An Interpretation Method of Gas–Water Two-Phase Production Profile in High-Temperature and High-Pressure Vertical Wells Based on Drift-Flux Model

Abstract

With the increasing demand for oil and gas, the depth of some vertical gas wells can reach 6000 m. At this time, the downhole fluid is in a state of high temperature and pressure, and interpretation of the production logging output profile faces the problem of inaccurate production calculations and difficulty judging the water-producing layer. The drift-flux model is usually used to calculate the gas–water two-phase flow. The drift-flux model is widely used to describe the two-phase flow in pipelines and wells because of its accuracy and simplicity. The constitutive correlations used in drift-flux models, which specify the relative motion between phases, have been extensively studied. However, most of the correlations are only extended by laboratory data of small pipe diameters at standard temperature and pressure and do not apply to high-temperature and high-pressure large-diameter gas wells. Therefore, we improved the distribution coefficient and drift velocity of drift-flux correlations in this study for high-temperature and high-pressure gas wells with large pipe diameters. Therefore, this study improved the distribution coefficient and drift velocity of the drift-flux correlations for high-temperature and high-pressure gas wells with large pipe diameters. In practical application, the coincidence rates of gas production and water production calculated by the new drift-flux model were 12.68% and 19.39%, respectively. For high-temperature and high-pressure deep wells, the measurement errors of production logging instruments are significant, and surface laboratory pipelines are challenging to configure and equip with actual high-temperature and high-pressure conditions. Therefore, this study used the method of numerical simulation to study the flow characteristics of the two phases of high-temperature and high-pressure gas and water to provide a basis for identifying the water layer. Combined with the proposed drift-flux correlations and the new method of determining the water-producing layer, a new method of production profile interpretation of high-temperature and high-pressure gas wells is obtained.

1. Introduction

In oilfield development and production, production logging is an indispensable key link. It is an important technology to monitor the physical properties of downhole fluids, as well as the production and injection of fluids after the casing is inserted into the wellbore. Production profile logging is one of the common production logging technologies, mainly monitoring the fluid flow and the output of each phase. In early production profile logging technology, only temperature logging was available. Fluid output from the production layer results in a change in the outlet temperature, so temperature logging can qualitatively judge the fluid output. Subsequently, PLT multi-parameter combination logging was developed, including temperature, pressure, density, gas holdup, turbine speed, completion gamma, and cable velocity, combined with the flow pattern and interpretation model, which can quantitatively complete the interpretation of the downhole flow of each phase [1,2].

In recent years, people’s demand for oil and natural gas has been increasing, and many oilfields have gradually explored deeper stratum, and the depth of some gas wells can reach 6000 m [3,4]. However, with the continuous increase in exploration depth, the temperature and pressure of the downhole fluid have increased, and the downhole fluid is in the state of 160 °C and 80 MPa. In the production process, water production in gas wells will seriously affect productivity, resulting in a significant decrease in gas production, so it is particularly important to judge the water production layer and accurately calculate water production in the logging interpretation of the vertical gas well production profile.

In the interpretation of the gas well output profile, changes in temperature, density, and water holdup curves are usually observed to comprehensively explain the fluid output. For example, if the mixed density decreases, the negative temperature anomaly indicates obvious gas production; on the contrary, if the mixed density increases, the positive temperature anomaly indicates liquid production [2]. However, in actual high-temperature and high-pressure gas lift wells, a small amount of liquid production makes the density and temperature changes not obvious. Moreover, the accuracy of capacitance water holdup instruments in gas wells is low, and they are difficult to use as a basis for judgment. In addition, surface multiphase flow experiments require the use of transparent pipelines for fluid observation, but the pipeline material has difficulty withstanding high temperature and high pressure, and it is difficult to reproduce the actual multiphase flow process under downhole conditions. Therefore, this study uses Ansys Fluent software to simulate the high-pressure downhole gas–water two-phase flow. Fluent software (2024 R1) is a powerful fluid dynamics simulation tool; it is the core solver in the Ansys CFD suite. Fluent supports the simulation of a variety of physical fields, including fluid flow, heat transfer, mass transfer, phase change, and multiphase flow, which makes it able to provide comprehensive solutions to complex engineering problems [5]. Fluent’s turbulence models have been at the forefront of commercial CFD software, offering a variety of models, including the Reynolds stress model, large eddy simulation, and separate eddy simulation to accurately simulate a variety of turbulence phenomena [6]. Fluent, with its broad user base and advanced simulation capabilities, is widely used in several industrial and research fields. In addition to judging the water production level of gas wells, the calculation of the gas–water flow of each phase is also particularly important. The drift-flux model has excellent performance in the flow calculation model of the output profile of gas lift wells. Due to its simple form, it has high calculation efficiency in production applications. The drift-flux model was originally proposed by Zuber [7]. Most scholars improved the model based on gas–water two-phase experiments under different conditions, and most scholars conducted experiments under normal temperature and pressure conditions. Taitel [8] completed gas–water two-phase experiments in a 5 cm vertical pipe under 25 °C and atmospheric pressure, divided the flow patterns according to different gas holdup rates, and gave the corresponding theoretical models of gas and liquid velocities under different gas holdup rates. In addition, Shi [9] believed that the relationship between experimental data under small pipe diameters was not applicable to actual wells with large pipe diameters. The existing drift-flux correlations for describing the flow patterns are all established for a single flow pattern, or a set of multiple correlations are used to express different flow patterns, which requires the experimenter to distinguish the flow patterns by visual observation in most cases [10]. Although some drift-flux correlations are considered to be independent of flow patterns, the existence of a large number of two-phase flow parameters greatly increases the complexity of drift-flux correlations, the difficulty of use, and the possibility of not covering real influencing factors.

In this study, the influence of water production velocity on mixture velocity in different formations is studied using a numerical simulation method, which provides a strong basis for judging the water production horizon. The advantages and disadvantages of various drift-flux models are compared, the model with excellent performance is improved, and a production profile interpretation method suitable for high-temperature and high-pressure gas wells is synthesized.

