OPEN ACCESS
This paper aims to identify the model that can accurately predict the liquid holdup under specific conditions. For this purpose, 431 sets of test data were obtained from a 60mmdiameter pipe and a 75mmdiameter pipe, and six existing models were evaluated against these data with the pipes in the horizontal direction or at small inclined angles. Three statistical parameters were introduced to select the bestperforming model. The test data were also adopted to explore the effects of pipe diameter, gasliquid ratio, liquid types and pipe inclination on liquid holdup. It is concluded that he modifiedHughmark correlation boasted the best accuracy for airwater mixture in the horizontal direction and at small inclination angles, while the BeggsBrill model outperformed the other models for white oilair flow; the increase in liquid holdup is proportional to pipe diameter at the same gasliquid ratio; as long as the gasliquid ratio is lower than 100, the inclined angle in the range of 0~30° had little effect on liquid holdup, and the effect gradually decreased with the increase in the gasliquid ratio; the liquid holdup is positively correlated with the viscosity and the content of heavy components, and negatively with density; the gasliquid ratio had a great impact on liquid holdup. The research findings provide a valuable reference for studies in similar fields.
liquid holdup, liquidgas twophase flow, horizontal and inclined pipe, gasliquid ratio, pipe diameter, liquid type, pipe inclination
Multiphase flow, a commonplace phenomenon in pipes, has attracted much attention from the petroleum industry thanks to its immense economic benefit. The existing research has been concentrated on the effect of the actual working conditions on such parameters as flow pattern, pressure gradient and liquid holdup. The pressure gradient prediction for multiphase flow Among them, the liquid holdup is essential to the design of multiphase flow pipe, due to its close correlation with the pressure gradient of the pipe, which is an important theoretical foundation for the design and analysis of oil and gas wells [12] .
This is particularly true to gasliquid twophase flow pipe. For this type of pipe, liquid holdup reflects the amount of effusion of the pipe, an existential threat to transmission reliability. The effusion not only drags down the gas transmission efficiency, but also pushes up the risk of pipe corrosion [3]. Therefore, it is of practical significance to explore the influencing factors of liquid holdup.
In most oil and gas gathering stations and the plain regions, most pipes are laid in the horizontal or nearhorizontal direction [4]. In this case, the liquid holdup varies with the gasliquid ratio, pipe diameter, liquid type and pipe inclination angle [5].
Since Lockhart and Martinelli (1949) established the first empirical liquid holdup equation [6], many scholars have proposed classical correlations for liquid holdup. For example, Hughmark (1962) tested the mixture of different liquid mediums and the air in vertical pipes (ID: 16mm~63.5mm), calculated liquid holdup in light of the test results, and applied the calculation method to horizontal pipes [7]. Hughmark’s method was modified by Garcia in 2005 [8].
In 1967, Guzhov et al. developed the correlation for liquid holdup based on the data for liquidgas mixture in pipes with an inclination angle between 9° to 9° [9]. Six year later, Beggs and Brill studied twophase flow in pipes tilted at 90° ~ +90°, presented the correlation between liquid holdup and pressure drop, and derived the correlation for different pipe inclination angles based on that for horizontal pipes [10].
In the past decades, many other scholars, such as Eaton et al. [11], AbdulMajeed [12], MinamiBrill [13], Ansari [14] and Xiao [15], have developed liquid holdup correlations for implementation in the petroleum industry. Despite these achievements, these liquid holdup correlations fail to realize desirable accuracy across different experimental or field conditions [16][17]. This conclusion is drawn through numerous evaluations.
In 1964, Dukler evaluated the correlations of Hoogeendoom, Hughmark and LockhartMartinelli with the AGA/API database, and discovered that even the bestperforming Hughmark’s correlation cannot achieve the desired accuracy [16].
In 1975, Vohra tested the correlations of BeggsBrill, Dukler, Eaton et al., Guzhov, Hughmark and LockhartMartinelli. The test data include 58 groups from Beggs’ research on a horizontal pipe and 238 groups from that of Eaton et al. The results show that the correlation of Eaton et al. enjoyed the highest accuracy with an average percent error (APE) of 5.9%, followed by the BeggsBrill model (18.9%) [17]. This is because most of the data came from Eaton et al.
In the same year, Mandhane et al. assessed the liquid holdup correlation presented by BeggsBrill, Eaton et al. and Hughmark based on 2,685 measured values of liquid holdup in horizontal gasliquid twophase flow, and recommended a calculation method in view of the flow pattern [18] (Table 1).
Table 1. Liquid holdup calculation method recommended by Mandhane et al.
Flow Pattern 
Recommended Method 
The MeanPercentage Absolute Error 
The MeanPercentage Error 
Bubble Flow, Elongated Bubble Flow 
Hughmark 
7.2% 
1.8% 
Stratified Flow 
Agrawal et al. 
34.8% 
26.8% 
Wavy Flow 
Jorah 
45.8% 
30.2% 
Slug Flow 
Hughmark 
62.2% 
0.2% 
Annular Flow, Mist Flow 
LockhartMartinelli 
6.0% 
0.4% 
Dispersed Bubble 
BeggsBrill 
29.2% 
5.7% 
In 1993, AbdulMajeed made some improvements to the BeggsBrill correlation, and compared the improved method with 11 related correlations (e.g. Eaton et al, MinamiBrill Ⅰ and Ⅱ) under horizontal, inclined and vertical conditions. The scholar proved that the modified method had the smallest APE of 6.8% [19].
Based on the data from various horizontal pipe experiments, Garcia et al. (2005) created a theoretical model for liquid holdup prediction in a horizontal pipe, and verified that the model is more accurate than 25 existing methods [20]
In 2008, Cheng et al. compared the Hughmark correlation and Garcia’s modifiedHughmark correlation based on the test data from the National Engineering Laboratory for Pipeline Safety in China University of Petroleum (Beijing). They recommended to forecast the liquid holdup in horizontal pipes with Garcia’s modified model [21].
The experiment was carried out in the Laboratiry of Multiphase Pipe Flow, Gas Lift Innovation Center, China National Petroleum Corp. The laboratory supports the dynamic analysis of singlephase flow, gasliquid twophase flow and oilgaswater threephase flow under different inclination angles (from horizontal to vertical), diameters and temperatures [22].
Figure 1. The test platform
As shown in Figure 1, the test platform consists of nine parts, including but not limited to a wellbore, an oilwater steady flow system, a gas steady flow system and a cooling water system. The platform has a gas flow metering module (accuracy: ±1%), a liquid flow metering module (accuracy: ±0.3%) and some piston metering devices. The test pipes are 40mm, 60mm and 75mm in diameter. During the test, the flow pattern was observed and recorded through a 7mlong transparent heatresistant and highvoltageresistance pipe section. The pipe section is installed with highprecision sensors of flow, moisture content, pressure, pressure difference and temperature, in addition to a highspeed camera system. The central control system monitors the temperature, liquid level and stirring device in the mixing tank, as well as the pressure, temperature, pressure gradient and velocity of the liquid and gas in the test section. It also controls the closing valves in the test section[23].
According to the test requirements, white oil or water was selected as the liquid phase input of the oilwater mixing tank. The input was pressurized by the liquid pump, stabilized by the regulator and measured. Then, the liquid was mixed with the air from the compressor unit before entering the pipe section. Finally, the gas was separated from the mixture in the gasliquid separator, and the liquid returned to the mixing tank to complete a cycle.
The liquid inflow was adjusted by the power of the liquid pump and the opening of the regulation valve. The air inflow was adjusted in the same manner. The average liquid holdup was measured in an 8.65mlong QCV pipe section.
The test medium and flow conditions are listed in Table 20.
Table 2. Test medium and flow conditions
Diameter(mm) 
Angle (°) 
Angle (°) 
liquid viscosity (cp) 
Liquid volume flow (m³/h) 
Air volume flow (m³/h) 
Temperature (℃) 
60 
0,15,30 
0,15,30 
1.2 
6.25~20 
200~2000 
8~14 
0 
10~11 
0.62~2.1 
30~600 
21~24 

