A Novel Prediction Model for Steam Temperature Field of Downhole Multi-Thermal Fluid Generator

Abstract

Aiming at the low efficiency of heavy-oil thermal recovery, a downhole multi-thermal fluid generator (DMTFG) can improve the viscosity reduction effect by reducing the heat loss of multi-thermal fluid in the process of wellbore transportation. The steam generated by the MDTFG causes damage to the packer and casing, owing to the return upwards along the annular space passage of the oil casing. To mitigate this damage, a heat transfer model for multi-channel coiled tubing wells and a prediction model for the upward return of the steam temperature field in the annulus were established with the basic laws of thermodynamics. Models were further verified by ANSYS. The results indicate the following four conclusions. First of all, when the surface pressure is constant, the deeper the located DMTFG, the shorter the distance for the steam to return would be. It is easier to liquefy the steam. Second, the higher the temperature of the steam produced by the downhole polythermal fluid generator, the larger the upward distance of the steam would be. Third, the higher the steam pressure at the outlet of the downhole polythermal fluid generator, the smaller the distance of steam upward return would be. Finally, the larger the diameter of the multi-channel conversion piping, the greater the distance of the steam return would be. It is meaningful to provide valuable theoretical guidance for packer position designing in the field. Meanwhile, the study also provides a modeling basis for the subsequent study of artificial intelligence in the downhole temperature field.

1. Introduction

Heavy oil represents a significant portion of the world’s oil and gas resources due to its abundant reserves, making it a crucial energy source in the 21st century [1,2]. Currently, heavy oil extraction primarily relies on conventional cold extraction methods, which appear to have a low efficiency [3]. Chinese researchers have developed a multi-thermal fluid injection thermal oil extraction method based on steam injection technology to enhance the efficiency of heavy oil extraction [4,5,6]. Multi-thermal fluid technology is a widely used method for heavy oil extraction. It utilizes the synergistic effect of N₂, CO₂, and steam to reduce the viscosity of heavy oil, minimize energy loss during steam injection, and significantly increase the amount of oil extracted [7,8,9,10,11,12,13]. The DMTFG is a device that generates multi-fluid heat, including steam, N₂, CO₂, and other mixed fluids at high temperatures and pressures. It is a crucial component for heat injection and extraction, and a prerequisite for achieving the multi-fluid heat extraction of heavy oil [14]. The process involves injecting fuel oil, air, and steam from the surface into the DMTFG through three pipelines. The DMTFG then injects the multi-fluid directly into the oil formation, reducing heat loss during injection from the wellhead to the oil formation [15,16]. The packer is a crucial downhole tool in heavy oil production, as it significantly impacts the effectiveness of steam injection and the lifespan of the casing [17]. However, the steam produced by the wellbore multi-thermal fluid generator has the phenomenon of upward return from the annulus, which can damage the packer and affect the effectiveness of steam injection in the casing. Therefore, it is crucial to accurately predict parameters such as the temperature change of the steam in the wellbore and the distance of upward return to improve the heavy oil production process [18,19,20].

