A Full-Scale Thermo-Hydro-Mechanical Coupled Numerical Model for Wellbore Injection Operations

Abstract

Injection operations are critical in subsurface energy engineering, where wellbores endure complex thermo-hydro-mechanical (THM) coupling under high-temperature and high-pressure conditions, impacting tubing string stability and wellbore long-term safety. Current tubing string THM research relies on simplified assumptions, focusing on single/dual-field coupling without full-scale modeling, failing to accurately characterize comprehensive multi-field behaviors or actual structural stress distributions. This paper presents a full-scale THM coupled numerical model for actual injection conditions, taking real wellbore structures as the object to realize unified modeling of tubing, packer, casing, cement sheath and formation, covering the entire well section and synergistically describing fluid flow, heat conduction and structural mechanical response. It considers fluid pressure/temperature effects on tubing axial load, thermal stress and deformation, as well as nonlinear boundary conditions like packer-casing contact and friction. The governing equations are discretized via the finite element method and solved by Newton iteration. Benchmark verification shows the maximum relative errors of casing inner/outer wall Mises stress vs. analytical solutions are 2.43% and 4.98%, confirming high accuracy. Systematic analysis of displacement, axial force, stress and temperature responses under typical conditions is conducted, providing reliable theoretical and technical support for wellbore structure optimization, injection parameter regulation and long-term wellbore integrity evaluation.

1. Introduction

Injection operations are widely applied in oil and gas recovery, geothermal energy utilization, underground gas storage, and other subsurface energy engineering fields. Wellbore stability and operational safety are crucial to the efficient and sustainable development of oil and gas resources [1,2]. As a key conduit for fluid transport between underground fluids and formations, the wellbore experiences complex fluid flow, significant temperature variations, and interactions between the tubing string and the surrounding rock throughout its service life. These factors significantly influence operational efficiency and the overall performance of the engineering system [3,4,5]. Under complex subsurface conditions, particularly those characterized by high temperature and high pressure, the coupling effects among fluid flow, heat transfer, and structural response become increasingly pronounced, constituting a typical THM multi-field coupling problem. With the advancement of underground energy exploitation toward deeper formations with higher temperature and pressure conditions, the fluid flow behavior, temperature evolution characteristics, and heat transfer processes involved in wellbore operations become progressively more complex, and the THM coupling effects become more significant. Therefore, conducting in-depth research on these coupling mechanisms is of considerable theoretical importance and practical engineering value for optimizing wellbore structural design, improving injection efficiency, and ensuring the long-term safe and stable operation of the wellbore [6,7,8].

During injection operations, the THM multi-field coupling effects play a crucial role in determining operational efficiency and structural safety. As the injected fluid flows through the wellbore, its pressure and temperature continuously vary along the well, influenced by intense heat exchange with the tubing string and surrounding media. These temperature changes cause the tubing string to undergo thermal expansion and contraction, generating additional axial forces and thermal stress concentrations that adversely affect the stability and performance of the tubing string [9]. Furthermore, fluid flow, heat transfer, and the structural deformation of the tubing string interact and feedback with each other, forming a highly nonlinear THM multi-field coupling system [10]. Under high-temperature and high-pressure operating conditions, the physical properties of the fluid in the wellbore change significantly with temperature and pressure, further intensifying the coupling effects between the fluid and the solid structure. These complex interactions not only affect fluid transport and heat exchange efficiency but may also lead to fatigue damage or even structural failure of the tubing string material, thereby threatening the safety and long-term stability of the injection operation [11,12]. Therefore, there is an urgent need for systematic and in-depth research on the THM multi-field coupling issues in wellbore injection operations.

Regarding the THM multi-field coupling problem of tubing strings, various analytical or numerical simulation methods have been developed to analyze the coupling among fluid flow, heat transfer, and structural response. Ramey et al. proposed a mathematical model describing the heat transfer process within the wellbore by dividing the wellbore into different regions and combining conservation laws [13]. Abdelhafiz et al. developed a model for predicting the temperature distribution of the wellbore under various operating conditions, considering transient temperature disturbances in the cement sheath, casing, and formation [14]. Sproull et al. established an analytical model describing the heat generation process under different boundary conditions for the infinite cylindrical heat source problem, providing an important reference for transient heat conduction problems in formations [15]. Gu et al. extended Ramey’s model to the working condition of superheated steam injection in horizontal wells, introducing the effects of phase change processes and variations in steam quality [16]. Dong et al. studied the characteristics of temperature distribution during fracturing and analyzed the effect of axial heat conduction [17]. Ma et al. established a fully coupled THM model for wellbore stability in deep unsaturated formations, emphasizing the significance of thermal–hydraulic–mechanical interactions under high-temperature and high-pressure conditions [18]. Zhai et al. developed a peridynamics-based finite element model to analyze sealing failure at the casing–cement interface, which is critical for long-term wellbore integrity in ultra-deep wells [19]. However, most of these models adopt simplified assumptions, analyzing only single-field or dual-field coupling problems, neglecting the complexity of THM three-field coupling, with limited success in comprehensively characterizing the coupling actions among fluid flow, temperature field evolution, and tubing string structural mechanical response. In addition, existing studies lack consideration of full-scale modeling of tubing string mechanics. They often reduce computational difficulty by reducing geometric scales and simplifying tubing string structural morphologies and boundary conditions. This leads to an inability to accurately reproduce the complete structural characteristics of the tubing string in actual wellbores, the stress distribution over the full length, and the multi-interface coupling effects among key components such as packers and joints with the tubing string body, fluid, and formation. Consequently, core mechanical responses such as axial deformation, thermal stress distribution, and contact nonlinearity of the tubing string are difficult to accurately characterize. Therefore, there is a pressing need to develop a more refined model to predict the mechanical characteristics and multi-field coupling effects of the tubing string at full scale, thereby providing precise theoretical support for wellbore structure optimization, injection process parameter regulation, and operation efficiency improvement, and ultimately ensuring the long-term safe and stable operation of the wellbore.

To overcome the above limitations, this paper proposes a full-scale THM multi-field coupled numerical model, developed for actual working conditions in wellbore injection operations. The model takes the real wellbore structure as the research object, providing a unified modeling of the tubing, packer, casing, cement sheath, and surrounding formation. In terms of spatial scale, it completely covers the entire well section from the wellhead to the bottom. In terms of multi-physical modeling, it synergistically describes key processes such as fluid flow, temperature conduction, and tubing string structural mechanical response. The model fully considers the influence of fluid pressure and temperature changes during the injection process on the tubing string axial load, thermal stress, and deformation behavior. Meanwhile, it introduces packer-casing contact, friction constraints, and nonlinear boundary conditions to simulate typical working conditions such as setting, liquid injection, and unloading. To ensure the accuracy and stability of solving the strongly coupled multi-field problem, this paper adopts the finite element method to uniformly discretize the governing equations and combines the Newton iteration strategy to solve the nonlinear coupled system. This full-scale, multi-field coupled modeling and numerical implementation provides a systematic and reliable theoretical basis and technical support for wellbore injection process optimization, packer structural innovative design, and long-term wellbore integrity evaluation from the perspective of practical engineering.

