Study on improving liquid carrying performance of annular jet pump gas well with static mixer

In the process of natural gas extraction, the phenomenon of liquid loading will affect the efficiency of gas well extraction and reduce the life of the well. Compared with conventional drainage gas extraction technology, the jet pump can not only reduce the bottom back pressure and ensure the stable production of gas reservoirs but also promote the final recovery rate. Since the jet pump relies on the interaction between fluid particles to transfer energy, the energy loss is large and the efficiency is low. To maximize the advantages of the gas‐driven jet pump, this study innovatively combines a static mixer with an annular jet pump. Utilizing the cyclonic effect produced by the static mixer, the original gas‐liquid axial motion is transformed into a stronger vortex motion, and the liquid droplets are changed into a liquid film that is easier to carry, which significantly improves the discharge efficiency of the jet pump. This study uses a combination of numerical simulation and experimental analysis to compare the associated effects of the new annular jet pump (NAJP) and the conventional annular jet pump (CAJP) on the liquid‐carrying performance of gas wells in terms of cyclonic effect, droplet breakage ratio, and pump efficiency. The results show that, compared with CAJP, NAJP increases the mass flow rate of the sucked fluid. The droplet breakage ratio increases by 15.4% year‐on‐year, while the critical liquid‐carrying flow rate is reduced by about 10.7%, resulting in a maximum pumping efficiency of 37%, an increase of about 30.7% year‐on‐year. At the same time, the reduction of the energy coefficient means lower energy consumption. In summary, NAJP is better than CAJP in terms of liquid‐carrying effect and efficiency.


| INTRODUCTION
Due to the gradual reduction of formation pressure, the selfinjection capacity of gas wells decreases significantly in the late stage of extraction.2][3] Gas lift technology is achieved by injecting high-pressure gas into the well from the surface and using the expansion effect of high-pressure gas to fully mix the gas-liquid, thus lifting the downhole liquid to the surface for liquid drainage production. 4,5The flow regime of gas-liquid two-phase flow in the wellbore is shown in Figure 1.When the gas content is low, a bubble flow will be formed and the gas will be dispersed in the continuous liquid phase, and when the gas content increases, the wellbore appears to have a slug flow or an agitated flow where the gas and liquid move alternately.Continuing to increase the gas content, the gas and liquid will form an annular flow, with some of the droplets dispersed in the gas and the other part forming a liquid film on the tubing wall and climbing upward.
The gas-driven jet pump combines gas lift and jet pump technology.The high-pressure gas flowing through the annular space of the oil jacket generates a high-speed jet at the nozzle outlet, which breaks up the large liquid mass and enters the throat.Field applications proved that gas-driven jet pumps can reduce the bottom flow pressure, improve the lifting effect, increase gas production, and extend the production cycle of gas wells. 6Gas-driven jet pumps can be divided into center jet pumps and annular jet pumps by the way the gas flow enters the jet pump.Compared with the center jet pump, the annular jet pump suction pipe is placed in the middle and the annular nozzle surrounds the outside of the suction pipe.It is characterized by high stability and simple maintenance 7 and can be used to reduce the differential pressure at the bottom of the well 8 and to pump the liquid loading. 9urrent research on annular jet pumps has focused on the effects of structural parameters such as throat length, area ratio, 10 nozzle diameter, 11 and suction chamber angle 12 on efficiency and performance.Xu et al. 13 found that the area ratio is the key factor affecting the efficiency of jet pumps through simulation studies, and the mixing effect at the outlet of the suction chamber can be improved by reducing the area ratio.Wang et al. 14 designed the conical suction chamber and diffusion chamber of the annular jet pump to be streamlined, which can improve the jet pump efficiency.Morrall et al. 15 studied annular jet pumps with multiple nozzles and showed that the performance and efficiency of the jet pump were improved when the diameter of the annular nozzle was increased to 3.5 mm.Asfora et al. 16 found through their study that the length of the throat affects the degree of gas-liquid mixing.When complete mixing occurs at the throat position, the higher the flow rate of the sucked fluid, the higher the efficiency of the jet pump.
The process of the annular jet pumps pumping lowvelocity fluids through high-velocity jets is a flow mixing under strong shear conditions.The degree of mixing between the working fluid and the sucked fluid has a significant impact on the performance of the jet pump. 17he liquid-gas jet mixes violently with the gas phase, forming a large contact area between the gas and liquid phases, which facilitates interphase mass transfer. 18,19herefore, improving the gas-liquid mixing capacity of the annular jet pump has a positive impact on improving the performance of the gas-driven jet pump.
Static mixers can improve the degree of mixing of gasliquid two-phase fluid in the pipe. 20,21The fluid flow through the static mixer will go through the process of splitting, position shifting, and remixing.In addition, the gas-liquid two-phase flow along the central position of the tube will generate a strong vortex, there is a strong shear force on the fluid, and the liquid fine part is further split thus enhancing the gas-liquid mixing capacity and improving the efficiency. 22This study innovatively combines a static mixer with the gas-driven annular jet pump to design a new jet pump.The effect of the mixer on the liquid-carrying performance of the annular jet pump was analyzed by comparing experiments and numerical simulations.The results show that NAJP can remarkably improve the flow rate of the sucked fluid, improve the pumping efficiency, and have a significant impact on the centrifugal velocity, droplets size, and critical liquid-carrying flow rate.

