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An improved model to predict liquid holdup in vertical gas wells
Journal of Petroleum Science and Engineering 184 (2020) 106491
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Journal of Petroleum Science and Engineering journal homepage: www.elsevier.com/locate/petrol
An improved model to predict liquid holdup in vertical gas wells a,∗
a
a
Chengcheng Luo , Liehui Zhang , Yonghui Liu , Yulong Zhao Pengbo Wua a b
a,∗∗
b
T a
, Chunyu Xie , Lang Wang ,
State Key Laboratory of Oil & Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu, Sichuan, 610500, China Southwest Oil & Gas Field Company, PetroChina, Chengdu, Sichuan, 610041, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Gas wells Liquid holdup Pressure gradient Mechanistic models Empirical models
Accurate prediction of the pressure gradient is crucial for gas-production design and gas well optimization. The most important role in modeling the pressure gradient is generally to calculate the liquid holdup accurately. Currently, numerous mechanistic and empirical models have been developed and modified to deal with all kinds of flow conditions. Due to the chaotic feature in the churn flow, it is difficult for mechanistic models to model this flow pattern accurately. Simplified assumptions for this pattern may lead to large deviations. For empirical models, the range of experimental parameters limits their application. In this study, a variation of liquid holdup with superficial gas velocity is analyzed in order to simplify the liquid-holdup correlation and improve its accuracy. Based on the annular/churn-flow and bubble/slug-flow transition points, a new empirical model is established using a power-law curve. Since the superficial gas velocity and liquid holdup corresponding to the two transition points are obtained from analytical or semi-analytical models, the proposed model is not subject to the experimental ranges, unlike other empirical models. In addition, it avoids dealing with modeling churn flow. The new model has been validated against the laboratory and field data from published studies. For comparison, some widely used models are also evaluated. The result shows that the proposed model, which is capable of predicting the pressure gradient at different pipe sizes, liquid velocities, and gas velocities, has better performance compared with other models.
1. Introduction When natural gas is produced from the reservoirs, the water and condensate are often carried upwards simultaneously in gas wells, leading to a two-phase flow in the wellbore. Due to its complex nature, the behavior of the two-phase flow is difficult to understand or model for researchers and field engineers in many aspects (Wu et al., 2017; Zhu et al., 2019a). Accurate prediction of the pressure gradient along the wellbore is crucial for production optimization and performance analysis in gas wells. Because the design of some artificial lift methods, such as electrical submersible pumps (Zhu et al., 2017; Zhu et al., 2018), surfactant injection (Liu et al., 2014), and plunger lift (Zhu et al., 2019bb) requires accurate pressure gradient and liquid holdup along the wellbore. For gas wells with low gas production in unconventional reservoirs, accurate pressure-drop calculation is particularly important for removing the liquid accumulated in the wellbore and for analyzing the well-production performance (Zhang et al., 2019). Generally, the most important parameter in modeling the pressure gradient is accurate
calculation of the liquid holdup, since the gravitational pressure gradient accounts for the majority of the total pressure gradient at low gas flow rate. Many parameters, including pipe size, fluid properties, and inclination, have remarkable effect on liquid-holdup prediction. Therefore, it is difficult to establish a unified model for dealing with all kinds of flow conditions. Currently, three types of models are widely used to predict the pressure gradient in the petroleum industry for optimizing the production of gas wells: empirical models, analytical models, and homogeneous models. The most widely used are empirical models, due to their accuracy. Unlike analytical models, these models perform fitting from experimental data or gas-well measured pressure data. The first commonly used empirical model, which is a flow-pattern dependent correlation, was proposed by Duns and Ros (1963). In their model, three flow patterns are divided depending on dimensionless gas and liquid velocities. In each region, different closure equations are correlated from the experimental data to determine the gas-liquid slip velocity. Similar
∗
Corresponding author. Corresponding author. E-mail addresses: [email protected] (C. Luo), [email protected] (L. Zhang), [email protected] (Y. Liu), [email protected] (Y. Zhao), [email protected] (C. Xie), [email protected] (L. Wang), [email protected] (P. Wu). ∗∗
https://doi.org/10.1016/j.petrol.2019.106491 Received 19 June 2019; Received in revised form 18 August 2019; Accepted 12 September 2019 Available online 16 September 2019 0920-4105/ © 2019 Elsevier B.V. All rights reserved.
Journal of Petroleum Science and Engineering 184 (2020) 106491
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experimental investigations considering pipe size, flow pattern, liquid properties, and inclination were subsequently conducted by Hagedorn and Brown (1965), Orkiszewski (1967), Beggs and Brill (1973), and Mukherjee and Brill (1983). One of the main common features of those models is that they deal with pressure scaling from experiment to gas wells by using dimensionless numbers. However, proper dimensionless numbers are lacking in a two-phase flow. Furthermore, most of the empirical models have complicated forms and calculation processes. Unlike empirical models, analytical models, which are unlimited in the ranges of experimental parameters, are applicable due to their establishment based on force balance at each flow pattern. Aziz et al. (1972) proposed a mechanistic model to predict the pressure gradient in wells producing oil and gas. They emphasized modeling of the bubble and slug flow. In churn and annular flow, they still adopted the Duns and Ros (1963) correlations. Later, Ansari et al. (1994) developed a more comprehensive analytical method, which includes the patterntransition prediction and independent mechanistic models at each corresponding flow pattern. According to their assumption, they simplified the churn flow and treated it as a part of the slug flow. Moreover, Chokshi (1994) established a new model in the vertical two-phase flow through a large-diameter pipe. Kaya et al. (1999) modeled the twophase flow considering the effect of inclination. However, there are two clear shortcomings that limit the accuracy of these analytical models. First, there are some discontinuities in the pressure-gradient curves when the flow pattern changes. Secondly, these analytical models have difficulty in dealing with churn flow due to its complexity. Simplification may lead to large deviations since churn flow is one of the most common flow patterns produced in gas wells. Homogeneous models are relatively simple and easy to calculate. When gas-liquid slip is considered in the models, they are called driftflux models. The concept of the drift-flux model was originally proposed by Zuber and Findlay (1965). Due to its flexibility, many researchers have modified this model and improved its accuracy. There are two important variables in the drift-flux model: distribution parameter and drift velocity. However, since these are obtained from low pressure, there is little application of the two variables in the field, probably due to their accuracy in high pressure. In the literature, one can easily conclude that numerous studies of two-phase modeling have been conducted. However, it is difficult to seek a balance between accuracy and flexibility. Especially for gas wells featuring high gas-liquid ratios, the existing experimental investigations, which often cover a much wider range of experimental parameters, such as gas velocity and liquid velocity, do not emphasize or even cover the entire gas well range, leading to lower accuracy in gas wells. In contrast, Gray (1974) attempted to develop a simple equation to predict the pressure gradient in vertical gas wells. However, this method was correlated from the measured pressure data of 108 gas wells with high gas rates. In low-production gas wells, the method could cause large deviations. Recent studies (Kumar and Lea, 2005; Pagan et al., 2017) have tried to simplify two-phase modeling to improve the accuracy, but the range of application still needs to be expanded. This paper proposes an improved model for predicting liquid holdup in a vertical two-phase flow. Since the superficial gas velocity and liquid holdup corresponding to the two transition points are obtained from analytical or semi-analytical models, the proposed model, unlike other empirical models, is capable of dealing with the effect of pipe size and fluid properties without limiting the experimental range.
2.1. Basic equation In a vertical two-phase flow, the total pressure gradient can be obtained from the momentum equation of mixture liquid, which is given by
−
ρ vm2 dp dv = ρm g + f m + ρm vm m . dz 2D dz
(1)
The mixture density, ρm , is expressed as
ρm = ρL αL + ρG (1 − αL ).
