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A new correlation for prediction of sub-critical two-phase flow pressure drop through large-sized wellhead chokes
Journal of Natural Gas Science and Engineering 26 (2015) 264e278
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Journal of Natural Gas Science and Engineering journal homepage: www.elsevier.com/locate/jngse
A new correlation for prediction of sub-critical two-phase flow pressure drop through large-sized wellhead chokes Siamak Seidi a, Tofigh Sayahi b, * a b
Department of Petroleum Engineering, Petroleum University of Technology, Ahwaz, Iran Department of Chemical Engineering, Faculty of Engineering, Ferdowsi University of Mashhad (FUM), Mashhad, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 4 April 2015 Received in revised form 9 June 2015 Accepted 10 June 2015 Available online xxx
Unacceptable errors resulted from utilizing the current correlations for investigating the behavior of subcritical two-phase flow regime through wellhead chokes of south Iranian gas condensate reservoirs led us toward establishing a new Gilbert-type correlation capable of fitting the production data with minimum errors. The proposed model, which is a modification of Nasriani and Kalantariasl model, is able to predict the high flow rates of gas condensate wells under sub-critical conditions particularly in case of large choke sizes. In order to validate the new correlation, Genetic Algorithm and non-linear regression analysis methods are implemented to sixty seven production datasets of fifteen wells with large wellhead choke sizes (40/64e192/64 inch) gathered from 10 different fields. Then the proposed correlation in addition to two other models (1.Osman and Dokla 2. Nasriani and Kalantariasl) are conducted to each choke size to investigate the applicability of the new formula in comparison with the existing ones. Moreover, in order to evaluate the new model in other field data, 39 data points gathered from gascondensate wells of Fars province of Iran are exposed to the proposed model, and the two other models. Finally, the main form of the new correlation is applied to dataset of each choke size. The results indicate that the non-linear regression technique is more accurate than Genetic algorithm in fitting the data to the proposed method. Furthermore, the new correlation has the minimum errors in comparison to other methods in both investigated areas. Finally, according to statistical error analysis for each choke size, the ability of the proposed correlation to predict gas flow rates of fluids passing through the wellhead chokes of gas-condensate wells under sub-critical conditions is found to be highly improved when applied to individual choke sizes. © 2015 Elsevier B.V. All rights reserved.
Keywords: Wellhead choke Sub-critical Gas-condensate Genetic algorithm Non-linear regression
1. Introduction Gas condensate reservoir is a particular kind of reservoir which has a behavior mediating between that of gas and volatile oil reservoirs. It contains low density liquid hydrocarbons which present as gaseous components in the raw natural gas and condense out of gas when the pressure is lowered below the dew point pressure of hydrocarbons. Due to changes in temperature and pressure, these types of reservoir are bound to instability of flow regime, fluctuation and phase change which may lead to liquid holdup and phase separation. Therefore, multiphase flow is common in gas condensate reservoirs. The flow rate of well is the most significant parameter for
* Corresponding author. Ferdowsi University of Mashhad (FUM) Campus, Azadi Sq., Mashhad, Khorasan Razavi, P.O. Box: 9177948974, Iran. E-mail address: tofi[email protected] (T. Sayahi). http://dx.doi.org/10.1016/j.jngse.2015.06.025 1875-5100/© 2015 Elsevier B.V. All rights reserved.
characterization of a reservoir and estimation of its behavior. Wellhead chokes (a configuration which an elbow is installed exactly upstream of chokes) normally are used in wells as controlling agents to adjust the flow rate and sustain adequate back pressure to avoid sand issues and water/gas coning. Therefore, estimation of choke performance by implementation and optimization of a relation between size of chokes and wellhead flow rates is only possible through a precise modeling and selection of an optimum choke size. Based on flow regime, the fluid flow through the choke can be characterized as either critical or sub-critical (sonic or subsonic, respectively). When Mach number is equal or more than unity (When the flow velocity is equal or greater than sonic velocity), any pressure disturbance wave from downstream cannot spread through upstream and mass flow rate reaches a maximum amount which only depends upon upstream conditions. This kind of flow is known as critical flow. Accordingly, to avoid any perturbation of
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Table 1 Different parameter ranges of south Iranian field data. Parameters
S (1/64inch)
LGR (bbl/MMscf)
Qg (MMscf/D)
Pu (psia)
Pd (psia)
DP (psia)
Tf ( F)
Minimum Maximum
40 192
0.688 32.215
11.3 113
1131 4452
824.84 3045.82
14.5 1407
109 211
Fig. 1. Derivation of the new correlation constants by Genetic Algorithm.
surface equipment, most chokes are usually managed to work in critical flow region. In contrast, the sub-critical flow is described as the flow which its highest possible flow rate is less than sonic velocity. In this case, the mass flow rate is a function of pressure drop across the choke and any downstream disturbance is able to influence the upstream region. Due to difficulty of estimating gas and sound velocities in the field (Nejatian et al., 2014), usually in literature, the value of 0.5 for downstream to upstream pressure ratio of chokes (Pd/Pu ¼ 0.5) has been selected as the boundary between critical and subcritical flow regimes. The flow which has a Pd/Pu less than 0.5 is considered to be a sub-critical flow and the one with the ratio greater than 0.5 is considered to be a critical flow (Omana et al., 1969; Nasriani and Kalantariasl, 2011). Empirical and Analytical methods are two categories of multiphase flow estimation approaches across the chokes. Tangren et al. (1949) were the first researchers to implement analytical investigation on two-phase flow regime across limitations of chokes. However, their method is only valid where the liquid is the continuous phase. Ros (1960) followed the Tangren et al. (1949)
method and converted it into a model where gas is considered to be the continuous phase. Ashford and Pierce (1975) extended Ros (1960) model and derived a correlation to estimate the critical flow rates. However, they assessed that uncertainty is introduced in their correlation due to difficulty of determining the downstream pressure accurately (Rahimzadeh and Mohammadmoradi, 2014). An extension was implemented to Ashford and Pierce (1975) theoretical model by Sachdeva et al. (1986) for a choke placed in a direct horizontal pipe section. They derived two different correlations, one for predicting critical pressure ratio and the other for making a distinction between sonic and subsonic flow. Brill and Beggs (1984) combined Bernoulli and continuity equations to develop a formula for calculation of sub-critical flow rates. This purely theoretical model (which is called Mechanistic model) is widely used by the industry with a proven accuracy. In another theory-based study, Perkins (1993) derived a theoretical multiphase equation based on mass, momentum and energy balance for description of isentropic flow across chokes. It is worth mentioning that some theoretical models established by Fortunati (1972), Ashford and Pierce (1975), Pilehvari (1981), Surbey et al. (1989), Sachdeva et al. (1986), and Perkins (1993) are applicable for both critical and sub-critical conditions (Nasriani and Kalantariasl, 2011; Rahimzadeh and Mohammadmoradi, 2014). Gilbert (1954) derived the earliest empirical relation between choke size and surface flow rate by utilizing 268 production datasets from Ten Section Kern County Oil Fields of California. For derivation of his correlation, Gilbert assumed that the flow passing through an edge knife choke is critical and assessed that the correlation is valid when the upstream pressure of the choke is at least 70% higher than the downstream pressure. Gilbert's formula has the following form (Gilbert, 1954):
Pwh ¼
cQRaGL Sb
Fig. 2. Comparison of real and predicted gas flow for all data points by Genetic Algorithm.
(1)
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Fig. 3. Comparison between real and predicted gas flow rates for all data points by non-linear regression analysis.
Fig. 4. Comparison between the new correlation, Model 1, and Model 2 for all data points.
Fig. 5. Comparison between real and predicted gas flow rates for all data points when d ¼ 1.
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Fig. 6. Comparison between real and predicted gas flow rates for the data gathered from gas-condensate wells located in Fars province of Iran.
a, b, and c: Empirical constants. Pwh: Wellhead pressure (m/Lt2), psia. Q: Volumetric liquid flow rate (L3/t), bbl/D. RGL: Gas-Liquid ratio (L3/L3), MScf/Stb. S: Choke size (L), 1/64 inch. Empirical formulas are generally valid over the range where field data are available and may lead to poor results when extrapolated to new conditions (Perkins, 1993), hence a lot of researchers developed similar correlations with different constants for different field data tests in critical region (Baxendell, 1958; Poettmann and Beck, 1963; Al-Attar and Abdul-Majeed, 1988; Enaibe and Ajienka, 1994; Lannom and Hatzignatiou, 1996; Mesallati et al., 2000; Alrumah and Bizanti, 2007; Beiranvand et al., 2012; Azin et al., 2014; Lak et al., 2014). By investigating data points collected from a gas-condensate well in Middle East under sonic conditions, Osman and Dokla (1990) showed that replacing upstream pressure by pressure drop across the choke in Gilbert formula can increase the accuracy of flow rate estimation. They used non-linear regression method in
description of the behavior of their datasets for choke size range of 28/64e72/64 inch and gas flow rate of 3.91e101.33 MMscf/D. There are few published works concerning the sub-critical twophase flow regime through wellhead chokes (Surbey et al., 1989; Elgibaly and Nashawi, 1998; Al-Attar, 2008; Nasriani and Kalantariasl, 2011; Nejatian et al., 2014). Al-Attar (2008) developed a new graphical approach in sub-critical condition for 97 data tests gathered from Middle East wells based on separate choke sizes (24/ 64 inch, 32/64 inch, 40/64 inch, 48/64 inch, 64/64 inch, 96/64 inch, 112/64 inch, and 128/64 inch). With the purpose of developing a general formula including the choke size as a variable, he also substituted pressure drop for upstream pressure in Gilbert formula (similar to Osman and Dokla (1990) work in critical conditions) and used a non-linear regression analysis to derive the constants of this modified Gilbert-type correlation under sub-critical conditions. The results indicated that the former approach is superior to the application of non-linear regression analysis method on the new Gilbert-type model. It should be noted that the maximum gas flow rate of his study was 43.7 MMscf/D. On the other hand, as indicated in a study by Ali Ghalambor (2002) on gas condensate reservoirs of
Fig. 7. Comparison of real and predicted gas flow for implementation of Equation (13) to data points of S ¼ 40/64 inch.
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Fig. 8. Comparison of real and predicted gas flow for implementation of Equation (13) to data points of S ¼ 64/64 inch.
Fig. 9. Comparison of real and predicted gas flow for implementation of Equation (13) to data points of S ¼ 128/64 inch.
Fig. 10. Comparison of real and predicted gas flow for implementation of Equation (13) to data points of S ¼ 144/64 inch.
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Fig. 11. Comparison of real and predicted gas flow for implementation of Equation (13) to data points of S ¼ 160/64 inch.
