Numerical Modeling and Simulation of Thermal Effect-Driven Bottom Hole Pressure Variation and Control Technology During Tripping-Out in HTHP Ultra-Deep Wells

Abstract

Controlling bottom hole pressure (BHP) during tripping-out is a key challenge in ultra-deep well drilling. Under high-temperature and high-pressure (HTHP) conditions, ultra-deep wells feature long tripping-out cycles, where thermal effects are prone to causing BHP reduction and increasing kick risk. However, existing pressure control technologies struggle to adapt to the requirements of narrow safe density windows in deep formations. This study establishes a transient tripping-out temperature field model, taking the PS6 ultra-deep vertical well as a case study to calculate the variations in temperature, equivalent static density (ESD), and BHP during tripping-out at 2910 m and 9026 m. A weighted drilling fluid supplementation method is presented, with supplementary parameters designed and its feasibility verified. The results indicate that during the entire tripping-out process, the bottom hole temperature at 2910 m increases by 17.5 °C and BHP rises by 0.016 MPa; at 9026 m, the temperature increases by 72.6 °C and BHP decreases by 2.410 MPa. Compared with the traditional “heavy mud cap” technology, the presented method can control BHP within a smaller fluctuation range (within 0.339 MPa) during tripping-out, better adapting to the safe tripping requirements of narrow safe density windows in deep formations and effectively mitigating kick risk.

1. Introduction

Deep and ultra-deep oil and gas resources are important replacement resources for China’s oil and gas reserve growth and production increase. Achieving high-quality and efficient drilling and completion of ultra-deep wells is of great significance for safeguarding national energy security [1,2]. Several challenges are associated with drilling operations; the impact of these challenges can be mitigated by proposing proactive solutions [3,4,5]. One of the problems is the variations in bottom hole pressure while tripping, especially in deep water wells. Compared with the widely studied BHP control during circulating drilling processes, during tripping-out, the drilling fluid ceases to circulate, and the downhole temperature gradually approaches the formation temperature as the operation proceeds [6]; detailed comparisons are presented in

Table 1. The differences between the studied working conditions and conventional drilling operations.

Drilling	Tripping-Out
Drilling fluid circulation carries away heat	Drilling fluid is stationary, causing wellbore temperature rise
Pump activation increases BHP	Swabbing pressure decreases BHP
Annular liquid level remains constant	Drill pipe tripping-out causes annular liquid level drop
BHP is obtainable	BHP acquisition is difficult

. For shallow wells, due to the low formation temperature and short tripping-out cycle, the impact of temperature changes on BHP fluctuation is limited. However, for ultra-deep wells, the deep formation temperature is high and the tripping-out cycle is long. During tripping-out, the downhole temperature undergoes significant changes, which in turn leads to remarkable BHP variations [7], as shown in Figure 1. Nevertheless, real-time monitoring of BHP during tripping-out is not feasible, posing potential risks to safe tripping-out. Therefore, establishing a transient tripping-out temperature field model to accurately predict the BHP variation law and proposing targeted pressure regulation methods are key technical requirements for realizing safe tripping-out of ultra-deep wells [8].

Ramey [9] first presented a wellbore fluid heat transfer model, laying the fundamental framework for downhole temperature field calculation. Keller et al. [10] developed a 2D transient heat transfer model describing the wellbore interior and its surrounding areas, incorporating the circulation process of drilling fluid from the drill pipe to the annulus, which enhances the authenticity of heat transfer calculation. Corre et al. [11] incorporated the thermal effects of bit rock-breaking and casing running into their wellbore temperature field model. Since the 1990s, research focus shifted to multi-operation adaptation and complex environment applications. Marshall and Lie [12] developed a transient temperature field model applicable to multiple scenarios, which breaks the application limitations of single-condition models and enhances technical versatility. Kabir et al. [13] developed an analytical model for circulating fluid temperature during drilling and workover, enabling rapid evaluation of wellbore temperature distribution under different operations. In recent years, with improved computational capabilities, temperature field models have achieved higher accuracy with more comprehensive considerations. Al Saedi et al. [14] presented analytical solutions for wellbore fluid temperature during drilling and cementing, enhancing calculation efficiency. Yang et al. [15,16] constructed a transient heat transfer model for deep wells, revealed the influence mechanism of drill string movement on annulus temperature distribution, and filled the research gap regarding the impact of drill string dynamic characteristics on the temperature field. Abdelhafiz et al. [17] combined numerical simulation and analytical methods to establish a transient temperature model for mud circulation, revealing temperature field distribution laws during circulation. Mao et al. [18] further integrated bit heat generation and dynamic mud thermophysical parameters, developing a more realistic wellbore temperature prediction model for HTHP environments. Dang et al. [19] established a temperature prediction model for complex geological conditions, revealing the influence of tripping speed and formation thermophysical properties on temperature fields. Chen et al. [20] presented a comprehensive prediction model for wellbore temperature changes by coupling multiphase flow and heat transfer in deepwater HTHP drilling, providing important references for temperature field analysis in ultra-deep well high-temperature environments. During the tripping-out process, the upward movement of the drill pipe causes a drop in the annulus liquid level, which leads to complex boundary conditions. To date, few studies have focused on temperature field distribution during tripping-out, especially considering the complex boundary conditions caused by annular liquid level drop. The construction of this model helps to better understand the changes in the wellbore temperature field during the tripping-out process.

Research on drilling fluid density under temperature–pressure effects began in 1982, when Hoberock et al. [21] studied the impact of pressure and temperature on bottom hole mud density, revealing that drilling fluid density models must integrate multiple factors and providing new insights into the relationship between drilling fluid and downhole environments. Peters et al. [22] constructed a density prediction model for HTHP oil-based drilling fluids, continuously improving model accuracy with experimental data to guide field density calculations. Zamora et al. [23] summarized ten technical challenges of deepwater drilling mud, defining key density-related challenges and research directions in deepwater environments. Demirdal et al. [24] studied downhole drilling fluid rheological changes, providing theoretical support for precise density model construction. The American Petroleum Institute (API) standardized research on drilling fluid rheology and hydraulics, covering density-related contents to promote knowledge sharing [25]. With increasing AI applications in the petroleum industry, Wang et al. [26] presented a support vector machine (SVM) method to predict drilling fluid density under HTHP conditions, applicable to both oil-based and water-based fluids and offering new approaches for complex operations. Tatar et al. [27] developed least squares support vector machine (LS-SVM) and multi-layer perceptron neural network (MLP-NN) models to determine brine density, focusing on estimating densities of brines with diverse salt compositions and concentrations to fill research gaps in special fluid densities. Agwu et al. [28] developed an artificial neural network model to predict downhole density of oil-based mud under HTHP conditions, eliminating the need for surface measurement equipment and enabling more accurate representation of downhole mud density at given pressure–temperature conditions. Alizadeh et al. [29] evaluated drilling fluid density under HTHP, introducing modeling and experimental techniques using intelligent models for prediction. Yan et al. [30] developed an anti-hydrogen sulfide fuzzy-ball kill fluid with excellent plugging performance, high-temperature resistance, and H₂S adsorption capacity, which can provide reference for the optimization of wellbore fluid performance in complex downhole environments. Liu et al. [31] investigated temperature–pressure coupling effects on drilling fluid density and rheology in deep HTHP wells, establishing an accurate wellbore temperature model to solve bottom hole pressure calculation deviations caused by ignoring thermo-pressure effects and providing theoretical support for safe drilling. Dabiri et al. [32] constructed a HTHP drilling fluid density prediction system using multi-model fusion, optimized algorithms, and parameter sensitivity quantification, providing calculation references for wellbore pressure control and drilling safety under extreme conditions [33]. These research foundations sufficiently support the density calculation requirements of this paper.