Researchers have done a lot of studies on the computational model of the temperature field in the steam in the wellbore, and the model has gradually matured [21,22]. In the 1960s, Ramey et al. were the first to establish a wellbore heat transfer model for thermal extraction wells, and carried out a systematic analysis of steam and hot water wellbore heat transfer. The heat transfer model divides the thermal wellbore heat transfer model into two parts: the heat transfer of the thermal fluid inside the wellbore along the vertical direction of the wellbore, and the heat transfer of the wellbore in the radial direction to the external environment layer [23]. Moreover, Ramey derived the temperature distribution equations of the thermal fluid with the time and depth of the well. Willhite suggested an integrated thermal conductivity coefficient based on the Ramey model, which attributes the natural convection coefficient and the radiative heat transfer coefficient in the steam injection annulus to the overall heat transfer coefficient. The calculation of the total heat transfer coefficient of the steam injection wellbore was proposed [24]. Subsequently, researchers analyzed and improved the traditional single-tube well heat transfer model, studied the heat transfer model of steam in both horizontal and vertical wells, and calculated the heat loss after steam injection into the well [25,26,27,28,29,30]. Xu and Nian analyzed the heat transfer of steam in horizontal and straight wellbores, respectively. Fan et al. coupled the vertical section of the wellbore with the horizontal section of the wellbore, on the basis of which a mathematical model of the distribution of the thermophysical parameters of the superheated steam injected into horizontal wells along the course of the wellbore was established [31,32,33]. For vertical and horizontal wellbores, Pang et al. studied the mathematical models of steam with different properties on this basis and calculated the heat transfer characteristics of supercritical steam in vertical wellbores [34]. However, in general, most of the models mentioned above are based on the single-tube steam injection method and may not be suitable for formations with highly heterogeneous oil reservoirs or long horizontal sections [35,36]. Therefore, in order to meet the requirements of the field, several scholars have suggested using parallel twin-tube and concentric-tube steam injection methods [37,38]. Sun et al. developed a concentric twin-tube well model and used finite difference and iterative methods to calculate pressure and temperature distributions within the well [39]. Dong established a mathematical model to calculate steam parameters in parallel twin-tube injection wells for heavy oil horizontal wells [40]. They introduced the concept of an equivalent tube to simplify the parallel twin-tube into a single tube, but this concept is not applicable to the three-tube model presented in this paper. G et al. also conducted research in this area [41]. A mathematical model of the pressure drop in the oil tubing annulus of the parallel twin tube was developed. The intermediate parameter for the tubing annulus pressure drop was simplified and the equivalent diameter was improved. The above models have laid the theoretical foundation for the study of heat transfer in the wellbore.

Moreover, the above research on wellbore heat transfer modeling also contributes to the development of artificial intelligence. Artificial intelligence (AI), a development that has taken the world by storm in the past few years, has been widely used in a variety of fields [42,43,44]. As a new technological science that researches and develops theories, methods, techniques, and application systems for modeling, extending, and expanding human intelligence, it has brought a lot of convenience to people. The application of artificial intelligence to mathematical modeling has been further studied by researchers. Further prediction of the model through artificial intelligence has improved the accuracy and speed of prediction [45,46,47,48,49].

This paper presented a new mathematical model for heat transfer in a parallel three-tube well, based on the fundamental theory of heat transfer. When the DMTFG injects the generated fluid into the oil formation, some of the steam returns upward through the oil casing annular space channel. At this point, the temperature of the steam in the annulus is higher than that of the steam injection pipeline. As the steam returns to a certain distance and cools down, it condenses into steam. Adding a packer at this position can effectively prevent the damage caused by constant contact between the packer and the steam, thereby improving economic efficiency. Moreover, this parallel three-pipe model can provide a theoretical basis for the subsequent prediction of temperature fields in heat transfer models by artificial intelligence.

2. Models

2.1. Assumption

The schematic diagram of the DMTFG is shown in Figure 1 and Figure 2, and the following basic assumptions were made in order to develop the mathematical model:

• (1). Stratigraphic thermophysical parameters do not vary with depth;

• (2). The multi-thermal fluid generator injection parameters are constant;

• (3). Heat transfer from the steam to the outer edge of the cement ring is in a steady state;

• (4). The steam flow in the ring air is a one-dimensional two-phase steady flow;

• (5). The thermophysical parameters of the formation remain constant regardless of depth variation.

2.2. Mathematical Model of Steam in the Annulus

According to Stefan Boltzmann’s law, the total ability of an object to radiate energy to the outside is proportional to the fourth power of its absolute temperature, so thermal radiation is negligible when the object is at low temperature. In the model, the water, air, and fuel are injected at room temperature, so thermal radiation between the three parallel injection tubes is negligible. The three tubes are filled with air between them, and since the convective heat transfer coefficient among the fluid in the injection tube and the injection tube is extremely large, the thermal resistance is profoundly large and thermal convection can be neglected [50]. For the parallel three-tube injection model, the whole heat transfer process consists of the following parts: the convective heat transfer, the casing heat conduction, and the cement ring heat conduction. The corresponding heat transfer process is shown in Figure 3:

For the eccentric injection problem of the three tubes, the concept of an equivalent tube column is introduced (Figure 2), where the heat transfer process of the column is equivalent to the combined effect of the heat transfer of the water injection tube, the air injection tube, and the fuel injection tube. The geometrical parameters of the equivalent tube are related to the dimensions of the water, air, and fuel injection tubes. According to the principle of equivalent heat transfer, the inner and outer dimensions of the equivalent tube column are defined as follows [51]:

(1)$\pi r_{\mathrm{ti}}^2=\pi r_1^2+\pi r_2^2+\pi r_3^2$

(2)$r_{to}\ln \frac{r_{to}}{r_{ti}}=r_{1o}\ln \frac{r_{1o}}{r_{1i}}+r_{2o}\ln \frac{r_{2o}}{r_{2i}}+r_{3o}\ln \frac{r_{3o}}{r_{3i}}$

where r_ti is the equivalent inner radius of the tube; r_1i is the inner radius of the tubing; r_2i is the inner radius of the air tube; r_3i is the inner radius of the tube; r_to is the equivalent outer radius of the tube; r_1o is the outer radius of the oil pipe; r_2o is the outer radius of the air pipe; and r_3o is the outer radius of the water pipe.

Thus, the mathematical model of the equivalent tube is established.

The formula for calculating the heat transfer coefficient of an equivalent tube is shown below:

(3)$\frac{1}{U_{to}}=\frac{1}{U_1}+\frac{1}{U_2}+\frac{1}{U_3}+\frac{1}{U_4}$

(4)$U_1={[\frac{r_{1o}}{{\lambda }_{tub}}\ln \frac{r_{1o}}{r_{1i}}]}^{-1}$

(5)$U_2={[\frac{r_{2o}}{{\lambda }_{tub}}\ln \frac{r_{2o}}{r_{2i}}]}^{-1}$

(6)$U_3={[\frac{r_{3o}}{{\lambda }_{tub}}\ln \frac{r_{3o}}{r_{3i}}]}^{-1}.$

(7)$U_4={[\frac{1}{h_r}+\frac{r_{to}}{{\lambda }_{cas}}\ln \frac{r_{co}}{r_{ci}}+\frac{r_{to}}{{\lambda }_{cem}}\ln \frac{r_h}{r_{co}}]}^{-1}$

where $U_{to}$ is the heat transfer coefficient of the equivalent pipe; $U_1$ is the heat transfer coefficient of the oil pipe; $U_2$ is the heat transfer coefficient of the air pipe; $U_3$ is the heat transfer coefficient of the water pipe; $U_4$ is the heat transfer coefficient from the center of the annulus to the outer edge of the cement ring; ${\lambda }_{tub}$ is the heat transfer coefficient of the oil/air/water pipe; $h_r$ is convective heat transfer coefficient of the annulus to the inner wall of the casing; ${\lambda }_{cas}$ is the heat transfer coefficient of the casing; ${\lambda }_{cem}$ is the heat transfer coefficient of the cement ring; $r_h$ is the radius of the outer edge of the cement ring; $r_{ci}$ is the radius of the inner edge of the casing; and $r_{co}$ is the radius of the outer edge of the casing.

The heat transfer coefficient is calculated as follows:

(8)$U_s={[\frac{1}{U_{to}}+\frac{1}{h_c}]}^{-1}$

where $h_c$ is the convective heat transfer coefficient of the equivalent tube to the annular air.

In an annular air flow heat transfer problem, the empirical formula for the Nusselt number of a cylinder can usually be used. The Nusselt number (Nu) is the ratio of convective to conductive heat across a boundary. For convective heat transfer over the surface of a cylinder, the formula for the Nusselt number can be expressed as:

(9)$Nu=\frac{hD}{k}$

where $h$ is the convective heat transfer coefficient on the surface of the cylinder, $D$ is the diameter of the cylinder, and $k$ is the thermal conductivity of the fluid.

The fluids injected in the equivalent tube are water, air, and fuel. The injected fluids cause forced convection heat loss to the annulus. Therefore, the forced convection heat transfer coefficient is calculated as follows [52]:

(10)$Nu=0.683R{e}^{0.486}{P}_r{}^{\frac{1}{3}}$

(11)$h_m=0.683R{e}^{0.466}{P}_r^{\frac{1}{3}}{C}_{\phi }\frac{{\lambda }_{tub}}{2r_i}$

where

(12)$Re=\frac{2v_i{r}_i}{{\mu }_i}$

(13)$P_r=\frac{1000C_i{\mu }_i}{{\lambda }_i}$

where $Re$ is the Reynolds number; $P_r$ is the Prandtl number; $C_{\phi }$ is the flow rate and wellbore angle correction factor (the vertical wellbore is used in this model, so the correction factor is 1); ${\lambda }_i$ is the thermal conductivity; ${\mu }_i$ is the kinetic viscosity; and $C_i$ is the specific heat.