2. Gas–Water Two-Phase Drift-Flux Model

Commonly used gas–water two-phase flow interpretation models include the slippage model, drift-flux model, and flow equalization model, among which the drift-flux model has better performance in practical applications. Zuber and Findlay [7] proposed a one-dimensional drift-flux model, as follows:

$$V_g=C_0{V}_m+V_d$$

(1)

where $V_g$ is the gas phase velocity, and $C_0$ is the flow distribution coefficient representing the influence of velocity distribution and concentration distribution. $V_d$ is the drift velocity considering the buoyancy effect. In the strongly coupled bubble flow and elastic flow, the drift velocity is quite large and relatively uniform, while in the annular flow, the drift velocity becomes small and negligible [10]. For bubble flow and slug flow, Zuber considers $C_0$ to be a constant with a value of 1.2 and proposes that drift velocity $V_d$ is related to gravity, the gas–liquid surface tension coefficient, and liquid density, and the specific expression is:

$$V_d=1.53(g\delta \Delta \rho /{\rho }_l^2{)}^{{\displaystyle \frac{1}{4}}}$$

(2)

where $g$ is the gravitational acceleration and $\delta$ is the gas–liquid surface tension coefficient.

The mean mixture velocity is the sum of the gas and liquid superficial velocities, as follows:

$$V_m=V_{sg}+V_{sl}={\alpha }_g{V}_g+\left(\begin{array}{c}1-{\alpha }_g\end{array}\right)V_l$$

(3)

where ${\alpha }_g$ is the gas holdup. The definition of gas holdup can be obtained by combining Equations (1) and (2). Therefore, Equation (4) can be used to predict the value of gas holdup in gas-liquid flow, which is a key parameter for characterizing two-phase flow:

$${\alpha }_g={\displaystyle \frac{V_{sg}}{C_0\left(V_{sg}+V_{sl}\right)+V_d}}$$

(4)

In the actual production interpretation, gas holdup is usually calculated using the homogeneous model, and the average fluid density is obtained by a PLT instrument during well measurement. The homogeneous model considers that the two phases of gas and water are the same system, and the mixed density is the sum of the proportion of each phase density, as follows:

$${\rho }_m={\displaystyle \frac{{\rho }_L{\nu }_{sL}}{{\nu }_m}}+{\displaystyle \frac{{\rho }_g{\nu }_{sg}}{{\nu }_m}}={\left(\begin{array}{c}1-{\alpha }_g\end{array}\right){\rho }_L+\alpha }_g{\rho }_g$$

(5)

where ${\rho }_m$ is the density of the mixed fluid, ${\rho }_L$ is the density of the liquid phase, and ${\rho }_g$ is the density of the gas phase.

The conventional output profile interpretation process is as follows:

1. Determine the perforation layer and identify the produced fluid according to the curves of temperature, pressure, density, and turbine speed.

2. Select the interpretation layer and complete the calculation of fluid physical property parameters, fluid apparent velocity, and gas holdup.

3. Calculate the fluid velocity.

In field applications, the downhole instrument measures the turbine flow rate in the whole hole in the gas lift well and pulls up and down to measure the turbine speed in the center of the wellbore at different cable speeds. Then, the cable speed and turbine speed are combined by the least squares method, and the apparent velocity $V_a$ is obtained. The apparent velocity shows the flow rate in the center of the pipeline, but the central velocity cannot represent the overall average velocity of the two phases of gas and water. To calculate the average velocity of the section, multiply the velocity correction coefficient $C_v$, as follows:

$$V_m=V_a{C}_v$$

(6)

The velocity correction coefficient can be used to scale the first production layer in the wellbore by measuring production at the wellhead, and other layers can be calculated according to the first layer coefficient and mixed Reynolds number:

$$C_{v1}={\displaystyle \frac{P_g{B}_g+P_w{B}_w}{V_{ai}{P}_c}}$$

(7)

$$R_{ei}={\displaystyle \frac{{\rho }_{mi}{V}_{ai}D}{{\mu }_{mi}}}$$

(8)

$$C_{vi}={\displaystyle \frac{R{e}_i}{R_{e1}}}{C}_{v1}$$

(9)

$$P_c=21600\pi \left({{\mathrm{D}}_h}^2-{{\mathrm{D}}_t}^2\right)$$

(10)

where $C_{v1}$ is the velocity profile correction coefficient of the first production layer, $P_g$ is the wellhead gas phase production, $P_w$ is the wellhead water phase production, $B_g$ is the gas volume coefficient, $B_w$ is the water volume coefficient, $V_{ai}$ is the apparent velocity of layer $i$, $R_{ei}$ is the Reynolds number of layer $i$, $C_{vi}$ is the velocity profile correction coefficient of layer $i$, $P_c$ is the pipe constant, $\pi$ is Pi, ${\mathrm{D}}_h$ is the inner diameter of the pipe, and ${\mathrm{D}}_t$ is the production logging tool string outside diameter.

4. Calculate the downhole flow rate:

$$Q_g=P_c{V}_{sg}$$

(11)

$$Q_w=P_c{V}_{sw}$$

(12)

2.1. Drift-Flux Model Review

In order to improve the accuracy and stability of DFM and describe the constitutive relationship of the relative motion between two phases, the distribution coefficient $C_0$ and drift velocity $V_d$ have attracted the attention of many researchers, and a large number of correlations have been developed, as shown in

Table 1. Drift-flux models.