| Model geometry
NAJP consists of two parts: the annular jet pump and the static mixer.The annular jet pump consists of an annular nozzle, suction chamber, throat, and diffuser tube.Lyu et al. 23 investigated the optimal combination of efficiencies for jet pumps with different configurations by varying the parameters such as area ratio, suction chamber angle, and diffuser angle, resulting in an efficiency of 35.62% when the area ratio is 1.75.Zeng et al. 24 combined experimental data to seek the largest combination of structural dimensions for the efficiency of the annular jet pump.The results showed that the test efficiency was 36.3% when the flow ratio was 0.6, the suction chamber angle was 15°, the relative throat length was 2.45, and the diffuser angle was 4°.By comparing the above studies and combining with the study of Xiao et al. 25 for different suction angles on the development process of the reflux region, the final selected structure of the annular jet pump is shown in Figure 2 and the dimensions are shown in Table 1.
The static mixer is composed of a spiral plate twisted at a certain angle.To further enhance the gas-liquid mixing effect, the mixer will be mounted as a welded section of the L s at the inlet of sucked fluid.The main parameters are the twist angle, aspect ratio, and vane thickness.The twist angle is taken as 180°according to the standard single-lobe Kenicstype static mixer Kumar et al. 26 The aspect ratio is selected according to the size of the inlet of the sucked fluid by the annular jet pump, and the vane thickness is taken as 1 mm based on the thin-walled assumption of Song et al. 27 The specific dimensions of the static mixer are shown in Figure 3, and the three-dimensional structure of NAJP is shown in Figure 4. F I G U R E 4 Geometric profile of new annular jet pump.

| Material
All solid surfaces in the model use the default materials with constant temperature and no-slip boundary conditions, specific fluid material properties are listed in Table 2.

| MATHEMATICAL MODELS
The following assumptions are made in this study for the gas-liquid two-phase flow problem: (1) The entire flow process is assumed to be adiabatic, and the effect of temperature on the gas-liquid two-phase flow is not considered.When the velocity and pressure distribution characteristics of gasliquid two-phase flow are examined, the two-phase flow is treated as a continuous medium.However, when the droplet fragmentation law is studied, it is assumed that the gas is a continuous phase and the liquid is a dispersed phase.Without considering the phase transition, the pressure and velocity are solved only by the equations of conservation of mass and momentum of the two phases.(2) Since the velocity of gas-liquid two-phase flow is not high and the density of each phase remains constant, the two-phase flow is regarded as incompressible flow.

| Eulerian model
Gas-liquid two-phase continuity equation, Momentum equation, where n represents the phase (n = g for gas phase, n = l for liquid phase, n = m for gas-liquid mixing), α n is the volume fraction of each phase, u n is the velocity of each phase, ρ n is the density of each phase, p n is the pressure of each phase, μ n is the dynamic viscosity coefficient, g is the acceleration of gravity and F is the external volume force.

| Turbulence model
Yang et al. 28 studied the effects of three models of k-ε, k-ω, and Reynolds Stress (RSM) on the internal flow field of jet pumps, and concluded that the combination of Realizable k-ε (RKE) and Standard Wall Functions (SWF) can obtain more accurate performance prediction and internal flow field details.In addition, RKE can maintain the consistency of the Reynolds stress with the turbulence, and can more accurately simulate the diffusion velocity of circular jets.0][31] Therefore, combined with the internal flow characteristics of NAJP, this study chooses to use the RKE turbulence model for simulation.The equations of turbulent kinetic energy and turbulent dissipation rate of this turbulence model can be expressed as, where u n,i is the different directional velocities of each phase, G n,k is the turbulent kinetic energy k generated by the average velocity gradient of each phase, C n,1ε , C n,2ε are the model constants, σ n,k , σ n,ε is the Prandtl number for k and ε respectively and μ t is the turbulent viscosity.