2.3. Temperature Field Modeling in the Annulus

Taking any section of pipe as a unit of analysis, part of the heat of the steam in the annulus is transferred to the equivalent pipe; the other part of the heat is transferred to the casing and then from the casing through the cement ring to the ground.

(14)$Q=Q_1+Q_2$

where $Q$ is the water steam heat in the center of the annulus; $Q_1$ is the heat transfer from the center of the annulus to the center of the equivalent pipe; and $Q_2$ is the heat transfer from the center of the annulus to the center of the equivalent pipe.

The heat transfer from the center of the annulus to the center of the equivalent tube is as follows:

(15)$\frac{d{Q}_1}{dz}=2\pi r_o{U}_s(T_s-T_o)$

where $r_o$ is the distance from the center of the annulus to the outer edge of the equivalent tube; $T_s$ is the temperature at the center of the equivalent tube; and $T_o$ is the temperature at the center of the annulus.

The heat transfer from the center of the ring void to the outer layer of the cement ring is as follows:

(16)$\frac{d{Q}_2}{dz}=2\pi r_h{U}_o(T_o-T_h)$

where $T_h$ is the temperature at the outer edge of the ring; and $r_h$ is the distance from the center of the ring to the outer edge of the ring.

Combining Equations (12)–(14), we can derive the annular air–steam temperature calculation formula as follows:

(17)$T_o=\frac{\frac{Q}{2\pi }-r_o{U}_s{T}_s+r_h{U}_o{T}_h}{r_h{U}_o-r_o{U}_s}$

The temperature of the steam in the annular air per unit length is the average temperature.

The energy equation for the steam contained in the annular air per unit wellbore length is obtained from the energy equation:

(18)$\frac{d{h}_m}{dz}=\frac{1}{w_t}Q-\frac{d\left(\frac{1}{2}{{\displaystyle v}}_m^2\right)}{dz}$

where $w_t$ is the steam flow rate; $h_m$ is the enthalpy of steam; and $Q$ is the heat of steam. The change of steam flow velocity in the annulus is small, so the accelerated pressure drop can be neglected in this model, meaning

(19)$\frac{d{h}_m}{dz}=\frac{1}{w_t}Q$

That is, the heat of steam in the annular air per unit wellbore length is

(20)$Q=W_{\mathrm{t}}\mathsf{\Delta }{h}_m$

2.4. Circumferential Air Pressure Loss Modelling

Owing to the fact that the enthalpy of steam is a function of temperature and pressure, the change in pressure should not be ignored when analyzing the distance of steam return in the annulus. The pressure change of steam in the well annulus is the result of the combined change of potential energy, kinetic energy, and friction loss [53]. According to the conservation of mass and momentum, the equation for the pressure drop in the well annulus is

(21)$\frac{dp}{dz}=\rho g-f\frac{{\rho v}^2}{2D}$

where

(22)$f=\left\{\begin{array}{ll}\frac{64}{Re}\hfill & Re\le 2000\hfill \\ {\left[1.14-2\log \left(\frac{\epsilon }{D}+\frac{21.25}{R{e}^{0.9}}\right)\right]}^{-2}\hfill & Re>2000\hfill \end{array}\right.$

(23)$D=r_{ci}-r_{ti}$

(24)$\epsilon ={\epsilon }_1\left(\frac{r_{ci}}{r_{ci}+r_{ti}}\right)+{\epsilon }_2\left(\frac{r_{ti}}{r_{ci}+r_{ti}}\right)$

where $dp$ is the pressure drop in the micro-segment; $f$ is the drag coefficient; $v$ is the fluid flow velocity; $r_{ci}$ is the equivalent pipe radius; $r_{ti}$ is the equivalent pipe radius; $R_e$ is the Reynolds number; and ${\epsilon }_1$ and ${\epsilon }_2$ for the roughness of the outer wall of the inner pipe and the outer pipe wall roughness, respectively.

Thus, the mathematical model of steam flow in annuli is established. The model can provide a model basis for future AI prediction of downhole temperature fields.

3. Numerical Solution of the Mathematical Model

In order to get accurate results for pressure and temperature, an iteration method is adopted.

Next, the accurate values of pressure and temperature at the outlet of the i_th segment are used as input values for the (i + 1)_th segment and another iteration begins. Finally, the pressure and temperature profiles in the annuli can be obtained.

