The Analysis of Transient Temperature in the Wellbore of a Deep Shale Gas Horizontal Well

Abstract

The transient temperature of the wellbore plays an important role in the selection of downhole tools during the drilling of deep shale gas horizontal wells. This study established a transient temperature field model of horizontal wells based on the convection heat transfer between wellbore and formation and the principle of energy conservation. The model verification shows that the root mean squared error (RMSE) between the measured annular temperature neat bit and the predicted value is 0.54 °C, indicating high accuracy. A well in Chongqing, China, is taken as an example to study the effects of bottom hole assembly (BHA), drill pipe size, drilling fluid density, flow rate, inlet temperature of drilling fluid, and drilling fluid circulation time on the temperature distribution in wellbore annulus. It is found that the increase in annular temperature is about 1 °C/100 m in the horizontal section when a positive displacement motor (PDM) is used. A Φ139.7 mm drill pipe is more favorable for cooling than Φ139.7 mm + Φ127 mm drill pipe. Reducing drilling fluid density and flow rate and inlet temperature is beneficial to reduce bottom hole temperature. Bit-breaking rock, bit hydraulic horsepower, and drill pipe rotation will increase the bottom hole temperature. The research results can provide theoretical guidance for temperature prediction, selection of proper drill tools, and adjustment of relevant parameters in deep shale gas horizontal wells.

1. Introduction

With the rapid development of technology, the exploration of oil and gas resources has greatly increased, and the exploration and development of deep shale gas has also made good progress [1,2]. However, with the continuous development of shale gas exploration and development to deep formation, complex geological conditions such as high temperature and high pressure have become challenges in deep shale gas horizontal well drilling engineering [3]. The high-temperature environment is a primary reason for the failure of downhole tools. Currently, tools are often selected based on formation temperature, with the empirical assumption that tool failure will not occur as long as the tool’s temperature limit equals the formation temperature. However, a significant amount of heat is generated during the drilling process, which can cause the bottom hole temperature to exceed the formation temperature [4]. Therefore, to ensure safe drilling in deep shale gas horizontal wells, it is necessary to conduct research on the temperature distribution of horizontal wellbores and influencing factors.

The temperature distribution in the wellbore during the drilling process is difficult to measure directly. Scholars often use computer simulations, employing analytical and numerical methods to explore the distribution rule of wellbore temperature. The analytical approach, represented by the research of Ramey [5], Holmes and Swift [6], Shiu and Beggs [7], Hasan and Kabir [8], and Li [9,10], is based on the steady-state heat transfer mechanism of the wellbore. It adopts a semi-analytical method to address the unsteady heat conduction within the formation. By analyzing the heat flow into and out of infinitesimal elements and applying the principle of energy conservation, an analytical mathematical model for steady-state heat conduction in the wellbore is established. However, this analytical model of the wellbore temperature field neglects factors such as heat source terms and wellbore structure, making it difficult to accurately predict the wellbore temperature field under complex operating conditions. The numerical method is rooted in the transient heat transfer mechanism of the wellbore. It takes into account factors such as thermal convection of the wellbore fluid, axial heat conduction of the drill pipe, convective heat transfer between the drill pipe and the fluid, as well as heat exchange among the casing, cement column, and formation. Based on the principle of energy conservation, differential equations of the wellbore temperature field are formulated. These equations are solved using methods such as finite difference, finite volume, and finite element [11,12,13,14,15]. Representative works include Raymond [11] and Marshall [12], who were the first to establish a numerical model for transient heat transfer in the wellbore during drilling fluid circulation.

Based on these studies, scholars have expanded the research scope and adaptation conditions of wellbore temperature fields. Their research has taken into account conditions such as fluid properties, drilling parameters, and heat sources [16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31]. Li et al. [21] studied the temperature profile of long horizontal wellbore sections using a three-dimensional stability model. Yang et al. [22] established a transient heat transfer model for the wellbore formation during the entire drilling process, analyzing the downhole temperature distribution during drilling fluid circulation and stop circulation. Li et al. [23] discussed the variation of wellbore and formation temperatures with shut-in time under overflow. Zhang et al. [24] studied the wellbore temperature distribution during the circulation phase when a continuous formation kick occurs at the bottom of the well. Al Saedi et al. [25] proposed two new models to solve the temperature distribution during drilling fluid circulation. Yang et al. [26] investigated transient temperature changes in the wellbore when the drill pipe maintains an eccentric position during deep well operations. Abdelhafiz et al. [27] established transient and steady-state numerical analysis models for wellbore temperature during drilling fluid circulation, analyzing the effects of flow rate and flow pattern on wellbore temperature distribution. Song and Guan [28] presented a complete transient temperature model for deepwater drilling fluid circulation, analyzing the influence of factors such as water depth, water temperature, and riser insulation on the temperature of drilling fluid circulation. Khan and May [29] developed a mathematical model for predicting transient bottomhole temperature during drilling, applicable to vertical wells, inclined wells, and horizontal wells. Yang et al. [30] established a reel well drilling temperature model considering bit heating. Mao et al. [31] created a wellbore temperature prediction model that accounts for bit heat generation and variations in mud thermophysical parameters, exploring the effects of mud system, density, flow rate, inlet temperature, Revolutions per minute (RPM), and weight on bit (WOB) on wellbore and downhole tool temperatures.

