Numerical study of temperature–pressure coupling model for the horizontal well with a slim hole

Horizontal wells with a slim hole (HW‐SH), characterized by high productivity, high environmental protection, and low cost are gradually being introduced into oil and gas extraction processes in high‐temperature and high‐pressure formations. However, the HW‐SH annulus is narrow, and the distributions of temperature and pressure in the wellbore are very different from those of conventional horizontal wells. Furthermore, the mud properties change under the influence of high temperature and pressure. In this study, a numerical model of the transient temperature‐pressure coupling field in an HW‐SH wellbore was developed based on high‐temperature and high‐pressure performance tests of mud, combined with the characteristics of a narrow annulus in HW‐SH. Subsequently, the effects of the ground temperature gradient, displacement, drill pipe speed, annulus size, and joint size on the temperature‐pressure coupling field were analyzed. Measured data from the HW‐SH drilled in the Mahu field in China were used to validate the developed model. The results showed that the ground temperature had a significant effect on the mud density and rheological properties. As the temperature increased, the mud density decreased, and its rheological properties improved. Although pressure also has a large effect on density, it has a negligible effect on the rheological properties of the mud. The coupled temperature‐pressure field model for HW‐SH considering the effects of the temperature and narrow annulus can predict the annulus temperature and pressure with an accuracy of up to 97%. The ground temperature gradient has the greatest influence on the HW‐SH annular temperature, larger than that of the displacement. In addition, the annular size, rotation speed of the drill pipe, and joint size have almost no influence on the annular temperature. Compared to those of conventional horizontal wells, the rotational speed, annulus size, and joint size of the HW‐SH have a greater impact on the annulus pressure.


