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New correlations to predict fracture conductivity based on the rock strength
Author’s Accepted Manuscript New Correlations to Predict Fracture Conductivity Based on the Rock Strength Mohammadreza Akbari, Mohammad Javad Ameri, Sina Kharazmi, Yaser Motamedi, Maysam Pournik www.elsevier.com/locate/petrol
PII: DOI: Reference:
S0920-4105(17)30361-3 http://dx.doi.org/10.1016/j.petrol.2017.03.003 PETROL3891
To appear in: Journal of Petroleum Science and Engineering Received date: 17 April 2016 Revised date: 25 September 2016 Accepted date: 1 March 2017 Cite this article as: Mohammadreza Akbari, Mohammad Javad Ameri, Sina Kharazmi, Yaser Motamedi and Maysam Pournik, New Correlations to Predict Fracture Conductivity Based on the Rock Strength, Journal of Petroleum Science and Engineering, http://dx.doi.org/10.1016/j.petrol.2017.03.003 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
New Correlations to Predict Fracture Conductivity Based on the Rock Strength Mohammadreza Akbaria, Mohammad Javad Ameria*, Sina Kharazmia, Yaser Motamedia,
Maysam Pournikb a
Faculty of Petroleum Engineering, Amirkabir University of Technology, Tehran, Iran
b
Mewbourne School of Petroleum & Geological Engineering, University of Oklahoma, OK, USA
*
Address correspondence to: Faculty of Petroleum Engineering, Amirkabir University of Technology, Tehran, Iran.
[email protected]
Abstract Acid fracturing is one of the applicable methods to stimulate oil and gas wells and increase the production rate in a carbonate reservoir. Acid fracture conductivity is an important parameter for the designing a fracture job. The amount of rock dissolved, fracture surface etching patterns, rock strength, and closure stress impact the resulting acid fracture conductivity. A model of acid fracture conductivity must accurately anticipate the fracture conductivity under closure stress. The rock strength affects substantially the fracture conductivity. A serious challenge of recent studies has been to predict the behavior of different formations under various closure stresses. This study developed a robust intelligent model based on genetic algorithm to precisely predict the fracture conductivity by incorporating experimental data from various formations, whereby resulted in a good match between the model predictions and the experimental data. The effect of rock strength was investigated on the fracture conductivity under various closure stresses. The results show the rock strength plays a significant role when anticipating fracture conductivity, as various formations have behaved differently under different closure stresses. Furthermore, due to the complexity of rock behavior at high closure stress, the results showed that as the closure stress increased the precision of all predictive correlations decreased considerably. Therefore, the fracture conductivity must be anticipated cautiously at high closure stresses by various predictive correlation particularly, in soft formation.
Nomenclature Dissolved rock equivalent conductivity (md-ft)
Closure stress (psi) φ
Formation porosity
hf
Fracture height (in)
n
Number of data (SRE)
Rock embedment strength (psi)
V
Volume of acid injected (in3)
w
Channel width (in) Fracture conductivity (md-ft)
X
Acid dissolving power
xf
Acid penetration distance (in)
Keywords: Acid Fracture Conductivity, Genetic Algorithms, Fracture Conductivity Correlations, Correlation Coefficient Optimization, Rock Strength
1. Introduction Acid fracturing is one of the preferred methods to stimulate the wells in oil and gas carbonate reservoirs. For this purpose, an acid is injected at high pressure to overcome compressive earth stresses and the formation’s tensile strength and propagate a two-wing crack away from the wellbore. Acid injected into the fracture reacts with the carbonate formation and this causes the fracture surface is etched (Economides, 2000). In a heterogeneous formation, this treatment has been proved to be a successful method to improve production rate (California Council, 2014). The created conductivity under the closure stress plays a significant role in the success of acid fracturing jobs. Since acid fracture conductivity is defined as a measure of the capacity for fluid flow through an acidized fracture, the resulting conductivity is influenced by the amount of rock dissolved, the pattern in which the rock is dissolved, the strength of pillars that prop the fracture open, and the amount of closure stress on the fracture (Pournik, 2008). Optimization of the live acid penetration distance and conductivity are the primarily targets of designing an acid fracturing treatment. Acid type and strength, acid injection volume, and injection rate are the key elements in such a design. While there are many studies for live acid penetration distance, few studies have been focused on the acid fracture conductivity, and more studies are necessary to improve precision of conductivity predictions (Nierode and Kruk, 1973, van Domselaar et al.,
1973; Coulter et al., 1974; Roberts and Guin, 1975; Anderson and Fredrickson, 1987, Beg, 1996, Ruffet & Fery, 1997, and Gong et. al., 1999). As the resulting conductivity inherently depends on a stochastic process and it is affected by a wide range of parameters, the prediction of fracture conductivity is too difficult (Mou, 2009). There have been many experimental and theoretical studies to improve the understanding of acid fracture performance. There has been also extensive work on the fluid flow in rough-walled natural fractures. However, very few studies have concentrated on fracture conductivity, especially on developing a precise correlation for the prediction of conductivity (Pournik, 2008).
2. Fracture Conductivity Models Williams et al. (1979) assumed fracture walls are uniformly dissolved and estimated the ideal fracture conductivity. In their model, the fracture width was calculated by equation 1. (1)
Where w is channel width (in), V is volume of acid injected (in3), X is Acid dissolving power, φ is formation porosity, xf is acid penetration distance (in) and hf is fracture height (in). Based on the cubic law (Zimmerman and Bodvarsson, 1996) for the channel width, the fracture conductivity is defined as equation 2. ( )
(2)
Where wkf is the conductivity in md-in. Since the equation doesn’t consider the effect closure stress on the conductivity, the model overestimates the conductivity measured in laboratory Nierode and Kruk (1973) developed an empirical correlation based on the experimental data on acid fracture conductivity from a laboratory test that is the most widely used in industry. The resulting conductivity is based on the volume of rock dissolved and the rock’s mechanical strength. Equations 3 to 5 were generated according to the experimental study conducted on different formation samples acidized with HCl acid (Nierode and Kruk, 1973). The amount of rock dissolved, rock strength and closure stress were the effective parameters on the estimation of conductivity. The correlation considers the role of closure stress exponentially in fracture conductivity.
(3) (4)
{
(5)
Where σc is the closure stress in psi, SRE is the rock embedment strength representing the rock strength in psi. The correlation calculates conductivity at zero closure stress using an empirical correlation as a function of ideal fracture width, which is the dissolved rock volume divided by the fracture area. DREC derived according to the cubic law indicates the acid dissolving power and the total volume of injected acid proportional to the geometry of the fracture created. The relationship between DERC and the fracture width is defined as equation 6 (Gomaa and Nasr-ElDin, 2009). (6)
By Substituting Eq.6 into Eq.4, the power of width is 2.466 and fracture conductivity is predicted in the lower bound. Anderson et al. (1989) found that the effect of formation characteristics such as hardness on the etched conductivities is very significant. For example, etched surfaces of chalk formations can be easily closed due to its natural softness. Nasr-El-Din et al. (2006) stated that the correlations developed by Nierode and Kruk (1973) were not categorized by lithology. The correlation was recalculated by graphing and evaluating the data again both as a lumped set and as individual sets by lithology. The power of width is 2.255 in their correlation for the lump set. The value is 2.623 and 1.677 for limestone and dolomite formations respectively. Their modified correlations kept the same form as the original ones, but made the constants different and were more precise (Gomaa & Nasr-El-Din, 2009). The objective of this work is to develop a precise correlation to estimate conductivity for acid fracturing treatment based on a new approach in evaluation algorithms. Due to the complication of predicting acid fracture conductivity and also the capability of genetic algorithm (GA) in optimization and modelling, in this paper, a robust intelligent model based on genetic algorithm is developed to predict acid fracture conductivity accurately. The input data are considered according to the rock strength to provide a better understanding of the rock strength effect. Ultimately, the resulting conductivity given by GA models is compared to the prior models.
