservoir recognition can be achieved by reservoir geological modeling. Unfortunately, most applications of stochastic modeling have been limited to reservoir properties that are not related to fluid saturation, such as porosity and permeability, and the spatial variation of the relative permeability curve is not taken into consideration.16 However, as one of the most important parameters in reservoir development, the relative permeability curve is the key parameter to describe wateroil two-phase flow characteristics and it affects the distribution law of oil and water greatly. With regard to the reservoir with a high degree of macroscopic heterogeneity and a low degree of microscopic heterogeneity, the heterogeneity can be reflected to a large extent by using different wateroil relative permeability curves for different depositional environments. Moreover, the production performance obtained with the above method has a good match with r 2011 American Chemical Society
the production realistic history. Nevertheless, with regard to the reservoir with a high degree of both macro- and microscopic heterogeneity, an ideal history match is still hard to achieve even if sedimentary facies are separated into sub- and microfacies and relative permeability curves are subdivided correspondingly. This will cause great problems for the remaining oil potential tapping and reservoir performance prediction.710 For these reasons, it is essential to build a prediction model for the wateroil relative permeability curve. On this basis, different relative permeability curves can be assigned to different grid cells in reservoir simulation, which gives a much more precise description of wateroil flow properties and provides a new technique for production history match for highly heterogeneous reservoirs. Many mathematical models have been proposed to describe the relative permeability curve. For the first time, Purcell11 established Received: June 3, 2011 Revised: August 18, 2011 Published: September 09, 2011 4405
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Table 1. Basic Data for WaterOil Relative Permeability Curve Experiments basic parameters
Table 2. Characteristic Parameters of the WaterOil Relative Permeability Curve
minimum
maximum
mean
characteristic
core length (cm)
4.38
7.14
5.46
core diameter (cm) core porosity (%)
2.31 23.2
2.54 39.5
2.46 33.7
irreducible water saturation (%) residual oil saturation (%)
core air permeability (μm2)
0.188
8.615
3.169
width of the two-phase flow region (%)
41.9
60.1
52.7
temperature (°C)
50
50
50
oil permeability at irreducible
0.128
3.040
1.634
oil viscosity (mPa s)
7.55
57.56
42.63
water saturation (μm2)
water viscosity (mPa s)
0.52
0.64
0.55
water permeability at residual
0.048
1.380
0.615
parameters
minimum
maximum
mean
15.6 17.4
25.0 35.6
20.3 27.0
oil saturation (μm2)
the relationship between the capillary force and relative permeability curve and derived a mathematical model to calculate relative permeability using capillary force data. Burdine12 modified Purcell’s model by introducing tortuosity to reflect the pore structure. On the basis of Purcell’s and Burdine’s models, Corey13 proposed a gasoil relative permeability model in imbibition process, and he found that relative permeability was a power function of dimensionless saturation. However, Brooks and Corey’s further research showed that this relationship was not always suitable for all kinds of pore structures. Thus, they proposed modifications to the relationship between relative permeability and dimensionless saturation and extended its use to the calculation of wateroil relative permeability.14 Thus far, the model of the relative permeability curve most commonly used is the modified BrooksCorey (MBC) model.15,16 However, the above-mentioned mathematical models only describe the correlation between relative permeability and dimensionless saturation quantitatively, while the mathematical functions between end-point values of the relative permeability curve and petrophysical properties are not established. As a result, the spatial distribution of the relative permeability curve cannot be generated according to petrophysical properties. Here, we list some common end-point values of the relative permeability curve: irreducible water saturation, residual oil saturation, oil relative permeability at irreducible water saturation, and water relative permeability at residual oil saturation. On the basis of 617 relative permeability curve data, Daqing oilfield17 found that there was a statistical relationship between irreducible water saturation and air permeability. Tjølsen et al.18 separated reservoirs into four different kinds according to their depositional environments, and for every one of them, they established correlations between endpoint values of the relative permeability curve and air permeability and between Corey exponents for oil and water and air permeability. Unfortunately, the end-point values of the relative permeability curve are calculated directly by petrophysical properties in the above-mentioned models, which may cause all predicted relative permeability curves to shift to the left or right unrealistically compared to the reservoir average relative permeability curve. As a result of the above-mentioned problems about the existing prediction models for the wateroil relative permeability curve, taking into consideration the correction effect of the average relative permeability curve on predicting relative permeability curves in different positions of the reservoir, this paper presents the relationships between petrophysical properties and end-point values of the relative permeability curve and between end-point values of the relative permeability curve and the relative permeability curve. Therefore, the correlation between petrophysical properties and the relative permeability curve is
developed, and at last, a prediction model for the wateroil relative permeability curve is established. From this model, the spatial distribution of the relative permeability curve can be obtained from petrophysical properties, which provides a basic calculation theory of the wateroil relative permeability curve for reservoir simulation using different relative permeability curves in different grid cells. On the basis of the above analysis, the outline of this paper is proposed as follows: (1) Establish the end-point models for the wateroil relative permeability curve, i.e., the relationships between end-point values of the relative permeability curve and petrophysical properties. (2) Establish the characterization model for the wateroil relative permeability curve, i.e., the correlation between end-point values of the relative permeability curve and the relative permeability curve. (3) Describe the wateroil relative permeability curve as a function of petrophysical properties by submitting the end-point models for the relative permeability curve into the characterization model for the relative permeability curve. In other words, the quantitative prediction model for the wateroil relative permeability curve is established.