It is not difficult to find that most existing models are aimed at predicting wellbore temperature distribution during vertical drilling. However, due to the different temperature distributions in the surrounding formations of the vertical section, oblique section, and horizontal section of horizontal wells, there are differences in wellbore temperature distribution compared to vertical wells. Therefore, this paper established a wellbore temperature field model for deep shale gas horizontal drilling, solved the model using the finite difference method, and validated it with field data. The results show that the model has sufficient accuracy to calculate the transient wellbore temperature distribution. Based on this, we investigated the effects of bottom hole assembly (BHA), drill pipe size, drilling fluid density, flow rate, inlet temperature of drilling fluid, and drilling fluid circulation time on the temperature distribution in the wellbore annulus. This research can provide a theoretical understanding of the prediction of transient wellbore temperature in deep shale gas horizontal wells and its application.

2. Wellbore Heat Transfer Model

2.1. Physical Model

During the drilling process, the drilling fluid discharged from the mud pump reaches the wellhead through the ground circulation system and then flows from the wellhead through the kelly, drill pipe, and drill collar to the drill bit, where it is sprayed through the drill nozzle. It then flows upward through the annulus between the drill pipe and the casing (or well wall) and returns to the ground, as shown in Figure 1. The flow of drilling fluid in the downhole circulation system is mainly divided into two stages: ① drilling fluid flows from the wellhead to the drill bit. ② drilling fluid returns from the bottom of the well to the ground.

2.2. Model Assumes

Based on the characteristics of drilling fluid circulation during horizontal drilling, the transient wellbore temperature field model is established based on the following basic assumptions:

• The formation temperature varies linearly with well depth, and the effects of overflow, loss, and fluid phase transitions during the drilling process on the wellbore temperature distribution are not considered.
• During rotary drilling, the friction heat generated by drill pipe buckling and rotation is not considered. The drill pipe is centered in the wellbore, and the wellbore trajectory has a regular geometric shape.
• Fluid conduction along the axial direction and radial temperature gradients within the fluid are neglected, while axial heat conduction of the drill pipe and convective heat transfer between the fluid and the drill pipe wall in the radial direction are considered.
• The thermophysical parameters of the fluid, formation and various heat transfer media in the wellbore remain constant, and the influence of fluid flow in the formation pores on the formation temperature distribution is neglected.

It is worth noting that the elimination of the effects of overflow, leakage, and fluid phase transition is mainly because these factors may not be the main influencing factors during drilling, and introducing these complex factors into the simulation will greatly increase the complexity and uncertainty of the model.

2.3. Mathematical Model

Based on the research of Raymond [11], Marshall & Bentsen [12], and Yang [32], etc., this paper takes a thermal equilibrium infinitesimal dz in the direction of the wellbore axis at depth z and time t. The heat exchange between various media downholes is shown in Figure 2. When the drilling fluid flows downward along the drill pipe from the wellhead at a certain temperature (T_p), its temperature change depends on the heat exchange with the annular drilling fluid in the radial direction (Q_ap), as well as the external work performed on it (dW₁), which includes friction heat generated during the downward movement of the drilling fluid (Q_fD), heat generated by the rotation of the drill pipe (Q_rd), heat from bit breaking rock (Q_rg), and heat generated by the drilling fluid passing through the nozzle (Q_np). When the drilling fluid enters the annulus after passing through the drill bit and returns upward, its temperature change depends on the heat exchange with the drilling fluid inside the drill pipe and the formation in the radial direction (Q_la), as well as the external work done on it (dW₂), such as the friction heat generated during the upward flow of the annular drilling fluid (Q_fa).

Taking the thermal equilibrium microfacies formation, fluid in the annulus, drill pipe wall, and fluid within the drill pipe as research objects, considering the influence of heat sources on the transient temperature field of the wellbore, and based on the conservation of energy, transient heat transfer mathematical models are established.

2.3.1. Temperature Model of Mud in Drill Pipe

There are two parts of heat change of drilling fluid in the drill pipe in the axial direction: ① inflow heat Q_p(z). ② outflow heat Q_p(z + dz).

In time dt, the net heat imported and exported by the drilling fluid in the drill pipe in the axial direction is

$$d{Q}_1=Q_{\mathrm{p}}\left(z\right)-Q_{\mathrm{p}}\left(z+dz\right)={\rho }_{\mathrm{l}}{c}_{\mathrm{l}}q\left[T_{\mathrm{p}}\left(z,t\right)-T_{\mathrm{p}}\left(z+dz,t\right)\right]dt$$

(1)

where ρ_l is the density of drilling fluid in the drill pipe, kg/cm³. c_l refers to the specific heat capacity of drilling fluid in the drill pipe, j/(kg·°C). q refers to the mass flow of drilling fluid, kg/s. T_p refers to the drilling fluid temperature in the drill pipe, °C. z is the well depth, m. t is time, s.