| The droplet breakage and critical liquid-carrying flow rate model
Droplets are subjected to gas phase traction during transport inside the jet pump and are further fragmented when they flow through the throat or collide with the jet pump wall.| 73 The droplet size has a great influence on the liquid-carrying performance of the gas well, and the specific droplet breakage model can be expressed as follows, where F, K, d, and m are the coefficients of the equation, derived from the Taylor analogy, where r is the radius of the droplet before deformation, μ l is the droplet viscosity, C F , C K , C d is a dimensionless constant derived from experiments and theory and is taken as 32 A significant theory to determine whether droplets can be carried out of the wellhead by the gas is the maximum droplet theory, which states that no liquid loading will occur at the bottom of the well as long as the gas can carry the largest diameter droplet to the surface. 33The minimum gas velocity that lifts the largest diameter droplet to the ground is called the critical liquid-carrying flow rate, based on Newton's second law and considering only the force in the vertical direction (Equations 9 and 10), when F D ≥ F G , the droplet can be carried to the ground, and the critical liquid-carrying flow rate of the gas can be obtained by taking the traction coefficient C D as 0.44. 34 The above equation is only for the analysis of the critical liquid-carrying flow rate in vertical pipes.When the pipe is installed with a vortex tool, the droplets will also be affected by the centrifugal force.The magnitude of the centrifugal force is shown in Equation (12).Ali 35 pointed out that the direction of the centrifugal force is away from the axis and the angle with the horizontal direction is equal to the cyclonic angle of the vortex tool.Analogous to the cyclonic angle of the vortex device, the cyclonic angle of the static mixer takes arctan (AR) as 45°, u θ is the tangential velocity, the combined force equation becomes where u t = u cf sinθ, u θ = u cf cosθ, u cf is the velocity after passing through the static mixer, then the critical liquidcarrying flow rate in the cyclonic field is the below equation.

| Jet pump efficiency
Define the dimensionless parameter mass flow ratio, which represents the ratio of the sucked fluid mass flow rate to the power gas mass flow rate (Equation 15).The dimensionless parameter head ratio represents the energy obtained by the sucked fluid to the energy consumed by the power gas (Equation 16).The parameter η represents the efficiency of the jet pump (Equation 17), Q m indicates mixed fluid volume flow rate, ρ m indicates mixed fluid density, Q g indicates power gas volume flow rate, ρ g indicates power gas density, H 3 indicates mixed fluid head at the outlet of the jet pump, H 2 indicates sucked fluid inlet head, H 1 indicates power gas inlet head.
The calculation of the head is derived from the variation form of Bernoulli's equation in Equations ( 18) and (19), which indicate the sum of the pressure head, dynamic head, and position head.0 indicates a certain overflow section of the annular jet pump. 1, 2, and 3 represent the power gas inlet, sucked fluid inlet, and outlet of the jet pump, respectively.The conditions of Equations ( 15)-( 17) are P 1 > P 3 > P 2 , indicating that the total pressure of the power gas inlet is greater than the outlet and the outlet pressure is greater than the sucked fluid inlet.
The drainage gas extraction method is accomplished by injecting high-pressure gas into a jet pump via a ground pump, where the pressure loss generated by the high-pressure gas is used to carry the sucked fluid and increase its head.Introducing the PEI (Pump Energy Indicator), PEI responds to the energy consumed by the pump to carry each cubic meter of fluid to produce a head, the larger the PEI value indicates that more energy is consumed to lift the same amount of fluid, the lower the energy efficiency, 36 (Equation 20), where W indicates the power of the ground pump, Q 2 indicates the flow rate of the sucked fluid, and H 3 indicates the outlet head.

| NUMERICAL SIMULATION AND EXPERIMENTAL VERIFICATION 4.1 | Meshes division
The annular jet pump has a large pipe diameter variation at the suction chamber and throat connection, and the static mixer also has a major distortion.When fitting boundary regions with large deformations, the model is delineated using a tetrahedral unstructured mesh since tetrahedral meshes are more advantageous than hexahedral meshes.The meshing of NAJP is shown in Figure 5.
To exclude the influence of the number of meshes on the calculation results requires checking the independence of the mesh because a reasonable number of meshes can make the calculation results accurate and improve efficiency.The total number of meshes in the computational domain model is controlled by adjusting the mesh size, and the mesh independence is verified by taking the volume fraction of water as the index.Last, the total number of meshes for CAJP is 0.83 million (Figure 6A) while the NAJP is 2.4 million (Figure 6B) after comprehensive consideration based on the results obtained from the simulation.