This study focuses on the BHP control challenge during tripping-out in HTHP ultra-deep wells. The traditional “heavy mud cap” technology is designed only for swab pressure and does not consider dynamic BHP changes caused by thermal effects. However, ultra-deep wells feature long tripping-out cycles and large formation temperature differences, leading to significant BHP losses due to thermal effects. Directly using the “heavy mud cap” to compensate for these losses will result in a sudden BHP rise, which is prone to exceeding the formation fracture pressure and cannot adapt to the narrow safe density window of deep formations. To address this, this study establishes a transient tripping-out temperature field model to quantify the influence law of thermal effects on BHP, proposes a weighted drilling fluid supplementation method with clarified key parameters, realizes precise BHP regulation during tripping-out, and provides technical support for the safe tripping-out of ultra-deep wells.

2. Study Objective

The main objectives of this study are as follows:

(1)

To establish a transient tripping-out temperature field model and quantify the influence law of thermal effects on bottom hole pressure (BHP) in high-temperature and high-pressure (HTHP) ultra-deep wells;

(2)

To analyze the variations in temperature, equivalent static density (ESD), and BHP during tripping-out at different depths (2910 m and 9026 m) by taking the PS6 ultra-deep vertical well as a case study;

(3)

To propose a weighted drilling fluid supplementation method for BHP control, and design the key parameters (supplementation timing, density, and volume) as well as verify its feasibility.

3. Methodology

This study adopts a combination of theoretical modeling, numerical calculation, and case verification, with the specific process as follows:

(1)

Data collection: The wellbore structure parameters of Well PS6 (an ultra-deep well in the Sichuan Basin) are obtained [34], while the material thermophysical properties and drilling operation parameters are selected from the on-site design schemes of similar wells;

(2)

Model establishment: A transient tripping-out temperature field model is constructed considering the thermal exchange between wellbore and formation, and coupled with the drilling fluid density calculation model based on Hoberock [21] and API [25] standards;

(3)

Numerical calculation: The wellbore and formation are discretized by grids, and the control equations are solved using the second-order upwind scheme and fully implicit time scheme; a calculation program is compiled to simulate the variations in temperature, ESD, and BHP during tripping-out;

(4)

Technology development and verification: Based on the simulation results, a weighted drilling fluid supplementation method is proposed, the supplementary parameters are designed, and the feasibility and superiority of this method are verified by calculating the theoretical changes in BHP.

4. Materials and Methods

4.1. Data Collection

The case is derived from Xu et al. [34]. Well PS6 in the Sichuan Basin, cited in this article, is an ultra-deep vertical well with a total depth (TD) of 9026 m. The wellbore structure of Well PS6 is comprehensively depicted in Figure 2, where all the required structural parameters to support subsequent calculations are provided.

To meet the requirements for temperature field calculation,

Table 2. Thermal physical parameters of materials.

Material	Density $(\mathbf{g}/\mathbf{c}{\mathbf{m}}^3$)	Specific Heat Capacity $(\mathbf{J}/\mathbf{k}\mathbf{g}·\mathbf{K}$)	Thermal Conductivity $(\mathbf{W}/(\mathbf{m}·\mathbf{K})$)
Drilling fluid 1	1.11	1600	1.75
Drilling fluid 2	2.18	1897	1.52
Drill pipe	7.84	400	43.75
Casing	7.84	400	43.75
Cement	2.09	2000	1
Formation rock	2.5	1414	2.41

provides the density and thermophysical parameters of the drilling fluid, drill tools, casing, and formation.

To meet the requirements of the overall calculation,

Table 3. Other parameters.

Parameter	Value	Units
Weight on the bit	75	$\mathrm{k}\mathrm{N}$
Rotary speed	100	$\mathrm{r}\mathrm{p}\mathrm{m}$
Rate of penetration	5	$\mathrm{m}/\mathrm{h}$
Mud flow rate	20	$\mathrm{l}/\mathrm{s}$
Mud inlet temperature	10	$\mathrm{℃}$
Geothermal gradient	2.7	$\mathrm{℃}/100 \mathrm{m}$
Surface temperature	10	$\mathrm{℃}$
Formation friction coefficient	0.4	Dimensionless

supplements all parameter values applied in the calculation process.

4.2. Analytical Work

During the tripping-out process, it is necessary to determine the initial tripping-out temperature field and the tripping-out process temperature field. The initial tripping-out temperature field refers to the temperature distribution when the temperature of the conventional circulating temperature field model is approximately stable. During the tripping-out process, however, the drilling fluid in the wellbore no longer circulates; the withdrawal of the drill pipe causes a drop in the annular liquid level, and drilling fluid is supplemented into the annulus at this point to fill the gap. In the space below the drill bit, the fluids in the annulus and the drill string mix, which leads to changes in the axial temperature distribution of the annulus. To facilitate calculations, the actual working conditions are abstracted into a mathematical model, and the following assumptions are presented:

(1)

Drilling fluid exhibits one-dimensional axial flow downhole.

(2)

The formation has a disturbance radius, beyond which the formation maintains its original undisturbed temperature.

(3)

The formation is isotropic, and the geothermal gradient remains constant.

(4)

Drilling fluid at a constant temperature is supplemented into the annulus from the wellhead during tripping to keep the annulus fluid level unchanged.

To analyze the downhole temperature model, a micro-element control volume P is selected. The axial upward direction of the control volume is defined as the north direction, and the radial axis direction is defined as the west direction, which are denoted by n, s, e, and w, respectively, as shown in Figure 3. For the convenience of formulating equations, the inner diameter of the wellbore drill pipe, outer diameter of the drill pipe, inner diameter of the casing, and outer diameter of the casing are denoted as r₁, r₂, r₃, and r₄, respectively.

4.2.1. Calculation of the Circulating Temperature Field

In accordance with the law of conservation of energy, the location of the control volume under drilling conditions and its energy analysis are shown in Figure 4. Among them, $\mathrm{Q}$ refers to the heat transfer rate with the unit of Watt ($\mathrm{W}$), and $\mathrm{S}$ denotes the friction-generated heat rate with the same unit of Watt ($\mathrm{W}$). For the detailed calculation method of the circulating temperature field, reference is made to Mao et al. [18].

4.2.2. Calculation of the Tripping-Out Temperature Field

During the tripping-out process, the annular liquid level drops as the drill pipe is pulled out; drilling fluid at a constant temperature is supplemented into the annulus from the wellhead to maintain the annulus volume balance. Based on the fluid flow and heat transfer characteristics, the wellbore is divided into two control regions, namely the fluid communication region below the drill bit and the fluid descending region above the drill bit, as shown in Figure 5.