In time dt, the heat exchanged between the drilling fluid in the drill pipe and the inner wall of the drill pipe in the radial direction by convection heat exchange is

$$d{Q}_2=2\pi r_{\mathrm{pi}}{h}_{\mathrm{pi}}\left[T_{\mathrm{d}}\left(z,t\right)-T_{\mathrm{p}}\left(z,t\right)\right]dzdt$$

(2)

where r_pi is the inner radius of the drill pipe, m. h_pi refers to the convective heat transfer coefficient of the inner wall of the drill pipe, W/(m²·°C). T_d refers to the temperature of the drill pipe wall, °C.

The change of heat in the drilling fluid inside the drill pipe in the radial direction is the heat transferred from the annular drilling fluid, and the change of heat caused by the internal heat source is the viscous dissipation heat generated by the flow of drilling fluid.

In time dt, the work performed by friction caused by fluid flow in the drill pipe is

$$dW=Q_{\mathrm{fp}}dzdt$$

(3)

where Q_fp refers to the heat generated by friction in the drill pipe, W/m.

In time dt, the increment of energy in the infinitesimal control body is

$$dE={\rho }_{\mathrm{l}}{c}_{\mathrm{l}}\frac{\partial T_{\mathrm{p}}}{\partial t}\mathsf{\pi }{r}_{\mathrm{pi}}^{2}dtdz$$

(4)

According to the first law of thermodynamics, the net heat imported and exported into and out of the infinitesimal element plus the heat generated by the heat source within the infinitesimal element equals the increase in internal energy of the infinitesimal element. Therefore, the temperature model of mud in drill pipe is as follows:

$$Q_{\mathrm{fp}}-{\rho }_{\mathrm{l}}q{c}_{\mathrm{l}}\frac{\partial T_{\mathrm{p}}}{\partial z}-2\mathsf{\pi }{r}_{\mathrm{pi}}{h}_{\mathrm{pi}}\left(T_{\mathrm{p}}-T_{\mathrm{d}}\right)={\rho }_{\mathrm{l}}{c}_{\mathrm{l}}\mathsf{\pi }{r}_{\mathrm{pi}}^2\frac{\partial T_{\mathrm{p}}}{\partial t}$$

(5)

2.3.2. Temperature Model of Drill Pipe and Drill Bit

The drill pipe wall exchanges heat through convection with the drilling fluid inside and outside the drill pipe radially while conducting heat axially through heat conduction. Based on the first law of thermodynamics, the temperature model of the drill pipe and drill bit can be established as follows:

$${\lambda }_{\mathrm{d}}\frac{{\partial }^2{T}_{\mathrm{d}}}{\partial z^2}+\frac{2r_{\mathrm{po}}{h}_{\mathrm{po}}}{r_{\mathrm{po}}^2-r_{\mathrm{pi}}^2}\left(T_{\mathrm{a}}-T_{\mathrm{d}}\right)+\frac{2r_{\mathrm{pi}}{h}_{\mathrm{pi}}}{r_{\mathrm{po}}^2-r_{\mathrm{pi}}^2}\left(T_{\mathrm{p}}-T_{\mathrm{d}}\right)+Q_{rg}={\rho }_{\mathrm{d}}{c}_{\mathrm{d}}\frac{\partial T_{\mathrm{d}}}{\partial t}$$

(6)

where ρ_d d is the density of the drill pipe, kg/m³. c_d refers to the specific heat capacity of the drill pipe, J/(kg·°C). λ_d is the thermal conductivity of the drill pipe, W/(m·°C). r_po is the outer radius of the drill pipe, m. T_a refers to the temperature of drilling fluid in the annulus, °C.

The mathematical expression of friction heat generated by the bit and the formation is [31,33]

$$Q_{rg}=\frac{\xi fWN\left(D^2+Dd+d^2\right)}{74.3(D+d)}$$

(7)

where ξ is the correction factor. Q_rg is the friction heat generated by the bit and the formation, W. f is the friction coefficient between the bit and the formation. W is the weight on the drill bit, kN. N is the RPM, r/min. D is the outer diameter of the drill bit, m. d is the inner diameter of the drill bit, m.

2.3.3. Temperature Model of Annular Mud

The heat inflow and outflow of the drilling fluid in the annulus in the axial direction are Q_a(z + dz) and Q_a(z). According to the first law of thermodynamics, the temperature model of annular mud is

$${\rho }_{l}q{c}_l\frac{\partial T_a}{\partial z}+2\pi r_w{U}_w\left(T_f-T_a\right)+2\pi r_{po}{h}_{po}\left(T_d-T_a\right)+Q_{fa}={\rho }_l{c}_l\pi \left(r_w^2-r_{po}^2\right)\frac{\partial T_a}{\partial t}$$

(8)

where h_po is the convective heat transfer coefficient of the outer wall of the drill pipe, W/(m²·°C). U_w is the comprehensive heat transfer coefficient.

