Partitioning model study of the traction coefficient in a droplet model in a wellbore

Gas wells need to face the problem of gas well outflow during production, and the main way to deal with gas well fluid accumulation is to calculate the critical liquid carrying velocity and other factors by using the prediction model of excess fluid accumulation. When the prediction results of the hydrops prediction model are biased, hydrops will be generated at the bottom of the wellbore, resulting in a decrease in gas well productivity. Aiming at the problem that the drag coefficient of the wellbore droplet movement model changes greatly in the process of natural gas production, which leads to the error of the wellbore effusion prediction, the commonly used droplet models and the common drag coefficient models are analyzed and evaluated. Compared with the model obtained by the partition and the existing drag model and the experimental value, it is found that the model can effectively reduce the average error rate between the calculated results and the experimental value. The findings of this study can help for better understanding of change of critical liquid carrying velocity of droplet in different Reynolds number region, which is of reference value for the subsequent drainage and mining process.


INTRODUCTION
With the continuous development of the gas field, 1,2 the pressure in the well changes, the formation water accumulates continuously, and the rising air cannot bring the formation water out of the wellhead, [3][4][5][6] which is easy to have a great impact on natural gas exploitation: first, the wellhead pressure decreases, and the gas well output decreases 7 ; The second is the accumulation of liquid at the bottom of the well, which leads to "water invasion" and "expansion of water-absorbing clay minerals" in the gas layer, 8 and ultimately the gas phase permeability is reduced. 9,10In China, only the Sichuan Basin has more than 80% of the total gas reserves in the form of water-bearing gas reservoirs.Therefore, water production from gas wells is an inevitable problem in the production process.With the gradual development of gas production engineering, more and more water-bearing gas reservoirs have been discovered, and the study of gas-liquid two-phase flow has become important in dealing with fluid-related issues in wellbore management.The motion of liquid droplets in the wellbore is one of the main research objects in the study of gas-liquid two-phase flow in the wellbore.The reasonableness of the drag coefficient value plays a crucial role in predicting the motion state of liquid droplets when analyzing the movement of the two-phase flow in the wellbore.In view of the errors that occur in practical working conditions, this article analyzes the problems that commonly occur in past liquid droplet models, explains the importance of drag models in gas well production, and then divides the models according to the relationship between Reynolds number and drag coefficient.The optimal model is selected, and the experimental data in the literature are used to verify the model in this article.This divided drag model can make the prediction results of the liquid droplet accumulation model in the wellbore more accurate, and provide theoretical guidance and reference for the production planning of water-producing gas wells.
The research method in this paper mainly analyzes the existing liquid accumulation model and the commonly used drag coefficient fitting method, selects the applicable interval of each drag coefficient calculation method, integrates it, and collects experimental data by referring to literature, so as to verify the calculation accuracy of the model proposed in this paper.
The innovation of this paper is the zoning thinking used to calculate the drag coefficient of the droplet model in the wellbore.Different from the practice of using the drag coefficient as a fixed value in the previous liquid accumulation model, or the current two-phase flow model to fit the drag coefficient in a small Reynolds number region, the drag coefficient in the range of 0 ∼ 10 6 is discussed in this paper.Compared with other models, the proposed partition model has better adaptability in larger Reynolds number intervals.At present, the limitation of this paper is that the lack of mathematical theory leads to the excessive complexity of the partition model, which cannot reflect the relationship between the drag coefficient of the drop and the Reynolds number.