As the drill bit rises, the original drilling fluid in the drill pipe and the annular drilling fluid in region (a) mix directly. In this region, there is no drill string, and the main heat transfer mechanisms are fluid mixing heat exchange and formation heat conduction. Equation (1) describes the temperature change in the fluid mixing region below the drill bit. The energy change in the control volume = axial conductive heat transfer energy of the fluid (inward/outward) + radial conductive heat transfer energy with the formation + mixed heat exchange energy between the descending annulus drilling fluid and the original fluid.

$$\begin{array}{c}\mathrm{c}\mathsf{\rho }\mathsf{\pi }{\mathrm{r}}_3^2\mathsf{\Delta }\mathrm{z}\left({\mathrm{T}}_{\mathrm{P}}^1-{\mathrm{T}}_{\mathrm{P}}^0\right)=\mathsf{\pi }{\mathrm{r}}_3^2\mathsf{\lambda }\mathsf{\Delta }\mathrm{t}{\left(\overline{{\mathrm{T}}_{\mathrm{N}}-{\mathrm{T}}_{\mathrm{S}}}\right)}^{0-1}\\ +2\mathsf{\pi }{\mathrm{r}}_3{\mathrm{k}}_{\mathrm{e}}\mathsf{\Delta }\mathrm{z}\mathsf{\Delta }\mathrm{t}{\left(\overline{{\mathrm{T}}_{\mathrm{E}}-{\mathrm{T}}_{\mathrm{P}}}\right)}^{0-1}+\mathrm{c}\mathsf{\rho }{\mathrm{V}}_{\mathrm{m}\mathrm{i}\mathrm{x}}{({\mathrm{T}}_{\mathrm{d}-\mathrm{l}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r}}-{\mathrm{T}}_{\mathrm{p}}^0)}^{0-1}\end{array}$$

(1)

where $\mathrm{c}$ is specific heat capacity, $\mathrm{J}/(\mathrm{k}\mathrm{g}·\mathrm{K})$; $\mathsf{\rho }$ is drilling fluid density, $\mathrm{k}\mathrm{g}/{\mathrm{m}}^3$; $\mathsf{\lambda }$ is thermal conductivity, $\mathrm{W}/(\mathrm{m}·\mathrm{K})$; ${\mathrm{k}}_{\mathrm{e}}$ is equivalent thermal conductivity, $\mathrm{W}/(\mathrm{m}·\mathrm{K})$.

In region (b), as the drill pipe moves upward, the annulus volume increases, causing the annular drilling fluid to descend. Equation (2) describes the energy equation of the drilling fluid in region (b). The energy change in the control volume = energy carried away by axial fluid flow—axially conducted heat transfer energy of the fluid (inward) + radial conductive heat transfer energy with the formation—radial convective heat transfer energy with the drill string.

$$\begin{array}{c}\mathrm{c}\mathsf{\rho }\mathsf{\pi }({\mathrm{r}}_3^2-{\mathrm{r}}_2^2)\mathsf{\Delta }\mathrm{z}\left({\mathrm{T}}_{\mathrm{P}}^1-{\mathrm{T}}_{\mathrm{P}}^0\right)=-\mathrm{c}\mathsf{\rho }\mathsf{\pi }\left({\mathrm{r}}_3^2-{\mathrm{r}}_2^2\right){\mathrm{v}}_{\mathrm{a}}\mathsf{\Delta }\mathrm{t}{\left(\overline{{\mathrm{T}}_{\mathrm{N}}-{\mathrm{T}}_{\mathrm{S}}}\right)}^{0-1}\\ +\mathrm{c}\mathsf{\rho }\mathsf{\pi }\left({\mathrm{r}}_3^2-{\mathrm{r}}_2^2\right){\mathrm{k}}_{\mathrm{m}}\mathsf{\Delta }\mathrm{t}{\left(\overline{{\mathrm{T}}_{\mathrm{N}}-{\mathrm{T}}_{\mathrm{S}}}\right)}^{0-1}+2\mathsf{\pi }{\mathrm{r}}_3{\mathrm{k}}_{\mathrm{e}}\mathsf{\Delta }\mathrm{z}\mathsf{\Delta }\mathsf{t}{\left(\overline{{\mathrm{T}}_{\mathrm{E}}-{\mathrm{T}}_{\mathrm{P}}}\right)}^{0-1}\\ -2\mathsf{\pi }{\mathrm{r}}_2{\mathrm{k}}_{\mathrm{c}\mathrm{a}}\mathsf{\Delta }\mathrm{z}\mathsf{\Delta }\mathrm{t}{\left(\overline{{\mathrm{T}}_{\mathrm{a}}-{\mathrm{T}}_{\mathrm{P}}}\right)}^{0-1}\end{array}$$

(2)

where $\mathrm{v}$ is drilling fluid velocity, $\mathrm{m}/\mathrm{s}$.

During the continuous upward movement of the drill pipe, heat exchange occurs between the drill pipe, the annular drilling fluid, and the fluid inside the drill pipe; meanwhile, the drill pipe itself also conducts heat axially. The energy equation for this process is shown in Equation (3). The energy change in the control volume = axial conductive heat transfer energy of the drill string (inward/outward) + radial convective heat transfer energy with the annulus drilling fluid—radial convective heat transfer energy with the fluid inside the drill pipe—energy carried away by axial movement of the drill string.

$$\begin{array}{c}{\mathrm{c}}_{\mathrm{p}}{\mathsf{\rho }}_{\mathrm{p}}\mathsf{\pi }{(\mathrm{r}}_2^2-{\mathrm{r}}_1^2)\mathsf{\Delta }\mathrm{z}\left({\mathrm{T}}_{\mathrm{P}}^1-{\mathrm{T}}_{\mathrm{P}}^0\right)={\mathrm{c}}_{\mathrm{p}}{\mathsf{\rho }}_{\mathrm{p}}\mathsf{\pi }\left({\mathrm{r}}_2^2-{\mathrm{r}}_1^2\right){\mathrm{k}}_{\mathrm{p}}\mathsf{\Delta }\mathrm{t}{\left(\overline{{\mathrm{T}}_{\mathrm{N}}-{\mathrm{T}}_{\mathrm{S}}}\right)}^{0-1}\\ +2\mathsf{\pi }{\mathrm{r}}_2{\mathrm{k}}_{\mathrm{c}\mathrm{a}}\mathsf{\Delta }\mathrm{z}\mathsf{\Delta }\mathrm{t}{\left(\overline{{\mathrm{T}}_{\mathrm{a}}-{\mathrm{T}}_{\mathrm{P}}}\right)}^{0-1}-2\mathsf{\pi }{\mathrm{r}}_1{\mathrm{k}}_{\mathrm{c}\mathrm{i}}\mathsf{\Delta }\mathrm{z}\mathsf{\Delta }\mathrm{t}{\left(\overline{{\mathrm{T}}_{\mathrm{p}}-{\mathrm{T}}_{\mathrm{d}-\mathrm{i}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{r}}}\right)}^{0-1}\\ -{\mathrm{c}}_{\mathrm{p}}{\mathsf{\rho }}_{\mathrm{p}}\mathsf{\pi }{(\mathrm{r}}_2^2-{\mathrm{r}}_1^2){\mathrm{v}}_{\mathrm{d}}\mathsf{\Delta }\mathrm{t}{\left(\overline{{\mathrm{T}}_{\mathrm{N}}-{\mathrm{T}}_{\mathrm{S}}}\right)}^{0-1}\end{array}$$