2.3.4. Temperature Model of Formation near Well

In the cylindrical coordinate system, the differential equation of formation heat conduction can be expressed as follows [22]:

$$\frac{{\partial }^2{T}_f}{\partial r^2}+\frac{1}{r}\frac{\partial T_f}{\partial r}+\frac{{\partial }^2{T}_f}{\partial z^2}=\frac{{\rho }_f{c}_f}{{\lambda }_f}\frac{\partial T_f}{\partial t}$$

(9)

where r is the radial distance, m. λ_f is the formation thermal conductivity, W/(m·K). c_f is the specific heat capacity of rock, J/(kg·°C). ρ_f is the rock density, kg/m³. T_f refers to the formation temperature, °C.

2.4. Initial Condition and Boundary Condition

Considering the characteristics of horizontal well drilling, definite solution conditions are given for the vertical section, oblique section, and horizontal section of the horizontal well.

2.4.1. Initial-Boundary Condition of Vertical Section

In the vertical section, the initial temperature of each heat transfer unit is assumed to be equal to the original formation temperature, as shown in Equation (10).

$${\left.T_p\right|}_{z,t=0}={\left.T_d\right|}_{z,t=0}={\left.T_a\right|}_{z,t=0}={\left.T_f\right|}_{r,z,t=0}=T_s+G_T\cdot z$$

(10)

At the wellhead (z = 0), the drilling fluid inlet temperature and the drilling fluid outlet temperature can be measured by wellhead instruments and equipment, which are as follows:

$$\left\{\begin{array}{l}{\left.T_{\mathrm{p}}\right|}_{z=0,t}=T_{\mathrm{in}}\\ {\left.T_{\mathrm{a}}\right|}_{z=0,t}=T_{\mathrm{out}}\end{array}\right.$$

(11)

At the kick-off point (z = H_kop), the temperature at the end of the vertical section is equal to the temperature at the beginning of the oblique section, which can be expressed as

$$\left\{\begin{array}{l}{\left.T_{\mathrm{p}}\right|}_{z=H_{\mathrm{kop}},t}=T_{\mathrm{p}\_\mathrm{kop}}\\ {\left.T_{\mathrm{a}}\right|}_{z=H_{\mathrm{kop}},t}=T_{\mathrm{a}\_\mathrm{kop}}\end{array}\right.$$

(12)

where T_{p_kop} refers to the drilling fluid temperature in the drill pipe at the kick-off point, °C. T_{a_kop} refers to the drilling fluid temperature in the annulus at the kick-off point, °C.

2.4.2. Initial-Boundary Value Condition of Oblique Section

In the oblique section, the initial temperature of each unit is the same as that of the formation, as shown in Equation (13).

$${\left.T_p\right|}_{z,t=0}={\left.T_d\right|}_{z,t=0}={\left.T_a\right|}_{z,t=0}={\left.T_f\right|}_{r,z,t=0}$$

(13)

The temperature at the end of the oblique section is the same as the beginning of the horizontal section, which can be expressed as

$$\left\{\begin{array}{l}{\left.T_{\mathrm{p}}\right|}_{z=H_{\mathrm{end}},t}=T_{\mathrm{p}\_\mathrm{end}}\\ {\left.T_{\mathrm{a}}\right|}_{z=H_{\mathrm{end}},t}=T_{\mathrm{a}\_\mathrm{end}}\end{array}\right.$$

(14)

where H_kop is the well depth of the kick-off point, m. H_end is the well depth at the beginning of the horizontal section, m. T_{p_end} is the drilling fluid temperature in the drill pipe at the beginning of the horizontal section, °C. T_{a_end} is the drilling fluid temperature in the annulus at the beginning of the horizontal section, °C.

2.4.3. Initial-Boundary Condition of Horizontal Segment

In the horizontal section, the initial temperature of each unit is the same as that of the formation, as shown in Equation (13).

At the far boundary of the formation, the formation temperature is not affected by the wellbore heat transfer, and the formation temperature is the original formation temperature, which is

$${\left.\frac{\partial T_f\left(r,z,t\right)}{\partial r}\right|}_{r\to \infty }=0$$

(15)

At the junction of the formation and the wellhole, the heat that the formation flows into the borehole through the borehole wall in the form of heat conduction is equal to the heat that the borehole wall exchanges with the annular drilling fluid in the radial direction through heat convection, which can be expressed as

$${\lambda }_f{\left|\frac{\partial T_f\left(r,z,t\right)}{\partial r}\right|}_{r=r_w}=h_w\left[T_f\left(r_w,z,t\right)-T_a\left(z,t\right)\right]$$

(16)

2.5. Model Discretization and Solution

The mathematical model of the transient temperature field in the wellbore is too complex. Despite a series of simplifications made during the modeling process and the determination of initial conditions and boundary conditions, the analytical solution of the control equation is still unavailable at present. Therefore, the finite difference method is adopted in this paper to solve the problem.

The solution domain of the control equations includes the wellbore and the formation. The entire two-dimensional surface area is discretized in the axial and radial directions (as shown in Figure 3), and difference equations are established at the nodes of each grid. For programming convenience, T_p, T_d, and T_a are denoted as T₁, T₂, and T₃, respectively. Symbols n, i, and j represent the time number, radial grid number, and axial grid number, respectively.

Therefore, the differential equations of the wellbore transient temperature field can be expressed as Equations (17) to (20).

