Document Type: Research Paper
Authors
^{1} Instructor, Department of Chemical Engineering, Faculty of Petroleum, Gas and Petrochemical Engineering, Persian Gulf University, Bushehr, Iran
^{2} Associate Professor, Department of Petroleum Engineering, Faculty of Petroleum, Gas and Petrochemical Engineering, Persian Gulf University, Bushehr, Iran
^{3} Associate Professor, Department of Chemical Engineering, Faculty of Petroleum, Gas and Petrochemical Engineering, Persian Gulf University, Bushehr, Iran
^{4} Associate Professor, Department of Mechanical Engineering, Faculty of Engineering, Persian Gulf University, Bushehr, Iran
Abstract
Keywords
Main Subjects
Gascondensate reservoirs are important sources of hydrocarbon considered as an intermediate of oil and gas reservoirs. As the hydrocarbon production starts, these reservoirs behave as a single phase gas reservoir. Initial fluid pressure is above the dew point curve, and as a result of reservoir production and pressure decrease, the condensate will form as a separate phase within the reservoir. Gasliquid ratio in gascondensate reservoirs changes between 3200 to 150000 SCF/STB (Danesh, 1998).
Wellhead choke is a type of valves installed to control the well flow and to protect surface facilities from damage due to pressure variation. Positive and adjustable chokes are the main types used on wellheads. Positive chokes has a fixed cross section, but the cross section of adjustable chokes can be controlled instrumentally. Fluid flow through chokes may be either critical or subcritical.
There are theoretical and empirical methods for choke modeling. In 1949, Tangren et al. (1949) presented the first theoretical investigation on twophase flow across the restrictions like chokes. Ashford and Pierce (Ashford et al., 1975) also presented a theoretical model for calculating pressure drop for multiphase flow through chokes and Sachdeva et al. (1986) extended the investigation of Ashford and Pierce. In 1954, Gilbert (1954) proposed a simple empirical correlation for critical flow through chokes and this correlation became a base for some other researchers such as Ros (1960), Baxendell (1957), Achong (1961) and Pilehvari (1981). AlAttar and AbdulMajeed (AlAttar et al., 1988) compared some choke flow correlations such as Gilbert and Ashford model with production data from 155 well tests. They showed that the Gilbert model had the minimum average error within the studied model for flow calculations through chokes. Osman and Dokla (Osman et al., 1990) analyzed field data to present empirical correlations for chokes in gascondensate wells. Perkins (1993) generated equations from general energy equation that described isentropic flow of multiphase mixtures through chokes. AlTowailib and AlMarhoun (AlTowailib et al., 1994) employed more than 3500 production test data from ten fields in the Middle East to present an empirical correlation for twophase critical flow through chokes. Elgibaly and Nashawi (Elgibaly et al., 1998) proposed empirical correlations for twophase flow through wellhead chokes based on data from the Middle East oil wells. Esmaeilzadeh et al. (Esmaeilzadeh et al., 2006) proposed different empirical forms of choke equation considering various parameters such as pressure drop and upstream temperature and verified them with data from five gascondensate reservoirs in Iran. Lak et al. (2014) made a comparison between some of experimental and theoretical models for well flow splitting calculations in a gascondensate field and showed that the theoretical mechanistic model has more accuracy in flow calculation.
In this paper, the Gilbert equation form is adjusted for a gascondensate reservoir fluid. The experimental data for this study is taken from DST data conducted for several production wells of this reservoir. Also, the effect of wellhead temperature on choke calculation is investigated for Gilbert form of choke equation.
2. The field and drill stem test data
The studied gascondensate field is located at the Persian Gulf offshore. It includes 11 operating platforms in the studied block. Each platform contains several wells. Well production streams pass through a flexible choke valve to reduce pressure, and then they are mixed together at the surface and sent to refinery.
A set of drill stem test (DST) data were collected from different wells of this gascondensate field in the time period from 1992 to 2013. This set of data contains wellhead pressure and temperature, choke diameter, separator pressure and temperature, and flow rates of gas, condensate, and water. A schematic of surface process is shown in Figure 1.
Figure 1
A schematic of surface process.