(3)

There is no axial flow of the fluid inside the drill pipe, and its energy change mainly stems from heat exchange with the drill pipe, which is given by Equation (4). The energy change in the control volume = axial conductive heat transfer energy of the fluid (inward/outward) + radial convective heat transfer energy with the drill string.

$$\begin{array}{c}\mathrm{c}\mathsf{\rho }\mathsf{\pi }{\mathrm{r}}_1^2\mathsf{\Delta }\mathrm{z}\left({\mathrm{T}}_{\mathrm{P}}^1-{\mathrm{T}}_{\mathrm{P}}^0\right)=\mathrm{c}\mathsf{\rho }\mathsf{\pi }{\mathrm{r}}_1^2{\mathrm{k}}_{\mathrm{d}-\mathrm{i}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{r}}\mathsf{\Delta }\mathrm{t}{\left(\overline{{\mathrm{T}}_{\mathrm{N}}-{\mathrm{T}}_{\mathrm{S}}}\right)}^{0-1}\\ +2\mathsf{\pi }{\mathrm{r}}_1{\mathrm{k}}_{\mathrm{c}\mathrm{i}}\mathsf{\Delta }\mathrm{z}\mathsf{\Delta }\mathrm{t}{\left(\overline{{\mathrm{T}}_{\mathrm{E}}-{\mathrm{T}}_{\mathrm{P}}}\right)}^{0-1}\end{array}$$

(4)

4.2.3. Discretization Solution

To ensure the consistency between the mathematical model and actual field operating conditions, the wellbore and formation are discretized into multiple grids (as shown in Figure 6), with the wellbore and formation divided into n grids axially and m layers radially. Grid sensitivity analysis was conducted in the program by batch modifying the grid step size (axial: 10 m, 5 m, 1 m). The results show that under different grids, the calculation deviation of bottom hole temperature (BHT) during tripping-out at 2910 m is ≤0.13%, and that at 9026 m is ≤0.07%. Meanwhile, time-step sensitivity analysis was performed by batch adjusting the time step (10 s, 5 s, 1 s). The results indicate that under different time steps, the calculation deviation of BHT during tripping-out at 2910 m is ≤0.17%, and that at 9026 m is ≤0.11%. The above analyses demonstrate that neither the grid nor the time step has a significant impact on the calculation results, ensuring the reliability of the solution. Finally, the time step is set to 10 s, the axial grid step size of the wellbore is set to 5 m, and the formation is radially extended outward at equal intervals of 0.5 m from the wellbore wall to the formation disturbance boundary (10 m), which guarantees the calculation accuracy for the long section of ultra-deep wells.

Considering that the convective term in the governing equation is much larger than the diffusive term, the second-order upwind scheme is adopted for the convective term, and the fully implicit scheme is used for the time term. By extending the energy equation of a single control volume to the entire space, the discrete equations of the wellbore and formation temperature model can be obtained. Aiming at the dynamic evolution of the wellbore temperature field during the tripping-out process, a multi-control-volume coupling model considering drill pipe upward movement is established.

Equation (5) is used to discretize the control volume of the drilling fluid inside the drill string.

$$\mathrm{c}\mathsf{\rho }\mathsf{\pi }{\mathrm{r}}_1^2\mathsf{\Delta }\mathrm{z}\left({\mathrm{T}}_{\mathrm{i},1}^1-{\mathrm{T}}_{\mathrm{i},1}^0\right)=2\mathsf{\pi }{\mathrm{r}}_1{\mathrm{h}}_{\mathrm{c}1}\mathsf{\Delta }\mathrm{z}\mathsf{\Delta }\mathrm{t}\left({\mathrm{T}}_{\mathrm{i},2}^1-{\mathrm{T}}_{\mathrm{i},1}^1\right)$$

(5)

where $\mathrm{h}$ is convective heat transfer coefficient, $\mathrm{W}/({\mathrm{m}}^2·\mathrm{K})$.

Equation (6) is used to discretize the control volume of the drill string.

$$\begin{gathered} \mathrm{c}\mathsf{\rho }\mathsf{\pi }\left({\mathrm{r}}_2^2-{\mathrm{r}}_1^2\right)\mathsf{\Delta }\mathrm{z}\left({\mathrm{T}}_{\mathrm{i},2}^1-{\mathrm{T}}_{\mathrm{i},2}^0\right)=\mathsf{\pi }\left({\mathrm{r}}_2^2-{\mathrm{r}}_1^2\right){\mathrm{k}}_{\mathrm{p}}\mathsf{\Delta }\mathrm{t}\left({\mathrm{T}}_{\mathrm{i}+\mathrm{1,2}}^1+{\mathrm{T}}_{\mathrm{t}-\mathrm{1,2}}^1-2{\mathrm{T}}_{\mathrm{i},2}^1\right)+ \\ 2\mathsf{\pi }\mathsf{\Delta }\mathrm{z}\mathsf{\Delta }\mathrm{t}\left[{\mathrm{r}}_2{\mathrm{h}}_{\mathrm{c}2}\left({\mathrm{T}}_{\mathrm{i},3}^1-{\mathrm{T}}_{\mathrm{i},2}^1\right)-{\mathrm{r}}_1{\mathrm{h}}_{\mathrm{c}1}\left({\mathrm{T}}_{\mathrm{i},2}^1-{\mathrm{T}}_{\mathrm{i},1}^1\right)\right]+\mathrm{c}\mathsf{\rho }\mathsf{\pi }\left({\mathrm{r}}_2^2-{\mathrm{r}}_1^2\right){\mathrm{v}}_{\mathrm{d}}\mathsf{\Delta }\mathrm{t}{\displaystyle \frac{{\mathrm{T}}_{\mathrm{i}+\mathrm{1,2}}^1-{\mathrm{T}}_{\mathrm{t}-\mathrm{1,2}}^1}{2\mathsf{\Delta }\mathrm{z}}} \end{gathered}$$

(6)

Equation (7) is used to discretize the control volume of the annular drilling fluid.

$$\begin{gathered} \mathrm{c}\mathsf{\rho }\mathsf{\pi }\left({\mathrm{r}}_3^2-{\mathrm{r}}_2^2\right)\mathsf{\Delta }\mathrm{z}\left({\mathrm{T}}_{\mathrm{i},3}^1-{\mathrm{T}}_{\mathrm{i},3}^0\right)=\mathrm{c}\mathsf{\rho }\mathsf{\pi }\left({\mathrm{r}}_3^2-{\mathrm{r}}_2^2\right){\mathrm{v}}_{\mathrm{a}}\mathsf{\Delta }\mathrm{t}{\displaystyle \frac{\left(3{\mathrm{T}}_{\mathrm{i}-\mathrm{1,3}}^1{-4{\mathrm{T}}_{\mathrm{i},3}^1+\mathrm{T}}_{\mathrm{i}+\mathrm{1,3}}^1\right)}{2\mathsf{\Delta }\mathrm{z}}} \\ +2\mathsf{\pi }\mathsf{\Delta }\mathrm{z}\mathsf{\Delta }\mathrm{t}\left[{\mathrm{r}}_3{\mathrm{h}}_{\mathrm{c}3}\left({\mathrm{T}}_{\mathrm{i},4}^1-{\mathrm{T}}_{\mathrm{i},3}^1\right)-{\mathrm{r}}_2{\mathrm{h}}_{\mathrm{c}2}\left({\mathrm{T}}_{\mathrm{i},3}^1-{\mathrm{T}}_{\mathrm{i},2}^1\right)\right] \end{gathered}$$