① Drilling fluid in drill string:

$$Q_{\mathrm{p}}-{\rho }_{\mathrm{l}}q{c}_{\mathrm{l}}\frac{T_{1,\mathrm{j}}^{n+1}-T_{1,\mathrm{j}-1}^{n+1}}{\Delta z}-2\mathsf{\pi }{r}_{\mathrm{pi}}{h}_{\mathrm{pi}}\left(T_{1,\mathrm{j}}^{n+1}-T_{2,\mathrm{j}}^{n+1}\right)={\rho }_{\mathrm{l}}{c}_{\mathrm{l}}\pi r_{\mathrm{pi}}^2\frac{T_{1,\mathrm{j}}^{n+1}-T_{1,\mathrm{j}}^n}{\Delta t}$$

(17)

② Drill string wall:

$$\begin{array}{l}{\lambda }_{\mathrm{d}}\frac{T_{2,\mathrm{j}+1}^{n+1}-2T_{2,\mathrm{j}}^{n+1}+T_{2,\mathrm{j}-1}^{n+1}}{{\left(\Delta z\right)}^2}+\frac{2r_{\mathrm{po}}{h}_{\mathrm{po}}}{r_{\mathrm{po}}^2-r_{\mathrm{pi}}^2}\left(T_{3,\mathrm{j}}^{n+1}-T_{2,\mathrm{j}}^{n+1}\right)\\ +\frac{2r_{\mathrm{pi}}{h}_{\mathrm{pi}}}{r_{\mathrm{po}}^2-r_{\mathrm{pi}}^2}\left(T_{1,\mathrm{j}}^{n+1}-T_{2,\mathrm{j}}^{n+1}\right)={\rho }_{\mathrm{d}}{c}_{\mathrm{d}}\frac{T_{2,\mathrm{j}}^{n+1}-T_{2,\mathrm{j}}^n}{\Delta t}\end{array}$$

(18)

③ Drilling fluid in annulus:

$$\begin{array}{l}{\rho }_{\mathrm{l}}q{c}_{\mathrm{l}}\frac{T_{3,\mathrm{j}}^{n+1}-T_{3,\mathrm{j}-1}^{n+1}}{\Delta z}+2\mathsf{\pi }{r}_{\mathrm{w}}{U}_{\mathrm{w}}\left(T_{4,\mathrm{j}}^{n+1}-T_{3,\mathrm{j}}^{n+1}\right)\\ +2\pi r_{\mathrm{co}}{h}_{\mathrm{co}}\left(T_{2,\mathrm{j}}^{n+1}-T_{3,\mathrm{j}}^{n+1}\right)+Q_{\mathrm{a}}={\rho }_{\mathrm{l}}{c}_{\mathrm{l}}\pi \left(r_{\mathrm{w}}^2-r_{\mathrm{po}}^2\right)\frac{T_{3,\mathrm{j}}^{n+1}-T_{3,\mathrm{j}}^n}{\Delta t}\end{array}$$

(19)

④ Formation:

$$\begin{array}{l}\frac{T_{\mathrm{i}+1,\mathrm{j}}^{n+1}-2T_{\mathrm{i}-1,\mathrm{j}}^{n+1}+T_{\mathrm{i}-1,\mathrm{j}}^{n+1}}{{\left(\Delta r\right)}^2}+\frac{1}{r_i}\frac{T_{\mathrm{i},\mathrm{j}}^{n+1}-T_{\mathrm{i},\mathrm{j}-1}^{n+1}}{\Delta r}\\ +\frac{T_{\mathrm{i},\mathrm{j}+1}^{n+1}-2T_{\mathrm{i},\mathrm{j}}^{n+1}+T_{\mathrm{i},\mathrm{j}-1}^{n+1}}{{\left(\Delta z\right)}^2}=\frac{{\rho }_{\mathrm{f}}{c}_{\mathrm{f}}}{{\lambda }_{\mathrm{f}}}\frac{T_{\mathrm{i},\mathrm{j}}^{n+1}-T_{\mathrm{i},\mathrm{j}}^n}{\Delta t}\left(\mathrm{i}\ge 4\right)\end{array}$$

(20)

Since Equations (17) to (20) adopt fully implicit difference discretization, T_p, T_d, and T_a cannot be directly solved. Typically, the above difference equations are integrated as follows [33]:

$$W_{\mathrm{i},\mathrm{j}}{T}_{\mathrm{i}-1,\mathrm{j}}^{n+1}+C_{\mathrm{i},\mathrm{j}}{T}_{\mathrm{i},\mathrm{j}}^{n+1}+E_{\mathrm{i},\mathrm{j}}{T}_{\mathrm{i}+1,\mathrm{j}}^{n+1}+N_{\mathrm{i},\mathrm{j}}{T}_{\mathrm{i},\mathrm{j}-1}^{n+1}+S_{\mathrm{i},\mathrm{j}}{T}_{\mathrm{i},\mathrm{j}+1}^{n+1}=B_{\mathrm{i},\mathrm{j}}$$

(21)

The naming rule for the coefficients in Equation (21) is shown in Figure 4. If the number of nodes along the axis is A_max, and the number of nodes in the radial direction is R_max, then all nodes are substituted into Equation (21). Integrating all the nodes, we can obtain the matrix Equation (22). Obviously, this coefficient matrix is a pentadiagonal sparse matrix, which can be solved using the chase method.

