Stress Evolution and Integrity Evaluation of Cement Sheath Under Alternating Temperature–Pressure Coupled Loads During Multi-Stage Fracturing in Shale Gas Wells

Abstract

Based on measured data from a shale gas well, this study develops a wellbore temperature cycle model and a temperature–pressure coupled finite element model to evaluate cement sheath stress during multi-stage fracturing. Dynamic temperature and pressure boundaries are applied to calculate radial and tangential stresses, while cumulative mechanical degradation and failure modes are assessed using the modified Mohr–Coulomb criterion. The results show that cement sheath temperature changes significantly, and stresses vary periodically with fracturing stages. The injection period is the most critical stage for cement sheath failure. Lower casing pressure and reduced fracturing fluid displacement can improve stress distribution and reduce damage. Higher initial fluid temperature increases radial stress but decreases tangential stress, while shallower horizontal well depth weakens temperature–pressure coupling. Optimizing these parameters can mitigate cement sheath damage, enhance structural integrity, and ensure safe fracturing operations.

1. Introduction

As a crucial clean energy source, shale gas holds substantial importance in enhancing energy structures and mitigating greenhouse gas emissions [1,2,3]. However, shale gas reservoirs typically exhibit extremely low permeability and porosity [4,5], making traditional extraction methods economically unviable. In shale gas exploitation, multi-stage fracturing has emerged as a pivotal technique for effectively releasing trapped shale gas [6]. Multi-stage fracturing employs a segmented fracturing approach, creating multiple independent and complex fracture networks within the reservoir. This markedly expands the interfacial contact area between the formation and the wellbore, improving hydrocarbon flow performance [7,8,9]. Practical engineering experience has verified that multi-stage fracturing not only maximizes reservoir potential—significantly improving the initial production rates of shale gas wells—but also enhances ultimate recovery, providing crucial support for the economic development of shale gas resources [10,11]. However, multi-stage fracturing technology has revealed various challenges with the continuous advancement of shale gas exploration and development. The most prominent issue is that during the casing fracturing operations, the cement sheath is influenced by extremely high internal casing pressures and rapid temperature fluctuations, resulting in significant temperature–pressure coupling effects that compromise the cement sheath [12,13,14].

Furthermore, cement sheath failure creates migration channels for shale gas, enabling its upward flow and subsequent accumulation in the wellhead region, which induces sustained casing pressure (SCP) phenomena [15,16]. Previous studies indicate that over 70% of wells in specific shale gas fields exhibit varying degrees of SCP after fracturing [17]. This problem becomes increasingly severe during long-term production due to cumulative casing pressure buildup, posing a significant threat to wellbore integrity [18,19,20]. Hence, characterizing cement sheath failure behaviour under multi-stage fracturing loads is critical.

Extensive research has been conducted to address cement sheath integrity failure during shale gas well fracturing operations. Cement sheath failure primarily manifests as debonding, radial cracking, and plastic deformation [21]. Interfacial debonding primarily stems from combined mechanical and thermal stresses acting on the casing–cement or formation–cement boundaries, with these effects being significantly amplified in high-pressure and high-temperature (HPHT) environments. Meanwhile, the substantial net pressures applied to the casing during stimulation treatments commonly generate radial fractures and induce permanent deformation in the cement sheath. These cracks and deformations progressively compromise the structural integrity of the cement sheath, thereby degrading its sealing capacity. In-depth analyses reveal that the cyclic variations in wellbore pressure and temperature during multi-stage fracturing operations constitute the primary driver of these failure mechanisms [22,23,24]. Specifically, in multi-stage fracturing injection cycles, the rapid pressure surge caused by high-pressure fluid injection is followed by an abrupt pressure decline during shut-in periods, subjecting the cement sheath to cyclic loading. Concurrently, the injection of low-temperature fracturing fluids induces sudden wellbore cooling, followed by temperature recovery due to geothermal effects, creating dynamic thermal fluctuations. This temperature–pressure coupling effect generates cumulative damage within the cement matrix, exacerbating failure risks. Ultimately, this progressive deterioration compromises cement sheath integrity and facilitates gas migration.

Scholars have thoroughly analyzed cement sheath failure mechanisms in coupled temperature–pressure fields. Zhao et al. (2019) [17] created an elastoplastic model to analyze the distribution of the cement sheath stress, revealing failure mechanisms induced by internal pressure and temperature variations. The research indicated that tensile failure frequently occurs at the inner surface of the cement sheath. It was also found that an increase in the thickness of cement sheaths would correspondingly raise the likelihood of failure, which in turn gives rise to sustained casing pressure (SCP) concerns [17]. Yang et al. (2021) [25] developed an elastic model for the casing–cement sheath-formation system. The investigation evaluated the impact of wellbore thermo-pressure variations during fracturing and flowback phases on cement sheath integrity. Taking Jimsar shale oil wells as a case study, their simulations revealed that it is easy for the cement sheath to sustain damage in the fracturing fluid injection period. At the same time, micro-annulus damage dominates during flowback periods [25]. Yang et al. (2023) [26] explored how temperature changes, formation creep, and pressure variations impact the integrity of cement sheaths in natural gas storage operations. By developing a creep model that incorporates both stress and temperature factors and conducting finite element simulation analysis, the results demonstrated that the interaction involving formation deformation and thermal variations notably heightens the likelihood of cement sheath failure [26]. Lian et al. (2023) [23] conducted a comprehensive study on SCP issues resulting from cement sheath integrity loss in challenging thermal and pressure conditions in gas wells. By developing a temperature calculation model and a comprehensive temperature–pressure coupled numerical model for the integrated structure made up of the casing, cement sheath, and the surrounding formation, they systematically investigated the elements impacting the cement sheath integrity, placing special emphasis on the impacts of maximum production rate and wellhead annular pressure [23]. The results demonstrate that: (1) energy transfer during gas production induces temperature fluctuations in the cement sheath, where higher production rates lead to more significant temperature variations and, consequently, more severe integrity deterioration; (2) reducing both production rate and wellhead annular pressure can effectively mitigate stress conditions, with wellhead pressure exhibiting more pronounced influence that must be carefully controlled within an optimal range to ensure cement sheath integrity. Lu et al. (2023) [27] investigated cement sheath integrity in vertical well sections concerning post-fracturing SCP phenomena frequently observed in shale oil/gas wells. Through a series of compression tests on cement specimens under various temperatures and confining pressure conditions, the mechanical parameters of the cement sheath at various well depths were acquired, and the integrity of the cement sheath was evaluated [27]. Zhang et al. (2024) [28] established a novel geological-wellbore integrity assessment model that integrates the combined influence of thermal variations and heterogeneous in situ stress. After experimental validation of the model’s accuracy, they carried out in-depth evaluations of the behaviour of cement sheath under fluctuating temperature and pressure scenarios. The results highlight substantial deterioration in the cement sheath’s mechanical characteristics and sealing effectiveness when exposed to elevated temperature settings [28].