Droplet model and traction coefficient
Droplet model is an important research object for gas-liquid two-phase flow in the wellbore.Through the analysis of gas-liquid two-phase flow in the wellbore, a droplet model is built to judge the liquid accumulation at the bottom of the well, which is of reference value for production planning.There are two sources of liquid in gas wells: one is free water or hydrocarbon condensate in the formation seeping into the wellbore together with gas, and the presence of liquid affects the flow characteristics of gas wells; the other is natural gas containing water vapor in the formation flowing into the wellbore, and condensate appears due to heat loss causing the temperature to gradually drop along the wellbore.Numerous scholars have optimized the build-up model of liquid droplets, such as the Turner model built by Turner in 1969, 11 the ellipsoidal model optimized by Li Min in 2001, 12 and the spherical cap model by Wang Yizhong in 2007 13 are the commonly used liquid droplet build-up models at present.At present, the accumulation model mainly used in each gas field considers that the droplet will be suspended in the wellbore when the gravity of the droplet is balanced with the buoyancy and traction forces applied to it.If the airflow can carry the largest droplet in the gas well out of the wellhead, the wellbore can continuously carry liquid production.Zhang Liehui et al. 14 summarized the critical liquid-carrying flow rate equation of the commonly used liquid droplet accumulation model as: where: C D -traction coefficient, dimensionless parameters; g-gravitational acceleration, m/s 2 ; v-critical fluid velocity, m/s; -liquid surface tension, N/m;  l -liquid phase density, kg/m 3 ;  g -gas phase density, kg/m 3 .In the same case, the density of the gas-liquid phase is fixed, therefore, it can be seen from Equation (1) that the value of the traction coefficient will greatly affect the calculation results of the critical liquid-carrying flow rate, and the critical liquid-carrying flow rate is inversely proportional to the size of the traction coefficient.
The traction coefficient is related to the deformation coefficient and internal flow, and the Eotvos number is usually used in this case Eo (=Δgd eq 2 /), Reynolds number Re (=ΔVd eq / l ), Weber number We (=ΔV 2 d eq /), Morton number Mo (=Δg l 4 / l 2  3 ) to represent the influence of droplets on the traction force during the ascent process, but the current technology cannot observe the four variables simultaneously; therefore, the traction coefficient is regarded as a function of Reynolds number in the vast majority of cases.In the past, scholars conducted a large number of experiments to obtain this functional relationship, and a total of 606 groups of smooth spherical correspondence were obtained.][18][19][20] According to the magnitude of Reynolds number in the CD-Re data point diagram in Figure 1, the curve can be divided into three parts: laminar flow zone, transition zone and turbulent flow zone.Laminar flow is a flow state in which the fluid flows in a tube with its masses moving in a smooth straight line in a direction parallel to the tube axis; turbulent flow is a fluid micro-cluster motion with randomness.The essential difference between laminar flow and turbulent flow is that there is no radial pulsation between fluid particles in the former.The latter has radial pulsation, and the greater the degree of turbulence, the greater the radial pulsation.On the other hand, the greater the flow rate, the greater the degree of turbulence, the thinner the boundary layer thickness, the smaller the heat transfer resistance.
According to the actual production data, the fluid Reynolds number during gas well production is usually greater than 1000, 11 while the traction coefficient in this part does not remain constant, and the relationship between Reynolds number and traction coefficient is significantly different between the parts 1000 < Re < 2 × 10 5 and 2 × 10 5 < Re < 10 6 , so it is necessary to divide this part into turbulent and highly turbulent zones, which makes it possible to study the The relationship between Reynolds number and traction coefficient is more accurate when studying the relationship between Reynolds number and traction coefficient in the wellbore.According to the characteristics of the curve, the Reynolds relationship diagram of the traction coefficient was finally divided into four regions: laminar flow region (Re < 1), transition region (1 < Re < 1000), turbulent region (1000 < Re < 2 × 10 5 ), and highly turbulent region (2 × 10 5 < Re < 10 6 ).