$${\left[\begin{array}{ccccccccccc}C& E& & \cdots & & S& & & & & \\ W& C& E& & & & S& & & & \\ & \cdots & \cdots & \cdots & & \cdots & & \cdots & & & \\ & & W& C& E& & & & S& & \\ & & & W& C& E& & & & S& \\ \cdots & & & & \cdots & \cdots & \cdots & & & & \cdots \\ & N& & \cdots & & W& C& E& & & \\ & & N& & \cdots & & W& C& E& & \\ & & & \cdots & & \cdots & & \cdots & \cdots & \cdots & \\ & & & & N& & \cdots & & W& C& E\\ & & & & & N& & \cdots & & W& C\end{array}\right]}_{A_{\max }{R}_{\max }\times A_{\max }{R}_{\max }}\left[\begin{array}{c}{T}_{1,1}^{n+1}\\ T_{2,1}^{n+1}\\ \cdots \\ T_{\mathrm{i}-1,\mathrm{j}}^{n+1}\\ T_{\mathrm{i},\mathrm{j}}^{n+1}\\ \cdots \\ T_{{\mathrm{i}}^{\ast },{\mathrm{j}}^{\ast }}^{n+1}\\ T_{{\mathrm{i}}^{\ast }+1,{\mathrm{j}}^{\ast }}^{n+1}\\ \cdots \\ T_{A_{\max }-1,R_{\max }}^{n+1}\\ T_{A_{\max },R_{\max }}^{n+1}\end{array}\right]=\left[\begin{array}{c}{B}_{1,1}\\ B_{2,1}\\ \cdots \\ B_{\mathrm{i}-1,1}\\ B_{\mathrm{i},\mathrm{j}}\\ \cdots \\ B_{{\mathrm{i}}^{\ast },{\mathrm{j}}^{\ast }}\\ B_{{\mathrm{i}}^{\ast }+1,{\mathrm{j}}^{\ast }}\\ \cdots \\ B_{A_{\max }-1,R_{\max }}\\ B_{A_{\max },R_{\max }}\end{array}\right]$$

(22)

3. Results and Discussion

3.1. Modeling Verification

To validate the model, we will verify its applicability through comparison with actual field measurement data. Three horizontal wells in Chongqing, China, were selected.

Table 1. The temperature parameters of the validation model.

	Geothermal Gradient/(°C/100 m)	Wellhead Temperature/°C
No.1 well	3.0	31
No.2 well	3.0	39
No.3 well	3.0	38

shows the temperature parameters of the validation model. The simulation parameters were consistent with the actual field data, and the simulation results are shown in Figure 5. The root mean square error (RMSE) of the predicted and measured results was calculated. The RMSE between the measured data and simulation results for No.1 well from 4300 m to 6320 m (horizontal section) was 1.05 °C. The RMSE for No.2 well from 5470 m to 6270 m (horizontal section) was 0.54 °C. The RMSE for No.3 well from 4330 m to 6510 m (horizontal section) was 0.99 °C. The results indicate that the temperature difference between the measured data and the predicted results is less than 6 °C, with an error of less than 5%. Therefore, this model demonstrates high prediction accuracy for wellbore temperature during horizontal well drilling.

3.2. Sensitivity Analysis of Key Parameters

This paper uses No.2 well, which has the highest prediction accuracy, as a field case to investigate the influence of BHA, drill pipe size, drilling fluid density, flow rate, inlet temperature of drilling fluid, and drilling fluid circulation time on the temperature distribution in the wellbore annulus. The relevant parameters of No.2 well are shown in

Table 2. Well structure data of No.2 horizontal well.

Number of Spudding	Borehole Size /mm	Well Depth /m	Casing Size /mm	Casing Depth /m
First-spudding	660.4	200	508	198
Second-spudding	406.4	2609	339.7	2607
Third-spudding	311.2	4008	244.47	4006
Four-spudding	215.9	6640	139.7	6638

,

Table 3. Drilling parameters.

Drilling Fluid Density/(g/cm3)	Plastic Viscosity/(mPa⋅s)	Stand Pipe Pressure/MPa	Flow Rate /(L/s)	WOB /KN	RPM /(r/min)
1.9~2.06	45–57	36–39	31–32	80	75–90

,

Table 4. BHA of No.2 horizontal well.

Drilling Tool Name	Outside Diameter /mm	Inner Diameter /mm	Length /m	Unit Weight /(N/m)
PDC bit	215.9	0	0.23	300
Drill collar	171.5	57.2	11.5	1582.7
Heavy-weight drill pipe	127	76.2	10	720.3
PDM	172	57.2	9	1582.7
Heavy-weight drill pipe	127	76.2	27	720.3
drilling jar	172	78	10	1580
Heavy-weight drill pipe	127	76.2	18	720.3
Drill pipe	127	108.61	2600	284.58
Drill pipe	139.7	118.62	3954.27	360.47

and

Table 5. Thermophysical parameters of heat transfer medium.