Most existing investigations have mainly focused on cement sheath integrity under relatively specific thermal or mechanical conditions. For example, some studies analyzed the stress distribution and failure mechanism of cement sheath under internal pressure or temperature variation, while others focused on thermo-pressure effects during fracturing-flowback, gas production, or gas storage processes. In addition, several studies evaluated cement sheath integrity in vertical well sections or under heterogeneous in situ stress conditions. These studies have provided important insights into cement sheath failure mechanisms. However, the dynamic characteristics of multi-stage fracturing have not been fully considered. In particular, the cyclic cooling–heating process induced by fracturing fluid injection and shut-in, the cyclic variation in casing pressure, and the progressive degradation of cement sheath mechanical properties with increasing fracturing stages have rarely been coupled in a unified stress evaluation framework.

Considering the preceding discussion, while numerous investigations have explored how temperature–pressure interactions affect the structural soundness of cement sheath, the majority of these studies have overlooked the temperature–pressure coupling effects that occur during multi-stage fracturing operations, as well as the effect of the fracturing stage on the cement sheath stress. They also failed to elucidate the stress variation pattern in cement sheaths when subjected to alternating loads. In reality, the periodic temperature variations in fracturing fluid and cyclic pressure changes in casing during multi-stage fracturing create complex temperature–pressure coupling loads. These coupling effects impact the casing directly and the cement sheath indirectly, leading to a gradual decline in the cement sheath’s mechanical properties as the fracturing stage increases. This situation significantly complicates the evaluation of cement sheath integrity in these circumstances. Consequently, the mechanism of cement sheath integrity failure during multi-stage fracturing is still poorly understood. There is an immediate need to develop an evaluation method for cement sheath integrity that considers the dynamic nature of multi-stage fracturing processes.

This research thoroughly examines the impact of fracturing fluid’s frictional heating and the effect of displacement changes on the casing wall’s convective heat transfer coefficient in the context of practical requirements for multi-stage fracturing engineering of horizontal wells. The finite difference approach developed a wellbore circulation temperature field model. Based on this, a numerical simulation method was formulated to calculate cement sheath stress under temperature–pressure coupling, and alternating loads were developed. By integrating analytical solutions with numerical computations, the dynamic temperature–pressure parameters output from the wellbore circulating temperature field model were applied as boundary conditions for the stress model. This approach enabled a systematic investigation into the evolution patterns of the stress dynamics, both radially and tangentially, within the cement sheath across various stages of multi-stage fracturing.

Additionally, sensitivity studies assessed how casing pressure, fracturing fluid displacement, fracturing fluid initial temperature, and well depth affect the cement sheath stress. The likelihood of cement sheath failure was evaluated quantitatively using a refined Mohr–Coulomb criterion. This research offers theoretical insights and practical recommendations for ensuring the cement sheath integrity during the multi-stage fracturing process.

2. Wellbore Circulation Temperature Field Model

During multi-stage hydraulic fracturing, the wellbore temperature field demonstrates transient evolution characteristics dominated by the combined thermal conduction between injected fluids and the formation. Specifically, the lower-temperature fracturing fluid is pumped into the wellbore during each injection stage and flows along the casing. The considerable thermal disparity between the injected fluid and the wellbore/surrounding formation results in ongoing heat transfer, causing a steady decline in wellbore temperature until injection ceases [29]. Once the injection halts, the wellbore temperature slowly regains warmth via heat transfer from the formation. This cyclical pattern of temperature variation persists through successive fracturing stages, concluding with the completion of the scheduled fracturing sequence.

To accurately characterize this complex heat transfer process, we developed a mathematical model for the wellbore circulating temperature field suitable for casing fracturing operations, which comprehensively considers the frictional heating effect generated by fracturing fluid flow, the influence of displacement variations on convective heat transfer coefficients, and the unique staged fracturing characteristics of multi-stage operations. The temperature–pressure profiles calculated by this horizontal well model serve as dynamic boundary conditions for the cement sheath stress calculation model, providing a theoretical basis for evaluating the cement sheath integrity [30,31].

Figure 1 shows the wellbore model and thermal profile for multi-stage fracturing, which illustrates the operational scheme for 20-stage fracturing. Each fracturing stage consists of two distinct phases: the fluid injection phase and the shut-in phase.

Throughout the injection phase, the fracturing fluid is introduced at the wellhead. It flows sequentially through the vertical segment, the heel of the horizontal section, and ultimately arrives at the toe of the horizontal section. When the first-stage fracturing begins, the fracturing fluid creates initial fractures in the formation near the toe area. This injection period lasts 4 h before transitioning into a 4 h shut-in period, after which the second-stage fracturing commences. Currently, the composite bridge plug separates the first-stage fracturing section from the second-stage fracturing section, and the fracturing fluid enters the target area of the second-stage fracturing while keeping the first-stage fracturing section closed. Similarly, during third-stage fracturing, both previous stages are isolated, resulting in a cumulative extension of shut-in time for the first-stage segment as the fracturing stage progresses. During injection periods, the low-temperature fracturing fluid causes formation temperature reduction, while during shut-in periods, heat conduction from the formation gradually restores temperatures in both the wellbore and near-wellbore zones. This cooling–heating cycle repeats with each additional fracturing stage until all 20 stages of fracturing operations are completed.

The cross-section of the heel assembly was selected as the research subject, with the cement sheath radially divided into five layers (i₁,i₂,i₃,i₄,i₅) to precisely analyze the dynamic temperature variations at the internal (i₁) and external (i₅) walls of the cement sheath. The casing fracturing heat transfer model demonstrates the temperature distribution and heat transfer direction between the fracturing fluid and the wellbore/near-well formation during the fracturing process. During casing fracturing operations, the key methods of heat transfer include forced convection caused by the movement of fracturing fluid and thermal conduction across the wellbore/near-well formation. Upon cessation of the pump during the well’s shut-in phase, with the fluid flow at a standstill, the heat transfer mechanism within the casing shifts from forced convection to natural convection.

Before conducting multi-stage hydraulic fracturing, the following fundamental assumptions must be established:

a. The wellbore is filled with fluid and has reached thermal balance before the initiation of injection, with the fluid temperature equaling the formation temperature;

b. Axial heat conduction is negligible during the injection period due to minimal heat transfer along the axial direction, allowing for the omission of both axial heat conduction and radial temperature variations;

c. The cement sheath forms a concentric, isotropic, and homogeneous structure centred around the casing axis;

d. The fracturing fluid is incompressible, undergoes no phase changes, and maintains constant thermodynamic properties regardless of temperature fluctuations;

e. During the injection period, the fracturing fluid maintains constant flow velocity and inlet temperature, with an instantaneous velocity drop to zero when shut-in occurs.