3. Methodology Genetic algorithms are applied as a type of optimization algorithm to find the optimal solution(s) to a given computational problem that maximizes or minimizes an objective function. Genetic Algorithms are direct, parallel, stochastic method for the global search and optimization which imitate the evolution of the living beings, Charles Darwin. This adaptive method may be used to solve search and optimization problems. Genetic Algorithm, which is based on the genetic process of biological organisms, is used to solve both constrained and unconstrained optimization problems. It could be applied to a variety of problems that are not well suited for the standard optimization algorithms particularly the problems with nonlinear objective function) (Kosters et al., 1999). Sometimes, genetic algorithms are applied as a process of optimization coefficients in the engineering and mathematics. Mathematical modelling to generate a functions is the first step to solve this type of problems. Then to find a solution, the coefficients that optimize the model or the function components that provide an optimal system performance are discovered (Goldberg, 1989). In this research, a genetic algorithm is used to optimize the coefficients of Nierode and Kruk correlation. Three modified correlations based on all input data, the samples in which the rock embedment strength is lower than 20000 psi and the samples in which rock embedment strength is higher than 20000 psi are developed. In the next step, three new correlations are generated according to the three data categories. In the new correlations the different functions were applied and then optimized their coefficients to predict precisely fracture conductivity. The quality of the fracture conductivity prediction is quantified in terms of the percentage of average absolute relative error (AARE) and correlation coefficient (R) by comparison with experimental data. Equations 7 and 8 are applied to calculate the performance criteria. ∑
[((
)
(̅̅̅̅̅̅)
)((
[ ((
)
(̅̅̅̅̅̅ )
) ((
(̅̅̅̅̅̅ )
)
)]
(7) √∑
∑
|(
)
( (
)
)
(̅̅̅̅̅̅)
)
|
) ]
(8)
3.1 Data analysis Nierode and Kruk experimental data sets are applied as modeling data. 106 experimental data set including dissolve rock equivalent conductivity as treatment parameter, rock embedment strength and closer stress for specific fracture conductivity are considered as input data. The data sets are divided into three groups; the first includes all 106 data sets to develop the correlations. The second group consists of 36 data sets in which the rock embedment strength is lower than 20000 psi. The third group includes 70 data set in which the rock embedment strength is upper than 20000 psi. Table 1 shows the properties and domain of all input variables in detail. The lithology 55 data sets are limestone, 51 data sets are dolomite.
Table 1. Domain of input data used in this study Parameter
Min
Max
STD
Mean
DREC (md-ft)
3.E+04
1.30E+08
107826007
6.E+07
Closure Stress (Psi)
0
7000
2379.809
2939
Rock Strength (Psi)
5600
88100
1802.509
35170
Fracture Conductivity (md-ft)
1.2
7400000
1264474.65
2.E+05
3.2. Development of a genetic algorithm to predict acid fracture conductivity The genetic algorithm is a space search method developed according to the mechanism of natural selection and survival of the fittest. In this algorithm, a random population of individuals start the evolution process (Holland, 1992). Each individual in the population is ranked based on the absolute relative error (ARE) between the experimental and the calculated fracture conductivity values. Operators including selection, crossover and mutation are applied to evolve the initial population by minimizing ARE. The selection operator selects individuals for crossover based on the fitness. Here, the stochastic uniform methods are used to select individuals for crossover. In this method, the probability of selection is proportional to the individuals fitness. The genetic material of the selected parents are recombined by the crossover operator. The elements from the first parent are selected by the crossover operator. In this process, the chromosomes from the parents exchange systematically using probabilistic decision. In other word, change occurs during reproduction. The mutation operator modifies individuals that have not been selected for
reproduction by randomly changing weights. In this work, the uniform mutation algorithm is used to select the weights for the mutation. Each one of the selected entries has 10 to 50% probability of being mutated according to the type of correlations. Mutated entries are replaced with a uniformly random value. After determination of a population as a new generation, the process continues until some criterion is met (Goldberg, 1989). Due to the stochastic nature of Genetic Algorithm, it is difficult to formally specify a convergence criterion. In this reseach, the algorithm stopped after 100 generations or if no improvement is observed over a pre-specified number of generations (in this case 50). The general flow chart of GA applied in this study is shown in figure 1. The parameters used to perform Genetic Algorithm and their description are listed in Table 2. The values of these parameters are shown in table 3.
7 Fig 1. GA procedure chart.
Table 2. The parameters used to set Genetic Algorithm Chromosome or Individual
A group of genes A part of chromosome; a gene contains a part of solution. It determines the
Gene
solution Number of individuals present with same length of chromosome
Population
The value assigned to an individual based on how far of close an individual is
Fitness
from the solution; greater the fitness value better the solution it contains Fitness function
A function that assigns fitness value to the individual
Selection
Selecting individuals for creating the next generation
Crossover
Exchanging the chromosomes from the parents using probabilistic decision.
Mutation
Changing a random gene in an individual
Table 3. The values of GA parameters for the modified and the new correlations Input Data
Correlation
N Pop
Pc
Pm
mu
Modified Correlation
3000
0.80
0.30
0.30
New Correlation
10000
0.90
0.10
0.30
Modified Correlation
4000
0.80
0.40
0.50
New Correlation
4000
0.80
0.50
0.70
Modified Correlation
4000
0.80
0.30
0.30
New Correlation
3000
0.80
0.30
0.30
All Data
RES<20000
RES>20000
4. Results and Discussion The GA optimization algorithm was run for three different categories. Six correlations including three modified and three new correlations were developed to predict the acid fracture conductivity. The results show rock strength is a significant parameter to determine acid fracture conductivity. The predicted values of fracture conductivity by the proposed correlations that were developed according to rock strength have a good match with the experimental results. The comparison of accuracy of Nierode & Kruk, Nasr-El-Din, the modified and the new correlations are shown in figure 2. As the results illustrate, the new correlation is more precise compared to the other correlations in different rock strength.
Fig 2: Average absolute relative error of the predictive correlation
Table 4 shows coefficients of the modified correlation in which all the experimental data were considered as input data. The correlation predicts the fracture conductivity with a high error. In the correlation the power of ideal width is 2.226. In order to develop a precise correlation for the category, the new functions were used to find the new correlation. GA optimization algorithm was run to find the best coefficients for the proposed functions. The new correlation and their coefficients are tabulated in table 5. The percentage of average absolute relative error of different correlations to predict fracture conductivity based on all input data were compared in table 6. Although the precision of the modified correlation has been improved, the AARE of the new correlation is better than common correlations (Nierode & Kruk and Nasr-El-Din’s correlations). Figure 3 illustrates the predicted values of conductivity using different correlations versus the measured values. As shown, Nierode & Kruk’s correlation overestimates the values of conductivity, while the modified and the new correlations underestimate these values.
Table 4: Coefficients of the modified correlation for all input Correlation
Coefficients 0.086 0.4311 0.742 0.3003 0.0246 0.5479
Table 5. Proposed correlation and their coefficients to predict fracture conductivity based on all input data Correlation
Coefficients 0.5
93
0.85
2.94
-14
-1.92
0.258
0.0004
0.27
2
0.05
Table 6. AARE of predictive correlations to predict fracture conductivity based on all input data Correlation
AARE (%)
R-Squared
N&K
60.52
0.77
Nasr El Din
53.55
0.81
Modified Correlation
40.18
0.62
New Correlation
34.55
0.85
Fig 3. Comparison between the measured and predicted values of conductivity by predictive correlations for all input data
The effect of rock strength was considered to provide a better understanding by running GA algorithm and applying the data of second and third categories. Table 7 shows coefficients of the modified correlation for rocks in which RES is less than 20000 Psi. The percentage of average absolute relative error of the correlation is 39.67%. The power of ideal width is 1.01 for the correlation. In this category, new functions were used to improve the accuracy of predictive correlation. Table 8 indicates the proposed correlation and their coefficients. The correlation has 14 coefficients. The results show the new correlation is more precise and complicated compared to the other correlations (Table 9). The comparison of the measured and calculated values of conductivity using predictive correlations has been depicted in figure 4. The predicted values by Nierode & Kruk’s correlation have the most deviation from the experimental data in this category, too.
Table 7. Coefficients of the modified correlation for rock with RES<20000 Psi Correlation
Coefficients 0.9124 100 0.337 0.5125 0.0412 0.0134
Table 8. Proposed correlation and their coefficients to predict fracture conductivity for rocks with RES<20000 Psi Correlation
Coefficients 0.2503 0.3926
0.1509 0.3444
-90 -2 0.7998 0.0341 0.4419
75 0.952 35 -24 0.0051
Table 9. Accuracy of predictive correlations to predict fracture conductivity for rock with RES<20000 Psi Correlation
AARE (%)
R-Squared
N&K
49.84
0.21
Nasr El Din
39.67
0.61
Modified Correlation
22.36
0.68
New Correlation
21.12
0.84
Fig 4. Comparison between the measured and predicted values of conductivity by predictive correlations for data with RES< 20000 psi.
Figure 5 illustrates the percentage of average absolute relative error of the different correlations versus closure stress for rocks in which RES is lower than 20000 Psi. There is an increasing trend for error as the closure stress increases. The results show that all correlations anticipate acid fracture conductivity relatively precisely at low and middle closure stresses. However, at high closure stresses, the predicted values have been deviated from the experimental data considerably. This reveals the significance and the intricate role of the closure stress on etching conductivity. In other words, predicting fracture conductivity is too difficult at high closure stresses and it is an uncertainty when seeking to control the acid fracturing process. Nevertheless, the new and the modified correlations are much more accurate compared to the previous correlations even at high closure stress.
Fig 5. The percentage of average absolute relative error of the predictive correlations versus closure stress for rocks with RES< 20000 psi
Table 10 shows the coefficients of the modified correlation for the samples in which RES is higher than 20000 Psi. The percentage of average absolute relative error of the correlation is 15.89%. Table 11 indicates the proposed correlation and their coefficients. In the correlation the power of ideal width is 2.475.