Medium	Density /(g/cm3)	Specific Heat Capacity /(J/kg·°C)	Thermal Conductivity /(W/(m·°C))
Drilling fluid	2.06	1510	0.75
Tubular column	7.80	400	43.75
Rock	2.64	837	2.25

.

3.2.1. Influence of BHA

Figure 6 shows the temperature change with well depth during drilling in the shale gas horizontal section with a PDM. In order to avoid the impact of single data on the analysis results, this section analyzes the temperature changes with well depth in the horizontal sections of three wells. It can be seen that the No.1 well increased from 104 °C to 123.6 °C in the well section from 4300 m to 6320 m (the length is 2020 m), with an increase of about 0.970 °C/100 m. No.2 well increased from 135.1 °C to 142.6 °C in the well section from 5470 m to 6270 m (the length is 800 m), with an increase of about 0.938 °C/100 m. No.3 well increased from 118.97 °C to 140.78 °C in the well section from 6510 m to 4330 m (the length is 2180 m), with an increase of about 1.000 °C/100 m. The results show that the temperature increase during drilling in the shale gas horizontal section with a PDM is about 1 °C/100 m.

Taking No.2 well as an example, the temperature changes in annular at 0 r/min, 50 r/min, and 100 r/min were analyzed, as shown in Figure 7. It can be seen that when the RPM of PDM is 0 r/min, the bottom hole temperature is 139.60 °C. When the RPM of PDM is 50 r/min, the bottom hole temperature is 140.26 °C. When the RPM of PDM is 100 r/min, the bottom hole temperature is 140.93 °C. The results indicate that as the PDM rotation speed decreases, the bottom hole temperature decreases.

3.2.2. Influence of Drill Pipe Size

Figure 8 shows the temperature changes when using Φ139.7 mm drill pipe and Φ139.7 mm + Φ127 mm drill pipe. It can be seen that the Φ139.7 mm drill pipe has a bottom hole temperature 2.7 °C lower than that of the Φ139.7 mm + Φ127 mm drill pipe. The turning point on the curve corresponds to the well depth where the wellbore curvature in the building section is the largest, indicating that the drilling pipe size at this well depth has a smaller impact on the wellbore temperature. However, in the horizontal section, as the well depth increases, the temperature difference increases, indicating that the influence of the drilling pipe size on the wellbore temperature is more significant in the horizontal section compared to the vertical and oblique sections. Figure 9 shows the frictional heat generation of the drill pipe. It can be observed that the Φ139.7 mm drill pipe generates more frictional heat both inside the pipe and in the annulus. Figure 10 illustrates the convective heat transfer coefficient of the drill pipe. It is evident that the Φ139.7 mm + Φ127 mm drill pipe has a higher convective heat transfer coefficient, indicating a greater amount of convective heat exchange and higher generated temperature. The results demonstrate that despite the higher frictional heat generation of the Φ139.7 mm drill pipe, its convective heat exchange is less, resulting in lower heat production. Therefore, using the Φ139.7 mm drill pipe is more beneficial for reducing bottom hole temperature compared to the Φ139.7 mm + Φ127 mm drill pipe.

3.2.3. Influence of Drilling Fluid Density

Figure 11 shows the curve of annular temperature variation with well depth under different drilling fluid densities. It can be seen that in the horizontal section, the annular temperature for a drilling fluid density of 1.85 g/cm³ is lower, indicating that reducing the drilling fluid density can lower the annular temperature. When the drilling fluid density is reduced by 0.17 g/cm³, the bottom hole temperature drops by 6.3 °C. The results show that reducing the drilling fluid density decreases the solid phase content in the drilling fluid. It reduces the circulation pressure loss and the amount of heat generated. In addition, it can effectively increase the fluid’s specific heat capacity and reduce thermal conductivity, which can effectively reduce temperature to a certain extent. However, reducing the density of drilling fluid may weaken its ability to carry drilling chips, thus affecting the borehole cleaning effect. The deterioration of drilling fluid performance may increase the wear of drilling equipment. Therefore, it is necessary to consider various factors to reasonably control the drilling fluid density.

3.2.4. Influence of Drilling Fluid Flow Rate

With a drilling fluid density of 2.02 g/cm³ and an inlet temperature of the drilling fluid of 58 °C, the variation curve of annular temperature with a well depth under different drilling fluid flow rates is shown in Figure 12. It can be seen that when the flow rate is 20.5 L/s, 23.5 L/s, 27.5 L/s, 31.5 L/s, and 34.5 L/s, the bottom hole temperatures are 135.2 °C, 136.2 °C, 138.2 °C, 140.9 °C, and 143.9 °C, respectively. As the drilling fluid flow rate increases, the bottom hole temperature rises. When the drilling fluid flow rate is reduced by 14 L/s, the bottom hole temperature drops by 8.7 °C. The results show that reducing the drilling fluid flow rate can lower the RPM of PDM. As mentioned in Section 3.2.1, reducing the RPM of PDM is beneficial for decreasing the bottom hole temperature. In addition, the circulation pressure loss increases with increasing drilling fluid flow rate. Therefore, reducing the drilling fluid flow rate can decrease the circulation pressure loss, resulting in less heat generation and, consequently, decreasing the bottom hole temperature.

