Evaluating the temperature profile and pressure drops in gas-lift wells via coupled temperature–pressure analysis

Abstract

The gas-lift technique is a widely implemented artificial lifting method that has been optimized through operational parameter adjustments and structural improvements. While accurately predicting the pressure drop and temperature distribution in the wellbore is critical for improving this process. However, current research on temperature–pressure coupling in gas-lift wellbores remains limited. This paper presents a comprehensive temperature–pressure coupled analysis to evaluate the temperature distribution and pressure drops in gas-lift wells. This study incorporates key factors such as gas injection temperature and volume, tubing diameter, and crude oil production into a fully coupled model that accounts for gas–water-oil three-phase flow regime within the tubing. By resolving the coupled heat transfer and hydrodynamic equations, the model accurately predicts liquid flow regimes and pressure drops under varying process parameters. The model was validated using data from published literature, and sensitivity analyses were conducted to ensure the model’s reliability and to identify critical parameters influencing temperature and pressure distributions. These findings provide theoretical guidance for the design and operation of gas-lift wells, enhance gas-lift efficiency, and offer significant practical implications for optimizing gas injection strategies and improving production rates in complex reservoirs.

Introduction

As an artificial lifting technology, the gas-lift technique only ranks second to sucker-rod pumping based on the number of installations throughout the world. This versatile technology has evolved and matured almost 160 years^1,2. In addition to crude oil lifting, the gas-lift technique is extensively used in drilling and well repair operations. For example, the gas-lift dual gradient drilling, as a part of managed pressure drilling (MPD) technology, addresses challenges encountered in deep water and also in the reservoirs with narrow drilling windows, where the pore pressure and fracture pressure are nearly the same³. Furthermore, the gas-lift technique is also employed to remove liquids from natural gas wells thereby enhancing natural gas production⁴. Recently, several oilfield services companies have attempted to use continuous gas-lift systems to unclog deep and low-pressure wells, specifically aiming to remove high-density condensate oil from the wellbore. The extensive use of gas-lift technology has made the optimization of operational parameters and the redesign of critical components, such as gas lift valves, a persistent area of interest in research^5,6,7. The gas-lift technique has the characteristics of high economic benefits, simple and reliable construction, free control of the depth of descent. However, the process encounters certain limitations, particularly in the cleaning of condensate oil at the bottom of the well. Field practice shows the cooling effect of mixing crude oil with low-temperature injection gas when using coiled tubing for gas lift. This will make the viscosity of heavy oil increase, leading to difficulties in draining the fluid, and cannot truly evaluate the production capacity. In addition, determining the optimal valve position in the annulus and predicting the best injection gas volume, among other critical factors, relies heavily on the temperature profile⁸. Blann and Williams⁹ demonstrated that gas injection pressure profoundly influences the depth of gas injection, compression requirements, and ultimate well productivity. Their work highlights that higher injection pressures enable deeper gas penetration near the formation, reducing fluid column density and maximizing reservoir drawdown, thereby minimizing compressor horsepower while boosting production rates. Janadeleh et al.¹⁰ demonstrated via PIPESIM simulations that gas lift optimization for a well in Iran’s Yaran field achieved a peak production of 2583 STB/D at 2000 psig injection pressure and 3 MMSCF/D injection rate. Exceeding these thresholds led to diminishing returns, while high compression costs and equipment pressure limits further reduced its economic feasibility in deep reservoirs. Abdollahi et al.¹¹ demonstrated that maintaining higher bottom-hole pressures (e.g., 1000–2000 psi) and optimizing gas injection rates (e.g., 7.5 MMSCFD) significantly improve cumulative oil production, supported by an artificial neural network (ANN) model with prediction errors as low as 2%. Collectively, these studies underscore the critical role of precise temperature and pressure management in enhancing gas lift efficiency and economic viability. Therefore, it is of great practical significance to study the temperature field of wellbore with the assisted lift of coiled tubing gas lift. In addition, in order to improve liquid production efficiency and enhance reservoir stimulation, it is necessary to maintain a sufficient pressure drop between the bottomhole flowing pressure and the reservoir pressure. Moreover, the flow regime inside the tubing plays a decisive role in determining the bottomhole flowing pressure. Therefore, evaluating the flow regime inside the tubing and accurately predicting pressure drops of gas-lift wells also plays a crucial role. The most critical issue is that the complex two-phase flow inside the oil pipe is coupled with its temperature field. In this paper, the temperature profile and pressure drop in gas-lift wells was studied to provide suggestions and guidance for field construction of gas lift process.

Many scholars^12,13 have studied the wellbore temperature field. A representative example is the work of Sagar et al.¹⁴ in which a simple model suitable for hand calculations is presented to predict temperature profiles in two-phase flowing wells. This model is developed with measured temperature data from 392 wells. Alves et al.¹⁵ developed a general model for predicting flowing temperature in wellbores and pipelines. The model is general and unified and can be applied to multiphase production systems over the entire inclination angle range, from horizontal to vertical, with compositional and black-oil fluid models. Yan Zhou et al.¹⁶ has proposed an efficient and accurate accelerated modeling method, offering a novel solution to the multiphase flow problems in oil and gas extraction. Hasan et al.¹⁷ developed a method that incorporate a new solution of the thermal diffusivity equation and the effect of both conductive and convective heat transport for the wellbore/formation system. For the multiphase flow in the wellbore, the Hasan-Kabir model has been adapted. These research works have laid the foundation for the study of temperature fields in gas lift wells. Hasan and Kabir¹⁸ present a mechanistic model that predicts the flowing temperature of the annular gas and the gas/liquid two-phase mixture in the tubing. This model is based on both well depth and production time, and it is applicable regardless of the well deviation angle. The main innovation of their work is the analytical solution, which allows for the flexibility to input any temperature distribution function. This enables the modeling of unsteady heat transfer in the formation. Their research results demonstrate that the temperature profiles in both the annulus and the tubing exhibit nonlinear characteristics. Building upon similar methodologies, Liao et al.¹⁹ also explored the temperature profile of gas lift wells. However, the gas temperature inside the annulus was assumed to be a linear gradient distribution, which simplifies the temperature field model. Jiang²⁰ and Mu et al.²¹ independently developed the same type of mechanistic model for computing flowing fluid temperature profiles in both conduits simultaneously during continuous-flow gas-lift operations. Their models assume steady heat transfer in the formation as well as in the conduits. They analyzed the heat transfer process in micro-units discretized from the wellbore and established the heat transfer equation according to the law of conservation of energy. Although the aforementioned research works treated the medium within the tubing as a gas–liquid two-phase fluid, they did not consider the changes in gas–liquid ratio, at different depths within the tubing. It needs to be studied by combining the pressure drop model of two-phase flow.