2. BASIC DATA OF THE WATEROIL RELATIVE PERMEABILITY CURVE In this study, strong effort has been made to collect a sufficient number of high-quality relative permeability curves. A total of 55 cores were selected from 5 wells in a high permeability sandstone reservoir of the Shengli oilfield. Then, the relative permeability curves are measured using an unsteady-state technique. The corresponding experimental data, as well as the range and mean of the characteristic parameters of these relative permeability curves, have been summarized in Tables 1 and 2. Because oil permeability at irreducible water saturation is regarded as the absolute permeability in all of the 55 relative permeability curves, oil relative permeability at irreducible water saturation is 1 all of the time.
3. END-POINT MODELS FOR THE WATEROIL RELATIVE PERMEABILITY CURVE To establish a prediction model between the wateroil relative permeability curve and petrophysical properties, the first step has been taken to build end-point models for the wateroil relative permeability curve. Considering the fact that oil relative permeability at irreducible water saturation equals a constant of 1, there are three end-point models left to be developed, including the irreducible water saturation model, residual oil saturation model, and water relative permeability model at residual oil saturation. 3.1. Irreducible Water Saturation Model. Recent research by Molinad et al.19 found that air permeability should be taken as 4406
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Figure 1. Crossplot between the real values and predicted values by the Tixier model of irreducible water saturation.
Figure 2. Crossplot between the real values and predicted values by the Timur model of irreducible water saturation.
one key factor in reservoir development, because permeability heterogeneity governs reservoir development performance. In well logging, irreducible water saturation is usually computed using empirical models among irreducible water saturation, air permeability, and porosity.20,21 On the basis of the above analysis, this paper selects air permeability and porosity as the main influencing factors. Then, a multivariate prediction model for irreducible water saturation is achieved by fitting and modifying the empirical models in well logging on the basis of the statistical analysis technique. There are three most widely used models in well logging: Tixier model, Timur model, and Coates model.22,23 These models adopt the following equations: Tixier model
Swir ¼
250ϕ3 k0:5
ð1Þ
Timur model
Swir ¼
100ϕ2:25 k0:5
ð2Þ
Coates model
Swir ¼
100ϕ2 k0:5 þ 100ϕ2
ð3Þ
where Swir is irreducible water saturation (fraction), ϕ is porosity (fraction), and k is air permeability (103 μm2). The above three models are validated using 55 relative permeability curves. Figures 13 compare the predicted values and the real values of irreducible water saturation. As can be seen from the crossplots of irreducible water saturation, most data points concentrate near the vicinity of the 45° line, while some others drop down to the region bounded by two straight lines,
Figure 3. Crossplot between the real values and predicted values by the Coates model of irreducible water saturation.
Figure 4. Crossplot between the real values and predicted values by the modified Tixier and Timur models of irreducible water saturation.