The detailed mathematical model solving process is shown as a flowchart in Figure 4.

4. Results and Discussion

4.1. Model Validation

4.1.1. Experimental Measurements

In order to verify the accuracy of the model, we set up a measurement experiment by applying the similarity criterion, as shown in Figure 5. It mainly consists of a steam generator, a plunger pump, an inner tube, an annular air tube, three injection tubes, and a temperature measurement system.

The main experimental steps are as follows:

• (1). Oil, water, and air are injected through a plunger pump into three tubes each, simulating the three tubes in the DMTFG.

• (2). The temperature of the water in the measuring container is measured using a temperature gauge that has been calibrated and then the probe is placed into the water. Compare the temperature measured by the probe with the known temperature to determine the deviation of the probe. Calibrate the probe.

• (3). High-temperature steam is injected into the annular air through a steam generator, and the inlet and outlet temperatures are measured using a temperature probe (Table 1).

The accuracy of the model was verified by comparing the calculated temperature with the experimental temperature. The results in Figure 6 show that the experimental temperatures are slightly lower than the calculated ones, which is due to excessive heat dissipation during the experiment.

4.1.2. Mathematical Simulation

The models were modeled and meshed using space claim as well as ICEM software version 2021, respectively, to further verify the accuracy of the models. Structured hexahedral meshing was performed using ICEM software (Figure 7). Analyze the temperature variation of steam in the annulus at different well depths. The inlet boundary condition is set as the velocity inlet and the outlet boundary condition is set as the pressure outlet. The standard model of the k-ε model is applied for processing, and the standard wall function method is used at the near-wall surface. For the coupling of velocity and pressure, the SIMPLEC method is used, and the first-order windward form is adopted for each discrete equation. During the iterative operation, the calculation convergence is judged based on the residuals falling below 10⁻⁶.

Based on the above model, a grid refinement study is carried out. Table 2 shows the grid refinements of the five schemes for grid-independent calculations. Structural stability is adopted as the assessment standard. From the results in the table, it can be found that when the number of grids exceeds 380,000, the minimum grid mass no longer varies with the number of grids. It provides a foundation for the subsequent simulation of the temperature field in the annular air.

The outlet temperature of the DMTFG is set to be 350 °C. When the multi-thermal fluid generator is located 1200 m below the ground, the temperature distribution of the wellbore is shown in Figure 8 when the upward-returning steam is stabilized. Each color in the figure represents a different temperature gradient, indicating that the temperature change is an inside-out heat dissipation process. The simulation results show that the steam generated by the multi-thermal fluid generator diffuses upward along the annulus, with the heat continuously dissipated toward the formation and the equivalent tube during the diffusion process until the steam condenses into liquid droplets. From the figure, it can be found that the temperature of the steam in the whole process decreases with the increase in the upward and backward distance.

Next, the results of the extrapolated model are compared with the predictions from the ANSYS simulations.

Calculations were performed using the following injection parameters (Table 3: the parameters are from an offshore oil field) for the DMTFG as the initial conditions:

The 1200 m, 1600 m, and 1800 m well models were created using ANSYS, and the simulation calculations were performed using the Fluent module. From Figure 9, it can be seen that the distance of the upward return of steam calculated by the DMTFG heat transfer model is basically the same as that obtained from the simulation, which verifies the accuracy of the model.

4.2. Heat Transfer Characteristics and Flow Properties of Annulus Flow in DMTFG

In this section, based on the validated model, a further discussion about the heat transfer characteristics is conducted.

A short section at the outlet end of the generator in the 1200 m model was selected to analyze the average temperature of the steam in the wellbore during this zone. It was found that the upward return of steam continued to dissipate heat to the tubing and casing as time progressed, which resulted in a decreasing average temperature of steam in the annulus.

As the temperature gradually decreases, the steam condenses into water. When water steam condenses to water, the enthalpy drops rapidly. From Figure 10, it can be found that the deeper the location downhole, the earlier the point of decline in the enthalpy of the steam. This is because the deeper the location in the well, the higher the pressure at the same temperature. When the pressure is higher, the density of the steam is higher, and the water is more likely to condense into liquid water.