| INTRODUCTION
As the global energy demand continues to increase, higher standards are being set for subsurface oil and gas recovery rates, drilling costs, and environmental protection. 1,2 Compared with those of conventional drilling methods, horizontal wells with a slim hole (HW-SH) have high productivity, high environmental protection, and low cost. As a result, HW-SH techniques are being gradually developed and used. With the deepening of oil and gas exploration and development, oil and gas extraction has also gradually expanded into deeper formations, where higher temperatures reduce the safety window of mud density. The mud performance under high-temperature and high-pressure environments varies greatly. Coupled with the narrow annulus of HW-SH, and the greatly different wellbore pressure distribution, compared with that of conventional horizontal wells, the HW-SH drilling process can easily lead to well wall instability, mud leakage, overflow, and other accidents, that seriously threaten drilling safety.
Scholars have conducted much research on conventional horizontal wellbore pressure in terms of experimental studies and theoretical derivations, [3][4][5][6][7][8][9] which laid the foundation for the prediction of horizontal wellbore pressure in HW-SH. Studies of wellbore pressure prediction in HW-SH are also increasing. Song et al. 10 concluded that the loss of annular pressure in HW-SH accounts for 30%-50%, or even up to 90%, of the circulating pressure loss in the entire system; thus, a more accurate model for calculating the hydraulic parameters needs to be designed. Delwiche et al. 11 concluded that the "Kurt effect" and "Crescent effect" exist in the drilling process of HW-SH and analyzed the effects of different drilling fluid rheology patterns, drill column rotation, and drill column eccentricity on the calculation of annular pressure consumption in HW-SH, which laid the theoretical foundation for the study of HW-SH hydraulics. McCann et al. 12 conducted an experimental study on the effect of drilling column rotation and eccentricity conditions on wellbore pressure patterns by drilling an HW-SH of 2500 ft. The study showed that the results calculated by the annular pressure dissipation model of conventional horizontal wells are within a reasonable error range when the drilling column is not rotating, but there is a large error when the drilling column is rotating. In turbulent flow, the faster the drilling column rotates, the higher the annular pressure dissipation. In both laminar and turbulent flows, the annular pressure dissipation tended to decrease gradually as the eccentricity of the drilling column increased. After analyzing a large amount of field and experimental data, Cartalos 13 concluded that the annular pressure depletion in HW-SH could not be predicted accurately, even though the drilling column rotation was considered. In addition to considering the drilling column rotation, he also considered the effect of the eccentricity of the drilling column and the variability of its eccentricity along the wellbore axial direction on the annular pressure dissipation and developed a new model for the annular pressure dissipation in HW-SH. Hansen 14 concluded that the mud type plays an important role in predicting HW-SH annular pressure depletion and proposed a model for calculating HW-SH annular pressure depletion for Newtonian, Bingham, and power-law fluids. However, the calculation process is complicated and cumbersome and cannot be extended to other applications. In 2005 and 2006, Hemphil 15 proposed a model for calculating wellbore pressure in HW-SH considering drilling column rotation and drilling column eccentricity, but the model did not consider the effect of drilling column joints on annular pressure depletion. Subsequently, Majed 16 proposed a model for calculating the annular pressure consumption considering a drilling tool joint by establishing a small borehole annulus with dimensions of 1.75" × 1.25" through an indoor experiment and indicated that the drilling tool joint could increase the annular pressure consumption by up to 30%.
However, the effect of temperature on the density and rheology of mud was not considered in the aforementioned studies. Owing to the small volume of the annulus, the effect of high temperature on the warming of the mud in the wellbore is obvious. In particular, if oilbased drilling fluids are used, the effect of high temperature on the mud temperature is more apparent because of their low specific heat capacity. Therefore, under the effect of high temperature and pressure at the wellbore, if the mud density and rheological parameters are still treated as constants, the calculated bottomhole pressure will be different from the actual one, which will easily lead to kick or well leak. [17][18][19][20][21][22][23] In response to this, Keelan et al. 17 specifically mentioned the effect of high temperature and pressure on the density of drilling fluids, arguing that this effect can substantially increase the risk of borehole instability. McMordie et al. 18 conducted an experimental study on the effects of temperature and pressure on the density of water and oil-based mud. The results showed that the variation in the density of mud is a function of temperature and pressure and is independent of the initial density. For oilbased and water-based muds with the same density under the same conditions, the former is more affected by temperature and pressure. Hoberock et al. 19 conducted a study of predictive models for water-based and oil-based mud densities based on their components. Kutasov, 20 on the other hand, proposed an empirical model for determining mud density and regressed the empirical coefficients in the model. Agwu et al. 21,22 summarized the rheological properties of drilling fluids under HP/HT conditions. The above studies show that temperature and pressure have a significant influence on mud performance and ignoring these effects in high-temperature and highpressure wells can have many negative effects and seriously threaten drilling safety. Wellbore temperature fields have been studied using both numerical and analytical methods. [23][24][25][26] For the analytical approach, it is necessary to simplify the model to obtain simpler equations. Ramey 27 was the first to propose an analytical approach to the transient heat transfer problem in a wellbore, where he approximated the heat transfer within the formation using a time-dependent total heat transfer coefficient. Subsequently, Willhite 28 provided a detailed derivation of the heat transfer thermal resistance between mud in the wellbore and formations. Among the numerical methods, Raymond 29 provided a method to predict the wellbore temperature field for steady-state and presented steady-state cases. Marshall and Bentsen 30 provided four partial differential energy equations applicable to the drilling column, within the drilling column, in the annulus, and in the formations to solve the non-steady state heat transfer in the wellbore. Zhu et al. [31][32][33] considered the influence of high temperature on wellbore drilling fluid circulation under different circumstances, and established a transient wellbore flow model suitable for dry hot rock, drilling fluid leakage or gas drilling. However, in the above methods, the physical properties of mud, such as density and viscosity, are treated as constants. This is not consistent with the actual situation of mud performance changes due to high temperature and pressure in the wellbore and can make the calculation results highly inaccurate.
In summary, wellbore temperature, pressure, and mud performance are closely related, and a change in one of the three values changes the other two. In this study, based on the characteristics of HW-SH with a narrow annulus, we established a coupled model of the temperature-pressure field applicable to HW-SH, considering the eccentricity of the drilling column, drilling column rotation, and drilling column joints, and combining the influence of mud performance by temperature and pressure. On this basis, the effects of the ground temperature gradient, discharge volume, and annulus size on the temperature and pressure fields of the wellbore are analyzed to provide a reference for the accurate prediction of safe drilling of HW-SH.

| MUD PERFORMANCE TESTING UNDER HIGH TEMPERATURE AND HIGH-PRESSURE CONDITIONS
Temperature and pressure have a direct impact on drilling fluid density, rheological parameters K, and n, and then indirectly affect Reynolds number, friction coefficient, pressure loss along the path, and so on. The response characteristics of oil-based mud to temperature-pressure changes were more apparent than those of water-based mud. Therefore, we used oil-based mud for testing. The initial mud density was 1.749 g/cm 3 , the initial flow index was 0.2081, and the initial consistency coefficient was 0.8381 at 15°C and 0.1 MPa. According to the test results, the variation law of the slurry density and rheological parameters with temperature and pressure was analyzed, and a mathematical model of the density and rheological parameters with temperature and pressure was established by multivariate nonlinear regression. Figure 1A shows the Anto Paar densitometer, and Figure 1B shows the FannIX77 fully automatic rheometer. The densitometer was used to assess the mud density; its working pressure range was 0.1-120 MPa, the working temperature range was 15-250°C, and measurement accuracy was 0.01 g/cm 3 . The rheometer was used for the mud rheological performance test, the test temperature range was −10 to 316°C, the pressure range was 0.1-206 MPa.