Louisiana, the sub-critical regimes are mostly occur in larger choke sizes. Taking this into consideration, Nasriani and Kalantariasl (2011) established the first empirical Gilbert-type equation for high flow rates through large choke sizes (with range of 40/64 inch to 192/64 inch) under sub-critical conditions. In their investigation, non-linear regression analysis of Excel solver was utilized for modeling 61 data tests gathered from fifteen wells of 10 different fields located in south-west of Iran. Moreover, they expanded AlAttar (2008) model for high rate gas condensate wells and large choke sizes under sub-critical flow regime. The error study revealed that Al-Attar graphical approach is more precise in estimation of high flow rates passing through the test restrictions in sub-critical conditions based on individual large choke sizes. Recently, Nejatian et al. (2014) implemented Least-Squares Support Vector Machine (LSSVM) algorithm for modeling a subcritical natural gas flow in orifice- and nozzle-type chokes by using 171 and 164 data points for each, respectively. The goodness of fitting data by aforementioned model is represented through average absolute relative deviation and squared correlation coefficient. Utilizing theoretical approaches for modeling the production data requires fluid properties which may not be available in many cases while empirical approaches were approved in practical production purposes by many researchers (Al-Attar, 2008; Nasriani and Kalantariasl, 2011; Beiranvand et al., 2012; Azin et al., 2014; Lak et al., 2014). Most of empirical models established for oil fields and under critical conditions while there are few ones for predicting the flow behavior through chokes of gascondensate wells in sub-critical flow regime. The objective of this investigation is to implement a simple modification to Nasriani and Kalantariasl (2011) method in order to present a more precise empirical model originated from Gilbert model in a subcritical flow regime across chokes of gas-condensate wells. This model is the second empirical Gilbert-type model for gascondensate wells which is applicable in high flow rates and large choke sizes under sub-critical conditions. In order to develop this newly proposed model, non-linear regression approach and Genetic Algorithm are conducted to find the best method for fitting the production data to the proposed model. Using Genetic Algorithm to optimize the empirical constants of a new empirical equation for two-phase flow of wellhead chokes has not been reported in literature. Afterward, the best optimization approach is
Table 2 Error analysis for all data points by implementing non-linear regression technique. Models
The new correlation
Model 1
Model 2
2.044 14.725 18.679 65.814 8.112 0.9204
9.637 34.861 47.894 986.117 31.402 0.6661
16.751 27.216 27.587 283.112 16.825 0.7993
Errors APD AAPD SD MSE RMSE R2
utilized to conduct a comparison between the new correlation and two other models derived in similar conditions, Osman and Dokla (Model 1), and Nasriani and Kalantariasl (Model 2). Both models are using pressure drop instead of upstream pressure to predict
Table 3 Error analysis for all data points when d ¼ 1 in Equation (2). Error type
APD
AAPD
SD
MSE
RMSE
R2
value
27.299
31.049
24.281
269.675
16.422
0.821
Table 4 Error analysis for the data gathered from gas-condensate wells located in Fars province of Iran. Models
The new correlation
Model 1
Model 2
9.345932 11.71358 37.249 61.78235 7.860175 0.8513
13.4963 22.84279 24.81883 137.2267 11.71438 0.7441
29.22089769 29.43477176 46.26681324 162.839 12.76084 0.7894
Errors APD AAPD SD MSE RMSE R2
Table 5 Number of field data in each choke size. Choke size Number of chokes Dataset Percent %
40 8 11.94
64 11 16.4
128 10 14.93
144 8 11.94
160 8 11.94
176 6 8.96
192 16 23.88
270
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Fig. 12. Comparison of real and predicted gas flow for implementation of Equation (13) to data points of S ¼ 176/64 inch.
the gas flow rate, however, the first one is developed for critical flow regime and the second for sub-critical one. Model 1 and Model 2 are selected to show the failure of existing empirical models (developed for both critical and sub-critical flow regimes) to predict the two-phase flow of south Iranian gas-condensate wells and the need to develop a new one. With the purpose of validating and evaluating the new model in other field data, 39 data points gathered from gas-condensate wells of Fars province of Iran are exposed to the proposed model, Model 1 and Model 2. Moreover, a correlation is derived for dataset of each choke size to minimize the error. Ultimately, statistical error analyses with various parameters are implemented to compare the aforementioned models. It is worth mentioning that in the industry, the total liquid and gas produced from some wells are usually directed to a separator unit and only one gaseliquid ratio is reported for all the wells. This value is not for a particular well. Therefore, finding field data for each gas-condensate well to develop the proposed model was very difficult.
2. Methodology Optimization is one of the most important areas of modern applied mathematics (Khishvand and Khamehchi, 2012). Nonlinear regression analysis and Genetic Algorithm are the two optimization methods used in this investigation. Non-linear regression is based on succeeding approximations which is able to fit the observational data with a function that nonlinearly depends on one or more model variables. This approach usually arises when there are physical evidences for believing that the data and the variables follows a particular functional form (Smyth, 2002). Non-linear regression method utilizes linear approximation of Taylor Series Expansion (the first two terms) to linearize a nonlinear combination of independent variables of model. In an iterative process, the estimation of parameters is conducted by minimizing the summation of squared errors. The iterations come to an end as the errors of estimates become sufficiently small. Nonlinear optimization approach of Excel Spreadsheet (a user friendly
Fig. 13. Comparison of real and predicted gas flow for implementation of Equation (13) to data points of S ¼ 192/64 inch.
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software) has been utilized to find the optimum empirical constants of the proposed model and establish the minimum error relationship between the dependent and independent variables in this study (Hemmati-Sarapardeh et al., 2013; Naseri et al., 2014). Genetic Algorithm (GA) is a heuristic method to optimize the data based on natural evolution through alteration and/or mutation in properties of the data. Genetic Algorithm have been used in many applications to find a solution in optimization problems, for example in property selection, recognizing patterns and understanding images. The key element of this method is the novel structure of the information processing system which is inspired by Darwin's evolution theory (Khishvand and Naseri, 2012). GA includes a main loop in which different data (individuals) are represented by their own chromosome. In this approach, the iterative procedure of selection typically begins with a population of gathered individuals at random. The population in every iteration is called a generation Each and every chromosome represents a point in the examination space and is a tentative solution to the optimization problem. Based on the fitness, the individuals have a chance to mate with each other and produce their offspring to compete in the next generation. For every generation, the fitness of each data to the goal function is calculated and then the data are ranked based on their fitness to the function. Next, in order to develop a new generation, the mating pool that is the set of chromosomes which are suitable to be utilized as the parents of next generation is built. The more fitted data from the existing population have more representatives in the mating pool. Modification is implemented to genome of each individual either through combination of properties or mutation. Then the candidates of the latest generation are utilized for the next iteration. The algorithm usually stops when either a maximum number of generations have been generated or an adequate point of fitness has been achieved for the population (Whitley, 1994; Mitchell, 1996; Ebrahimi et al., 2013). Predictive models, regarding of their algorithms, suffer from some errors and they are more regional in some cases. One of the errors associated with nonlinear regression analysis of Excel is the use of only first two terms of Taylor Series Expansion and ignoring the other terms. Another drawback of this method emerges when the function is highly nonlinear; in this case, the Spreadsheet optimizer might not derive the optimum solution, even though it
Table 8 Error analysis for data points of choke size S ¼ 128/64 inch.
Table 6 Error analysis for data points of choke size S ¼ 40/64 inch.
Table 10 Error analysis for data points of choke size S ¼ 160/64 inch.
Error type
APD AAPD SD MSE RMSE R2
The new correlation General form
Specific form
15.223 31.659 35.132 39.417 6.278 0.8552
6.274 9.759 26.783 21.815 4.671 0.9674
Model 1
50.188 78.072 72.038 1087.738 32.981 0.7873
Model 2
16.298 37.759 36.834 75.244 8.674 0.817
Table 7 Error analysis for data points of choke size S ¼ 64/64 inch. Error type
APD AAPD SD MSE RMSE R2
The new correlation General form
Specific form
12.769 16.485 9.590 61.378 7.834 0.9251
1.661 6.472 6.783 14.831 3.851 0.9669
Model 1
42.471 58.728 62.371 3807.686 61.706 0.6768
Error type
APD AAPD SD MSE RMSE R2
The new correlation General form
Specific form
0.144 12.127 21.675 34.935 5.911 0.8811
0.749 8.362 11.736 16.101 4.013 0.9451
Model 1
Model 2
5.097 23.133 26.568 138.619 11.774 0.8198
21.955 24.499 21.752 164.507 12.826 0.7608
says that a solution is found and all the conditions and the constraints are satisfied. Furthermore, due to the fact that there are limitations on the size of decision variables that Excel can solve, this method is not suitable for models with too many variables (usually more than 200 variables). On the other hand, one of the significant drawback of GA method is the large overproduction of individuals. GA applies to populations of chromosomes, and large amounts of data are utilized to solve even the easiest problems (Braha, 2001). Based on Adriaans and Zantinge (1996), the end-user of a GA method does not really see how the algorithm is creating and selecting individuals for finding patterns in large amounts of data. The relation between the pressure and the liquid flow rate is not always a straight line but may be concave to the origin (Gilbert, 1954). Al-Towailib and Al-Marhoun (1994) and Alrumah and Bizanti (2007) took this into consideration and used an exponent for pressure to obtain a more precise predictive model for their oil
Table 9 Error analysis for data points of choke size S ¼ 144/64 inch. Error type
APD AAPD SD MSE RMSE R2
Error type
APD AAPD SD MSE RMSE R2
The new correlation General form
Specific form
5.098 14.359 14.194 94.692 9.731 0.9619
2.297 11.088 14.179 27.462 5.240 0.9794
The new correlation General form
Specific form
3.665 14.527 13.750 76.386 8.740 0.8602
1.181 14.185 13.605 44.420 6.665 0.925
Model 1
Model 2
7.6175 29.142 32.644 484.239 22.005 0.9161
23.936 29.543 26.909 181.869 13.486 0.9143
Model 1
Model 2
17.637 22.376 23.174 193.615 13.914 0.8414
28.691 29.236 19.926 220.372 14.845 0.8434
Table 11 Error analysis for data points of choke size S ¼ 176/64 inch. Model 2
7.601 31.546 38.399 636.680 25.232 0.695
Error type
APD AAPD SD MSE RMSE R2
The new correlation General form
Specific form
6.189 7.0629 6.754 85.673 9.256 0.7564
0.860 4.117 4.150 16.604 4.075 0.9277
Model 1
Model 2
9.994 13.069 16.085 367.339 19.166 0.6905
4.355 14.789 13.874 181.797 13.483 0.6739
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Equations (3) and (4), these two formulas have the same forms with different constants due to different production data:
Table 12 Error analysis for data points of choke size S ¼ 192/64 inch. Error type
APD AAPD SD MSE RMSE R2
The new correlation General form
Specific form
0.780 11.230 12.665 74.189 8.613 0.8445
0.602 10.572 11.982 71.148 8.435 0.8326
Model 1
Model 2
Model 1 (Osman and Dokla, 1990) 1.872 21.452 26.871 404.382 20.109 0.7407
15.099 23.152 25.173 338.081 18.387 0.7252
Qg ¼
S1:8587 DP
(3)
302ðLGRÞ0:4038
Model 2 (Nasriani and Kalantariasl, 2011) fields. In this study, an exponent is added to the upstreamdownstream pressure difference (pressure drop) of Model 1 to consider this curvature in the flow passing through the chokes of gas condensate wells. Sixty seven production datasets collected from fifteen gas-condensate wells of more than ten different south Iranian gas fields are fitted to the proposed model through nonlinear regression analysis and Genetic Algorithm methods. Then the best method was implemented to three different models (including Model 1, Model 2 and the newly proposed model) to examine the applicability of the proposed model in comparison to the current ones. Moreover, a correlation was derived for dataset of each choke size to minimize the error. The downstream-upstream pressure ratio for all data tests are less than 0.5 which means that the flowing fluid is in subcritical condition. Table 1 represents the different range of parameters utilized in this investigation. The proposed model is as follows:
Qg ¼
S1:9 DP
Seven statistical error analyses (Equations (5)e(11)) including percent deviation (PD), average percent deviation (APD), absolute average percent deviation (AAPD), standard deviation (SD), mean square error (MSE), root mean square error (RMSE) and R-squared (R2) are implemented to evaluate the accuracy of the aforementioned choke models (Ghaffari et al., 2006; Sin et al., 2006; Nasriani and Kalantariasl, 2011):
PDi ¼
Qg;real Qg;predicted Qg;real Pn i¼1
APD ¼
(2)
(5)
PDi
(6)
n Pn
Qg ¼ aðLGRÞb Sc DP d
(4)
9350ðRGL Þ0:65
jPDi j n
i¼1
AAPD ¼
(7)
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u n . uX SD ¼ t ðPDi APDÞ2 ðn 1Þ a, b, c and d: Empirical constants. DP: Pressure drop (m/Lt2), psia. Qg: Volumetric gas flow rate (L3/t), MMscf/D. LGR: Liquid-Gas ratio (L3/L3), Stb/MMscf. S: Choke size (L), 1/64 inch. Finally, Model 1 and Model 2 in addition to the proposed correlation are utilized to fit the data of each choke size. As shown in
(8)
i¼1
Pn MSE ¼ RMSE ¼
i¼1
Qg;predict Qg;real
2
n pffiffiffiffiffiffiffiffiffiffi MSE
Fig. 14. Comparison of real and predicted gas flow for specific implementation of the main form to data points of S ¼ 40/64 inch.