The structure of this article is arranged as follows: Section 2 systematically describes the proposed model. First, the overall mechanical model of the wellbore tubing string is developed, and the typical working condition evolution of the tubing string during the service process (including stages such as setting, setting stop, liquid injection, and injection stop) is outlined. Subsequently, the governing equations for the solid mechanics field, thermodynamics field, and wellbore internal fluid flow are established, respectively. Among them, the thermodynamics part describes the heat transfer processes successively inside the tubing, in the annular fluid, radially through the tubing string, in the casing, and between the cement sheath and the formation. The fluid part provides the calculation model for pressure drop and flow characteristics within the wellbore. On this basis, the end of Section 2 introduces the unified solution method and its numerical implementation for the THM coupled equation system. Section 3 presents the numerical calculation results and corresponding analysis. First, the rationality and accuracy of the established model are verified through classical benchmarks. Subsequently, based on given wellbore parameters and analysis procedures, the displacement, axial force, stress, and temperature response characteristics of the tubing string under various working conditions such as setting, setting stop, liquid injection, and injection stop are systematically analyzed. Section 4 summarizes the work of the whole paper, outlines the main characteristics of the proposed multi-field coupled modeling method, and summarizes the key conclusions regarding the mechanical and thermal responses of the tubing string throughout the wellbore injection process.

2. Model Description

The tubular mechanical model for wellbore injection operations constructed in this paper is shown in Figure 1. This model completely covers the casing, tubing, packer, tubing string, and annular space, accurately reconstructing the topological structure and positional relationships of the multi-layer media within the wellbore. The tubing string exhibits significant multi-stage evolution characteristics throughout its service cycle. With the switching of working conditions, the pressure field, temperature field, and fluid medium types within the wellbore undergo dynamic changes. These environmental loads superimpose with the nonlinear contact behavior between the packer and the casing, all of which have a coupled impact on the mechanical response of the tubing string, thereby determining the overall stability of the wellbore and operational safety. Based on the engineering operation process, the service process of the tubing string can be divided into four typical stages: setting, setting stop, liquid injection, and injection stop. The mechanical behaviors and analysis focus of each stage are as follows.

In the setting stage, the packer rubber element gradually establishes contact with the casing. This stage involves large deformation of the rubber elements and the onset of contact nonlinearity. The wellhead pressure slowly rises from 0 to 12 MPa and is further loaded to 15 MPa. During this process, the focus is on analyzing the frictional force generated by the rubber element-casing contact and the impact of the piston effect on the deformation and stress distribution of the packer. As the setting pressure is removed, the system enters the setting stop stage. The elastic potential energy accumulated in the tubing string is released, inducing axial rebound. The displacement and stress state of the packer are jointly controlled by residual friction, temperature changes, and thermal expansion effects. Simulating this rebound process helps evaluate the stability of the packer under unloading conditions. Subsequently, the liquid injection stage begins, where liquid is injected into the wellbore under high-pressure conditions of 0–25 MPa. The influences of pressure load, frictional resistance effects, and thermal effects on the mechanical response of the tubing string are particularly significant and require comprehensive analysis based on the fluid flow model. When the injection stops, the pressure within the wellbore gradually decreases while the temperature begins to rise, causing the tubing string to undergo axial rebound again. This involves the redistribution of residual stress in the tubing string under variable temperature environments and its potential impact on the long-term integrity of the wellbore.

2.1. Solid Mechanics of Tubing String

A 3D full-scale finite element model is established according to the actual geometric dimensions and spatial morphology of the tubular to uniformly describe its overall and local structures. Meanwhile, geometric nonlinearity, material nonlinearity, as well as contact, friction, and multi-load coupling effects are considered to truly reflect the actual stress state and deformation behavior of the tubing string under complex working conditions. Based on the basic principles of continuum mechanics, the full-scale tubing string is modeled as an elastic body continuously distributed along the well depth direction. The coupling effects of axial tension/compression and torsional loads, as well as the nonlinear boundary control effects of multi-point constraints such as the wellbore wall and packer on the mechanical behavior of the tubing string, are comprehensively considered. Aiming at the axial response of the tubing string under distributed loads, the quasi-static equilibrium equation establishing the tubing string equilibrium relationship is as follows [16]:

$$EA{\displaystyle \frac{{\mathrm{d}}^{2}u}{\mathrm{d}{z}^2}}+f\left(z\right)=0$$

(1)

where $E$ is the elastic modulus, $A$ is the cross-sectional area, $u\left(z\right)$ is the axial displacement, and $f\left(z\right)$ is the distributed load.

Under the action of torsional loads, the torsional mechanical behavior of the tubing string can be described by the following governing equation:

$$GJ{\displaystyle \frac{{\mathrm{d}}^2\theta }{\mathrm{d}{z}^2}}+m\left(z\right)=0$$

(2)

where $G$ is the shear modulus, $J$ is the polar moment of inertia, $\theta \left(z\right)$ is the angle of twist, and $m\left(z\right)$ is the distributed torque.

Further considering the contact constraints and friction effects between the wellbore wall and the packer, as well as the potentially nonlinear constitutive characteristics of the tubing string material under high-temperature and high-pressure environments, a contact potential energy term is introduced in the numerical modeling.

For the stick case:

$${\pi }_c={\displaystyle \frac{1}{2}}{\int }_{{\tau }_c}{(ϵ}_N{g_{Ns}}^2+{ϵ}_T{\mathit{g}}_T·{\mathit{g}}_T)dl$$

(3)

$$\delta {\pi }_c={\int }_{{\tau }_c}{(ϵ}_N{g}_{Ns}\delta g_{Ns}+{ϵ}_T{\mathit{g}}_T·{\delta \mathit{g}}_T)dl$$

(4)

$$∆\delta {\pi }_c={\int }_{{\tau }_c}{(ϵ}_N{∆g}_{Ns}\delta g_{Ns}+{ϵ}_N{g}_{Ns}∆\delta g_{Ns}+{ϵ}_T∆{\mathit{g}}_T·{\delta \mathit{g}}_T+{ϵ}_T{\mathit{g}}_T·{∆\delta \mathit{g}}_T)dl$$

(5)

For the slip case:

$$\delta {\pi }_c={\int }_{{\tau }_c}{(ϵ}_N{g}_{Ns}\delta g_{Ns}+{\mathit{t}}_T·{\delta \mathit{g}}_T)dl$$

(6)

$$∆\delta {\pi }_c={\int }_{{\tau }_c}{(ϵ}_N{∆g}_{Ns}\delta g_{Ns}+{ϵ}_N{g}_{Ns}∆\delta g_{Ns}+∆{\mathit{t}}_T·{\delta \mathit{g}}_T+{\mathit{t}}_T·{∆\delta \mathit{g}}_T)dl$$

(7)

where ${\pi }_c$ is the contact potential energy, $\delta {\pi }_c$ is the virtual work contribution of contact forces that corresponds to the right-hand side of the system, $∆\delta {\pi }_c$ corresponds to the contribution of contact to the system stiffness matrix, $g_{Ns}$ is the normal penetration amount, ${\mathit{g}}_T$ is the tangential sticking displacement vector, and ${\mathit{t}}_T$ is the tangential friction force vector. Coulomb’s friction law is used to judge the slip/stick states, and the relative positional relationship between slave surface nodes and master surface line elements is used to judge whether contact occurs. The contact system adopts master–slave contact element, where the casing surface is defined as the master surface and the packer rubber surface as the slave surface.