| Response characteristics of density to temperature and pressure changes
The densities of 20 groups of oil-based mud were assessed under different temperature and pressure conditions, and the results are shown in Figure 2. As the temperature increased, the mud expanded due to heat, and the density decreased. As the pressure increased, the mud was compressed, and the density increased. The temperature increased from 20°C to 140°C, the pressure increased from 0.1 to 52 MPa, and the mud density increased from 1.749 to 1.824 g/cm 3 , indicating that high temperature and pressure did have a significant effect on the drilling fluid density.
2.2.2 | Characteristics of rheology response to temperature and pressure changes Figure 3 shows the shear rate-shear stress intersection of mud under different temperature and pressure conditions. The shear stress of the mud decreased significantly when the temperature increased, indicating that the rheological properties of the mud had improved. However, the increase in the shear stress of the mud was small when the pressure increased, which indicates that the flow properties of the mud became poor; however, this effect was small.

| Mud performance regression model
A multivariate nonlinear regressor was used to regress the mud density and rheological parameters and establish a mathematical model for the variation in mud density and rheological parameters with changes in temperature and pressure.

| Mud density regression model
During the drilling process, the wellbore temperature and pressure gradually increase in the vertical direction, and the mud density changes accordingly. The relationship between the three can be expressed by a partial differential equation, as shown in Equation (1). 33 where dh denotes the spatial step size. The effect of pressure on mud density is called the elastic compression effect. The strength of this effect is expressed by the elastic compression coefficient C e of mud, which is calculated using Equation (2).
The volume of the mud changes at high temperatures and pressures, and the water, oil, and solid phases in the mud have different compression coefficients and corresponding content changes. Therefore, the compression coefficient of the drilling fluid is a function of temperature and pressure. According to Peters 34 and others, this function can be expressed as: Combining (2) and (3), Equation (4) is obtained: The effect of temperature on mud density is called the thermal expansion effect and it can be expressed by Equation (5).
Combining Equations (1), (4), and (5), the relationship between the drilling fluid density, pressure, and temperature can be obtained using Equation (6). where, And where, P 0 is the surface pressure, MPa, and takes the value of 0.1 MPa; T 0 is the surface temperature,°C, and takes the value of 15°C; ρ P T ( , ) is the mud density with temperature-pressure variation, g/m 3 ; T is the wellbore temperature,°C; P is the wellbore pressure, MPa; ρ is the initial mud density, g/cm 3 and B 2 are the correlated fitting coefficients; ξ P , ξ PP , ξ T , ξ TT , and ξ PT are regression coefficients. Table 1 shows the regression coefficients of the regression model, where the correlation coefficient (R 2 ) of the regression model was 0.9281. Figure 4 shows the mud density variation with temperature-pressure variation.

| Mud rheological parameters regression model
Because the power-law model is based on two parameters (consistency coefficient and flowability index) to determine the relationship between mud shear stress and shear rate, and the two parameters do not respond to changes in temperature and pressure with a single trend, it is assumed that temperature and pressure only affect the drilling fluid consistency coefficient and have no effect on the flowability index in the analysis process; thus, a simple prediction model for the consistency coefficient is provided. 35,36 To avoid the errors caused by this method, in this study, we first used the power-law model to fit the regression of mud rheological parameter-specific equations, as shown in (8) and (9), and their correlation coefficients were 0.9598 and 0.9662, respectively. where, where K P T ( , ) is the mud consistency coefficient with temperature-pressure variation, n P T ( , ) is the mud flowability index with temperature-pressure variation, and ζ P , ζ PP , ζ TT , ζ PT , ω P , ω PP , ω TT , and ω PT are regression coefficients. Table 2 lists the regression coefficients of the regression model of the mud rheological parameters, and Figure 5 shows the variation law of the mud rheological parameters with temperature and pressure.
In addition, the response characteristics of the rheological properties of the slurry to temperature and pressure variations showed that temperature dominated the rheological parameters rather than pressure. It can be seen that when drilling into deeper formations, although the decrease in mud viscosity caused by high temperatures is partially compensated by the increase in viscosity due to increased pressure, the former decrease far exceeds the latter increase. In summary, owing to the synergistic effect of temperature and pressure, the viscosity of the drilling fluid gradually decreases with an increase in well depth.