(7)

Equation (8) is used to discretize the control volumes of the casing and formation.

$$\begin{gathered} \mathrm{c}\mathsf{\rho }\mathsf{\pi }\left({\mathrm{r}}_{\mathrm{j}}^2-{\mathrm{r}}_{\mathrm{j}-1}^2\right)\mathsf{\Delta }\mathrm{z}\left({\mathrm{T}}_{\mathrm{i},\mathrm{j}}^1-{\mathrm{T}}_{\mathrm{i},\mathrm{j}}^0\right)=\mathsf{\pi }\left({\mathrm{r}}_{\mathrm{j}}^2-{\mathrm{r}}_{\mathrm{j}-1}^2\right){\mathrm{k}}_{\mathrm{p}}\mathsf{\Delta }\mathrm{t}\left({\mathrm{T}}_{\mathrm{i}+1,\mathrm{j}}^1+{\mathrm{T}}_{\mathrm{i}-1,\mathrm{j}}^1-2{\mathrm{T}}_{\mathrm{i},\mathrm{j}}^1\right) \\ +2\mathsf{\pi }\mathsf{\Delta }\mathrm{z}\mathsf{\Delta }\mathrm{t}\left[{\mathrm{r}}_{\mathrm{j}}{\mathrm{k}}_{\mathrm{j}}\left({\mathrm{T}}_{\mathrm{i},\mathrm{j}+1}^1-{\mathrm{T}}_{\mathrm{i},\mathrm{j}}^1\right)-{\mathrm{r}}_{\mathrm{j}-1}{\mathsf{\lambda }}_{\mathrm{j}-1}\left({\mathrm{T}}_{\mathrm{i},\mathrm{j}}^1-{\mathrm{T}}_{\mathrm{i},\mathrm{j}-1}^1\right)\right] \end{gathered}$$

(8)

Equation (9) provides a summary of Equations (5)–(8).

$$\begin{gathered} {\mathsf{\alpha }}_{\mathrm{i},\mathrm{j}}{\mathrm{T}}_{\mathrm{i},\mathrm{j}}^{\mathrm{t}+\mathsf{\Delta }\mathrm{t}}-{\mathsf{\beta }}_{\mathrm{i},\mathrm{j}}{\mathrm{T}}_{\mathrm{i},\mathrm{j}}^{\mathrm{t}}={\mathsf{\delta }}_{\mathrm{i}-2\mathrm{j}}{\mathrm{T}}_{\mathrm{i}-2\mathrm{j}}^{\mathrm{t}+\mathsf{\Delta }\mathrm{t}}+{\mathsf{\phi }}_{\mathrm{i}-1,\mathrm{j}}{\mathrm{T}}_{\mathrm{i}-1,\mathrm{j}}^{\mathrm{t}+\mathsf{\Delta }\mathrm{t}}+{\mathsf{\gamma }}_{\mathrm{i}+1,\mathrm{j}}{\mathrm{T}}_{\mathrm{i}+1,\mathrm{j}}^{\mathrm{t}+\mathsf{\Delta }\mathrm{t}}+{\mathsf{\epsilon }}_{\mathrm{i}+2\mathrm{j}}{\mathrm{T}}_{\mathrm{i}+2,\mathrm{j}}^{\mathrm{t}+\mathsf{\Delta }\mathrm{t}} \\ +{\mathsf{\eta }}_{\mathrm{i},\mathrm{i}-2}{\mathrm{T}}_{\mathrm{i},\mathrm{j}-2}^{\mathrm{t}+\mathsf{\Delta }\mathrm{t}}+{\mathsf{\theta }}_{\mathrm{i},\mathrm{j}-1}{\mathrm{T}}_{\mathrm{i},\mathrm{j}-1}^{\mathrm{t}+\mathsf{\Delta }\mathrm{t}}+{\mathsf{\sigma }}_{\mathrm{i},\mathrm{j}+1}{\mathrm{T}}_{\mathrm{i},\mathrm{j}+1}^{\mathrm{t}+\mathsf{\Delta }\mathrm{t}}+{\mathsf{\xi }}_{\mathrm{i},\mathrm{j}+2}{\mathrm{T}}_{\mathrm{i},\mathrm{j}+2}^{\mathrm{t}+\mathsf{\Delta }\mathrm{t}}+{\mathrm{b}}_{\mathrm{i},\mathrm{j}} \end{gathered}$$

(9)

Equation (10) transforms Equation (9) into the SOR successive over-relaxation iteration format.

$$\begin{gathered} {\mathrm{T}}_{\mathrm{i},\mathrm{j}}^{\mathrm{t}+\mathsf{\Delta }\mathrm{t}\left(\mathrm{k}+1\right)}={\mathrm{T}}_{\mathrm{i},\mathrm{j}}^{\mathrm{t}+\mathsf{\Delta }\mathrm{t}\left(\mathrm{k}\right)} \\ +{\displaystyle \frac{\mathsf{\omega }}{{\mathsf{\alpha }}_{\mathrm{i},\mathrm{j}}}}\left[\begin{array}{l}{\mathsf{\beta }}_{\mathrm{i},\mathrm{j}}{\mathrm{T}}_{\mathrm{i},\mathrm{j}}^{\mathrm{t}}+{\mathsf{\delta }}_{\mathrm{i}-2,\mathrm{j}}{\mathrm{T}}_{\mathrm{i}-2,\mathrm{j}}^{\mathrm{t}+\mathsf{\Delta }\mathrm{t}\left(\mathrm{k}+1\right)}+{\mathsf{\phi }}_{\mathrm{i}-1,\mathrm{j}}{\mathrm{T}}_{\mathrm{i}-1,\mathrm{j}}^{\mathrm{t}+\mathsf{\Delta }\mathrm{t}\left(\mathrm{k}+1\right)}+{\mathsf{\gamma }}_{\mathrm{i}+1,\mathrm{j}}{\mathrm{T}}_{\mathrm{i}+1,\mathrm{j}}^{\mathrm{t}+\mathsf{\Delta }\mathrm{t}\left(\mathrm{k}\right)}\\ +{\mathsf{\epsilon }}_{\mathrm{i}+2,\mathrm{j}}{\mathrm{T}}_{\mathrm{i}+2,\mathrm{j}}^{\mathrm{t}+\mathsf{\Delta }\mathrm{t}\left(\mathrm{k}\right)}+{\mathsf{\eta }}_{\mathrm{i},\mathrm{j}-2}{\mathrm{T}}_{\mathrm{i},\mathrm{j}-2}^{\mathrm{t}+\mathsf{\Delta }\mathrm{t}\left(\mathrm{k}+1\right)}+{\mathsf{\theta }}_{\mathrm{i},\mathrm{j}-1}{\mathrm{T}}_{\mathrm{i},\mathrm{j}-1}^{\mathrm{t}+\mathsf{\Delta }\mathrm{t}\left(\mathrm{k}+1\right)}\\ +{\mathsf{\sigma }}_{\mathrm{i},\mathrm{j}+1}{\mathrm{T}}_{\mathrm{i},\mathrm{j}+1}^{\mathrm{t}+\mathsf{\Delta }\mathrm{t}\left(\mathrm{k}\right)}+{\mathsf{\xi }}_{\mathrm{i},\mathrm{j}+2}{\mathrm{T}}_{\mathrm{i},\mathrm{j}+2}^{\mathrm{t}+\mathsf{\Delta }\mathrm{t}\left(\mathrm{k}\right)}+{\mathrm{b}}_{\mathrm{i},\mathrm{j}}-{\mathsf{\alpha }}_{\mathrm{i},\mathrm{j}}{\mathrm{T}}_{\mathrm{i},\mathrm{j}}^{\mathrm{t}+\mathsf{\Delta }\mathrm{t}\left(\mathrm{k}\right)}\end{array}\right] \end{gathered}$$

(10)

where $\mathsf{\omega }$ is the relaxation factor for the successive over-relaxation (SOR) iteration, dimensionless.