The penalty parameter ${ϵ}_N$ is not only a mathematical approximation of the Lagrange multiplier; its physical essence can be regarded as a “virtual spring stiffness” applied to the contact interface. The selection of this parameter faces a trade-off between numerical stability and computational accuracy: a value that is too small leads to excessive non-physical penetration, while a value that is too large may cause ill-conditioning of the stiffness matrix, leading to iteration non-convergence. The penalty parameters ${ϵ}_N$ (normal stiffness) and ${ϵ}_T$ (tangential stiffness) are defined as:

$${ϵ}_N={ϵ}_T=s{\displaystyle \frac{E{A}_c}{V}}$$

(8)

where $s$ is a scaling factor, taken as 1000 in this paper, $E$ is Young’s modulus, $A_c$ is the contact area, and $V$ is the finite element volume.

Based on the above theoretical model and numerical solution strategy, we have established a full-scale three-dimensional tubing string finite element model suitable for complex wellbore injection working conditions. This model not only systematically reveals the stress distribution and response of the tubing string under various loads and boundary actions but also lays a solid foundation for subsequently introducing fluid and temperature fields to carry out fluid-thermal-solid multi-field coupling analysis, thereby providing a reliable theoretical basis for tubing string structural optimization design and wellbore safety evaluation.

2.2. Thermodynamics of Tubing String

The flow and heat transfer model of the fluid in the wellbore is shown in Figure 2. In this model, the heat transfer process is partitioned into three distinct regions: Region I corresponds to the heat transfer area of the fluid inside the tubing string; Region II encompasses the heat transfer zone across the tubing, annulus, casing and cement sheath; Region III involves the heat transfer area within the formation.

2.2.1. Heat Conduction Inside the Tubing

Region I is the flow and heat transfer region of the fluid inside the tubing. Its main characteristics are the high-speed flow of the fluid along the well in the depth direction and the significant axial heat transport caused thereby. In this region, the fluid presents a forced convection state driven by injection pressure, with axial energy transfer dominating and radial temperature gradients being relatively small. Therefore, the heat transfer inside the tubing can be approximately described as a one-dimensional problem dominated by the coupling of axial convection and axial heat conduction.

The heat transfer inside the tubing is mainly affected by the heat carried by the inflow and outflow of the fluid in the axial direction and the heat transferred by convection. Friction between the fluid and the pipe wall during flow also generates heat. The heat transfer model inside the tubing is expressed as follows [17]:

$$v_m{{\rho }_{m}c}_1{\displaystyle \frac{d{T}_1}{dz}}={\displaystyle \frac{2h_1\left(T_2-T_1\right)}{r_1}}+{\displaystyle \frac{Q_{f_1}}{\pi r_1^2}}+{\lambda }_1{\displaystyle \frac{d^2{T}_1}{d{z}^2}}-p_1{\displaystyle \frac{d{v}_m}{dz}}+c_1{\alpha }_J{\rho }_m{v}_m{\displaystyle \frac{d{p}_1}{dz}}$$

(9)

where $T_1$ refers to the inner wall temperature of the tubing, $T_2$ refers to the outer wall temperature of the tubing, ${\lambda }_1$ is the thermal conductivity of the fluid, ${\alpha }_J$ is the Joule-Thomson coefficient of the fluid, ${\rho }_m$, $v_m$, and $c_1$ are the density, flow velocity, and specific heat capacity at constant pressure of the fluid, respectively. The assumption of one-dimensional heat conduction inside the tube is valid under specific engineering boundaries and practical injection conditions. To illustrate, the assumption is applicable to vertical and deviated wells with small well deviation and high injection rates.

$Q_{f_1}$ and $d$ are defined as follows:

$$Q_{f_1}={\displaystyle \frac{{\rho }_m{v_m}^2}{4r_1}}dqf$$

(10)

$$d={\displaystyle \frac{\sqrt{\frac{{{\rho }_m\lambda }_1{c}_1}{{{\rho }_2\lambda }_2{c}_2}}+1}{\sqrt{\frac{{{\rho }_m\lambda }_1{c}_1}{{{\rho }_2\lambda }_2{c}_2}}}}$$

(11)

where $q$ is the fluid injection rate, and $f$ is the friction coefficient.

2.2.2. Heat Conduction Inside the Annulus Fluid

Heat transfer in the annular fluid is primarily caused by axial heat conduction and convective heat transfer. Since the fluid velocity in the annulus is far lower than that in the tubing, its axial energy transport capability is relatively weak, but it cannot be ignored under high temperature difference conditions. Therefore, the annular fluid is regarded as a continuous medium undergoing coupled axial heat conduction and convection along the well depth direction. The heat transfer model of the annular fluid is [18]:

$$\begin{array}{c}{\lambda }_3{\displaystyle \frac{d^2{T}_3}{d{z}^2}}+{\displaystyle \frac{2r_3\left(T_4-T_3\right)h_3}{r_3^2-r_2^2}}+{\displaystyle \frac{2r_2\left(T_2-T_3\right)h_2}{r_3^2-r_2^2}}=0\end{array}$$

(12)

where $T_3$ refers to the inner wall temperature of the casing, $r_2$ and $r_3$ are the outer radius of the tubing and the inner radius of the casing, respectively. $h_2$ and $h_3$ are the heat transfer coefficients at corresponding positions. In the axial direction, there is a temperature gradient between the annular fluid and the upper and lower adjacent nodes, forming an additional axial heat conduction flux.

2.2.3. Radial Heat Conduction of the Tubing String

Radial heat conduction of the tubing string is the core channel for heat exchange between the fluid inside the wellbore and the annulus, playing a key role in bridging the temperature fields of Region I and Region II. Due to the high thermal conductivity of the tubing string material, the heat conduction process can establish a stable temperature gradient in a short amount of time; therefore, this process can be approximately regarded as quasi-steady-state heat transfer. Meanwhile, to more accurately reflect the wellbore injection working conditions, axial heat transfer inside the tubing string is also considered, so that the temperature field within the tubing string wall is jointly controlled by radial conduction and axial heat conduction.