According to this figure, wellhead flow production enters a test separator after passing across a choke valve with a specific opening diameter. Gas, condensate, and water separation is performed in the test separator. Then, separator condensate enters a stock tank in standard conditions (60 °F and 1 atm), and condensate and gas leave the stock tank. Total gas flow rate (q_{g}) is the sum of separator gas and stock tank gas flow rates. Gasliquid ratio (GLR) is calculated by dividing total gas flow rate by total liquid (water and condensate) flow rate (q_{c}+q_{w}) as shown in Equation 1:
(1) 
A sample of surface DST data is given in Table 1. Totally, 864 DST datasets are collected and studied in this work. Plot of GLR versus wellhead pressure is shown in Figure 2. As seen in this figure, wellhead pressure range is 1580 to 4180 psi, and GLR changes between 13900 and 43300 SCF/STB. In average, water content is 5% of total liquid flow rate. Ranges of operational variables are presented in Table 2.
Figure 2
Values of GLR versus wellhead pressure of DST data.
The data presented in Tables 1 and 2 show that the water flow rate is recorded as zero in 231 data entries or 26.7% of all the data. It is due to the uncertainty of water flowmeter because sometimes the flow transmitter does not operate properly and results in a water flow rate of zero in recording DST data. This makes the mean absolute error of 5% in the measuring of total liquid phase flow rate.
Table 1
A sample of surface DST data.
P_{wh}(psi) 
T_{wh}(°F) 
s 
P_{sep}_{.} (psi) 
T_{sep}_{. }(°F) 
q_{g}(MMSCFD) 
q_{c }(STBD) 
q_{w }(STBD) 
R (SCF/STB) 
3906.0 
111.0 
24 
554.0 
92.0 
11.622 
426.0 
18.0 
26176 
3921.0 
115.0 
24 
551.0 
91.0 
11.507 
427.0 
9.0 
26392 
3923.0 
115.0 
24 
543.0 
31.0 
11.324 
440.0 
2.0 
25620 
3927.0 
118.0 
24 
543.0 
90.0 
11.348 
412.0 
17.0 
26452 
3542.0 
125.0 
32 
532.0 
70.0 
17.848 
754.0 
8.0 
23423 
3544.0 
128.0 
32 
525.0 
73.0 
17.666 
680.0 
12.0 
25529 
3548.0 
128.0 
32 
526.0 
75.0 
17.803 
665.0 
14.0 
26219 
3547.0 
131.0 
32 
521.0 
75.0 
17.761 
547.0 
35.0 
30517 
3001.0 
139.0 
40 
535.0 
80.0 
24.097 
917.0 
5.0 
26136 
3006.0 
141.0 
40 
536.0 
80.0 
24.075 
916.0 
0.0 
26283 
2993.0 
141.0 
40 
543.0 
80.0 
24.265 
914.0 
0.0 
26548 
Table 2
Ranges of operational variables in DST data.
Variable 
Range 
P_{wh} (psi) 
15804180 
T_{wh} (°F) 
95151 
s (^{1}/_{64} in) 
2464 
P_{sep}_{.} (psi) 
2651095 
T_{sep}_{.} (°F) 
31126 
q_{g} (MMSCFD) 
10.747.3 
q_{c} (STBD) 
2812520 
q_{w} (STBD) 
0195 
R (SCF/STB) 
1390043300 
3. Choke modeling
As mentioned before, the wellhead chokes are installed to control the well flow or downstream pressure. If flow velocity through the choke is less than the sound speed, it is called subcritical flow. Otherwise, flow in the choke is critical when flowing velocity is greater than the sound speed. In a critical flow, the flow rate is not dependent on downstream conditions. For a compressible fluid, the flow rate increases when pressure ratio decreases. Once critical pressure ratio has been reached at the sonic velocity, the flow becomes choked and the flow rate remains constant (Holland et al., 1995).
Several empirical models have been developed for critical choke flow modeling. The general form of these models is presented in Equation 2 that is based on Gilbert investigations on chokes (Guo et al., 2007).
(2) 
where, C, m, and n are empirical constants related to fluid properties in this equation. Gilbert calculated the values of 10, 0.546, and 1.89 for C, m, and n respectively on the basis of production data from the ten section field in California (Gilbert, 1954).
Other values for the constants were proposed by other researchers such as Ros (1960), Baxendell (1957), Achong (1961), and Pilehvari (1981). These values are presented in Table 3. The optimized values of the constants are not unique, and there is rather a large variation in the constants, mostly for C, and to a less extent for m and n.
Table 3
Empirical parameters of Equation 2.