The challenge of accurately predicting pressure drops in either flowing or gas lift wells has increased²². Many particular solutions for specific conditions are available but none is generally accepted as a comprehensive solution for any condition. The reason is that the analysis of the two-phase flow is very complex and difficult even for a particular condition due to a large number of variables involved. The difference in velocity and the geometry of the two phases strongly influence pressure drop. For calculating the pressure gradient of two-phase flow in the wellbore, several well-known methods include the Beggs-Brill Correction Method²³, Mukherjee Brill Method²⁴, Hasan Method²⁵, Orkiszewski Method²⁶, Hagedorn-Brown Method²⁷, Aziz Method²⁸, etc. Building on previous work, Liao et al.²⁴ proposed a novel calculation method tailored for conditions with high oil-to-gas ratios. Measured data have been utilized to demonstrate the high accuracy of this approach. The literature²⁹ shows that the method developed by Orkiszewski is relatively high adaptable and accurate, due to the synthesis of research from multiple scholars and the refinement of the pressure drop calculation model. Dong et al.³⁰ developed an improved Orkiszewski’s model by correcting the threshold value of liquid distribution coefficient using particle swarm optimization method, with the aim of reducing the prediction errors of annular-mist, slug and annular-slug transition flow regime. However, a common limitation of these methods is the assumption of a linear gradient distribution for the temperature field within the computational domain.

Few scholars have delved into the study of temperature–pressure coupling in gas lift wells, yet their contributions have significantly advanced our understanding of multiphase flow regime and their implications for well operations. For example, Zhong et al.³¹ presented a coupling model for wellbore pressure and temperature gradients in gas lift operations, where light oil is injected from the annulus. Despite the presence of gas–liquid two-phase flow in both the annulus and the tubing, their applied pressure gradient model simplifies the scenario by neglecting variations in mixture fluid density, viscosity, and surface tension, which may affect the model’s accuracy and applicability.

Gao et al.³² developed a temperature calculation model that considers a pure gas phase inside the coiled oil pipe and a gas–liquid phase outside. This model takes into account the effects of temperature and pressure on physical parameters such as density and viscosity, thereby enhancing the precision of the results. However, it does not account for the influence of the gas–liquid two-phase flow regime in the annulus on the pressure drop, representing a notable limitation. Dong Xiao et al.³³ established a coupling model of wellbore heat transfer and cuttings bed height during the drilling process and analyzed the impacts of circulation flow rate and inlet temperature on wellbore heat transfer.

The wellbore temperature and the flow pressure at the well bottom significantly influence the operational parameters of gas-lift assisted processes, including oil recovery, condensate oil backflow, and reservoir dredging. To enhance the accuracy of analyzing temperature and pressure distribution in gas-lift wells, this paper develops a fully coupled temperature–pressure model. This model integrates bidirectional coupling between multiphase flow regime, and heat transfer mechanisms, to predict the fluid flow regimes and pressure drops in gas-lift wells. This model considers gas‒water‒oil three phase flow within the tubing, as well as the effects of temperature and pressure on the physical parameters and flow regime. Utilizing the established model, the influence of gas injection temperature, gas injection volume, tubing diameter, and crude oil production on wellbore temperature distribution and flow pressure are systematically analyzed. These findings offer valuable theoretical guidance for optimizing gas-lift-assisted production processes.

Establishing the temperature‒pressure coupled model

Overview of gas-lift assisted production and model assumptions

Figure 1 shows a classic configuration of a gas lift well with valves installed on the tubing string³⁴. It implies that the basic components of gas lift wells are tubing string with gas injection valves, downhole chamber, gas injection chokes and controllers, and gas compression station.

Fig. 1

Configuration of a gas lift well with valve installation.

Calculation of temperature

Basic assumptions

During gas lift operation, gas is injected down the annulus to aerate the liquid to be lifted up. According to the system architecture of gas lift well, the simplified wellbore profile is shown in Fig. 2a, and Fig. 2b shows the subsurface equipment configuration of the wellbore. Following a comprehensive review of the existing literature on wellbore heat transfer, the assumptions are as shown below.

• (a) The heat transfer within the wellbore is in a steady state.

• (b) Because the thermal resistance of the casing is minimal compared with that of the cement sheath, it is justifiably ignored in our analysis.

• (c) The temperature gradient is not affected by the fluids in the wellbore.

• (d) Only radial heat transfer, without considering axial heat transfer and heat radiation is considered.

• (e) The heat generated by friction between the heavy oil flow and the pipe wall is considered.

Fig. 2

Schematic of the wellbore profile and heat transfer.

Modeling of the annular temperature field

For an annular control volume with a axial length $\Delta L$, the heat transfer is depicted in Fig. 2c. The gas medium, as it enters and exits the control volume, brings in and carries out heat. Heat conduction in the annular region is primarily composed of two components: conduction between the annulus and the tubing, and conduction between the annulus and the surrounding formation. Additionally, friction between the gas and the inner wall of the casing, as well as the outer wall of the tubing, will generate heat. By analyzing the heat transfer of the control volume, the following heat balance equation can be obtained:

$$Q_{{\text{a,in}}} + q_{{\text{f}}} + q_{{\text{t}}} + q_{{\text{a,Fri}}} - Q_{{\text{a,out}}} = 0$$

(1)

where $Q_{{\text{a,in}}}$ represents the heat input by incoming gas, J; $Q_{{\text{a,out}}}$ represents the heat output by outgoing gas, J; $q_{{\text{f}}}$ represents the heat transferred from the formation, J; and $q_{{\text{t}}}$ represents the heat transferred from the tubing, J; $q_{{\text{a,Fri}}}$ represents the heat generated by friction, J. Each term can be expressed as follows:

$$\left\{ {\begin{aligned}& {Q_{{\text{a,in}}} = C_{{\text{a}}} \dot{m}_{{\text{a}}} T_{{{\text{a,}}L}} \Delta t} \hfill \\ &{q_{{\text{f}}} = {\uppi }D_{{\text{c}}} \Delta LK_{{\text{w}}} \left( {\frac{{\partial T_{{\text{c}}} }}{\partial r}} \right)\Delta t} \hfill \\ &{q_{{\text{t}}} = {\uppi }D_{{\text{t}}} \Delta LK_{{\text{t}}} \left( { - \frac{{\partial T_{{\text{t}}} }}{\partial r}} \right)\Delta t} \hfill \\ &Q_{{\text{a,out}}} = C_{{\text{a}}} \dot{m}_{{\text{a}}} T_{{{\text{a,}}L + \Delta L}} \Delta t \hfill \\& q_{{\text{a,Fri}}} = \frac{{\lambda_{{\text{a}}} \dot{m}_{{\text{a}}} v_{{\text{a}}}^{2} }}{{4r_{{\text{c}}} }}\Delta L\Delta t \hfill \\ \end{aligned} } \right.$$