(9) (10)
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Fig. 15. Comparison of real and predicted gas flow for specific implementation of the main form to data points of S ¼ 64/64 inch.
Pn
i¼1
R2 ¼ 1
0
Pn
i¼1
Qg;predict Qg;real
2 12
(11)
Pn B C Qg;real;i C B BQg;real i¼1n C @ A
where i ¼ 1,2,3,..,n.
3. Result and discussion As shown in Table 1, the two-phase data gathered from south Iranian gas-condensate reservoirs cover high flow rates (from 11.3 to 113 MMscf/D) and a broad range of choke sizes (from 40/64 to 192/64 inch).
There are few simple empirical formulas for predicting the behavior of flow in chokes under sub-critical conditions, and especially for gas-condensate wells. The failure in our efforts to utilize the current equations to fit the data of sub-critical two-phase fluid flow across wellhead chokes led us to correlate a new Gilberttype formula (the results of estimating the field data with Model 1 and Model 2 are shown in Figs. 4e11 and Table 2). As mentioned in previous section, DP was substituted for Pwh in Gilbert formula (Equation (1)) and an exponent was added to DP to develop the proposed model (Equation (2)). Then Genetic Algorithm and nonlinear regression analysis methods were used to obtain the optimum values of the empirical constants for the collected gascondensate field data under sub-critical conditions. 3.1. Genetic Algorithm method The MATLAB Genetic Algorithm (GA) toolbox is utilized in this
Fig. 16. Comparison of real and predicted gas flow for specific implementation of the main form to data points of S ¼ 128/64 inch.
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Fig. 17. Comparison of real and predicted gas flow for specific implementation of the main form to data points of S ¼ 144/64 inch.
study. This toolbox initializes a random sample of individuals (field data) with various variables to be optimized (Ebrahimi et al., 2013). Equation (2) is the objective function in this optimization problem and the variables are Qg, LGR, S, and DP. The range of this variables are listed in Table 1. The optimum solutions (the best empirical constants) achieved for the proposed model are shown in Fig. 1 and a comparison between measured and predicted gas flow rates of all data pointes are presented in Fig. 2. The obtained solutions were 0.0164, 0.3931, 1.2624, and 0.556 for a, b, c, and d, respectively and the new equation turns into the following form:
Qg ¼
0:0164S1:2624 DP 0:5566 ðLGRÞ0:3931
(12)
One can see that the data are comparatively in good agreement with the proposed model through this algorithm (MSE ¼ 58.6106 MMscf/D and R2 ¼ 0.9189).
3.2. Non-linear regression analysis method In this section, with the purpose of finding the optimum or the most effective way of using our limited resources (field data) to obtain the objective of the optimization problem (optimum values for empirical constants of the newly proposed model), the entire sixty seven production datasets are fitted with the proposed model through non-linear regression analysis approach of Excel Spreadsheet and Solver. A mathematical model which conducted in a spreadsheet of Excel is known as a Spreadsheet model. The Solver need to identify the location of the Spreadsheet model (Equation (2)), decision variables (Qg, LGR, S and DP), the essence of the optimization (maximizing or minimizing) and the problem limitations (the range of field data, Table 1). Therefore, the data must be organized in the spreadsheet to represent the model and then the solver was used to find the best solution (Chandrakantha, 2008). The result of choke modeling with non-linear regression
Fig. 18. Comparison of real and predicted gas flow for specific implementation of the main form to data points of S ¼ 160/64 inch.
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Fig. 19. Comparison of real and predicted gas flow for specific implementation of the main form to data points of S ¼ 176/64 inch.
method are shown in Fig. 3. The obtained general formula is as follows:
Qg ¼
0:015S1:27 DP 0:56 ðLGRÞ0:4
(13)
As one can see from Table 2, this method is able to fit the data slightly better than Genetic Algorithm and the obtained constants in both methods are approximately the same. Therefore, non-linear regression method which has the better results is used in the following sections. Fig 4 illustrates a comparison between real and predicted flow rates of all three abovementioned models for all data points. Based on seven statistical measures of accuracy (PD, APD, AAPD, SD, MSE, RMSE, R2) in Table 2, it can be concluded that the gas flow rates simulated using the newly proposed correlation are in the best agreement with those measured data. As mentioned before, Model 1 and Model 2 have the same form. In both formulas, the exponent of DP is considered to be
unity but the first formula is derived under critical conditions and the other under sub-critical ones. The same consideration was implemented to fit all data points (d ¼ 1) and the results are shown in Fig. 5 and Table 3. One can see that this form of correlation (R2 ¼ 0.821) has less accuracy in comparison with the proposed one (R2 ¼ 0.9204) in the investigated fields. Taking the results of Table 2 into account, it can be concluded that even though both models are developed to predict the gas flow rate of gas-condensate wells, the accuracy of Model 2 (R2 ¼ 0.7993) is more than Model 1 (R2 ¼ 0.6662). This can show that the empirical models developed for critical flow regime cannot predict the field data of sub-critical one even with the same form. Consequently, the two-phase flow regime through wellhead chokes of gascondensate wells under sub-critical conditions needs more attention and more investigations. Unfortunately, there is no available data in literature about the fluid passing through wellhead chokes of gas condensate wells to evaluate the new model with other researcher's data. Therefore, in
Fig. 20. Comparison of real and predicted gas flow for specific implementation of the main form to data points of S ¼ 192/64 inch.
276
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Table 13 Values of empirical constants of Equation (2) when applied to each choke size. Choke size (1/64 inch)
a
b
c
d
40 64 128 144 160 176 192
0.0177 0.0174 0.0173 0.0172 0.0179 0.018 0.0174
0.43 0.26 0.44 0.59 1.07 0.7 0.3
1.5 1.38 1.33 1.24 1.45 1.48 1.28
0.39 0.44 0.49 0.64 0.6 0.4 0.48
order to show that the newly proposed model is able to capture other field data better than available models (Model 1 and Model 2), non-linear regression analysis is implemented to 39 data points of seven gas condensate wells located in Fars province of Iran. Based on the results which are shown in Fig. 6 and Table 4, the new correlation is also able to predict the flow rates passing through the chokes of gas-condensate wells in other fields better than the existing models (R2 ¼ 0.8513, 0.7441, and 0.7894 for the new model, Model 1, and Model 2 respectively). For the future researchers, all the original data based on which the correlation is developed in addition to the data collected from the gascondensate wells of Fars province are tabulated in Appendix as Table A1 and Table A2. Finally, it is highly recommended to verify the precision of the newly proposed model with more field data of sub-critical two-phase flow regime through wellhead chokes of gas-condensate wells.