| Assumptions
Based on the mud flow characteristics and the heat transfer law, the following assumptions were made: (1) Isotropic specific heat and thermal conductivity of the stratigraphic rocks and neglecting axial heat transfer within the stratigraphic rocks. (2) The radial temperature gradient in the wellbore annulus and drill column is zero. (3) The effect of rock chip beds was ignored. (4) The effects of the drill column eccentricity and drill column joints on the heat transfer process were ignored.

|
process, the drilling fluid exchanges heat through fluid heat convection and heat transfer ( Figure 6). The entire heat transfer system is divided into three areas: the area inside the drill column, the annular space area, and the formation area, which are independent and interrelated.
(1) Heat transfer inside drill column. As shown in the figure, for the fluid inside the drill column, the main sources of heat are fluid axial heat transfer Q p , frictional heat generation Q cp , and heat transfer Q cp between the outer and inner walls of the drill column.
Neglecting the effect of thermal conductivity of the drill column, that is, the temperature of the outer wall of the drill column is equal to that of the inner wall. According to the energy conservation theorem, where, Q ap1 is the energy transferred from the annular fluid to the outer wall of the drill column; Q ap2 is the energy transferred from the outer wall of the drill column to the inner wall of the drill column; Q ap3 is the energy transferred from the inner wall of the drill column to the fluid inside the drill column.
Where, where h po is the convection heat transfer coefficient on the outer surface of the drill pipe, W/(m 2 ·K); h pi is the convection heat transfer coefficient on the inner surface of the drill pipe, W/(m 2 ·K); d po is the outer diameter of the drill pipe, m; dpi is the inner diameter of the drill pipe, m; λ dp is the thermal conductivity of the drill pipe, W/(m·K); is the drilling fluid temperature inside the annular unit,°C; T z t ( , ) p is the drilling fluid temperature,°C; is the outer wall temperature of the drill pipe,°C ; T z t ( , ) pi is the inner wall temperature of the drill pipe,°C ; ∆t is the time step, s; ∆z is the space step, m. The heat transfer from the annular fluid to the fluid inside the drill column can be expressed as po pi dp (12) Defining U as the total heat transfer coefficient from the outside of the drill column to the inside of the drill column, and for the inside of the drill column, The reciprocal of the total heat transfer coefficient from the outside of the drill column to the inside of the drill column can be expressed as pi pi po po pi po pi dp (14) For the area inside the drill column, the heat sources were fluid axial heat transfer, frictional heat generation, and heat transfer from the outer to the inner wall of the drill column. The increment in the fluid axial heat transfer ∆t is where q m is the displacement (L/s); c m is the mud specific heat capacity (J/(kg·K). The heat gain due to friction is ∆ ∆ Q z t cp . The total energy increment within the system is.
According to the energy conservation theorem, the sum of the incremental heat transfer in the axial direction of the fluid, incremental heat transfer due to friction, and heat transfer from the outer wall of the drill column to the inner wall is the energy increment inside the system. Combining Equations (13)- (16) and replacing the differential with the differential, we can obtain the equation for the energy within the drill pipe.  (17) (2) Heat transfer in the wellbore annulus. As shown in Figure 7, for the annular region of the stratigraphic section, heat is transferred from the formation to the annulus in the form of convection heat transfer versus heat transferred from the fluid into the micro-element of the annulus in ∆t, with heat sources generated by friction and heat transfer to the interior of the drill column.
The heat flow from the stratum to the annular micro-element is where T f is the rock temperature in the formation unit°C ; h f is the convective heat transfer coefficient at the surface of the borehole wall in the bare borehole section W/(m 2 ·K); and d w is the borehole diameter m.
The heat of the fluid entering the annular microelement is Similarly, the heat generated by friction is The heat transfer to the interior of the drill column is The total energy increment within the system is Similarly, according to the law of conservation of energy, the heat flowing into the system unit and the AN ET AL. heat generated by friction within the system unit minus the heat flowing out of the system unit are equal to the total energy increment within the system.
Combining Equations (18)-(11) and replacing the differential with the differential, the energy equation in the annulus can be obtained as follows: (3) Heat transfer within the formations. As shown in Figure 8, the formation temperature was higher than the wellbore mud temperature, and the heat within the formation was transferred to the wellbore.
Considering only the variation in the temperature profile with the radial distance of the stratum, the difference between the energy flowing into and out of the radial unit of time is equal to the value of the variation in energy within the unit. , where ρ f is the rock density (g/cm 3 ); λ f is the thermal conductivity of the stratum rock (W/(m·K).
The energy equation for the heat transfer within the formation can be obtained by taking the boundary point within the formation as a reference.
(4) Initial and boundary conditions. Initially, at the same depth of the annulus fluid, the fluid temperature F I G U R E 7 Schematic diagram of heat transfer in the wellbore annulus F I G U R E 8 Schematic diagram of heat transfer within the formations inside the drill pipe is equal to the formation temperature and is the original formation temperature. The specific expressions are as follows: .
At the wellhead, the temperature is the surface temperature.
The original stratigraphic temperature does not vary with time and can be expressed as