4.2.4. Drilling Fluid Density Calculation

For the calculation of drilling fluid density, reference is made to the model presented by Hoberock [21], which uses a component material balance model to predict the downhole density of water-based and diesel-based muds. This model is based on the change in mud volume caused by the variation in formation depth where the drilling fluid is located, as shown in Equation (11).

$$\mathsf{\rho }\left(\mathrm{P},\mathrm{T}\right)={\displaystyle \frac{{\mathsf{\rho }}_{\mathrm{o}\mathrm{i}}{\mathrm{f}}_{\mathrm{v}\mathrm{o}}+{\mathsf{\rho }}_{\mathrm{w}\mathrm{i}}{\mathrm{f}}_{\mathrm{v}\mathrm{w}}+{\mathsf{\rho }}_{\mathrm{s}}{\mathrm{f}}_{\mathrm{v}\mathrm{s}}+{\mathsf{\rho }}_{\mathrm{c}}{\mathrm{f}}_{\mathrm{v}\mathrm{c}}}{1+{\mathrm{f}}_{\mathrm{v}\mathrm{o}}\left(\frac{{\mathsf{\rho }}_{\mathrm{o}\mathrm{i}}}{{\mathsf{\rho }}_{\mathrm{o}}}-1\right)+{\mathrm{f}}_{\mathrm{v}\mathrm{w}}\left(\frac{{\mathsf{\rho }}_{\mathrm{w}\mathrm{i}}}{{\mathsf{\rho }}_{\mathrm{w}}}-1\right)}}$$

(11)

where ${\mathsf{\rho }}_{\mathrm{o}\mathrm{i}}$ and ${\mathsf{\rho }}_{\mathrm{w}\mathrm{i}}$ are the densities of oil and water under initial conditions, kg/m³; ${\mathsf{\rho }}_{\mathrm{o}}$ and ${\mathsf{\rho }}_{\mathrm{w}}$ are the densities of oil and water under specific pressure and temperature conditions, kg/m³; ${\mathrm{f}}_{\mathrm{v}\mathrm{o}}$, ${\mathrm{f}}_{\mathrm{v}\mathrm{w}}$, ${\mathrm{f}}_{\mathrm{v}\mathrm{s}}$, and ${\mathrm{f}}_{\mathrm{v}\mathrm{c}}$ are the volume fractions of oil, water, solid substances, and chemical additives, respectively, dimensionless.

For the density prediction of the aqueous phase and oil phase in drilling fluid, reference is made to Equation (12) recommended by the [25].

$${\mathsf{\rho }}_{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{e}-\mathrm{P},\mathrm{T}}\mathrm{o}\mathrm{r}{\mathsf{\rho }}_{\mathrm{b}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{e}-\mathrm{P},\mathrm{T}}=\left({\mathrm{a}}_1+{\mathrm{b}}_1\mathrm{P}+{\mathrm{c}}_1{\mathrm{P}}^2\right)+\left({\mathrm{a}}_2+{\mathrm{b}}_2\mathrm{P}+{\mathrm{c}}_2{\mathrm{P}}^2\right)\left(1.8\mathrm{T}+32\right)$$

(12)

The correlation coefficients of Equation (12) refer to

Table 4. A reference table for correlation coefficients used in formulas.

Multiply Coefficients by ρ Density in kg/m3	Mineral Oils	CaCl2 Brine19.3%wt
${\mathrm{a}}_1,\mathrm{k}\mathrm{g}/{\mathrm{m}}^3$	$\mathsf{\rho }\times 1.03$ × 100	$\mathsf{\rho }\times 1.02$ × 100
${\mathrm{b}}_1,\left(\mathrm{k}\mathrm{g}/{\mathrm{m}}^3\right)/\mathrm{P}\mathrm{a}$	$\mathsf{\rho }\times 4.55$ × 10−6	$\mathsf{\rho }\times 1.71$ × 10−6
${\mathrm{c}}_1,\left(\mathrm{k}\mathrm{g}/{\mathrm{m}}^3\right)/\mathrm{P}{\mathrm{a}}^2$	$\mathsf{\rho }\times -3.63$ × 10−11	$\mathsf{\rho }\times 1.13$ × 10−11
${\mathrm{a}}_2,\left(\mathrm{k}\mathrm{g}/{\mathrm{m}}^3\right)/\mathrm{℃}$	$\mathsf{\rho }\times -4.13$ × 10−4	$\mathsf{\rho }\times -3.15$ × 10−4
${\mathrm{b}}_2,\left[\left(\mathrm{k}\mathrm{g}/{\mathrm{m}}^3\right)/\mathrm{P}\mathrm{a}\right]/\mathrm{℃}$	$\mathsf{\rho }\times 9.53$ × 10−9	$\mathsf{\rho }\times 3.50$ × 10−9
${\mathrm{c}}_2,\left[\left(\mathrm{k}\mathrm{g}/{\mathrm{m}}^3\right)/\mathrm{P}{\mathrm{a}}^2\right]/\mathrm{℃}$	$\mathsf{\rho }\times -1.41$ × 10−13	$\mathsf{\rho }\times -6.49$ × 10−14
$\mathsf{\rho },\mathrm{k}\mathrm{g}/{\mathrm{m}}^3$	803.55	1174.2948

.

4.3. Calculation Process

Based on C# language (.NET Framework 4.8) with Visual Studio 2022 as the development environment, a calculation program for wellbore temperature distribution, equivalent static density (ESD), and bottom hole pressure (BHP) throughout the tripping-out process was developed, and the program flow is shown in Figure 7. By coupling the wellbore-stratum transient temperature field with the drilling fluid density model, through grid discretization, iterative solution, and other steps, the quantitative simulation of temperature, ESD, and BHP changes under different working conditions is realized, providing technical support for the analysis of thermal effect influence laws and the parameter design of the weighted drilling fluid supplementation method.

4.3.1. Initial Conditions

(1)

Before the operation begins, the mud is in a static state, and its temperature distribution is consistent with that of the formation, i.e., the original geothermal gradient, which is given by Equation (13).

$${\mathrm{T}}_{(\mathrm{r},\mathrm{z})\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}}^0={\mathrm{T}}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{e}}+\mathrm{G}\mathrm{z}$$

(13)

where ${\mathrm{T}}_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}}$ is initial static temperature, $\mathrm{℃}$; ${\mathrm{T}}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{e}}$ is surface temperature, $\mathrm{℃}$; $\mathrm{G}$ is geothermal gradient, $\mathrm{℃}/100 \mathrm{m}$.