eration, the following conclusion can be made. 1. The non-linear regression analysis showed more accuracy in fitting the data to the proposed model in comparison to Genetic Algorithm. 2. Seven statistical measures of accuracy (PD, APD, AAPD, SD, MSE, RMSE, and R2) indicate that the presented correlation is the most precise one in predicting the gas flow rate flowing through the wellhead chokes in the investigated fields. 3. The application of the new correlation should be used for the range of field data covered in this investigation (gas flow rate of 11.3e113 MMscf/D and choke size range of 40/64e192/64 inch) 4. The newly proposed model can also be used to predict gas flow rates in conditions where measuring the flow rates of gascondensate wells under sub-critical conditions is not accessible or difficult to perform. 5. In similar condition, the new model can help the production engineer to optimize the production rate by determining the optimum choke size. 6. The proposed model is also able to capture other field data better than the available models. 7. Based on individual choke size analyses, among wide range of choke sizes used in this investigation, the one with the size of 144/64 in has the minimum error. 8. The precision of the newly proposed method has been found to be greatly improved when applied to individual choke sizes. Acknowledgment
3.3. Choke size analysis Three different predictive models, (Model 1, Model 2, and the new correlation) are utilized to determine the best operating wellhead choke size for the data gathered form high rate gascondensate reservoirs of south of Iran. Choke sizes and number of chokes in each choke size are listed in Table 5. Implementation results of the three models (Equations (3), (4) and (13)) for each choke size are shown in Figs. 7e13. As shown in the figures, higher fluid flow rates leads to reduction in accuracy of the models. In AlAttar (2008) model, where the exponent d is considered to be unity, the ability of his formula to predict gas flow rates deteriorates at larger choke sizes, however, according to error analyses (Tables 6e12), the newly proposed model has high accuracy even in large choke sizes. Moreover, the best wellhead choke size in the gathered data is 144/64 inch. Finally, the main form of the proposed model (Equation (2)) was separately conducted for each choke size and the constants were specifically derived for each one. The results are shown in Figs. 14e20 and Table 13. Error analyses indicate that applying the main form (Equation (2)) to each choke size increases the accuracy of fitting in comparison with general form (Equation (13)). In other words, the ability of the newly proposed model to predict gas flow rates of fluids passing through the wellhead chokes of gas-condensate wells under sub-critical conditions is found to be highly enhanced when subjected to individual choke sizes. 4. Conclusion A new correlation was established for sub-critical two-phase flow through wellhead chokes and the validity of proposed formula was tested based on 67 data point gathered from south Iranian high rate gas-condensate reservoirs utilizing Genetic Algorithm and non-linear regression analysis. Taking all the results into consid-
The authors would like to thank Dr. Jamshid Moghaddasi (Petroleum Department of Petroleum University of Technology) for his useful suggestions. Nomenclature a empirical constant AAPD absolute average percent deviation APD average percent deviation b empirical constant c empirical constant d empirical constant LGR liquid-gas ratio, Stb/MMscf MSE mean square error, MMscf/D Pd downstream pressure, psia Pu upstream pressure, psia Pwh wellhead pressure, psia PDi percent deviation of data i Q volumetric liquid flow rate bbl/D Qg volumetric gas flow rate, MMscf/D Qg, predicted predicted values of volumetric gas flow rate, MMscf/D Qq, real real values of volumetric gas flow rate, MMscf/D R2 R-squared error RGL gas-liquid ratio, MScf/Stb RMSE root mean square error, (MMscf/D)2 S choke size, 1/64 inch SD standard deviation, MMscf/D Tf wellhead flowing temperature, F Appendix
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Table A 1 The sub-critical data gathered from South Iranian gas-condensate wells. Number
Pu(psia)
Pd(psia)
Qg(MMscf/D)
S(1/64inch)
LGR(bbl/MMscf)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67
1501.978 1518.076 2422 2103 1827 1406.5 1450 2741 1827.57 2059 1319.5 1334 1305 2016.216 1636.156 1493.5 3901.499 1957.5 1232.5 2102.5 1247 1972 2073.5 1783.5 1160 1605.094 1914 2044.5 1290.5 1131 1145.5 1957.5 1885 1986.5 2451 2393 2393.314 2480.337 2407.818 1624 1348.5 1870.5 1334 1392 1174.5 1841.5 1841.5 3785.475 1682 1624 1798 3350.385 4452.613 1783.5 1740 1595 1595 1493.5 1566 1653 1653 1537 1653 1653 1972 2175 1653
824.83258 1484.13896 1740 1726 1638 1319.5 1334 1798 1653.534 2044.5 1290.5 1290.5 1290.5 1900.45559 1195.5025 1435.5 2944.301 1885 1218 2030 1232.5 1899.5 2001 1566 1131 902.85872 1827 2001 1276 1102 1131 1870.5 1566 1899.5 1769 1682 1726.41031 1755.41786 1697.40276 1551.5 1290.5 1537 1290.5 1348.5 1145.5 1667.5 1667.5 2987.81 1595 1508 1624 2016.109 3045.822 1653 1595 1508 1566 1406.5 1464.5 1479 1479 1435.5 1493.5 1522.5 1696.5 1899.5 1435.5
11.3008 12.7134 13.1 15.1 15.641 19.811715 23.696365 26.84 27.73 28.1637125 28.711095 28.852355 28.852355 31.78320005 32.48949338 33.019525 38.72 38.881815 39.5528 44.14375 44.4969 45.238515 45.521035 45.55635 45.9095 46.5564 47.67525 50.041355 52.2662 52.9725 54.3851 56.398055 56.85715 56.963095 57.1043 57.5674 58.26920009 58.26920009 58.26920009 58.5769905 60.0355 63.567 63.743575 63.743575 64.62645 66.74535 67.063185 77.5 82.531155 83.738928 83.873125 84.4 86.76 87.59709175 88.57002 89.91199 90.2474825 95.27987 96.2863475 101.389365 101.6542275 102.06035 103.8261 104.8414063 109.702516 111.2952225 113.008
40 40 40 40 40 128 128 40 64 160 144 144 144 64 64 128 40 160 192 160 192 144 160 128 192 40 144 160 192 192 192 160 128 160 64 64 64 64 64 176 128 128 192 128 192 128 128 64 192 176 192 64 64 192 176 192 192 192 176 144 192 176 176 192 144 144 160
12.46244655 9.564575959 16.1 16.1 4.81 19.52099674 32.21498797 13.536 5.556 6.144483081 4.452782104 4.809004672 4.630893388 5.428 27.49134896 14.99697013 4.374 7.355996036 2.388472321 6.236566615 1.220062296 7.008679032 6.434982585 14.89366558 2.130210959 1.537100382 5.582007645 5.708110435 1.540662608 1.976322809 2.440124593 7.32749823 13.27107178 5.64505904 8.193 8.193 8.193196393 8.193196393 8.193196393 5.555555556 1.920039643 14.62115532 2.128429846 1.246778989 1.923601869 5.042330454 5.443080844 4.374 5.184819482 5.586592179 8.506594931 6.3 4.374 6.319388362 5.120521308 4.239048563 0.688222002 3.419736656 3.875701543 3.961551182 5.03894634 3.526603426 4.198082968 5.681749965 5.566690075 5.413158148 5.713809996
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Table A 2 The sub-critical data gathered from gas-condensate wells of Fars province of Iran. Number
Pu(psia)
Pd(psia)
Qg(MMscf/D)
S(1/64inch)
LGR(bbl/MMscf)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39
2103.238 2335.299 2074.231 1929.193 2175.757 2161.254 2306.291 2335.299 2277.284 2219.269 2248.276 2233.772 2248.276 2262.78 2233.772 2422.321 2378.81 2364.306 2451.329 2349.803 2335.299 2422.321 2480.337 2451.329 2335.299 2393.314 2364.306 2335.299 2378.81 2277.284 2407.818 2349.803 2306.291 2262.78 2306.291 2248.276 2219.269 2088.735 2074.231
1885.682 1769.652 1827.667 1726.14 1842.17 1842.17 1668.125 1813.163 1784.155 1682.629 1813.163 1682.629 1813.163 1769.652 1682.629 1958.201 1755.148 1639.118 1871.178 1827.667 1784.155 1871.178 1755.148 1784.155 1784.155 1932.704 1900.186 1648.976 1972.704 1929.193 2001.712 1842.17 1900.186 1856.674 1885.682 1871.178 1842.17 1885.682 1827.667
13.15733 17.65733 10.5944 10.21658 17.75733 15.81957 28.25173 26.486 33.31467 27.95173 24.92027 28.25173 28.886 32.31467 28.25173 38.84613 45.90907 49.44053 38.84613 42.3776 43.10907 44.14333 42.92445 43.43704 52.972 54.6212 42.90907 52.44053 51.972 45.90907 49.44053 51.772 49.44053 53.00627 47.20627 48.72508 51.20627 43.6748 47.6748
32 32 32 32 40 40 48 48 48 48 48 48 48 48 48 56 56 56 56 56 56 56 56 56 64 64 64 64 64 64 64 64 64 68 68 68 68 72 72
8.155251818 8.155251818 9.816290462 10.16429765 8.159153296 8.122463891 9.583173087 9.19353791 12.36184011 10.63976638 8.191659193 9.597138551 8.529570772 9.836195229 10.14937554 6.45991707 6.014062572 6.171049925 8.460166818 6.325900765 7.255890711 6.937610995 7.374252268 7.374252268 8.039172638 8.039172638 8.200994417 7.846533459 7.684255496 6.647521148 8.300264718 7.620984015 6.958300819 6.273129561 7.857927247 6.642739111 8.192356652 5.274135026 5.718364109
References Adriaans, P., Zantinge, D., 1996. Data Mining. Harlow. Addison Wesley, England. Al-Attar, H., 2008. Performance of wellhead chokes during sub-critical flow of gas condensates. J. Petroleum Sci. Eng. 60 (3), 205e212. Al-Attar, H., Abdul-Majeed, G., 1988. Revised bean performance equation for East Baghdad oil wells. SPE Soc. Petroleum Eng. Prod. Eng. (United States) 3 (1). Al-Towailib, A.I., Al-Marhoun, M., 1994. A new correlation for two-phase flow through chokes. J. Can. Petroleum Technol. 33 (5), 40e43. Ali Ghalambor, S., 2002. Applicability of Sachdeva's Choke Flow Model in Southwest Louisiana Gas Condensate Wells. Alrumah, M.K., Bizanti, M.S., 2007. New Multiphase Choke Correlations for Kuwait Fields. SPE Middle East Oil and Gas Show and Conference. Society of Petroleum Engineers. Ashford, F., Pierce, P., 1975. Determining multiphase pressure drops and flow capacities in down-hole safety valves. J. Petroleum Technol. 27 (9), 1145e1152. Azin, R., Chahshoori, R., Osfouri, S., Lak, A., Sureshjani, M.H., Gerami, S., 2014. Integrated analysis of choke performance and flow behaviour in high-rate, deviated gas-condensate wells. Gas Process. J. (1), 8e18. Publisher, College of engineering, university of Isfahan, I.R. Iran. Baxendell, P., 1958. Producing Wells on Casing Flow-an Analysis of Flowing Pressure Gradients. Beiranvand, M.S., Mohammadmoradi, P., Aminshahidy, B., Fazelabdolabadi, B., Aghahoseini, S., 2012. New multiphase choke correlations for a high flow rate