(2)

where $C_{{\text{a}}}$ represents the heat capacity of medium in annulus in point L, J/kg °C; $\dot{m}_{{\text{a}}}$ represents the mass flow rate of medium in annulus in point L, kg/s. $T_{{{\text{a,}}L}}$ represents the temperature in annulus in point L, °C; $\Delta t$ represents the flow time, s; $D_{{\text{c}}}$ represents the outer diameter of casing, m. $K_{{\text{w}}}$ represents the thermal conductivity of cement, W/m·°C; $D_{{\text{t}}}$ represents the outer diameter of tubing, m. $K_{{\text{t}}}$ represents the thermal conductivity of tubing, W/m·°C; $T_{{{\text{a,}}L + \Delta L}}$ represents the temperature in annulus in point $L + \Delta L$, °C; $\lambda_{{\text{a}}}$ represents the friction coefficient between annular fluid and pipe wall; $v_{{\text{a}}}$ represents the velocity of the annular fluid, m/s; $r_{{\text{c}}}$ represents the equivalent inner diameter of casing, m; $\frac{{\partial T_{{\text{t}}} }}{\partial r}$ represents the temperature gradient of tubing, °C/m; $\frac{{\partial T_{{\text{c}}} }}{\partial r}$ represents the temperature gradient of cement, °C/m. The expressions for these two derivative terms are as follows:

$$\left\{ {\begin{aligned}& {\frac{{\partial T_{{\text{t}}} }}{\partial r} = \frac{{T_{{{\text{a}},L}} - T_{{{\text{t}},L}} }}{{\frac{{D_{{\text{t}}} - d_{{\text{t}}} }}{2}}}} \\& {\frac{{\partial T_{{\text{c}}} }}{\partial r} = \frac{{T_{{{\text{g}},L}} - T_{{{\text{a}},L}} }}{{\frac{{D_{{\text{w}}} - D_{{\text{c}}} }}{2}}}} \\ \end{aligned} } \right.$$

(3)

where $T_{{{\text{t}},L}}$ represents the temperature in tubing in point L, °C; $d_{{\text{t}}}$ represents the inner diameter of tubing, m; $T_{{{\text{g}},L}}$ represents the temperature in formation in point L, °C; $D_{{\text{w}}}$ represents the outer diameter of cement, m; $d_{{\text{c}}}$ represents the inner diameter of casing, m. By substituting Eqs. (2) and (3) into Eq. (1), the thermal equilibrium equation for the annular control volume can be obtained as follows:

$$\begin{aligned} & \left( {\frac{1}{\Delta L} - \frac{{2{\uppi }D_{{\text{c}}} K_{{\text{w}}} }}{{C_{{\text{a}}} \dot{m}_{{\text{a}}} \left( {D_{{\text{w}}} - D_{{\text{c}}} } \right)}} - \frac{{2{\uppi }D_{{\text{t}}} K_{{\text{t}}} }}{{C_{{\text{a}}} \dot{m}_{{\text{a}}} \left( {D_{{\text{t}}} - d_{{\text{t}}} } \right)}}} \right)T_{{{\text{a}},L}} - \frac{1}{\Delta L}T_{{{\text{a,}}L + \Delta L}} + \frac{{2{\uppi }D_{{\text{t}}} K_{{\text{t}}} }}{{C_{{\text{a}}} \dot{m}_{{\text{a}}} \left( {D_{{\text{t}}} - d_{{\text{t}}} } \right)}}T_{{{\text{t}},L}} \\ & \quad + \frac{{2{\uppi }D_{{\text{c}}} K_{{\text{w}}} }}{{C_{{\text{a}}} \dot{m}_{{\text{a}}} \left( {D_{{\text{w}}} - D_{{\text{c}}} } \right)}}T_{{{\text{g}},L}} + \frac{{\lambda_{{\text{a}}} v_{{\text{a}}}^{2} }}{{4r_{{\text{c}}} C_{{\text{a}}} }} = 0 \\ \end{aligned}$$

(4)

Modeling of the tubing temperature field

For a tubing control volume with a depth of L and an infinitesimal length of $\Delta L$, the heat transfer is depicted in Fig. 2d. As the fluid medium enters and exits the control volume, it brings in and carries out heat. The primary mode of heat transfer within this volume is conduction between the tubing and the annulus. Additionally, heat is generated due to friction between the fluid medium and the inner wall of the tubing. The heat balance equation of the control volume can be obtained as follows:

$$Q_{{\text{t,in}}} - q_{{\text{t}}} - Q_{{\text{t,out}}} + q_{{\text{t,Fri}}} = 0$$

(5)

where $Q_{{\text{t,in}}}$ represents the heat input by incoming gas, J; $Q_{{\text{t,out}}}$ represents the heat output by outgoing gas, J; $q_{{\text{t}}}$ represents the heat transferred to the annulus, J; and $q_{{\text{t,Fri}}}$ represents the heat generated by friction, J;. Each term can be expressed as follows:

$$\left\{ {\begin{aligned}& {Q_{{\text{t,in}}} = C_{{\text{t}}} \dot{m}_{{\text{t}}} T_{{{\text{t,}}L + \Delta L}} \Delta t} \hfill \\ & {q_{{\text{t}}} = {\uppi }D_{{\text{t}}} \Delta LK_{{\text{t}}} \left( { - \frac{{\partial T_{{\text{t}}} }}{\partial r}} \right)\Delta t} \hfill \\ & {Q_{{\text{t,out}}} = C_{{\text{t}}} \dot{m}_{{\text{t}}} T_{{{\text{t,}}L}} \Delta t} \hfill \\ &{q_{{\text{t,Fri}}} = \frac{{\lambda_{{\text{t}}} \dot{m}_{{\text{t}}} v_{{\text{t}}}^{2} }}{{4d_{{\text{t}}} }}\Delta L\Delta t} \hfill \\ \end{aligned} } \right.$$