Gas well fluid accumulation model study
In the past, Turner's model, Li Min's model and many other models referred to the relationship diagram in Figure 1 when selecting the traction coefficient, which was taken as an approximation of the flow region as well as the highly turbulent region, and multiplied by a correction value according to the droplet shape, as shown in Table 1.
The existing droplet traction models are obtained by fitting previous experimental values in various ways, and the commonly used ones are mainly the Clift model validated by Brown and Lawler, 21 the GP model, 22 the Flemmer & Banks model, 23 the Haider & Levenspiel model, 18 the Khan & Richardson model, 24 the Brauer model, 25 and the Shao Mingwang model, 26 and so forth.The applicability of these models varies, and a comparative analysis and evaluation of each model is required.Aspects of comparative model studies.
1.The basis for judging whether a gas well with a low water-to-gas ratio can rely on its own ability to carry liquids continuously.
The Yang Chuandong model, the Turner model, the Li Min model, and the ball cap model are based on the liquid to gas ratio, but they differ only in the limits of judgment.as the lower limit for classifying fog flow, is not representative of the universality of gas field development technology, and according to the different geological conditions of gas fields in the Surig South area and the inability to single calculate the single water production of each gas well, it is also not suitable to use the kinetic energy factor method as the basis for judging gas well development.
2. In the application of continuous liquid-carrying model for gas wells with low water-to-gas ratio, the model for liquid droplets (i.e., resistance factor) is selected.
The Turner model, the Li Min model and the ball cap model use spherical, ellipsoidal and conical shapes as the droplet models, respectively, that is, they use different resistance factor values, which are actually improvements of the spherical model (Table 2).And the drag coefficient value is selected in the special case of standard drag coefficient, which can be understood more clearly according to the currently published information, that is, the following Figure 2.
Due to the limitations of experimental conditions and techniques, the curve drawn by experimental data has not been applied to the actual technical research.The reasons for this phenomenon are as follows: a.It is caused by the error of the drag coefficient.No matter what model is assumed in the calculation, in the real case, the liquid particles formed after the liquid is broken are a variety of small regular strip, ribbon liquid units.Therefore, the diversity of droplet shapes affects the accuracy of the drag coefficient, leading to the fact that various current critical liquid carrying flow models are only suitable for some cases rather than applicable in all cases.b.Caused by particle population effect.In the calculation, we only consider that there is one particle in the flow field.In practice, the liquid particles are broken into a large number of very small liquid droplets under the combined action of aerodynamic force and surface tension, and then the small droplets that fall off the droplets are mixed with the larger droplets.The whole atomization field is composed of liquid particles with different sizes.The number of particles entering the air flow after the droplet is broken is very large, and the movement of such particle groups in the flow field is much more complex than that of a single particle.Most particle group resistance coefficient formulas show that particle group effect will lead to the increase of the resistance coefficient.Therefore, the assumption of a single particle will also lead to a decrease in the possibility of drag coefficient, an increase in the critical flow velocity of droplets, and a larger critical liquid carrying flow rate.c.It is caused by the assumption that liquid particles will not be broken in motion in the numerical calculation of these models.Therefore, the assumption that the liquid particles will not be broken during the movement will also make the maximum velocity calculated by numerical calculation inconsistent with the actual situation.

TA B L E 3
Calculation methods of drag coefficients of different particle shapes and Re.

Scope of Re application
Spherical

99
Note: : Bed porosity, C D,0 : Drag coefficient of a single particle, : The sphericity of a particle.
3. The selection of pressure.
The purpose of using gas well to carry liquid is to use the energy of the gas to move the droplets from the bottom of the well to the wellhead.So the wellhead pressure should be selected when studying the critical liquid carrying flow of gas well.
Although the research on the resistance coefficient of a single particle has been relatively complete, the research on the resistance coefficient of particle groups under various conditions (different particle shapes, different particle concentrations, etc.) and considering the interaction between particles basically began in the 20th century.The resistance coefficient of spherical particle group and that of non-spherical particle group are widely used.The application conditions and expressions of the two are shown in Table 3.
The following is a comparative study on the accuracy of the resistance coefficient of spherical particle group and that of non-spherical particle group, that is, the droplet shape is assumed to be spherical (sphericity is 1) and nearly ellipsoidal (sphericity is 0.7) respectively, and the bed porosity in the laboratory can be regarded as 1 in the gas well.The comparison results are shown in Table 4.
After analyzing the above data, it can be clearly seen that the spherical particle group model is more accurate than the laboratory model, so the spherical particle group model is selected for the calculation of the resistance coefficient of the spherical particle group below.