3.2.5. Influence of Inlet Temperature of Drilling Fluid

With a drilling fluid density of 2.02 g/cm³ and a drilling fluid flow rate of 31.5 L/s, the variation curve of annular temperature with well depth under different inlet temperatures is shown in Figure 13. It can be seen that when the inlet temperatures are 28 °C, 38 °C, 48 °C, and 58 °C, the bottom hole temperatures are 139.1 °C, 140.9 °C, 142.3 °C, and 143.4 °C, respectively. As the inlet temperature decreases, the bottom hole temperature decreases. When the drilling fluid inlet temperature is reduced by 30 °C, the bottom hole temperature drops by 4.3 °C. It is worth noting that when the well is deep, and the formation temperature is high, despite decreasing the inlet temperature, the cold slurry can easily be heated during its flow in the pipe, resulting in a smaller impact of ground cooling on the bottom hole temperature.

3.2.6. Influence of Drilling Fluid Circulation Time

Figure 14 shows the curve of the wellbore temperature varying with time during the drilling. It can be seen that as time increases, the wellbore temperature shows a slow upward trend. When the drilling fluid is circulated without drilling, the relationship curve between drilling fluid circulation time and annular temperature in horizontal wells is shown in Figure 15. It can be seen that as the circulation time increases, the annular temperature decreases, but the degree of decrease diminishes. According to Figure 15, at well depths of 5751 m and 6251 m, the circulation times required for the bottom hole temperature to drop by 3 °C are 25 min and 15 min, respectively. Theoretically, the higher the temperature, the shorter the circulation time required to achieve the same temperature drop, and the flow rate should not be too large during circulation cooling. Figure 16 shows the relationship between different drilling fluid circulation flow rates and circulation times at a well depth of 5751 m. The larger the circulation flow rate, the lower the cooling efficiency for the same circulation time.

3.2.7. Influence of Heat Sources

Figure 17 shows the temperature distribution of wellbore annulus with well depth in the horizontal well when considering different heat source terms. Obviously, when considering bit-breaking rock, bit hydraulic horsepower, and drill pipe rotation, the annulus temperature at the same well depth is higher than when there are no heat source terms. Moreover, as the well depth increases, the difference is more pronounced. Figure 18 shows the heat generated by different heat sources. The temperature produced by the bit-breaking rock is 2.4 °C. Bit hydraulic horsepower generates a temperature of 1.82 °C, top drive rotation generates a maximum temperature of 2.95 °C, and top drive rotation (80 r/min) and PDM rotation (100 r/min) generate a temperature of 4.28 °C. Therefore, near the critical temperature point of the rotary steering drilling tool, measures such as reducing the top drive rotation speed and removing the PDM can reduce the downhole annulus temperature.

4. Conclusions

This paper establishes a wellbore temperature field model during the drilling process of deep shale gas horizontal wells. The finite difference method is used to solve the model, and field data are utilized for verification. Based on this, the study explores the influence of BHA, drill pipe size, drilling fluid density, flow rate, inlet temperature of drilling fluid, and drilling fluid circulation time on the temperature distribution of the wellbore annulus. Key findings are as follows:

(1)

Based on the established wellbore temperature field model, the RMSE of the prediction results for three wells on site are 1.05 °C, 0.54 °C, and 0.99 °C. The temperature difference between the measured data and the prediction results is less than 6 °C, and the relative error is less than 5%, indicating that the model has high accuracy.

(2)

When drilling in the shale gas horizontal section with a PDM, the annular temperature increases by approximately 1 °C/100 m. As the RPM of PDM decreases, the bottom hole temperature decreases. In addition, using Φ139.7 mm drill pipe is favorable for cooling than Φ139.7 mm + Φ127 mm drill pipe.

(3)

In the horizontal well section, when the drilling fluid density is reduced by 0.17 g/cm³, the bottom hole temperature decreases by 6.3 °C. When the drilling fluid flow rate is reduced by 14 L/s, the bottom hole temperature drops by 8.7 °C. Additionally, decreasing the drilling fluid inlet temperature by 30 °C results in a 4.3 °C reduction in the bottom hole temperature. These results show that reducing the density, flow rate, and inlet temperature of drilling fluid is beneficial for decreasing the bottom hole temperature in horizontal well sections.

(4)

When the drilling fluid is circulating without drilling, at a depth of 5751 m and 6251 m, the bottom hole temperature decreases by 3 °C, and the corresponding circulation times are 25 min and 15 min, respectively. The higher the temperature, the shorter the circulation time required to reduce the same degree of temperature, and the circulation flow rate should not be too large.

(5)

Bit-breaking rock, bit hydraulic horsepower, and drill pipe rotation affect the wellbore annular temperature. Therefore, it is recommended to reduce the downhole circulation temperature by measures such as decreasing the top drive rotation speed and removing the PDM near the critical temperature point of the rotary steering drilling tool.

This paper explores the influence of seven factors on wellbore annular temperature. However, due to certain assumptions made during the model-building process, the model has some limitations. The future study should focus on refining the model by determining the relationship between some parameters and temperature based on experimental results, in order to improve the prediction accuracy of the model.