(6)

where $C_{{\text{t}}}$ represents the heat capacity of the medium in the tubing, J/kg·°C; and $\dot{m}_{{\text{t}}}$ represents the mass flow rate of the medium in the tubing, kg/s. $T_{{{\text{t,}}L}}$ represents the temperature in the tubing at point L, °C; $T_{{{\text{t,}}L + \Delta L}}$ represents the temperature in the annulus at point $L + \Delta L$, °C; $\lambda_{{\text{t}}}$ represents the friction coefficient between the fluid and pipe wall; $v_{{\text{t}}}$ represents the velocity of the fluid in the tubing, m/s; and $d_{{\text{t}}}$ represents the inner diameter of the tubing, m. By substituting Eqs. (3) and (6) into Eq. (5), the thermal equilibrium equation for the tubing control volume can be obtained as follows:

$$\frac{1}{\Delta L}T_{{{\text{t,}}L + \Delta L}} - \left( {\frac{1}{\Delta L} + \frac{{2{\uppi }D_{{\text{t}}} K_{{\text{t}}} }}{{C_{{\text{t}}} \dot{m}_{{\text{t}}} \left( {D_{{\text{t}}} - d_{{\text{t}}} } \right)}}} \right)T_{{{\text{t,}}L}} + \frac{{2{\uppi }D_{{\text{t}}} K_{{\text{t}}} }}{{C_{{\text{t}}} \dot{m}_{{\text{t}}} \left( {D_{{\text{t}}} - d_{{\text{t}}} } \right)}}T_{{{\text{a}},L}} + \frac{{\lambda_{{\text{t}}} v_{{\text{t}}}^{2} }}{{4d_{{\text{t}}} C_{{\text{t}}} }} = 0$$

(7)

Notably, the length of the control volume is considered relatively small, resulting in minimal variation in the physical properties of the medium within this segment. Consequently, the mass flow rate and specific heat capacity within the control volume are approximated using the average values for this segment.

Boundary conditions and formation temperature

The temperature at the inlet of the annulus is the same as the temperature of the injected gas, which can be expressed as:

$$T_{{{\text{a,}}L = 0}} = T_{{{\text{inlet}}}}$$

(8)

When the gas in the annulus enters the oil pipe through the gas lift valve and mixes with the liquid in the tubing, it can be expressed as:

$$T_{{{\text{t,}}L = L_{\max } }} = T_{{{\text{mix}}}}$$

(9)

Since a sudden pressure drop occurs at the gas injection valve, it is necessary to account for the temperature decrease resulting from expansion, as described by the Joule-Thompson effect. Guo et al.³⁴ considered this effect and established an approximate calculation expression for the mixture, as shown in Eq. (10).

$$T_{{{\text{mix}}}} = \frac{{0.84C_{{\text{a}}} \dot{m}_{{\text{a}}}^{{}} \left( {T_{{\text{a}}} - 43.7} \right) + C_{liquid} \dot{m}_{liquid}^{{}} T_{liquid} }}{{C_{{\text{a}}} \dot{m}_{{\text{a}}}^{{}} + C_{liquid} \dot{m}_{liquid}^{{}} }}$$

(10)

in which, the subscript ‘liquid’ refers to the produced fluid from the formation.

The formation temperature is assumed to be

$$T_{{{\text{g}},L}} = T_{{{\text{g0}}}} + GL$$

(11)

where $T_{{{\text{g0}}}}$ is the surface temperature, °C; G is the temperature gradient of the formation, generally assumed to be 0.025 °C/m.

Calculation of pressure drop

Calculation method for pressure drop

The Orkiszewski method, developed through an analysis and synthesis of existing models, offers enhanced accuracy. Therefore, this method is employed to calculate the pressure drop. The expression for pressure drop is

$$\Delta p_{k} = \frac{{\overline{\rho }_{{{\text{mix}}}} g + \tau_{{{\text{Fri}}}} }}{{1 - \frac{{\dot{m}_{{\text{t}}} q_{{\text{g}}} }}{{A_{{\text{t}}}^{2} \overline{p}}}}}\Delta L$$

(12)

where $\Delta p_{k}$ is the pressure drop for the targeted segment, Pa; $\overline{p}$ is the average pressure of targeted segment, Pa; $\overline{\rho }_{{{\text{mix}}}}$ is the density of the mixture, kg/m³; $\tau_{{{\text{Fri}}}}$ is the pressure drop due to friction, Pa/m; $q_{{\text{g}}}$ is the volumetric flow rate of the gas phase, m³/s; g is the acceleration due to gravity, which is 9.8 m/s². $A_{{\text{t}}}$ is the cross-sectional area of the tubing, m². $\overline{\rho }_{{{\text{mi}} {\text{x}}}}$ and $\tau_{{{\text{Fri}}}}$ need to be determined based on the flow regime of the fluid. The flow regime and their discrimination criteria are shown in Tab. 1.

Table 1 Orkiszewski method-based criteria for determining the flow regime.

In the above table, $\overline{v}_{{\text{g}}}$ is the dimensionless gas flow rate; $L_{{\text{B}}}$ is the thresholds of Bubble flow; $L_{{\text{S}}}$ is the thresholds of Slug flow; $L_{{\text{M}}}$ is the thresholds of Annular-Mist flow. The detailed calculation methods for these variables are as follows:

$$\overline{v}_{{\text{g}}} = \frac{{q_{{\text{g}}} }}{{A_{{\text{t}}} }}\left( {\frac{{\rho_{{\text{l}}} }}{g\sigma }} \right)^{0.25}$$

(13)

$$L_{{\text{B}}} = \left\{ {\begin{array}{*{20}c} {1.071 - 0.727\frac{{v_{{\text{t}}}^{2} }}{D}} & {{\text{for }}L_{{\text{B}}} \ge 0.13} \\ {0.13} & {{\text{for }}L_{{\text{B}}} < 0.13} \\ \end{array} } \right.$$

(14)

$$L_{{\text{S}}} = 50 + 36\overline{v}_{{\text{g}}} \frac{{q_{{\text{l}}} }}{{q_{{\text{g}}} }}$$

(15)

$$L_{{\text{M}}} = 75 + 84\left( {\overline{v}_{{\text{g}}} \frac{{q_{{\text{l}}} }}{{q_{{\text{g}}} }}} \right)^{0.75}$$

(16)

In the above equations, D is the inner diameter of the tubing, m; $v_{{\text{t}}}$ is the flow velocity of gas–liquid mixture, m/s; $\sigma$ is the surface tension of the liquid, N/m; $q_{{\text{g}}}$, $q_{{\text{l}}}$ and $q_{{\text{t}}}$ represent the volumetric flow rates of the gas phase, liquid phase, and the gas–liquid mixture, m³/s. $\rho_{{\text{l}}}$ is the density of the liquid phase, kg/m³. It should be noted that the parameters related to physical properties are all related to the average temperature and pressure of the target segment.