75 
0 
0,15, 30 
1.2 
6.25~20 
200~2000 
7~9.5 
0,15, 30 
Angle (°) 
10~11 
0.62~2.1 
30~600 
28~29 
Overall, the test was designed such that the effect of pipe diameter, liquid type and inclination angle can be easily investigated. The liquid holdups under different conditions were obtained for further analysis.
4.1 Evaluation of liquid holdup correlations based on airwater mixture
The six most popular models, including BeggsBrill, MukherjeeBrill, Eaton et al., modifiedHughmark, MinamiBrill I and MinamiBrill II, were evaluated against 261 sets of test data on airwater mixture at the inclination angles of 0°, 15° and 30°.
To predict the liquid holdup at the varied inclinations, the BeggsBrill inclination modifications were applied to the Eaton et al. and the modifiedHughmark correlations.
The accuracy of each method was tested by three statistical parameters: the average relative error ε_{1}, the absolute relative average error ε_{2} and the standard deviation of relative error ε_{3}:
$e_{i}=\frac{\left(H_{L}\right)_{i p r e}\left(H_{L}\right)_{i e x p}}{\left(H_{L}\right)_{i e x p}} \times 100 \%$ (1)
$\varepsilon_{1}=\frac{1}{N} \sum_{i=1}^{N}\left(e_{i}\right)$ (2)
$\varepsilon_{2}=\frac{1}{N} \sum_{i=1}^{N}\lefte_{i}\right$ (3)
$\varepsilon_{3}=\sqrt{\frac{\sum_{i=1}^{N}\left(e_{i}\varepsilon_{1}\right)^{2}}{N1}}$ (4)
The average relative error ε_{1 }reflects the difference between the predicted value and the measured value. A positive value indicates overprediction and the inverse is also true. However, the true average error might be concealed as the individual errors could offset each other in the summation process. That is why the absolute average relative error ε_{2} was introduced. The standard deviation of relative error ε_{3} reveals the degree of dispersion between the predicted value and the measured value. The verification results are shown in Table 3.
Figure 2. Accuracy of the existing models at liquid flow rates of 6.25 m³/h and 8.33 m³/h
From Table 3, it can be seen that the modifiedHughmark model outperformed the other models with the lowest absolute average relative errors at 0°, 15° and 30° for the 60mmdiameter pipe, while MinamiBrill I correlation boasted the best accuracy for the 75mmdiameter pipe. Except for these two models, the other four models underwent significant increases in liquid holdup prediction errors with the expansion of the pipe diameter. When the liquid flow rate grew from 6.25m^{3}/h to 8.33m^{3}/h, MukherjeeBrill, Minami I, Minami II and modifiedHughmark correlations all enjoyed better accuracy under horizontal and inclined conditions (Figure 2). Therefore, the accuracy of liquid holdup correlation is yet to be improved at low liquid flow rates. It is also observed that the accuracy of BeggsBrill model and Eaton et al. model degenerated with the increase in liquid flow rate. The error difference was more pronounced in Eaton et al. model, indicating the need for improvement at high liquid flow rates.
Table 3. Verification results of the six models based on airwater mixture
Models 
Error types 
Errors (%) 