Figure 5. Crossplot between the real values and predicted values by the modified Coates model of irreducible water saturation.
which represents an absolute error of 0.1. For the sake of improving the fitting effect, these three models are modified. At first, these models are log-transformed. The transformed Tixier and Timur models are shown as eq 4, and the transformed Coates model is shown as eq 5 logðSwir Þ ¼ a logðkÞ þ b logðϕÞ þ c log
1 1 Swir
ð4Þ
¼ a logðkÞ þ b logðϕÞ þ c
ð5Þ
where a, b, and c are the correction coefficients (fraction). The least-squares regression analysis has been used to calculate a, b, and c; thereby, three modified models are acquired and shown as eqs 6 and 7. On the basis of these modified models, comparisons between the predicted values and the real values of irreducible water saturation are summarized in Figures 4 and 5. 4407
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Table 3. Error Analysis of the Original and Modified Irreducible Water Saturation Models AAE
RMSE
irreducible water saturation models original modified original modified Tixier model
0.0371
0.0181
0.0463
0.0216
Timur model
0.0428
0.0181
0.0493
0.0216
Coates model
0.0325
0.0179
0.0383
0.0213
As can be seen from the crossplots of irreducible water saturation, all data points concentrate in the vicinity of the 45° line.
Figure 6. Statistical relationship of residual oil saturation and irreducible water saturation.
modified Tixier and Timur models Swir ¼
1:3138ϕ0:5656 k0:1589
ð6Þ
modified Coates model Swir ¼
2:6389ϕ0:7056 þ 2:6389ϕ0:7056
k0:1989
ð7Þ
To compare the fitting effect of modified models and original models, eqs 8 and 9 are used to calculate the average absolute error (AAE) and root-mean-square error (RMSE) between predicted values and real values of irreducible water saturation for the original and modified models, respectively. The comparison results are concluded in Table 3. From Table 3, it can be found that errors caused by modified models are much smaller than those caused by original models and the error caused by the modified Coates model is the smallest among the three modified models AAE ¼
1 n jεi jðεi ¼ xi x0 i Þ n i¼1
RMSE ¼
∑
vffiffiffiffiffiffiffiffiffiffiffiffi u n u u εi 2 ti ¼ 1
∑
n
ð8Þ
ð9Þ
where xi is the real value of the ith sample (fraction), x0 i is the predicted value of the ith sample (fraction), n is the total number of samples, and εi is the absolute error of the ith sample (fraction). 3.2. Residual Oil Saturation Model. Statistical analysis on the basic data of the relative permeability curve has been performed, which indicates that there is a strong statistical correlation between the ratio of residual oil saturation to irreducible water saturation and irreducible water saturation, as shown in Figure 6. By fitting this correlation using the power function, the residual oil saturation model is established and takes the form of eq 10 Sor ¼ 0:1472Swir 1:3717 Swir
Figure 7. Crossplot between the real values and predicted values of residual oil saturation.
ð10Þ
where Sor is residual oil saturation (fraction). A crossplot between real values and predicted values of residual oil saturation is shown as Figure 7. It demonstrates that data points concentrate in the vicinity of the 45° line and AAE is
Figure 8. Statistical relationship of water permeability at residual oil saturation and oil permeability at irreducible water saturation.
0.0306. Therefore, the residual oil saturation model established is reliable. 3.3. Water Relative Permeability Model at Residual Oil Saturation. Oil permeability at irreducible water saturation is selected as the absolute permeability. Then, water relative permeability at residual oil saturation is calculated by the following equation: krwor ¼
kwor kowc
ð11Þ
where krwor is water relative permeability at residual oil saturation (fraction), kwor is water permeability at residual oil saturation (103 μm2), and kowc is oil permeability at irreducible water saturation (103 μm2). On the basis of statistical analysis on the basic data of the relative permeability curve, strong statistical relationships between kwor and kowc and between kowc and k are found, as shown in Figures 8 and 9. By fitting these two correlations with the power 4408
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Figure 9. Statistical relationship of oil permeability at irreducible water saturation and air permeability.