Model 
C 
m 
n 
Gilbert (Gilbert, 1954) 
10.00 
0.546 
1.89 
Ros (Ros, 1960) 
17.40 
0.500 
2.00 
Baxendell (Baxendell, 1957) 
9.56 
0.546 
1.93 
Achong (Achong, 1961) 
3.82 
0.650 
1.88 
Pilehvari (Pilehvari, 1981) 
46.67 
0.313 
2.11 
Because of different ranges of operational parameters such as flow rate, pressure, and GLR, the optimized constants of a certain dataset cannot be used to make predictions from another dataset, and this can lead to a considerable error. For example, if the original Gilbert equation (Gilbert, 1954) is used to predict wellhead pressure for sample surface DST data of Table 1, a mean absolute error (ME%) of 28.2% will be resulted, as shown in Table 4. The relative error and mean absolute error percent are calculated from Equations 3 and 4.
(3) 

(4) 
Table 4
Wellhead pressure calculations with original Gilbert (Gilbert, 1954) equation (C=10.00, m=0.546, and n=1.89).
P_{wh,rep}(psi) 
s (^{1}/_{64} in) 
q_{L }(STBD) 
GLR (SCF/STB) 
P_{wh,cal }(psi) 
RE (%) 
3906 
24 
444 
26176 
2825 
27.7 
3921 
24 
436 
26392 
2786 
28.9 
3923 
24 
442 
25620 
2779 
29.2 
3927 
24 
429 
26452 
2745 
30.1 
3542 
32 
762 
23423 
2649 
25.2 
3544 
32 
692 
25529 
2521 
28.9 
3548 
32 
679 
26219 
2510 
29.3 
3547 
32 
582 
30517 
2337 
34.1 
3001 
40 
922 
26136 
2232 
25.6 
3006 
40 
916 
26283 
2224 
26.0 
2993 
40 
914 
26548 
2231 
25.4 
ME (%) 
28.2 
Standard deviation from the mean absolute error is defined by Equation 5. This parameter shows the dispersion of error values around the mean error.
(5) 
Mean absolute error is calculated for the entire DST data with the mentioned models. The calculation results are presented in Table 5.
Table 5
The mean absolute error of models for the entire DST data.
Model 
ME (%) 
Gilbert (Gilbert, 1954) 
27.7 
Ros (Ros, 1960) 
46.3 
Baxendell (Baxendell, 1957) 
40.0 
Achong (Achong, 1961) 
18.5 
Pilehvari (Pilehvari, 1981) 
85.1 
As these calculations show, the error of the studied models is between 18.5 to 85.1%. Respectively, Achong and Pilehvari have the minimum and maximum errors between these models. Also, the parameter C of the choke Equation 2 is minimum and maximum for these two models (respectively 3.82 and 46.67). Generally, it appears that high C values cause a higher error in choke calculations for this gascondensate fluid. In other words, lower values of C should be used for the modeling of wellhead chokes in this gascondensate reservoir.
4. Method of DST data regression
In Gilbert equation and similar equations, parameters such as gasliquid ratio and choke size appear in power form. Therefore, after taking logarithm, equation converts to a linear equation to obtain constants with the minimization of the calculation error.
Taking logarithm from both sides of Equation 2 results in Equation 6:
(6) 
The error of pressure calculation for a given point (e_{i}) and the sum of square of errors (S) are defined as Equations 7 and 8:
(7) 
Then, S is defined as the summation of error squares:
(8) 
Parameter S will be minimized provided that all partial derivatives are equal to zero. In other words:
(9) 
Writing the equations for three derivatives leads to a set of equations with C, m, and n as unknowns. These equations are shown with Equations 10 to 12.
(10) 

(11) 

(12) 
By solving this set of equations, the optimum values for parameters C, m, and n are calculated.
In order to investigate the effect of temperature on choke calculations, Gilbert equation was changed and wellhead temperature was considered in the equation, as follows:
(13) 
For this case, S is defined with Equation 14. 
(14) 
Taking derivatives of Equation 14 with respect to constants n, m, C, and k results in set of Equations 15 to 18:
(15) 

(16) 

(17) 

(18) 
5. Results and discussion
In this study, the calculations are performed for two sets of DST data. The first dataset includes all 864 data points. The second dataset involves those points for which water flow rate is available; this set includes 633 data points. The results are presented in Table 6.
Table 6
Regression results.
Dataset 
N 
C 
m 
N 
ME (%) 
SD 
All Data 
864 
0.25 
0.899 
1.76 
5.1 
5.1 
Data with specified water flow rate 
633 
0.72 
0.811 
1.80 
3.0 
3.0 
Figure 3 compares the calculated versus reported wellhead pressure for all the data. The 45° line is also shown for better comparison. The reported wellhead pressure versus the calculated wellhead pressure for data with a specified water flow rate is shown in Figure 4.