Correlation	Distribution	Drift Velocity
Shi [9]	$$\begin{gathered} C_0={\displaystyle \frac{A}{1+(A-1){\gamma }^2}} \\ \gamma ={\displaystyle \frac{\beta -B}{1-B}} \\ \beta =\max \left({\alpha }_g,F_{\nu }{\displaystyle \frac{{\alpha }_g\left|V_m\right|}{V_{sgf}}}\right) \\ {V}_{sgf}=K_u\sqrt{{\rho }_l/{\rho }_g}(g\delta \Delta \rho /{\rho }_l^2{)}^{1/4} \end{gathered}$$	$V_d={\displaystyle \frac{(1-{\alpha }_g{C}_0)C_{0}K\left({\alpha }_g\right)V_c}{{\alpha }_g{C}_0\sqrt{\frac{{\rho }_g}{{\rho }_l}}+1-{\alpha }_g{C}_0}}$ $V_c={\left({\displaystyle \frac{\delta g({\rho }_l-{\rho }_g)}{{\rho }_l^2}}\right)}^{1/4}$ $K_u\left(\widehat{D}\right)={\left({\displaystyle \frac{g\left({\rho }_l-{\rho }_g\right)}{\delta }}\right)}^{1/2}D,{\alpha }_g\ge 0.21$    $K\left({\alpha }_g\right)=1.53/C_0{,\alpha }_g\le 0.06$
Hibiki and Ishii [11]	(1) $C_0=1.2-0.2\sqrt{{\displaystyle \frac{{\rho }_g}{{\rho }_l}}}\left(1-\mathrm{e}\mathrm{x}\mathrm{p}(-18{\alpha }_g)\right)$	$V_d=1.41(g\delta \Delta \rho /{\rho }_l^2{)}^{1/4}(1-{\alpha }_g{)}^{7/4}$
	(2) $C_0=1.2-0.2\sqrt{{\displaystyle \frac{{\rho }_g}{{\rho }_l}}}$	$V_d=0.35\sqrt{gD({\rho }_l-{\rho }_g)/{\rho }_l}$
	(3) $C_0=1+(1-{\alpha }_g)/({\alpha }_g+4\sqrt{{\rho }_g/{\rho }_l})$	$V_d=(1-{\alpha }_g)\left({\alpha }_g+4{\displaystyle \frac{\sqrt{{\rho }_g/{\rho }_l}(\sqrt{gD({\rho }_l-{\rho }_g)/{\rho }_l})}{0.015{\rho }_g}}\right)$
Greskovich and Cooper [12]	$C_0=1$	$V_d=0.671\sqrt{gD}(sin\theta {)}^{0.263}$
Liao [13]	$C_0=1.2-0.2\sqrt{{\displaystyle \frac{{\rho }_g}{{\rho }_l}}}\left(1-\mathrm{e}\mathrm{x}\mathrm{p}(-18{\alpha }_g)\right)$	$V_d=0.33{\left({\displaystyle \frac{g\delta \Delta \rho }{{{\rho }_l}^2}}\right)}^{1/4}$
Jowitt [14]	$C_0=1+0.796\mathrm{e}\mathrm{x}\mathrm{p}\left(-0.061\sqrt{{\displaystyle \frac{{\rho }_g}{{\rho }_l}}}\right)$	$V_d=0.034\left(\sqrt{{\displaystyle \frac{{\rho }_g}{{\rho }_l}}}-1\right)$
Sonnenburg [15]	$C_0=1+\left(0.32-0.32\sqrt{{\displaystyle \frac{{\rho }_g}{{\rho }_l}}}\right)$	$V_d={\displaystyle \frac{C_0(1-C_0{\alpha }_g)}{\left(C_0{\alpha }_g/\sqrt{g\delta \Delta \rho /{\rho }_g}\right)+\left(1-C_0{\alpha }_g/\sqrt{g\delta \Delta \rho /{\rho }_l}\right)}}$
Bestion [16]	$C_0=1$	$V_d=0.188\sqrt{{\displaystyle \frac{gd\Delta \rho }{{\rho }_g}}}$
Ishii [17]	$C_0=1.2-0.2\sqrt{{\displaystyle \frac{{\rho }_g}{{\rho }_l}}}\left(1-exp(-18{\alpha }_g)\right)$	$V_d=\left(C_0-1\right)u_M+\sqrt{2}{\left({\displaystyle \frac{g\delta \Delta \rho }{{{\rho }_l}^2}}\right)}^{1/4}$
Gomez [18]	$C_0=1.15$	$V_d=1.53(g\delta \Delta \rho /{\rho }_l^2{)}^{1/4}(1-{\alpha }_{\mathrm{g}}{)}^{1/2}$
Choi [19]	$C_0={\displaystyle \frac{2}{1+{\left({Re}_m/1000\right)}^2}}+{\displaystyle \frac{1.2-0.2\sqrt{({\rho }_g/{\rho }_l)}(1-\mathrm{e}\mathrm{x}\mathrm{p}(-18{\alpha }_g))}{1+{\left(1000/R{e}_m\right)}^2}}$	$V_d=0.0246\times cos\theta +1.606\times {(g\delta \Delta \rho /{\rho }_l^2)}^{1/4}sin\theta$
Liu [20]	$C_0={\displaystyle \frac{2-1.07\times {\alpha }_g^{1.3}}{1+{\left(R{e}_m/1000\right)}^2}}+{\displaystyle \frac{1.2-0.205\times {\alpha }_g^{5.8}}{1+{\left(1000/R{e}_m\right)}^2}}$	$V_d=(0.35sin\theta +0.45cos\theta )\sqrt{g\mathrm{D}}(1-{\alpha }_g{)}^{3/4}$

. Hibiki and Ishii [11] proposed a simple flow pattern correlation. The distribution coefficients of bubble flow, elastic flow, and annular flow were calculated by considering the density ratio of two phases (${\rho }_g$/${\rho }_l$) and the gas content. Shi [9] established a set of distribution coefficient correlations, the maximum value of which is A, and the addition of the quadratic term makes $C_0$ tend toward 1 when the gas holdup or two-phase mixture velocity is high. For bubble and slug flow patterns, A defaults to 1.2, and B is the gas content or mixture velocity expressed in $V_{sgf}$.

Based on the studies of Ishii [17] and Fabre and Line [21], Choi [19] proposed a simple correlation of distribution coefficients expressed in terms of gas content and mixture Reynolds number for a wide range of fluid viscosity data. For the distribution coefficient correlation, two-phase characteristics such as two-phase flow and density are taken into account. Liu [20] optimized various coefficients on the model proposed by Bhagwat and Ghajar [10] and finally concluded that the distribution coefficient $C_0$ is related to the mixture Reynolds number and gas holdup, and the drift velocity is related to gravity, the pipe inner diameter, inclination angle, and gas holdup.

For the study of drift velocity, ${\left({\displaystyle \frac{g\delta \Delta \rho }{{{\rho }_l}^2}}\right)}^{1/4}$ appears in the correlations given by Ishii, Liao, and Shi, and the coefficients given are all different. Shi pointed out in the study that this is the characteristic coefficient $V_c$. According to the study by Harmathy [22], the coefficient before this expression in bubble flow is 1.53. The drift velocity correlation proposed by Holmes [23] is a function of the critical Kutateladze number ($K_u$), which has a split form according to the value of the gas holdup and is denoted as $K_u\left(a_g\right)$. For the Kutateladze number, the drift velocity model proposed by Shi [9] gives different calculation methods for $K_u\left(a_g\right)$ and $K_u\left(\widehat{D}\right)$. $K_u\left(\widehat{D}\right)$ is related to the pipe diameter. Figure 1 shows the relationship between the critical Kutateladze number and the dimensionless pipe diameter given by Shi [9]. The density term is ignored in the correlation given by Liu [20] because the density of gas is much less than that of liquid at normal temperature and pressure. In high-temperature and high-pressure gas wells, the density of underground natural gas is affected by temperature and pressure and can reach 250 kg/m³. Therefore, a density term should be added to the drift velocity correlation.