(2)
On the right of Eq. (1), the three terms represent the gravitational, frictional, and accelerational pressure gradients, respectively. Normally, the acceleration term can be neglected in the fully developed flow, due to its much smaller value. When the gas production in the gas wells is high enough, the flow becomes annular/mist flow in the wellbore. In most cases, the flow is very regular, and the pressure gradient can be predicted by assuming there is no slip between the gas and liquid phases. This is because the fraction of liquid is very small and the effect of the no-slip assumption on the pressure gradient is negligible. From Eqs. (1) and (2), it is clear that gas density and friction play important roles in calculating the pressure gradient. Owing to the regular flow and low calculated difference in frictional pressure correlations, one can easily predict the pressure gradient accurately in the annular flow. As gas velocity decreases, the flow becomes irregular. According to the studies of Liu (2014), Al-Ruhaimani et al. (2016), and Liu et al. (2018), the frictional pressure gradient may be negative in a non-annular flow, due to the liquid-film reversal. Although measurement uncertainty and limited experiment-facility length may affect the accuracy of measurement, it can still be concluded that liquid holdup dominates the total pressure gradient in a non-annular flow. Therefore, accurate prediction of liquid holdup is crucial for calculating the pressure gradient. One of the greatest challenges in predicting the pressure gradient in gas wells in particular is dealing with the complicated flow in low gas production. In this study, the proposed liquid-holdup model is correlated based on two points: the liquid-film-reversal point and the bubble/slug transition point. Due to the low calculated difference from the different frictional pressure correlations under annular flow and the small fraction in the total pressure gradient under non-annular flow, we adopted the friction factor of Mukherjee (1979) to calculate the frictional pressure gradient in this study. 2.2. Model establishment Al-Sarkhi and Sarica (2010) proposed a simplified correlation for the pressure gradient in horizontal pipes. Based on the momentum equations of the gas and liquid phases, they derived a unique relationship between a dimensionless pressure gradient and gas velocity. Then a new power-law equation was proposed, which is correlated from published experimental data, despite the quadratic relationship in the theoretical modeling. In the vertical pipes, many experimental studies (Skopich et al., 2015; Liu et al., 2018) have shown that the curve of the liquid holdup with gas velocity is smooth and regular, which suggests that the behavior of the liquid holdup can be followed by a simple correlation. Some researchers (Mukherjee and Brill, 1983; Lumban-Gaol and Valkó, 2014) have tried to establish exponential correlations to model the relationship between gas velocity and liquid holdup. However, there are two points which can't be dealt with by these models: First, the models are correlated from low-pressure experimental data. Highpressure conditions are difficult to deal with in gas wells. Although many dimensionless numbers were proposed for a widely applicable solution (Wu et al., 2017), there is no universal form to address the high
2. Proposed model The primary goal of this paper is to propose a power-law correlation for predicting liquid holdup, thus establishing an improved two-phase model.
2
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Fig. 1. The variation trend of liquid holdup with superficial gas velocity.
pressure in gas wells. In addition, non-dimensional coefficients should be used to include the effects of liquid physical properties, pipe size, and liquid velocity. Existing models, however, have trouble in dealing with situations beyond the range of experimental data. Furthermore, exponential correlations may not capture the behavior of liquid holdup at low gas velocity. In the liquid-holdup curve, which covers slug and bubble flow (Owen, 1986; Godbole et al., 2011), it's clear that the liquid holdup increases gently with the decrease of gas at high gas velocity. When gas velocity decreases to a given point, liquid holdup starts to increase dramatically. The tendency for dramatic change in the liquid holdup is more obvious in large-diameter pipes (Zabaras et al., 2013; Skopich et al., 2015). As shown in Fig. 1, liquid holdup never experiences dramatic increase if the variation tendency obeys the exponential curve. Moreover, some model evaluations of gas wells indicate that Mukherjee's exponential correlation sometimes overestimates the pressure gradient in the wellbore (Ansari et al., 1994; Li and Du, 1996). According to the analysis above, we have proposed a new powerlaw correlation to fit the variation of liquid holdup with gas velocity. The power-law form is more capable of capturing the change tendency of liquid holdup at low gas velocity, especially for large-diameter pipes in real gas wells, which is written as:
αL = avsgb .
Fig. 2. Schematic of force balance in liquid film and gas core.
The interfacial and wall friction force can be given, respectively, as
and
Eqs. (4)–(7) can be combined to eliminate the pressure gradient, resulting in the combined momentum equation for annular flow, which is given by:
1 1 ⎞ τL SL τi Si ⎛⎜ + − (ρL − ρG ) g = 0. ⎟ − A A AL g⎠ ⎝ L
(3)
Si = π (D − 2δ ),
(10)
AL = π (Dδ − δ 2),
(11)
2 D AG = π ⎛ − δ ⎞ . 2 ⎝ ⎠
(12)
The wall shear stress can be written as:
τL = fL
ρL v L2 2
(13)
and −n
ρ Dv L ⎞ fL = CL ⎜⎛ L . ⎟ ⎝ μL ⎠
(14)
Then, the interfacial shear stress can be obtained based on the liquid-film superficial velocity, which is expressed as
~ ~2 ~ τi = g (ρL − ρG ) D (δ − δ )(1 − 2δ ) −n ~ ρ D 1 ⎡ (1 − 2δ ) ⎤ CL ρL ⎜⎛ L ⎟⎞ (vSL )2 − n ⎢ ~ ~2 2 ⎥, 32 ⎝ μL ⎠ ⎣ (δ − δ ) ⎦ ~ where the dimensionless film thickness δ is defined as
+
δ ~ δ = . D
(4)
(15)
(16)
For a fixed superficial liquid velocity, the interfacial shear stress is a ~ function of the dimensionless film thickness. In the curve of δ − τi , the minimum point indicates the liquid-film reversal, i.e., the transition from annular to churn flow. Hence, we can get the interfacial shear
and
dp − αG ρG g − Fi = 0. dz
(9)
SL = πD,
In this study, the transition from annular to churn flow in vertical pipes is calculated based on the Barnea (1986) model for predicting liquid-film reversal. The schematic of force balance in the liquid film and gas core is presented in Fig. 2. We assume that the gas and the liquid flow separately. The liquid and gas momentum equations for the vertical flow can be obtained, respectively, as
− αG
(8)
The geometric parameters are defined in terms of the uniform liquid-film thickness and the pipe diameter:
2.3. Annular/churn-flow transition point
dp − αL ρL g − FL + Fi = 0 dz
(7)
Fi = τi Si/ A.
Unlike the exponential correlations in previous studies, the parameters a and b in Eq. (3) were not obtained by fitting the experimental data in this work. For fixed pipe diameters, liquid properties, and, velocity, the parameters should be constant; thus, we determine a and b by using two key points instead of by correlating them from experimental data. In the vertical two-phase flow, annular and bubble flow are regular and stable flow patterns, due to the gas or liquid phases becoming continuously dominant. Therefore, the annular/churn-flow and bubble/slug-flow transition points are used to determine the curve.
− αL
(6)
FL = τL SL / A
(5) 3
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this study. When the gas velocity is high enough to cause the calculated liquid holdup to become lower than the no-slip liquid holdup, the pressure gradient is predicted by assuming there is no gas-liquid slip. Lowering the gas velocity to a given point, the liquid holdup may reach 1. Further decrease in the gas velocity would lead to an unrealistic value in the liquid holdup. Thus, the liquid holdup is set to 1 when the gas velocity is lower than this given point.
stress corresponding to the minimum point τi,CR by differentiating Eq. (15) with respect to dimensionless film thickness: −n
ρ D 1 ~ ~2 g (ρL − ρG ) D (1 − 6δ − 6δ ) − CL ρL ⎜⎛ L ⎟⎞ (vSL )2 − n 16 ⎝ μL ⎠ ~ ~2 ~2 ⎡ (δ − δ ) + (1 − 2δ ) ⎤ ⎢ ⎥ = 0. ~ ~2 3 (δ − δ ) ⎣ ⎦
(17)
At this time, we can also obtain the corresponding critical film ~ thickness, δcr . Hence liquid holdup at the transition point from annular to churn flow can be expressed as:
~ ~ 2 αL,A−C = 4(δcr − δcr ).
3. Discussion and model validation In order to evaluate the performance of the proposed model, a sensitivity analysis was conducted. The accuracy of the proposed model was then evaluated against laboratory data (Skopich et al., 2015; Liu et al., 2018) and field data (Govier and Fogarasi, 1975; Rendeiro and Kelso, 1988; Liao, 2007). For comparison, some widely used pressuregradient models, including those of Gray (1974), Mukherjee and Brill (1983), Hagedorn and Brown (1965); Ansari et al. (1994), and Zhang et al. (2003), were also evaluated.
(18)
The critical gas velocity uG, A − C can also be expressed as
1 ρG vSG, A − C 2 f , 2 i (1 − 2δ~cr ) 4
(19)
~ fi = 0.005(1 + 300δcr ).