Average density and the pressure drop due to friction

• I. Bubble flow

The average density of gas–liquid mixture is

$$\overline{\rho }_{{{\text{Mix}}}} = (1 - H_{{\text{g}}} )\rho_{{\text{l}}} + H_{{\text{g}}} \rho_{{\text{g}}}$$

(17)

$$H_{{\text{g}}} = \frac{1}{2}\left[ {1 + \frac{{q_{{\text{t}}} }}{{v_{{\text{s}}} A_{{\text{t}}} }} - \sqrt {\left( {1 + \frac{{q_{{\text{t}}} }}{{v_{{\text{s}}} A_{{\text{t}}} }}} \right)^{2} - \frac{{4q_{{\text{g}}} }}{{v_{s} A_{t} }}} } \right]$$

(18)

in which, $v_{{\text{s}}}$ is the slip velocity of bubble, taken as 0.244 m/s. $H_{{\text{g}}}$ represents the gas volume fraction. The pressure drop due to friction is

$$\tau_{{{\text{Fri}}}} = f\frac{{\rho_{{\text{l}}} v_{{{\text{lh}}}}^{2} }}{2D}$$

(19)

$$v_{{{\text{lh}}}} = \frac{{q_{{\text{l}}} }}{{A_{{\text{t}}} \left( {1 - H_{{\text{g}}} } \right)}}$$

(20)

in which, $v_{{{\text{lh}}}}$ represents the actual flow velocity of the liquid phase, m/s; $f$ denotes the frictional resistance coefficient, which are detailed in reference ²⁶.

• II. Slug flow

The average density of gas–liquid mixture is

$$\overline{\rho }_{{{\text{Mix}}}} = \frac{{W_{{\text{t}}} + \rho_{{\text{l}}} v_{{\text{b}}} A_{{\text{t}}} }}{{q_{{\text{t}}} + v_{{\text{b}}} A_{{\text{t}}} }} + \delta \rho_{{\text{l}}}$$

(21)

where $\delta$ represents the distribution coefficient of gas in liquid; $v_{{\text{b}}}$ represents the bubble rise velocity, as shown below

$$v_{{\text{b}}} = C_{1} C_{2} \sqrt {gD}$$

(22)

in which, $C_{1}$ is a function of bubble Reynolds number, and $C_{2}$ is a function of both bubble Reynolds number $N_{{\text{b}}}$ and liquid Reynolds number $N_{{{\text{Re}}}}$. Details can be found in Reference²⁶. Due to the correlation between $N_{{\text{b}}}$ and $v_{{\text{b}}}$, it is necessary to assume an initial value of $v_{{\text{b}}}$ and then use iterative calculations to refine it.

The value of $\delta$ needs to be calculated using a formula based on the type of liquid phase and the flow velocity of the gas–liquid mixture. The details are shown in Table 2.

Table 2 Selection of calculation formula for $\delta$.

Supplementary conditions

$$\left\{ {\begin{aligned}& {\delta \ge - 0.2132v_{t} } \quad {{\text{for }}v_{t} < 3.048} \hfill \\& {\delta \ge \frac{{ - v_{b} A_{t} }}{{q_{t} + v_{b} A_{t} }}} \quad\,\,{{\text{for }}v_{t} > 3.048} \hfill \\ \end{aligned} } \right.$$

(24)

The wall friction-loss term is expressed as

$$\tau_{{{\text{Fri}}}} = \frac{{f\rho_{{\text{l}}} v_{{\text{t}}}^{2} }}{2D}\left( {\frac{{q_{{\text{l}}} + v_{{\text{b}}} A_{{\text{t}}} }}{{q_{{\text{t}}} + v_{{\text{b}}} A_{{\text{t}}} }} + \delta } \right)$$

(25)

• III. Transition flow

The average density of the gas–liquid mixture and wall friction-loss are initially computed individually for slug flow and mist flow scenarios. Subsequently, these values are refined using an interpolation method to ascertain the final results.

$$\overline{\rho }_{{{\text{Mix}}}} = \frac{{L_{{\text{M}}} - \overline{v}_{{\text{g}}} }}{{L_{{\text{M}}} - L_{{\text{S}}} }}\rho_{{{\text{Slug}}}} + \frac{{\overline{v}_{{\text{g}}} - L_{{\text{S}}} }}{{L_{{\text{M}}} - L_{{\text{S}}} }}\rho_{{{\text{Mist}}}}$$

(26)

$$\tau_{{{\text{Fri}}}} = \frac{{L_{{\text{M}}} - \overline{v}_{{\text{g}}} }}{{L_{{\text{M}}} - L_{{\text{S}}} }}\tau_{{{\text{Slug}}}} + \frac{{\overline{v}_{{\text{g}}} - L_{{\text{S}}} }}{{L_{{\text{M}}} - L_{{\text{S}}} }}\tau_{{{\text{Mist}}}}$$

(27)

$\rho_{{{\text{Slug}}}}$ and $\tau_{{{\text{Slug}}}}$ are the average density of gas–liquid mixture and wall friction-loss calculated based on slug flow, respectively. $\rho_{{{\text{Mist}}}}$ and $\tau_{{{\text{Mist}}}}$ are the average density of gas–liquid mixture and wall friction-loss calculated based on annular-mist flow, respectively.

• VI. Annular-Mist flow

The average flowing density for mist flow closely resembles that of bubble flow, as shown bellow

$$\overline{\rho }_{{{\text{Mix}}}} = (1 - H_{{\text{g}}} )\rho_{{\text{l}}} + H_{{\text{g}}} \rho_{{\text{g}}}$$

(28)

In mist flow, the gas–liquid slip velocity is nearly zero, therefore the gas volume fraction is

$$H_{{\text{g}}} = \frac{{q_{{\text{g}}} }}{{q_{{\text{g}}} + q_{{\text{l}}} }}$$

(29)

The wall friction-loss term is calculated based on gas phase

$$\tau_{{{\text{Fri}}}} = \frac{{f\rho_{{\text{g}}} v_{{{\text{sg}}}}^{2} }}{2D}$$

(30)

where $v_{{{\text{sg}}}}$ is apparent flow velocity of gas, which is calculated as $v_{{{\text{sg}}}} = {{q_{{\text{g}}} } \mathord{\left/ {\vphantom {{q_{{\text{g}}} } {A_{{\text{g}}} }}} \right. \kern-0pt} {A_{{\text{g}}} }}$, m/s.

Physical properties related to temperature and pressure

Within the framework of the model we’ve constructed, the key variables affected by temperature and pressure encompass the volume of the gas, the surface tension of the liquid, and its viscosity. To begin with, leveraging the gas flow rate at the wellhead ($q_{{{\text{outlet}}}}$), we can estimate the gas capacity (q) under the specific conditions of pressure (P) and temperature (T) inside the tubing by employing the ideal gas law.

$$q = \frac{{P_{{{\text{outlet}}}} \left( {T + T_{{{\text{sc}}}} } \right)}}{{P\left( {T_{{{\text{outlet}}}} + T_{{{\text{sc}}}} } \right)}}q_{{{\text{outlet}}}}$$

(31)

where $T_{{{\text{sc}}}}$ is the reference standard temperature, expressed in Kelvins, with a value of 273.15 K; $q_{{{\text{outlet}}}}$ is the volumetric flow rate of gas at the outlet, m³/s; $P_{{{\text{outlet}}}}$ is the surface pressure, which is closely aligned with the standard pressure $P_{{{\text{sc}}}}$, amounting to 101.325 kPa.