2.2. Distribution Coefficient Analysis

The distribution coefficient is deformed based on the original drift-flux model, and the relationship between the distribution coefficient and the mixture Reynolds number is obtained. Liu [20] proved in the study that the distribution coefficient is correlated with the Reynolds number and gas holdup and obtains the effects of the Reynolds number and void fraction on the distribution coefficient more intuitively. Equation (13) can be written as follows:

$$C_0={\displaystyle \frac{\left(\begin{array}{c}\frac{V_{sg}}{{\alpha }_g}-V_d\end{array}\right){\rho }_m{\mathrm{D}}_h}{{\mu }_{m}R{e}_m}}$$

(13)

Figure 2 shows the variation of the distribution coefficient with actual gas holdup under different pipe diameters. The overall trend in the figure shows that the distribution coefficient has little change in the range of 0–0.2 gas holdup, and the value is about 2, while the distribution coefficient decreases significantly in the range of 0.2–0.5 gas holdup. The distribution coefficient varies between 1.18 and 1.12; the increase to 0.85 is the second turning point. The distribution coefficient decreases significantly again and finally approaches 1.0. The gas holdup studied in this paper ranges from 0.5 to 1, and the research object is vertical wells with high gas production, so this paper focuses on the distribution coefficient and drift velocity changes in this region. It is found that the overall value of the distribution coefficient in a 125 mm pipe diameter is less than that in a 165 mm pipe diameter, and the distribution coefficient changes relatively smoothly in the range of 0.55~0.85 gas holdup. The data show that the gas flow in small pipe diameters is more stable than in large pipe diameters.

According to the above points, the models containing gas holdup in the expression of the distribution coefficient were selected from several models in Table 1 for comparison. Ishii, Liao, and Hibiki (3) used the same model, taking density ratio and gas holdup into consideration. Choi’s model was similar to Liu’s, taking gas holdup and the mixture Reynolds number as the main variables. Figure 3 is the diagram of the relationship between the distribution coefficient and Reynolds number given by Liu [20]. Choi [19] also considered the density ratio. Figure 4 compares multiple models considering gas holdup factors. The horizontal coordinate is the gas holdup, and the vertical coordinate is the distribution coefficient. When the gas holdup approaches 0, the model distribution coefficient values of Choi [19] and Liu [20] approach 2, and those of Hibiki and Ishii approach 1.5 and 1.2, respectively. When the gas holdup increases and approaches 1, only Liu’s model distribution coefficient is 1.

Overall, the distribution coefficients of the four models all decrease with an increase in gas holdup. The calculated values of the Choi [19] and Liu [20] models are the same in the 0–0.1 gas holdup range, and the changing trend is similar. They decline rapidly in the range of 0–0.2, slow down in the range of 0.2–0.7, the slope becomes more extensive in the range of 0.7–1.0, and the values decrease rapidly.

Comparing the calculated values of the distribution coefficients of each model with the measured values, it is found that Liu’s model has the same trend and similar values with the measured values within the range of the measured data. However, it is slightly higher than the measured values in the range of 0.5–0.85 and more accurate in the range of 0.85–1.0. Therefore, the optimization based on Liu’s model has a better effect for field applications. The distribution coefficient model proposed by Liu is as follows:

$$C_0={\displaystyle \frac{a-b\times {\alpha }_g^c}{1+{\left(R{e}_m/1000\right)}^2}}+{\displaystyle \frac{d-e\times {\alpha }_g^f}{1+{\left(1000/R{e}_m\right)}^2}}$$

(14)

There are six empirical parameters to be optimized contained in vector $X_P$, given by:

$$X_P=[abcdef{]}^T$$

(15)

The actual well database shown in

Table 2. Actual well parameters.

Sources	Data Points	Property
		Diameter (m)	Temperature (°C)	Pressure (MPa)	Gas Density (kg/m3)	Water Density (kg/m3)	Gas Holdup Range	Gas Production Rate (m3/d)	Water Production Rate (m3/d)
A-8-1	8	0.125	171.61	79.41	261.72	1113.43	0.92~0.97	1657.21	9.31
A-8-13	6	0.125	146.33	83.93	257.31	1268.12	0.91~0.96	1844.64	4.51
A-6	6	0.125	180.53	88.65	250.76	1076.91	0.57~0.91	1827	7.83
A-9	9	0.125	153.74	74.21	229.55	1030.22	0.81~0.95	1936.91	8.25
B-24-5	7	0.125	154.05	80.54	237.47	1031.42	0.83~0.96	414.62	2.61
B-24	8	0.125	154.05	85.33	245.58	1103.18	0.63~0.93	292.93	2.83
B-242	8	0.125	157.39	94.68	245.59	1103.14	0.82~0.95	2577.43	6.81
B-2-1	10	0.165	97.58	33.14	180.52	1010.32	0.51~0.97	1876.89	6.43
B-2-3	6	0.165	99.41	49.56	209.84	1041.16	0.64~0.98	4747.59	10.15
B-2-4	9	0.165	97.98	35.56	171.51	1016.27	0.73~0.96	4188.88	5.84
B-2-5	4	0.165	100.12	39.12	182.52	1019.58	0.71~0.89	5621.75	15.64
B-2-6	8	0.165	100.25	38.22	180.65	1019.36	0.69~0.89	7916.72	12.47
B-2-7	7	0.165	98.51	42.69	192.66	1050.37	0.75~0.91	7538.45	10.51
B-2-8	9	0.165	100.68	40.61	187.47	1051.85	0.58~0.81	1331.70	27.95
C-9	4	0.165	98.18	36.91	177.89	1047.61	0.81~0.93	1038.27	15.87
C-2-2	8	0.165	100.83	33.73	165.51	1047.11	0.86~0.92	1764.80	35.72
C-2-10	7	0.165	101.13	23.97	133.44	1043.72	0.77~0.89	1774.22	37.38
C-2-11	6	0.165	170.53	74.26	253.31	1109.58	0.51~0.68	954.59	60.12

was applied to the parameterization of the regression analysis and the parameters were optimized by minimizing the mean difference between the measured and calculated gas holdup, which was defined as follows:

$${\epsilon }_1={\displaystyle \frac{1}{N}}{\sum }_{i=1}^N\left(a_{gi,cal}-a_{gi,actual}\right)$$

(16)

This study uses a genetic algorithm to optimize Equation (16). The genetic algorithm was first proposed by John Holland in the United States in the 1970s. The algorithm was designed and proposed according to the law of evolution of organisms in nature. It is a computational model of biological evolution that simulates Darwinian biological evolution’s natural selection and genetic mechanism. It is a method to search for the global optimal solution by simulating the natural evolution process. The fitting values of these parameters are 36.41, 1.07, 1.3, −35.21, 0.19, and 4.42, respectively.