| Multiphase flow model setup
For CAJP, the power gas inlet is named inlet-gas, the sucked fluid inlet is named inlet-water, the pump outlet is named outlet, and the rest of the pump faces are named wall.NAJP performs the same naming, with the difference that the static mixer faces are also named wall.
Since the Eulerian model can calculate the gas-liquid mixed flow more accurately than the mixing model, the Eulerian model is selected for the multiphase flow model the RKE model is selected for the turbulent flow model, and the SWF method is used.
The model is established along the x-axis, representing the vertical direction of the pump, and the acceleration is −9.81 m/s 2 .The initial operating parameters of the annular jet pump are obtained from the engineering example solution, as shown in Table 3.
To obtain the actual flow rate value of the downhole gas and liquid, the actual flow rate at the site needs to be transformed by solving the actual downhole gas density (Equation 21) and introducing the dimensionless volume coefficient (Equation 22).B g is the ratio of the actual downhole gas volume to the standard atmosphere at a wellhead temperature of 20°C, where ρ 0 indicates the relative density of natural gas, Z indicates the actual gas compression coefficient, T is the actual downhole temperature, P is the downhole pressure, Z sc indicates the gas compression coefficient in the standard condition, T sc is the temperature in the standard condition, P sc is the standard atmospheric pressure.
The standard condition wellhead T sc = 20°C (293.15K), P sc = 0.101325 MPa.Calculated by increasing the formation temperature by 4°C for every 100 m increase, the downhole temperature T = 51.2°C(324.35K).The compression coefficients were taken as Z sc = 0.99, Z = 0.92, and the volume coefficient was calculated as B g = 0.017.
Power gas flow, q q B = .
Sucked fluid velocity, Water volume fraction, The initial value of the water volume fraction is set to 10% as shown by the above Equations ( 22)- (25).The velocity value can be derived from the actual downhole gas-liquid mixing flow rate, so the inlet velocity of the sucked fluid is set to be 9 m/s.Considering the effect of well depth on pressure, the operating pressure is set to 6 MPa because the liquid loading has been produced and the flow pressure is slightly lower than the normal formation pressure.The power gas inlet is a pressure inlet, which is 6.5 MPa, and the outlet is set as a pressure outlet.
The whole simulation is based on the transient solution, and the Phase Coupled SIMPLE format is chosen to solve the pressure-velocity coupling, and the Second Order Upwind format is used for the momentum, turbulent kinetic energy, and turbulent dissipation rate, and the Quick format is used for the volume fraction, taking into account the accuracy and stability of the solution.
Before running the calculations, the time for the fluid to pass through the jet pump is initially estimated, setting the step size to 0.01 s and the total number of steps to 200, making the time greater than the estimated value to ensure the accuracy of the calculations.The residual value is set to 10 −4 and the results are considered to converge when all values are below 10 −4 and the monitored values have stabilized.

| Discrete phase model setup
The discrete phase model (DPM) setup is similar to the multiphase flow model, and the final result is the droplet distribution of different particle sizes obtained from the model.For this model, the action between continuous and discrete phases is considered, in addition to which the droplet may be subjected to Saffman lift F SL , virtual mass force F VM , and pressure gradient force F PG , as shown in Figure 7.Although the three forces may be exerted on the droplet, their effect on the droplet is minimal compared to the gravity or drag force. 37The droplet injection time is 0.5 s, and the inlet and outlet conditions are set to escape, while all walls are set to trap conditions.The distribution of droplet mass fraction at the sucked fluid inlet is assumed to obey a normal distribution with a particle size distribution ranging from 10 to 70 μm, as shown in Table 4.In addition, to obtain the average droplet size as well as the diffusion coefficient it is also necessary to calculate the cumulative percentage of the droplet mass fraction, as shown in Table 5.
Using the Rosin-Rammler distribution in DPM (Equation 26), where d l is the droplet size, d ̅ is the average droplet size.When d l = d ̅ , the average particle size can be determined by finding the range corresponding to D = 0.368, which obviously falls within the range of 40-50 μm particle sizes, and d ̅ = 44.4 by interpolation.
where s is the diffusion coefficient, which can be calculated by Equation ( 27) as shown in Table 5.To find the average value s̅ = 3.782.