The percentage of average absolute relative error of the
correlations to estimate fracture conductivity for the category were compared in table 12. The AARE of the new correlation is 14.74% that shows the accuracy of the correlation has been improved slightly. In these categories, the values predicted by Nierode & Kruk’s correlation has the most error in comparison with the other correlations. Figure 6 illustrates the predicted values of conductivity using different correlations versus the measured values. As shown, N&K’s correlation overestimates the values of conductivity, while the modified and the new correlations underestimate these values with limited error.
Table 10. Coefficients of the modified correlation for rock with RES>20000 Psi Correlation
Coefficients 0.037 0.1664 0.825 -78 -10
0.0008
Table 11. Proposed correlation and their coefficients to predict fracture conductivity for rocks with RES>20000 Psi Correlation
(
Coefficients
)
2.068
52
0.9851
33
13
-4
2.2663
-0.5915
0.0879
0.7526
-40
0.9063
Table 12. Accuracy of predictive correlations to predict fracture conductivity for rock with RES>20000 Psi Correlation
AARE (%)
R-Squared
N&K
18.45
0.82
Nasr El Din
16.57
0.83
Modified Correlation
15.89
0.85
New Correlation
14.74
0.88
Fig 6. Comparison between the measured and predicted values of conductivity by predictive correlations for data with RES> 20000 psi.
The results of the second and the third category reveal that formation strength plays a substantial role in acid fracture conductivity prediction and universal correlations are not recommended to anticipate the fracture conductivity. On the other hand, the new correlations consider the role of DREC exponentially in fracture conductivity. The relationships differ from the modified correlations. DREC behaves as power-law and the power of DREC is less than 1 similar to the previous empirical correlations in the modified correlations. In other words, the relation between the average width and fracture conductivity is demonstrated to be different by the new and the modified correlations. Furthermore, the accuracy of the modified and the new correlations that were developed for the second and the third categories are the same nearly. Although the new correlations predict the fracture conductivity slightly more accurate, these correlations are too complicated and it is not recommended for initially and fast estimation of fracture conductivity. Moreover, as the rock strength increases (RES), the accuracy of predictive correlations improve dramatically. It means the behaviour of rocks in which RES is high are more reasonable under different closure stresses. In other word, an acid fracturing treatment must be designed cautiously, particularly at high closure stresses in soft formations.
5. Conclusion 1- GA is a powerful tool to precisely and rapidly predict the acid fracture conductivity at different closure stresses as compared to the prior well-known correlations. The fracture conductivity predicted by GA correlation closely matched with the experimental data. It is recommended that the GA correlations is applied in the commercial software instead of previous empirical correlations (i.e. Nierode & Kruk correlations) to precisely predict acid fracture conductivity. 2- Rock strength plays a significant role when anticipating fracture conductivity, as various formations have behaved differently under different closure stresses. The behaviour of strong rocks are more reasonable under different closure stresses. While the prediction of the behaviour soft rocks are much more complicated. Therefore, it is not possible to present a comprehensive precise model to predict acid fracture conductivity for all rock types. It is recommended that the predictive fracture conductivity correlations are developed based on rock strength.
3- Although the modified correlations show a good match with the experimental data, the new correlations are more precise than them. It also means that the new correlations developed in this study, give a better approximation of acid fracture conductivity than the available correlations of Nierode & Kruk and Nasr el Din. However, new correlations are more complicated compared to the previous correlations. Nonetheless, in order to estimate fracture conductivity rapidly, the modified correlations are recommended. 4- The new correlations consider the role of DREC exponentially in fracture conductivity. While in the modified correlations, DREC behaves as power- law and the power of DREC is less than 1. 5- The prediction of fracture conductivity is too difficult at high closure stresses due to the complexity of rock behaviour under these tensions. As the closure stress increases, the accuracy of all predictive correlations decreases dramatically. Therefore, the fracture conductivity must be anticipated with more cautious at high closure stresses by every predictive correlation. Furthermore, as the rock strength increases (RES), the precision of correlations improve considerably. It indicates the strong formations behave reasonably under different closure stresses.
Appendix A. Nierode and Kruk’s experimental data and the predicted values by GA modified correlations The fracture conductivity predicted by GA modified correlations for rock with RES<20000 Psi Closure Stress psi
Fracture Conductivity (Experimental), md-in
11800
0
9.13
Fracture Conductivity (GA modified correlation), mdin 8.11
11800
1000
7.37
6.56
3.30E+04
11800
3000
3.80
3.48
3.30E+04
11800
5000
0.26
0.39
Clearfork Dolomite
3.90E+06
12200
0
12.25
9.72
6
Clearfork Dolomite
3.90E+06
12200
1000
11.27
8.19
7
Clearfork Dolomite
3.90E+06
12200
3000
9.21
5.1
8
Clearfork Dolomite
3.90E+06
12200
5000
7.31
2.08
9
Clearfork Dolomite
3.90E+06
12200
7000
5.29
0.96
10
Capps Limestone
3.20E+05
13000
0
9.18
8.87
Rock Type
DREC, md-in
RES, psi
1
Clearfork Dolomite
3.30E+04
2
Clearfork Dolomite
3.30E+04
3
Clearfork Dolomite
4
Clearfork Dolomite
5
No.
11
Capps Limestone
3.20E+05
13000
1000
8.34
7.38
12
Capps Limestone
3.20E+05
13000
3000
6.63
4.39
13
Capps Limestone
3.20E+05
13000
5000
4.94
1.40
14
Capps Limestone
3.20E+05
13000
7000
3.21
1.58
15
Indiana Limestone
3.10E+08
14300
0
15.81
11.19
16
Indiana Limestone
3.10E+08
14300
1000
14.50
9.74
17
Indiana Limestone
3.10E+08
14300
3000
11.85
6.85
18
Indiana Limestone
3.10E+08
14300
5000
9.21
3.96
19
Indiana Limestone
3.10E+08
14300
7000
6.58
1.07
20
Clearfork Dolomite
8.30E+06
14400
0
12.43
9.97
21
Clearfork Dolomite
8.30E+06
14400
1000
10.57
8.53
22
Clearfork Dolomite
8.30E+06
14400
3000
6.90
5.64
23
Clearfork Dolomite
8.30E+06
14400
5000
3.21
2.76
24
Cisco Limestone
3.00E+05
14800
0
8.85
8.85
25
Cisco Limestone
3.00E+05
14800
1000
8.13
7.42
26
Cisco Limestone
3.00E+05
14800
3000
6.68
4.56
27
Cisco Limestone
3.00E+05
14800
5000
5.24
1.71
28
Cisco Limestone
3.00E+05
14800
7000
3.78
1.14
29
Clearfork Dolomite
3.20E+06
16600
0
11.28
9.65
30
Clearfork Dolomite
3.20E+06
16600
1000
9.61
8.28
31
Clearfork Dolomite
3.20E+06
16600
3000
6.17
5.53
32
Clearfork Dolomite
3.20E+06
16600
5000
2.77
2.7
33
San Andres Dolomite
3.40E+06
17300
0
9.14
9.67
34
San Andres Dolomite
3.40E+06
17300
1000
7.93
8.32
35
San Andres Dolomite
3.40E+06
17300
3000
5.52
5.62
36
San Andres Dolomite
3.40E+06
17300
5000
3.13
2.93
The fracture conductivity predicted by GA modified correlations for rock with RES>20000 Psi Closure Stress psi
Fracture Conductivity (Experimental), md-in
Fracture Conductivity (GA modified correlation), md-in
21500
0
13.58
12.35
21500
1000
12.61
11.67
2.80E+07
21500
3000
10.66
10.33
Indiana Limestone
2.80E+07
21500
5000
8.75
8.98
Indiana Limestone
2.80E+07
21500
7000
6.80
7.63
6
Indiana Limestone
4.50E+06
22700
0
13.03
10.84
7
Indiana Limestone
4.50E+06
22700
1000
11.91
10.15
8
Indiana Limestone
4.50E+06
22700
3000
9.61
8.77
9
Indiana Limestone
4.50E+06
22700
5000
7.31
7.39
10
Indiana Limestone
4.50E+06
22700
7000
5.01
6.01
Rock Type
DREC, md-in
RES, psi
1
Indiana Limestone
2.80E+07
2
Indiana Limestone
2.80E+07
3
Indiana Limestone
4 5
No.