(20)
τi,CR = where
3.1. Sensitivity analysis It is clear that some variables, such as pipe diameter, liquid velocity, and pressure, determine the liquid-holdup curve from the previous models and from the proposed model. Thus, the effect of these parameters on liquid holdup was analyzed based on the new model first. Fig. 3 presents the calculated liquid holdup as a function of superficial gas velocity at superficial liquid velocities of 0.01 m/s, 0.05 m/s, and 0.1 m/s. The pressure is set to 0.1 MPa, and the pipe diameter is 62 mm. It can be inferred that the calculated liquid holdup increases with the superficial liquid velocity at a fixed gas velocity. As the gas velocity decreases, the difference in liquid holdup due to change of liquid velocity first increases and subsequently decreases. It can also be seen that a steeper liquid-holdup curve is available at a lower liquid velocity. At high gas velocity, the liquid holdup is very low; liquid velocity has a great effect on liquid holdup. In contrast, liquid holdup is high at low gas velocity, while the effect of liquid velocity is relatively small, leading to a more rapid increase of liquid holdup at low liquid velocity. Fig. 4 shows the variation of calculated liquid holdup with superficial gas velocity at different superficial liquid velocities. The pressure is set to 0.1 MPa, and the superficial liquid velocity is 0.05 m/s. It is clear that the calculated liquid holdup is larger in smaller pipe diameters. The difference in liquid holdup due to pipe diameter seems to be nearly the same at different gas velocities. Fig. 5 illustrates that the calculated liquid holdup varies with the
2.4. Bubble/slug-flow transition point In this study, the drift-flux model is used to predict the bubble-slugtransition gas velocity. The general equation of the drift-flux closure relationship is expressed as: (21)
vG = C0 vm + vd.
Here, vG is the gas velocity; vm is the mixture velocity of gas and liquid; C0 is the distribution parameter; and vd is drift velocity. The gas velocity is defined as:
vG = vSG/(1 − αL ).
(22)
The mixture velocity of gas and liquid is defined as the sum of gas and liquid phase superficial velocities, (23)
vm = vSG + vSL.
In the drift-flux model, the distribution parameter and drift velocity are related to the flow pattern. In this study, Hasan and Kabir, 1988 correlation is used to predict the transition criterion of bubble and slug flow. In bubble flow, the distribution parameter C0 is equal to 1.2, and the drift velocity is given as: 0.25
vd = 1.53 ⎡ ⎢ ⎣
gσ (ρL − ρG ) ⎤ ⎥ ρL ⎦
. (24)
The transition to slug flow was experimentally verified at a liquid holdup of about αL,B−S = 0.75. Thus, the superficial gas velocity corresponding to the transition can be written as: 0.25
vSG,B−S = 0.429vSL + 0.546 ⎡ ⎢ ⎣
gσ (ρL − ρG ) ⎤ ⎥ ρL ⎦
. (25)
2.5. Model solution When the annular-churn and bubble-slug transitions have been determined, the parameters b and a in Eq. (3) can be obtained, respectively,
⎧b =
log(α L,B−S) − log(α L,A−C) log(v SG,B−S) − log(v SG,A−C)
⎨a = α b L,B−S/ vSG,B−S ⎩
. (26)
Although the transitions were established based on analytical or semi-analytical analysis, the proposed model is unable to deal with some extreme conditions at high or low gas velocities, leading to an unphysical liquid holdup. Therefore, some assumptions are added in
Fig. 3. The effect of superficial liquid velocity on liquid holdup. 4
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Brown, 1965; Gray, 1974; and Mukherjee and Brill, 1983) have smooth curves. This is because the mechanistic models are based on different assumptions at different flow patterns. Flow transition causes the discontinuities. Furthermore, they overestimate the liquid holdup at annular flow, perhaps due to some improper empirical closure relationships. Although the Gray (1974) model shows a good prediction at liquid velocities of 0.01 m/s in 50.8-mm and 101.6-mm pipes, it greatly underestimates liquid holdup at a liquid velocity of 0.05 m/s, the same as Mukherjee and Brill (1983) model. The large deviation from these empirical models might be attributed to the data collected beyond the range of experimental data for the models. Fig. 7 shows the comparison between the measured and calculated holdups. It includes all 35 experimental data points of Skopich et al. (2015). It's clear that the proposed model has better performance than other models. The new model predicted most of the data points within ± 25% error lines. As shown in Table 1, the proposed model has an average relative error (ARE) of 7.50% and an absolute average relative error (AARE) of 28.92%, both of which are minimum in all the models. The comparison between the liquid holdups measured by Liu et al. (2018) and calculated by these models is presented in Fig. 8. In the study of Liu et al. (2018), the pipe diameter is 30 mm, and the liquid superficial velocities are set as 0.01, 0.03, and 0.05 m/s, respectively. It can be found from Fig. 8 that the new model still performs best in all of these models at different liquid velocities. In Fig. 9, this model shows a good prediction of liquid holdup in most of the data points with an error under 25%. Table 2 presents the comparison results of all the models. The ARE and AARE of the new model are −5.89% and 17.41%, respectively. From the comparison between the liquid holdups measured and calculated by the proposed model, it can be found that a larger deviation occurs at lower gas velocities. It might be because the new model underestimates the uptrend of liquid holdup with gas velocity. Although the deviation seems to be dramatic in an atmospheric pressure condition, it can be inferred from Fig. 5 that the increased pressure will reduce the gas velocity corresponding to liquid film reversal and make the curve much steeper, thus causing the deviation of this model to become much smaller.
Fig. 4. The effect of pipe diameter on liquid holdup.
3.3. Model validation against field data
Fig. 5. The effect of pressure on liquid holdup.
In gas wells, only the pressure gradient can be measured. As mentioned in Section 2.1, liquid holdup is the controlling variable for accurate prediction of the pressure gradient, especially with lower gas velocities. Hence, we can evaluate the proposed model by calculating the pressure gradient against field data. In order to evaluate the performance of the proposed model at the high pressure of real gas wells, this study has collected some published pressure-gradient data from previous studies (Govier and Fogarasi, 1975; Rendeiro and Kelso, 1988; Liao, 2007), which cover low, intermediate, and high gas flow rates. The data set of Govier and Fogarasi (1975) and Rendeiro and Kelso (1988), which consist of 102 wells (94 wells available) and 50 wells, are identified to cover intermediate and high gas flow rate, with gas flow rate ranging from 1.3 to 77.6 × 104 m3/d, liquid flow rate between 1.0 and 347.8 m3/d, and wellhead pressure between 3.1 and 84.9 MPa. The data set of Liao (2007) covers low and intermediate gas flow rates, ranging from 3393 to 27,056 m3/d. Note that Liao (2007) conducted the pressure-gradient experiment in an artificial well with a depth of 970 m and measured the pressure at different depths. No liquid-loading phenomenon occurred. A total of 38 experimental tests were available in his study. The tubing head pressure of Liao (2007) experiment ranges from 0.898 to 4.658 MPa. Figs. 10 and 11 show the prediction results for the data set of Govier and Fogarasi (1975) and Rendeiro and Kelso (1988) using the proposed model and other widely used models. Fig. 10 shows that all the models have good performance in predicting bottomhole pressure. This is
superficial gas velocity in different pressure conditions. The pipe diameter is set to 50 mm, and the liquid superficial velocity is 0.05 m/s. The pressure has dramatic effect on liquid-film reversal, due to its effect on gas density. Hence, the film-reversal points are also marked with the purple dotted line in Fig. 5. It is clear that the increased pressure narrows the range between the two key points of the proposed model, of which the trending change is indicated in Fig. 5 by the solid yellow arrow. This change can lead to a much steeper trend in the liquidholdup curve. Furthermore, the curves become much closer as the pressure increases. At gas velocities above 15 m/s, it can be seen that the liquid holdup is irrelevant to gas velocities at pressures of 10 and 20 MPa. This is due to the no-slip assumption of the proposed model.
3.2. Model validation against laboratory data In this study, the experimental data was collected from the studies of Skopich et al. (2015) and Liu et al. (2018). Fig. 6 presents the liquid-holdup-model comparisons with the experimental results of Skopich et al. (2015), who measured liquid holdup at liquid velocities of 0.01 m/s and 0.05 m/s and pipe diameters of 50.8 mm and 101.6 mm. On the whole, the proposed modelin this study shows a better performance and change trend than other models. Note that the mechanistic models (Ansari et al., 1994; Zhang et al., 2003) display discontinuities, while the empirical models (Hagedorn and 5
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Fig. 6. Comparison between liquid holdups measured by Skopich et al. (2015) and calculated by the proposed and previous models at different pipe diameters and liquid velocities. (a) D = 50.8 mm; vSL = 0.01 m/s, (b) D = 50.8 mm; vSL = 0.05 m/s, (c) D = 101.6 mm; vSL = 0.01 m/s, and (d) D = 101.6 mm; vSL = 0.05 m/s. Table 1 Prediction results of these models for the experimental data of Skopich et al. (2015).
Ansari Zhang Gray Mukherjee-Brill Hagedorn-Brown Current model
Average Relative Error %
Absolute Average Relative Error %
246.99 98.14 −39.23 −87.34 31.95 7.50
247.58 103.63 46.57 87.34 58.86 28.92
because the wells in this data set have intermediate and high gas flow rates and high wellhead pressure, leading to a smaller fraction of gravitational pressure loss in the total pressure gradient. Thus, even larger deviations in the calculated liquid holdup may have a smaller effect on predicting bottomhole pressure. Overall, it is clear that the proposed model has the best agreement with measured bottomhole pressure, followed by Gray (1974) model and Ansari et al. (1994) model. All three models have ARE and AARE less than 10%, which
Fig. 7. Comparison between liquid holdups measured by Skopich et al. (2015) and calculated by the proposed and previous models.