2.1. Multi-Stage Fracturing Injection Period

(1) The heat transfer equation within the casing

• where the subscripts f and c denote fracturing fluid and casing, respectively.

$$Q_f-{\rho }_f{q}_f{c}_f{\displaystyle \frac{\mathsf{\partial }{T}_f}{\mathsf{\partial }z}}-2\pi r_f{h}_f(T_f-T_c)={\rho }_f{c}_f\pi {r_f}^2{\displaystyle \frac{\mathsf{\partial }{T}_f}{\mathsf{\partial }t}}$$

(1)

On the basis of the assumption, the heat source comes from the viscous dissipation of fracturing fluid, and the heat flux generated during fluid injection into the wellbore can be expressed as:

$$Q_f=\Delta p_f{q}_f={\displaystyle \frac{F{\rho }_f{v_f}^2}{r_f}}$$

(2)

The friction factor is determined by the fluid state of fracturing fluid:

Under laminar flow conditions,

$$F=\mathrm{Re}/16$$

(3)

Under turbulent flow conditions,

$${\displaystyle \frac{1}{\sqrt{F}}}=-2\log ({\displaystyle \frac{\epsilon /2r_f}{3.7}}+{\displaystyle \frac{2.51}{\mathrm{Re}\cdot \sqrt{F}}})$$

(4)

The fracturing fluid is injected into the wellbore under controlled wellhead pressure conditions, leading to forced convection heat transfer within the casing. The coefficient for forced convection heat transfer ($h_f$) is calculated based on the Nusselt number ($Nu$), which is calculated based on the Reynolds number ($\mathrm{Re}$) and Prandtl number ($\mathrm{Pr}$). The values of these parameters are fully determined by known factors.

$$\begin{array}{l}{h}_f={\displaystyle \frac{Nu\cdot {\lambda }_f}{2r_f}}\\ Nu=0.023{\mathrm{Re}}^{0.8}{\mathrm{Pr}}^{1/3}\\ \mathrm{Pr}={\displaystyle \frac{c_f\cdot m{u}_f}{{\lambda }_f}}\\ \mathrm{Re}={\displaystyle \frac{2r_f{v}_f{\rho }_f}{m{u}_f}}\end{array}$$

(5)

As shown in Figure 1, the wellbore–formation system is simplified as a one-dimensional radial heat-transfer model, including the fracturing fluid, casing, cement sheath, and formation control volumes. The radial boundaries are denoted by r_f, r_c, r_ce, r_r1, r_r2, …, r_ri, while T_f, T_c, T_ce, and T_ri represent the average temperatures of the corresponding control volumes.

Accordingly, Equations (6)–(9) correspond to the radial heat-transfer structure shown in Figure 1.

(2) Casing wall heat transfer equation

$${\displaystyle \frac{4r_c{\lambda }_{c,ce}(T_{ce}-T_c)}{({r_c}^2-{r_f}^2)(r_{ce}-r_f)}}-{\displaystyle \frac{2r_f{h}_f(T_c-T_f)}{{r_c}^2-{r_f}^2}}={\rho }_c{c}_c{\displaystyle \frac{\mathsf{\partial }{T}_c}{\mathsf{\partial }t}}$$

(6)

(3) Cement sheath heat transfer equation

$${\displaystyle \frac{4r_{ce}{\lambda }_{ce,r1}(T_{r1}-T_{ce})}{({r_{ce}}^2-{r_c}^2)(r_{r1}-r_c)}}-{\displaystyle \frac{4r_c{\lambda }_{c,ce}(T_{ce}-T_c)}{({r_{ce}}^2-{r_c}^2)(r_{ce}-r_f)}}={\rho }_{ce}{c}_{ce}{\displaystyle \frac{\mathsf{\partial }{T}_{ce}}{\mathsf{\partial }t}}$$

(7)

(4) Formation heat transfer equation

$${\displaystyle \frac{4r_{r1}{\lambda }_e(T_{r2}-T_{r1})}{({r_{r1}}^2-{r_{ce}}^2)(r_{r2}-r_{ce})}}-{\displaystyle \frac{4r_{ce}{\lambda }_{ce,r1}(T_{r1}-T_{ce})}{({r_{r1}}^2-{r_{ce}}^2)(r_{r1}-r_c)}}={\rho }_e{c}_e{\displaystyle \frac{\mathsf{\partial }{T}_{r1}}{\mathsf{\partial }t}}$$

(8)

$${\displaystyle \frac{4r_{ri}{\lambda }_e(T_{ri+1}-T_{ri})}{({r_{ri}}^2-{r_{ri-1}}^2)(r_{ri+1}-r_{ri-1})}}-{\displaystyle \frac{4r_{ri}{\lambda }_e(T_{ri}-T_{ri-1})}{({r_{ri}}^2-{r_{ri-1}}^2)(r_{ri}-r_{ri-2})}}={\rho }_e{c}_e{\displaystyle \frac{\mathsf{\partial }{T}_{ri}}{\mathsf{\partial }t}}$$

(9)

Among them,

$$\begin{array}{l}{\lambda }_{c,ce}={\displaystyle \frac{{\lambda }_c{\lambda }_{ce}\ln (\frac{r_c+r_{ce}}{r_c+r_f})}{{\lambda }_c\ln (\frac{r_c+r_{ce}}{2r_c})+{\lambda }_c\ln (\frac{2r_c}{r_c+r_f})}}\\ {\lambda }_{ce,r1}={\displaystyle \frac{{\lambda }_{ce}{\lambda }_e\ln (\frac{r_{ce}+r_{r1}}{r_{ce}+r_c})}{{\lambda }_{ce}\ln (\frac{r_{ce}+r_{r1}}{2r_{ce}})+{\lambda }_e\ln (\frac{2r_{ce}}{r_{ce}+r_c})}}\end{array}$$

(10)

2.2. Multi-Stage Fracturing Shut-In Period

(1) The heat transfer equation within the casing

In the non-operational phase of multi-stage fracturing, fracturing fluid flow halts, and temperature changes are exclusively driven by radial natural convection between the fluid and the internal wall of the casing. As a result, the forced convection heat transfer coefficient ($h_f$) transitions to a natural convection coefficient (${h_f}^{\ast }$). This transformation simplifies the heat transfer equation to the following form:

$$-2\pi r_f{h_f}^{\ast }(T_f-T_c)={\rho }_f{c}_f\pi {r_f}^2{\displaystyle \frac{\mathsf{\partial }{T}_f}{\mathsf{\partial }t}}$$

(11)

(2) Casing wall heat transfer equation

As the dominant heat transfer mechanism in the casing shifts from forced convection to natural convection, the corresponding wall heat transfer equation reduces to:

$${\displaystyle \frac{4r_c{\lambda }_{c,ce}(T_{ce}-T_c)}{({r_c}^2-{r_f}^2)(r_{ce}-r_f)}}-{\displaystyle \frac{2r_f{h_f}^{\ast }(T_c-T_f)}{{r_c}^2-{r_f}^2}}={\rho }_c{c}_c{\displaystyle \frac{\mathsf{\partial }{T}_c}{\mathsf{\partial }t}}$$

(12)

During shut-in, it is only necessary to modify the heat transfer equation in the casing and the heat transfer equation of the casing wall, while all other heat transfer equations remain the same as during injection. Specifically:

$$\begin{array}{l}{h_f}^{\ast }={\displaystyle \frac{0.049{(Gr\mathrm{Pr})}^{(1/3)}{\mathrm{Pr}}^{0.074}{\lambda }_f}{2r_f}}\\ Gr={\displaystyle \frac{{r_f}^3{\rho }_{f}g{\beta }_f(T_f-T_c)}{m{u_f}^2}}\\ \mathrm{Pr}={\displaystyle \frac{c_f\cdot m{u}_f}{{\lambda }_f}}\end{array}$$

(13)

2.3. Model Solution

The first step in solving the wellbore circulating temperature field model involves discretization of the model. A fully implicit finite difference formulation was implemented to numerically solve the coupled heat transfer equations throughout the multi-region system to enhance numerical stability and accuracy. This approach effectively eliminates strict limitations on time step sizes. A numerical solution programme was developed based on the discretized wellbore circulating temperature field model. Figure 2 presents the computational domain, which incorporates the complete wellbore media and its surrounding formation zone.