| Wellbore pressure model
According to Cartalos and Song et al., 10,13 the drill column rotation, drill column eccentricity, and drill column joint size significantly affect the pressure dissipation in HW-SH. However, their study was based only on indoor experiments ignoring temperature and pressure, that have a strong influence on the drilling fluid density and rheology. Therefore, in the case of high pressure and high temperature in HW-SH, failure to consider the effects of temperature and pressure is bound to cause errors. To improve the calculation accuracy of the cyclic pressure consumption in HW-SH, the cyclic pressure depletion ∆P P T ( , ) in Song's model is replaced in this study by the cyclic pressure depletion after considering the temperature and pressure. Accordingly, Equation (28) is obtained.
Where ∆P P T ( , ) is the circulating pressure depletion considering the temperature and pressure factors, F t is the drill column rotation factor, R is the drill column eccentricity factor, F con is the drill column joint influence factor, and ∆P c is the additional pressure depletion of annular rock chips.
The ∆P P T ( , ) is calculated as follows.
where, f is the friction coefficient, which is calculated as follows: When the flow state of the mud is laminar.
When the flow state of the mud is turbulent, f must be solved using an iterative method.
where R e is the Reynolds number. R e can be calculated for power-law fluids flowing in the annulus as follows: where, R P T ( , ) e is the Reynolds number considering the temperature-pressure effect.
The F t varies with the Taylor number (T a ) and R e . When T < 41 a , F t decreases with the drill column rotation but is close to 1. When T > 41 a , F t increases with an increase in the drill column speed. When the mud is in a transition flow, F t is maximum, and when the mud is in a turbulent flow, the drill column rotation has almost no effect on the circulating pressure consumption. The computational model for when R e is another value, F t is calculated by linear interpolation, but it is only applicable where, ω is the angular velocity, rad/s. According to Delwiche et al., 11 the annular pressure depletion decreases as the eccentricity of the drilling AN ET AL.
| 1069 column increases. However, R can be affected by mud properties, eccentricity of the drilling column, and mud flow status. A sinusoidally curved wellbore annulus provides an accurate description of the actual wellbore annulus shape, considering the rotation and drilling pressure of the drill column. Additionally, the concept of average eccentricity (λ avg ) was introduced to facilitate the calculation.
( ) When calculating R, the effect of the mud flow status must be considered first. When the mud is in the laminar flow regime, the drill column eccentricity factor is expressed as R lam and the Reynolds number is R ecl . When the mud is in the turbulent flow regime, the drill column eccentricity factor is expressed as R tur and the Reynolds number is R ec2 . When the mud is in the laminar flow to the excessive flow critical point, the drill column eccentricity factor is maximum and is expressed as R max .
When the mud is at the critical point of the laminar to transition flow, R ecl is R lam and R tur are calculated as follows.
Therefore, R is calculated as.  (40) In the process of calculating the pressure depletion of conventional horizontal wells, the influence of drill pipe joints on the pressure depletion, especially the pressure depletion inside the drill pipe, is often ignored. The HW-SH has a very small internal diameter of the drill pipe, and the pressure depletion is larger than that of conventional horizontal wells. If the effect of drill pipe joints is not considered, it will have a significant impact on the accuracy of the calculation of the circulating pressure depletion. 37 Therefore, when calculating the circulating pressure consumption in HW-SH, it is important to consider the effect of the drilling column joint on the circulating pressure depletion to calculate the circulating pressure depletion more accurately.
where L con is the length of the drill-pipe joint, L p is the length of the drill pipe, D p is the outer diameter of the drill pipe, and D j is the outer diameter of the joint. When calculating the internal pressure consumption of the drill column, these two diameters were replaced by the internal diameter of the drill pipe and the internal diameter of the drill pipe joint.
In the drilling process of HW-SH, wellbore cleanliness is good because of the good centering of the drill pipe and the high return velocity of the annulus during the drilling process. Therefore, the effect of the rock chip bed on the annular pressure depletion can be ignored, and only the effect on the annular pressure depletion when the concentration of the annular rock chip is stabilized can be considered. Then, the ∆P c can be calculated as follows: where, g is the gravitational acceleration; H is the spatial step length; ρ S and ρ m are the rock chip and mud densities, respectively; C a is the annular rock chip concentration.