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Fig. 8. Comparison between liquid holdups measured by Liu et al. (2018) and calculated by the proposed and previous models at different liquid velocities. (a) vSL = 0.01 m/s, (b) vSL = 0.03 m/s, and (c) vSL = 0.05 m/s. Table 2 Prediction results of these models for the experimental data of Liu et al. (2018).
Ansari Zhang Gray Mukherjee-Brill Hagedorn-Brown Current model
Average Relative Error %
Absolute Average Relative Error %
19.17 −7.84 −73.79 −90.59 29.71 −5.89
36.68 20.76 73.79 90.59 53.89 17.41
means that the proposed model is capable of dealing with the gas wells at high pressure and gas flow rates. The performance of the proposed and previous models against the data of Liao (2007) is presented in Fig. 12. This figure shows that the new model has relatively good prediction, as well as the model of Mukherjee and Brill (1983). However, the Ansari et al. (1994) and the Gray (1974) models underestimate the bottomhole pressure, although they have good performance for the data set of Govier and Fogarasi (1975). From the prediction results in Fig. 13, it can be concluded that
Fig. 9. Comparison between liquid holdups measured by Liu et al. (2018) and calculated by the proposed and previous models.
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Fig. 10. Comparison between the calculated and measured bottomhole pressure for the data set of Govier and Fogarasi (1975) and Rendeiro and Kelso (1988). (a) Mukherjee and Brill (1983), (b) Gray (1974), (c) Hagedorn and Brown (1965), (d) Ansari et al. (1994), (e) Zhang et al. (2003), and (f) Current model.
accurately predict the bottomhole pressure at different ranges of gas velocities from the data of Govier and Fogarasi (1975), Rendeiro and Kelso (1988) and Liao (2007).
the proposed model still has good prediction in low and intermediate gas flow rates with an ARE of −6.39% and an AARE of 8.67%, only lowering the accuracy of the Mukherjee and Brill (1983) model. As a whole, the proposed correlation is the only model that is able to 8
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(2) Since the superficial gas velocity and liquid holdup corresponding to the two transition points are obtained from analytical or semianalytical models, the proposed model is capable of dealing with the effect of pipe size and fluid properties with no limit on experimental range, unlike other empirical models. (3) The sensitivity analysis shows that increased pressure leads to a much steeper trend in the liquid-holdup curve. (4) By comparing the proposed model of this study with other widely used models using laboratory data and field data, it is clear that this new model preforms well at different ranges of pressures and gas velocities. Acknowledgement This work is supported by the National Natural Science Foundation of China (Key Program) (Grant No. 51534006) and the National Natural Science Foundation of China (Grant No. 51704247,51874251).
Fig. 11. Prediction results of these models for the data set of Govier and Fogarasi (1975) and Rendeiro and Kelso (1988).
Nomenclature
A AG AL a b C0 CL D FL Fi f g n p SL Si vd vm vSG vG vSG, A−C
Fig. 12. Comparison between the calculated and measured bottomhole pressure for the data set of Liao (2007).
vSG, B−S vSL z αL αL,B−S αL,A−C
αG δ ~ δ δcr ~ δcr μL ρm ρG ρL σ τi τi,CR τw
Fig. 13. Prediction results of these models for the data set of Liao (2007).
4. Conclusions In this paper, an improved power-law two-phase model is developed to predict the pressure gradient in a vertical upward two-phase flow. The following conclusions can be reached: (1) In this model, only annular/churn-flow and bubble/slug-flow transition points are required for determining the unsolved coefficient.
pipe area, m2 area of occupied by gas, m2 area of occupied by liquid, m2 coefficient, dimensionless coefficient, dimensionless distribution parameter coefficient pipe diameter, m frictional fraction force, N interfacial fraction force, N frictional fraction factor, dimensionless acceleration of gravity, m/s2 Constant pressure, Pa wetted perimeter by liquid film, m2/s wetted perimeter by gas film, m2/s drift velocity, m/s mixture velocity, m/s gas superficial velocity, m/s gas velocity, m/s gas superficial velocity of annular-churn-transition point, m/ s gas superficial velocity of bubble-slug-transition point, m/s liquid superficial velocity, m/s depth, m liquid holdup, dimensionless liquid holdup of bubble-slug-transition point, dimensionless liquid holdup of annular-churn-transition point, dimensionless void fraction, dimensionless film thickness, m dimensionless film thickness, m critical liquid-film thickness, m critical dimensionless liquid-film thickness, m liquid viscosity, Pa·s mixture density, kg/m3 gas density, kg/m3 liquid density, kg/m3 liquid surface tension, N/m interfacial shear stress, N/m2 critical interfacial shear stress, N/m2 wall shear stress, N/m2
References Al-Ruhaimani, F., Pereyra, E., Sarica, C., Al-Safran, E.M., Torres, C.F., 2016. Experimental
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investigations of the circumferential liquid film distribution of air-water annular twophase flow in a horizontal pipe. J. Nat. Gas Sci. Eng. 57, 349–358. Lumban-Gaol, A., Valkó, Peter P., 2014. Liquid holdup correlation for conditions affected by partial flow reversal. Int. J. Multiph. Flow 67, 149–159. Mukherjee, H., 1979. Liquid Holdup Correlations for Inclined Two-phase Flow. PhD dissertation. University of Tulsa, Tulsa. Mukherjee, H., Brill, J.P., 1983. Liquid holdup correlations for inclined two-phase flow. J. Pet. Technol. 22 (3), 1003–1008 SPE-184401-PA. Orkiszewski, J., 1967. Predicting two-phase pressure drops in vertical pipe. J. Pet. Technol. 19 (6), 829–838 SPE-1546-PA. Owen, D.G., 1986. An Experimental and Theoretical Analysis of Equilibrium Annular Flow. PhD dissertation. University of Birmingham, Birmingham, UK. Pagan, E., Williams, W.C., Kam, S., Waltrich, P.J., 2017. A simplified model for churn and annular flow regimes in small- and large-diameter pipes. Chem. Eng. Sci. 162, 309–321 2017. Rendeiro, C. M., Kelso, C. M., 1988. An investigation to improve the accuracy of calculating bottomhole pressures in flowing gas wells producing liquids. In: Proceedings of Permian Basin Oil and Gas Recovery Conference, Midland, Texas, 10-11 March. Skopich, A., Pereyra, E., Sarica, C., Kelkar, M., 2015. Pipe diameter effect on liquid loading in vertical gas wells. SPE Prod. Oper. 30 (2), 164–176. Wu, B., Firouzi, M., Mitchell, T., Rufford, T.E., Leonardi, C., Towler, B., 2017. A critical review of flow maps for gas-liquid flows in vertical pipes and annuli. Chem. Eng. J. 326, 350–377. Zabaras, G., Menon, R., Schoppa, W., Wicks III, M., 2013. Large diameter riser laboratory gas-lift tests. In: Offshore Technology Conference, Houston, TX, 6–9 May. Zhang, H.-Q., Wang, Q., Sarica, C., Brill, J.P., 2003. Unified model for gas-liquid pipe flow via slug dynamics–Part 2: model validation. J. Energy Resour. Technol. 125 (4), 274–283. Zhang, L.H., Chen, Z.X., Zhao, Y.L., 2019. Well Production Performance Analysis for Shale Gas Reservoirs. Elsevier Press, Calgary. Zhu, H. W., Zhu, J. J., Zhang, H.-Q., 2017. Efficiency and critical velocity analysis of gravitational separator through CFD simulation. In: the International Mechanical Engineering Congress and Exposition, Florida, 3-9 November. Zhu, H. W., Zhu, J. J., Rutter, R., Zhang, J. C., Zhang, H.-Q., 2018, Sand erosion model prediction, selection and comparison for Electrical Submersible Pump (ESP) using CFD method In: the 5th Joint US-European Fluids Engineering Division Summer Meeting, Montreal, Quebec, Canada, 15-20 July. Zhu, J.J., Zhu, H.W., Zhang, J.C., Zhang, H.Q., 2019a. A numerical study on flowpatterns inside an electrical submersible pump (ESP) and comparison with visualization experiments. J. Petrol. Sci. Eng. 173, 339–350. Zhu, J. J. , Zhu, H. W., Zhao, Q. Q., Fu, W. Q., Shi, Y., Zhang, H. Q., 2019b. Atransient plunger lift model for liquid unloading from gas wells. In: the International Petroleum Technology Conference, Beijing, 26-28 March. Zuber, N., Findlay, J.A., 1965. Average volumetric concentration in two-phase flow systems. J. Heat Transf. 87, 453–468.