Iranian oil field. Mech. Sci. 3 (1), 43e47. Braha, D., 2001. Data Mining for Design and Manufacturing: Methods and Applications. Kluwer Academic Publishers. Brill, J.P., Beggs, H.D., 1984. Two-phase Flow in Pipes (publisher not identified). Chandrakantha, L., 2008. Using Excel Solver in Optimization Problems. Mathematics and Computer Science Department, New York. Ebrahimi, A.N., Jamshidi, S., Iglauer, S., Boozarjomehry, R.B., 2013. Genetic algorithm-based pore network extraction from micro-computed tomography images. Chem. Eng. Sci. 92, 157e166. Elgibaly, A., Nashawi, I., 1998. New correlations for critical and subcritical two-phase flow through wellhead chokes. J. Can. Petroleum Technol. 37, 36e43. Enaibe, J., Ajienka, O., 1994. MuItiphase Flow Metering: an Evaluation of Discharge Coefficients. Fortunati, F., 1972. In: . This Paper was Prepared for the SPE-European Spring Meeting I972 of the Society of Petroleum 13tigineers of AIME, Held in Amsterdam, The Netherlands, May 16e18, 1972. Permission to Copy Isrestricted to an Abstract of. Ghaffari, A., Abdollahi, H., Khoshayand, M., Bozchalooi, I.S., Dadgar, A., RafieeTehrani, M., 2006. Performance comparison of neural network training algorithms in modeling of bimodal drug delivery. Int. J. Pharm. 327 (1), 126e138. Gilbert, W., 1954. Flowing and gas-lift well performance. API Drill. Prod. Pract. 20 (1954), 126e157. Hemmati-Sarapardeh, A., Khishvand, M., Naseri, A., Mohammadi, A.H., 2013. Toward reservoir oil viscosity correlation. Chem. Eng. Sci. 90, 53e68. Khishvand, M., Khamehchi, E., 2012. Nonlinear risk optimization approach to gas lift allocation optimization. Industrial Eng. Chem. Res. 51 (6), 2637e2643. Khishvand, M., Naseri, A., 2012. An artificial neural network approach to predict asphaltene deposition test result. Fluid Phase Equilibria 329, 32e41. Lak, A., Azin, R., Osfouri, S., Gerami, S., Chahshoori, R., 2014. Choke modeling and flow splitting in a gas-condensate offshore platform. J. Nat. Gas Sci. Eng. 21, 1163e1170. Lannom, D.A., Hatzignatiou, D., 1996. Multiphase-flow choke correlation limits analyzed. Oil Gas J. 94 (15). Mesallati, A., Biznati, M., Mansouri, N., 2000. Multiphase-flow choke correlations for offshore Bouri oil field. In: Proceedings of International Gas Union 21st World Gas Conference, Nice, France, June. Mitchell, M., 1996. An Introduction to Genetic Algorithms, 1996. PHI Pvt. Ltd., New Delhi, p. 2. Naseri, A., Khishvand, M., Sheikhloo, A., 2014. A correlations approach for prediction of PVT properties of reservoir oils. Petroleum Sci. Technol. 32 (17), 2123e2136. Nasriani, H.R., Kalantariasl, 2011. Two-phase flow choke performance in high rate gas condensate wells. In: SPE Asia Pacific Oil and Gas Conference and Exhibition. Society of Petroleum Engineers. Nejatian, I., Kanani, M., Arabloo, M., Bahadori, A., Zendehboudi, S., 2014. Prediction of natural gas flow through chokes using support vector machine algorithm. J. Nat. Gas Sci. Eng. 18, 155e163. Omana, R., Houssiere Jr., C., Brown, K.E., Brill, J.P., Thompson, R.E., 1969. Multiphase flow through chokes. In: Fall Meeting of the Society of Petroleum Engineers of AIME. Society of Petroleum Engineers. Osman, M.E., Dokla, M.E., 1990. Gas Condensate Flow through Cho~ Es. Perkins, T.K., 1993. Critical and subcritical flow of multiphase mixtures through chokes. SPE Drill. Complet. 8 (04), 271e276. Pilehvari, A.A., 1981. Experimental Study of Critical Two-phase Flow through Wellhead Chokes. University of Tulsa. Poettmann, F., Beck, R., 1963. New charts developed to predict gas-liquid flow through chokes. World Oil 184 (3), 95e100. Rahimzadeh, A., Mohammadmoradi, P., 2014. Development of choke multiphase flow modeler. In: International Petroleum Technology Conference, International Petroleum Technology Conference. Ros, N., 1960. An analysis of critical simultaneous gas/liquid flow through a restriction and its application to flowmetering. Appl. Sci. Res. 9 (1), 374e388. Sachdeva, R., Schmidt, Z., Brill, J., Blais, R., 1986. Two-phase Flow through Chokes. paper SPE 15657, pp. 5e8. Sin, H.N., Yusof, S., Hamid, N.S.A., Rahman, R.A., 2006. Optimization of enzymatic clarification of sapodilla juice using response surface methodology. J. Food Eng. 73 (4), 313e319. Smyth, G.K., 2002. Nonlinear regression. Encycl. Environ. 3, 1405e1411. Surbey, D., Kelkar, B., Brill, J., 1989. Study of multiphase critical flow through wellhead chokes. SPE Prod. Eng. 4 (02), 142e146. Tangren, R., Dodge, C., Seifert, H., 1949. Compressibility effects in two-phase flow. J. Appl. Phys. 20 (7), 637e645. Whitley, D., 1994. A genetic algorithm tutorial. Stat. Comput. 4 (2), 65e85.
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Journal of Natural Gas Science and Engineering journal homepage: www.elsevier.com/locate/jngse
A new correlation for prediction of sub-critical two-phase flow pressure drop through large-sized wellhead chokes Siamak Seidi a, Tofigh Sayahi b, * a b
Department of Petroleum Engineering, Petroleum University of Technology, Ahwaz, Iran Department of Chemical Engineering, Faculty of Engineering, Ferdowsi University of Mashhad (FUM), Mashhad, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 4 April 2015 Received in revised form 9 June 2015 Accepted 10 June 2015 Available online xxx
Unacceptable errors resulted from utilizing the current correlations for investigating the behavior of subcritical two-phase flow regime through wellhead chokes of south Iranian gas condensate reservoirs led us toward establishing a new Gilbert-type correlation capable of fitting the production data with minimum errors. The proposed model, which is a modification of Nasriani and Kalantariasl model, is able to predict the high flow rates of gas condensate wells under sub-critical conditions particularly in case of large choke sizes. In order to validate the new correlation, Genetic Algorithm and non-linear regression analysis methods are implemented to sixty seven production datasets of fifteen wells with large wellhead choke sizes (40/64e192/64 inch) gathered from 10 different fields. Then the proposed correlation in addition to two other models (1.Osman and Dokla 2. Nasriani and Kalantariasl) are conducted to each choke size to investigate the applicability of the new formula in comparison with the existing ones. Moreover, in order to evaluate the new model in other field data, 39 data points gathered from gascondensate wells of Fars province of Iran are exposed to the proposed model, and the two other models. Finally, the main form of the new correlation is applied to dataset of each choke size. The results indicate that the non-linear regression technique is more accurate than Genetic algorithm in fitting the data to the proposed method. Furthermore, the new correlation has the minimum errors in comparison to other methods in both investigated areas. Finally, according to statistical error analysis for each choke size, the ability of the proposed correlation to predict gas flow rates of fluids passing through the wellhead chokes of gas-condensate wells under sub-critical conditions is found to be highly improved when applied to individual choke sizes. © 2015 Elsevier B.V. All rights reserved.
Keywords: Wellhead choke Sub-critical Gas-condensate Genetic algorithm Non-linear regression
1. Introduction Gas condensate reservoir is a particular kind of reservoir which has a behavior mediating between that of gas and volatile oil reservoirs. It contains low density liquid hydrocarbons which present as gaseous components in the raw natural gas and condense out of gas when the pressure is lowered below the dew point pressure of hydrocarbons. Due to changes in temperature and pressure, these types of reservoir are bound to instability of flow regime, fluctuation and phase change which may lead to liquid holdup and phase separation. Therefore, multiphase flow is common in gas condensate reservoirs. The flow rate of well is the most significant parameter for
* Corresponding author. Ferdowsi University of Mashhad (FUM) Campus, Azadi Sq., Mashhad, Khorasan Razavi, P.O. Box: 9177948974, Iran. E-mail address: tofi[email protected] (T. Sayahi). http://dx.doi.org/10.1016/j.jngse.2015.06.025 1875-5100/© 2015 Elsevier B.V. All rights reserved.
characterization of a reservoir and estimation of its behavior. Wellhead chokes (a configuration which an elbow is installed exactly upstream of chokes) normally are used in wells as controlling agents to adjust the flow rate and sustain adequate back pressure to avoid sand issues and water/gas coning. Therefore, estimation of choke performance by implementation and optimization of a relation between size of chokes and wellhead flow rates is only possible through a precise modeling and selection of an optimum choke size. Based on flow regime, the fluid flow through the choke can be characterized as either critical or sub-critical (sonic or subsonic, respectively). When Mach number is equal or more than unity (When the flow velocity is equal or greater than sonic velocity), any pressure disturbance wave from downstream cannot spread through upstream and mass flow rate reaches a maximum amount which only depends upon upstream conditions. This kind of flow is known as critical flow. Accordingly, to avoid any perturbation of
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Table 1 Different parameter ranges of south Iranian field data. Parameters
S (1/64inch)
LGR (bbl/MMscf)
Qg (MMscf/D)
Pu (psia)
Pd (psia)
DP (psia)
Tf ( F)
Minimum Maximum
40 192
0.688 32.215
11.3 113
1131 4452
824.84 3045.82
14.5 1407
109 211
Fig. 1. Derivation of the new correlation constants by Genetic Algorithm.
surface equipment, most chokes are usually managed to work in critical flow region. In contrast, the sub-critical flow is described as the flow which its highest possible flow rate is less than sonic velocity. In this case, the mass flow rate is a function of pressure drop across the choke and any downstream disturbance is able to influence the upstream region. Due to difficulty of estimating gas and sound velocities in the field (Nejatian et al., 2014), usually in literature, the value of 0.5 for downstream to upstream pressure ratio of chokes (Pd/Pu ¼ 0.5) has been selected as the boundary between critical and subcritical flow regimes. The flow which has a Pd/Pu less than 0.5 is considered to be a sub-critical flow and the one with the ratio greater than 0.5 is considered to be a critical flow (Omana et al., 1969; Nasriani and Kalantariasl, 2011). Empirical and Analytical methods are two categories of multiphase flow estimation approaches across the chokes. Tangren et al. (1949) were the first researchers to implement analytical investigation on two-phase flow regime across limitations of chokes. However, their method is only valid where the liquid is the continuous phase. Ros (1960) followed the Tangren et al. (1949)
method and converted it into a model where gas is considered to be the continuous phase. Ashford and Pierce (1975) extended Ros (1960) model and derived a correlation to estimate the critical flow rates. However, they assessed that uncertainty is introduced in their correlation due to difficulty of determining the downstream pressure accurately (Rahimzadeh and Mohammadmoradi, 2014). An extension was implemented to Ashford and Pierce (1975) theoretical model by Sachdeva et al. (1986) for a choke placed in a direct horizontal pipe section. They derived two different correlations, one for predicting critical pressure ratio and the other for making a distinction between sonic and subsonic flow. Brill and Beggs (1984) combined Bernoulli and continuity equations to develop a formula for calculation of sub-critical flow rates. This purely theoretical model (which is called Mechanistic model) is widely used by the industry with a proven accuracy. In another theory-based study, Perkins (1993) derived a theoretical multiphase equation based on mass, momentum and energy balance for description of isentropic flow across chokes. It is worth mentioning that some theoretical models established by Fortunati (1972), Ashford and Pierce (1975), Pilehvari (1981), Surbey et al. (1989), Sachdeva et al. (1986), and Perkins (1993) are applicable for both critical and sub-critical conditions (Nasriani and Kalantariasl, 2011; Rahimzadeh and Mohammadmoradi, 2014). Gilbert (1954) derived the earliest empirical relation between choke size and surface flow rate by utilizing 268 production datasets from Ten Section Kern County Oil Fields of California. For derivation of his correlation, Gilbert assumed that the flow passing through an edge knife choke is critical and assessed that the correlation is valid when the upstream pressure of the choke is at least 70% higher than the downstream pressure. Gilbert's formula has the following form (Gilbert, 1954):
Pwh ¼
cQRaGL Sb
Fig. 2. Comparison of real and predicted gas flow for all data points by Genetic Algorithm.
(1)
266
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Fig. 3. Comparison between real and predicted gas flow rates for all data points by non-linear regression analysis.
Fig. 4. Comparison between the new correlation, Model 1, and Model 2 for all data points.