The surface tension of oil water gas mixtures is indeed a complex issue, as the surface tension of the mixture depends not only on the surface tension of oil and water, but also on factors such as gas composition, concentration of each component, temperature, pressure, and their interactions. Katz et al.³⁵ developed a formula for computing the surface tension of hydrocarbon mixtures has been developed based upon recent measurements of surface tension for the methane-propane system. Data on n-butane gasoline mixtures have been used to demonstrate that the formula applies to complex mixtures.

Viscosity is mainly used to calculate the Reynolds number of fluids. An increase in temperature generally leads to a significant reduction in the stability and viscosity of most crude oil emulsions. This is primarily due to the decrease in crude oil viscosity, which facilitates easier flow. Additionally, the reduction in oil density results in a greater disparity between the densities of oil and water, which can destabilize the emulsion. The solubility of interfacially active substances within the oil phase is also increased, which can weaken the interfacial film, making it less stable and more prone to disruption. Furthermore, the increased likelihood of droplet collisions promotes coagulation, coalescence, and sedimentation, speeding up the emulsion’s breakdown. However, under certain conditions, a temperature rise can unexpectedly enhance the construction of the interfacial film, resulting in a stronger structure that paradoxically improves emulsion stability and viscosity. This complex interplay of factors illustrates the nuanced relationship between temperature and the properties of crude oil emulsions. Luo et al.³⁶ used the stirring viscosity measurement method to determine and study the apparent viscosity of an oil–water mixture containing water, and provided the following empirical formula

$$\mu = 474.242\dot{\gamma }^{ - 0.658} \varphi_{{\text{w}}}^{ - 2.021} T^{ - 0.905} c_{{{\text{sur}}}}^{0.51}$$

(32)

$\mu$ is the viscosity of the oil–water mixture, mPa·s; $\dot{\gamma }$ stands for the shear rate, s^-1; $\varphi_{{\text{w}}}^{{}}$ is the water content; T denotes the temperature, °C; $c_{{{\text{sur}}}}^{{}}$ is the combined content of asphaltenes, resins, wax, and mechanical impurities, %.

Specific implementation method for the temperature–pressure coupling calculation

To simplify calculations and ensure model applicability, the following assumptions were made for the coupled temperature–pressure model of gas–water-oil three-phase flow:

• (1) The gas, oil, and water phases are assumed to be in local thermodynamic equilibrium, with no explicit modeling of phase transitions (e.g., gas dissolution in oil or water vaporization).

• (2) The water–oil mixture is treated as a homogeneous fluid with volume-averaged properties (e.g., density, viscosity), neglecting slip velocity between phases.

• (3) Fluid properties (e.g., oil viscosity) are temperature-dependent but pressure-independent, and water is treated as incompressible.

• (4) Transient thermal effects (e.g., time-dependent heat accumulation) are neglected; heat transfer is modeled via steady-state conduction–convection.

• (5) The flow regime transitions dynamically in the wellbore, and the Orkiszewski correlation is adopted to calculate pressure drops by accounting for varying flow regime (e.g., bubbly, slug, churn, and annular-mist flow).

Employing the finite element method, the tubing and annular space are discretized into n equal-length segments, each with a length of $\Delta L$. Consequently, this discretization results in n + 1 nodes, as illustrated in Fig. 3. Then, based on this assumption, an equation system for the wellbore temperature field can be established. Furthermore, when calculating the pressure drop across each segment, it is assumed that the flow regime of the fluid within the micro-element segment is uniform, and its physical properties are correlated with the average temperature and pressure of that segment.

$$\left\{ {\begin{aligned}& {\left( {\frac{1}{\Delta L} - \frac{{2{\uppi }D_{{\text{c}}} K_{{\text{w}}} }}{{C_{{\text{a}}} \dot{m}_{{\text{a}}} \left( {D_{{\text{w}}} - D_{{\text{c}}} } \right)}} - \frac{{2{\uppi }D_{{\text{t}}} K_{{\text{t}}} }}{{C_{{\text{a}}} \dot{m}_{{\text{a}}} \left( {D_{{\text{t}}} - d_{{\text{t}}} } \right)}}} \right)T_{{{\text{a}},i}} - \frac{1}{\Delta L}T_{{{\text{a,}}i + 1}} + \frac{{2{\uppi }D_{{\text{t}}} K_{{\text{t}}} }}{{C_{{\text{a}}} \dot{m}_{{\text{a}}} \left( {D_{{\text{t}}} - d_{{\text{t}}} } \right)}}T_{{{\text{t}},i}} = - \frac{{2{\uppi }D_{{\text{c}}} K_{{\text{w}}} }}{{C_{{\text{a}}} \dot{m}_{{\text{a}}} \left( {D_{{\text{w}}} - D_{{\text{c}}} } \right)}}T_{{{\text{g}},i}} - \frac{{\lambda_{{\text{a}}} v_{{\text{a}}}^{2} }}{{4r_{{\text{c}}} C_{{\text{a}}} }}} \hfill \\ & {\frac{{2{\uppi }D_{{\text{t}}} K_{{\text{t}}} }}{{C_{{{\text{t}},i}} \dot{m}_{{{\text{t}},i}} \left( {D_{{\text{t}}} - d_{{\text{t}}} } \right)}}T_{{{\text{a}},i}} - \left( {\frac{1}{\Delta L} + \frac{{2{\uppi }D_{{\text{t}}} K_{{\text{t}}} }}{{C_{{{\text{t}},i}} \dot{m}_{{{\text{t}},i}} \left( {D_{{\text{t}}} - d_{{\text{t}}} } \right)}}} \right)T_{{{\text{t,}}i}} + \frac{1}{\Delta L}T_{{{\text{t,}}i + 1}} = - \frac{{\lambda_{{\text{t}}} v_{{{\text{t}},i}}^{2} }}{{4d_{{\text{t}}} C_{{{\text{t}},i}} }}} \hfill \\ & {T_{{{\text{a,}}i{ = 1}}} = T_{{{\text{inlet}}}} } \hfill \\ & {T_{{{\text{t,}}i = n}} = \frac{{0.84C_{{\text{a}}} \dot{m}_{{\text{a}}}^{{}} \left( {T_{{\text{a}}} - 43.7} \right) + C_{liquid} \dot{m}_{liquid}^{{}} T_{liquid} }}{{C_{{\text{a}}} \dot{m}_{{\text{a}}}^{{}} + C_{liquid} \dot{m}_{liquid}^{{}} }}} \hfill \\ \end{aligned} } \right.$$

(33)

Fig. 3

Mesh division of the tubing and annulus.