The heat transfer inside the tubing mainly includes convective heat transfer between the pipe and the internal fluid as well as the annular fluid [19], and also includes axial heat conduction and the additional heat generated by friction between the fluid and the pipe wall. The heat transfer model inside the pipe is:

$${\displaystyle \frac{2r_1\left(T_1-T_2\right)h_1}{r_2^2-r_1^2}}+{\lambda }_2{\displaystyle \frac{d^2{T}_2}{d{z}^2}}+{\displaystyle \frac{2r_2\left(T_3-T_2\right)h_2}{r_2^2-r_1^2}}+{\displaystyle \frac{Q_{f_2}}{\pi \left(r_2^2-r_1^2\right)}}=0$$

(13)

where $T_4$ refers to the outer wall temperature of the casing. $Q_{f_2}$ is defined as follows:

$$Q_{f_2}={\displaystyle \frac{{\rho }_m{v}_m^2}{4r_1}}{\displaystyle \frac{q_f}{\sqrt{\frac{{{\rho }_m\lambda }_1{c}_1}{{{\rho }_2\lambda }_2{c}_2}}+1}}$$

(14)

2.2.4. Casing Heat Conduction

The casing is located between the tubing string and the cement sheath and is an important heat transfer and isolation layer in the wellbore structure. Its heat transfer process has an important impact on the overall temperature distribution of the wellbore. In the model, the casing is regarded as a continuous, uniform cylindrical solid, and its heat transfer is mainly achieved through two ways: axial heat conduction and radial heat conduction.

In the radial direction, the inner wall of the casing receives heat from the annular fluid and transfers it outward to the cement sheath through the casing wall. In the axial direction, the temperature inside the casing changes along the well depth direction due to the influence of the injection fluid and the geothermal gradient, thereby forming an additional axial heat flux. The heat conduction model of the casing can be described as [20]:

$${\displaystyle \frac{2r_3{h}_3\left(T_3-T_4\right)}{r_4^2-r_3^2}}{+\lambda }_4{\displaystyle \frac{d^2{T}_4}{d{z}^2}}+{\displaystyle \frac{4\left(T_5-T_4\right)r_4{\lambda }_5}{\left(r_4^2-r_3^2\right)\left(r_5-r_3\right)}}=0$$

(15)

where $T_5$ refers to the outer wall temperature of the cement sheath, $r_4$ is the outer radius of the casing. ${\lambda }_{45}$ is defined as follows:

$${\lambda }_{45}={\displaystyle \frac{\left(r_5-r_3\right){\lambda }_4{\lambda }_5}{{\lambda }_5\left(r_4-r_3\right)+{\lambda }_4\left(r_5-r_4\right)}}$$

(16)

where $r_5$ is the outer diameter of the cement sheath, and ${\lambda }_i$ is the thermal conductivity at the corresponding position.

2.2.5. Heat Transfer Model of Cement Sheath and Formation

Region III includes the cement sheath and the formation outside it, which is the main region for heat diffusion and dissipation outward in the wellbore-formation system. Due to the huge volume and large heat capacity of the formation, its temperature change has obvious hysteresis relative to the internal wellbore process. As a transition medium between the casing and the formation, the cement sheath mainly undertakes radial and axial heat conduction roles. After heat is transferred from the outer wall of the casing into the cement sheath, it continues to diffuse into the interior of the formation [21]. The heat transfer model for the cement sheath and formation is expressed as follows:

$$\left\{\begin{array}{c}{\displaystyle \frac{4r_{j-1}{\lambda }_{j-0.5}\left(T_{j-1}-T_j\right)}{\left(r_j^2-r_{j-1}^2\right)\left(r_j-r_{j-2}\right)}}+{\lambda }_j{\displaystyle \frac{d^2{T}_j}{d{z}^2}}+{\displaystyle \frac{4r_j{\lambda }_{j+0.5}\left(T_{j+1}-T_j\right)}{\left(r_j^2-r_{j-1}^2\right)\left(r_{j+1}-r_{j-1}\right)}}=0,j=5, 6\\ {\lambda }_{j-0.5}={\displaystyle \frac{\left(r_j-r_{j-2}\right){\lambda }_{j-1}{\lambda }_j}{\left(r_{j-1}-r_{j-2}\right){\lambda }_j+\left(r_j-r_{j-1}\right){\lambda }_{j-1}}}\\ {\lambda }_{j+0.5}={\displaystyle \frac{\left(r_{j+1}-r_{j-1}\right){\lambda }_j{\lambda }_{j+1}}{\left(r_{j+1}-r_j\right){\lambda }_j+\left(r_j-r_{j-1}\right){\lambda }_{j+1}}}\end{array}\right.$$

(17)

The formation is assumed to be an infinite medium. Its initial temperature satisfies the geothermal gradient distribution along the well depth direction. The formation temperature far away from the wellbore remains unchanged during the calculation process, thereby providing stable external boundary conditions for heat transfer.

2.3. Fluid Flow Inside the Tubing String

During the wellbore injection operation, the fluid flows continuously along the axial direction of the tubing string from the wellhead to the bottom. Its flow state directly determines the pressure distribution within the wellbore, temperature evolution, and the form of loads generated on the tubing string structure. Therefore, it is necessary to establish a mechanical model of fluid flow inside the tubing string to describe the pressure changes and flow characteristics of the fluid during the injection process [22].

Fluid flow is governed by the mass conservation equation, whose one-dimensional axial form is:

$${\rho }_m{\displaystyle \frac{\mathsf{\partial }{v}_m}{\mathsf{\partial }z}}+v_m{\displaystyle \frac{\mathsf{\partial }{\rho }_m}{\mathsf{\partial }z}}=0$$

(18)

Considering one-dimensional energy conservation, the pressure drop for fluid flowing in the tubing string is [23,24,25]:

$${\displaystyle \frac{dp}{dz}}={\rho }_{m}g-{\displaystyle \frac{{\rho }_m{v}_{m}d{v}_m}{dz}}-{\displaystyle \frac{f{\rho }_m{v_m}^2}{2d_t}}$$

(19)

where ${\rho }_m$ is the density of the fluid in the tubing string, $v_m$ is the flow rate of the fluid in the tubing string, $z$ is the axial coordinate of the tubing string, $f$ is the friction coefficient, and $d_t$ is the inner diameter of the tubing.