The complete proposed correlation distribution coefficient is as follows:

$$C_0={\displaystyle \frac{36.41-1.07\times {\alpha }_g^{1.3}}{1+{\left(R{e}_m/1000\right)}^2}}+{\displaystyle \frac{-35.21-0.19\times {\alpha }_g^{4.42}}{1+{\left(1000/R{e}_m\right)}^2}}$$

(17)

2.3. Drift Velocity Analysis

Figure 5 shows the variation curves of different drift velocity models with gas holdup. In the figure, the horizontal coordinate is gas holdup, and the vertical coordinate is drift velocity. The models in the figure all consider gas holdup and fluid density. In addition, when the gas holding rate approaches 1, the drift velocity does not approach 0, and the overall prediction of Hibiki and Ishii (3) is low, which is too different from the actual value. The values of the other three models in the range of 0.5–0.55 are consistent with the actual values, but with the increase in gas holdup, only the Shi model has the same trend as the actual trend, and the error is small.

Combined with the optimized distribution coefficient model and Shi’s drift velocity model, it is better to apply the drift-flux model in medium and high gas holdup, and the final drift-flux model is given as follows:

$$C_0={\displaystyle \frac{36.41-1.07\times {\alpha }_g^{1.3}}{1+{\left(R{e}_m/1000\right)}^2}}+{\displaystyle \frac{-35.21-0.19\times {\alpha }_g^{4.42}}{1+{\left(1000/R{e}_m\right)}^2}}$$

(18)

$$V_d={\displaystyle \frac{\left(1-{\alpha }_g{C}_0\right)C_{0}K\left({\alpha }_g\right)V_c}{{\alpha }_g{C}_0\sqrt{\frac{{\rho }_g}{{\rho }_l}}+1-{\alpha }_g{C}_0}}$$

(19)

3. Gas–Water Two-Phase Numerical Simulations

To understand the state and parameter characteristics of gas–water two-phase flow in high-temperature and high-pressure gas wells, this study conducted numerical simulations using Ansys Fluent, a commercial CFD software. Unsteady numerical simulations of the two-phase turbulence in the inlet section of a vertical pipe were carried out using the fluid volume method.

For the study of complex two-phase flows, Taha and Cui [24] and Yang et al. [25] adopted the VOF method combined with the turbulence model. The VOF model has more advantages than the Euler model and the mixture model in solving the phase volume fraction and tracking the phase interface change. The mixture model is mainly suitable for simulating relatively simple multiphase flow, while the Euler model is more effective in simulating complex behavior such as fluid force and collisions received by particles, droplets, and bubbles in the flow process. This study selected the gas–liquid two-phase flow transient based on the previous results, and gravity was considered. The turbulence model uses the RNG k−ε model in the standard wall function method. Therefore, the multiphase flow model in this paper uses the VOF method to treat the two-phase mixture as a single fluid for which the properties change with the local volume fraction, and the scalar values of the turbulent kinetic energy k and the dissipation rate are shared by the two phases throughout the domain. The working pressure, outlet pressure, and inlet pressure were set to 50 MPa, 0 Pa, and 100 Pa, respectively, and a constant velocity and 10% turbulence intensity were used as boundary conditions at the liquid and gas inlet.

The second order upwind scheme was used for the two turbulence equations, and the PRESTO! scheme was used for the pressure interpolation. The format is used for pressure interpolation, and the pressure–velocity coupling was processed by the PISO algorithm. The method of surface meshing and expansion in the meshing module was used to complete the mesh division of the model. The calculation domain of the simulation was similar to that of Enrico Da Riva [26]. The Enrico Da Riva [26] model has an axisymmetric liquid inlet, while in this study, a one-sided liquid inlet was used to simulate the flow state when formation water was produced through perforation, as shown in Figure 6.

In the initial simulation case, the pipe is filled with gas, and the gas inlet and liquid inlet flow rates are set according to

Table 3. Fluid physical property parameters.

Mixture	Gas Superficial Velocity (m/s)	Diameter (m)	Water Superficial Velocity (m/s)	Temperature (°C)	Pressure (MPa)	Surface Tension (N/m)	Gas Density (kg/m3)	Gas Viscosity (Pa·s)	Water Density (kg/m3)	Water Viscosity (Pa·s)
Gas–Water	1	0.125	0.01/0.03/0.05/0.07/0.1	100	50	0.032	300	0.0000328	1100	0.000224
Gas–Water	1	0.165	0.01/0.03/0.05/0.07/0.1	100	50	0.032	300	0.0000328	1100	0.000224

. In this study, only gas carrying liquid upward flow is considered, and gas flows in the center, carrying entrained droplets. The liquid film overcomes gravity to flow upward due to the force exerted by the fast-moving gas core, and the liquid moves upward due to interface shear and the formation of “resistance” to the wave and resistance to the droplet. Turner et al. [27] believed that the gas velocity in the gas core in a gas lift well needs to be high enough to drive the entrained droplets. Otherwise, annular flow is impossible. Hinze [28] proposed that the minimum gas velocity required to suspend liquid droplets in gas lift well production was determined by the balance between gravity and resistance acting on the liquid droplets. At this time, Hinze [28] ignored the friction between the pipe wall and liquid, so the minimum gas velocity carrying liquid in the vertical pipe was obtained as follows:

$$U_g={\left({\displaystyle \frac{4K}{3C_d}}\right)}^{1/4}{\displaystyle \frac{\left[\delta g\right({\rho }_l-{\rho }_g){]}^{1/4}}{{\rho }_g^{1/2}}}$$

(20)

where $K$ is the critical Weber number, Turner (1969) [27] suggested 30 to 40; $C_d$ is the drag coefficient 0.44; and $g$ is gravity, 9.81. $\delta$ indicates the value of gas–water surface tension (0.032 N/m), ${\rho }_l$ indicates the value of water density (1100 kg/m³), and ${\rho }_g$ indicates the value of gas density (300 kg/m³). In this case, the $U_g$ value is calculated as 0.7 m/s.