| Gas-liquid two-phase flow experiment
To verify the reliability of the simulation results, the gasliquid two-phase flow experiment was designed, and the specific experimental system flow is shown in Figure 8.
The main devices are CAJP, NAJP, gas and water supply systems, a console, and connecting pipes.At the same time, the velocity of the sucked fluid outlet and the jet pump pressure were measured and combined as a criterion.
The working condition parameters selected for the experiment are consistent with the simulation and the process is described as follows: (1) Connected the experimental devices according to the flow chart, started the gas compressor 1, and then opened valves 4 and 8, so that the gas filled the jet pump to remove the impurities and droplets that may exist in the annular jet pump.| 77 14 to inject the mixer 18 with water.After that, the opening of valve 14 was adjusted to bring the flow meter 15 to the set value.Finally, the valve 8 was gradually adjusted upward so that the velocity of the flow meter 19 was displayed as 9 m/s.The mixer 18 served to fully mix the gas and water.(3) Reopened valve 4 and kept it at a smaller opening.
After the system had stabilized, gradually increased the gas volume so that pressure gauge 7 reached the set value of 6.5 MPa.(4) After the whole system had stabilized in operation, the velocity detection device 21 was activated and then the radial position of the probe was adjusted to record velocities.(5) After a set of data had been completed, the inlet velocity of sucked fluid could be changed by regulating valves 8 and 14.Finally, replaced NAJP and repeated the above steps after the test of CAJP was completed.(6) At the end of the experiment, the piston pump 13 and valve 14 were closed.Closed the compressor and valves after maintaining the ventilated state for some time.The mixture at the jet pump outlet could be separated and reused through separator 23 in the experiment, and the separated gas would need to go through the process of secondary separation, drying, and compression, and finally stored in gas canister 24.Pressure gauges 3, 11, and 17 were used to detect pipe pressure to provide protection.
Figure 9 shows the comparison of the experimental and simulated values of velocity and pressure drop for CAJP and NAJP.It can be concluded that the same points of both are: with the increasing velocity of the sucked fluid, the jet pump sucked outlet velocity is on the rise.In the case of constant inlet pressure of power gas, the outlet mixture can obtain more energy and the pressure drop tends to decrease.The difference is that the pressure drop of CAJP is smaller than that of NAJP, while a smaller flow velocity will be obtained.On the contrary, NAJP will obtain a larger pressure drop but will increase the flow velocity.It is noteworthy that the experimental values of both velocities are smaller than the simulated values, and the pressure drop is larger than the simulated values, which is presumed to be partly due to the friction between the fluid and the device.
The calculated error between the experimental and simulated values is controlled within 5%, and the curves of both values are in terrific agreement, indicating that the simulation results of the selected model are in line with the actual fluid motion.

| RESULTS AND DISCUSSION
In this section, the effect of the static mixer on the liquidcarrying performance of the annular jet pump will be comprehensively evaluated by the cyclonic effect, droplet breakage ratio, and jet pump efficiency.