analysis and model evaluation of high-liquid-viscosity two-phase upward vertical pipe flow. SPE J. 22 (3), 1–24. Al-Sarkhi, A., Sarica, C., 2010. Power-law correlation for two-phase pressure drop of gas/ liquid flows in horizontal pipelines. SPE Proj. Facil. Constr. 5 (04), 176–182. Ansari, A.M., Sylvester, N.D., Sarica, C., Shoham, O., Brill, J.P., 1994. A comprehensive mechanistic model for upward two-phase flow in wellbores. SPE Prod. Facil. 9 (2), 143–151 SPE-20630-PA. Aziz, K., Govier, G.W., Fogarasi, M., 1972. Pressure drop in wells producing oil and gas. J. Can. Pet. Technol. 11 (3), 38–48. Barnea, D., 1986. Transition from annular flow and from dispersed bubble flow-unified models for the whole range of pipe inclinations. Int. J. Multiph. Flow 12 (5), 733–744. Beggs, D.H., Brill, J.P., 1973. A study of two-phase flow in inclined pipes. J. Pet. Technol. 25 (5), 607–617 SPE-4007-PA. Chokshi, R.N., 1994. Prediction of Pressure Drop and Liquid Holdup in Vertical Twophase Flow through Large Diameter Tubing. PhD dissertation. University of Tulsa, Tulsa. Duns Jr., H., Ros, N.C.J., 1963. Vertical flow of gas and liquid mixtures in wells. In: 6th World Petroleum Congress, 19–26 June. Frankfurt Am Main, Germany. Godbole, P.V., Tang, C.C., Ghajar, A.J., 2011. Comparison of void fraction correlations for different flow patterns in upward vertical two-phase flow. Heat Transf. Eng. 32 (10), 843–860. Govier, G.W., Fogarasi, M., 1975. Pressure drop in wells producing gas and condensate. J. Can. Pet. Technol. 14 (04), 28–41. Gray, H.E., 1974. Vertical Flow Correlation in Gas Wells. User's Manual for API 148 Subsurface Controlled Safety Valve Sizing Computer Program. (Appendix B). Hagedorn, A.R., Brown, K.E., 1965. Experimental study of pressure gradients occurring during continuous two-phase flow in small-diameter vertical conduits. J. Pet. Technol. 17 (04), 475–484. Hasan, A.R., Kabir, C.S., 1988. A study of multiphase flow behavior in vertical wells. SPE Prod. Eng. 3 (2), 263–272 SPE-15138-PA. Kaya, A. S., Sarica, C., and Brill, J. P. 1999. Comprehensive mechanistic modeling oftwophase flow in deviated wells. In: presented at the SPE Annual Technical Conference and Exhibition, Houston, 3–6 October. SPE-56522-MS. Kumar, N., Lea, J. F., 2005. Improvements for flow correlations for gas wells experiencing liquid loading. In: SPE Western Regional Meeting, 30 March - 1 April. Irvine, USA. Li, Y., Du, Z., 1996. A new numerical model for two-phase flow in liquid-cut gas wells. In: presented at the Gas Technology Conference, Calgary, 28 April - 1May. SPE 35612. Liao, K.G., 2007. Study on Flow Pattern and Pressure Drop Models of Gas-Water Twophase Pipe Flow in 930m Experiment Well. PhD dissertation. Southwest Petroleum University, Chengdu. Liu, L., 2014. The phenomenon of negative frictional pressure drop in vertical two-phase flow. Int. J. Heat Fluid Flow 45, 72–80. Liu, L., Li, X., Tong, L., Liu, Y., 2014. Effect of surfactant additive on vertical two-phase flow. J. Pet. Sci. Eng. 115, 1–10. Liu, Y.H., Luo, C.C., Zhang, L.H., Liu, Z.,B., Xie, C.Y., Wu, P.B., 2018. Experimental
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Contents lists available at ScienceDirect
Journal of Petroleum Science and Engineering journal homepage: www.elsevier.com/locate/petrol
An improved model to predict liquid holdup in vertical gas wells a,∗
a
a
Chengcheng Luo , Liehui Zhang , Yonghui Liu , Yulong Zhao Pengbo Wua a b
a,∗∗
b
T a
, Chunyu Xie , Lang Wang ,
State Key Laboratory of Oil & Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu, Sichuan, 610500, China Southwest Oil & Gas Field Company, PetroChina, Chengdu, Sichuan, 610041, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Gas wells Liquid holdup Pressure gradient Mechanistic models Empirical models
Accurate prediction of the pressure gradient is crucial for gas-production design and gas well optimization. The most important role in modeling the pressure gradient is generally to calculate the liquid holdup accurately. Currently, numerous mechanistic and empirical models have been developed and modified to deal with all kinds of flow conditions. Due to the chaotic feature in the churn flow, it is difficult for mechanistic models to model this flow pattern accurately. Simplified assumptions for this pattern may lead to large deviations. For empirical models, the range of experimental parameters limits their application. In this study, a variation of liquid holdup with superficial gas velocity is analyzed in order to simplify the liquid-holdup correlation and improve its accuracy. Based on the annular/churn-flow and bubble/slug-flow transition points, a new empirical model is established using a power-law curve. Since the superficial gas velocity and liquid holdup corresponding to the two transition points are obtained from analytical or semi-analytical models, the proposed model is not subject to the experimental ranges, unlike other empirical models. In addition, it avoids dealing with modeling churn flow. The new model has been validated against the laboratory and field data from published studies. For comparison, some widely used models are also evaluated. The result shows that the proposed model, which is capable of predicting the pressure gradient at different pipe sizes, liquid velocities, and gas velocities, has better performance compared with other models.
1. Introduction When natural gas is produced from the reservoirs, the water and condensate are often carried upwards simultaneously in gas wells, leading to a two-phase flow in the wellbore. Due to its complex nature, the behavior of the two-phase flow is difficult to understand or model for researchers and field engineers in many aspects (Wu et al., 2017; Zhu et al., 2019a). Accurate prediction of the pressure gradient along the wellbore is crucial for production optimization and performance analysis in gas wells. Because the design of some artificial lift methods, such as electrical submersible pumps (Zhu et al., 2017; Zhu et al., 2018), surfactant injection (Liu et al., 2014), and plunger lift (Zhu et al., 2019bb) requires accurate pressure gradient and liquid holdup along the wellbore. For gas wells with low gas production in unconventional reservoirs, accurate pressure-drop calculation is particularly important for removing the liquid accumulated in the wellbore and for analyzing the well-production performance (Zhang et al., 2019). Generally, the most important parameter in modeling the pressure gradient is accurate
calculation of the liquid holdup, since the gravitational pressure gradient accounts for the majority of the total pressure gradient at low gas flow rate. Many parameters, including pipe size, fluid properties, and inclination, have remarkable effect on liquid-holdup prediction. Therefore, it is difficult to establish a unified model for dealing with all kinds of flow conditions. Currently, three types of models are widely used to predict the pressure gradient in the petroleum industry for optimizing the production of gas wells: empirical models, analytical models, and homogeneous models. The most widely used are empirical models, due to their accuracy. Unlike analytical models, these models perform fitting from experimental data or gas-well measured pressure data. The first commonly used empirical model, which is a flow-pattern dependent correlation, was proposed by Duns and Ros (1963). In their model, three flow patterns are divided depending on dimensionless gas and liquid velocities. In each region, different closure equations are correlated from the experimental data to determine the gas-liquid slip velocity. Similar
∗
Corresponding author. Corresponding author. E-mail addresses: [email protected] (C. Luo), [email protected] (L. Zhang), [email protected] (Y. Liu), [email protected] (Y. Zhao), [email protected] (C. Xie), [email protected] (L. Wang), [email protected] (P. Wu). ∗∗
https://doi.org/10.1016/j.petrol.2019.106491 Received 19 June 2019; Received in revised form 18 August 2019; Accepted 12 September 2019 Available online 16 September 2019 0920-4105/ © 2019 Elsevier B.V. All rights reserved.