Figure 10. Crossplot between the real values and predicted values of water relative permeability at residual oil saturation.
function, the following eqs 12 and 13 are obtained.
of the relative permeability curve predicted can be obtained successfully. The improved end-point models should satisfy the following constraint: if the input petrophysical properties are the same as those reflected by the average relative permeability curve, then the predicted relative permeability curve should be identical to the average curve. To meet this constraint, the end-point values calculated by improved end-point models should be equal to those of the average relative permeability curve. 4.1. Irreducible Water Saturation Model. The ratio form is taken to describe the deviations of petrophysical properties and end-point values of the relative permeability curve. Their specific equations are listed below
kwor ¼ 0:1548kowc 1:1138
ð12Þ
kowc ¼ 12:6120k0:6003
ð13Þ
Now, the water relative permeability model at residual oil saturation is achieved using eqs 1113. The crossplot of real values and predicted values for water relative permeability at residual oil saturation is drawn in Figure 10. AAE is 0.0779. It is certain that the established model is of high precision and can meet the engineering requirements. In conclusion, the paper has established the end-point models for the relative permeability curve on the basis of the statistical analysis technique. However, there are still some weaknesses of these models listed below. In conventional reservoir simulation, only one average relative permeability curve is usually adopted to describe the reservoir flow properties. If the spatial heterogeneity of relative permeability curves is taken into consideration, actual relative permeability curves representing different reservoir positions should fluctuate with the average relative permeability curve as the center. Nevertheless, the above models calculate the end-point values of the relative permeability curve directly by porosity and permeability, which may cause all of the predicted relative permeability curves to shift to the left or right compared to the average relative permeability curve, which is unrealistic.
4. IMPROVED END-POINT MODELS FOR THE WATER OIL RELATIVE PERMEABILITY CURVE Considering the shortages of end-point models mentioned above, this paper introduces the conception of deviation to describe the correction effect of the average relative permeability curve on predicting relative permeability curves in different positions of the whole reservoir. Furthermore, the improved endpoint models for the relative permeability curve are established. Here are the specific outlines. First, input the end-point values of the average relative permeability curve, as well as the reference petrophysical properties, such as porosity and permeability, which are reflected by the average relative permeability curve. Then, calculate the deviation of some petrophysical properties compared to the reference value using the deviation equation. On this basis, the deviation direction and extent of the end-point values of the relative permeability curve under the petrophysical property relative to that of the average relative permeability curve will be described quantitatively. Then, the end-point values
RSwir ¼
Swir S̅ wir
ð14Þ
Rk ¼
k k̅
ð15Þ
Rϕ ¼
ϕ ϕ
ð16Þ
where RSwir is the deviation of irreducible water saturation (fraction), Rk is the deviation of air permeability (fraction), Rϕ is the deviation of porosity (fraction), Swir is the reference value of irreducible water saturation (fraction), k is the reference value of air permeability (103 μm2), and ϕ is the reference value of ̅ porosity (fraction). The methods used to obtain reference values of irreducible water saturation, air permeability, and porosity are summarized as follows: (1) If the average relative permeability curve and corresponding petrophysical properties are available, they can be taken directly as the reference values. (2) If the average relative permeability curve is not available but there are many groups of relative permeability curves obtained by core analysis, then the arithmetic mean values of these parameters can be regarded as the reference values. The mean values of irreducible water saturation, air permeability, and porosity are obtained by arithmetic average operation over the basic data of 55 relative permeability curves. Their values are 0.203, 3169 103 μm2, and 33.7% respectively. Considering that the log function of the Coates model under ratio form may be meaningless, this study only establishes the irreducible water saturation model on the basis of the Tixier and Timur models. On the basis of the statistical analysis technique, the irreducible water saturation model is established and takes the 4409
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Figure 11. Crossplot between the real values and predicted values by the ratio-formed Tixier and Timur models of irreducible water saturation.
Figure 13. Crossplot between the real values and predicted values of water relative permeability at residual oil saturation under ratio form.
4.3. Water Relative Permeability Model at Residual Oil Saturation. The calculation formula for the deviation of water
relative permeability at residual oil saturation is given below Rk rwor ¼
Figure 12. Crossplot between the real values and predicted values of residual oil saturation under ratio form.