Fig. 5. Comparison between real and predicted gas flow rates for all data points when d ¼ 1.
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Fig. 6. Comparison between real and predicted gas flow rates for the data gathered from gas-condensate wells located in Fars province of Iran.
a, b, and c: Empirical constants. Pwh: Wellhead pressure (m/Lt2), psia. Q: Volumetric liquid flow rate (L3/t), bbl/D. RGL: Gas-Liquid ratio (L3/L3), MScf/Stb. S: Choke size (L), 1/64 inch. Empirical formulas are generally valid over the range where field data are available and may lead to poor results when extrapolated to new conditions (Perkins, 1993), hence a lot of researchers developed similar correlations with different constants for different field data tests in critical region (Baxendell, 1958; Poettmann and Beck, 1963; Al-Attar and Abdul-Majeed, 1988; Enaibe and Ajienka, 1994; Lannom and Hatzignatiou, 1996; Mesallati et al., 2000; Alrumah and Bizanti, 2007; Beiranvand et al., 2012; Azin et al., 2014; Lak et al., 2014). By investigating data points collected from a gas-condensate well in Middle East under sonic conditions, Osman and Dokla (1990) showed that replacing upstream pressure by pressure drop across the choke in Gilbert formula can increase the accuracy of flow rate estimation. They used non-linear regression method in
description of the behavior of their datasets for choke size range of 28/64e72/64 inch and gas flow rate of 3.91e101.33 MMscf/D. There are few published works concerning the sub-critical twophase flow regime through wellhead chokes (Surbey et al., 1989; Elgibaly and Nashawi, 1998; Al-Attar, 2008; Nasriani and Kalantariasl, 2011; Nejatian et al., 2014). Al-Attar (2008) developed a new graphical approach in sub-critical condition for 97 data tests gathered from Middle East wells based on separate choke sizes (24/ 64 inch, 32/64 inch, 40/64 inch, 48/64 inch, 64/64 inch, 96/64 inch, 112/64 inch, and 128/64 inch). With the purpose of developing a general formula including the choke size as a variable, he also substituted pressure drop for upstream pressure in Gilbert formula (similar to Osman and Dokla (1990) work in critical conditions) and used a non-linear regression analysis to derive the constants of this modified Gilbert-type correlation under sub-critical conditions. The results indicated that the former approach is superior to the application of non-linear regression analysis method on the new Gilbert-type model. It should be noted that the maximum gas flow rate of his study was 43.7 MMscf/D. On the other hand, as indicated in a study by Ali Ghalambor (2002) on gas condensate reservoirs of
Fig. 7. Comparison of real and predicted gas flow for implementation of Equation (13) to data points of S ¼ 40/64 inch.
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Fig. 8. Comparison of real and predicted gas flow for implementation of Equation (13) to data points of S ¼ 64/64 inch.
Fig. 9. Comparison of real and predicted gas flow for implementation of Equation (13) to data points of S ¼ 128/64 inch.
Fig. 10. Comparison of real and predicted gas flow for implementation of Equation (13) to data points of S ¼ 144/64 inch.
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Fig. 11. Comparison of real and predicted gas flow for implementation of Equation (13) to data points of S ¼ 160/64 inch.
Louisiana, the sub-critical regimes are mostly occur in larger choke sizes. Taking this into consideration, Nasriani and Kalantariasl (2011) established the first empirical Gilbert-type equation for high flow rates through large choke sizes (with range of 40/64 inch to 192/64 inch) under sub-critical conditions. In their investigation, non-linear regression analysis of Excel solver was utilized for modeling 61 data tests gathered from fifteen wells of 10 different fields located in south-west of Iran. Moreover, they expanded AlAttar (2008) model for high rate gas condensate wells and large choke sizes under sub-critical flow regime. The error study revealed that Al-Attar graphical approach is more precise in estimation of high flow rates passing through the test restrictions in sub-critical conditions based on individual large choke sizes. Recently, Nejatian et al. (2014) implemented Least-Squares Support Vector Machine (LSSVM) algorithm for modeling a subcritical natural gas flow in orifice- and nozzle-type chokes by using 171 and 164 data points for each, respectively. The goodness of fitting data by aforementioned model is represented through average absolute relative deviation and squared correlation coefficient. Utilizing theoretical approaches for modeling the production data requires fluid properties which may not be available in many cases while empirical approaches were approved in practical production purposes by many researchers (Al-Attar, 2008; Nasriani and Kalantariasl, 2011; Beiranvand et al., 2012; Azin et al., 2014; Lak et al., 2014). Most of empirical models established for oil fields and under critical conditions while there are few ones for predicting the flow behavior through chokes of gascondensate wells in sub-critical flow regime. The objective of this investigation is to implement a simple modification to Nasriani and Kalantariasl (2011) method in order to present a more precise empirical model originated from Gilbert model in a subcritical flow regime across chokes of gas-condensate wells. This model is the second empirical Gilbert-type model for gascondensate wells which is applicable in high flow rates and large choke sizes under sub-critical conditions. In order to develop this newly proposed model, non-linear regression approach and Genetic Algorithm are conducted to find the best method for fitting the production data to the proposed model. Using Genetic Algorithm to optimize the empirical constants of a new empirical equation for two-phase flow of wellhead chokes has not been reported in literature. Afterward, the best optimization approach is
Table 2 Error analysis for all data points by implementing non-linear regression technique. Models
The new correlation
Model 1
Model 2
2.044 14.725 18.679 65.814 8.112 0.9204
9.637 34.861 47.894 986.117 31.402 0.6661
16.751 27.216 27.587 283.112 16.825 0.7993
Errors APD AAPD SD MSE RMSE R2
utilized to conduct a comparison between the new correlation and two other models derived in similar conditions, Osman and Dokla (Model 1), and Nasriani and Kalantariasl (Model 2). Both models are using pressure drop instead of upstream pressure to predict
Table 3 Error analysis for all data points when d ¼ 1 in Equation (2). Error type
APD
AAPD
SD
MSE
RMSE
R2
value
27.299
31.049
24.281
269.675
16.422
0.821
Table 4 Error analysis for the data gathered from gas-condensate wells located in Fars province of Iran. Models
The new correlation
Model 1
Model 2
9.345932 11.71358 37.249 61.78235 7.860175 0.8513
13.4963 22.84279 24.81883 137.2267 11.71438 0.7441
29.22089769 29.43477176 46.26681324 162.839 12.76084 0.7894
Errors APD AAPD SD MSE RMSE R2
Table 5 Number of field data in each choke size. Choke size Number of chokes Dataset Percent %
40 8 11.94
64 11 16.4
128 10 14.93
144 8 11.94
160 8 11.94
176 6 8.96
192 16 23.88
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Fig. 12. Comparison of real and predicted gas flow for implementation of Equation (13) to data points of S ¼ 176/64 inch.
the gas flow rate, however, the first one is developed for critical flow regime and the second for sub-critical one. Model 1 and Model 2 are selected to show the failure of existing empirical models (developed for both critical and sub-critical flow regimes) to predict the two-phase flow of south Iranian gas-condensate wells and the need to develop a new one. With the purpose of validating and evaluating the new model in other field data, 39 data points gathered from gas-condensate wells of Fars province of Iran are exposed to the proposed model, Model 1 and Model 2. Moreover, a correlation is derived for dataset of each choke size to minimize the error. Ultimately, statistical error analyses with various parameters are implemented to compare the aforementioned models. It is worth mentioning that in the industry, the total liquid and gas produced from some wells are usually directed to a separator unit and only one gaseliquid ratio is reported for all the wells. This value is not for a particular well. Therefore, finding field data for each gas-condensate well to develop the proposed model was very difficult.
2. Methodology Optimization is one of the most important areas of modern applied mathematics (Khishvand and Khamehchi, 2012). Nonlinear regression analysis and Genetic Algorithm are the two optimization methods used in this investigation. Non-linear regression is based on succeeding approximations which is able to fit the observational data with a function that nonlinearly depends on one or more model variables. This approach usually arises when there are physical evidences for believing that the data and the variables follows a particular functional form (Smyth, 2002). Non-linear regression method utilizes linear approximation of Taylor Series Expansion (the first two terms) to linearize a nonlinear combination of independent variables of model. In an iterative process, the estimation of parameters is conducted by minimizing the summation of squared errors. The iterations come to an end as the errors of estimates become sufficiently small. Nonlinear optimization approach of Excel Spreadsheet (a user friendly
Fig. 13. Comparison of real and predicted gas flow for implementation of Equation (13) to data points of S ¼ 192/64 inch.
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software) has been utilized to find the optimum empirical constants of the proposed model and establish the minimum error relationship between the dependent and independent variables in this study (Hemmati-Sarapardeh et al., 2013; Naseri et al., 2014). Genetic Algorithm (GA) is a heuristic method to optimize the data based on natural evolution through alteration and/or mutation in properties of the data. Genetic Algorithm have been used in many applications to find a solution in optimization problems, for example in property selection, recognizing patterns and understanding images. The key element of this method is the novel structure of the information processing system which is inspired by Darwin's evolution theory (Khishvand and Naseri, 2012). GA includes a main loop in which different data (individuals) are represented by their own chromosome. In this approach, the iterative procedure of selection typically begins with a population of gathered individuals at random. The population in every iteration is called a generation Each and every chromosome represents a point in the examination space and is a tentative solution to the optimization problem. Based on the fitness, the individuals have a chance to mate with each other and produce their offspring to compete in the next generation. For every generation, the fitness of each data to the goal function is calculated and then the data are ranked based on their fitness to the function. Next, in order to develop a new generation, the mating pool that is the set of chromosomes which are suitable to be utilized as the parents of next generation is built. The more fitted data from the existing population have more representatives in the mating pool. Modification is implemented to genome of each individual either through combination of properties or mutation. Then the candidates of the latest generation are utilized for the next iteration. The algorithm usually stops when either a maximum number of generations have been generated or an adequate point of fitness has been achieved for the population (Whitley, 1994; Mitchell, 1996; Ebrahimi et al., 2013). Predictive models, regarding of their algorithms, suffer from some errors and they are more regional in some cases. One of the errors associated with nonlinear regression analysis of Excel is the use of only first two terms of Taylor Series Expansion and ignoring the other terms. Another drawback of this method emerges when the function is highly nonlinear; in this case, the Spreadsheet optimizer might not derive the optimum solution, even though it
Table 8 Error analysis for data points of choke size S ¼ 128/64 inch.
Table 6 Error analysis for data points of choke size S ¼ 40/64 inch.
Table 10 Error analysis for data points of choke size S ¼ 160/64 inch.