11
Cisco Limestone
2.00E+06
25300
0
11.85
10.17
12
Cisco Limestone
2.00E+06
25300
1000
11.03
9.45
13
Cisco Limestone
2.00E+06
25300
3000
9.47
8.01
14
Cisco Limestone
2.00E+06
25300
5000
7.90
6.55
15
Cisco Limestone
2.00E+06
25300
7000
6.34
5.108
16
Capps Limestone
2.90E+05
30100
0
9.79
8.58
17
Capps Limestone
2.90E+05
30100
1000
8.82
7.80
18
Capps Limestone
2.90E+05
30100
3000
6.84
6.25
19
4.86
4.69
Capps Limestone
2.90E+05
30100
5000
20
Capps Limestone
2.90E+05
30100
7000
2.89
3.13
21
Canyon Limestone
4.60E+07
30700
0
13.59
12.76
22
Canyon Limestone
4.60E+07
30700
1000
12.87
11.98
23
Canyon Limestone
4.60E+07
30700
3000
11.45
10.41
24
Canyon Limestone
4.60E+07
30700
5000
10.04
8.84
25
Canyon Limestone
4.60E+07
30700
7000
8.59
7.27
26
Clearfork Dolomite
3.60E+04
35000
0
8.13
6.86
27
Clearfork Dolomite
3.60E+04
35000
1000
7.43
6.03
28
Clearfork Dolomite
3.60E+04
35000
3000
6.01
4.38
29
Clearfork Dolomite
3.60E+04
35000
5000
4.60
2.73
30
Clearfork Dolomite
3.60E+04
35000
7000
3.17
1.09
31
Canyon Limestone
2.70E+08
46400
0
14.28
14.22
32
Canyon Limestone
2.70E+08
46400
1000
13.42
13.31
33
Canyon Limestone
2.70E+08
46400
3000
11.77
11.48
34
Canyon Limestone
2.70E+08
46400
5000
10.04
9.66
35
Canyon Limestone
2.70E+08
46400
7000
8.39
7.84
36
San Andres Dolomite San Andres Dolomite San Andres Dolomite San Andres Dolomite San Andres Dolomite San Andres Dolomite San Andres Dolomite San Andres Dolomite San Andres Dolomite San Andres Dolomite San Andres Dolomite San Andres Dolomite San Andres Dolomite San Andres Dolomite
1.00E+06
46500
0
11.32
9.60
1.00E+06
46500
1000
10.59
8.69
1.00E+06
46500
3000
9.16
6.86
1.00E+06
46500
5000
7.6
5.04
1.00E+06
46500
7000
6.25
3.21
1.90E+07
62700
0
12.25
12.03
1.90E+07
62700
1000
11.45
11.02
1.90E+07
62700
3000
9.85
9.01
1.90E+07
62700
5000
8.21
7.008
1.90E+07
62700
7000
6.58
4.99
5.10E+08
63800
0
13.99
14.74
5.10E+08
63800
1000
13.52
13.73
5.10E+08
63800
3000
12.61
11.71
5.10E+08
63800
5000
11.69
9.69
37 38 39 40 41 42 43 44 45 46 47 48 49
50
San Andres Dolomite
5.10E+08
63800
7000
10.75
7.67
51
Cisco Limestone
1.20E+05
67100
0
7.82
7.85
52
Cisco Limestone
1.20E+05
67100
1000
7.17
6.83
53
Cisco Limestone
1.20E+05
67100
3000
5.82
4.77
54
Cisco Limestone
1.20E+05
67100
5000
4.47
2.72
55
Cisco Limestone
1.20E+05
67100
7000
3.13
0.67
56
San Andres Dolomite San Andres Dolomite San Andres Dolomite San Andres Dolomite San Andres Dolomite San Andres Dolomite San Andres Dolomite San Andres Dolomite San Andres Dolomite San Andres Dolomite
2.40E+06
76500
0
9.85
10.32
2.40E+06
76500
1000
8.82
9.26
2.40E+06
76500
3000
6.74
7.12
2.40E+06
76500
5000
4.60
4.99
2.40E+06
76500
7000
2.56
2.86
2.70E+06
76600
0
9.30
10.42
2.70E+06
76600
1000
8.57
9.35
2.70E+06
76600
3000
7.09
7.22
2.70E+06
76600
5000
5.59
5.08
2.70E+06
76600
7000
1.79
2.95
57 58 59 60 61 62 63 64 65 66
Canyon Limestone
1.30E+08
88100
0
14.07
13.62
67
Canyon Limestone
1.30E+08
88100
1000
13.54
12.50
68
Canyon Limestone
1.30E+08
88100
3000
12.64
10.28
69
Canyon Limestone
1.30E+08
88100
5000
10.77
8.06
70
Canyon Limestone
1.30E+08
88100
7000
8.82
5.84
Appendix B. Genetic algorithm parameters in Initial and general conditions The required primary pieces of information for starting GA is: 1. Population size (Npop): A group of interbreeding individuals 2. Crossover rate (Pc): Crossover probability 3. Mutation rate (Pm): Mutation probability The general stages in genetic algorithm are as follows: 1. Initialization 1.1 Set the parameters (Npop, Pc, Pm, stopping criteria, selection strategy, crossover operation, mutation operation, and number of generation) 1.2 Generate an initial population randomly 2. Compute and save the fitness for each individual in the current population 3. Define selection probabilities for each individual based on fitness criteria
4. Generate the next population by selecting individuals from current population randomly to produce offspring via GA operators such as crossover and mutation operators 5. Repeat step 2 until stopping criteria is satisfied. According to what follows, the proposed GA is described in details.
References Anderson, M.S., and Fredrickson, S.E. 1989. Dynamic Etching Tests Aid Fracture- Acidizing Treatment Design. SPE Production Engineering 4(4): 443-449. SPE-16452. Beg, M.S., Kunak, A.O., Gong, M., Zhu, D., and Hill, A.D. 1998. A Systematic Experimental Study of Acid Fracture Conductivity. SPEPF 13(4): 267-271. SPE-52402. California Council on Science and Technology Lawrence Berkeley National Laboratory Pacific Institute, 2014. Advanced Well Stimulation Technologies in California An Independent Review of Scientific and Technical Information. Coulter, A.W., Alderman, E.N., Cloud, J.E., and Crowe, C.W. 1974. Mathematical Model Simulates Actual Well Conditions in Fracture Acidizing Treatment Design. Paper SPE-5004 presented at the SPE_AIME 49th Annual Fall Meeting, Houston, Texas, USA, 6-9 October 1974. Economides, M. J. and Nolte, K. G. 2000. Reservoir Stimulation, 3rd ed., John Wiley & Sons, Hoboken, N.J. and Chichester. Gomaa, A.M., and Nasr-El-Din, H.A. 2009. Acid Fracturing: the Effect of Formation Strength on Fracture Conductivity. Paper SPE-119623 presented at the 2009 SPE Hydraulic Fracturing Technology Conference, Woodlands, Texas, U.S.A. Gong, M., Lacote, S., and Hill, A.D. 1999. New Model of Acid-Fracture Conductivity Based on Deformation of Surface Asperities. SPEJ 4(3): 206-214. SPE-57017.
Holland, J.H., 1992. Adaptation in Natural and Artificial Systems, MIT Press, Cambridge, MA. Kosters W.A., Kok J.N. and Floreen P., 1999. Fourier analysis of Genetic Algorithms, Theoretical Computer Science, Elsevier, 229, 199, pp. 143-175. Mou, J. 2009. Modeling Acid Transport and Non-Uniform Etching in a Stochastic Domain in Acid Fracturing. PhD dissertation. Texas A&M University, College Station. Nasr-El-Din, H.A., Al-Driweesh, S.M., Metcalf, A.S., and Chesson, J.B. 2008. Fracture Acidizing: What Role Does Formation Softening Play in Production Response? SPE Production & Operations 23(2): 184-191. SPE-103344. Nierode, D.E., and Kruk, K.F. 1973. An Evaluation of Acid Fluid Loss Additives, Retarded Acids, and Acidized Fracture Conductivity. Paper SPE-4549 presented at the 48th Annual Fall Meeting of Society of Petroleum Engineering of AIME, Las Vegas, Nevada, USA, 1973. Goldberg, D.E. 1989, Genetic algorithms in search, optimization and machine learning, AddisonWesley, Reading, MA, Pournik, M. 2008. Laboratory-Scale Fracture Conductivity Created by Acid Etching. PhD dissertation. Texas A&M University, College Station, Texas, USA. Roberts, L.D., and Guin, J.A. 1975. The Effects of Surface Kinetics in Fracture Acidizing. SPEJ 8: 385-395. SPE-4349. Ruffet, C.S., Fery, J.J., and Onaisi, A. 1997. Acid-Fracturing Treatment: A Surface- Topography Analysis of Acid-Etched Fractures to Determine Residual Conductivity. Paper SPE-38175 presented at the SPE European Formation Damage Conference, The Hague, The Netherlands, 2-3 June 1997. Valdo F. Rodrigues, SPE, Wellington Campos, SPE, Ana C. R. Medeiros, SPE, North Fluminense State University; and Rodolfo A. Victor, Petrobras/RH/UP, 2011. Acid-Fracture Conductivity Correlations for a Specific Limestone Based on Surface Characterization. Van Domselaar, H.R., Schols, R.S., and Visser, W. 1973. An Analysis of the Acidizing Process in Acid Fracturing. SPEJ 8: 239-250. SPE-3748.
Zimmerman, R. W. and G. S. Bodvarsson (1996). "Hydraulic conductivity of rock fractures." Transport in Porous Media 23(1): 1-30.
Highlights
A Novel Correlations to Predict Acid Fracture Conductivity Based on the Rock Strength are developed.
The rock strength plays a significant role when anticipating fracture conductivity.