(2)

Since the drilling state is maintained before tripping-out, the wellbore temperature distribution at the initial moment of tripping-out is the same as that at the end of drilling.

4.3.2. Boundary Conditions

(1)

At the drill bit location, the temperature of the annular drilling fluid at the bottom hole is the same as that of the drilling fluid inside the drill string. In this study, the thermal boundary is set at a radius of 10 m, which is given by Equation (14).

$$\left\{\begin{array}{c}{\mathrm{T}}_{(\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g},\mathrm{e}\mathrm{n}\mathrm{d},\mathrm{a}\mathrm{l}\mathrm{l})}={\mathrm{T}}_{(\mathrm{a}\mathrm{n}\mathrm{n}\mathrm{u}\mathrm{l}\mathrm{u}\mathrm{s},\mathrm{e}\mathrm{n}\mathrm{d},\mathrm{a}\mathrm{l}\mathrm{l})}\\ {\mathrm{T}}_{(\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t},\mathrm{z},\mathrm{t})\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}={\mathrm{T}}_{(\mathrm{r},\mathrm{z},0)\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}\end{array}\right.$$

(14)

(2)

During the tripping-out process, drilling fluid at a constant temperature is continuously supplemented into the annulus. For the wellhead of the annulus, this is described by Equation (15).

$${\mathrm{T}}_{(\mathrm{a}\mathrm{n}\mathrm{n}\mathrm{u}\mathrm{l}\mathrm{u}\mathrm{s},0,\mathrm{a}\mathrm{l}\mathrm{l})}={\mathrm{T}}_{\mathrm{i}\mathrm{n}}$$

(15)

(3)

The formation has a fixed disturbance radius, and the formation outside the disturbance radius is not affected by wellbore heat exchange, maintaining a constant original temperature.

(4)

The thermophysical parameters (density, specific heat capacity, thermal conductivity) of materials such as drill pipe, casing, and formation are known fixed values, and the initial density and thermophysical parameters of drilling fluid are set according to the design values.

5. Research Limitations

During the establishment of the tripping-out temperature field model, idealized assumptions are adopted for convenience of calculation. Complex actual downhole conditions such as non-one-dimensional flow of drilling fluid and formation heterogeneity are not considered, which may lead to minor deviations in the calculation results of the temperature field.

Fixed values are used for the thermophysical parameters of materials such as drilling fluid and formation rock, but these parameters are affected by temperature and pressure, especially in the high-temperature and high-pressure (HTHP) downhole environment of ultra-deep wells.

This study conducts analysis with Well PS6 (an ultra-deep vertical well) as a case study, without considering the relevant calculations for directional wells and horizontal wells. As a result, the proposed weighted drilling fluid supplementation method has not yet covered complex well type scenarios, leading to a limited scope of application.

The mud supplementation strategy is designed based on idealized operating conditions (fixed mud supplementation rate, precise density control), without considering the impact of on-site operating errors such as pump pressure fluctuations and deviations in mud supplementation timing on the bottom hole pressure control effect. There may be differences between the model prediction results and field practical applications.

6. Results and Discussion

We performed calculations for the tripping-out scenarios when Well PS6 was drilled to depths of 2910 m and 9026 m.

Table 5. Calculation results of tripping-out time at 2910 m.

Operation	Tripping Speed	Cumulative Time (h)
Tripping-out at 2910 m	400 m/h	7.275
	500 m/h	6
	600 m/h	5
Break Out Stands	1 min/30 m	1.5
Filling the Annulus	1.5 min/90 m	1

and

Table 6. Calculation results of tripping-out time at 9026 m.

Operation	Tripping Speed	Cumulative Time (h)
Tripping-out at 9026 m	400 m/h	22.56
	500 m/h	18
	600 m/h	15
Break Out Stands	1 min/30 m	5
Filling the Annulus	1.5 min/90 m	2.5

estimate the time required for operations such as drill string disassembly, annular mud replenishment, and other operations during the tripping-out process. Taking a tripping-out speed of 400 m/h as an example, the calculations show that 10 h are needed to complete the tripping-out operation at 2910 m, and 30 h at 9026 m. Subsequent calculations are all based on the time required to complete the tripping-out operation at a tripping-out speed of 400 m/h as an example.

Figure 8 shows the wellbore temperature distribution at 0 h, 2.5 h, 5 h, 7.5 h, and 10 h of continuous tripping-out from 2910 m. As time increases, the downhole temperature exhibits the characteristic of decreasing in the shallow formation and increasing in the deep formation. Compared with the initial moment, the bottom hole temperature rises by 17.5 °C at the end of tripping-out. Figure 8a presents a comparative verification of the circulating temperature field with the Drillbench software, and the deviation of the calculated bottom hole temperature is less than 1 °C.

Figure 9 shows the overall change in ESD during the tripping-out process from 2910 m and the detailed variation in bottom hole ESD with time. Throughout the tripping-out process, at tripping-out rates of 400 m/h, 500 m/h, and 600 m/h (i.e., with the required tripping-out times being 10 h, 8.5 h, and 7.5 h, respectively), the ESD at the bottom hole increases by 0.0005 g/cm³, 0.0007 g/cm³, and 0.0008 g/cm³ respectively. The increase in the amplitude of ESD is within an extremely small range and slightly increases with the acceleration in tripping-out speed.

Figure 10 shows the pressure change in partial well sections and the detailed variation in bottom hole pressure with time during the tripping-out process from 2910 m. Throughout the tripping-out process, at tripping-out rates of 400 m/h, 500 m/h, and 600 m/h (i.e., with the required tripping-out times being 10 h, 8.5 h, and 7.5 h, respectively), the pressure at the bottom hole increases by 0.016 MPa, 0.018 MPa, and 0.02 MPa, respectively. It can be observed that at 2910 m, the BHP shows a slight upward trend and is positively correlated with the tripping-out speed. During the tripping-out process, the temperature drop in the shallow well section counterbalances the temperature rise in the deep well section, resulting in a wellbore pressure slightly higher than the initial value at the end of tripping-out.

Figure 11 shows the wellbore temperature distribution at 0 h, 5 h, 10 h, 15 h, 20 h, 25 h, and 30 h of continuous tripping-out from 9026 m. Compared with the initial moment, the bottom hole temperature rises by 72.6 °C at the end of tripping-out. Figure 11a presents a comparative verification of the circulating temperature field with the Drillbench software, and the deviation of the calculated bottom hole temperature is less than 3 °C.

Figure 12 shows the overall change in ESD during the tripping-out process from 9026 m and the detailed variation in bottom hole ESD with time. Throughout the tripping-out process, at tripping-out rates of 400 m/h, 500 m/h, and 600 m/h (i.e., with the required tripping-out times being 30 h, 25.5 h, and 22.5 h, respectively), the ESD at the bottom hole decreases by 0.0273 g/cm³, 0.0223 g/cm³, and 0.0204 g/cm³, respectively. The decrease in the amplitude of ESD is significant and decreases with the acceleration in tripping-out speed.