Pipe Diameter ID=60mm 
Pipe Diameter ID=75mm 

0° 
15° 
30° 
0° 

BeggsBrill 
ε_{1} 
 14.443 
 23.551 
 19.433 
 53.434 
ε_{2} 
28.830 
30.702 
29.702 
57.257 

ε_{3} 
4.065 
3.166 
3.706 
0.125 

MukherjeeBrill 
ε_{1} 
 44.414 
 33.019 
 29.619 
 49.366 
ε_{2} 
44.414 
33.019 
29.670 
49.366 

ε_{3} 
1.447 
1.520 
1.770 
1.625 

Eaton et al 
ε_{1} 
 60.558 
 23.396 
 28.351 
 64.701 
ε_{2} 
60.558 
40.572 
45.417 
64.701 

ε_{3} 
0.932 
6.601 
8.315 
0.793 

Modified  Hughmark 
ε_{1} 
5.424 
 10.382 
 20.021 
1.208 
ε_{2} 
20.065 
20.604 
25.275 
22.386 

ε_{3} 
3.128 
2.351 
1.521 
3.492 

MinamiBrill Ⅰ 
ε_{1} 
 20.511 
 19.554 
 21.844 
 27.692 
ε_{2} 
24.400 
26.144 
27.018 
20.821 

ε_{3} 
2.103 
2.771 
2.36 
1.710 

MinamiBrill Ⅱ 
ε_{1} 
17.511 
 16.628 
19.454 
 29.060 
ε_{2} 
24.098 
25.761 
26.522 
29.578 