As the closure stress increases the precision of all predictive correlations decreases considerably.
The fracture conductivity must be anticipated cautiously at high closure stresses by various predictive correlation particularly in soft formation.
PII: DOI: Reference:
S0920-4105(17)30361-3 http://dx.doi.org/10.1016/j.petrol.2017.03.003 PETROL3891
To appear in: Journal of Petroleum Science and Engineering Received date: 17 April 2016 Revised date: 25 September 2016 Accepted date: 1 March 2017 Cite this article as: Mohammadreza Akbari, Mohammad Javad Ameri, Sina Kharazmi, Yaser Motamedi and Maysam Pournik, New Correlations to Predict Fracture Conductivity Based on the Rock Strength, Journal of Petroleum Science and Engineering, http://dx.doi.org/10.1016/j.petrol.2017.03.003 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
New Correlations to Predict Fracture Conductivity Based on the Rock Strength Mohammadreza Akbaria, Mohammad Javad Ameria*, Sina Kharazmia, Yaser Motamedia,
Maysam Pournikb a
Faculty of Petroleum Engineering, Amirkabir University of Technology, Tehran, Iran
b
Mewbourne School of Petroleum & Geological Engineering, University of Oklahoma, OK, USA
*
Address correspondence to: Faculty of Petroleum Engineering, Amirkabir University of Technology, Tehran, Iran.
[email protected]
Abstract Acid fracturing is one of the applicable methods to stimulate oil and gas wells and increase the production rate in a carbonate reservoir. Acid fracture conductivity is an important parameter for the designing a fracture job. The amount of rock dissolved, fracture surface etching patterns, rock strength, and closure stress impact the resulting acid fracture conductivity. A model of acid fracture conductivity must accurately anticipate the fracture conductivity under closure stress. The rock strength affects substantially the fracture conductivity. A serious challenge of recent studies has been to predict the behavior of different formations under various closure stresses. This study developed a robust intelligent model based on genetic algorithm to precisely predict the fracture conductivity by incorporating experimental data from various formations, whereby resulted in a good match between the model predictions and the experimental data. The effect of rock strength was investigated on the fracture conductivity under various closure stresses. The results show the rock strength plays a significant role when anticipating fracture conductivity, as various formations have behaved differently under different closure stresses. Furthermore, due to the complexity of rock behavior at high closure stress, the results showed that as the closure stress increased the precision of all predictive correlations decreased considerably. Therefore, the fracture conductivity must be anticipated cautiously at high closure stresses by various predictive correlation particularly, in soft formation.
Nomenclature Dissolved rock equivalent conductivity (md-ft)
Closure stress (psi) φ
Formation porosity
hf
Fracture height (in)
n
Number of data (SRE)
Rock embedment strength (psi)
V
Volume of acid injected (in3)
w
Channel width (in) Fracture conductivity (md-ft)
X
Acid dissolving power
xf
Acid penetration distance (in)
Keywords: Acid Fracture Conductivity, Genetic Algorithms, Fracture Conductivity Correlations, Correlation Coefficient Optimization, Rock Strength
1. Introduction Acid fracturing is one of the preferred methods to stimulate the wells in oil and gas carbonate reservoirs. For this purpose, an acid is injected at high pressure to overcome compressive earth stresses and the formation’s tensile strength and propagate a two-wing crack away from the wellbore. Acid injected into the fracture reacts with the carbonate formation and this causes the fracture surface is etched (Economides, 2000). In a heterogeneous formation, this treatment has been proved to be a successful method to improve production rate (California Council, 2014). The created conductivity under the closure stress plays a significant role in the success of acid fracturing jobs. Since acid fracture conductivity is defined as a measure of the capacity for fluid flow through an acidized fracture, the resulting conductivity is influenced by the amount of rock dissolved, the pattern in which the rock is dissolved, the strength of pillars that prop the fracture open, and the amount of closure stress on the fracture (Pournik, 2008). Optimization of the live acid penetration distance and conductivity are the primarily targets of designing an acid fracturing treatment. Acid type and strength, acid injection volume, and injection rate are the key elements in such a design. While there are many studies for live acid penetration distance, few studies have been focused on the acid fracture conductivity, and more studies are necessary to improve precision of conductivity predictions (Nierode and Kruk, 1973, van Domselaar et al.,
1973; Coulter et al., 1974; Roberts and Guin, 1975; Anderson and Fredrickson, 1987, Beg, 1996, Ruffet & Fery, 1997, and Gong et. al., 1999). As the resulting conductivity inherently depends on a stochastic process and it is affected by a wide range of parameters, the prediction of fracture conductivity is too difficult (Mou, 2009). There have been many experimental and theoretical studies to improve the understanding of acid fracture performance. There has been also extensive work on the fluid flow in rough-walled natural fractures. However, very few studies have concentrated on fracture conductivity, especially on developing a precise correlation for the prediction of conductivity (Pournik, 2008).
2. Fracture Conductivity Models Williams et al. (1979) assumed fracture walls are uniformly dissolved and estimated the ideal fracture conductivity. In their model, the fracture width was calculated by equation 1. (1)
Where w is channel width (in), V is volume of acid injected (in3), X is Acid dissolving power, φ is formation porosity, xf is acid penetration distance (in) and hf is fracture height (in). Based on the cubic law (Zimmerman and Bodvarsson, 1996) for the channel width, the fracture conductivity is defined as equation 2. ( )
(2)
Where wkf is the conductivity in md-in. Since the equation doesn’t consider the effect closure stress on the conductivity, the model overestimates the conductivity measured in laboratory Nierode and Kruk (1973) developed an empirical correlation based on the experimental data on acid fracture conductivity from a laboratory test that is the most widely used in industry. The resulting conductivity is based on the volume of rock dissolved and the rock’s mechanical strength. Equations 3 to 5 were generated according to the experimental study conducted on different formation samples acidized with HCl acid (Nierode and Kruk, 1973). The amount of rock dissolved, rock strength and closure stress were the effective parameters on the estimation of conductivity. The correlation considers the role of closure stress exponentially in fracture conductivity.
(3) (4)
{
(5)
Where σc is the closure stress in psi, SRE is the rock embedment strength representing the rock strength in psi. The correlation calculates conductivity at zero closure stress using an empirical correlation as a function of ideal fracture width, which is the dissolved rock volume divided by the fracture area. DREC derived according to the cubic law indicates the acid dissolving power and the total volume of injected acid proportional to the geometry of the fracture created. The relationship between DERC and the fracture width is defined as equation 6 (Gomaa and Nasr-ElDin, 2009). (6)
By Substituting Eq.6 into Eq.4, the power of width is 2.466 and fracture conductivity is predicted in the lower bound. Anderson et al. (1989) found that the effect of formation characteristics such as hardness on the etched conductivities is very significant. For example, etched surfaces of chalk formations can be easily closed due to its natural softness. Nasr-El-Din et al. (2006) stated that the correlations developed by Nierode and Kruk (1973) were not categorized by lithology. The correlation was recalculated by graphing and evaluating the data again both as a lumped set and as individual sets by lithology. The power of width is 2.255 in their correlation for the lump set. The value is 2.623 and 1.677 for limestone and dolomite formations respectively. Their modified correlations kept the same form as the original ones, but made the constants different and were more precise (Gomaa & Nasr-El-Din, 2009). The objective of this work is to develop a precise correlation to estimate conductivity for acid fracturing treatment based on a new approach in evaluation algorithms. Due to the complication of predicting acid fracture conductivity and also the capability of genetic algorithm (GA) in optimization and modelling, in this paper, a robust intelligent model based on genetic algorithm is developed to predict acid fracture conductivity accurately. The input data are considered according to the rock strength to provide a better understanding of the rock strength effect. Ultimately, the resulting conductivity given by GA models is compared to the prior models.
3. Methodology Genetic algorithms are applied as a type of optimization algorithm to find the optimal solution(s) to a given computational problem that maximizes or minimizes an objective function. Genetic Algorithms are direct, parallel, stochastic method for the global search and optimization which imitate the evolution of the living beings, Charles Darwin. This adaptive method may be used to solve search and optimization problems. Genetic Algorithm, which is based on the genetic process of biological organisms, is used to solve both constrained and unconstrained optimization problems. It could be applied to a variety of problems that are not well suited for the standard optimization algorithms particularly the problems with nonlinear objective function) (Kosters et al., 1999). Sometimes, genetic algorithms are applied as a process of optimization coefficients in the engineering and mathematics. Mathematical modelling to generate a functions is the first step to solve this type of problems. Then to find a solution, the coefficients that optimize the model or the function components that provide an optimal system performance are discovered (Goldberg, 1989). In this research, a genetic algorithm is used to optimize the coefficients of Nierode and Kruk correlation. Three modified correlations based on all input data, the samples in which the rock embedment strength is lower than 20000 psi and the samples in which rock embedment strength is higher than 20000 psi are developed. In the next step, three new correlations are generated according to the three data categories. In the new correlations the different functions were applied and then optimized their coefficients to predict precisely fracture conductivity. The quality of the fracture conductivity prediction is quantified in terms of the percentage of average absolute relative error (AARE) and correlation coefficient (R) by comparison with experimental data. Equations 7 and 8 are applied to calculate the performance criteria. ∑
[((
)
(̅̅̅̅̅̅)
)((
[ ((
)
(̅̅̅̅̅̅ )
) ((
(̅̅̅̅̅̅ )
)
)]
(7) √∑
∑
|(
)
( (
)
)
(̅̅̅̅̅̅)
)
|
) ]
(8)
3.1 Data analysis Nierode and Kruk experimental data sets are applied as modeling data. 106 experimental data set including dissolve rock equivalent conductivity as treatment parameter, rock embedment strength and closer stress for specific fracture conductivity are considered as input data. The data sets are divided into three groups; the first includes all 106 data sets to develop the correlations. The second group consists of 36 data sets in which the rock embedment strength is lower than 20000 psi. The third group includes 70 data set in which the rock embedment strength is upper than 20000 psi. Table 1 shows the properties and domain of all input variables in detail. The lithology 55 data sets are limestone, 51 data sets are dolomite.