Under injection working conditions, the fluid in the tubing string can be regarded as a one-dimensional flow medium along the well depth direction. Its pressure change mainly comes from the combined effects of gravity, flow friction, and fluid acceleration [26,27,28]. The pressure drop caused by the fluid’s own weight is:

$$D_1={\rho }_{m}g$$

(20)

The pressure drop caused by viscous friction between the fluid and the inner wall of the tubing string is:

$$D_2=-f{\displaystyle \frac{{\rho }_m{v_m}^2}{2d_t}}$$

(21)

The pressure drop in vertical direction caused by changes in fluid density or flow velocity is:

$$D_3=-{\displaystyle \frac{{\rho }_m{v}_{m}d{v}_m}{dz}}={\displaystyle \frac{dp}{dz}}{\displaystyle \frac{{\rho }_m{v_m}^2}{p}}$$

(22)

Substitute Equation (22) into Equation (19), the pressure drop for the fluid along the well depth direction in the tubing string is given as follows:

$${\displaystyle \frac{dp}{dz}}={\displaystyle \frac{{\rho }_{m}g-f\frac{{\rho }_m{v_m}^2}{2d_t}}{1-\frac{{\rho }_m{v_m}^2}{p}}}$$

(23)

2.4. Solution of THM Coupled Equation System

The wellbore injection process is essentially a highly nonlinear multi-physics and strongly coupled problem. During the wellbore injection process, there are significant mutual coupling effects among fluid flow inside the tubing string, tubing string structural deformation, and temperature field evolution, as shown in Figure 3. Changes in fluid pressure and temperature directly affect the loads borne by the tubing string and the material’s thermal expansion behavior, while the axial deformation and torsion of the tubing string in turn change the fluid flow channel and heat transfer path, thereby forming a strongly nonlinear THM coupling problem. As shown in Figure 4, based on the aforementioned fluid flow model, tubing string mechanical model, and heat transfer model, this study integrates the three types of physical fields into a unified solution framework, constructs a THM coupled governing equation system, and adopts the Newton iteration method for numerical solution [29,30,31,32].

In the numerical solution, the finite element method is adopted to discretize spatial variables. At time step $t_{i+1}$, the THM coupled system can be uniformly written in residual form:

$$\mathbf{R}\left(\mathbf{x}\right)=\left[\begin{array}{c}{\mathbf{R}}_u\left(u,p,T\right)\\ {\mathbf{R}}_p\left(u,p,T\right)\\ {\mathbf{R}}_T\left(u,p,T\right)\end{array}\right]=\mathbf{0}$$

(24)

where $\mathit{x}=[\mathit{u},\mathit{p},\mathit{T}{]}^T$ is the unknown variable vector. The corresponding tangent matrix is:

$${\mathbf{K}}_{\mathrm{t}\mathrm{a}\mathrm{n}}^{\left(i\right)}={\left.{\displaystyle \frac{\mathsf{\partial }\mathbf{R}}{\mathsf{\partial }\mathit{x}}}\right|}_{{\mathbf{x}}^{\left(i\right)}}$$

(25)

By solving the linearized equation:

$${\mathbf{K}}_{\mathrm{t}\mathrm{a}\mathrm{n}}^{\left(i\right)}\mathsf{\Delta }{\mathit{x}}^{\left(i\right)}=-{\mathbf{R}}^{\left(i\right)}$$

(26)

Finally, update the solution vector:

$${\mathit{x}}^{\left(i+1\right)}={\mathit{x}}^{\left(i\right)}+\mathsf{\Delta }{\mathit{x}}^{\left(i\right)}$$

(27)

The mark of iteration convergence is that the displacement, pressure, and temperature variables between two adjacent iterations satisfy [33,34,35,36]:

$$\left\{\begin{array}{c}{‖{\mathit{u}}^{\left(i+1\right)}-{\mathit{u}}^{\left(i\right)}‖}_2\le {\delta }_u\\ {‖{\mathit{p}}^{\left(i+1\right)}-{\mathit{p}}^{\left(i\right)}‖}_2\le {\delta }_p\\ {‖{\mathit{T}}^{\left(i+1\right)}-{\mathit{T}}^{\left(i\right)}‖}_2\le {\delta }_T\end{array}\right.$$

(28)

where the value of ${\delta }_u$, ${\delta }_p$ and ${\delta }_T$ are set to 1 × 10⁻⁶. The values of 1 × 10⁻⁶ for displacement, pressure, and temperature are sufficiently small to fully capture the subtle THM coupled response characteristics of the tubing string and packer system, ensuring high fidelity of numerical results. If the tolerance is set too small (e.g., 1 × 10⁻⁸ or smaller), the Newton iteration will require excessive steps, leading to a sharp increase in computing time and resource consumption.

To improve the numerical stability of time integration and suppress high-frequency oscillations, the generalized-α scheme is introduced to update the field variables [37,38,39,40]:

$$\left\{\begin{array}{c}{\mathit{u}}_j^{n+1}=\left(1-{\alpha }_m\right){\mathit{u}}_j^n+{\alpha }_m{\mathit{u}}_j^{n-1}\\ {\mathit{p}}_j^{n+1}=\left(1-{\alpha }_p\right){\mathit{p}}_j^n+{\alpha }_p{\mathit{p}}_j^{n-1}\\ {\mathit{T}}_j^{n+1}=\left(1-{\alpha }_T\right){\mathit{T}}_j^n+{\alpha }_T{\mathit{T}}_j^{n-1}\end{array}\right.$$

(29)

When $t_j+\mathsf{\Delta }t\ge t_{\mathrm{e}\mathrm{n}\mathrm{d}}$, the numerical calculation terminates. Figure 4 shows the finite element solution process.

In summary, by coupling and solving the energy conservation equation, fluid mass conservation equation, and tubing string structural equation within a unified framework, and combining Newton iteration and generalized-α time integration methods, synchronous calculation of fluid flow, temperature evolution, and tubing string structural response during the wellbore injection process can be achieved, providing a reliable numerical basis for subsequent working condition analysis and safety assessment.

3. Numerical Results

3.1. Model Verification

3.1.1. Verification Example 1

To verify the accuracy and reliability of the THM coupled model established in this study in describing the thermal–mechanical response of the tubing string under wellbore injection working conditions, a typical wellbore structure is selected as a verification example for comparative analysis of calculation results. In the finite element modeling process, a parametric approach is used to construct the physical model of the wellbore cross-section. From inside to outside, it includes the production casing, technical casing, conductor pipe, and cement sheath. The geometric dimensions and material parameters of each layer are set according to the actual wellbore structure, with specific parameters shown in

Table 1. Parameters for the Verification Cases.

Parameter	Tubing	Casing	Cement Sheath
Outer Radius (m)	0.07302	0.1397	0.3233
Thickness (m)	0.00551	0.00917	0.0389
Length (m)	3900	3900	3900
Elastic Modulus (GPa)	206	200	14
Poisson’s Ratio	0.29	0.3	0.26
Density (kg/m3)	7850	7850	3150
Thermal Conductivity (W/m·K)	50	44.77	0.71
Thermal Expansion Coefficient (K−1)	1.2 × 10−5	1.19 × 10−5	10 × 10−5

.

Regarding boundary conditions, an axial load is applied at the upper end of the model to simulate the effect of tubing string self-weight and additional fluid forces, and an axial displacement constraint is applied at the lower end. A static pressure load is applied to the inner wall of the casing to reflect the influence of injection pressure, and the cement sheath provides continuous radial support to the outer wall of the casing. Thermal boundary conditions are introduced through the steady-state thermal analysis module. The wellbore temperature is distributed along the well depth direction according to the geothermal gradient, and the temperature of the upper end section of the current well section is input into the model as the reference temperature. Geometric nonlinear effects are also considered to more truly reflect the structural response under high-temperature and high-pressure conditions.