Error type
APD AAPD SD MSE RMSE R2
The new correlation General form
Specific form
15.223 31.659 35.132 39.417 6.278 0.8552
6.274 9.759 26.783 21.815 4.671 0.9674
Model 1
50.188 78.072 72.038 1087.738 32.981 0.7873
Model 2
16.298 37.759 36.834 75.244 8.674 0.817
Table 7 Error analysis for data points of choke size S ¼ 64/64 inch. Error type
APD AAPD SD MSE RMSE R2
The new correlation General form
Specific form
12.769 16.485 9.590 61.378 7.834 0.9251
1.661 6.472 6.783 14.831 3.851 0.9669
Model 1
42.471 58.728 62.371 3807.686 61.706 0.6768
Error type
APD AAPD SD MSE RMSE R2
The new correlation General form
Specific form
0.144 12.127 21.675 34.935 5.911 0.8811
0.749 8.362 11.736 16.101 4.013 0.9451
Model 1
Model 2
5.097 23.133 26.568 138.619 11.774 0.8198
21.955 24.499 21.752 164.507 12.826 0.7608
says that a solution is found and all the conditions and the constraints are satisfied. Furthermore, due to the fact that there are limitations on the size of decision variables that Excel can solve, this method is not suitable for models with too many variables (usually more than 200 variables). On the other hand, one of the significant drawback of GA method is the large overproduction of individuals. GA applies to populations of chromosomes, and large amounts of data are utilized to solve even the easiest problems (Braha, 2001). Based on Adriaans and Zantinge (1996), the end-user of a GA method does not really see how the algorithm is creating and selecting individuals for finding patterns in large amounts of data. The relation between the pressure and the liquid flow rate is not always a straight line but may be concave to the origin (Gilbert, 1954). Al-Towailib and Al-Marhoun (1994) and Alrumah and Bizanti (2007) took this into consideration and used an exponent for pressure to obtain a more precise predictive model for their oil
Table 9 Error analysis for data points of choke size S ¼ 144/64 inch. Error type
APD AAPD SD MSE RMSE R2
Error type
APD AAPD SD MSE RMSE R2
The new correlation General form
Specific form
5.098 14.359 14.194 94.692 9.731 0.9619
2.297 11.088 14.179 27.462 5.240 0.9794
The new correlation General form
Specific form
3.665 14.527 13.750 76.386 8.740 0.8602
1.181 14.185 13.605 44.420 6.665 0.925
Model 1
Model 2
7.6175 29.142 32.644 484.239 22.005 0.9161
23.936 29.543 26.909 181.869 13.486 0.9143
Model 1
Model 2
17.637 22.376 23.174 193.615 13.914 0.8414
28.691 29.236 19.926 220.372 14.845 0.8434
Table 11 Error analysis for data points of choke size S ¼ 176/64 inch. Model 2
7.601 31.546 38.399 636.680 25.232 0.695
Error type
APD AAPD SD MSE RMSE R2
The new correlation General form
Specific form
6.189 7.0629 6.754 85.673 9.256 0.7564
0.860 4.117 4.150 16.604 4.075 0.9277
Model 1
Model 2
9.994 13.069 16.085 367.339 19.166 0.6905
4.355 14.789 13.874 181.797 13.483 0.6739
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Equations (3) and (4), these two formulas have the same forms with different constants due to different production data:
Table 12 Error analysis for data points of choke size S ¼ 192/64 inch. Error type
APD AAPD SD MSE RMSE R2
The new correlation General form
Specific form
0.780 11.230 12.665 74.189 8.613 0.8445
0.602 10.572 11.982 71.148 8.435 0.8326
Model 1
Model 2
Model 1 (Osman and Dokla, 1990) 1.872 21.452 26.871 404.382 20.109 0.7407
15.099 23.152 25.173 338.081 18.387 0.7252
Qg ¼
S1:8587 DP
(3)
302ðLGRÞ0:4038
Model 2 (Nasriani and Kalantariasl, 2011) fields. In this study, an exponent is added to the upstreamdownstream pressure difference (pressure drop) of Model 1 to consider this curvature in the flow passing through the chokes of gas condensate wells. Sixty seven production datasets collected from fifteen gas-condensate wells of more than ten different south Iranian gas fields are fitted to the proposed model through nonlinear regression analysis and Genetic Algorithm methods. Then the best method was implemented to three different models (including Model 1, Model 2 and the newly proposed model) to examine the applicability of the proposed model in comparison to the current ones. Moreover, a correlation was derived for dataset of each choke size to minimize the error. The downstream-upstream pressure ratio for all data tests are less than 0.5 which means that the flowing fluid is in subcritical condition. Table 1 represents the different range of parameters utilized in this investigation. The proposed model is as follows:
Qg ¼
S1:9 DP
Seven statistical error analyses (Equations (5)e(11)) including percent deviation (PD), average percent deviation (APD), absolute average percent deviation (AAPD), standard deviation (SD), mean square error (MSE), root mean square error (RMSE) and R-squared (R2) are implemented to evaluate the accuracy of the aforementioned choke models (Ghaffari et al., 2006; Sin et al., 2006; Nasriani and Kalantariasl, 2011):
PDi ¼
Qg;real Qg;predicted Qg;real Pn i¼1
APD ¼
(2)
(5)
PDi
(6)
n Pn
Qg ¼ aðLGRÞb Sc DP d
(4)
9350ðRGL Þ0:65
jPDi j n
i¼1
AAPD ¼
(7)
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u n . uX SD ¼ t ðPDi APDÞ2 ðn 1Þ a, b, c and d: Empirical constants. DP: Pressure drop (m/Lt2), psia. Qg: Volumetric gas flow rate (L3/t), MMscf/D. LGR: Liquid-Gas ratio (L3/L3), Stb/MMscf. S: Choke size (L), 1/64 inch. Finally, Model 1 and Model 2 in addition to the proposed correlation are utilized to fit the data of each choke size. As shown in
(8)
i¼1
Pn MSE ¼ RMSE ¼
i¼1
Qg;predict Qg;real
2
n pffiffiffiffiffiffiffiffiffiffi MSE
Fig. 14. Comparison of real and predicted gas flow for specific implementation of the main form to data points of S ¼ 40/64 inch.
(9) (10)
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Fig. 15. Comparison of real and predicted gas flow for specific implementation of the main form to data points of S ¼ 64/64 inch.
Pn
i¼1
R2 ¼ 1
0
Pn
i¼1
Qg;predict Qg;real
2 12
(11)
Pn B C Qg;real;i C B BQg;real i¼1n C @ A
where i ¼ 1,2,3,..,n.
3. Result and discussion As shown in Table 1, the two-phase data gathered from south Iranian gas-condensate reservoirs cover high flow rates (from 11.3 to 113 MMscf/D) and a broad range of choke sizes (from 40/64 to 192/64 inch).
There are few simple empirical formulas for predicting the behavior of flow in chokes under sub-critical conditions, and especially for gas-condensate wells. The failure in our efforts to utilize the current equations to fit the data of sub-critical two-phase fluid flow across wellhead chokes led us to correlate a new Gilberttype formula (the results of estimating the field data with Model 1 and Model 2 are shown in Figs. 4e11 and Table 2). As mentioned in previous section, DP was substituted for Pwh in Gilbert formula (Equation (1)) and an exponent was added to DP to develop the proposed model (Equation (2)). Then Genetic Algorithm and nonlinear regression analysis methods were used to obtain the optimum values of the empirical constants for the collected gascondensate field data under sub-critical conditions. 3.1. Genetic Algorithm method The MATLAB Genetic Algorithm (GA) toolbox is utilized in this
Fig. 16. Comparison of real and predicted gas flow for specific implementation of the main form to data points of S ¼ 128/64 inch.
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Fig. 17. Comparison of real and predicted gas flow for specific implementation of the main form to data points of S ¼ 144/64 inch.
study. This toolbox initializes a random sample of individuals (field data) with various variables to be optimized (Ebrahimi et al., 2013). Equation (2) is the objective function in this optimization problem and the variables are Qg, LGR, S, and DP. The range of this variables are listed in Table 1. The optimum solutions (the best empirical constants) achieved for the proposed model are shown in Fig. 1 and a comparison between measured and predicted gas flow rates of all data pointes are presented in Fig. 2. The obtained solutions were 0.0164, 0.3931, 1.2624, and 0.556 for a, b, c, and d, respectively and the new equation turns into the following form:
Qg ¼
0:0164S1:2624 DP 0:5566 ðLGRÞ0:3931
(12)
One can see that the data are comparatively in good agreement with the proposed model through this algorithm (MSE ¼ 58.6106 MMscf/D and R2 ¼ 0.9189).
3.2. Non-linear regression analysis method In this section, with the purpose of finding the optimum or the most effective way of using our limited resources (field data) to obtain the objective of the optimization problem (optimum values for empirical constants of the newly proposed model), the entire sixty seven production datasets are fitted with the proposed model through non-linear regression analysis approach of Excel Spreadsheet and Solver. A mathematical model which conducted in a spreadsheet of Excel is known as a Spreadsheet model. The Solver need to identify the location of the Spreadsheet model (Equation (2)), decision variables (Qg, LGR, S and DP), the essence of the optimization (maximizing or minimizing) and the problem limitations (the range of field data, Table 1). Therefore, the data must be organized in the spreadsheet to represent the model and then the solver was used to find the best solution (Chandrakantha, 2008). The result of choke modeling with non-linear regression
Fig. 18. Comparison of real and predicted gas flow for specific implementation of the main form to data points of S ¼ 160/64 inch.
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Fig. 19. Comparison of real and predicted gas flow for specific implementation of the main form to data points of S ¼ 176/64 inch.
method are shown in Fig. 3. The obtained general formula is as follows:
Qg ¼
0:015S1:27 DP 0:56 ðLGRÞ0:4
(13)
As one can see from Table 2, this method is able to fit the data slightly better than Genetic Algorithm and the obtained constants in both methods are approximately the same. Therefore, non-linear regression method which has the better results is used in the following sections. Fig 4 illustrates a comparison between real and predicted flow rates of all three abovementioned models for all data points. Based on seven statistical measures of accuracy (PD, APD, AAPD, SD, MSE, RMSE, R2) in Table 2, it can be concluded that the gas flow rates simulated using the newly proposed correlation are in the best agreement with those measured data. As mentioned before, Model 1 and Model 2 have the same form. In both formulas, the exponent of DP is considered to be
unity but the first formula is derived under critical conditions and the other under sub-critical ones. The same consideration was implemented to fit all data points (d ¼ 1) and the results are shown in Fig. 5 and Table 3. One can see that this form of correlation (R2 ¼ 0.821) has less accuracy in comparison with the proposed one (R2 ¼ 0.9204) in the investigated fields. Taking the results of Table 2 into account, it can be concluded that even though both models are developed to predict the gas flow rate of gas-condensate wells, the accuracy of Model 2 (R2 ¼ 0.7993) is more than Model 1 (R2 ¼ 0.6662). This can show that the empirical models developed for critical flow regime cannot predict the field data of sub-critical one even with the same form. Consequently, the two-phase flow regime through wellhead chokes of gascondensate wells under sub-critical conditions needs more attention and more investigations. Unfortunately, there is no available data in literature about the fluid passing through wellhead chokes of gas condensate wells to evaluate the new model with other researcher's data. Therefore, in
Fig. 20. Comparison of real and predicted gas flow for specific implementation of the main form to data points of S ¼ 192/64 inch.