| Cyclonic effect
Based on the initial boundary conditions, the pressure of the power gas is ensured to be constant, and the inlet velocity of the sucked fluid is changed to 9-19 m/s to observe the change of water volume fraction and centrifugal velocity.
To clarify the internal transport characteristics and volume distribution of water in the annular jet pump, the sucked fluid outlet section (Figure 10), the shaft center section (Figure 11), and sucked fluid outlet centerline (Figure 12) were taken for comparative analysis and study of CAJP and NAJP.
For CAJP, the water is mainly concentrated in the inlet section circle range of sucked fluid, the content is F I G U R E 8 Experimental system working flow chart.1-gas compressor, 2gas tank, 3 (7, 11, 17, 20, 22)-pressure gauge, 4(8, 14)-valve, 5 (9, 15, 19) high and uniformly distributed.The water volume fraction in the power gas inlet area is extremely little, almost zero.A transitional partition layer appears between the power gas and the sucked fluid, and the volume fraction of water gradually increases, as shown in Figure 10.In addition, Figure 11A shows that the water gradually concentrates towards the center position of the pump under the action of the contraction of the suction chamber section and the partial velocity of the power gas.As the velocity increases, the water passes through the position of the throat to form a sharp point, and the range of forward extension gradually increases, indicating that the water mainly moves along the center of the jet pump, and the gas moves along the wall.Apparently, the water volume fraction at the centerline is not sensitive to changes in the velocity of the sucked fluid (Figure 11A).
For NAJP, due to the addition of the static mixer, the water will be thrown near the wall by centrifugal force and strongly mixed with the power gas, which improves the mixing effect.The section gradually generates a water ring (Figure 10) and continues to move in the form of an annular flow, while the gas continues to move along the center of the pump as shown in Figure 11B.Figure 12B shows that the volume fraction curve gradually becomes M-shaped as the velocity of the sucked fluid increases.Although the value of the volume fraction at the center decreases from 0.06 to 0.03, the volume fraction at the water ring tends to increase.It makes the water that would have been located in the middle of the jet pump to be separated, and the closer to the center of the pump, position C, the less water there is.As it gradually moves away from the center position C-B, the volume fraction of water gradually increases and peaks at position B. As the power gas enters from the annular nozzle, the velocity and pressure are extremely high, and the separated water is isolated by the power gas, so that during the change of position B-A, the further away from the center of the pump, the smaller the volume fraction of water.During the NAJP drainage process, it transforms the gas-liquid slug flow or agitated flow into the annular flow which is desired to be maintained or present in engineering compared to other flow patterns and is necessary for gas wells to carry fluids continuously. 38ang 39 showed that it is easier for gas flow to carry a liquid film than to carry droplets.If the disordered motion of the droplet is allowed to transform into the form of liquid film, the chance of it colliding with the wall must be increased.However, the centrifugal velocity of the droplet is the key factor that makes it close to the wall and transforms into liquid film.In the annular jet pump to take a-f sections Figure 13, respectively, representing the distance from the x-axis 50 mm (nozzle outlet), 90 mm (throat inlet), 200 (throat outlet), 400 mm (diffusion tube outlet), 500 and 600 mm of the position.Figure 14 shows the y-axis velocity contour diagrams for each section of CAJP and NAJP when the velocity of the sucked fluid is 19 m/s.The blue area represents the negative direction of the y-axis while the red area represents the positive direction.
For CAJP, the reasons for y-axis velocity generation are due to the rapid change in the diameter of the suction chamber leading to the original axial velocity decomposition into centripetal velocity in a and b sections.The generation of centripetal velocity drives the water mainly along the center of the pump.Although CAJP generates a centrifugal velocity along the y-axis in the D-section under the diffusion effect of the diffuser, the magnitude of the centrifugal velocity at this point is not sufficient to convert the droplet into the form of a liquid film (Figure 15 and Figure 11A).The outlet gas-liquid mixture continues to move forward in the form of a disordered linear flow (Figure 16A).
For NAJP, after the gas-liquid mixture passes through the static mixer, the fluid will produce a mandatory centrifugal motion, which generates a large centrifugal velocity in a and b sections.Combining Figure 15 and Figure 11B the droplets in the gas stream are thrown to  the wall at a large centrifugal velocity to form a liquid film during the cyclonic maintenance phase, while the gas is located in the center of the pump.It makes the previously disordered turbulent motion change to an ordered annular flow, and the gas-liquid mixture at the pump outlet moves spirally upward along the wellbore (Figure 16B).