Discretization of the full computational domain was achieved through a structured, non-uniform grid configuration to accommodate different regions’ physical characteristics and computational accuracy requirements [32].

Wellbore Region: Given its relatively simple structure and small radial dimensions, the radial direction was discretized into a single layer of grids. The grid spacing was appropriately determined based on the wellbore’s structural and flow characteristics. This discretization strategy effectively captures heat transfer processes within the wellbore while reducing computational complexity.

Near-Wellbore Formation Region: The radial grid spacing increases exponentially with distance from the wellbore. Specifically, starting from the centre of the fracturing string, the grid step size expands following an exponential function as the radial distance increases. This continues until a sufficient distance is reached, allowing for reasonable computational simplification in far-field regions.

This non-uniform mesh refinement approach enhances numerical simulation efficiency while preserving adequate grid density in the wellbore vicinity to solve the dynamic temperature of the formation accurately. The discretized equations are detailed in Appendix A. A numerical solution program was developed based on the discretized wellbore circulating temperature field model.

After the computational domain has been meshed, the next step is to set up the initial and boundary conditions for the model.

For the initial condition setting, the entire heat conduction system is assumed to have reached thermal equilibrium before the first fracturing operation, with the initial temperature field consistent with the geothermal temperature. For multi-stage fracturing operations, the temperature field after each fracturing stage serves as the primary condition for the subsequent stage.

$$T(z,r,t=0)=T_s+T_G{\displaystyle {\int }_0^z\cos }\alpha \mathrm{d}z$$

(14)

The boundary conditions include the fracturing fluid inlet temperature, surface temperature, a “no-heat-transfer” condition for formation temperatures sufficiently distant from the wellbore to remain unaffected by fracturing operations and the bottom of the calculation domain. The expressions for the boundary conditions are as follows:

$$\left\{\begin{array}{l}{T}_m(z=0,t)=T_{in}\\ T_e(z=0,t)=T_s\\ {\left.{\displaystyle \frac{\mathsf{\partial }T(z,r,t)}{\mathsf{\partial }r}}\right|}_{r\to \infty }=0\\ T(z=z_b,r,t)=T_s+T_G{\displaystyle {\int }_0^{z_b}\cos }\alpha \mathrm{d}z\end{array}\right.$$

(15)

The numerical solution procedure begins by inputting fundamental parameters of the simulated well, including the wellbore geometric parameters and thermophysical properties of each zone. Subsequently, the program allocates grid parameters and initializes the temperature field. The set of linear equations about the computational domain throughout the injection and shut-in periods is resolved through the Gauss–Seidel technique. The wellbore temperature field is refreshed at each iteration step until all the specified fracturing stages have been accomplished.

3. Temperature–Pressure Coupling Effect on Cement Sheath Stress Calculation Model

3.1. Numerical Model

During multi-stage fracturing operations, the wellbore system in the reservoir section undergoes complex temperature–pressure coupling effects: on the one hand, the internal casing pressure exhibits significant cyclical variations during the injection period, high-pressure conditions are created by the combined effects of wellhead injection pressure, frictional resistance from fracturing fluid flow, and hydrostatic pressure, while during the shut-in period, with injection cessation and friction disappearance, the pressure rapidly decreases to hydrostatic pressure levels. On the other hand, as a result of dynamic thermal interaction among the formation, fluid, and wellbore, the wellbore temperature field also undergoes cyclical changes with increasing fracturing stage. This alternating temperature–pressure environment leads to continuous degradation of cement sheath material properties, with key mechanical parameters such as tensile strength, compressive strength and elastic modulus gradually deteriorating as the fracturing stage progresses.

Particularly for horizontal sections of shale oil wells, the varying temperature–pressure conditions across different fracturing stages, combined with the time-dependent characteristics of material properties, lead to considerable spatial inconsistencies and changes in stress states within the cement sheath over time. This complexity in stress computation and analysis is quite challenging. To address this, a finite element model was constructed to calculate the stress in the cement sheath under coupled temperature–pressure effects based on representative multi-stage fracturing operating conditions.

The assumption is that homogeneous thermomechanical properties are present across all wellbore components (casing, cement sheath, and formation) in the horizontal orientation, reducing the problem to a two-dimensional heat conduction and stress–strain analysis. A finite element model measuring 3 m × 3 m was established for the heel assembly cross-section based on Saint-Venant’s principle, utilizing variable-density meshing to minimize computational interference, as depicted in Figure 3.

The numerical simulation model contains the subsequent boundary and loading conditions: far-field in situ stresses and initial wellbore assembly temperatures are applied using the finite element Predefined Field function; the formation at infinite distance serves as a constant-temperature heat source equal to reservoir temperature during fracturing; the casing internal surface temperature and pressure act as dynamic boundary conditions calculated by the wellbore circulating temperature field model and input as time-dependent functions; boundary displacements are fixed at 0 m; and surface-to-surface contact is defined for the wellbore assembly components.

3.2. Criterion for the Failure of Cement Sheath

The failure of cement sheath mainly takes the forms of tensile failure and compressive failure. Tensile failure may lead to fracture of the cement sheath under tension, while compressive failure may cause crushing or plastic yielding under compression. Under pure tensile loading conditions, the maximum tensile stress criterion is employed to evaluate failure, where tensile fracture initiates when the cement sheath stress surpasses its ultimate tensile strength.

However, during actual multi-stage fracturing operations, the loading conditions on the cement sheath are more complex. At each fracturing stage, the cement sheath may not only experience either pure compression or tension, but also combined compressive–tensile loading. Therefore, the Mohr–Coulomb failure criterion has been introduced for analysis to accurately assess cement sheath failure under such complex loading conditions [33,34].

Extensive trial calculations and previous research results confirm that in cylindrical coordinates ${{\sigma }_1}^t={\sigma }_{\theta }$, ${{\sigma }_3}^t={\sigma }_r$, where ${\sigma }_{\theta }$ and ${\sigma }_r$ represent the current tangential and radial stresses of the cement sheath, respectively [35]. Under initial conditions, the cement sheath exhibits a compressive strength (${\sigma }_c$) of 61.71 MPa and a tensile strength (${\sigma }_t$) of 4.89 MPa [36].

Considering the progressive degradation of the cement sheath’s mechanical properties (including compressive and tensile strength) with increasing fracturing stage during the fracturing process, detailed mechanical performance parameters were obtained through degradation experiments for each of the 20 fracturing stages to quantify this deterioration behaviour. Figure 4 illustrates the variations in compressive strength (${\sigma }_c$) and tensile strength (${\sigma }_t$) of the cement sheath after each fracturing stage. These experimental data provide precise mechanical performance indicators for the cement sheath at different fracturing stages and have been incorporated into the original Mohr–Coulomb failure criterion to ensure accurate and reliable failure assessment [37].

To adapt to the multi-stage fracturing process, the modified Mohr–Coulomb failure criterion optimizes the parameters of the original Mohr–Coulomb criterion: on one hand, it supplements time subscripts to the radial and tangential stress variables; on the other hand, it adds fracturing stage subscripts to the compressive strength and tensile strength parameters. The modified Mohr–Coulomb failure criterion is presented in

Table 1. The modified Mohr–Coulomb failure criterion [38].