Figure 3
The reported wellhead pressure versus the calculated wellhead pressure for all the data.
Figure 4
The reported wellhead pressure versus the calculated wellhead pressure for the data with a specified water flow rate.
According to the results, data without a water flow rate are scattered at the top of the graph. Uncertain data have a mean absolute error of 8.6%, while the error of the remained data is 3.9%; the mean absolute error is 5.1% for all the data. It is between two mentioned errors. In this case, for 38% of the uncertain data, the mean absolute error is more than 10%, which indicates that these data are the major factor of the error. In other words, uncertain data increase the regression error of all the data.
The results show that the regression for the data containing water flow rates is more accurate with a mean absolute error of 3.0%. In this case, 3% of the total data has more than 10% error. Better regression is due to the elimination of the uncertain data. In other words, 5% error of the liquid phase flow measurement, which includes 26.7% of all the data, cause an increase of 2.1% in the error of calculations. This implies that the elimination of uncertain data causes a considerable reduction of error in calculations, and removing uncertain data in data filtering will result in more accurate calculations. The parameters obtained from this regression can be used for the choke calculation of gascondensate fluid with proper accuracy.
Therefore, although the watercondensate ratio is not high, and in average water forms 5% of the total liquid phase, choke calculation accuracy decreases due to neglecting water flow rate.
The results of the regression of Equation 13 for data with a specified water flow rate are shown in Table 7. Also, the wellhead pressure calculated by Equation 2 versus the wellhead pressure calculated by Equation 13 for this regression is shown in Figure 5.
Table 7
Regression results for Equation 13.
Dataset 
N 
C 
m 
n 
k 
ME (%) 
SD 
Data with specified water flow rate 
633 
0.40 
0.821 
1.82 
0.115 
3.0 
2.9 
Figure 5
The wellhead pressure calculated by Equation 2 versus the wellhead pressure calculated by with Equation 13.
The regression results for Equation 13 show that the mean absolute error of 3.0% does not change, and there is not a considerable difference between the wellhead pressures calculated by Equations 2 and 13. Therefore, considering the wellhead temperature in the Gilbert equation form has no tangible effect on choke calculation accuracy.
6. Conclusions
In this work, critical flow calculations through chokes were studied based on DST data from a gascondensate reservoir. The water flow rate of 26.7% of these data was zero because of uncertainty in water flowmeter. Therefore, DST data was divided into two sets; the first dataset contained all the data, and, in the second, the uncertain data was eliminated. The base form for choke calculation was Gilbert model. Using DST data, the equation parameters was modified.
The results showed that uncertainty in water flow rate decreases the accuracy of regression. Uncertain data had a mean absolute error of 8.6%. The mean absolute error was 5.1% for all the data and 3.0% for data with a specified water flow rate. This shows that the elimination of uncertain data causes a remarkable reduction in error.
For studying the effect of wellhead temperature on calculations, temperature was considered in the calculation equation, and specified data and parameters were modified again. The results showed that the mean absolute error is 3.0%, and there is no significant difference between the wellhead pressure values calculated by the new equation and those calculated by Gilbert form equation. Therefore, wellhead temperature has no considerable effect on choke calculation.
Nomenclature
C 
: Choke equation constant 
e 
: Pressure calculation error [psi] 
GLR 
: Gasliquid ratio [] 
k 
: Choke equation constant [] 
m 
: Choke equation constant [] 
ME (%) 
: Mean absolute error percent [] 
n 
: Choke equation constant [] 
N 
: Number of data [] 
P_{sep.} 
: Separator pressure [psi] 
P_{wh} 
: Wellhead pressure [psi] 
P_{wh,cal} 
: Calculated wellhead pressure [psi] 
P_{wh,rep} 
: Reported wellhead pressure [psi] 
q_{c} 
: Condensate flow rate [STBD] 
q_{g} 
: Total gas flow rate [MMSCFD] 
q_{gs} 
: Stock tank gas flow rate [MMSCFD] 
q_{L} 
: Liquid flow rate; summation of water and condensate flow rates [STBD] 
q_{w} 
: Water flow rate [STBD] 
R 
: Gasliquid ratio [SCF/STB] 
RE% 
: Relative error percent [] 
s 
: Choke diameter [^{1}/_{64} in] 
S 
: Summation of error squares [] 
SD 
: Standard Deviation [] 
T_{sep.} 
: Separator temperature [°F] 
T_{wh} 
: Wellhead temperature [°F] 