ε_{3} 
2.623 
3.314 
2.978 
1.929 
4.2 Evaluation of liquid holdup correlations based on airoil mixture
The six models were also tested with 170 sets of data on airwhite oil twophase flow. According to the absolute average relative errors, all models other than the Beggs and Brill model exceeded 100% in terms of the error (Figure 3).
As shown in Figure 4, the absolute average relative errors of the modifiedHughmark model rocketed up with the increase in liquid flow rate, while the accuracy of BeggsBrill correlation was enhanced when the liquid flow rate climbed up from 15m³/h to 30m³/h. Comparing Figure 3 and Figure 4, it is clear that the BeggsBrill model outshined the other correlations for the white oilair two phase flow. The statistical errors of liquid holdup by this method are shown in Table 4. It can be seen that the accuracy of BeggsBrill correlation under the horizontal condition differed greatly from that under the inclined condition. Meanwhile, when the predicted liquid holdups for waterair flow (Table 3) were contrasted with those of white oilair flow (Table 4), the model was subject to less error for the inclined pipe. Furthermore, the model was less accurate for the 75mmdiameter pipe than the 60mmdiameter pipe at all inclination angles.
To sum up, the modifiedHughmark model performs well in predicting liquid holdup of waterair twophase flow in the horizontal direction and at small inclination angles (0~30°), while the BeggsBrill correlation is applicable to the cases that the measurement accuracy is inadequate for the physical properties of the liquid.
Figure 3. Absolute average relative errors of the existing models
Figure 4. Absolute average relative errors of the existing models at different liquid flow rates
Table 4. Statistical errors of liquid holdup by BeggsBrill method

Error types 
Errors 

Pipe Diameter ID=60mm 
Pipe Diameter ID=75mm 

0° 
15° 
30° 
0° 
15° 
30° 

BeggsBrill 
ε_{1} 
38.741 
28.134 
8.7179 
63.972 
38.711 
27.999 
ε_{2} 
53.390 
36.657 
28.8495 
62.016 
37.021 
41.982 