Stress Interval	Description of Interval	Relation Among Principal Stresses	Failure Standard
1	Tension–tension–tension	${{\sigma }_1}^t\ge {{\sigma }_3}^t\ge 0$	${{\sigma }_1}^t\ge {{\sigma }_t}^N$
2	Compression–compression–compression	$0\ge {{\sigma }_1}^t\ge {{\sigma }_3}^t$	$-{{\sigma }_3}^t\ge {{\sigma }_c}^N$
3	Tension–compression–compression; Tension–tension–compression	${{\sigma }_1}^t\ge 0\ge {{\sigma }_3}^t$	${\displaystyle \frac{{{\sigma }_1}^t}{{{\sigma }_t}^N}}-{\displaystyle \frac{{{\sigma }_3}^t}{{{\sigma }_c}^N}}\ge 1$

, which lists the principal stress relationships and their corresponding failure criteria under different stress intervals:

a. Tension–tension–tension interval: Tensile failure occurs when the maximum principal stress ${{\sigma }_1}^t$ (current tangential stress) exceeds the tensile strength ${{\sigma }_t}^N$ under the current fracturing stage ($N$). If ${{\sigma }_1}^t\ge {{\sigma }_t}^N$, failure will occur.

b. Compression–compression–compression interval: Compressive failure occurs when the negative value of the minimum principal stress ${{\sigma }_3}^t$ (current radial stress) exceeds the compressive strength ${{\sigma }_c}^N$ under the current fracturing stage ($N$), if ${{\sigma }_3}^t\ge {{\sigma }_c}^N$, failure will occur.

c. Tension–compression–compression and tension-tension–compression intervals: Failure occurs if the condition ${\displaystyle \frac{{{\sigma }_1}^t}{{{\sigma }_t}^N}}-{\displaystyle \frac{{{\sigma }_3}^t}{{{\sigma }_c}^N}}\ge 1$ is satisfied.

3.3. Geometry and Mechanics Parameters

The modelling was carried out using a shale gas well in Sichuan Province, where substantial SCP was noted following multiple fracturing stages. The well extends to a total depth of 5300 m, with a vertical depth of 3650 m and a geothermal gradient of 3 °C per 100 m. At the reservoir site, the maximum horizontal principal stress measures 90 MPa, the minimum horizontal principal stress is 72 MPa, and the vertical principal stress amounts to 100 MPa. The casing pressure is set at 75 MPa (comprising the combined effects of wellhead pressure, hydrostatic pressure at 3650 m vertical depth, and fluid friction). The fracturing fluid displacement rate was 12 m³/min with an initial fluid temperature of 25 °C. The fracturing design employed staged fracturing technology with 20 fracturing stages distributed along the horizontal section. Each fracturing stage consisted of a 4 h fluid injection period followed by a 4 h shut-in period, forming an 8 h operational cycle, resulting in a cumulative fracturing operation time of 160 h.

Table 2. Wellbore assembly parameters.

	Out Diameter (mm)	Poisson’s Ratio	Coefficient of Heat Conduction (W.(m.°C)−1)	Density (kg.m−3)	Specific Heat (J. (kg.°C)−1)	Coefficient of Thermal Expansion (10−6.°C−1)
Casing	139.7	0.3	45.34	7849	460.9	13
Cement sheath	215.9	0.16	0.98	3100	837	11
Formation	\\	0.23	1.59	2242.6	1256.9	10.5

presents the specific geometric, material, and thermodynamic parameters of the wellbore.

4. Results and Discussion

4.1. Analysis of Transient Temperature Field in the Wellbore

4.1.1. Cement Sheath Temperature Field

Based on the established wellbore circulation temperature field model, temperature profiles of the cement sheath were analyzed versus well depth at sequential injection time points (0 h, 1 h, 2 h, 3 h, 4 h) during a single-stage fracturing operation. As depicted in Figure 5, the axial temperature variation in the cement sheath exhibits two distinct phases: in the 0–4300 m vertical depth section, the temperature continuously increases with prolonged fracturing time; beyond 4300 m into the horizontal section, the temperature at 0 h (initial geothermal temperature) remains constant at 124.38 °C, while temperatures at other injection periods gradually approach stable values.

Notably, the cement sheath’s axial temperature exhibits a significant sudden drop during the initial fracturing stage (0–1 h) (ΔT_max = 56.21 °C, where ΔT_max denotes the maximum temperature difference during 0–1 h), with the cooling rate gradually decreasing in subsequent periods. This phenomenon occurs because, during the 4 h fracturing operation, the fracturing fluid enters the wellbore at a relatively low temperature and comes into direct contact with the internal casing wall. As the fluid flows through the wellbore and undergoes heat exchange with it, combined with the casing’s good thermal conductivity, which promotes the radial heat conduction within the wellbore, the temperature of the cement sheath undergoes rapid changes under the coupled temperature–pressure effects of the wellbore, resulting in its gradual temperature decrease.

To further analyze the temperature difference patterns and the difference between the internal and external walls of the cement sheath (Figure 6), we calculate the cement sheath temperature field at 4300 m depth throughout 20 fracturing stages. The data reveal two key findings. First, the maximum temperature difference between internal and external walls shows a monotonic decreasing trend with increasing fracturing stage (ΔT_max declines from 36.94 °C to 3.29 °C, representing a 91.09% reduction). Second, within individual fracturing cycles, the temperature difference exhibits pulsed variations, with peak difference consistently occurring within the first 4 h of each fracturing stage.

4.1.2. Radial Temperature Field in the Wellbore

A quantitative investigation was performed on the temperature distribution in various radial sections of the wellbore to elucidate the evolutionary characteristics of the temperature field in the cement sheath. Figure 7 presents the dynamic response of the radial temperature field at the heel of the horizontal well (4300 m depth) during the 0–240 min fracturing process. The results demonstrate that during the initial fracturing stage (0–60 min), the cement sheath temperature exhibits a rapid decline, plummeting from the initial 124.38 °C to 45.19 °C, representing a dramatic temperature drop of 63.7%, with the most intense temperature variations occurring in this phase. Subsequently (120–240 min), the rate of temperature decrease slows significantly, and the cooling trend becomes more gradual.

4.2. Temperature–Pressure Coupling and Alternating Conditions Impact on Cement Sheath Stress

To explore how temperature–pressure coupling and alternating conditions impact the cement sheath stress, the numerical simulation parameters are set with a casing pressure of 75 MPa, fracturing fluid displacement of 12 m³/min, initial fracturing fluid temperature of 25 °C, and a well depth of 4300 m, analyzing the changing features of the cement sheath stresses (radial and tangential). As depicted in Figure 8, during the first two fracturing stages, the combined action of temperature–pressure coupling effects and alternating loads causes the cement sheath stress to exhibit distinct periodic variation patterns. These stress variations demonstrate cycles perfectly synchronized with the fracturing operation cycles (4 h injection phase + 4 h shut-in phase), a pattern that remains consistent throughout all 20 fracturing stages.