4.3. Effect of the Depth of the DMTFG on the Upward Return Distance of the Steam

There is an upward return process for the steam generated by the DMTFG. In this paper, the outlet of the DMTFG is taken as the starting point, and the upward direction of the steam is taken as the positive direction. The distance from the outlet end of the generator until the steam condenses into liquid drops is the upward return distance of the steam. The temperature distribution of the steam in the annulus and the enthalpy change with the distance of the steam upward and backward were calculated for the DMTFG at different depths from the subsurface, which is located at different locations downhole, as shown in Figure 11:

It is found that the steam in the annulus will lose heat in the direction of the floor and in the direction of the equivalent tube, which leads to a decrease in the temperature of the steam and a decrease in the enthalpy. When the steam returns to a certain height, the steam will be condensed into liquid drops and the enthalpy will change significantly, i.e., the enthalpy change in Figure 11 is the height at which the steam returns to freeze. As shown in Figure 10, when the DMTFG is located at 1000 m, the steam returns at 700 m to liquid drops. When the DMTFG is located at 1200 m underground, the distance of steam return is 400 m, and the distance of steam return decreases. When the DMTFG is located at 1800 m below ground level, the distance of steam return is reduced to 100 m. The above results are caused by the fact that when the DMTFG is located deeper in the subsurface, the formation pressure is higher, the enthalpy of the steam is lower, and the enthalpy of the steam changes more rapidly, which results in a smaller distance for the steam to return upwards, and the steam is more likely to be liquefied. From this, it can be concluded that when the DMTFG is located deeper downhole, the smaller the steam return distance. If a packer is installed at the location where the gas phase is converted to liquid phase, it can effectively prevent the steam from escaping upwards and reduce the contact of the steam with the packer, which protects the packer very well.

4.4. Effect of Multi-Thermal Fluid Generator Outlet Temperature on Upward Return Distance

The variation in the steam upward return distance was calculated for different outlet temperatures of the DMTFG, as shown in Figure 12:

The results show that the upward return distance of the steam changes when the exit temperature of the DMTFG is different, namely when the initial temperature of the steam in the annulus is different. As shown in Figure 12, when the DMTFG is located at 1200 m under the ground and the initial pressure of the steam in the annulus is 14.5 MPa, the upward return distance of the steam increases with the increase in the outlet temperature of the generator. This is due to the fact that when the pressure is constant, the enthalpy of the steam increases with the temperature, which leads to a slower heat transfer rate of the steam, thus increasing the distance of steam return. From this, it can be found that selecting a suitable generator outlet temperature can reduce the effect of steam on the packer.

4.5. Effect of Outlet Pressure of Multi-Thermal Fluid Generator on the Upward Return Distance

The variation in the upward return distance of the steam was calculated for different outlet pressures of the DMTFG, as shown in Figure 13:

The results show that when the initial temperature of the steam in the annulus is constant at 340 °C, the steam return distance decreases as the pressure at the outlet of the generator increases. This is due to the fact that when the steam temperature is constant, the enthalpy of steam decreases as the steam pressure increases. The steam is liquefied more easily, which leads to a decrease in the steam return distance.

4.6. Effect of Equivalent Pipe Size on Steam Upward Return Distance

The variation of the upward return distance of the steam is calculated for different equivalent tube sizes, as shown in Figure 14:

When the diameters of fuel, water, and air injection pipes in the DMTFG were changed, the equivalent pipe diameters were also changed. The results show that the upward return distance of the steam constantly increases when the diameter of the equivalent pipe becomes larger. This is because the amount of steam generated by the DMTFG changes when the diameter of the equivalent pipe changes, and the size of the annular space also changes. When the equivalent tube diameter becomes larger, the heat transfer distance increases. At the same time, the mass flow rate of the steam in the annulus becomes larger, which ultimately leads to a decrease in the heat transfer rate of the steam and an increase in the upward return distance.

5. Conclusions

In this paper, a new method for calculating the temperature field of a parallel three-pipe has been presented. The method allows for individual adjustment of injection volume and calculation of the distance of steam up and back in the annulus. At the same time, physical and numerical modeling methods were used to validate the model and to analyze the changes in the properties of the fluid during the upward return process. It is observed that the distance of steam upward return decreases as the DMTFG is located deeper in the well. When the temperature of the steam produced by the DMTFG increases, the upward return distance of the steam also increases. Additionally, as the steam pressure at the outlet of the multi-thermal fluid generator increases, the steam is more likely to liquefy, resulting in a shorter steam return distance. To minimize damage to the packer, it is recommended to adjust the outlet temperature and pressure accordingly during practical application. As the equivalent pipe diameter becomes larger, the annular air area increases, and the heat transfer distance increases, while the steam mass flow in the annulus becomes larger, which ultimately leads to a reduction in the heat transfer rate of steam and an increase in the upward return distance.