| The droplet breakage ratio and critical liquid-carrying flow rate
This section will analyze the impact of the static mixer on the liquid-carrying performance of the annular jet pump from a "microscopic" point of view.The internal flow pattern of the jet pump can be divided into three stages, as shown in Figure 17.
In the first stage, the gas will carry the low-velocity mixed fluid through the suction chamber and then into the throat, at this time the mixed fluid from the slug flow into the agitated flow, and the large area of the liquid block will be divided and broken.
The second stage is the droplets motion stage when the gas-liquid mixture enters the throat, the liquid appears to pulsate under the interference of the high-velocity gas.The fluctuation amplitude of the surface of the liquid jet is increasing, and the continuous liquid will be dispersed into droplets of different particle sizes under the action of shear.Collisions occur between the gas molecules moving at high speed and the liquid molecules moving at low speed, and energy is transferred to the liquid.
The third stage is the annular flow stage, after the gasliquid mixture reaches the diffuser, the deceleration and pressurization are realized under the diffusion effect, and the liquid droplets are further broken to form annular flow.It will be more conducive to carrying the fluid to the wellhead, and the more gas content in the center, the smaller the droplets and the easier it will be to drain the fluid.
Compared with CAJP, NAJP will break up the liquid faster to form smaller droplets under the effect of shear and collision when flowing through the mixer.It reduces the particle size of the droplets in the gas core thereby reducing the density and making it easier for the fluid to be discharged from the diffuser of the jet pump.
Figure 18 shows the total number of droplets with different particle size ranges in the period from 0.4 to 0.5 s.The results prove that both CAJP and NAJP can obtain droplets with smaller particle sizes in the initial condition particle size range from 10 to 70 μm.The droplets are mainly concentrated in 1-7 and 10-40 μm after being broken by CAJP, while the droplets are mainly concentrated in 0.7-4 μm after being broken by NAJP, and among them, 1-2.5 μm particle size accounted for a larger proportion, about 50%.
From Figure 19, it is concluded that CAJP still retains about 25% of the droplets in the initial size range, compared to NAJP where only 1.3% of the droplets are retained in that range.At the same time, CAJP can obtain about 7% of droplets below 1 μm, while NAJP can  obtain about 23% of droplets below 1 μm.Overall, NAJP makes it easier to break up large droplets.
To better evaluate the effect of both in terms of droplet breakage performance, the characteristic parameter η b is introduced, representing the ratio of the number of droplets broken to the total number of injected droplets at a particular time.Figure 20 displays the change curve of the droplet breakage ratio in 0.4-0.5 s, from which it can be illustrated that the droplet breakage ratio of NAJP in the whole period is significantly larger than that of CAJP.Averaging the droplet breakage ratio for both jet pumps gives an average breakage ratio of 40.15% for CAJP and 46.31% for NAJP, with a 15.4% year-over-year increase in the NAJP breakage ratio.
Turner et al. 40 concluded from a large number of studies that it is possible to determine whether a gas well is liquid loading by using a droplet model.Based on the droplet model, if the particle size of the droplet is known, the critical flow rate of the annular jet pump can be calculated.Taking the droplet with the largest particle size at the outlet section of the annular jet pump (CAJP d max = 7e −5 m, NAJP d max = 3e −5 m) within 0.4-0.5 s.According to Equations ( 11) and ( 14) derived in Section 3.4, the critical liquid-carrying flow rates of CAJP and NAJP can be calculated respectively.The result shows that the critical liquid-carrying flow rate is 0.223 m/s for CAJP and 0.199 m/s for NAJP which is about 10.7% lower than CAJP.It can be concluded that the droplets are broken obviously after passing through the static mixer, and it plays the role of atomizing droplets, the smaller the droplets will be more easily carried to the ground by gas.