As depicted in Figure 8, the pattern of change in the cement sheath’s radial stress is complex under cyclic fracturing. The specific manifestations are as follows:

During the injection phase of the first fracturing stage, the radial stress first decreases and then increases (the negative sign only indicates that the cement sheath is under compression). When fracturing fluid initially enters the casing, the original completion fluid in the wellbore moves downward, and the casing contracts due to cooling, reducing its compressive effect on the cement sheath and consequently decreasing radial stress. Under thermal conduction effects, the temperature of the internal wall of the cement sheath gradually becomes almost the same as that of the external wall of the casing, and simultaneously, the temperature of the external wall of the cement sheath starts to decline, leading to overall contraction. Combined with the mechanical load from formation pressure, the radial stress continuously increases until the end of the injection phase. During the shut-in phase of the first fracturing stage, the fluid stops flowing, and the internal pressure reduces to hydrostatic pressure, lessening the compressive force acting on the cement sheath, which results in an immediate drop in radial stress. As fracturing time progresses, the cement sheath temperature gradually rises due to thermal conduction, leading to thermal expansion. However, constrained by both the casing and formation, this expansion is restricted, resulting in increased radial stress that superimposes the original compressive load, further compressing the cement sheath.

During the injection phase of the second fracturing stage, when fluid re-enters the wellbore, the cement sheath’s radial stress instantly increases. In this stage, the compressive load borne by the cement sheath is mainly controlled by the casing pressure and formation pressure. Notably, the formation pressure fully manifested its influence during the first-stage injection phase. For the following fracturing procedures, the stress condition of the cement sheath is analyzed, taking into consideration the impact of the already established formation pressure. As the second fracturing stage progresses, the casing undergoes cooling and contraction, which diminishes its compressive impact on the cement sheath. Consequently, the radial stress diminishes until the conclusion of the injection phase. Upon re-entering the shut-in phase, the radial stress instantly decreases and then gradually increases with continued fracturing time.

As illustrated in Figure 9, the cement sheath’s radial stress variation pattern during subsequent fracturing stages follows a similar trend to that observed in the second stage. However, due to the progressive degradation of compressive strength of the cement sheath with increasing fracturing stage, the absolute value of radial compressive stress gradually decreases.

As shown in Figure 10, the cement sheath tangential stress variations are described as follows:

During the injection phase of the first fracturing stage, the tangential stress initially increases and then decreases. This trend is obtained from the numerical simulation results, although the local peak is not very obvious in the figure due to the short duration of the initial stress variation. When the fracturing fluid first enters the casing, the original completion fluid within the wellbore moves downward, causing the cement sheath temperature to decrease and the tangential stress to increase. Subsequently, due to thermal conduction, the temperature difference between the internal and external walls of the cement sheath decreases, and the overall wellbore temperature continues to decline, leading to contraction of the cement sheath and a reduction in tangential stress until the end of the injection phase. During the shut-in phase, with fluid flow ceased and internal casing pressure reduced, the tangential stress direction reverses (negative values). As the cement sheath temperature gradually rises and thermal expansion occurs, constrained by both casing and formation, this restricted expansion causes tangential stress to increase.

At the beginning of the second fracturing stage, when fracturing fluid re-enters the casing, the tangential stress direction reverses again (positive values). In this stage, the compressive load imposed on the cement sheath is mainly determined by the casing pressure and the formation pressure, with the formation pressure having fully acted during the first fracturing stage. In the subsequent fracturing stages, the cement sheath’s stress state is analyzed according to this established mechanical condition. Under thermal conduction effects, overall cooling-induced contraction of the wellbore causes tangential stress to rise until the injection phase ends. Upon re-entering the shut-in phase, the tangential stress direction reverses once more and gradually increases with extended fracturing time.

The fracturing patterns of subsequent fracturing stages are similar to those of the second. However, with the increase in the fracturing stage, the cement sheath tensile strength gradually deteriorates. Meanwhile, the overall trend of the tangential stress is that the positive tangential stress gradually increases as the fracturing stage progresses. The negative tangential stress decreases slowly, as depicted in Figure 11.

Throughout the multi-stage fracturing procedure, the distribution of the cement sheath stresses (radial and tangential) after varying fracturing stages is shown in Figure 12. As the compressive and tensile strengths of the cement sheath deteriorate with the increasing number of fracturing stages, the peak radial and tangential stresses it can bear show a decreasing trend. A quantitative analysis reveals that when the count of the fracturing stage is 5, the peak radial stress within the cement sheath is −38.04 MPa, and the maximum tangential stress is −13.76 MPa. When the fracturing stage increases to 20, the corresponding stress values decrease to −25.91 MPa and −11.91 MPa, with a reduction of 31.89% and 13.44%, respectively.

Figure 13 defines $\eta$ as the Mohr–Coulomb damage value, which is dimensionless. Under the temperature–pressure coupling and alternation conditions, the cement sheath integrity is evaluated in accordance with the modified Mohr–Coulomb failure criterion. As is noticeable from the figure, during the 20-stage fracturing process, the peak value of $\eta$ occurs during the injection phase of the first fracturing stage. In this instant, the cement sheath tangential stress has exceeded its tensile strength, which theoretically poses a risk of failure. However, in practice, because the strength of the cement material has not yet significantly deteriorated in the initial stage, the structural integrity is still maintained.

Provided that the cement sheath remains intact during the injection period of the first stage, its tensile strength will gradually deteriorate as the fracturing stage increases. In particular, from the shut-in phase of the first fracturing stage until the shut-in phase of the seventh stage, the damage level of the cement sheath consistently stays beneath the threshold value. This suggests that no failure risk exists within this time frame. However, from the eighth fracturing stage until the operation comes to an end, the cement sheath faces failure risks during the injection phases of subsequent stages.

4.3. Sensitivity Analysis

Considering the practicality of multi-stage fracturing operations and combining the numerical model established previously, the cement sheath stresses (radial and tangential) under temperature–pressure coupling and its integrity are analyzed. During the analysis, the cement sheath stress under various casing pressures, fracturing fluid displacements, initial temperatures of fracturing fluid, and well depths (fracturing stages) is examined. The purpose is to analyze the influence of these factors on the structural failure of the cement sheath and manage them appropriately to minimize the failure risk.

4.3.1. Casing Pressure Sensitivity Analysis

In shale gas exploitation by staged hydraulic fracturing, the cyclical change in casing pressure with each fracturing stage directly affects the stress evolution patterns within the cement sheath. To elucidate the mechanistic interplay between cyclic fracturing-induced dynamic casing pressure and cement sheath stress evolution, calculations are performed at a well depth of 4300 m with initial fracturing parameters set at 12 m³/min displacement, 25 °C fluid temperature, and 20 fracturing stages, examining dynamic radial and tangential stress variations under varying casing pressures (75 MPa, 85 MPa, 95 MPa, 105 MPa). Throughout the shut-in phase of each fracturing stage, when the casing pressure reduces to the hydrostatic pressure at 4300 m depth, the cement sheath stress variations under varying casing pressures become primarily thermally dominated.

Figure 14 shows how the radial stress changes with varying casing pressures as the fracturing stage advances, from which the following observations can be made:

(a) Under identical casing pressure conditions, during the first-stage fracturing injection period, the cement sheath radial stress initially decreases before increasing. In contrast, the radial stress progressively declines with extended injection time in subsequent injection periods. During all shut-in periods of fracturing operations, the radial stress instantly decreases as the casing pressure drops to hydrostatic pressure. Subsequently, as the cement sheath temperature gradually recovers, its radial stress increases with prolonged shut-in time.