DROPLET DRAG FORCE MODEL ANALYSIS IN WELLBORE
When Turner et al. 17 constructed the droplet accumulation model, considering that in actual gas well production, wellbore fluid Reynolds number was 1 × 10 4 ∼ 2 × 10 5 , the drag coefficient of Turner model was 0.44.After calculation, 14 scholars found that the actual critical liquid carrying capacity was greater than the theoretical critical liquid carrying capacity of the model.Therefore, in consideration of safety, the coefficient of 1.2 is added to the model.Guo believes that this is caused by the critical stress state.When the object is in force equilibrium, its motion state is static or uniform, that is, the droplet cannot judge its motion state under the condition of force equilibrium. 27Therefore, he added a flow coefficient of 1.2 on the basis of Turner model.According to the calculation formula of critical velocity in Equation 1, the critical liquid carrying capacity is inversely proportional to the drag coefficient.In the same volume, spherical projection area is also the smallest, 28 and according to Table 1, the drag coefficient of this kind of droplets is the smallest, and spherical droplets are the most difficult to be taken out.When spherical droplets can be taken out of the wellbore, it is difficult to generate liquid accumulation in the wellbore.Therefore, the accumulation model spherical droplets as the research object.
In the practical application of Turner model, the manufacturer found that the calculation result of Turner model was often larger than the actual working condition.For example, when Turner model was used in the early gas field production of our country, it was found that the critical flow obtained is higher than the actual value.So we took 1/3 of the Turner calculation result as the applied value.In 1991, Coleman used Turner to calculate the fluid accumulation in a gas well with wellhead flow lower than 500 psi, and found that not using the safety factor could better conform to the working condition, 25 and the cause of this phenomenon could not be explained by the traditional reversal model force analysis.Although Guo et al. 27 tried to explain the unreasonable correction from the perspective of force and movement, it was difficult to reach a complete conclusion.The reason is that in the Reynolds number range commonly used in gas well production, the drag coefficient in turbulent zone did not change significantly, but the drag coefficient in highly turbulent zone varies greatly with the Reynolds number.If the drag coefficient is fixed in this area and supplemented by the correction value, the calculated results are often difficult to agree with the actual situation.
Therefore, in order to make the calculated results consistent with the actual situation, scholars began to study the accurate solution method of drag coefficient.In 2019, Pan Jie compared the global accuracy of various commonly used models, as shown in Figure 3.By calculating the error values of each model and comparing them with the commonly used experimental values of drag force, he concluded that Brauer model had the highest global accuracy in the turbulent region. 29Considering the physical property difference between the rigid sphere and the droplet, Panje multiplied the model by 1.2 and applied it to the droplet, namely: According to the data of model comparison by Pan Jie et al., it can be seen that Shao Mingwang model is suitable for non-turbulent region in many cases.In this comparison, when applied to gas well experimental data, the accuracy of the calculation results can be maintained at a low Reynolds number, but the accuracy of the calculation results decreases rapidly after entering the turbulent zone.
With the increasing of Reynolds number, the calculated results of this model always remain between 0.40 and 0.41 in the highly turbulent region, which well explains the reason why the drag coefficient of the conventional spherical model is usually around 0.4.However, after many years of actual data verification, this coefficient has a large difference with the  experimental value, so it is difficult to conform to the actual situation.Therefore, common models were compared in the highly turbulent region, as shown in Figure 4.
As can be seen from the figure above, there is a large gap between other models and data points except GP model.When establishing GP model, Barati focused on studying the variation law of drag coefficient in high turbulence region, and used logarithm data points of hyperbolic tangent function to fit, so as to improve the accuracy of this region.The model is applicable to 2 × 10 5 ∼ 10 6 .
The drag coefficient of the above droplet inversion model is usually fixed.However, in the production of gas Wells with high gas-water ratio, the Reynolds number of gas-liquid two-phase flow is usually greater than 1000, and the drag coefficient of highly turbulent section in this region changes greatly.Therefore, the method of taking a fixed value and adding a correction coefficient in the past liquid accumulation model is difficult to apply to each working condition.However, when the drag force model is used to calculate the drag force coefficient, due to the differences in the fitting methods of each model, the application conditions of each model are different, and the same model cannot be applied to the flow modes with different Reynolds numbers.