RSwir
Rϕ ¼ 0:1453 Rk
Rk rwor ¼ 1:0864Rk owc 0:1205
ð18Þ
where RSor is the deviation of residual oil saturation (fraction) and Sor is the reference value of residual oil saturation (fraction). The average residual oil saturation is 0.270. Under the constraint that, if Rϕ = 1 and Rk = 1, then RSor = 1, the ratio-formed residual oil saturation model is achieved. Equation 19 describes this model. RSor ¼ RSwir 1:365 RSwir
ð21Þ
ð17Þ
Figure 11 presents the crossplot of real values and predicted values for irreducible water saturation. This figure indicates that all data points concentrate in the vicinity of the 45° line and AAE is 0.0191. In addition, the established model meets the constraint that, if Rϕ = 1 and Rk = 1, then RSwir = 1. Furthermore, it is highly precise; therefore, the ratio-formed Tixier and Timur models are selected as the ultimate irreducible water saturation model. 4.2. Residual Oil Saturation Model. Here is the calculation formula for deviation of residual oil saturation Sor RSor ¼ S̅ or
ð20Þ
where Rkrwor is the deviation of water relative permeability at residual oil saturation (fraction) and krwor is the reference value of water relative permeability at residual oil saturation (fraction). The mean value of water relative permeability at residual oil saturation is 0.355. Owing to the fact that water relative permeability at residual oil saturation has a strong correlation with air permeability and is not nearly affected by porosity, the constraint is simplified as, if Rk = 1, then Rkrwor = 1. Under this constraint, the ratio-formed water relative permeability model at residual oil saturation is developed and is described as eqs 21 and 22
form of eq 17. 0:5310
krwor k̅ rwor
ð19Þ
Figure 12 presents the crossplot of real values and predicted values for residual oil saturation. This figure indicates that data points concentrate in the vicinity of the 45° line and AAE is 0.0308. Obviously, the fitting effect is satisfactory.
Rk owc ¼ 0:5028Rk 0:6003 ,
Rk owc ¼
kowc k̅
ð22Þ
where Rkowc is the deviation of oil permeability at irreducible water saturation (fraction). Figure 13 presents the crossplot of real values and predicted values for water relative permeability at residual oil saturation. AAE is 0.0772. Therefore, this model is precise and can meet the engineering requirements.
5. CHARACTERIZATION MODEL FOR THE WATEROIL RELATIVE PERMEABILITY CURVE After the end-point values of the relative permeability curve are obtained, the characterization model between end-point values of the relative permeability curve and the relative permeability curve can be established using the normalization method of the reservoir average relative permeability curve. This method has been widely used in the calculation of the relative permeability curve under different temperatures during the thermal development process of heavy oil reservoirs.24 Here are the basic theories. First, take the average relative permeability curve as a reference standard. Then, shift the average relative permeability curve as a whole to the left or right, according to the deviation rules for end-point values of the relative permeability curve predicted relative to those of the average relative permeability curve. Consequently, the relative permeability curve predicted is obtained. During the above movement process, the shape of the 4410
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Figure 15. Normalized average relative permeability curve. Figure 14. Calculation procedure of the quantitative prediction model for the relative permeability curve.
relative permeability curve is kept the same. The main procedures of this method are explained as follows: 5.1. Calculation of the Normalized Average Relative Permeability Curve. The calculation procedures are given below: (1) If the average relative permeability curve is available, eq 23 is used to normalize this curve. Thus, the normalized curve can be obtained directly. (2) If the average relative permeability curve is not available but there are many groups of relative permeability curves obtained by core analysis, the following method should be adopted. First, normalize every relative permeability curve using eq 23. Then, interpolate each normalized curve to obtain kro*(Sw*) and krw*(Sw*) at the same Sw*. Finally, calculate the arithmetic means of kro*(Sw*) and krw*(Sw*) at the same Sw* using eq 24 Sw Swir kro ðSw Þ krw ðSw Þ Sw ¼ kro ðSw Þ ¼ krw ðSw Þ ¼ 1 Swir Sor krowc krwor
ð23Þ
where Sw* is the normalized water saturation (fraction), kro(Sw*) is oil relative permeability (fraction), krw(Sw*) is water relative permeability (fraction), kro*(Sw*) is the normalized oil relative permeability (fraction), krw*(Sw*) is the normalized water relative permeability (fraction), and krowc is oil relative permeability at irreducible water saturation (constant value of 1 in this paper). m
k̅ ro ðSw Þ ¼
∑ ½kro ðSw Þi i¼1 m
Figure 16. Relative permeability curves predicted from typical petrophysical data.
where k0 rowc and k0 rwor are oil relative permeability at irreducible water saturation and water relative permeability at residual oil saturation of the relative permeability curve predicted (fraction), respectively, k0 ro(Sw*) and k0 rw(Sw*) are oil relative permeability and water relative permeability predicted (fraction), respectively. A prediction model for the wateroil relative permeability curve is obtained by submitting the end-point models into the characterization model for the wateroil relative permeability curve. Figure 14 shows the calculation procedures of this model.
