Pressure changes in two-phase flow of gas and liquid in a wellbore

The phenomenon of high gas-liquid ratio has an impact during the lifting process of oil wells, and it is necessary to study the variation of the two-phase flow of gas and liquid in the wellbore. In this paper, the pressure losses along the wellbore and the coupling pressure losses in the process of gas-liquid two-phase flow in the wellbore were studied through experiments. It was found that with the increase in gas content, the pressure losses gradually decreased, and there was a turning point in the pressure losses within the gas content range of 10-15%. With the increase in flow rate, the pressure losses also increased. The theoretical model was used to calculate the bottom hole pressure values of actual oil wells under different gas contents and flow rates. It was found that in the calculation process, the dominant flow regime is often slug flow. Through experiments and theoretical calculations, the influence of gas content and flow rate on the pressure inside the wellbore was determined, providing guidance for understanding the changes in pressure during the lifting process of oil wells.


Introduction
At present, in certain domestic oil fields, during the extraction process, there is a phenomenon of high gas-liquid ratio that affects the lifting of oil wells, and it can even lead to gas lock phenomena.Therefore, it is necessary to research the two-phase flow of gas and liquid in the wellbore, particularly focusing on the study of pressure changes within the wellbore [1][2] .Currently, researchers mostly study the flow patterns inside wellbores through theoretical calculations, with only a few scholars employing a combination of experiments and simulations to investigate the two-phase flow patterns within wellbores [3] .Dong Yong and his colleagues addressed the issue of relatively high relative error in the Orkiszewski model's predictions by conducting experiments.The experimental results showed that the improved model reduced the average prediction error to 34.98% [4][5][6] .Chen and Sun [7] , in response to gas wells with significant variations in gas-oil ratio, utilized numerical simulation methods to determine the fluid flow patterns in the wellbore of such wells.They found that the factors influencing fluid flow patterns under different conditions were primarily wellhead pressure and wellbore depth.Luo et al. [8] , focusing on a specific gas field, calculated the pressure gradient in the wellbore and studied the flow behavior of two-phase flow under different gas and liquid flow rates through experiments.Building upon the Mukherjee-Brill model, they established a new holdup model capable of calculating pressure gradients within various pressure and gas flow ranges.Gao et al. [9] employed the theory of thermal boundary to investigate heat transfer processes under different flow patterns.Based on experimental data, they introduced the concept of gas volume fraction, modified the Dittus-Boelter equation, and proposed new expressions for the heat transfer coefficient under various flow patterns.This study, based on experiments involving gas-liquid two-phase flow, determined the changes in pressure loss within a gas-containing pipe.Subsequently, a theoretical model was used to calculate the bottomhole pressure, allowing the determination of the impact of gas fraction and flow rate on pressure.

Experimental equipment
The experimental setup consists of the following components: a liquid supply system (including an agitation device), a gas supply system, a temperature control system, and a parameter measurement system.The configuration of the column and the temperature control system during the experimental process are mainly shown in Figure 1: A pump was used in the liquid supply system to deliver the prepared solution to the experimental pipeline, with flow regulated by valves.The agitation device is used to stir the experimental solution to prevent stratification of the water-crude oil mixture.The gas supply system is used to transport gas to the experimental pipeline, utilizing gas cylinders for gas supply, with an initial cylinder pressure of 16 MPa.The temperature control system is employed to maintain the temperature of the measured samples and includes a temperature controller, heating device, cooling device, and heat tracing system at measurement points.The parameter measurement system primarily involves determining the fluid flow velocity through the vertical experimental pipeline and the corresponding pressure differentials at specific points (in the centimeters of water column) under stable flow conditions, with the pressure differentials read from pressure sensors.The overall experimental process is illustrated in Figure 2.