| Solution method
To accurately predict annular pressure depletion in horizontal wells, Yang et al. 38 established a horizontal wellbore pressure prediction model applicable to conventional well types. However, the effect of temperature on mud rheology was not considered in their calculations. Song et al. 10 considered the narrow annulus characteristics of HW-SH and established a model to calculate the pressure consumption of HW-SH; however, they did not consider the effect of temperature on mud. In fact, mud performance, temperature, and pressure interact with each other, and none of the above models consider the coupling effect between them. Therefore, based on their research results, we used the finite difference method to couple the three methods for the calculation. The model discretization and computational flow are shown in Figures 8 and 9, respectively. From the perspective of stability, the differential equations established above were treated in a fully implicit form. The first-order spatial derivatives in the differential equation are in a first-order windward format, and the first-order time derivatives are in a two-point backward differential. The number of nodes in the well depth direction is denoted by m and the number of nodes in the radial direction is denoted by n. Figure 10 shows all the calculation steps in one-time step. Where, P, T, ρ, K , and n are matrices representing pressure, temperature, drilling fluid density, consistency factor, and flowability index, respectively. All iterations are calculated using the Gauss-Seidel method. k represents the iteration step and t represents the time step. After obtaining the mud density and pressure in the entire wellbore for the next time step, the temperature and rheology parameters are automatically obtained until the end of all time steps.

| SOLUTION RESULTS AND VALIDATION
We used data from an HW-SH (#A) in the Mahu oil field in China to verify the accuracy of the model. The well depth of #A was 4921 m, and the depth of the kickoff point was 3225 m. Between them, the average rate of penetration (ROP) of the horizontal section was 8.9 m/h. The combination of drilling tools used was "Φ165.1mm drill bit (0.21 m) + 1.25°screw (7.4 m) + floating valve (0.50 m) + Φ120.65mm directional joint (0.85 m) + Φ311mm Φ120.65mm nonmagnetic drill collar (9.49 m) + Φ311mm DSTJ40 female joint (0.48 m) + Φ101.6mm weighted drill pipe (3 * 28.25 m) + Φ101.6mm slope drill pipe," other drilling parameters are shown in Table 3. Figure 11 shows the temperature distribution in the annulus for different cycle times. There is a "neutralization point" in the temperature profile, where the temperature remains constant around this point. Above this point, the annular temperature increases with increasing mud circulation time, whereas below this point, the annular temperature decreases with increasing mud circulation time. This is because the fluid in the annulus above the neutralization point is heated, while the fluid in the annulus below this point continuously removes heat from the bottom of the well. The "neutralization point" remains in a constant-heat state. After 7 h of drilling, the downhole drilling fluid temperature and the formation temperature reached a dynamic equilibrium, and the drilling fluid temperature in the horizontal section was 96.52-102.78°C, which was 81.52-87.78°C, higher than the surface temperature of 15°C. This temperature difference causes the mud performance to change significantly, and if the surface mud density and rheological parameters are used to calculate the wellbore pressure, large errors will occur. Therefore, it is necessary to consider the effect of temperature. Figure 12 shows the variation in mud density with the well depth in the wellbore annulus. In the developed models, we used the equivalent viscosity (μ e ) to represent the rheological properties of mud. 39 Figure 13 shows the variation pattern of the equivalent viscosity of the mud in the annulus with the well depth. As the drilling time increased, the mud temperature in the annulus gradually decreased and the density gradually increased. After 7 h of drilling, the mud density in the horizontal section was 0.032-0.033 g/cm 3 lower than that of the surface mud. In addition, the mud density jumped at the casing shoe owing to the increase in the annular pressure caused by the change in the annular size. Moreover, as the mud temperature in the horizontal section decreased with drilling time, the pressure gradually increased, and the mud equivalent viscosity increased. The effects of temperature and pressure on mud density and rheological properties are evident from the graph.