Under the above working conditions, the tubing string models at different well depths are solved to obtain the von Mises stress distribution cloud map and the displacement distribution cloud map in the depth direction of the production casing in the presence of defects in the cement ring, as shown in Figure 5a,b. Based on the results, the maximum equivalent stress at key positions is compared and analyzed with the calculation results of the aforementioned THM coupling theoretical model. Figure 6a,b show the comparison of the Mises stress on the inner and outer walls of the casing varying with well depth against the analytical solution. The obtained stress distribution laws and variation trends are highly consistent with the analytical solution. The maximum relative errors of the Mises stress on the inner and outer walls of the casing compared to the analytical solution are controlled within 2.43% and 4.98%, respectively.

Overall, the error level is within the allowable range for engineering. The verification results indicate that the THM coupling model can accurately describe the temperature-mechanical response characteristics of the tubing string during the wellbore injection process, possessing good engineering applicability and computational reliability. It can provide a credible theoretical basis for subsequent complex working condition analysis and parameter sensitivity studies.

3.1.2. Verification Example 2

This benchmark case investigates the contact between an elastic straight pipe and a rigid flat plane, aiming to validate the accuracy and robustness of the pipe-wall contact numerical model under pure mechanical loading conditions. The geometric and material parameters of the model are specified as follows: outer diameter of the pipe D = 114.3 mm, pipe thickness t = 6.0 mm, Young’s modulus E = 210 GPa, Poisson’s ratio ν = 0.30, and a vertical compressive load F = 100 kN applied along the vertical direction. The penalty-based contact algorithm is adopted for the simulation.

As shown in Figure 7b, the normalized contact pressure distribution obtained from the present numerical simulation is compared against the classical Hertz analytical contact solution [41]. The numerical results exhibit a perfect parabolic pressure profile that closely overlaps with the analytical curve across the entire contact region (−1.0 ≤ x/a ≤ 1.0), with the peak normalized pressure occurring at the contact center x/a = 0. This excellent consistency verifies that the established contact algorithm can accurately reproduce the fundamental contact stress field and contact mechanics behavior of the elastic tubular structure under normal compression.

Figure 7c demonstrates the evolution of normal contact penetration $\left|g_{Ns}\right|$ at the position x = 0 with respect to the logarithmic scaling factor lg(s). When the scaling factor s is relatively small, the contact penetration remains at a high level. As lg(s) increases, the magnitude of constraint violation decreases rapidly in a monotonic trend. Once the scaling factor exceeds a critical threshold, the penetration converges to an extremely low, near-stable residual value, indicating that a larger scaling factor significantly enhances constraint enforcement and effectively eliminates unphysical interface interpenetration.

Figure 7d further quantifies the computational performance variation with the scaling factor. It is observed that the total number of nonlinear solution iterations increases as the scaling factor grows. Higher scaling factor values improve contact accuracy and constraint satisfaction, but simultaneously increase the ill-conditioning of the global system stiffness matrix and contact nonlinearity, which leads to a gradual rise in iterative computational cost. This result clearly reveals the inherent trade-off between contact constraint precision and numerical computational efficiency for the adopted penalty-scaling contact approach. This confirms that s = 1000 is the appropriate choice for this full-scale THM coupled model.

3.2. Results Analysis

Based on the constructed model, the overall calculation process of the results analysis part is shown in Figure 8. The calculation process strictly follows the actual engineering operation sequence, and the specific implementation strategy is as follows:

First, basic wellbore and formation parameters such as well deviation, layer thickness, and geothermal gradient are input as initial and boundary conditions. In the packer setting stage, pressure loads of 0–12 MPa and 12–15 MPa are applied step-by-step, considering the casing contact effect and piston effect, respectively, and calculating the resulting tubing string displacement and axial force changes. Subsequently, entering the fluid injection stage, within the injection pressure range of 0–25 MPa, the fluid thermal expansion effect, as well as the spring rebound and bulging effects caused during the injection process, are also considered. Throughout the entire analysis process, the THM-coupled governing equations are used to synchronously solve the pressure field, temperature field, and structural displacement field. The Newton-Raphson algorithm is utilized to perform nonlinear iterations until the convergence criteria are met. Through this analysis process, the evolution laws of pressure, temperature, and tubing string mechanical response during the entire wellbore injection and packing process can be obtained within a unified numerical framework, laying the foundation for subsequent working condition analysis and parameter sensitivity studies.

To ensure the accuracy of the numerical simulation, the key structural dimensions and material physical property parameters are shown in

Table 2. Wellbore Structure and Material Parameters.

Parameter	Tubing	Casing	Cement Sheath
Inner Radius (m)	0.0889	0.1525	0.2455
Thickness (m)	0.00952	0.0253	0.0389
Length (m)	4000	4000	4000
Elastic Modulus (GPa)	206	200	14
Poisson’s Ratio	0.29	0.3	0.26
Density (kg/m3)	7850	7850	3150
Thermal Conductivity (W/m·K)	50	44.77	0.71
Thermal Expansion Coefficient (K−1)	1.2 × 10−5	1.19 × 10−5	10 × 10−5

and

Table 3. Formation and single-phase Carbon Dioxide Parameters.

Parameter Group	Input Parameter	Value
Formation	Surface Temperature (K)	281
	Geothermal Gradient (K/km)	30
	Formation Thermal Diffusivity (m2/s)	1.0 × 10−6
Carbon Dioxide	Injection Temperature (K)	300
	Injection Pressure (MPa)	25
	Density (kg/m3)	600
	Thermal Conductivity (W/m·K)	0.1
	Thermal Expansion Coefficient (K−1)	2 × 10−4

, constructing a unified THM coupling analysis model. Throughout this study, single-phase carbon dioxide is adopted as the injected fluid for the entire fracturing process, with its physical parameters treated as constant values. The employment of temperature- and pressure-dependent (T–P dependent) CO₂ parameters would introduce significant nonlinearity into the numerical model, which is prone to induce convergence challenges in full-scale finite element simulations involving contact nonlinearity. By contrast, the use of constant CO₂ parameters ensures robust iterative convergence and enables reliable characterization of the THM coupling evolution trends. Given that the core focus of this paper lies in investigating the tubing string’s displacement, axial force, and stress distributions, as well as the contact mechanical behavior of the packer under thermal and pressure loads—rather than achieving high-precision prediction of CO₂ physical parameters—constant parameters are deemed sufficient to capture both the qualitative characteristics and engineering-level quantitative trends of the aforementioned mechanical responses. The thermal model is valid for single-phase CO₂ within a narrow T–P range as used in this study. For water, the variation in thermophysical properties is negligible, and thus the constant-property assumption incurs only a marginal error. The parameters in the table can simultaneously reflect the mechanical deformation characteristics of the wellbore structure under pressure and heating conditions, as well as heat conduction and thermal expansion behaviors, providing a unified physical property basis for THM coupling calculations.