(b) For the equivalent fracturing stage, the cement sheath radial stress demonstrates a consistent upward trend with increasing casing pressure.

(c) Higher casing pressures correlate with expanded ranges of the fracturing stages where radial stress exceeds the compressive strength. Specifically, at 75 MPa, 85 MPa, 95 MPa and 105 MPa casing pressures, the radial stress surpasses the stage-specific compressive strength during stages 13–20, 9–20, 6–20, and 1–20 injection periods, respectively. However, during shut-in periods across all stages, the radial stress consistently remains below the compressive strength threshold.

To further quantify the overall trend of the radial stress with changes in casing pressures during the multi-stage fracturing process, the radial stress changes under varying casing pressures are compared. The fracturing stages are 5th, 10th, 15th, and 20th, respectively, as depicted in Figure 15. Based on this figure, the following information is clearly discernible.

(a) For a constant casing pressure, the radial stress diminishes as the fracturing stage increases. According to quantitative analysis, with the casing pressure being 75 MPa, the peak radial stresses at the fifth, 10th, 15th, and 20th fracturing stages are −38.04 MPa, −33.06 MPa, −29.16 MPa, and −25.91 MPa, respectively. From the fifth to the 20th fracturing stage, the radial stress decreases cumulatively by 31.89%.

(b) For an identical fracturing stage, the radial stress rises as the casing pressure escalates. When the count of fracturing stages is five, the maximum radial stresses at casing pressures of 75 MPa, 85 MPa, 95 MPa, and 105 MPa measure −38.04 MPa, −39.9 MPa, −41.77 MPa, and −43.63 MPa, respectively. When the casing pressure starts from 75 MPa and ultimately reaches 105 MPa, the cement sheath radial stress experiences a cumulative growth of 14.70%.

With the increasing fracturing stage, the patterns of tangential stress variation under various casing pressures are depicted in Figure 16. From this figure, the following observations can be made.

(a) At a constant casing pressure, throughout the initial fracturing stage’s injection period, the positive tangential stress exhibits an increasing trend followed by a reduction. Throughout the injection phases of subsequent fracturing stages, the positive tangential stress increases with the continuous increase in the fracturing fluid injection time. In contrast, during all shut-in periods of fracturing, the casing pressure falls to hydrostatic levels, causing the tangential stress to invert. Subsequently, as the cement sheath’s temperature slowly returns to normal, the negative tangential stress intensifies with extended shut-in time.

(b) At a constant number of fracturing stages, the cement sheath tangential stress demonstrates a consistent upward trend with increasing casing pressure.

(c) During the initial fracturing stage’s injection period, the tangential stress surpasses the tensile strength of the cement sheath for all casing pressures. In subsequent fracturing stages, the tangential stress never surpasses the tensile strength that is specific to each stage across all casing pressure levels.

To further quantify the overall trend of tangential stress with changes in casing pressures during the multi-stage fracturing process, the tangential stress changes under varying casing pressures are compared when the fracturing stages are fifth, 10th, 15th, and 20th, respectively, as depicted in Figure 17. Based on this figure, the following information can be discerned.

(a) Under the same casing pressure, tangential stress displays a decreasing trend as the fracturing stage increases. Quantitative analysis indicates that when the casing pressure is 75 MPa, the peak tangential stresses at the fifth, 10th, 15th, and 20th fracturing stages are −13.76 MPa, −12.84 MPa, −12.27 MPa, and −11.91 MPa, respectively. During the fracturing process from the fifth to the 20th fracturing stage, the cumulative reduction in the cement sheath tangential stress is 13.44%.

(b) Under the same fracturing stage, the tangential stress positively correlates with casing pressure, exhibiting a progressive increase as the casing pressure rises. In the fifth fracturing stage, the peak tangential stresses at casing pressures of 75 MPa, 85 MPa, 95 MPa, and 105 MPa are −13.76 MPa, −13.75 MPa, −13.73 MPa, and −13.71 MPa, respectively. As the casing pressure increases from 75 MPa to 105 MPa, the cumulative increase in the tangential stress is 0.36%.

Figure 18 illustrates the dynamic variation in failure criterion values under varying casing pressures during fracturing. The analysis results based on the modified Mohr–Coulomb failure criterion are as follows:

(a) Under consistent casing pressure, the damage value of the cement sheath exhibits a distinctly non-uniform distribution across various fracturing stages: it reaches its peak during the first-stage fracturing, while in subsequent stages, the peak damage value shows a monotonically increasing trend with the progression of fracturing stages.

(b) With a fixed number of fracturing stages, the damage value demonstrates a continuous growth pattern as the casing pressure gradually increases.

(c) During the injection phase of the first-stage fracturing, the damage values under varying casing pressures all exceed the failure criterion threshold. In the injection phases of other fracturing stages, as the casing internal pressure rises, the range of fracturing stages in which the cement sheath damage value surpasses the threshold gradually expands. Specifically, when the casing pressures are 75 MPa, 85 MPa, 95 MPa, and 105 MPa, the cement sheath damage values exceed the threshold during the injection phases of stages 8-20, 4-20, 2-20, and 2-20, respectively. However, during the shut-in phases of all fracturing stages, the cement sheath damage values remain below the threshold.

When the cement sheath damage value exceeds the critical threshold, it reveals that the cement sheath integrity has been compromised. Higher casing pressures subject the cement sheath to more significant mechanical stress, increasing the risk of failure. Therefore, in fracturing operations, reasonably reducing the casing pressure is a critical factor in guaranteeing the integrity of the cement sheath.

4.3.2. Displacement Sensitivity Analysis

Throughout the hydraulic fracturing operation, changes in fracturing fluid displacements significantly impact the frictional resistance and heat transfer coefficients inside the wellbore, which further alter the temperature and pressure conditions inside the wellbore, ultimately exerting a significant influence on the cement sheath’s stress condition. To handle this situation, the casing pressure is 95 MPa, the fracturing fluid temperature is 25 °C, and the fracturing stage is set at 20. The cement sheath at a depth of 4300 m was chosen as the research target to analyze the variation rules of the radial and tangential stresses under varying fracturing fluid displacements (8 m³/min, 12 m³/min, 16 m³/min, and 20 m³/min).

As the fracturing stage increases, Figure 19 shows the variation laws of the radial stress under varying fracturing fluid displacements.

(a) Under the same fracturing fluid displacements, during the injection phase of the first fracturing stage, the radial stress initially shows a decreasing trend and subsequently undergoes an increase. During the injection periods of the other fracturing stages, the radial stress exhibits a downward trend as the injection duration of the fracturing fluid extends. In contrast, during all shut-in periods of fracturing, the radial stress instantaneously decreases as the casing pressure reduces to the hydrostatic pressure value. Subsequently, as the temperature of the cement sheath gradually recovers, its radial stress increases as the duration of shut-in expands.

(b) Under the same fracturing stage, the radial stress keeps on diminishing as the displacement of the fracturing fluid increases.