The detailed calculation process of the coupled wellbore temperature and pressure field is shown in Fig. 4. Initially, when computing the temperature field, it is reasonable to assume that the medium within the tubing and annulus is uniformly distributed. Subsequently, this derived temperature distribution is employed to calculate the pressure field within the tubing. The subsequent pressure field computation will, in turn, refine the flow velocity and mass flow rate of the gas–liquid mixture traversing the tubing. The pressure gradient within the tubing must be meticulously calculated for each segment, proceeding from the top to the bottom of the wellbore. The termination criterion for the calculation is defined by either the global iteration count reaching the predetermined threshold of $N_{\max }$, or the pressure fluctuation at the base of the oil pipe falling below a 10% tolerance.

Fig. 4

Coupled wellbore temperature and pressure field calculation flowchart.

Model validation and sensitivity analysis

Model validation

To validate the accuracy of the proposed model, the predicted results are corroborated against measured data. The measured data are obtained from reference²⁶. The model utilizes the pertinent parameters of injected gases and production as inputs, as detailed in Table 3. The calculated results and measured values of the wellbore temperature profile are shown in Fig. 5a. The pressure drop within the tubing is shown in Fig. 5b. The distribution of flow regime within the tubing is shown in Fig. 5c.

Table 3 Input data of the proposed model.

Fig. 5

Comparison between calculated results and measured data.

From Fig. 5a, it can be seen that for the vast majority of wellbores, the temperature inside the tubing is relatively highest. And the overall trend of change is basically consistent with the formation temperature. And also, it is in good agreement with the measured data. It should be noted that temperature profiles of tubing and annulus are below the profile of formation near the gas injection point. This can be explained by the Joule-Thompson effect. At gas lift injection valve, the gas from annulus is pushed into the tubing with a sudden pressure drop. According to the Joule-Thompson effect, the sudden gas expansion will cause a significant temperature drop between two ends of the valve (downstream and upstream). The temperatures inside tubing and annulus have a high chance to drop below the formation temperature. Figure 5b shows that the calculated pressure gradient is in good agreement with the measured data. The pressure at the bottom of the well is about 8.2 MPa, which reflects that the injection of gas significantly reduces the pressure gradient inside the tubing. Figure 5c shows a simplified wellbore cross-section structure, with the middle pipe being an oil pipe. It can be seen that the oil pipe is green, indicating that the flow regime of the fluid medium in the entire pipe section of the oil pipe is slug flow.

Sensitivity analysis

The application of gas lift technology is extensive, encompassing various aspects of the oil and gas industry such as drilling, oil extraction, and well maintenance. Understanding the influence of gas lift operational parameters on wellbore conditions is crucial for optimizing field operations. This paper aims to elucidate these effects by conducting a sensitivity analysis of key gas lift process parameters within the context of the temperature–pressure coupling model developed herein. All analyses are based on the data listed in Tab. 3 of the previous section.

Gas injection temperature

The gas injection temperature is a pivotal parameter in the gas lift process, significantly impacting its overall efficiency. It exerts its influence primarily by altering the viscosity of the wellbore fluid, which is essential for the fluid’s flow characteristics and the effectiveness of the lift. In scenarios where the objective is to increase the viscosity of the wellbore fluid or facilitate the removal of wax deposits from the pipe wall through gas injection, the role of the gas injection temperature becomes particularly pronounced. The calculation results for different gas injection temperatures are shown in Fig. 6.

Fig. 6

Influence of the gas injection temperature on the wellbore pressure and temperature.

Figure 6 presents data that reveals a relatively minor influence of the injected gas temperature on the tubing temperature. This phenomenon can be primarily attributed to the low specific heat of the gas, coupled with its relatively low mass flow rate, which limits the amount of heat it can transport. Consequently, when the gas is commingled with the liquid, the resulting temperature of the gas–liquid mixture remains relatively stable without significant fluctuations. To achieve a more pronounced increase in oil pipe temperature, enhancing the thermal properties of the injected medium, such as using high-temperature water vapor, is advisable. One effective approach is to increase both the temperature and the specific heat capacity of the medium being injected. This can be accomplished by opting for mediums such as high-temperature water vapor, high-temperature water, or even oil. By selecting mediums with higher thermal retention capabilities, a more pronounced increase in the tubing temperature can be realized, thereby optimizing the thermal management within the wellbore. This strategy underscores the importance of selecting the appropriate injection medium to achieve the desired thermal effects in well operations. It highlights the need for a nuanced understanding of the thermodynamic properties of various mediums and their potential impact on wellbore temperature profiles.

Gas injection volume

In the realm of gas lift systems, the gas injection volume is recognized as a pivotal parameter that significantly influences operational efficiency, thus necessitating a thorough examination. As depicted in Table 3, the gas–liquid ratio is initially established at a value of 103. Utilizing this benchmark, the study explores the implications of varying the gas–liquid ratio (GLR), adjusting it to levels of 50, 75, 125, and 150, with the liquid flow rate being meticulously maintained at a constant value throughout the experimentation. The outcomes of these computational explorations are meticulously presented in Fig. 7, offering a visual representation of the impact that alterations in the gas injection volume have on the system’s performance.

Fig. 7

Influence of gas injection volume on the wellbore pressure and temperature.

The data in Fig. 7a reveal a subtle yet significant interplay between the Gas–Liquid Ratio (GLR) and the system’s thermal and hydraulic behavior. An increase in GLR results in a minor temperature rise at the injection point, predominantly due to the Joule–Thomson effect, which accounts for the temperature change during the rapid expansion of the gas. This effect, while notable, has a muted impact on the overall temperature profile along the tubing, suggesting a robust thermal stability within the wellbore. However, the influence of GLR on pressure drop is more pronounced. As GLR decreases, a significant escalation in pressure drop is observed. This increase can be attributed to several factors. The reduced gas volume leads to decreased momentum transfer, potentially inducing liquid loading as the gas struggles to entrain the heavier liquid column. Additionally, the diminished gas presence impedes effective bubble formation, crucial for reducing the liquid’s density and hydrostatic pressure. The heightened continuity of the liquid phase escalates frictional pressure drop due to its higher viscosity relative to gas. Moreover, the gas may reach a saturation point within the liquid, beyond which it can no longer aid in pressure reduction. These interconnected factors culminate in the observed increase in pressure drop with decreasing GLR, highlighting the imperative of meticulously managing GLR to uphold well efficiency and mitigate operational challenges. The balance between thermal stability and hydraulic efficiency, as dictated by GLR, is pivotal in optimizing gas lift operations.