Establishment of zone drag coefficient model
Based on the above analysis of the existing whole-domain fitting value methods of drag coefficients, it is suggested in this paper to adopt the partition model for calculation according to the applicable conditions of each model under the premise that the calculation methods of drag models in different Reynolds number regions cannot be applied at present.With the increase of Reynolds number, the accuracy of Shao Mingwang model decreases rapidly, which is inconsistent with the experimental value.However, this model has a high accuracy in laminar flow region and transition region.Therefore, this paper adopts Shao Mingwang model in the calculation of laminar flow region and transition region.According to the experimental analysis of Pan Jie et al., 30 the Barati model has a high accuracy in the turbulent region, and this paper also verifies that the model has better applicability than other models in the turbulent region after calculation, so this paper adopts the Brauer model in the turbulent region.The fitting method of GP model can better target the change of Reynolds number in the highly turbulent region.Therefore, in the highly turbulent region, the fitting method of GP model is adopted in this paper to make it more accurate in this region.By referring to the evaluation of common droplet drag force models in various literatures, it is found that global fitting is difficult to satisfy the relationship between Reynolds number and drag coefficient under four flow modes at the same time, mainly because the fitting calculation method and focus point adopted by each model are different.Table 2 shows the model obtained by the partition optimization in this paper.The most appropriate droplet model is adopted in each Reynolds region to make the calculated results closer to the experimental values.both sides of the droplet caused by the actual production cylinder airflow, this paper suggests that the model should also add a correction value according to the droplet deformation in practical application, so that the model can accurately reflect the corresponding droplet drag coefficient of each region and reasonably predict the liquid accumulation condition.
The partitioning model in this paper is shown in Table 5.

Method of partition drag force
This paper compares the zoned drag coefficient model with other non-zoned drag coefficient models, and verifies the feasibility and superiority of this zoned model according to the literature. 29Table 6 shows the calculated values of each model in the non-highly turbulent region.
For the non-highly turbulent region, the average error value of the Brauer model is 8.83%, the average error value of the Shao Mingwang model is 18.41%, and the average error value of the proposed model is 6.17%.Therefore, this paper is superior in this partition.
For highly turbulent areas, according to the literature, 16 as shown in Table 7, the average error rate of Brauer model is 170.76%, the average error rate of Clift model is 201.58%, and the average error rate of Shao Mingwang model is 190.9%.Only GP model accords with the experimental value, and the average error rate is 5.83%.The other models obviously do not conform to the highly turbulent region, so the GP model is the most accurate in this region.
In summary, in the laminar flow area, Shao Mingwang area has higher accuracy; In the transition region and turbulent region, the error of Brauer model is small.The GP model adopts a fitting method for highly turbulent areas, so it has high accuracy in this area.Using the partition method in this paper can make the calculated value maintain accuracy in more intervals.

SUMMARY AND CONCLUSION
In this paper, the method of the drag force system of the existing fluid model is analyzed, and the calculation and verification of the fitting method of the drag force is obtained, and the following conclusion is obtained.Each commonly used model according to the shape of the droplet and its force analysis to determine the drag coefficient of the droplet, supplemented by the correction value, the main shape of each model is round spherical droplet, ellipsoidal droplet and spherical cap droplet, the drag coefficient increases in turn, because the critical liquid carrying flow rate is inversely proportional to the drag coefficient, so the maximum critical flow rate in the wellbore should be used as the research object.
The relationship between Reynolds number and drag coefficient in the high turbulence region is quite different from that in the turbulent region, so it is difficult to accurately calculate the drag coefficient through a fitting model.It is necessary to calculate and analyze the applicable scope of each model.Based on the analysis of the whole paper, after summarizing and sorting out the commonly used drag force models, by comparing the application scope of each model, it is suggested to use Shao Ming Wang model suitable for laminar flow area and transition area, Brauer model suitable for turbulent flow area, GP model suitable for highly turbulent area to calculate the drag coefficient in the partition, and obtain the partition drag force model with wider application scope and higher accuracy.Compared with the single fitting model, the results obtained after partition calculation can better reflect the relationship between Reynolds number and drag coefficient.

2
Comparison of standard and resistance coefficient curves.

F I G U R E 3
Comparison of common drag models.

F I G U R E 4
Schematic diagram of the high turbulence zone of each model.
TA B L E 1 Comparison of domestic and foreign gas well fluid-carrying model.
TA B L E 2s Z s P Comparative study on the accuracy of the resistance coefficient of spherical particle group and non-spherical particle group.
TA B L E 4

model Gp model Reynolds number Experimental value Calculated values Absolute error rate Calculated value Absolute error rate Calculated value Absolute error rate Calculated value
TA B L E 6 compares the non-altitude turbulence area.common models are compared in high turbulence area error.