Steps of the experiment (1) Preparation work
a.The process was checked, the opening and closing status of each valve were checked, and the valve was functioning properly.
b.The gas supply system was activated, and the gas pressure inside the cylinder was checked.c.The parameter testing system was turned on, and it was ensured that the initial readings of each instrument were correct and error-free.
d.The liquid storage tank's water level was adjusted to the level required for the experiment.e.The gas flow in the experimental pipeline was checked.f.The temperature control system was turned on, and the functionality of each device was checked.
(2) Steps of the experiment a.The liquid supply system was opened, and the outlet valve was adjusted to ensure that the liquid supply could achieve the experimental flow rate.
b.The gas supply system was turned on, and the outlet pressure of the gas supply system was obtained through the digital display pressure sensor.At the same time, pressure measurement points throughout the experimental pipeline were observed, and it was confirmed that the internal pressure of the experimental pipeline reached the experimental pressure.
c. Different injection gas volumes were altered, and the flow patterns in the experimental pipeline were observed.The fluid flow state in the experimental pipeline was determined by observing the flow patterns.After the flow of fluid inside the experimental pipeline stabilized, experimental data was collected.Pressure differentials at various pressure measurement points along the experimental pipeline, pressure values at the outlets of the liquid supply system and gas supply system, as well as pressure differentials at various joints or couplings along the experimental pipeline, were included。 d.The data for the current injected gas volume were recorded for the completion of a set of gas injection experiments.After completing a set of gas injection experiments, gas injection was ceased, the injection flow rate was modified, and Step c was repeated.
(3) Post-testing work a.The experimental process valve was closed, and the purge valve was opened.b.The gas supply system was turned on, and the liquid in the experimental pipeline was blown back into the buffer tank.c.The gas supply system was turned off, and the buffer tank valve was opened to transfer the liquid inside the buffer tank to the liquid storage tank.The water level inside the liquid storage tank was monitored to prevent it from becoming too high.
d.The temperature control system was turned off, and the gas storage tank's emptying valve was opened, allowing all gases within the experimental pipeline to be purged.

The experimental measurement results.
(1) The impact of gas content on pressure.
The pressure loss caused by the flow of two-phase mixed fluid at different gas contents along the pipeline and at the junction was measured in the experiment.The total pressure drop loss in the experimental pipeline's two-phase flow comprises gravitational pressure drop loss, frictional pressure drop loss, and acceleration pressure drop loss.Gravitational and frictional pressure drop losses are significantly greater than acceleration pressure drop losses.Therefore, the influence of acceleration pressure drop was neglected in the experimental data analysis.The total pressure drop losses and the variations in pressure differences along the pipeline and at the junction under different injection flow conditions were tested in the experiment.The gas contents are 6%, 8%, 10%, 12%, 14%, 16%, 18%, and 20%, as shown in Figures 3 and 4   From Figure 3 and Figure 4, it can be observed that the pressure drop in the wellbore decreases with an increase in gas content, but the rate of decrease gradually diminishes.Within the gas content range of 10-15%, a turning point in pressure loss becomes evident.This is because, with the increase in gas injection, the density of the mixed fluid in the wellbore continuously decreases, leading to a gradual reduction in gravitational pressure drop.It can be observed that within a gas content range of below 20%, gravitational pressure drop dominates.At the same gas content, with an increase in liquid phase displacement, the total pressure drop in the vertical wellbore gradually increases.An increase in the liquid phase content affects the density of the mixture, thereby increasing the gravitational pressure difference.The trend of increased pressure difference loss becomes predominant.