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experimental investigations considering pipe size, flow pattern, liquid properties, and inclination were subsequently conducted by Hagedorn and Brown (1965), Orkiszewski (1967), Beggs and Brill (1973), and Mukherjee and Brill (1983). One of the main common features of those models is that they deal with pressure scaling from experiment to gas wells by using dimensionless numbers. However, proper dimensionless numbers are lacking in a two-phase flow. Furthermore, most of the empirical models have complicated forms and calculation processes. Unlike empirical models, analytical models, which are unlimited in the ranges of experimental parameters, are applicable due to their establishment based on force balance at each flow pattern. Aziz et al. (1972) proposed a mechanistic model to predict the pressure gradient in wells producing oil and gas. They emphasized modeling of the bubble and slug flow. In churn and annular flow, they still adopted the Duns and Ros (1963) correlations. Later, Ansari et al. (1994) developed a more comprehensive analytical method, which includes the patterntransition prediction and independent mechanistic models at each corresponding flow pattern. According to their assumption, they simplified the churn flow and treated it as a part of the slug flow. Moreover, Chokshi (1994) established a new model in the vertical two-phase flow through a large-diameter pipe. Kaya et al. (1999) modeled the twophase flow considering the effect of inclination. However, there are two clear shortcomings that limit the accuracy of these analytical models. First, there are some discontinuities in the pressure-gradient curves when the flow pattern changes. Secondly, these analytical models have difficulty in dealing with churn flow due to its complexity. Simplification may lead to large deviations since churn flow is one of the most common flow patterns produced in gas wells. Homogeneous models are relatively simple and easy to calculate. When gas-liquid slip is considered in the models, they are called driftflux models. The concept of the drift-flux model was originally proposed by Zuber and Findlay (1965). Due to its flexibility, many researchers have modified this model and improved its accuracy. There are two important variables in the drift-flux model: distribution parameter and drift velocity. However, since these are obtained from low pressure, there is little application of the two variables in the field, probably due to their accuracy in high pressure. In the literature, one can easily conclude that numerous studies of two-phase modeling have been conducted. However, it is difficult to seek a balance between accuracy and flexibility. Especially for gas wells featuring high gas-liquid ratios, the existing experimental investigations, which often cover a much wider range of experimental parameters, such as gas velocity and liquid velocity, do not emphasize or even cover the entire gas well range, leading to lower accuracy in gas wells. In contrast, Gray (1974) attempted to develop a simple equation to predict the pressure gradient in vertical gas wells. However, this method was correlated from the measured pressure data of 108 gas wells with high gas rates. In low-production gas wells, the method could cause large deviations. Recent studies (Kumar and Lea, 2005; Pagan et al., 2017) have tried to simplify two-phase modeling to improve the accuracy, but the range of application still needs to be expanded. This paper proposes an improved model for predicting liquid holdup in a vertical two-phase flow. Since the superficial gas velocity and liquid holdup corresponding to the two transition points are obtained from analytical or semi-analytical models, the proposed model, unlike other empirical models, is capable of dealing with the effect of pipe size and fluid properties without limiting the experimental range.
2.1. Basic equation In a vertical two-phase flow, the total pressure gradient can be obtained from the momentum equation of mixture liquid, which is given by
−
ρ vm2 dp dv = ρm g + f m + ρm vm m . dz 2D dz
(1)
The mixture density, ρm , is expressed as
ρm = ρL αL + ρG (1 − αL ).
(2)
On the right of Eq. (1), the three terms represent the gravitational, frictional, and accelerational pressure gradients, respectively. Normally, the acceleration term can be neglected in the fully developed flow, due to its much smaller value. When the gas production in the gas wells is high enough, the flow becomes annular/mist flow in the wellbore. In most cases, the flow is very regular, and the pressure gradient can be predicted by assuming there is no slip between the gas and liquid phases. This is because the fraction of liquid is very small and the effect of the no-slip assumption on the pressure gradient is negligible. From Eqs. (1) and (2), it is clear that gas density and friction play important roles in calculating the pressure gradient. Owing to the regular flow and low calculated difference in frictional pressure correlations, one can easily predict the pressure gradient accurately in the annular flow. As gas velocity decreases, the flow becomes irregular. According to the studies of Liu (2014), Al-Ruhaimani et al. (2016), and Liu et al. (2018), the frictional pressure gradient may be negative in a non-annular flow, due to the liquid-film reversal. Although measurement uncertainty and limited experiment-facility length may affect the accuracy of measurement, it can still be concluded that liquid holdup dominates the total pressure gradient in a non-annular flow. Therefore, accurate prediction of liquid holdup is crucial for calculating the pressure gradient. One of the greatest challenges in predicting the pressure gradient in gas wells in particular is dealing with the complicated flow in low gas production. In this study, the proposed liquid-holdup model is correlated based on two points: the liquid-film-reversal point and the bubble/slug transition point. Due to the low calculated difference from the different frictional pressure correlations under annular flow and the small fraction in the total pressure gradient under non-annular flow, we adopted the friction factor of Mukherjee (1979) to calculate the frictional pressure gradient in this study. 2.2. Model establishment Al-Sarkhi and Sarica (2010) proposed a simplified correlation for the pressure gradient in horizontal pipes. Based on the momentum equations of the gas and liquid phases, they derived a unique relationship between a dimensionless pressure gradient and gas velocity. Then a new power-law equation was proposed, which is correlated from published experimental data, despite the quadratic relationship in the theoretical modeling. In the vertical pipes, many experimental studies (Skopich et al., 2015; Liu et al., 2018) have shown that the curve of the liquid holdup with gas velocity is smooth and regular, which suggests that the behavior of the liquid holdup can be followed by a simple correlation. Some researchers (Mukherjee and Brill, 1983; Lumban-Gaol and Valkó, 2014) have tried to establish exponential correlations to model the relationship between gas velocity and liquid holdup. However, there are two points which can't be dealt with by these models: First, the models are correlated from low-pressure experimental data. Highpressure conditions are difficult to deal with in gas wells. Although many dimensionless numbers were proposed for a widely applicable solution (Wu et al., 2017), there is no universal form to address the high
2. Proposed model The primary goal of this paper is to propose a power-law correlation for predicting liquid holdup, thus establishing an improved two-phase model.
2
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Fig. 1. The variation trend of liquid holdup with superficial gas velocity.
pressure in gas wells. In addition, non-dimensional coefficients should be used to include the effects of liquid physical properties, pipe size, and liquid velocity. Existing models, however, have trouble in dealing with situations beyond the range of experimental data. Furthermore, exponential correlations may not capture the behavior of liquid holdup at low gas velocity. In the liquid-holdup curve, which covers slug and bubble flow (Owen, 1986; Godbole et al., 2011), it's clear that the liquid holdup increases gently with the decrease of gas at high gas velocity. When gas velocity decreases to a given point, liquid holdup starts to increase dramatically. The tendency for dramatic change in the liquid holdup is more obvious in large-diameter pipes (Zabaras et al., 2013; Skopich et al., 2015). As shown in Fig. 1, liquid holdup never experiences dramatic increase if the variation tendency obeys the exponential curve. Moreover, some model evaluations of gas wells indicate that Mukherjee's exponential correlation sometimes overestimates the pressure gradient in the wellbore (Ansari et al., 1994; Li and Du, 1996). According to the analysis above, we have proposed a new powerlaw correlation to fit the variation of liquid holdup with gas velocity. The power-law form is more capable of capturing the change tendency of liquid holdup at low gas velocity, especially for large-diameter pipes in real gas wells, which is written as:
αL = avsgb .
Fig. 2. Schematic of force balance in liquid film and gas core.
The interfacial and wall friction force can be given, respectively, as
and
Eqs. (4)–(7) can be combined to eliminate the pressure gradient, resulting in the combined momentum equation for annular flow, which is given by:
1 1 ⎞ τL SL τi Si ⎛⎜ + − (ρL − ρG ) g = 0. ⎟ − A A AL g⎠ ⎝ L
(3)
Si = π (D − 2δ ),
(10)
AL = π (Dδ − δ 2),
(11)
2 D AG = π ⎛ − δ ⎞ . 2 ⎝ ⎠
(12)
The wall shear stress can be written as:
τL = fL
ρL v L2 2
(13)
and −n
ρ Dv L ⎞ fL = CL ⎜⎛ L . ⎟ ⎝ μL ⎠
(14)
Then, the interfacial shear stress can be obtained based on the liquid-film superficial velocity, which is expressed as
~ ~2 ~ τi = g (ρL − ρG ) D (δ − δ )(1 − 2δ ) −n ~ ρ D 1 ⎡ (1 − 2δ ) ⎤ CL ρL ⎜⎛ L ⎟⎞ (vSL )2 − n ⎢ ~ ~2 2 ⎥, 32 ⎝ μL ⎠ ⎣ (δ − δ ) ⎦ ~ where the dimensionless film thickness δ is defined as
+
δ ~ δ = . D
(4)
(15)
(16)
For a fixed superficial liquid velocity, the interfacial shear stress is a ~ function of the dimensionless film thickness. In the curve of δ − τi , the minimum point indicates the liquid-film reversal, i.e., the transition from annular to churn flow. Hence, we can get the interfacial shear
and
dp − αG ρG g − Fi = 0. dz
(9)
SL = πD,
In this study, the transition from annular to churn flow in vertical pipes is calculated based on the Barnea (1986) model for predicting liquid-film reversal. The schematic of force balance in the liquid film and gas core is presented in Fig. 2. We assume that the gas and the liquid flow separately. The liquid and gas momentum equations for the vertical flow can be obtained, respectively, as
− αG
(8)
The geometric parameters are defined in terms of the uniform liquid-film thickness and the pipe diameter:
2.3. Annular/churn-flow transition point
dp − αL ρL g − FL + Fi = 0 dz
(7)
Fi = τi Si/ A.