3.2.1. Setting Process Analysis

The setting process is a typical evolution process coupling large geometric deformation and contact nonlinearity. In the initial stage, where the setting pressure is 0–12 MPa, the packer rubber element gradually undergoes axial compression and radial expansion but has not yet formed effective contact with the inner wall of the casing. When the setting pressure is below approximately 5 MPa, the rubber element is in a state of free expansion. The overall displacement of the packer is mainly determined by the axial compressive deformation of the tubing string and the fluid static pressure load. In this stage, the axial force borne by the packer mainly comes from the piston effect generated by the pressure applied at the wellhead and the fluid self-weight term in the pipe. There is no obvious frictional force action between the rubber element and the casing. Packer vertical displacement increases linearly by 491 mm as setting pressure rises, as shown in Figure 8 and Figure 9a.

When the setting pressure further increases and approaches 5 MPa, the system enters the contact establishment period. The outer surface of the rubber element begins to make initial contact with the inner wall of the casing. At this time, the contact area is small, and the frictional force has not been fully established, but local stress has obviously concentrated in the contact area of the rubber element. As shown in Figure 10b,c, as the pressure gradually rises to 12 MPa, the contact area between the rubber element and the casing continues to expand, and the axial displacement, contact force, and local equivalent stress of the packer continue to increase. At this time, the tubing string undergoes a bulging effect, producing radial expansion, accompanied by significant changes in axial force and axial displacement, as shown in Figure 9c. Since the rubber element has just entered the contact state in this stage, the frictional force is still in the process of gradual establishment; therefore, the overall force and displacement changes still exhibit relatively smooth evolution characteristics. The overall force and displacement response show nonlinear characteristics of a smooth transition from “free deformation” to “constrained deformation”.

When the setting pressure enters the reinforced setting stage of 12–15 MPa, the rubber element and the inner wall of the casing have formed full contact, and the frictional force increases rapidly and becomes one of the dominant force factors. In this stage, the axial force of the packer is composed of two parts: one is the piston effect caused by the wellhead pressure; the other is the axial frictional force converted from the radial contact pressure between the rubber element and the casing. Since the space between the two packers is filled with liquid, as the setting pressure continues to rise, the piston effect is significantly enhanced, leading to a rapid accumulation of axial force in the packer. As shown in Figure 11c, under a setting pressure of 15 MPa, the resultant force at the packer can reach approximately 617 kN.

From the perspective of displacement characteristics, the increase in axial displacement of the packer in the 12–15 MPa stage is significantly reduced, as shown in Figure 12a. This indicates that the rubber element has entered a strong constraint state, and the overall stiffness of the system has significantly improved. Meanwhile, the force and temperature at the packer grow with the setting pressure to 615 kN and 366.5 K, respectively, as shown in Figure 12b,c, indicating that the mechanical response in this stage has completely transformed into a strong coupling mode controlled by contact friction, fluid pressure, and thermal effects.

In summary, the 0–12 MPa stage mainly reflects the setting establishment process of the rubber element from free expansion to initial contact, while the 12–15 MPa stage corresponds to the reinforced setting process dominated by frictional force, hydraulic force, and thermal effects after the rubber element is fully contacted, providing a reliable initial state for the mechanical response analysis of the subsequent fluid injection stage.

3.2.2. Setting Stop Process Analysis

After completing the 15 MPa setting pressure loading and forming a stable setting state, the system enters the setting stop stage. The main characteristic of this stage is the gradual removal of the active setting pressure at the wellhead, while the contact and friction relationships already established between the rubber element and the casing still exist. The tubing string system will produce an obvious rebound response during the unloading process. This process involves the combined effects of pressure unloading, friction constraints, and temperature and volume effects, and is a key link in evaluating the long-term stability of the packer and wellbore safety.

When the setting pressure is removed from 15 MPa, the piston effect applied at the wellhead rapidly weakens, and the tubing string undergoes axial retraction driven by the accumulated elastic potential energy. However, since the rubber element has undergone significant radial expansion and formed high-stress contact with the casing during the high-pressure setting stage, its contact pressure does not disappear with the removal of the wellhead pressure, thereby retaining a large frictional resistance at the rubber element-casing interface. This frictional force forms a strong axial constraint on the axial rebound of the tubing string, making the unloading process exhibit obvious nonlinear characteristics. As shown in Figure 13, after removing the 15 MPa setting pressure, the resultant force at the packer can reach approximately 669 kN, marking that the system has formed a stable “self-locking anchoring” state.

From the perspective of displacement response results, the packer and the tubing string as a whole show an axial rebound trend to 400 mm in the setting stop stage, but the amplitude of the rebound displacement is significantly smaller than the cumulative displacement increment in the setting loading stage (Figure 14a), exhibiting the characteristic of “limited unloading rebound.” The axial force evolution results indicate that during the unloading process, the axial force at the packer decays rapidly to −670 kN with the setting pressure, mainly reflecting the influence of wellhead pressure release and the gradual disappearance of the piston effect (Figure 14b). From the perspective of the contact force between the packer and the casing, the radial contact force decreases somewhat with the removal of the setting pressure during the setting stop process (Figure 14c). However, since the rubber element has undergone full radial expansion during the setting loading stage and has formed a stable contact state with the casing, the contact force does not drop to zero synchronously. As the unloading is completed, the contact force tends to stabilize, and the system enters a mechanical equilibrium state dominated by friction constraints.

Comprehensive analysis shows that the setting stop stage is not a simple load removal process but a process where the tubing string undergoes limited rebound and re-achieves force-thermal equilibrium under the condition of rubber element-casing friction constraints. The model successfully captures this “limited rebound” mechanism and can capture key characteristics such as displacement recovery, residual axial force, and stress redistribution during the unloading process, providing a reliable initial state and mechanical basis for the packer stability analysis under subsequent injection working conditions.

3.2.3. Injection Process Analysis

In the liquid injection stage, the wellhead injection pressure gradually increases from 0 MPa to 25 MPa. Under the combined action of high-pressure instantaneous loading and injection cooling, the mechanical and thermal responses of the packer and tubing string exhibit significant THM coupling characteristics. Overall, the response laws in this stage are obviously different from the setting loading and unloading processes, with liquid pressure loads and temperature disturbances becoming the dominant factors.

As known from Figure 15, during the high-pressure liquid injection stage, the von Mises stress level in the contact area of the tubing string and packer increases significantly. The high-stress areas are still mainly concentrated in the contact section between the packer rubber element and the casing. Compared with the setting process, the stress distribution in this stage is more controlled by internal pressure loads and thermal shrinkage effects, forming obvious axial-hoop stress coupling characteristics under radial constraint conditions.

As known from Figure 16, after liquid injection, the wellbore temperature shows an overall downward trend, and the magnitude of the temperature drop is from 388 K to 375 K. This is the result of the combined effects of sudden fluid pressure changes, local flow acceleration, and heat exchange between the tubing string and the fluid. The temperature decrease not only directly affects the thermal deformation behavior of the tubing string but also indirectly strengthens the axial force and stress response by changing the material thermal expansion effect and fluid physical property parameters, reflecting typical THM strong coupling characteristics.