Table 1. Domain of input data used in this study Parameter
Min
Max
STD
Mean
DREC (md-ft)
3.E+04
1.30E+08
107826007
6.E+07
Closure Stress (Psi)
0
7000
2379.809
2939
Rock Strength (Psi)
5600
88100
1802.509
35170
Fracture Conductivity (md-ft)
1.2
7400000
1264474.65
2.E+05
3.2. Development of a genetic algorithm to predict acid fracture conductivity The genetic algorithm is a space search method developed according to the mechanism of natural selection and survival of the fittest. In this algorithm, a random population of individuals start the evolution process (Holland, 1992). Each individual in the population is ranked based on the absolute relative error (ARE) between the experimental and the calculated fracture conductivity values. Operators including selection, crossover and mutation are applied to evolve the initial population by minimizing ARE. The selection operator selects individuals for crossover based on the fitness. Here, the stochastic uniform methods are used to select individuals for crossover. In this method, the probability of selection is proportional to the individuals fitness. The genetic material of the selected parents are recombined by the crossover operator. The elements from the first parent are selected by the crossover operator. In this process, the chromosomes from the parents exchange systematically using probabilistic decision. In other word, change occurs during reproduction. The mutation operator modifies individuals that have not been selected for
reproduction by randomly changing weights. In this work, the uniform mutation algorithm is used to select the weights for the mutation. Each one of the selected entries has 10 to 50% probability of being mutated according to the type of correlations. Mutated entries are replaced with a uniformly random value. After determination of a population as a new generation, the process continues until some criterion is met (Goldberg, 1989). Due to the stochastic nature of Genetic Algorithm, it is difficult to formally specify a convergence criterion. In this reseach, the algorithm stopped after 100 generations or if no improvement is observed over a pre-specified number of generations (in this case 50). The general flow chart of GA applied in this study is shown in figure 1. The parameters used to perform Genetic Algorithm and their description are listed in Table 2. The values of these parameters are shown in table 3.
7 Fig 1. GA procedure chart.
Table 2. The parameters used to set Genetic Algorithm Chromosome or Individual
A group of genes A part of chromosome; a gene contains a part of solution. It determines the
Gene
solution Number of individuals present with same length of chromosome
Population
The value assigned to an individual based on how far of close an individual is
Fitness
from the solution; greater the fitness value better the solution it contains Fitness function
A function that assigns fitness value to the individual
Selection
Selecting individuals for creating the next generation
Crossover
Exchanging the chromosomes from the parents using probabilistic decision.
Mutation
Changing a random gene in an individual
Table 3. The values of GA parameters for the modified and the new correlations Input Data
Correlation
N Pop
Pc
Pm
mu
Modified Correlation
3000
0.80
0.30
0.30
New Correlation
10000
0.90
0.10
0.30
Modified Correlation
4000
0.80
0.40
0.50
New Correlation
4000
0.80
0.50
0.70
Modified Correlation
4000
0.80
0.30
0.30
New Correlation
3000
0.80
0.30
0.30
All Data
RES<20000
RES>20000
4. Results and Discussion The GA optimization algorithm was run for three different categories. Six correlations including three modified and three new correlations were developed to predict the acid fracture conductivity. The results show rock strength is a significant parameter to determine acid fracture conductivity. The predicted values of fracture conductivity by the proposed correlations that were developed according to rock strength have a good match with the experimental results. The comparison of accuracy of Nierode & Kruk, Nasr-El-Din, the modified and the new correlations are shown in figure 2. As the results illustrate, the new correlation is more precise compared to the other correlations in different rock strength.
Fig 2: Average absolute relative error of the predictive correlation
Table 4 shows coefficients of the modified correlation in which all the experimental data were considered as input data. The correlation predicts the fracture conductivity with a high error. In the correlation the power of ideal width is 2.226. In order to develop a precise correlation for the category, the new functions were used to find the new correlation. GA optimization algorithm was run to find the best coefficients for the proposed functions. The new correlation and their coefficients are tabulated in table 5. The percentage of average absolute relative error of different correlations to predict fracture conductivity based on all input data were compared in table 6. Although the precision of the modified correlation has been improved, the AARE of the new correlation is better than common correlations (Nierode & Kruk and Nasr-El-Din’s correlations). Figure 3 illustrates the predicted values of conductivity using different correlations versus the measured values. As shown, Nierode & Kruk’s correlation overestimates the values of conductivity, while the modified and the new correlations underestimate these values.
Table 4: Coefficients of the modified correlation for all input Correlation
Coefficients 0.086 0.4311 0.742 0.3003 0.0246 0.5479
Table 5. Proposed correlation and their coefficients to predict fracture conductivity based on all input data Correlation
Coefficients 0.5
93
0.85
2.94
-14
-1.92
0.258
0.0004
0.27
2
0.05
Table 6. AARE of predictive correlations to predict fracture conductivity based on all input data Correlation
AARE (%)
R-Squared
N&K
60.52
0.77
Nasr El Din
53.55
0.81
Modified Correlation
40.18
0.62
New Correlation
34.55
0.85
Fig 3. Comparison between the measured and predicted values of conductivity by predictive correlations for all input data
The effect of rock strength was considered to provide a better understanding by running GA algorithm and applying the data of second and third categories. Table 7 shows coefficients of the modified correlation for rocks in which RES is less than 20000 Psi. The percentage of average absolute relative error of the correlation is 39.67%. The power of ideal width is 1.01 for the correlation. In this category, new functions were used to improve the accuracy of predictive correlation. Table 8 indicates the proposed correlation and their coefficients. The correlation has 14 coefficients. The results show the new correlation is more precise and complicated compared to the other correlations (Table 9). The comparison of the measured and calculated values of conductivity using predictive correlations has been depicted in figure 4. The predicted values by Nierode & Kruk’s correlation have the most deviation from the experimental data in this category, too.
Table 7. Coefficients of the modified correlation for rock with RES<20000 Psi Correlation
Coefficients 0.9124 100 0.337 0.5125 0.0412 0.0134
Table 8. Proposed correlation and their coefficients to predict fracture conductivity for rocks with RES<20000 Psi Correlation
Coefficients 0.2503 0.3926
0.1509 0.3444
-90 -2 0.7998 0.0341 0.4419
75 0.952 35 -24 0.0051
Table 9. Accuracy of predictive correlations to predict fracture conductivity for rock with RES<20000 Psi Correlation
AARE (%)
R-Squared
N&K
49.84
0.21
Nasr El Din
39.67
0.61
Modified Correlation
22.36
0.68
New Correlation
21.12
0.84
Fig 4. Comparison between the measured and predicted values of conductivity by predictive correlations for data with RES< 20000 psi.
Figure 5 illustrates the percentage of average absolute relative error of the different correlations versus closure stress for rocks in which RES is lower than 20000 Psi. There is an increasing trend for error as the closure stress increases. The results show that all correlations anticipate acid fracture conductivity relatively precisely at low and middle closure stresses. However, at high closure stresses, the predicted values have been deviated from the experimental data considerably. This reveals the significance and the intricate role of the closure stress on etching conductivity. In other words, predicting fracture conductivity is too difficult at high closure stresses and it is an uncertainty when seeking to control the acid fracturing process. Nevertheless, the new and the modified correlations are much more accurate compared to the previous correlations even at high closure stress.
Fig 5. The percentage of average absolute relative error of the predictive correlations versus closure stress for rocks with RES< 20000 psi
Table 10 shows the coefficients of the modified correlation for the samples in which RES is higher than 20000 Psi. The percentage of average absolute relative error of the correlation is 15.89%. Table 11 indicates the proposed correlation and their coefficients. In the correlation the power of ideal width is 2.475.
The percentage of average absolute relative error of the
correlations to estimate fracture conductivity for the category were compared in table 12. The AARE of the new correlation is 14.74% that shows the accuracy of the correlation has been improved slightly. In these categories, the values predicted by Nierode & Kruk’s correlation has the most error in comparison with the other correlations. Figure 6 illustrates the predicted values of conductivity using different correlations versus the measured values. As shown, N&K’s correlation overestimates the values of conductivity, while the modified and the new correlations underestimate these values with limited error.