Unlike the exponential correlations in previous studies, the parameters a and b in Eq. (3) were not obtained by fitting the experimental data in this work. For fixed pipe diameters, liquid properties, and, velocity, the parameters should be constant; thus, we determine a and b by using two key points instead of by correlating them from experimental data. In the vertical two-phase flow, annular and bubble flow are regular and stable flow patterns, due to the gas or liquid phases becoming continuously dominant. Therefore, the annular/churn-flow and bubble/slug-flow transition points are used to determine the curve.
− αL
(6)
FL = τL SL / A
(5) 3
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this study. When the gas velocity is high enough to cause the calculated liquid holdup to become lower than the no-slip liquid holdup, the pressure gradient is predicted by assuming there is no gas-liquid slip. Lowering the gas velocity to a given point, the liquid holdup may reach 1. Further decrease in the gas velocity would lead to an unrealistic value in the liquid holdup. Thus, the liquid holdup is set to 1 when the gas velocity is lower than this given point.
stress corresponding to the minimum point τi,CR by differentiating Eq. (15) with respect to dimensionless film thickness: −n
ρ D 1 ~ ~2 g (ρL − ρG ) D (1 − 6δ − 6δ ) − CL ρL ⎜⎛ L ⎟⎞ (vSL )2 − n 16 ⎝ μL ⎠ ~ ~2 ~2 ⎡ (δ − δ ) + (1 − 2δ ) ⎤ ⎢ ⎥ = 0. ~ ~2 3 (δ − δ ) ⎣ ⎦
(17)
At this time, we can also obtain the corresponding critical film ~ thickness, δcr . Hence liquid holdup at the transition point from annular to churn flow can be expressed as:
~ ~ 2 αL,A−C = 4(δcr − δcr ).
3. Discussion and model validation In order to evaluate the performance of the proposed model, a sensitivity analysis was conducted. The accuracy of the proposed model was then evaluated against laboratory data (Skopich et al., 2015; Liu et al., 2018) and field data (Govier and Fogarasi, 1975; Rendeiro and Kelso, 1988; Liao, 2007). For comparison, some widely used pressuregradient models, including those of Gray (1974), Mukherjee and Brill (1983), Hagedorn and Brown (1965); Ansari et al. (1994), and Zhang et al. (2003), were also evaluated.
(18)
The critical gas velocity uG, A − C can also be expressed as
1 ρG vSG, A − C 2 f , 2 i (1 − 2δ~cr ) 4
(19)
~ fi = 0.005(1 + 300δcr ).
(20)
τi,CR = where
3.1. Sensitivity analysis It is clear that some variables, such as pipe diameter, liquid velocity, and pressure, determine the liquid-holdup curve from the previous models and from the proposed model. Thus, the effect of these parameters on liquid holdup was analyzed based on the new model first. Fig. 3 presents the calculated liquid holdup as a function of superficial gas velocity at superficial liquid velocities of 0.01 m/s, 0.05 m/s, and 0.1 m/s. The pressure is set to 0.1 MPa, and the pipe diameter is 62 mm. It can be inferred that the calculated liquid holdup increases with the superficial liquid velocity at a fixed gas velocity. As the gas velocity decreases, the difference in liquid holdup due to change of liquid velocity first increases and subsequently decreases. It can also be seen that a steeper liquid-holdup curve is available at a lower liquid velocity. At high gas velocity, the liquid holdup is very low; liquid velocity has a great effect on liquid holdup. In contrast, liquid holdup is high at low gas velocity, while the effect of liquid velocity is relatively small, leading to a more rapid increase of liquid holdup at low liquid velocity. Fig. 4 shows the variation of calculated liquid holdup with superficial gas velocity at different superficial liquid velocities. The pressure is set to 0.1 MPa, and the superficial liquid velocity is 0.05 m/s. It is clear that the calculated liquid holdup is larger in smaller pipe diameters. The difference in liquid holdup due to pipe diameter seems to be nearly the same at different gas velocities. Fig. 5 illustrates that the calculated liquid holdup varies with the
2.4. Bubble/slug-flow transition point In this study, the drift-flux model is used to predict the bubble-slugtransition gas velocity. The general equation of the drift-flux closure relationship is expressed as: (21)
vG = C0 vm + vd.
Here, vG is the gas velocity; vm is the mixture velocity of gas and liquid; C0 is the distribution parameter; and vd is drift velocity. The gas velocity is defined as:
vG = vSG/(1 − αL ).
(22)
The mixture velocity of gas and liquid is defined as the sum of gas and liquid phase superficial velocities, (23)
vm = vSG + vSL.
In the drift-flux model, the distribution parameter and drift velocity are related to the flow pattern. In this study, Hasan and Kabir, 1988 correlation is used to predict the transition criterion of bubble and slug flow. In bubble flow, the distribution parameter C0 is equal to 1.2, and the drift velocity is given as: 0.25
vd = 1.53 ⎡ ⎢ ⎣
gσ (ρL − ρG ) ⎤ ⎥ ρL ⎦
. (24)
The transition to slug flow was experimentally verified at a liquid holdup of about αL,B−S = 0.75. Thus, the superficial gas velocity corresponding to the transition can be written as: 0.25
vSG,B−S = 0.429vSL + 0.546 ⎡ ⎢ ⎣
gσ (ρL − ρG ) ⎤ ⎥ ρL ⎦
. (25)
2.5. Model solution When the annular-churn and bubble-slug transitions have been determined, the parameters b and a in Eq. (3) can be obtained, respectively,
⎧b =
log(α L,B−S) − log(α L,A−C) log(v SG,B−S) − log(v SG,A−C)
⎨a = α b L,B−S/ vSG,B−S ⎩
. (26)
Although the transitions were established based on analytical or semi-analytical analysis, the proposed model is unable to deal with some extreme conditions at high or low gas velocities, leading to an unphysical liquid holdup. Therefore, some assumptions are added in
Fig. 3. The effect of superficial liquid velocity on liquid holdup. 4
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Brown, 1965; Gray, 1974; and Mukherjee and Brill, 1983) have smooth curves. This is because the mechanistic models are based on different assumptions at different flow patterns. Flow transition causes the discontinuities. Furthermore, they overestimate the liquid holdup at annular flow, perhaps due to some improper empirical closure relationships. Although the Gray (1974) model shows a good prediction at liquid velocities of 0.01 m/s in 50.8-mm and 101.6-mm pipes, it greatly underestimates liquid holdup at a liquid velocity of 0.05 m/s, the same as Mukherjee and Brill (1983) model. The large deviation from these empirical models might be attributed to the data collected beyond the range of experimental data for the models. Fig. 7 shows the comparison between the measured and calculated holdups. It includes all 35 experimental data points of Skopich et al. (2015). It's clear that the proposed model has better performance than other models. The new model predicted most of the data points within ± 25% error lines. As shown in Table 1, the proposed model has an average relative error (ARE) of 7.50% and an absolute average relative error (AARE) of 28.92%, both of which are minimum in all the models. The comparison between the liquid holdups measured by Liu et al. (2018) and calculated by these models is presented in Fig. 8. In the study of Liu et al. (2018), the pipe diameter is 30 mm, and the liquid superficial velocities are set as 0.01, 0.03, and 0.05 m/s, respectively. It can be found from Fig. 8 that the new model still performs best in all of these models at different liquid velocities. In Fig. 9, this model shows a good prediction of liquid holdup in most of the data points with an error under 25%. Table 2 presents the comparison results of all the models. The ARE and AARE of the new model are −5.89% and 17.41%, respectively. From the comparison between the liquid holdups measured and calculated by the proposed model, it can be found that a larger deviation occurs at lower gas velocities. It might be because the new model underestimates the uptrend of liquid holdup with gas velocity. Although the deviation seems to be dramatic in an atmospheric pressure condition, it can be inferred from Fig. 5 that the increased pressure will reduce the gas velocity corresponding to liquid film reversal and make the curve much steeper, thus causing the deviation of this model to become much smaller.
Fig. 4. The effect of pipe diameter on liquid holdup.
3.3. Model validation against field data
Fig. 5. The effect of pressure on liquid holdup.