This study fixed the apparent velocity of the gas flow velocity at 1 m/s, higher than that of gas–liquid at minimum speed, and the liquid velocity inlet apparent velocity was set at 0.01/0.03/0.05/0.07/0.1 m/s.

3.1. Wellbore Model

When the local formation water production reaches a specific value, the gas output of gas lift wells is significantly affected, reducing the economic benefits. Therefore, accurately finding water-producing layers and timely plugging can improve productivity. This paper uses Ansys Fluent numerical simulation software to conduct simulation experiments to observe the flow state of gas–water two phases during water production in the perforated layer, from which the basis for judging water production is obtained. The model established is shown in the figure. The gas phase inlet is at the bottom of the vertical pipe, and the 45 mm diameter water inlet is constructed on the side of the vertical pipe with a height of 250 mm from the bottom gas inlet. The physical models of bottom inlet diameters of 165 mm and 125 mm were established, respectively, and the simulation experiments were carried out under different gas and liquid phase velocities. The experimental scheme is shown in Table 3.

3.2. Fluid Parameter Calculation

The theoretical correlation of gas density can be obtained according to the gas state equation, as follows:

$$PV=ZnRT=Z{\displaystyle \frac{m}{M}}RT$$

(21)

$${\rho }_g={\displaystyle \frac{pM}{ZRT}}$$

(22)

where $M$ is the molar mass of a substance, $\mathrm{k}\mathrm{g}/\mathrm{m}\mathrm{o}\mathrm{l}$; $R$ is universal gas constant, $\mathrm{J}/(\mathrm{m}\mathrm{o}\mathrm{l}·\mathrm{K})$; $Z$ is compressibility factor of gas, which can be obtained in PVT experiment; and $T$ is the temperature of gas, $K$.

The downhole viscosity of natural gas can be calculated by the following correlation, which is related to density:

$${\mu }_g={10}^{-7}K{e}^{x{\rho }_g^y}$$

(23)

$$K={\displaystyle \frac{(9.4+0.02M_g)(1.8T{)}^{1.5}}{209+19M_g+1.8T}}$$

(24)

$$x=3.5+{\displaystyle \frac{986}{1.8T}}+0.01M_g$$

(25)

$$y=2.4-0.2x$$

(26)

where $T$ is the absolute temperature of gas, $K$; ${\mathsf{\mu }}_{\mathrm{g}}$ is the viscosity of natural gas, $\mathrm{P}\mathrm{a}·\mathrm{s}$; and $M_g$ is the molar mass of gas.

The density of formation water is mainly affected by temperature, pressure, and the salinity of formation water. Dissolved gas can reduce the density of surface water, but the effect is not significant because dissolved gas in water is relatively small. The commonly used correlation for calculating ${\rho }_w$ is the following:

$${\rho }_l={\rho }_{l_1}/(62.4{\rho }_{l_2}\ast {\rho }_{l_3})$$

(27)

$${\rho }_{l_1}={10}^{\left(3.05\times {10}^{-7}{K}_w+1.745\right)}$$

(28)

$${\rho }_{l_2}=1-1.063\times {10}^{-6}{T}^2-1.87\times {10}^{-5}T$$

(29)

where $T$ is fluid temperature, $K$; $K_w$ is formation water salinity, $\mathrm{p}\mathrm{p}\mathrm{m}$.

The viscosity of downhole formation water is related to temperature, pressure, and the salinity of formation water, as follows:

$${\mu }_w={\mu }_{w_1}{\mu }_{w_2}/T$$

(30)

$${\mu }_{w_1}=58.4+0.00022K_w$$

(31)

$${\mu }_{w_2}=1+3\times {10}^{-11}\left(T-40\right)P^{1.755}$$

(32)

where $P$ is the fluid pressure, MPa.

Based on the above correlation and field conditions, the physical property parameters of the fluid were obtained by calculation, as shown in Table 3.

3.3. Gas–Water Two-Phase Flow and Velocity Analysis

Figure 7 and Figure 8 show the volume fraction cloud maps at different formation water apparent velocities in the 125 mm and 165 mm boreholes, respectively. In the cloud image, red is the water phase, and blue is the gas phase. Formation water with different apparent velocities enters the wellbore from the side inlet. Due to friction and gravity between the gas and liquid, natural gas brings water formation upward. During the upward movement, the water phase is dispersed by the gas phase with higher velocity. With the increase in formation water inlet velocity, the flow rate of the produced formation water becomes higher and higher, the proportion of the integral water phase in the wellbore becomes higher and higher, and the overall flow pattern is mist flow.

By comparing the gas–water two-phase flow state under the conditions of 125 mm pipe diameter and 165 mm pipe diameter, it is found in the figure that the simulated flow conditions of the two groups are similar when the apparent velocity of the formation water inlet is 0.01 m/s. The gas carries water upward and is distributed randomly in the wellbore. When the apparent velocity of the water inlet increases to 0.03 m/s or higher, the simulated flow conditions of the two groups are similar. In the 165 mm wellbore, a small amount of water accumulates and moves downward. By comparison, in the 125 mm wellbore, this phenomenon occurs only when the water phase velocity inlet reaches 0.1 m/s, indicating that the larger the pipe diameter, the more likely formation water is to accumulate and fall back. In addition, the water phase in the 125 mm pipe diameter is more likely to gather into clusters or strips during upward movement. By comparison, the water phase in the 165 mm pipe diameter is relatively dispersed and moves upward in a small blister state.

Figure 9 and Figure 10 are the cloud images of gas–water phase mixture velocity under different apparent formation water velocities in a 125 mm wellbore and 165 mm wellbore, respectively. When the gas rises at 1 m/s and reaches 0.25 m of the wellbore, the water phase enters the wellbore at a certain speed and contacts with the gas, resulting in obvious changes in the gas–liquid junction velocity and a larger gas phase velocity. The water phase at the entrance still moves at the original speed, and some liquid phase increases in speed after being carried upward by the gas phase, but it does not reach the same speed as the gas phase. When some liquid phase passes through the center of the wellbore during upward movement, the mixture speed will decrease. Therefore, when a full-hole turbine flowmeter is used in the actual field to measure in the center, the turbine speed value will fluctuate from side to side rather than increasing steadily.