Author Contributions

Conceptualization, Y.H. and Z.H.; methodology, Y.H. and X.D.; data analysis, Z.H. and Y.Z.; Writing—Reviewing and Editing, Y.H. and X.D.; investigation, L.H. and J.G. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

Data is contained within the article.

Conflicts of Interest

The authors declare no conflicts of interest.

Nomenclature

$r_{ti}$	the equivalent inner radius of the tube	$\mathrm{m}$
$r_{1i}$	the inner radius of the tubing	$\mathrm{m}$
$r_{2i}$	the inner radius of the air tube	$\mathrm{m}$
$r_{3i}$	the inner radius of the tube	$\mathrm{m}$
$r_{to}$	the equivalent outer radius of the tube	$\mathrm{m}$
$r_{1o}$	the outer radius of the oil pipe	$\mathrm{m}$
$r_{2o}$	the outer radius of the air pipe	$\mathrm{m}$
$r_{3o}$	the outer radius of the water pipe	$\mathrm{m}$
$r_h$	radius of the outer edge of the cement ring	$\mathrm{m}$
$r_{ci}$	radius of the inner edge of the casing	$\mathrm{m}$
$r_{co}$	the radius of the outer edge of the casing	$\mathrm{m}$
$D$	the diameter of the cylinder	$\mathrm{m}$
$U_{to}$	heat transfer coefficient of the equivalent pipe	$\mathrm{W}\cdot {\mathrm{m}}^{-2}\cdot {\mathrm{K}}^{-1}$
$U_1$	heat transfer coefficient of the oil pipe	$\mathrm{W}\cdot {\mathrm{m}}^{-2}\cdot {\mathrm{K}}^{-1}$
$U_2$	heat transfer coefficient of the air pipe	$\mathrm{W}\cdot {\mathrm{m}}^{-2}\cdot {\mathrm{K}}^{-1}$
$U_3$	heat transfer coefficient of the water pipe	$\mathrm{W}\cdot {\mathrm{m}}^{-2}\cdot {\mathrm{K}}^{-1}$
$U_4$	heat transfer coefficient	$\mathrm{W}\cdot {\mathrm{m}}^{-2}\cdot {\mathrm{K}}^{-1}$
${\lambda }_{tub}$	heat transfer coefficient of the oil/air/water pipe	$\mathrm{W}\cdot {\mathrm{m}}^{-1}\cdot {\mathrm{K}}^{-1}$
$h_r$	convective heat transfer coefficient	$\mathrm{W}\cdot {\mathrm{m}}^{-2}\cdot {\mathrm{K}}^{-1}$
${\lambda }_{cas}$	heat transfer coefficient of the casing	$\mathrm{W}\cdot {\mathrm{m}}^{-1}\cdot {\mathrm{K}}^{-1}$
${\lambda }_{cem}$	heat transfer coefficient of the cement ring	$\mathrm{W}\cdot {\mathrm{m}}^{-1}\cdot {\mathrm{K}}^{-1}$
${\lambda }_i$	the thermal conductivity	$\mathrm{W}\cdot {\mathrm{m}}^{-1}\cdot {\mathrm{K}}^{-1}$
$h$	convective heat transfer coefficient on the surface of the cylinder	$\mathrm{W}\cdot {\mathrm{m}}^{-2}\cdot {\mathrm{K}}^{-1}$
$k$	he thermal conductivity of the fluid	$\mathrm{W}\cdot {\mathrm{m}}^{-1}\cdot {\mathrm{K}}^{-1}$
${\mu }_i$	the kinetic viscosity	$\mathrm{m}\cdot \mathrm{Pa}\cdot \mathrm{s}$
$C_i$	the specific heat	$\mathrm{J}\cdot {\mathrm{kg}}^{-1}$
$Re$	the Reynolds number
$P_r$	the Prandtl number
$C_{\phi }$	the flow rate and wellbore angle correction factor
$Q$	the water steam heat in the center of the annulus	$\mathrm{W}$
$Q_1$	the heat transfer from the center of the annulus to the center of the equivalent pipe	$\mathrm{W}$
$Q_2$	is the heat transfer from the center of the annulus to the center of the equivalent pipe	$\mathrm{W}$
$T_s$	the temperature at the center of the equivalent tube	$℃$
$T_o$	the temperature at the center of the annulus	$℃$
$T_h$	the temperature at the outer edge of the ring	$℃$
$w_t$	the steam flow rate	$\mathrm{kg}\cdot {\mathrm{s}}^{-1}$
$h_m$	the enthalpy of steam	$\mathrm{kJ}\cdot {\mathrm{kg}}^{-1}$
$dp$	the pressure drop in the micro-segment	$\mathrm{MPa}$
$f$	the drag coefficient
$v$	the fluid flow velocity	$\mathrm{m}\cdot {\mathrm{s}}^{-1}$