Building upon the previous analysis, the model presented in this paper was further employed to examine the effects of increasing the Gas–Liquid Ratio (GLR) to 700, with a wellhead pressure of 1,106 kPa. The results, as depicted in Fig. 8, reveal a distinct stratification of flow regimes within the tubing: an upper section of approximately 50 m exhibits a mist flow, a middle section displays a transitional flow, and a lower section of about 300 m transitions into a slug flow. Notably, this configuration is accompanied by a more pronounced reduction in pressure drop. The implications of these findings are multifaceted. The identification of flow regime transitions provides critical insights into the fluid dynamics within the wellbore under varied GLR and pressure conditions. The observed stratification of mist, transitional, and slug flows deepens our comprehension of the behavior of gas–liquid two-phase flow, especially under elevated GLR values. The significant decrease in pressure drop underscores the potential for optimizing gas lift efficiency and operational costs through strategic adjustments of GLR and wellhead pressure.

Fig. 8

The flow regime of fluid and pressure drop in tubing.

Liquid production rate

In the operation of gas-lift wells, the liquid production rate (LPR) is a pivotal operational parameter that directly influences the flow characteristics and overall thermodynamic behavior within the wellbore. To delve into the effects of LPR variation on the performance of gas-lift wells, this study incrementally increased the liquid flow rate while maintaining a constant gas flow rate, starting from a reference point of 294.12 m³/D. Specifically, we increased or decreased the liquid flow rate to levels of 120 m³/D, 210 m³/D, 400 m³/D, and 500 m³/D to explore the resultant changes in well performance. The results are shown in Fig. 9.

Fig. 9

Influence of liquid production rate on the wellbore pressure and temperature.

The analysis captured in Fig. 9a delineates a clear trend: with an escalation in LPR, there is a corresponding rise in temperature throughout the tubing’s length. This temperature increase is primarily due to the additional thermal load introduced by the heightened liquid flow, which results in a general warming of the mixed fluid within the wellbore. While an increase in temperature can favorably impact fluid viscosity and flow conditions, it is imperative to contemplate the possible thermal effects on wellbore equipment and production infrastructure. Moreover, Fig. 9b illustrates that an upsurge in LPR is paralleled by a notable increase in pressure drop across the tubing. This finding is in harmony with the principles of fluid dynamics, where an augmentation in flow rate engenders higher flow velocities, leading to increased frictional resistance and, consequently, greater pressure drop. Furthermore, as the liquid flow rate intensifies, the gas’s capacity to lift the liquid may become saturated, presenting a more formidable challenge to effectively elevate the liquid, which in turn, exacerbates the pressure drop. Expanding upon our preceding analyses, a novel set of parameters is explored through the established model. The outcomes are articulated in Fig. 10, which presents a scenario with an LPR of 2694.12 m³/day and a GLR of 33.4. Under these conditions, the tubing exhibits a pronounced stratification of flow regime, with slug flow dominating the upper 800 m and bubble flow prevailing in the lower sections, as depicted in Fig. 10a. Concurrently, Fig. 10b exhibits a marked increase in pressure drop, indicative of the complex interplay between flow rates, fluid dynamics, and the well’s operational parameters. This comprehensive examination underscores the intricate relationship between LPR, GLR, and their collective influence on wellbore temperature and pressure dynamics. The insights gleaned from Figs. 9 and 10 are invaluable for optimizing gas-lift strategies, ensuring operational efficiency, and mitigating potential challenges associated with variations in wellbore conditions.

Fig. 10

The flow regime of fluid and pressure drop in tubing.

Inner diameter of the tubing

The impacts of the inner diameter of the tubing on flow regime, pressure gradients, and overall well performance are not fully understood and need to be investigated. By meticulously examining the effects of altering the tubing size while keeping all other parameters constant, this analysis highlights the implications of scaling the diameter up by factors of 51.2%, 75.8%, 124.2%, and 148.4%, as well as reducing it by the same proportions. An increase factor greater than 1 signifies an enlargement of the tubing, whereas a factor less than 1 indicates a reduction in size, both of which are critical to understanding their respective impacts on well dynamics. The results are shown in Fig. 11.

Fig. 11

Influence of the inner diameter of the tubing on the wellbore pressure and temperature.

The findings presented in Fig. 11a reveal an intriguing phenomenon: a reduction in tubing size is associated with a slight increase in the overall temperature of the tubing. This subtle elevation in temperature can be attributed to the increased surface area to volume ratio that results from the smaller diameter, which enhances heat transfer between the fluid and the tubing wall. Additionally, the reduced flow area in smaller tubing can lead to higher fluid velocities, which may result in greater mixing and thus a more uniform temperature distribution within the fluid. Conversely, Fig. 11b depicts a more pronounced effect: as the tubing size decreases, there is a significant rise in pressure drop. This increase in pressure drop is primarily due to the increased fluid velocity in the constricted tubing, which leads to higher frictional losses. The smaller cross-sectional area of the tubing results in a greater flow velocity for the same volumetric flow rate, according to the principle of continuity in fluid dynamics. This increased velocity exacerbates the friction between the fluid and the tubing wall, thereby increasing the pressure gradient required to maintain the flow. Linking these results to our previous discussions, it can infer that while an increase in temperature with reduced tubing size may initially seem beneficial for reducing fluid viscosity, the concomitant increase in pressure drop must be carefully managed. The higher pressure drop can impose additional energy requirements for the operation of the well and may necessitate stronger pumping or lifting equipment to overcome the increased resistance.

Conclusions

The developed temperature–pressure coupled model accurately predicts the temperature distribution and pressure gradients in gas-lift wells, providing a theoretical foundation for well design and operational optimization. The sensitivity analysis of the factors, such as gas injection temperature, gas injection volume, liquid production rate and inner diameter of tubing, affecting the temperature distribution of wellbore is carried out by using the established model. The main conclusions are as follows:

• (1) The temperature field model was solved using a numerical difference method, which effectively accounts for the non-uniform distribution of the medium within the computational domain and the influence of temperature and pressure on fluid properties. The model’s predictions showed excellent agreement with measured data, thereby validating its accuracy and reliability for practical applications.

• (2) The analysis revealed that the impact of gas injection temperature on tubing heat is relatively minor, primarily due to the low specific heat and flow rate of the gas. To enhance wellbore thermal management, it is recommended to use injection mediums with higher thermal retention capabilities, such as steam.

• (3) Both GLR and LPR significantly influence wellbore dynamics. Increasing GLR leads to minor temperature rises and reduced pressure drop, enhancing thermal stability and hydraulic efficiency. Conversely, higher LPR results in increased temperatures and pressure drop, affecting fluid viscosity and posing challenges for lifting capacity. Strategic adjustments of GLR and LPR are critical for optimizing gas-lift operations and effectively managing wellbore conditions.

• (4) Reducing tubing size slightly increases temperature due to enhanced heat transfer but significantly raises pressure drop due to increased velocity and friction, necessitating careful management to balance thermal benefits with operational energy demands.