Selection of stress models
When calculating the pressure of a two-phase flow, the first step is to determine the temperature variation within the wellbore.The temperature calculation model for an oil well needs to consider the flow regimes of the fluid at different locations and the fluid velocities at these positions.The overall calculation process is as follows: A differential element of length dz on the wellbore was taken, with the oil wellhead as the coordinate origin.The positive direction of the z-coordinate is oriented vertically downward, opposite to the direction of fluid flow.
The temperature distribution formula for the produced fluid within the wellbore of an oil well is: where K is the heat transfer coefficient from the fluid in the tubing to the formation per unit length of the tubing, W/(m.℃).
T is the temperature of the oil well's produced fluid, ℃.Ts is the temperature of the formation, ℃. ql is the heat released during the wax deposition of crude oil, W/m; W is the water equivalent of the produced liquid, W/℃.T0 is the original formation temperature at the surface, ℃. gT is the geothermal gradient, ℃/m, generally, g=0.03-0.035℃/m.z is the vertical distance from the wellhead to a certain depth in the well.
Go is the mass flow rate of crude oil, kg/s.Co is the specific heat capacity of crude oil, J/(kg.℃).
Gg is the mass flow rate of natural gas, kg/s.Cg is the specific heat capacity of natural gas, J/(kg.℃).
Gw is the mass flow rate of water, kg/s.Cw is the specific heat capacity of water, J/(kg.℃).The calculated variation in wellbore temperature, as determined above, is used for pressure calculations.The methods for calculating frictional pressure drop differ under different flow regimes, while the gravitational pressure drop remains the same under various flow regimes.The process is as follows.
1) Calculation method for frictional pressure difference in bubbly flow: Calculation of liquid holdup: H1-Liquid holdup, dimensionless.vbf-The rising speed of bubbles in a liquid flow, m/s.vm-Average flow rate of gas-liquid mixture, m/s.vbs-The rising speed of bubbles in a stationary liquid, m/s.g-Gravitational acceleration, m/s 2 .Therefore, the density of the gas-liquid mixture is: ( ) 3) Calculation method of frictional pressure difference during transition state: When the flow pattern is in a transitional state, and the pressure gradient needs to be calculated, the formulas for slug flow, annular flow, and mist flow must be used simultaneously.
4) Calculation method for frictional pressure difference in annular flow and mist flow: Calculation of frictional pressure difference:

Calculation of wellbore pressure changes
The above model is used to calculate the pressure changes during the gas content range of 6-20%, and the liquid production rates are 13 m 3 /d, 15 m 3 /d, and 18 m 3 /d.
(1) Under the premise of determining the wellhead oil pressure, the calculated wellbore depth is 800 m, and the settlement results are as follows: From Figure 5, it can be observed that with an increase in gas content, the calculated bottomhole pressure increases.Under constant gas content conditions, the bottom hole pressure gradually rises with an increase in liquid production rate.The calculated pressure variations align with the observed pressure changes within the pipeline during the experiments.
(2) Under the premise of setting the bottom hole pressure at 7 MPa, the wellhead oil pressure is calculated, and the settlement result is as follows: According to Figure 6, it can be observed that, under the same bottomhole flowing pressure conditions, the calculated wellhead pressure gradually increases with the increase in the gas-liquid ratio.Conversely, with an increase in flow rate, the calculated wellhead pressure gradually decreases.This indicates a consistent trend of losses in the wellbore with the experimental results.Under the assumption of identical fluid properties, the pressure variation of fluid flow in the wellbore is directly proportional to the gas-liquid ratio and inversely proportional to the liquid production rate.

Conclusion
(1) Through two-phase gas-liquid flow experiments, it was discovered that as the gas content increases, the pressure loss decreases.Additionally, as the flow rate increases, the pressure loss also increases.
(2) With the increase in gas content, the magnitude of pressure loss variations gradually decreases.Within the range of 10-15%, the pressure loss tends to stabilize.

Figure 3 .
Figure 3. Change curve of coupling pressure loss.

Figure 4 .
Figure 4. Pressure loss variation curve along the way.
-Friction pressure gradient, Pa/m.D-The diameter of the pipe, m.2) Calculation method of frictional pressure difference during slug flow: 14) Calculation of gravity pressure difference under various flow states:The true density of gas-liquid mixtures can be expressed as: pressure difference at the center of gravity: of gas-liquid mixture, kg/m 3 .ρg-Density of gas phase, kg/m 3 .Φ-Void rate, m 3 /m 3 .1-Φ-Liquid holdup, m 3 /m 3 .Δp-Gravity pressure difference, Pa.Δz-Position height difference, m.

Figure 5 .
Figure 5. Bottom hole pressure variation curve under different gas content.

Figure 6 .
Figure 6.Wellhead pressure change curve under different gas content.