| Pressure field
The wellbore pressure was represented by the equivalent circulating pressure (ECD). Figure 14 shows the ECD for the full-well section. As the mud circulation time increased, the temperature of the annular mud  gradually decreased, leading to an increase in mud density and viscosity and a gradual increase in the annular pressure depletion and ECD. After 7 h of circulation, the pressure in the annulus almost reached a steady state. In the straight section, as the mud temperature increased with an increase in well depth, this lead to a corresponding decrease in mud density, viscosity, and ECD. In the slanting section, due to the effect of drilling fluid static column pressure and annular pressure depletion, the ECD shows a decrease followed by an increase. In the horizontal section, the ECD of the drilling fluid gradually increased along the horizontal section because the vertical depth was constant, the density of the drilling fluid changed less, while the annular pressure depletion increased with an increase in the horizontal section length. The ECD at the bottom of the horizontal section was 0.097 g/cm 3 higher than that at the beginning of the horizontal section. Therefore, when designing the mud density in the narrow safety window, the change in ECD at the same depth should also be considered so that the mud safety window is maintained to achieve safe drilling. After 7 h of drilling, the ECD at the bottom of the well was 0.145 g/cm 3 higher than the surface mud density, and this additional pressure consumption must be considered when designing the mud density in leakprone formations.

| Model validation
When the bottomhole temperature pressure reached a stable value, the bottomhole pressure calculated using the developed model was 25 MPa, while the measured   Figure 15 shows the differences in ECD calculations for the different models using data from #A for analysis. If the effect of temperature is not considered, the calculations will show a constant drilling fluid density and viscosity, making the unit annular pressure consumption constant throughout the straight section of the well. Furthermore, this will result in the ECD not changing with well depth. Ignoring the effect of temperature will result in larger ECD calculation results, which will lead to accidents such as well surges and gas intrusion (2800-3000 m section). If we do not consider the characteristics of the narrow annulus of HW-SH, although the ECD of the straight section will show a decreasing trend with an increase in well depth, neglecting the influence of drill pipe rotation, joint, and eccentricity on ECD will still lead to smaller ECD calculation results, which may lead to well leakage and other accidents (2600-2800 m section). Similarly, ignoring the effect of temperature in the slope making and horizontal sections leads to large ECD calculation results. Ignoring the characteristics of a narrow annulus leads to smaller ECD calculation results. When the wellhead was not supplemented with pressure, the actual ECD at the bottom of the well was 1.72 g/cm 3 . From the calculation results, if the temperature is not considered, the calculated ECD at the bottom of the well is 1.783 g/cm 3 with an error of 0.063 g/cm 3 . If both the temperature and characteristics of the narrow annulus are not considered, the calculated bottomhole ECD is 1.754 g/cm 3 , which reduces the bottomhole error but still causes a large error in the straight section of the well. If the temperature and the characteristics of the narrow annulus are also considered, the calculated bottomhole ECD is 1.716 g/ cm 3 , and the error is only 0.004 g/cm 3 , which is very accurate and meets the engineering requirements.

| CHARACTERISTICS OF TEMPERATURE-PRESSURE COUPLING FIELD IN HW-SH
Using the established model, the effects of the ground temperature gradient, annular size, displacement, rotational speed, and joint size on the annular temperature and pressure fields of HW-SH were analyzed to determine the main influencing factors and provide references for safe and efficient drilling in the actual drilling process. Figure 16 shows the annulus temperature and pressure fields under different ground temperature gradient conditions. The ground temperature gradient directly affects the distribution of the annulus temperature. Undoubtedly, a larger ground temperature gradient results in a higher temperature in the annulus. As the ground temperature gradient increases, ECD at the bottom of the well decreases significantly. This is due to the increase in temperature, decrease in mud density, increase in shear dilution, and decrease in viscosity. In the bottomhole, the ECD decrease accelerates gradually with an increase in ground temperature gradient. This shows that it is essential to consider the effect of temperature on the ECD. Figure 17 shows the annular temperature and pressure fields for different annular sizes. In the case of a fixed drilling tool, the smaller annular size was due to the narrowing of the borehole, whereas the larger annular space size was due to the expansion of the borehole. The effect of the annular size change owing to either the expansion or narrowing of the annular temperature field is small and can be ignored. However, the change in annular size had a significant effect on the bottomhole ECD. The smaller the annulus size, the larger was the bottomhole ECD. Moreover, the rate of increase in the bottomhole ECD increased gradually as the annulus size decreased. On the one hand, this is because the mud flow rate in the annulus becomes larger when the annulus becomes smaller, and on the other hand, it is because the effect of drill pipe rotation on the growth of annular pressure dissipation becomes F I G U R E 16 Effect of ground temperature gradient on the temperature and pressure fields in the annulus F I G U R E 17 Effect of annulus size on the annulus temperature and pressure fields AN ET AL.