In summary, the mechanical response of the packer in the liquid injection stage is mainly controlled by the combined effects of high internal pressure, temperature drop, and fluid flow effects. Among them, the pressure effect dominates the overall force level, while the temperature effect plays a significant modification role in displacement and stress distribution. The above results indicate that the THM coupling model established in this study can reasonably characterize the multi-physics field coupling response characteristics of the packer and tubing string during the liquid injection process, providing a reliable basis for safety assessment and parameter optimization under high-pressure injection working conditions.

3.2.4. Injection Stop Process Analysis

During the injection stop process, the high-pressure injection condition at the wellhead is removed, the liquid flow inside the wellbore gradually weakens and finally stops, the internal pressure load borne by the tubing string decreases significantly, and at the same time, due to the absence of continuous liquid injection cooling, the wellbore temperature field begins to slowly rise. In this stage, the tubing string structural response mainly manifests as two physical processes with vastly different time scales: the rebound process under the combined action of transient pressure unloading and quasi-static temperature rise. The mechanical response characteristics in this stage are mainly determined by the elastic rebound induced by pressure unloading and the thermal stress release caused by temperature hysteresis.

From the perspective of displacement response results, after the liquid injection stops, the tubing string as a whole shows an axial rebound trend. It can be seen from the simulation results in Figure 17 that after the liquid injection stops, a total axial tensile force of approximately 752 kN is still maintained at the packer.The tubing string undergoes axial retraction, and the displacement at the packer gradually decreases to 400 mm, as shown in Figure 18a. In terms of axial force evolution (Figure 18b), although the unloading of internal pressure eliminates most of the fluid piston force, leading to an overall downward trend in axial force, the packer still retains the above-mentioned axial tensile force. The physical cause lies in the fact that at this time, the tubing string is still in a low-temperature shrinkage state (Figure 18d shows that the temperature has not fully recovered), and the huge cold-shrinkage thermal stress is “locked” inside the tubing string by the friction anchoring effect of the rubber element, forming high-intensity residual thermal tension.

From the perspective of the contact force between the packer and the casing, the radial contact force decreases somewhat with the removal of the wellhead pressure during the liquid injection stop process (Figure 18c). The maximum value of the contact stress is 7.7 MPa. In terms of temperature response, after the liquid injection stops, the wellbore is no longer subjected to the continuous action of cold liquid, and the temperature field begins to slowly rise towards the formation temperature, as shown in Figure 18d. However, due to the huge heat capacity of the formation, the thermal recovery process exhibits obvious hysteresis, which also explains why the residual thermal stress can be maintained for a long time.

Comprehensive analysis shows that the liquid injection relief stage is a critical period for the reconstruction of the mechanical state of the tubing string. Pressure unloading dominates the macroscopic displacement rebound behavior, while the lagging recovery of the temperature field dominates the high-level maintenance of residual stress. Under the combined action of the two, the tubing string reaches a new stress equilibrium state at the packer, verifying that under the liquid injection stop working condition, the packer still possesses reliable mechanical stability and sealing integrity.

4. Conclusions

This paper focuses on the THM multi-field coupling problem of the tubing string under wellbore injection working conditions and constructs a multi-field coupled numerical model based on full-scale geometric characteristics. This model achieves the synergistic solution of fluid flow, temperature field evolution, and tubing string structural mechanics within a unified computational framework. Fluid behavior is described through equations such as mass conservation and energy conservation, while tubing string mechanical response is characterized by structural equations considering axial deformation, thermal stress, and contact nonlinear boundary conditions. In terms of numerical implementation, the finite element method is adopted to discretize spatial variables, and combined with the Newton iteration strategy, the nonlinear THM coupled equation system is solved stably and efficiently. This method can synchronously obtain key response quantities such as temperature within the wellbore as well as tubing string displacement and stress, providing reliable theoretical and numerical tools for mechanical and thermal analysis of the wellbore under complex injection and setting working conditions.

Based on the above model and case study analysis, the following main conclusions can be drawn:

(1)

Accuracy and Applicability of the Model: The established THM multi-field coupled full-scale model can reasonably describe the interactions among fluid pressure, temperature field, and tubing string structural response during processes such as wellbore setting and injection. The high degree of agreement between numerical results and theoretical solutions proves the accuracy and applicability of the model in handling complex working condition analysis.

(2)

Contact Nonlinearity and Anchoring Mechanism: In the setting loading stage, strong contact and high friction constraints are gradually formed between the packer rubber element and the casing, significantly improving the overall stiffness of the system. In the setting stop stage, although the wellhead pressure is removed, the residual contact force and frictional force can still effectively limit the tubing string rebound, anchoring the packer to maintain a stable sealing state.

(3)

THM Strong Coupling Effects: During the liquid injection loading process, the combined action of high-pressure injection and cooling effects causes a significant increase in the axial force and stress levels of the tubing string. The temperature effect has an important modulation effect on the structural response, which is particularly obvious under high-pressure working conditions. After the liquid injection stops, the tubing string as a whole exhibits a gentle rebound characteristic, and the axial force and stress levels gradually decrease. Residual contact force and friction constraints are key factors in maintaining the stability of the wellbore structure.

(4)

Necessity of Full-Scale Modeling: Research results indicate that the contact behavior between the packer and the casing and its evolution laws have a decisive impact on wellbore safety. Full-scale analysis can restore boundary effects and cumulative deformation along the path that are ignored by simplified models. It is a necessary means to evaluate the working reliability of the packer and the long-term service performance of the tubing string.

(5)

The proposed model adopts Newton iteration combined with the generalized-α time integration scheme, which ensures strong convergence and good computational efficiency. Under the given convergence tolerance (1 × 10⁻⁶), typical working condition simulations converge within 10–15 iteration steps. The full-scale modeling does not cause excessive computational cost because the contact algorithm and finite element discretization are optimized for wellbore structures. The proposed method achieves high engineering realism while maintaining acceptable computing time and resource consumption, making it suitable for practical wellbore injection analysis.

The series of conclusions and numerical analysis of the THM multi-field coupled full-scale model provides a systematic and reliable theoretical basis and technical support for wellbore injection process optimization, packer structural innovative design, and long-term wellbore integrity evaluation from the perspective of engineering practice. Specifically, the research conclusions can directly guide the precise matching of key process parameters such as injection pressure, injection rate, and setting load, helping to avoid risks such as tubing string stress concentration and rebound instability caused by high-pressure injection and cooling effects. In addition, it provides targeted design directions for packer rubber element material selection, structural morphology optimization, and contact interface performance improvement, ensuring the long-term sealing reliability of the packer under complex working conditions. Furthermore, it simultaneously provides core theoretical support for the improvement of the wellbore full-life-cycle integrity evaluation system, laying a solid technical foundation for the safe and efficient implementation of wellbore engineering in fields such as oil and gas development and carbon sequestration. It has important theoretical value and engineering practical significance for promoting the development of wellbore engineering towards refinement and safety.