m
k̅ rw ðSw Þ ¼
∑ ½krw ðSw Þi i¼1 m
ð24Þ
where kro*(Sw*) is the normalized average oil relative permeability (fraction), krw*(Sw*) is the normalized average water relative permeability (fraction), and m is the number of relative permeability curves. 5.2. Prediction of a Certain Relative Permeability Curve. Here are the prediction procedures: (1) Calculate the end-point values of the relative permeability curve according to air permeability and porosity. (2) Compute the corresponding normalized water saturation Sw* of the arbitrary water saturation Sw using eq 23. Then, carry out interpolation in the normalized average relative permeabiliy curve to obtain kro*(Sw*) and krw*(Sw*). (3) Obtain oil relative permeability k0 ro(Sw*) and water relative permeability k0 rw(Sw*) at the above water saturation Sw using the inverse transformation formula shown as eq 25 k0 ro ðSw Þ ¼ k̅ ro ðSw Þk0 rowc k0 rw ðSw Þ ¼ k̅ rw ðSw Þk0 rwor
ð25Þ
6. VALIDATION OF THE QUANTITATIVE PREDICTION MODEL FOR THE WATEROIL RELATIVE PERMEABILITY CURVE With regard to the 55 relative permeability curves, normalization work has been carried out and then the normalized average relative permeability curve is obtained, which is shown as Figure 15. On this basis, typical petrophysical data are selected and the corresponding relative permeability curves are predicted, as shown in Figure 16. To validate the prediction model, AAE and RMSE between predicted values by this model and real values of relative permeability are calculated using eqs 8 and 9. The calculation results are shown in Table 4. For each relative permeability curve, both AAE of oil relative permeability and that of water relative permeability are calculated, then the histogram and probability density curve of AAE for 55 relative permeability curves are obtained. The results are shown as Figure 17. As can be seen from Table 4 and Figure 17, both AAE of oil relative permeability and that of water relative permeability for 4411
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Table 4. Validation Results of the Quantitative Prediction Model for the Relative Permeability Curve AAE
RMSE
kro
0.0811
0.1030
krw
0.0394
0.0478
different positions of the reservoir, this paper presents improved end-point models for the wateroil relative permeability curve, which makes the prediction procedure more realistic. (3) After end-point models for the relative permeability curve are obtained, the characterization model for the relative permeability curve is established using the normalization method of the reservoir average relative permeability curve. Finally, a prediction model for the wateroil relative permeability curve is obtained. The validation results show that the average absolute errors of oil and water relative permeability are 0.0811 and 0.0394, respectively. Therefore, this model is highly precise and can meet the engineering requirements. In addition, this model provides a calculation theory of the wateroil relative permeability curve for further reservoir simulation using different relative permeability curves in different grid cells.
’ AUTHOR INFORMATION Corresponding Author
*Telephone: 86-546-8395660. Fax: 86-546-8395660. E-mail:
[email protected].
’ ACKNOWLEDGMENT The authors greatly appreciate the financial support of the National Natural Science Foundation of China (Grants 10772200 and 10972237), the Important National Science and Technology Specific Projects of China (Grant 2011ZX05011), the Fundamental Research Funds for the Central Universities (Grant 10CX03002A), and the Graduate Innovation Fund of the China University of Petroleum (Grant S10-07). ’ REFERENCES Figure 17. Histogram and probability density curve of AAE of relative permeability for 55 relative permeability curves.
55 relative permeability curves are in accordance with log-normal probability distribution; therefore, AAE corresponding to the peak value of the frequency number distribution is small. These results indicate that, for most relative permeability curves, the established prediction model is precise enough and can meet the engineering requirements.
7. CONCLUSION (1) The empirical models in well logging have been introduced and modified. On the basis of the statistical analysis technique, end-point models for the relative permeability curve have been established. The residual oil saturation shows a power function relationship with the irreducible water saturation. They are both affected by air permeability and porosity, while water relative permeability at residual oil saturation is only dependent upon air permeability. (2) The above end-point models predict the end-point values of the relative permeability curve directly by petrophysical properties. This may cause all predicted relative permeability curves to shift to the left or right compared to the average relative permeability curve. For these reasons, taking into consideration the correction effect of the average relative permeability curve on predicting relative permeability curves in
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