Figure 13 shows the pressure change in partial well sections and the detailed variation in bottom hole pressure with time during the tripping-out process from 9026 m. Throughout the tripping-out process, at tripping-out rates of 400 m/h, 500 m/h, and 600 m/h (i.e., with the required tripping-out times being 30 h, 25.5 h, and 22.5 h, respectively), the pressure at the bottom hole decreases by 2.410 MPa, 1.972 MPa, and 1.799 MPa, respectively. It can be observed that at 9026 m, the BHP exhibits a significant decrease in amplitude, which decreases with the acceleration in tripping-out speed. During the tripping-out process, the temperature rise in the deep well section dominates, resulting in a wellbore pressure at the end of tripping-out that is significantly lower than the initial value.

To address the issue where thermal effects reduce bottom hole pressure (BHP) during tripping operations under high-temperature and high-pressure (HTHP) conditions, a safe tripping-out control technology is presented. The method involves adding a weighted mud tank on the surface before the start of tripping-out operations. Based on subsequent calculation results, two types of drilling fluids are alternately supplemented into the wellhead annulus during the tripping-out process to achieve precise control of downhole pressure, as shown in Figure 14.

Equation (16) describes the calculation method for the density of the supplemented weighted drilling fluid during the tripping-out process.

$$\left\{\begin{array}{c}{\mathsf{\rho }}_{\left(\mathrm{i}\right)\mathrm{a}\mathrm{d}\mathrm{d}}={\displaystyle \frac{\mathsf{\rho }{\mathrm{H}}_{\mathrm{i}}}{{\mathrm{L}}_{\mathrm{i}}}}={\displaystyle \frac{\mathsf{\rho }{\mathrm{H}}_{\mathrm{i}}}{{\mathrm{H}}_{\mathrm{i}}-{\mathrm{l}}_{\mathrm{i}}}}\\ {\mathrm{H}}_{\mathrm{i}}={\displaystyle \frac{\mathrm{v}{\mathrm{t}}_{\mathrm{i}}{\mathrm{r}}_2^2}{\left({\mathrm{r}}_3^2-{\mathrm{r}}_2^2\right)}}\\ {\mathrm{l}}_{\mathrm{i}}={\displaystyle \frac{{\mathrm{V}}_{\mathrm{i}}-{\mathrm{V}}_{\mathrm{i}-1}}{({\mathrm{r}}_3^2-{\mathrm{r}}_2^2)\mathsf{\pi }}}={\displaystyle \sum _{\mathrm{j}=1}^0}{\displaystyle \frac{{\mathsf{\rho }}_{\mathrm{i},\mathrm{j}}}{{\mathsf{\rho }}_{\mathrm{i}-1,\mathrm{j}}}}{\mathrm{d}}_{\mathrm{j}}\end{array}\right.$$

(16)

where ${\mathrm{H}}_{\mathrm{i}}$is the annulus fluid level drop height during tripping-out, m; ${\mathrm{L}}_{\mathrm{i}}$ is the flow channel length, m; ${\mathrm{l}}_{\mathrm{i}}$ is the fluid level rise due to thermal expansion, m.

The current pressure control method for the tripping-out process involves circulating weighted drilling fluid through the drill string into the annulus to form a “weighted mud cap” prior to tripping-out operations. This method is used to balance downhole pressure changes during tripping-out, but it fails to account for pressure losses caused by downhole temperature variations during the process. If this portion of the loss is incorporated into the method, it will lead to an excessive increase in downhole pressure, thereby elevating the risk of well leakage during the formation of the “weighted mud cap”. In contrast, the technology presented in this paper can maintain bottom hole pressure fluctuations within a narrower range, which is more suitable for the “narrow safe density window” in deep formations.

Figure 15 calculates the cumulative volume of drilling fluid required to be supplemented into the annulus after considering the expanded volume of drilling fluid during the tripping-out process. Figure 16 calculates the density of the weighted drilling fluid that needs to be supplemented at different times. For the convenience of on-site practical operations, weighted drilling fluid with a constant density of 2.800 g/cm³ and initial drilling fluid are alternately supplemented to control the bottom hole pressure.

Figure 17 calculates the alternating supplementation time of the two types of drilling fluids with different densities. Based on the design results of the supplemented drilling fluid, Figure 18 calculates the variation in bottom hole pressure with tripping-out time before and after the application of the tripping-out pressure control technology. It can be observed that after adopting the tripping-out pressure control technology, the bottom hole pressure can be controlled to fluctuate within the range of 0.339 MPa during the tripping-out process, which solves the problem of overflow risk caused by the decrease in bottom hole pressure due to temperature recovery during the tripping-out process.

7. Conclusions

A weighted drilling fluid supplementation method is proposed to address the problem of bottom hole pressure (BHP) reduction during tripping operations caused by thermal effects under high-temperature and high-pressure (HTHP) conditions. This study establishes a transient tripping-out temperature field model. Calculations of temperature, equivalent static density (ESD), and BHP during tripping-out at 2910 m and 9026 m in Well PS6 reveal that at 2910 m, the bottom hole temperature rises by 17.5 °C and BHP increases slightly by 0.016 MPa, while at 9026 m, the bottom hole temperature surges by 72.6 °C and BHP drops significantly by 2.410 MPa. This discrepancy arises from the fact that the annular temperature in the shallow section is higher than the formation temperature, whereas the opposite holds true for the deep section. At 2910 m, the cooling effect of the shallow annular region dominates; in contrast, the tripping cycle at 9026 m is longer and the formation temperature difference is larger, intensifying the phenomenon of BHP reduction during tripping-out induced by thermal effects in the deep section.

Targeting the risk of significant BHP drop in ultra-deep well sections like 9026 m, this study proposes the technical concept of “alternate supplementation of weighted drilling fluid”. Through optimized design of mud supplementation timing, constant heavy mud density, and supplementation volume via numerical simulation, the technical feasibility is verified: this concept can control BHP fluctuation within 0.339 MPa during tripping-out, effectively mitigating the risk of pressure loss of control caused by thermal effects. Compared with the existing “weighted mud cap” technology (which involves circulating and injecting heavy mud before tripping-out without considering pressure loss due to thermal effects), the proposed technology avoids the well leakage risk potentially caused by one-time heavy mud injection through dynamic alternate supplementation. Additionally, it offers higher BHP control accuracy, making it more suitable for the operational requirements of narrow safe density windows in deep formations. It should be noted that this technical concept has only been verified for feasibility through numerical simulation and has not undergone field testing yet. Subsequent optimization of operational procedures and parameter adaptability through field experiments is required.

8. Future Work

While this study successfully establishes a transient temperature–pressure coupled model for ultra-deep wells and proposes a dual-density fluid control method, there are several aspects that warrant further investigation:

The current model has been validated against the commercial software Drillbench with a high degree of agreement. Nevertheless, due to the scarcity of real-time downhole measurement data during tripping operations in ultra-deep wells, direct validation of the model using field data remains limited. Future research will focus on further calibrating the model through validation by comparison with actual high-temperature downhole data.

The current study focuses on specific case analyses. To systematically quantify the impact of key parameters—such as geothermal gradient, formation thermal conductivity, and tripping speed—on the wellbore temperature and pressure fields, a comprehensive sensitivity analysis is required.

The simulation scenarios can be further expanded. Future research will consider more complex operational conditions, such as simulation analysis under multi-gradient drilling environments and the application of the model in extended-reach wells and horizontal wells.