ε_{3} 
9.446 
4.5499 
5.52790 
1.525 
2.607 
7.7427 
4.3 Effect of pipe diameter
Figure 5. Measured liquid holdup at different pipe diameters and gasliquid ratios
Figure 6. Measured liquid holdup at different pipe diameters and gasliquid ratios at the inclination angle of 15°
Figure 7. Measured liquid holdup at different pipe diameters and gasliquid ratios at the inclination angle of 30°
In this section, three different inner diameters (40mm, 60mm and 75mm) of the pipe were selected to investigate the effect of pipe diameter on liquid holdup. As shown in Figure 5, the liquid holdup is positively correlated with the pipe diameter at the same gasliquid ratio. This means the liquid flow area increases with the diameter, and occupies a greater portion in the pipe section. In addition, with the increase of gasliquid ratio, the influence of pipe diameter on liquid holdup was rather small at the same liquid flow rate. As the gas picked up speed, the liquid carrying capacity increased, and the liquid holdup at each pipe diameter gradually moved to zero.
Similar patterns were observed in the tests at the inclination angles of 15° and 30° (Figures 6 and 7). Therefore, a smaller pipe diameter or higher gasliquid ratio is recommended to reduce liquid holdup in gas transmission pipe and to balance the throughput and flow rate.
4.4 Effect of liquid medium
Before the test on the effect of liquid medium, a theoretical analysis was conducted on the assumption that the physical properties of the liquid medium vary independently. The analysis was made in reference to the modifiedHughmark correlation. According to the correlation, Equations (5)~(10) are true:
$H_{L}=1K \frac{U_{s g}}{U_{s g}+U_{s l}}$ (5)
$K=0.17460.1301(\ln Z)+0.7508(\ln Z)^{2}0.4308(\ln Z)^{3}$$+0.09553(\ln Z)^{4}0.007452(\ln Z)^{5}$ (6)
$Z=\frac{R e^{1 / 6} F r^{1 / 8}}{\lambda_{L}^{1 / 4}}$ (7)
$F r=\frac{\left(U_{s g}+U_{s l}\right)^{2}}{g D}$ (8)
$R e=\frac{\left(\rho_{g} U_{s g}+\rho_{l} U_{s l}\right) D}{\lambda_{L} \mu_{L}+\left(1\lambda_{L}\right) \mu_{g}}$ (9)
$\lambda_{L}=\frac{U_{s g}}{U_{s g}+U_{s l}}$ (10)
Equation (6) increases monotonically when the value of ln(Z) is greater than 1. According to the test data, the values of ln(Z) in this research always exceed 1. Following this equation, any increase in viscosity will lead to the decline in Reynolds number, provided that all the other parameters are constant. In this scenario, the Z in Equation (7) and K in Equation (6) will also decrease. As can be seen from Equation (5), the liquid holdup will grow in magnitude, revealing that viscosity is proportional to liquid holdup. Diffusion is retarded by an increase in viscosity ratio at a fixed fluidity for the dispersed phase[24]. It is also deduced that density is negatively correlated with liquid holdup. Moreover, the liquid holdup in the pipe will increase with the proportion of heavy components [25]. Under the same gasliquid volume flow, it is expected that liquid holdup will be relatively high if the liquid medium has a small density, high viscosity and high content of heavy components. According to the test results in Figure 8, it is clear that the white oilair twophase flow had a higher liquid holdup than the waterair mixture under the same gasliquid volume flow. Since the former has a smaller density, higher viscosity and higher content of heavy components than the latter, the test results are consistent with the predicted results.
Then, the relationship between the gasliquid ratio and the liquid holdup at the inclined angle of 15° was compared with that at 30°. The comparison shows that the liquid holdups varied in a similar pattern with those for the horizontal pipe at the same liquid flow conditions (Figure 9 and Figure 10).
Figure 8. Measured liquid holdups at different gasliquid ratios for white oilair twophase flow and waterair mixture in the horizontal pipe
Figure 9. Measured liquid holdups at different gasliquid ratios for white oilair twophase flow and waterair mixture in a pipe with the inclined angle of 15°
Figure 10. Measured liquid holdups at different gasliquid ratios for white oilair twophase flow and waterair mixture in a pipe with the inclined angle of 30°
4.5 Effect of pipe inclination
According to the theoretical liquid holdup calculated by Beggs and Brill, the correlations for horizontal pipe cannot be directly applied to the inclined pipe. This implies the importance of inclination on liquid holdup.