In the case of 0.01 m/s water phase inlet apparent velocity, due to the small flow rate of the water phase and its small proportion in the wellbore, the gas phase flows faster in the center of the wellbore. The water phase influences it less, so the overall flow velocity of the gas and water phases is relatively stable. With the increasing flow rate of formation water entering the wellbore, the mixture velocity becomes more extensive at the first contact position of the gas and water phases. Secondly, the upper part of the gas–water two-phase flow (wellbore longitudinal 1 m~2 m) is affected. More water phase enters the wellbore, carried by the gas phase, and rises in the upper part of the wellbore. Due to the slow speed of the water phase, the overall mixture speed is affected and cannot rise at a stable speed.

In addition, we also evaluated the velocity of the wellbore center. Figure 9 and Figure 10 show the wellbore center’s mixture velocity under different apparent water phase velocities in 125 mm and 165 mm pipe diameter, respectively. Similar to the apparent velocity measured by downhole instruments in the center, there is a common point in different pipe diameters: when the water phase starts to enter the wellbore at 0.25 m of the wellbore longitudinal axis, the central velocity increases obviously, and then the gas–liquid phase moves to 0.55 m, and the apparent velocity decreases significantly. At the distance of 0.55 m~2 m, the apparent velocity value fluctuates continuously back and forth, and a jumping curve is shown in the image.

By comparing the cloud image of the mixture velocities of 125 mm and 165 mm bores, we can find that the influence of the water relative mixture velocity in the 125 mm bore is more significant than that in the 165 mm bore. The apparent velocity of the 0.25 m–0.55 m longitudinal wheelbase in the 125 mm bore is significantly greater than that in the 165 mm bore. Figure 11 and Figure 12 show that the apparent velocity fluctuates more significantly between 0.55 and 2 m. This result corresponds to the cloud map of phase volume fraction, and the water phase in the 125 mm wellbore is more concentrated, resulting in a more chaotic overall mixture velocity.

In summary, when the water phase enters the wellbore and contacts the gas phase, the mixture velocity increases significantly; with continuous movement of the water phase and its dispersion in the wellbore, the mixture velocity around the water phase decreases. Therefore, in the actual measurement, the turbine curve shows left and right fluctuations; the apparent velocity is unstable, shown as jagged in the log image. It is difficult for density and water retention instruments to measure water phase characteristics in wells with low water production, so the velocity curve can assist in judging whether formation water is produced, which is of great significance for field interpretation and guidance.

4. Results and Discussion

4.1. Comparison of Drift-Flux Model Performance

Equation (4) for predicting gas holdup is obtained by combining the distribution coefficient $C_0$, mixture velocity, and drift velocity. In this paper, 130 data points were selected from actual well conditions with diameters of 125 mm and 165 mm, and gas holdup ranged from 0.5 to 1.0 (Table 1). Based on these data, the accuracy of the parameters of the drift-flux model proposed by scholars was verified, and the quantitative evaluation performance of the proposed model and the mature drift-flux model was compared with the example well interpretation. The correlations proposed by Hibiki and Ishii [11], Choi [19], and Liu [20] were chosen as comparative correlations for performance estimation, all of which claim to capture the $C_0$ behavior as the flow pattern changes. In addition to the mean difference ${\epsilon }_1$ in Equation (33), statistical parameters such as the mean relative absolute difference ${\epsilon }_2$ and the standard deviation ${\epsilon }_3$ between the measured and calculated porosity values were used.

Mean difference ${\epsilon }_1$:

$${\epsilon }_1={\displaystyle \frac{1}{N}}{\sum }_{i=1}^N\left(a_{gi,cal}-a_{gi,actual}\right)$$

(33)

Mean absolute relative difference ${\epsilon }_2$:

$${\epsilon }_2={\displaystyle \frac{1}{N}}{\sum }_{i=1}^N\left|{\displaystyle \frac{{\alpha }_{gi,cal}-{\alpha }_{gi,actual}}{{\alpha }_{gi,actual}}}\right|$$

(34)

Standard deviation ${\epsilon }_3$:

$${\epsilon }_3=\sqrt{{\displaystyle \frac{1}{N-1}}{\sum }_{i=1}^N{\left[\left({\alpha }_{gi,cal}-{\alpha }_{gi,actual}\right)-{\epsilon }_1\right]}^2}$$

(35)

where $N$ and $i$ are the total number of experimental data points and the i-th data point, respectively.

Figure 13 illustrates the performance of the proposed and compared correlations concerning the measurement of gas holdup at different pipe diameter data points. The long and short dashed lines are the ±20% and ±10% error bands, respectively. Overall, the proposed correlations showed the best performance in the comparative correlations of different pipe diameters. In the case of a 125 mm pipe diameter, the data points of correlation prediction proposed in this paper were 92.3% and 80.5% within the error bands of ±20% and ±10%, respectively. Liu’s correlation (2020) was better than the other correlations. More than 85% of the data points were in the ±20% range, which was slightly lower than that of the proposed correlation, while only 60% of the data points in Choi and Hbiki’s correlation were in the ±20% range. In the case of a pipe diameter of 165 mm, due to the more complex changes of gas–water two-phase flow patterns in a large pipe diameter, the prediction accuracy of all relations was lower than that of a small pipe diameter.

The predicted value of the correlation proposed in this paper was low in the range of 0.5~0.75 gas holdup and high in the range of 0.75~1.0. The correlation of Liu also had the same trend, which was low in the range of 0.5~0.85 and was more accurate only in the range of 0.85~1.0 high gas holdup because the distribution coefficient of Liu was high in the range of 0.5~0.85, resulting in low gas holdup prediction. The overall accuracy of Hbiki and Ishii was low, and the predicted value was lower in the high gas holdup range, but it was more accurate than that in the low gas holdup range. Choi’s prediction values tended to increase first and then decrease, with a significant error between 0.5 and 0.75 and 0.9 and 1.0, with only 60.4% of the data points in the ±20% range. The reason is that the predicted value of the Choi distribution coefficient was low in the range of 0.5~0.75. With the increase in gas holdup, the distribution coefficient did not approach 1.0 but 1.1, resulting in a low prediction of gas holdup between 0.9 and 1.0. Figure 14 shows the histogram distribution of the ±20% and ±10% errors of 125 mm and 165 mm pipe diameters. It is evident that the prediction accuracy of the proposed equation for gas holdup is significantly higher than that of other equations, and the effect is remarkable for practical applications.

Table 4. The quantitative performance of different correlations in terms of pipe diameter.