| Wellbore annulus size
| 1075 more obvious when the annulus becomes smaller, and the two aspects together lead to a larger change of pressure depletion. Therefore, during the drilling process of HW-SH, especially in creep-prone formations (e.g., paste salt rock formations), attention should be paid to the effect of annulus shrinkage on ECD to avoid more complex downhole accidents, such as fracture expansion and well leakage due to the surge of ECD caused by shrinkage. Figure 18 shows the temperature and pressure fields of the annulus under different displacement conditions. The most intuitive phenomenon of increasing the displacement is the increase in the mud flow rate in the annulus. In the section of the well below the "neutralization point," the faster flowing mud removes the heat faster, resulting in a lower temperature in the annulus, that is, the higher the displacement, the lower the temperature in the annulus. In the section of the well above the "neutralization point," the mud flowing faster will bring the depth heat faster, resulting in a higher annular temperature, that is, the higher the displacement, the higher the annular temperature. Like the effect of annulus size on the bottomhole ECD, the higher the displacement, the higher the flow rate of mud in the annulus, the higher the pressure depletion along the way, and the higher the bottomhole ECD. Although the increase in displacement and the increase in mud density and viscosity in the section below the "neutralization point" can lead to a reduction in circulating pressure depletion, the significant effect of the increased flow rate completely overrides this effect. In practice, although a high displacement is good for keeping the bottom of the well clean, it leads to high bottomhole ECD and is prone to downhole complications such as wall ruptures and well leaks, especially in formations with narrow density windows.

| Rotational speed of drilling column
The rotational speed of the drill pipe is one of the biggest differences between HW-SH and conventional horizontal wells in terms of pressure consumption calculation algorithms. Figure 19 shows the annular temperature and pressure fields for a drilling column with rotational speeds ranging from 0 to 160 r/min. Although the annular temperature increases with an increase in the rotational speed, the increase is very small. Because this effect is caused by increased friction between the drill pipe and mud, the heat from friction is very limited. For the calculation of pressure depletion, when the rotational speed is low (less than 20 r/min), the rotational speed of the drill column has no effect on the calculation of annular pressure depletion. When the rotational speed increases from 20 to 160 r/min, the bottomhole ECD increases significantly with an increase in the rotational speed. The rate of increase is gradual in the segment from 20 to 60 r/min and decreases in the segment from 60 to 160 r/min. This law is consistent with the results found by Mccann et al. 11 and Wang et al. 40 in their indoor experiments, and confirms that the present model can successfully F I G U R E 18 Effect of displacement on the temperature and pressure fields in the annular air characterize the effect of the rotational speed of the drill pipe on the pressure depletion in HW-SH. It is worth noting that the bottomhole ECD increased from 1.682 to 1.744 g/cm 3 when the rotational speed of the drill column increased from 0 to 160 r/min, an increase of 0.062 g/cm 3 . Therefore, ignoring the effect of drill column rotation will lead to large errors when predicting the bottomhole ECD in HW-SH. Figure 20 shows the annular air temperature and pressure fields for different joint sizes. The effect of the change in the joint size was like that of the annulus size. As the joint size increases, the annulus size decreases. The difference is that the change in annulus size due to reduction and expansion is within a well section, whereas the change in annulus size due to joint size change is only within a short well section that can be considered as a point. The change in joint size has almost no effect on the annular temperature field but has a large effect on the annular pressure field. As the joint size increases, the annulus size decreases and the ECD at the bottom of the well increases. Therefore, the selection of small joints can effectively reduce the bottomhole ECD and the risk of well wall destabilization while ensuring sufficient strength.

| Joint size
F I G U R E 19 Influence of the rotational speed of the drilling column on the temperature and pressure fields in the annulus F I G U R E 20 Effect of joint size on the temperature and pressure fields in the annulus AN ET AL.

| CONCLUSION
A wellbore temperature-pressure coupled field model applicable to the HW-SH was developed, based on the characteristics of the narrow annulus of the HW-SH, combined with the influence of temperature and pressure on mud performance. The developed model was validated using actual measurement data. On this basis, the effects of factors such as the ground temperature gradient, displacement, and annulus size on the annulus temperature and pressure fields were analyzed. The results are as follows: (1) Temperature has a significant effect on the density and rheology of mud. The density of the mud decreased and the rheological performance improved when the temperature increased. The pressure also has a significant effect on the density of the mud, and the density of the mud increases when the pressure increases. The effect of pressure on the rheological properties of the mud can be ignored. (2) The coupled temperature-pressure model of the HW-SH considering the effects of temperature and narrow annulus can accurately predict the annulus temperature and pressure fields. (3) The ground temperature gradient has the greatest influence on the HW-SH annular temperature, larger than that of the displacement. In addition, the annular size, drill pipe drilling speed, and joint size have almost no influence on the annular temperature field. (4) Compared to those of conventional horizontal wells, the drilling speed of the drill pipe, annular size, and joint size of HW-SH have a greater influence on the annular ECD.