Footnotes

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Figures and Tables

Figure 1. Schematic diagram of wellbore structure, 1: ground, 2: casing pipe, 3: oil filler pipe, 4: air injection pipe, 5: water injection pipe, 6: downhole multi-thermal fluid generator.

Figure 2. Schematic diagram of parallel three-tube structure.

Figure 3. Heat transfer process diagram.

Figure 4. Numerical solutions of the mathematical model.

Figure 5. Experimental setups.

Figure 6. Comparison between the results of the experimental data and calculation result.

Figure 7. Mesh.

Figure 8. Wellbore temperature distribution.

Figure 9. Steam temperature distribution diagram of downhole multi-thermal fluid generator at different depth of borehole: (a) Downhole at 1200 m depth; (b) Downhole at 1600 m depth; (c) Downhole at 1800 m depth.

Figure 10. Chart of steam heat enthalpy variation when downhole multi-thermal fluid generator is located at different depths of borehole.

Figure 11. Chart of steam heat enthalpy and temperature variation when downhole multi-thermal fluid generator is located at different depths of borehole: (a) Downhole at 1000 m depth; (b) Downhole at 1200 m depth; (c) Downhole at 1400 m depth; (d) Downhole at 1600 m depth; (e) Downhole at 1800 m depth.

Figure 12. Variation diagram of steam up-return distance of multi-thermal fluid generator at different outlet temperatures.

Figure 13. Variation diagram of steam up-return distance of multi-thermal fluid generator under different outlet pressure.

Figure 14. Variation diagram of steam upturn distance with different equivalent tube sizes.

Table 1

Experimental results.

Well Depth/m	Inlet Temperature/°C	Outlet Temperature/°C	Average/°C
100	248	241	243
		245
		243
150	253	246	246
		248
		244
200	258	253	250.33
		248
		250
250	263	256	256.33
		255
		258
300	268	260	260
		262
		258
350	273	266	266.33
		264
		269
400	278	266	267.33
		267
		269

Table 2

Grid-independent verification scheme.

Type	Mesh Number	The Minimum Mesh Quality
1	16,124	0.05
2	32,451	0.23
2	102,990	0.47
3	275,320	0.58
4	385,983	0.76
5	433,673	0.75

Table 3

Basic parameter table.

Parameter	Value	Parameter	Value
Inner radius of oil pipe/m	0.007	Equivalent tube thermal conductivity/W·(m·°C)−1	50
Outer radius of oil pipe/m	0.009	Casing thermal conductivity/W·(m·°C)−1	50
Inner radius of air pipe/m	0.021	Thermal conductivity of cement rings/W·(m·°C)−1	1
Outer radius of air pipe/m	0.025	Convective heat transfer coefficient of the annulus to the casing/W·(m2·°C)−1	1.7
Inner radius of water pipes/m	0.01	Convective heat transfer coefficients for equivalent tube-to-annular air/W·(m2·°C)−1	1.7
Outer radius of water pipes/m	0.012	Surface temperature/°C	30
Equivalent inner radius of the tube/m	0.0735	Temperature gradient/°C·m−1	0.032
Equivalent outer radius of the tube/m	0.0805	Distance from the center of the annulus to the inner edge of the casing/m	0.020875
Equivalent tube depth/m	500	Distance from the center of the annulus to the outer edge of the equivalent tube/m	0.020875
Radius of inner edge of casing/m	0.12225	Radius of outer edge of cement ring/m	0.19225
Radius of outer edge of casing/m	0.14225	Steam flow/kg·h−1	200

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