Table 10. Coefficients of the modified correlation for rock with RES>20000 Psi Correlation
Coefficients 0.037 0.1664 0.825 -78 -10
0.0008
Table 11. Proposed correlation and their coefficients to predict fracture conductivity for rocks with RES>20000 Psi Correlation
(
Coefficients
)
2.068
52
0.9851
33
13
-4
2.2663
-0.5915
0.0879
0.7526
-40
0.9063
Table 12. Accuracy of predictive correlations to predict fracture conductivity for rock with RES>20000 Psi Correlation
AARE (%)
R-Squared
N&K
18.45
0.82
Nasr El Din
16.57
0.83
Modified Correlation
15.89
0.85
New Correlation
14.74
0.88
Fig 6. Comparison between the measured and predicted values of conductivity by predictive correlations for data with RES> 20000 psi.
The results of the second and the third category reveal that formation strength plays a substantial role in acid fracture conductivity prediction and universal correlations are not recommended to anticipate the fracture conductivity. On the other hand, the new correlations consider the role of DREC exponentially in fracture conductivity. The relationships differ from the modified correlations. DREC behaves as power-law and the power of DREC is less than 1 similar to the previous empirical correlations in the modified correlations. In other words, the relation between the average width and fracture conductivity is demonstrated to be different by the new and the modified correlations. Furthermore, the accuracy of the modified and the new correlations that were developed for the second and the third categories are the same nearly. Although the new correlations predict the fracture conductivity slightly more accurate, these correlations are too complicated and it is not recommended for initially and fast estimation of fracture conductivity. Moreover, as the rock strength increases (RES), the accuracy of predictive correlations improve dramatically. It means the behaviour of rocks in which RES is high are more reasonable under different closure stresses. In other word, an acid fracturing treatment must be designed cautiously, particularly at high closure stresses in soft formations.
5. Conclusion 1- GA is a powerful tool to precisely and rapidly predict the acid fracture conductivity at different closure stresses as compared to the prior well-known correlations. The fracture conductivity predicted by GA correlation closely matched with the experimental data. It is recommended that the GA correlations is applied in the commercial software instead of previous empirical correlations (i.e. Nierode & Kruk correlations) to precisely predict acid fracture conductivity. 2- Rock strength plays a significant role when anticipating fracture conductivity, as various formations have behaved differently under different closure stresses. The behaviour of strong rocks are more reasonable under different closure stresses. While the prediction of the behaviour soft rocks are much more complicated. Therefore, it is not possible to present a comprehensive precise model to predict acid fracture conductivity for all rock types. It is recommended that the predictive fracture conductivity correlations are developed based on rock strength.
3- Although the modified correlations show a good match with the experimental data, the new correlations are more precise than them. It also means that the new correlations developed in this study, give a better approximation of acid fracture conductivity than the available correlations of Nierode & Kruk and Nasr el Din. However, new correlations are more complicated compared to the previous correlations. Nonetheless, in order to estimate fracture conductivity rapidly, the modified correlations are recommended. 4- The new correlations consider the role of DREC exponentially in fracture conductivity. While in the modified correlations, DREC behaves as power- law and the power of DREC is less than 1. 5- The prediction of fracture conductivity is too difficult at high closure stresses due to the complexity of rock behaviour under these tensions. As the closure stress increases, the accuracy of all predictive correlations decreases dramatically. Therefore, the fracture conductivity must be anticipated with more cautious at high closure stresses by every predictive correlation. Furthermore, as the rock strength increases (RES), the precision of correlations improve considerably. It indicates the strong formations behave reasonably under different closure stresses.
Appendix A. Nierode and Kruk’s experimental data and the predicted values by GA modified correlations The fracture conductivity predicted by GA modified correlations for rock with RES<20000 Psi Closure Stress psi
Fracture Conductivity (Experimental), md-in
11800
0
9.13
Fracture Conductivity (GA modified correlation), mdin 8.11
11800
1000
7.37
6.56
3.30E+04
11800
3000
3.80
3.48
3.30E+04
11800
5000
0.26
0.39
Clearfork Dolomite
3.90E+06
12200
0
12.25
9.72
6
Clearfork Dolomite
3.90E+06
12200
1000
11.27
8.19
7
Clearfork Dolomite
3.90E+06
12200
3000
9.21
5.1
8
Clearfork Dolomite
3.90E+06
12200
5000
7.31
2.08
9
Clearfork Dolomite
3.90E+06
12200
7000
5.29
0.96
10
Capps Limestone
3.20E+05
13000
0
9.18
8.87
Rock Type
DREC, md-in
RES, psi
1
Clearfork Dolomite
3.30E+04
2
Clearfork Dolomite
3.30E+04
3
Clearfork Dolomite
4
Clearfork Dolomite
5
No.
11
Capps Limestone
3.20E+05
13000
1000
8.34
7.38
12
Capps Limestone
3.20E+05
13000
3000
6.63
4.39
13
Capps Limestone
3.20E+05
13000
5000
4.94
1.40
14
Capps Limestone
3.20E+05
13000
7000
3.21
1.58
15
Indiana Limestone
3.10E+08
14300
0
15.81
11.19
16
Indiana Limestone
3.10E+08
14300
1000
14.50
9.74
17
Indiana Limestone
3.10E+08
14300
3000
11.85
6.85
18
Indiana Limestone
3.10E+08
14300
5000
9.21
3.96
19
Indiana Limestone
3.10E+08
14300
7000
6.58
1.07
20
Clearfork Dolomite
8.30E+06
14400
0
12.43
9.97
21
Clearfork Dolomite
8.30E+06
14400
1000
10.57
8.53
22
Clearfork Dolomite
8.30E+06
14400
3000
6.90
5.64
23
Clearfork Dolomite
8.30E+06
14400
5000
3.21
2.76
24
Cisco Limestone
3.00E+05
14800
0
8.85
8.85
25
Cisco Limestone
3.00E+05
14800
1000
8.13
7.42
26
Cisco Limestone
3.00E+05
14800
3000
6.68
4.56
27
Cisco Limestone
3.00E+05
14800
5000
5.24
1.71
28
Cisco Limestone
3.00E+05
14800
7000
3.78
1.14
29
Clearfork Dolomite
3.20E+06
16600
0
11.28
9.65
30
Clearfork Dolomite
3.20E+06
16600
1000
9.61
8.28
31
Clearfork Dolomite
3.20E+06
16600
3000
6.17
5.53
32
Clearfork Dolomite
3.20E+06
16600
5000
2.77
2.7
33
San Andres Dolomite
3.40E+06
17300
0
9.14
9.67
34
San Andres Dolomite
3.40E+06
17300
1000
7.93
8.32
35
San Andres Dolomite
3.40E+06
17300
3000
5.52
5.62
36
San Andres Dolomite
3.40E+06
17300
5000
3.13
2.93
The fracture conductivity predicted by GA modified correlations for rock with RES>20000 Psi Closure Stress psi
Fracture Conductivity (Experimental), md-in
Fracture Conductivity (GA modified correlation), md-in
21500
0
13.58
12.35
21500
1000
12.61
11.67
2.80E+07
21500
3000
10.66
10.33
Indiana Limestone
2.80E+07
21500
5000
8.75
8.98
Indiana Limestone
2.80E+07
21500
7000
6.80
7.63
6
Indiana Limestone
4.50E+06
22700
0
13.03
10.84
7
Indiana Limestone
4.50E+06
22700
1000
11.91
10.15
8
Indiana Limestone
4.50E+06
22700
3000
9.61
8.77
9
Indiana Limestone
4.50E+06
22700
5000
7.31
7.39
10
Indiana Limestone
4.50E+06
22700
7000
5.01
6.01
Rock Type
DREC, md-in
RES, psi
1
Indiana Limestone
2.80E+07
2
Indiana Limestone
2.80E+07
3
Indiana Limestone
4 5
No.