Figure 11. Measured liquid holdups at different inclined angles and gasliquid ratios under the liquid flow of 10.42 m³/h
Figure 12. Measured liquid holdup at different inclined angles and gasliquid ratios under the liquid flow of 12.5 m³/h
As shown in Figure 11, under the liquid flow rate of 10.42m³/h and the gasliquid ratio of 40, the liquid hold decreased as the inclined angle shifted from 0° to 15°, and increased as the angle expanded from 15° to 30°. By contrast, when the liquid flow rate increased to 12.5 m³/h in Figure 12, the liquid holdup rose slightly as the inclined angle increased from 0° to 30°. To sum up, the liquid holdup varied insignificantly when the inclined angle fell between 0° and 30°.
Figure 13. Measured liquid holdups at different gasliquid ratios and inclined angles at the liquid flow of 10.42 m³/h
Figure 14. Measured liquid holdups at different gasliquid ratios and inclined angles at the liquid flow of 12.5 m³/h
As can be seen in Figures 13 and 14, with the increase in the liquid flow rate, the liquid holdup curves of three inclined angles overlapped each other. This phenomenon reveals that the liquid holdup is not heavily affected by small variation in the inclined angle under a low gasliquid ratio (<100), and the influence will further decrease with the growth of the gasliquid ratio.
4.6 Effect of gasliquid ratio
It can be seen from Figure 13 and 14 that the liquid holdup decreased with increase in gasliquid ratio. The rate of decrease was rapid at low gasliquid ratios, and slows down as the ratio increased. Eventually, the liquid holdup stabilized at a certain value and grew with the increase in the liquid flow. Hence, the gasliquid ratio has a great impact on liquid holdup in a certain range.
In this test, it is observed that when the gasliquid ratio was less than 200, the liquid holdup decreased at a rather fast pace; when the ratio fell between 200 and 300, the decrease rate gradually slowed down; when the ratio was greater than 300, the liquid holdup curves became asymptotic.
Through the simulation of the actual working conditions (gas volume: 4×10^{5 }m^{3}/d; liquid volume: 8~200 m^{3}/d), the author discovered that the effect of the gasliquid ratio on liquid holdup was minimal when the former exceeded 20,000 [26]. The simulation results under other working conditions show that the effect was no longer obvious after the gasliquid ratio surpassed 1,000 [27]. Therefore, the range of gasliquid ratio has a significant impact on the liquid holdup, and the exact impact depends on the specific conditions.
This paper experimentally explores the liquid holdup of liquidgas twophase flow in horizontal pipes with small inclined angles. The following results can be highlighted:
Six existing liquid holdup models were evaluated against the test data from the Laboratory of Multiphase Pipe Flow, Gas Lift Innovation Center, China National Petroleum Corp. The evaluation results show that the modifiedHughmark correlation boasted the best accuracy for airwater mixture in the horizontal direction and at small inclination angles, while the BeggsBrill model outperformed the other models for white oilair flow.
The increase in liquid holdup is proportional to pipe diameter at the same gasliquid ratio. To reduce the pipe effusion, the liquid holdup should be suppressed by reducing the pipe diameter if conditions permit.
As long as the gasliquid ratio is lower than 100, the inclined angle in the range of 0~30° had little effect on liquid holdup, and the effect gradually decreased with the increase in the gasliquid ratio.
The liquid holdup is influenced by such physical properties as viscosity, density and content of heavy components. Specifically, the liquid holdup is positively correlated with the viscosity and the content of heavy components, and negatively with density. This is proved by the test results that the white oilair flow had a greater liquid holdup than the waterair mixture at the same gas and liquid flow rates.
It is also observed that the gasliquid ratio had a great impact on liquid holdup. When the ratio fell between 0 and 300, the liquid holdup declined rapidly; when the ratio exceeded 300, the impact was weakened and the liquid holdup curves became asymptotic.
Thanks for Luo Wei of the corresponding author for the article. This work is supported by the National Natural Science Found Project (NO. 61572084) and National Key Scientific and Technological Project (2016ZX05056004002, 2016ZX05046004003)
e 
the absolute error, dimensionless 
H_{L} 
liquid holdup, dimensionless 
N