In gas wells, only the pressure gradient can be measured. As mentioned in Section 2.1, liquid holdup is the controlling variable for accurate prediction of the pressure gradient, especially with lower gas velocities. Hence, we can evaluate the proposed model by calculating the pressure gradient against field data. In order to evaluate the performance of the proposed model at the high pressure of real gas wells, this study has collected some published pressure-gradient data from previous studies (Govier and Fogarasi, 1975; Rendeiro and Kelso, 1988; Liao, 2007), which cover low, intermediate, and high gas flow rates. The data set of Govier and Fogarasi (1975) and Rendeiro and Kelso (1988), which consist of 102 wells (94 wells available) and 50 wells, are identified to cover intermediate and high gas flow rate, with gas flow rate ranging from 1.3 to 77.6 × 104 m3/d, liquid flow rate between 1.0 and 347.8 m3/d, and wellhead pressure between 3.1 and 84.9 MPa. The data set of Liao (2007) covers low and intermediate gas flow rates, ranging from 3393 to 27,056 m3/d. Note that Liao (2007) conducted the pressure-gradient experiment in an artificial well with a depth of 970 m and measured the pressure at different depths. No liquid-loading phenomenon occurred. A total of 38 experimental tests were available in his study. The tubing head pressure of Liao (2007) experiment ranges from 0.898 to 4.658 MPa. Figs. 10 and 11 show the prediction results for the data set of Govier and Fogarasi (1975) and Rendeiro and Kelso (1988) using the proposed model and other widely used models. Fig. 10 shows that all the models have good performance in predicting bottomhole pressure. This is
superficial gas velocity in different pressure conditions. The pipe diameter is set to 50 mm, and the liquid superficial velocity is 0.05 m/s. The pressure has dramatic effect on liquid-film reversal, due to its effect on gas density. Hence, the film-reversal points are also marked with the purple dotted line in Fig. 5. It is clear that the increased pressure narrows the range between the two key points of the proposed model, of which the trending change is indicated in Fig. 5 by the solid yellow arrow. This change can lead to a much steeper trend in the liquidholdup curve. Furthermore, the curves become much closer as the pressure increases. At gas velocities above 15 m/s, it can be seen that the liquid holdup is irrelevant to gas velocities at pressures of 10 and 20 MPa. This is due to the no-slip assumption of the proposed model.
3.2. Model validation against laboratory data In this study, the experimental data was collected from the studies of Skopich et al. (2015) and Liu et al. (2018). Fig. 6 presents the liquid-holdup-model comparisons with the experimental results of Skopich et al. (2015), who measured liquid holdup at liquid velocities of 0.01 m/s and 0.05 m/s and pipe diameters of 50.8 mm and 101.6 mm. On the whole, the proposed modelin this study shows a better performance and change trend than other models. Note that the mechanistic models (Ansari et al., 1994; Zhang et al., 2003) display discontinuities, while the empirical models (Hagedorn and 5
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Fig. 6. Comparison between liquid holdups measured by Skopich et al. (2015) and calculated by the proposed and previous models at different pipe diameters and liquid velocities. (a) D = 50.8 mm; vSL = 0.01 m/s, (b) D = 50.8 mm; vSL = 0.05 m/s, (c) D = 101.6 mm; vSL = 0.01 m/s, and (d) D = 101.6 mm; vSL = 0.05 m/s. Table 1 Prediction results of these models for the experimental data of Skopich et al. (2015).
Ansari Zhang Gray Mukherjee-Brill Hagedorn-Brown Current model
Average Relative Error %
Absolute Average Relative Error %
246.99 98.14 −39.23 −87.34 31.95 7.50
247.58 103.63 46.57 87.34 58.86 28.92
because the wells in this data set have intermediate and high gas flow rates and high wellhead pressure, leading to a smaller fraction of gravitational pressure loss in the total pressure gradient. Thus, even larger deviations in the calculated liquid holdup may have a smaller effect on predicting bottomhole pressure. Overall, it is clear that the proposed model has the best agreement with measured bottomhole pressure, followed by Gray (1974) model and Ansari et al. (1994) model. All three models have ARE and AARE less than 10%, which
Fig. 7. Comparison between liquid holdups measured by Skopich et al. (2015) and calculated by the proposed and previous models.
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Fig. 8. Comparison between liquid holdups measured by Liu et al. (2018) and calculated by the proposed and previous models at different liquid velocities. (a) vSL = 0.01 m/s, (b) vSL = 0.03 m/s, and (c) vSL = 0.05 m/s. Table 2 Prediction results of these models for the experimental data of Liu et al. (2018).
Ansari Zhang Gray Mukherjee-Brill Hagedorn-Brown Current model
Average Relative Error %
Absolute Average Relative Error %
19.17 −7.84 −73.79 −90.59 29.71 −5.89
36.68 20.76 73.79 90.59 53.89 17.41
means that the proposed model is capable of dealing with the gas wells at high pressure and gas flow rates. The performance of the proposed and previous models against the data of Liao (2007) is presented in Fig. 12. This figure shows that the new model has relatively good prediction, as well as the model of Mukherjee and Brill (1983). However, the Ansari et al. (1994) and the Gray (1974) models underestimate the bottomhole pressure, although they have good performance for the data set of Govier and Fogarasi (1975). From the prediction results in Fig. 13, it can be concluded that
Fig. 9. Comparison between liquid holdups measured by Liu et al. (2018) and calculated by the proposed and previous models.
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Fig. 10. Comparison between the calculated and measured bottomhole pressure for the data set of Govier and Fogarasi (1975) and Rendeiro and Kelso (1988). (a) Mukherjee and Brill (1983), (b) Gray (1974), (c) Hagedorn and Brown (1965), (d) Ansari et al. (1994), (e) Zhang et al. (2003), and (f) Current model.
accurately predict the bottomhole pressure at different ranges of gas velocities from the data of Govier and Fogarasi (1975), Rendeiro and Kelso (1988) and Liao (2007).
the proposed model still has good prediction in low and intermediate gas flow rates with an ARE of −6.39% and an AARE of 8.67%, only lowering the accuracy of the Mukherjee and Brill (1983) model. As a whole, the proposed correlation is the only model that is able to 8
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(2) Since the superficial gas velocity and liquid holdup corresponding to the two transition points are obtained from analytical or semianalytical models, the proposed model is capable of dealing with the effect of pipe size and fluid properties with no limit on experimental range, unlike other empirical models. (3) The sensitivity analysis shows that increased pressure leads to a much steeper trend in the liquid-holdup curve. (4) By comparing the proposed model of this study with other widely used models using laboratory data and field data, it is clear that this new model preforms well at different ranges of pressures and gas velocities. Acknowledgement This work is supported by the National Natural Science Foundation of China (Key Program) (Grant No. 51534006) and the National Natural Science Foundation of China (Grant No. 51704247,51874251).
Fig. 11. Prediction results of these models for the data set of Govier and Fogarasi (1975) and Rendeiro and Kelso (1988).
Nomenclature
A AG AL a b C0 CL D FL Fi f g n p SL Si vd vm vSG vG vSG, A−C
Fig. 12. Comparison between the calculated and measured bottomhole pressure for the data set of Liao (2007).
vSG, B−S vSL z αL αL,B−S αL,A−C
αG δ ~ δ δcr ~ δcr μL ρm ρG ρL σ τi τi,CR τw
Fig. 13. Prediction results of these models for the data set of Liao (2007).
4. Conclusions In this paper, an improved power-law two-phase model is developed to predict the pressure gradient in a vertical upward two-phase flow. The following conclusions can be reached: (1) In this model, only annular/churn-flow and bubble/slug-flow transition points are required for determining the unsolved coefficient.
pipe area, m2 area of occupied by gas, m2 area of occupied by liquid, m2 coefficient, dimensionless coefficient, dimensionless distribution parameter coefficient pipe diameter, m frictional fraction force, N interfacial fraction force, N frictional fraction factor, dimensionless acceleration of gravity, m/s2 Constant pressure, Pa wetted perimeter by liquid film, m2/s wetted perimeter by gas film, m2/s drift velocity, m/s mixture velocity, m/s gas superficial velocity, m/s gas velocity, m/s gas superficial velocity of annular-churn-transition point, m/ s gas superficial velocity of bubble-slug-transition point, m/s liquid superficial velocity, m/s depth, m liquid holdup, dimensionless liquid holdup of bubble-slug-transition point, dimensionless liquid holdup of annular-churn-transition point, dimensionless void fraction, dimensionless film thickness, m dimensionless film thickness, m critical liquid-film thickness, m critical dimensionless liquid-film thickness, m liquid viscosity, Pa·s mixture density, kg/m3 gas density, kg/m3 liquid density, kg/m3 liquid surface tension, N/m interfacial shear stress, N/m2 critical interfacial shear stress, N/m2 wall shear stress, N/m2
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