| Jet pump efficiency
In this section, the performance of CAJP and NAJP will be evaluated in terms of the pumping capacity of the gasliquid mixed fluid and the energy conversion efficiency.
Figure 21 displays the velocity curves of the sucked fluid outlet and the power gas.It can be concluded that the power gas velocity of NAJP is less than CAJP and the sucked fluid outlet velocity is greater than CAJP in the velocity variation ranges.For annular jet pumps, the greater the outlet velocity of the sucked fluid, the smaller the velocity of the power gas, the greater the velocity ratio, and the greater the q, the better the mass transfer | 83 performance of the jet pump.Obviously, NAJP is superior to CAJP for mass transfer performance.
Figure 22 shows the variation of q versus h with the velocity of the sucked fluid for a certain inlet pressure of the power gas for CAJP and NAJP.When the velocity of the sucked fluid increases, the q of both shows an upward trend, and the difference of q also raises, while the change of h is the opposite of q.The larger the velocity of the sucked fluid, the smaller the value of h.The reason for this difference stems from the fact that the addition of the static mixer enhances the velocity of the sucked fluid outlet, thus growing the mass flow rate.However, the static mixer will raise the pressure drop (Section 4.3), resulting in lower pressure at its outlet than CAJP, which declines the head of the outlet mixture and h.
Evaluation of the performance of the jet pump requires comprehensive consideration of the effects of q and h. Figure 23 demonstrates the variation of the pump efficiency of CAJP and NAJP with the sucked fluid velocity, and both efficiencies are trending upward, with a maximum efficiency of 28.3% for CAJP and 37% for NAJP, which is an increase of 30.7% year-on-year.It can be indicated that although the static mixer raises the pressure drop of the jet pump, it also promotes the velocity of the sucked fluid.q improves significantly more than the decrease in h, leading to the efficiency increase continuously.
At a certain mass flow ratio, the necessary condition for the effective action of an annular jet pump is a positive head ratio.And the head ratio is determined by the head of each section (the total pressure), which reflects the size of the dynamic and static pressure on the section.According to the jet pump fluid-carrying principle, only when the pressure of the power gas is greater than the mixed fluid outlet pressure, as well as the mixed fluid outlet pressure is greater than the sucked fluid inlet pressure (P 1 > P 3 > P 2 > 0), the annular jet pump can be realized the energy conversion between the fluids and operated effectively. 41However, the decisive conditions for the operation of jet pumps are no longer fulfilled when the velocity of the sucked fluid is greater than 20 m/s or less than 7 m/s.
Figure 24 indicates the variation patterns of CAJP and NAJP at 1/(Q 2 •H 3 ) for different sucked fluid velocities.When both pumps have the same injection pressure, which corresponds to the same power of the ground pump, the magnitude of PEI depends on 1/ (Q 2 •H 3 ) as shown in Equation (20).It can be concluded that when increasing the suctioned fluid velocity, the PEI of both pumps tends to decrease, but the energy coefficient of NAJP is always smaller than that of CAJP.It proves that NAJP consumes less energy while improving efficiency and is more conducive to draining wellbore fluid than CAJP.In this study, 3D CFD simulations of CAJP and NAJP were conducted and the reliability of the simulated data was verified through experiments.The effects of CAJP and NAJP on the liquid-carrying performance of gas wells were evaluated in terms of cyclonic effect, the droplet breakage ratio, and jet pump efficiency: (1) Compared with CAJP, due to the addition of the static mixer, NAJP made the original gas-liquid axial motion into a stronger vortex motion.The cyclonic effect enhanced the centrifugal velocity of the droplets so that the droplets at the center position were thrown to the wall to be converted into a liquid film and formed an annular flow, which was more conducive to the recovery of the liquid loading in the wellbore.(2) Compared with CAJP, NAJP could better break the large-size droplets and achieve the full mixing of gasliquid two-phase flow.In the same period when both droplet's breakage reached a steady state, the average breakage ratio of CAJP was 40.15%, while NAJP was 46.31%.The crushing efficiency of NAJP increased by 15.4% year-on-year.At the same time, based on obtaining smaller droplets, the critical carrying flow rate of NAJP was also reduced by about 10.7%.Overall, NAJP was more likely to carry droplets to the ground.(3) Compared with CAJP, NAJP could significantly increase the velocity of the sucked fluid, which could result in a larger mass flow rate ratio.However, due to the addition of the static mixer NAJP raised the pressure drop of the jet pump and obtained a smaller head ratio than CAJP.Taking into account the effects of q and h, the efficiency of NAJP was higher than that of CAJP, with a maximum of 37%, an increase of 30.7% year-on-year, in addition, NAJP reduced the energy coefficients and energy consumption.

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I G U R E 2 Two-dimensional schematic diagram of annular jet pump.T A B L E 1 Annular jet pump geometry.D 0 /mm D s /mm D t /mm L s /mm L t /mm L/mm α/°β/°w/mm I G U R E 3 Three-dimensional diagram of static mixer.

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Partial demonstration of the mesh division of new annular jet pump.

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Annular jet pump meshes independence test.(a) Conventional annular jet pump.(b) New annular jet pump.T A B L E 3 XX-to well two engineering parameters.

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Closed valve 4 and turned down the opening of valve 8. Then started the piston pump 13 and opened valve F I G U R E 7 Forces diagram of the droplet in the flow field.

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I G U R E 11 Volume fraction contour plot of the axial section from the annular jet pump, (a) conventional annular jet pump, (b) new annular jet pump.F I G U R E 12 Volume fraction distribution on the centerline, (a) conventional annular jet pump, (b) new annular jet pump.

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I G U R E 13 Take a-f sections on the annular jet pump.F I G U R E 14 Contour diagram of centrifugal velocity in the y-axis for each section, conventional annular jet pump-left, new annular jet pump-right.

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I G U R E 15 Velocity in the y-axis for each section, "-" denotes centripetal velocity, "+" denotes centrifugal velocity.F I G U R E 16 Outlet speed vector diagram, (A) conventional annular jet pump, (B) new annular jet pump.

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Internal flow pattern of annular jet pumps.(A) Conventional annular jet pump, (B) new annular jet pump.

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I G U R E 18 Number of droplets in different size ranges.

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I G U R E 19 Proportion of droplets in different size ranges.F I G U R E 20 Droplet breakage ratio variation graph.F I G U R E 21 Variation of two velocities with sucked fluid velocity.(A) conventional annular jet pump, (B) new annular jet pump.LIANG ET AL.

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I G U R E 22 Variation of q and h with sucked fluid velocity.(A) Conventional annular jet pump, (B) new annular jet pump.F I G U R E 23 Variation of jet pump efficiency with sucked fluid velocity.