the number of the whole group under horizontal or inclinations respectively 
U_{s} 
superficial velocity, m.s^{1} 
Greek Symbols 

ε_{1} 
the average relative error 
ε_{2} 
the absolute relative average error 
ε_{3} 
the standard deviation of relative error 
$\lambda$ 
liquid holdup of noslippage 
$\rho$ 
density, kg.m^{31} 
$\mu$ 
dynamic viscosity, kg. m^{1}.s^{1} 
Subscripts 

i 
No. i set data,dimensionless 
ipre 
the predict values 
iexp 
the experimental values 
g 
gas 
[1] Osman E.S.A. (2001). Flow regimes and liquid holdup in horizontal multiphase flow, Journal of Petroleum Technology, Vol. 53, No. 10, pp. 4242. DOI: 10.2118/10010042JPT
[2] Lei Y., Liao R.Q., Li M.X., Li Y., Luo W. (2017). Modified Mukherjeebrill prediction model of pressure gradient for multiphase flow in wells, International Journal of Heat and Technology, Vol. 35, No. 1, pp. 103108. DOI: 10.18280/ijht.350114
[3] Qin J. (2011). Detection of gas pipeline effusion and analysis of safe operation. M.S., Dissertation, College of Mechanical and Electronic Engineering, China University of Petroleum (East China), Qingdao, China.
[4] Chen J.L., (2010). The gasliquid twophase flow in horizontal pipes, Gasliquid twophase pipe flow in Petroleum Industry, Chen J.L., Chen T.P. (Eds.), Petroleum Industry Press, Beijing, pp. 170170.
[5] Sarica C., Pereyra E.J., Brito R. (2013). Effect of medium oil viscosity on two phase oil gas flow behavior in horizontal pipes, Offshore Technology Conference, Huston, pp. S1S18
[6] Lockhart R.W. (1949). Proposed correlation of data for isothermal twophase, twocomponent flow in pipes, Chem.eng.prog , Vol. 45, pp. 3948.
[7] Hughmark G.A. (1962). Holdup in gas–liquid flow, Chem. Eng. Prog, Vol. 58, pp. 62–65.
[8] García F., García R., Joseph D.D. (2005). Composite power law holdup correlations in horizontal pipes, International Journal of Multiphase Flow, Vol. 31, No. 12, pp. 1276~1303. DOI: 10.1016/j.ijmultiphaseflow.2005.07.007.
[9] Guzhov A.I., Mamaev V.A., Odisharii︠a︡ G.E. (1967). Some study of transportation in gasliquid systems: Une étude sur le transport des systèmes gazliquides, International Gas Union.
[10] Beggs D.H., Brill J.P. ( 1973). An experimental study of twophase flow in inclined pipes, Journal of Petroleum Technology, Vol. 25, No. 5, pp. 607617. DOI: 10.2118/4007PA
[11] Eaton A.B., (1967). The prediction of flow patterns, liquid holdup and pressure losses occurring during continuous twophase flow in horizontal pipelines, Journal of Petroleum Technology, Vol. 19, No. 6, pp. 815828.
[12] AbdulMajeed G.H. (1996). Liquid holdup in horizontal twophase gasliquid flow, Journal of Petroleum Science & Engineering, Vol. 15, No. 24, pp. 271280. DOI: 10.1016/09204105(95)000690
[13] Minami K., Brill J.P. (1987). Liquid holdup in wetgas pipelines, Spe Production Engineering, Vol. 2, No. 1, pp. 3644. DOI: 10.2118/14535PA
[14] Ansari A.M., Sylvester N.D., Shoham O. (1990). A comprehensive mechanistic model for upward twophase flow in wellbores, The 65th SPE Annual Technical Conference and Exhibition, New orleans, pp. S1S8.
[15] Xiao J., Shoham O., Brill J. (1990). A comprehensive mechanistic model for twophase flow in pipelines, The 65th SPE Annual Technical Conference and Exhibition, New Orleans, LA, Paper SPE 20631, pp. 167–180.
[16] Dukler A.E., Iii M.W., Cleveland R.G. (1964). Frictional pressure drop in twophase flow: A. A comparison of existing correlations for pressure loss and holdup, Aiche Journal, Vol. 10, No. 1, pp. 3843. DOI: 10.1002/aic.690100117
[17] Vohra I.R., Hernandez F., Marcano N. (1975). Comparison of liquidholdup and frictionfactor correlations for gasliquid flow, Journal of Petroleum Technology, Vol. 27, No. 5, pp. 564568. DOI: 10.2118/4690PA
[18] Mandhane J.M., Gregory G.A., Aziz K. (1975). Critical evaluation of holdup prediction methods for gasliquid flow in horizontal pipes, Journal of Petroleum Technology, Vol. 29, No. 10, pp. 13481358. DOI: 10.2118/5140PA
[19] AbdulMajeed G.H. (1993). Liquid holdup correlation for horizontal, vertical and inclined twophase flow, Society of Petroleum Engineers.
[20] García F., García R., Joseph D.D. (2005). Composite power law holdup correlations in horizontal pipes, International Journal of Multiphase Flow, Vol. 31, No. 12, pp. 12761303. DOI: 10.1016/j.ijmultiphaseflow.2005.07.007
[21] Cheng X.J., Yu D., Gong J. (2008). Research on calculation method of liquid holdup of ga liquid two  phase flow in horizontal, OilGasfield Surface Engineering, Vol. 11, pp. 1415.
[22] Fu T.T., Liu J., Liao R.G. (2017). Water holdup in noslip oilwater twophase stratified flow, International Journal of Heat and Technology, Vol. 35, No. 2, pp. 306312. DOI: 10.18280/ijht.350211.
[23] Liao R.Q., Li Y., Song J.P. (2014). The eestablishment of mutiphase flow exiperiment platform in gas lift test base, Journal of Oil and Gas Technology, Vol. 36, No. 9, pp. 129131.
[24] Lamorgese A., Mauri R. (2017). Effect of viscosity ratio on structure evolution during mixing/demixing of regular binary mixtures, Chemical Engineering Transactions, Vol 57, pp. 12251230 DOI: 10.3303/CET1757205
[25] Li Y.X., Feng S.C. (1998). Study on distribution law of fluid composition in wet natural gas transmission pipeline, Oil & Gas Storage and Transportation, Vol. 17, No. 2, pp. 15.
[26] Yao L.Y., Lei W., Chen H.L. (2013). Analysis of the influence of liquid holdup of the gasliquid mixing pipeline under the undulating terrain, China Oil and Gas Field Ground Engineering Technology Conference Proceedings, pp. 15281532.
[27] Zhuang Z.H. (2016). Study on liquid holdup prediction and the law of pigging of undulating natural gas pipeline. M.S., Dissertation, School of petroleum and gas engineering, Southwest Petroleum University, Chengdu, China.