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Table 13 Values of empirical constants of Equation (2) when applied to each choke size. Choke size (1/64 inch)
a
b
c
d
40 64 128 144 160 176 192
0.0177 0.0174 0.0173 0.0172 0.0179 0.018 0.0174
0.43 0.26 0.44 0.59 1.07 0.7 0.3
1.5 1.38 1.33 1.24 1.45 1.48 1.28
0.39 0.44 0.49 0.64 0.6 0.4 0.48
order to show that the newly proposed model is able to capture other field data better than available models (Model 1 and Model 2), non-linear regression analysis is implemented to 39 data points of seven gas condensate wells located in Fars province of Iran. Based on the results which are shown in Fig. 6 and Table 4, the new correlation is also able to predict the flow rates passing through the chokes of gas-condensate wells in other fields better than the existing models (R2 ¼ 0.8513, 0.7441, and 0.7894 for the new model, Model 1, and Model 2 respectively). For the future researchers, all the original data based on which the correlation is developed in addition to the data collected from the gascondensate wells of Fars province are tabulated in Appendix as Table A1 and Table A2. Finally, it is highly recommended to verify the precision of the newly proposed model with more field data of sub-critical two-phase flow regime through wellhead chokes of gas-condensate wells.
eration, the following conclusion can be made. 1. The non-linear regression analysis showed more accuracy in fitting the data to the proposed model in comparison to Genetic Algorithm. 2. Seven statistical measures of accuracy (PD, APD, AAPD, SD, MSE, RMSE, and R2) indicate that the presented correlation is the most precise one in predicting the gas flow rate flowing through the wellhead chokes in the investigated fields. 3. The application of the new correlation should be used for the range of field data covered in this investigation (gas flow rate of 11.3e113 MMscf/D and choke size range of 40/64e192/64 inch) 4. The newly proposed model can also be used to predict gas flow rates in conditions where measuring the flow rates of gascondensate wells under sub-critical conditions is not accessible or difficult to perform. 5. In similar condition, the new model can help the production engineer to optimize the production rate by determining the optimum choke size. 6. The proposed model is also able to capture other field data better than the available models. 7. Based on individual choke size analyses, among wide range of choke sizes used in this investigation, the one with the size of 144/64 in has the minimum error. 8. The precision of the newly proposed method has been found to be greatly improved when applied to individual choke sizes. Acknowledgment
3.3. Choke size analysis Three different predictive models, (Model 1, Model 2, and the new correlation) are utilized to determine the best operating wellhead choke size for the data gathered form high rate gascondensate reservoirs of south of Iran. Choke sizes and number of chokes in each choke size are listed in Table 5. Implementation results of the three models (Equations (3), (4) and (13)) for each choke size are shown in Figs. 7e13. As shown in the figures, higher fluid flow rates leads to reduction in accuracy of the models. In AlAttar (2008) model, where the exponent d is considered to be unity, the ability of his formula to predict gas flow rates deteriorates at larger choke sizes, however, according to error analyses (Tables 6e12), the newly proposed model has high accuracy even in large choke sizes. Moreover, the best wellhead choke size in the gathered data is 144/64 inch. Finally, the main form of the proposed model (Equation (2)) was separately conducted for each choke size and the constants were specifically derived for each one. The results are shown in Figs. 14e20 and Table 13. Error analyses indicate that applying the main form (Equation (2)) to each choke size increases the accuracy of fitting in comparison with general form (Equation (13)). In other words, the ability of the newly proposed model to predict gas flow rates of fluids passing through the wellhead chokes of gas-condensate wells under sub-critical conditions is found to be highly enhanced when subjected to individual choke sizes. 4. Conclusion A new correlation was established for sub-critical two-phase flow through wellhead chokes and the validity of proposed formula was tested based on 67 data point gathered from south Iranian high rate gas-condensate reservoirs utilizing Genetic Algorithm and non-linear regression analysis. Taking all the results into consid-
The authors would like to thank Dr. Jamshid Moghaddasi (Petroleum Department of Petroleum University of Technology) for his useful suggestions. Nomenclature a empirical constant AAPD absolute average percent deviation APD average percent deviation b empirical constant c empirical constant d empirical constant LGR liquid-gas ratio, Stb/MMscf MSE mean square error, MMscf/D Pd downstream pressure, psia Pu upstream pressure, psia Pwh wellhead pressure, psia PDi percent deviation of data i Q volumetric liquid flow rate bbl/D Qg volumetric gas flow rate, MMscf/D Qg, predicted predicted values of volumetric gas flow rate, MMscf/D Qq, real real values of volumetric gas flow rate, MMscf/D R2 R-squared error RGL gas-liquid ratio, MScf/Stb RMSE root mean square error, (MMscf/D)2 S choke size, 1/64 inch SD standard deviation, MMscf/D Tf wellhead flowing temperature, F Appendix
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Table A 1 The sub-critical data gathered from South Iranian gas-condensate wells. Number
Pu(psia)
Pd(psia)
Qg(MMscf/D)
S(1/64inch)
LGR(bbl/MMscf)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67
1501.978 1518.076 2422 2103 1827 1406.5 1450 2741 1827.57 2059 1319.5 1334 1305 2016.216 1636.156 1493.5 3901.499 1957.5 1232.5 2102.5 1247 1972 2073.5 1783.5 1160 1605.094 1914 2044.5 1290.5 1131 1145.5 1957.5 1885 1986.5 2451 2393 2393.314 2480.337 2407.818 1624 1348.5 1870.5 1334 1392 1174.5 1841.5 1841.5 3785.475 1682 1624 1798 3350.385 4452.613 1783.5 1740 1595 1595 1493.5 1566 1653 1653 1537 1653 1653 1972 2175 1653
824.83258 1484.13896 1740 1726 1638 1319.5 1334 1798 1653.534 2044.5 1290.5 1290.5 1290.5 1900.45559 1195.5025 1435.5 2944.301 1885 1218 2030 1232.5 1899.5 2001 1566 1131 902.85872 1827 2001 1276 1102 1131 1870.5 1566 1899.5 1769 1682 1726.41031 1755.41786 1697.40276 1551.5 1290.5 1537 1290.5 1348.5 1145.5 1667.5 1667.5 2987.81 1595 1508 1624 2016.109 3045.822 1653 1595 1508 1566 1406.5 1464.5 1479 1479 1435.5 1493.5 1522.5 1696.5 1899.5 1435.5
11.3008 12.7134 13.1 15.1 15.641 19.811715 23.696365 26.84 27.73 28.1637125 28.711095 28.852355 28.852355 31.78320005 32.48949338 33.019525 38.72 38.881815 39.5528 44.14375 44.4969 45.238515 45.521035 45.55635 45.9095 46.5564 47.67525 50.041355 52.2662 52.9725 54.3851 56.398055 56.85715 56.963095 57.1043 57.5674 58.26920009 58.26920009 58.26920009 58.5769905 60.0355 63.567 63.743575 63.743575 64.62645 66.74535 67.063185 77.5 82.531155 83.738928 83.873125 84.4 86.76 87.59709175 88.57002 89.91199 90.2474825 95.27987 96.2863475 101.389365 101.6542275 102.06035 103.8261 104.8414063 109.702516 111.2952225 113.008
40 40 40 40 40 128 128 40 64 160 144 144 144 64 64 128 40 160 192 160 192 144 160 128 192 40 144 160 192 192 192 160 128 160 64 64 64 64 64 176 128 128 192 128 192 128 128 64 192 176 192 64 64 192 176 192 192 192 176 144 192 176 176 192 144 144 160
12.46244655 9.564575959 16.1 16.1 4.81 19.52099674 32.21498797 13.536 5.556 6.144483081 4.452782104 4.809004672 4.630893388 5.428 27.49134896 14.99697013 4.374 7.355996036 2.388472321 6.236566615 1.220062296 7.008679032 6.434982585 14.89366558 2.130210959 1.537100382 5.582007645 5.708110435 1.540662608 1.976322809 2.440124593 7.32749823 13.27107178 5.64505904 8.193 8.193 8.193196393 8.193196393 8.193196393 5.555555556 1.920039643 14.62115532 2.128429846 1.246778989 1.923601869 5.042330454 5.443080844 4.374 5.184819482 5.586592179 8.506594931 6.3 4.374 6.319388362 5.120521308 4.239048563 0.688222002 3.419736656 3.875701543 3.961551182 5.03894634 3.526603426 4.198082968 5.681749965 5.566690075 5.413158148 5.713809996
278
S. Seidi, T. Sayahi / Journal of Natural Gas Science and Engineering 26 (2015) 264e278
Table A 2 The sub-critical data gathered from gas-condensate wells of Fars province of Iran. Number
Pu(psia)
Pd(psia)
Qg(MMscf/D)
S(1/64inch)
LGR(bbl/MMscf)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39
2103.238 2335.299 2074.231 1929.193 2175.757 2161.254 2306.291 2335.299 2277.284 2219.269 2248.276 2233.772 2248.276 2262.78 2233.772 2422.321 2378.81 2364.306 2451.329 2349.803 2335.299 2422.321 2480.337 2451.329 2335.299 2393.314 2364.306 2335.299 2378.81 2277.284 2407.818 2349.803 2306.291 2262.78 2306.291 2248.276 2219.269 2088.735 2074.231
1885.682 1769.652 1827.667 1726.14 1842.17 1842.17 1668.125 1813.163 1784.155 1682.629 1813.163 1682.629 1813.163 1769.652 1682.629 1958.201 1755.148 1639.118 1871.178 1827.667 1784.155 1871.178 1755.148 1784.155 1784.155 1932.704 1900.186 1648.976 1972.704 1929.193 2001.712 1842.17 1900.186 1856.674 1885.682 1871.178 1842.17 1885.682 1827.667
13.15733 17.65733 10.5944 10.21658 17.75733 15.81957 28.25173 26.486 33.31467 27.95173 24.92027 28.25173 28.886 32.31467 28.25173 38.84613 45.90907 49.44053 38.84613 42.3776 43.10907 44.14333 42.92445 43.43704 52.972 54.6212 42.90907 52.44053 51.972 45.90907 49.44053 51.772 49.44053 53.00627 47.20627 48.72508 51.20627 43.6748 47.6748
32 32 32 32 40 40 48 48 48 48 48 48 48 48 48 56 56 56 56 56 56 56 56 56 64 64 64 64 64 64 64 64 64 68 68 68 68 72 72
8.155251818 8.155251818 9.816290462 10.16429765 8.159153296 8.122463891 9.583173087 9.19353791 12.36184011 10.63976638 8.191659193 9.597138551 8.529570772 9.836195229 10.14937554 6.45991707 6.014062572 6.171049925 8.460166818 6.325900765 7.255890711 6.937610995 7.374252268 7.374252268 8.039172638 8.039172638 8.200994417 7.846533459 7.684255496 6.647521148 8.300264718 7.620984015 6.958300819 6.273129561 7.857927247 6.642739111 8.192356652 5.274135026 5.718364109
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