Correlations	D = 0.125 (70 Data)					D = 0.165 (60 Data)
	$\pm 20\%$	$\pm 10\%$	${\epsilon }_1$	${\epsilon }_2$	${\epsilon }_3$	$\pm 20\%$	$\pm 10\%$	${\epsilon }_1$	${\epsilon }_2$	${\epsilon }_3$
Proposed correlation	92.3%	80.5%	−0.003	0.009	0.014	90.7%	83.0%	−0.004	0.09	0.018
Hibiki and Ishii	68.4%	54.2%	−0.028	0.027	0.034	65.2%	50.4%	0.085	0.31	0.075
Choi et al. [19]	60.4%	35.1%	−0.011	0.01	0.059	55.8%	33.9%	−0.098	0.62	0.081
Liu et al. [20]	85.1%	74.2%	0.010	0.048	0.021	82.4%	70.5%	0.0078	0.14	0.068

shows the quantitative performance of different correlations in terms of pipe diameter.

4.2. Analysis of Field Well Production Profile

In this study, well A-1, with a formation temperature of 146.33 °C, formation pressure of 83.93 MPa, and pipe diameter of 165 mm in A specific area, was selected, which produced gas–water two-phase fluids. Figure 15 shows the output profile log diagram of well A-1. The measuring instrument was a multi-parameter logging instrument, which completed continuous measurements at 5, 10, 15, 20, 25, and 30 m/min speeds, respectively. The measured parameters included depth, temperature, pressure, CCL, GR, capacitance water retention, fluid density, and turbine speed. The figure has seven curves, from the first to the seventh, respectively: depth, production level, perforation level, base curve, production curve, mixture speed, and turbine speed. The wellhead of well A-1 produced two phases of gas and water. The numerical simulation of downhole high-temperature and high-pressure fluid flow characteristics and the proposed model was applied to the actual well interpretation, and the flow interpretation accuracy of multiple models was compared.

In interpreting the gas well production profile, GR is used to measure depth correction in the basic curve, and CCL is used to judge the position of the casing collar. The production interval is usually determined by combining the changes in temperature, density, capacitance, and turbine speed in the production curve.

(1)

From 5637 m to 5642 m, the temperature curve did not change significantly, the pressure curve was trapezoidal, and the pressure gradient did not increase significantly. The water retention curve was straight, and the value was stable at about 6098.48 cps. Meanwhile, the density curve was also straight, and the value was stable at 1.15 g/cm³. Moreover, the turbine curve did not increase significantly, which shows that no fluid was produced in this section, and water was still at the bottom of the well.

(2)

The temperature curve significantly rose from 5635 m to 5637 m, with the value increasing from 146.37 °C to 146.61 °C. The gradient of the pressure curve was significantly larger, indicating liquid output because when the liquid entered the wellbore from the formation, the capacitance water holding rate increased significantly. The value increased from 6098.48 cps to 8770.79 cps, and the fluid density curve decreased significantly from 1.15 g/cm³ to 0.35 g/cm³, indicating a large amount of gas output. The mixture velocity curve fluctuated significantly from left to right, showing a zigzag shape. This phenomenon was characterized by the upward movement of the two gas and water phases, and the fluid produced in this layer was synthetically analyzed as the two phases of gas and water.

(3)

From 5633 m to 5635 m, the temperature curve fluctuated slightly, and the mixture velocity curve zigzagged because the output of the two gas and water phases in the lower layer moved upward, and the rest of the curve did not change significantly, which showed that no fluid was produced in this section.

(4)

From 5631 m to 5633 m, the temperature curve decreased slightly. However, the density curve showed an increase in “peak.” Then, it decreased to the original value, indicating that this layer may have been a situation where gas and water flowed together. The mixture velocity in this layer first increased significantly and then decreased, showing a zigzag phenomenon, and the other curves had no prominent fluid output characteristics. Comprehensive analysis showed that formation water entered the wellbore at a certain speed, and the PLT instrument in the middle measured contact with formation water, resulting in a sudden increase in the density response value. At the same time, natural gas production was much higher than water formation, and the density value decreased to 0.3 g/cm³. In addition, combined with the numerical simulation results, formation water entered the wellbore and contacted upward-moving natural gas, increasing the mixture speed. The mixture velocity curve of this layer was consistent with the numerical simulation conclusion, so the produced fluid in this layer was the two phases of gas and water.

To interpret the output profile of high-temperature and high-pressure gas wells, conventional production logging curves (temperature, pressure, density, capacitance water holdup) are not enough to judge the type of produced fluid in some intervals. In this case, numerical simulation software is used to observe the fluid flow status and velocity changes, which can better help interpreters to complete the interpretation of the output profile.

Table 5. Production comparison.

Correlation	Calculated Gas Production (m3/d)	Calculated Water Production (m3/d)	Wellhead Metering Gas Production (m3/d)	Relative Error of Gas Production	Wellhead Metering Water Production (m3/d)	Relative Error of Water Production
Proposed correlation	698,451	12.68	683,258	0.22%	10.62	19.39%
Hibiki and Ishii	633,251	65.15		7.31%		513.46%
Choi et al. [19]	611,548	80.21		10.49%		655.27%
Liu et al. [20]	653,184	30.15		4.41%		183.89%

shows the comparison results between the calculation of each model’s gas–water flow and surface metering flow. The calculation accuracy of the model proposed in this paper was relatively high, and the relative error of gas production was 0.22%, while that of water production was 19.39%. The calculation error of the other models was enormous compared with the actual results, so the model proposed in this paper was more suitable for calculating the flow of high-temperature and high-pressure gas wells.

5. Conclusions

Improved drift-flux correlations for gas–liquid two-phase flow conditions in high-temperature and high-pressure vertical wells are presented in this paper. The following conclusions can be drawn:

1. Compared with drift-flux models studied by many scholars, the distribution coefficient relationship with excellent performance was optimized based on the actual logging data points in the field. The drift velocity relationship suitable for field applications was selected, and the drift-flux model, suitable for interpreting high-temperature and high-pressure gas wells, was finally obtained.

2. Using numerical simulation software, the gas–water two-phase numerical simulations with different pipe diameters and different water yields were carried out under the conditions of 100 °C and 50 MPa, and the gas–water flow state when different formation water flows entered the wellbore and the change in mixing velocity were given, which provided a strong basis for judging the water level in the actual production well section.

3. Compared with the other three mature drift-flux correlations, the proposed correlation shows equivalent and better performance in predicting void fraction. In the actual field gas-water two-phase interpretation, the relative errors of each phase flow in the gas-water two-phase quantitative calculation reach 12.68% and 19.39% respectively, which are far lower than the other drift-flux correlations.