11
Cisco Limestone
2.00E+06
25300
0
11.85
10.17
12
Cisco Limestone
2.00E+06
25300
1000
11.03
9.45
13
Cisco Limestone
2.00E+06
25300
3000
9.47
8.01
14
Cisco Limestone
2.00E+06
25300
5000
7.90
6.55
15
Cisco Limestone
2.00E+06
25300
7000
6.34
5.108
16
Capps Limestone
2.90E+05
30100
0
9.79
8.58
17
Capps Limestone
2.90E+05
30100
1000
8.82
7.80
18
Capps Limestone
2.90E+05
30100
3000
6.84
6.25
19
4.86
4.69
Capps Limestone
2.90E+05
30100
5000
20
Capps Limestone
2.90E+05
30100
7000
2.89
3.13
21
Canyon Limestone
4.60E+07
30700
0
13.59
12.76
22
Canyon Limestone
4.60E+07
30700
1000
12.87
11.98
23
Canyon Limestone
4.60E+07
30700
3000
11.45
10.41
24
Canyon Limestone
4.60E+07
30700
5000
10.04
8.84
25
Canyon Limestone
4.60E+07
30700
7000
8.59
7.27
26
Clearfork Dolomite
3.60E+04
35000
0
8.13
6.86
27
Clearfork Dolomite
3.60E+04
35000
1000
7.43
6.03
28
Clearfork Dolomite
3.60E+04
35000
3000
6.01
4.38
29
Clearfork Dolomite
3.60E+04
35000
5000
4.60
2.73
30
Clearfork Dolomite
3.60E+04
35000
7000
3.17
1.09
31
Canyon Limestone
2.70E+08
46400
0
14.28
14.22
32
Canyon Limestone
2.70E+08
46400
1000
13.42
13.31
33
Canyon Limestone
2.70E+08
46400
3000
11.77
11.48
34
Canyon Limestone
2.70E+08
46400
5000
10.04
9.66
35
Canyon Limestone
2.70E+08
46400
7000
8.39
7.84
36
San Andres Dolomite San Andres Dolomite San Andres Dolomite San Andres Dolomite San Andres Dolomite San Andres Dolomite San Andres Dolomite San Andres Dolomite San Andres Dolomite San Andres Dolomite San Andres Dolomite San Andres Dolomite San Andres Dolomite San Andres Dolomite
1.00E+06
46500
0
11.32
9.60
1.00E+06
46500
1000
10.59
8.69
1.00E+06
46500
3000
9.16
6.86
1.00E+06
46500
5000
7.6
5.04
1.00E+06
46500
7000
6.25
3.21
1.90E+07
62700
0
12.25
12.03
1.90E+07
62700
1000
11.45
11.02
1.90E+07
62700
3000
9.85
9.01
1.90E+07
62700
5000
8.21
7.008
1.90E+07
62700
7000
6.58
4.99
5.10E+08
63800
0
13.99
14.74
5.10E+08
63800
1000
13.52
13.73
5.10E+08
63800
3000
12.61
11.71
5.10E+08
63800
5000
11.69
9.69
37 38 39 40 41 42 43 44 45 46 47 48 49
50
San Andres Dolomite
5.10E+08
63800
7000
10.75
7.67
51
Cisco Limestone
1.20E+05
67100
0
7.82
7.85
52
Cisco Limestone
1.20E+05
67100
1000
7.17
6.83
53
Cisco Limestone
1.20E+05
67100
3000
5.82
4.77
54
Cisco Limestone
1.20E+05
67100
5000
4.47
2.72
55
Cisco Limestone
1.20E+05
67100
7000
3.13
0.67
56
San Andres Dolomite San Andres Dolomite San Andres Dolomite San Andres Dolomite San Andres Dolomite San Andres Dolomite San Andres Dolomite San Andres Dolomite San Andres Dolomite San Andres Dolomite
2.40E+06
76500
0
9.85
10.32
2.40E+06
76500
1000
8.82
9.26
2.40E+06
76500
3000
6.74
7.12
2.40E+06
76500
5000
4.60
4.99
2.40E+06
76500
7000
2.56
2.86
2.70E+06
76600
0
9.30
10.42
2.70E+06
76600
1000
8.57
9.35
2.70E+06
76600
3000
7.09
7.22
2.70E+06
76600
5000
5.59
5.08
2.70E+06
76600
7000
1.79
2.95
57 58 59 60 61 62 63 64 65 66
Canyon Limestone
1.30E+08
88100
0
14.07
13.62
67
Canyon Limestone
1.30E+08
88100
1000
13.54
12.50
68
Canyon Limestone
1.30E+08
88100
3000
12.64
10.28
69
Canyon Limestone
1.30E+08
88100
5000
10.77
8.06
70
Canyon Limestone
1.30E+08
88100
7000
8.82
5.84
Appendix B. Genetic algorithm parameters in Initial and general conditions The required primary pieces of information for starting GA is: 1. Population size (Npop): A group of interbreeding individuals 2. Crossover rate (Pc): Crossover probability 3. Mutation rate (Pm): Mutation probability The general stages in genetic algorithm are as follows: 1. Initialization 1.1 Set the parameters (Npop, Pc, Pm, stopping criteria, selection strategy, crossover operation, mutation operation, and number of generation) 1.2 Generate an initial population randomly 2. Compute and save the fitness for each individual in the current population 3. Define selection probabilities for each individual based on fitness criteria
4. Generate the next population by selecting individuals from current population randomly to produce offspring via GA operators such as crossover and mutation operators 5. Repeat step 2 until stopping criteria is satisfied. According to what follows, the proposed GA is described in details.
References Anderson, M.S., and Fredrickson, S.E. 1989. Dynamic Etching Tests Aid Fracture- Acidizing Treatment Design. SPE Production Engineering 4(4): 443-449. SPE-16452. Beg, M.S., Kunak, A.O., Gong, M., Zhu, D., and Hill, A.D. 1998. A Systematic Experimental Study of Acid Fracture Conductivity. SPEPF 13(4): 267-271. SPE-52402. California Council on Science and Technology Lawrence Berkeley National Laboratory Pacific Institute, 2014. Advanced Well Stimulation Technologies in California An Independent Review of Scientific and Technical Information. Coulter, A.W., Alderman, E.N., Cloud, J.E., and Crowe, C.W. 1974. Mathematical Model Simulates Actual Well Conditions in Fracture Acidizing Treatment Design. Paper SPE-5004 presented at the SPE_AIME 49th Annual Fall Meeting, Houston, Texas, USA, 6-9 October 1974. Economides, M. J. and Nolte, K. G. 2000. Reservoir Stimulation, 3rd ed., John Wiley & Sons, Hoboken, N.J. and Chichester. Gomaa, A.M., and Nasr-El-Din, H.A. 2009. Acid Fracturing: the Effect of Formation Strength on Fracture Conductivity. Paper SPE-119623 presented at the 2009 SPE Hydraulic Fracturing Technology Conference, Woodlands, Texas, U.S.A. Gong, M., Lacote, S., and Hill, A.D. 1999. New Model of Acid-Fracture Conductivity Based on Deformation of Surface Asperities. SPEJ 4(3): 206-214. SPE-57017.
Holland, J.H., 1992. Adaptation in Natural and Artificial Systems, MIT Press, Cambridge, MA. Kosters W.A., Kok J.N. and Floreen P., 1999. Fourier analysis of Genetic Algorithms, Theoretical Computer Science, Elsevier, 229, 199, pp. 143-175. Mou, J. 2009. Modeling Acid Transport and Non-Uniform Etching in a Stochastic Domain in Acid Fracturing. PhD dissertation. Texas A&M University, College Station. Nasr-El-Din, H.A., Al-Driweesh, S.M., Metcalf, A.S., and Chesson, J.B. 2008. Fracture Acidizing: What Role Does Formation Softening Play in Production Response? SPE Production & Operations 23(2): 184-191. SPE-103344. Nierode, D.E., and Kruk, K.F. 1973. An Evaluation of Acid Fluid Loss Additives, Retarded Acids, and Acidized Fracture Conductivity. Paper SPE-4549 presented at the 48th Annual Fall Meeting of Society of Petroleum Engineering of AIME, Las Vegas, Nevada, USA, 1973. Goldberg, D.E. 1989, Genetic algorithms in search, optimization and machine learning, AddisonWesley, Reading, MA, Pournik, M. 2008. Laboratory-Scale Fracture Conductivity Created by Acid Etching. PhD dissertation. Texas A&M University, College Station, Texas, USA. Roberts, L.D., and Guin, J.A. 1975. The Effects of Surface Kinetics in Fracture Acidizing. SPEJ 8: 385-395. SPE-4349. Ruffet, C.S., Fery, J.J., and Onaisi, A. 1997. Acid-Fracturing Treatment: A Surface- Topography Analysis of Acid-Etched Fractures to Determine Residual Conductivity. Paper SPE-38175 presented at the SPE European Formation Damage Conference, The Hague, The Netherlands, 2-3 June 1997. Valdo F. Rodrigues, SPE, Wellington Campos, SPE, Ana C. R. Medeiros, SPE, North Fluminense State University; and Rodolfo A. Victor, Petrobras/RH/UP, 2011. Acid-Fracture Conductivity Correlations for a Specific Limestone Based on Surface Characterization. Van Domselaar, H.R., Schols, R.S., and Visser, W. 1973. An Analysis of the Acidizing Process in Acid Fracturing. SPEJ 8: 239-250. SPE-3748.
Zimmerman, R. W. and G. S. Bodvarsson (1996). "Hydraulic conductivity of rock fractures." Transport in Porous Media 23(1): 1-30.
Highlights
A Novel Correlations to Predict Acid Fracture Conductivity Based on the Rock Strength are developed.
The rock strength plays a significant role when anticipating fracture conductivity.
As the closure stress increases the precision of all predictive correlations decreases considerably.
The fracture conductivity must be anticipated cautiously at high closure stresses by various predictive correlation particularly in soft formation.