(c) As the fracturing fluid displacement increases, the range of fracturing stages in which the cement sheath radial stress exceeds its compressive strength gradually narrows. Specifically, when the fracturing fluid displacements are 8 m³/min, 12 m³/min, 16 m³/min, and 20 m³/min, the radial stress surpasses the compressive strength corresponding to the respective fracturing stage during the injection periods of fracturing stages 1-20, 1-20, 3-20, and 4-20, respectively. During the shut-in phases of all fracturing stages, the radial stress remains below its compressive strength.

To further quantify the radial stress under varying fracturing fluid displacements, the radial stress was compared at varying fracturing fluid displacements and at the fifth, 10th, 15th, and 20th fracturing stages, as depicted in Figure 20. Based on this figure, the following is evident:

(a) Under the same fracturing fluid displacements, the cement sheath radial stress displays a decreasing trend as the fracturing stage increases. Quantitative analysis indicates that when the fracturing fluid displacement is 8 m³/min, the maximum radial stresses at the fifth, 10th, 15th, and 20th fracturing stages are −44.51 MPa, −39.91 MPa, −36.06 MPa, and −32.66 MPa, respectively. During the fracturing process from the fifth to 20th stages, the cumulative reduction in the radial stress is 26.62%.

(b) Under the same number of fracturing stages, the radial stress rises as the fracturing fluid displacement increases. When there are five fracturing stages, the maximum radial stresses at fracturing fluid displacements of 8 m³/min, 12 m³/min, 16 m³/min, and 20 m³/min are −44.51 MPa, −43.04 MPa, −41.77 MPa, and −41.12 MPa, respectively. As the fracturing fluid displacement increases from 8 m³/min to 20 m³/min, the cumulative reduction in the radial stress is 7.62%.

As the fracturing stage progresses, the variation patterns of the tangential stress under varying fracturing fluid displacements are depicted in Figure 21. According to this figure, we can notice the following.

(a) Under the same fracturing fluid displacements, during the injection period of the first fracturing stage, the positive cement sheath tangential stress initially increases and subsequently declines. During the injection periods of the other fracturing stages, the positive tangential stress increases with the injection time of the fracturing fluid. In contrast, during all shut-in periods of fracturing, since the casing pressure reduces to the hydrostatic pressure, the tangential stress reverses direction. Subsequently, as the cement sheath’s temperature gradually recovers, the negative tangential stress demonstrates a positive correlation with shut-in time.

(b) Under the same number of fracturing stages, the cement sheath tangential stress continuously keeps decreasing as the displacement of the fracturing fluid increases.

(c) During the injection phase of the first fracturing stage, the cement sheath tangential stress exceeds the tensile strength corresponding to the first fracturing stage for all fracturing fluid displacements. In the other fracturing stages, the tangential stress remains below the tensile strength corresponding to the respective fracturing stage.

To further quantify the tangential stress under varying fracturing fluid displacements, the tangential stress was compared under the conditions of varying fracturing fluid displacements and at the fifth, 10th, 15th, and 20th fracturing stages, as depicted in Figure 22. According to this figure, it is noticeable that:

(a) Under the same fracturing fluid displacements, the tangential stress decreases with the increase in the fracturing stage. Quantitative analysis indicates that when the fracturing fluid displacement is 8 m³/min, the maximum tangential stresses at the fifth, 10th, 15th, and 20th fracturing stages are −14.87 MPa, −14.24 MPa, −13.85 MPa, and −13.61 MPa, respectively. From the fifth to the 20th stages, the cumulative reduction in the tangential stress is 8.47%.

(b) Under the same number of fracturing stages, the tangential stress rises as the fracturing fluid displacement increases. When there are five fracturing stages, the maximum tangential stresses at fracturing fluid displacements of 8 m³/min, 12 m³/min, 16 m³/min, and 20 m³/min are −14.87 MPa, −14.29 MPa, −13.73 MPa, and −13.53 MPa, respectively. As the fracturing fluid displacement increases from 8 m³/min to 20 m³/min, the cumulative reduction in the tangential stress is 9.01%.

Figure 23 illustrates the variation in the failure determination values under varying fracturing fluid displacements as the process continues. The determination results relying on the modified Mohr–Coulomb criterion demonstrate the following.

(a) Under the same fracturing fluid displacements, the cement sheath’s damage value shows a significant non-uniform distribution characteristic in each fracturing stage. It reaches the peak value during the first fracturing stage, and the peak damage value during the successive fracturing stages rises in a monotonous manner as the fracturing stage increases.

(b) Under the same number of fracturing stages, during the 20-stage fracturing process, the damage value keeps increasing as the fracturing fluid displacement grows.

(c) When the fracturing fluid displacement is 8 m³/min, the damage value exceeds the determination value in the whole fracturing stages except the injection period of the second stage. When the fracturing fluid displacement is 12 m³/min, 16 m³/min, and 20 m³/min, the damage value exceeds the determination value in all injection periods of the fracturing stages. However, during the shut-in periods of all fracturing stages, the damage value remains below the determination value.

An increase in fracturing fluid displacements causes a considerable volume of low-temperature fracturing fluid to enter the well rapidly, causing a rapid decline in the temperature of the cement sheath and intensifying the damaging effect resulting from thermal stress. Although the damage values in all injection periods of the fracturing stages exceed the critical determination value under varying fracturing fluid displacements, altering the displacements fails to bring about an effective improvement in the failure situation of the cement sheath. However, decreasing the displacements of the fracturing fluid is capable of reducing the damage value. As demonstrated by the analysis presented previously, moderately decreasing the displacement of fracturing fluid can effectively lower the probability of cement sheath failure.

4.3.3. Initial Temperature Sensitivity Analysis

The initial temperature of the fracturing fluid shows a significant seasonal variation throughout the hydraulic fracturing operation, characterized by lower initial temperatures in winter and higher initial temperatures in summer. This temperature difference can significantly alter the temperature and pressure in the wellbore, thereby directly affecting the thermal stress condition and the mechanical performance of the cement sheath. To address this, the initial fracturing fluid displacement is determined to be 12 m³/min, the fracturing stage is 20, and the casing pressure is 95 MPa. The cement sheath at a depth of 4300 m is chosen as the research target to analyze the variation rules of the radial and tangential stresses under varying initial fracturing fluid temperatures (5 °C, 25 °C, and 45 °C).

As the fracturing stage goes up, the variation laws of the radial stress under varying initial fracturing fluid temperatures are presented in Figure 24. According to this figure, we can see the following.

(a) Under the same initial temperature, during the injection phase of the first fracturing stage, the radial stress initially shows a decreasing trend and subsequently undergoes an increase. During the injection periods of the other fracturing stages, the radial stress exhibits a downward trend as the injection duration of the fracturing fluid extends. In contrast, during all shut-in periods of fracturing, the radial stress instantaneously decreases as the casing pressure reduces to the hydrostatic pressure value. Subsequently, as the temperature of the cement sheath gradually recovers, its radial stress increases as the duration of shut-in expands.

(b) Under the same fracturing stage, the cement sheath radial stress continuously keeps increasing as the initial temperature of the fracturing fluid rises.

(c) As the initial temperature increases, the radial stress of the cement sheath surpasses its compressive strength and gradually broadens. Specifically, when the initial temperature is 5 °C, 25 °C, and 45 °C, the radial stress surpasses the compressive strength corresponding to the respective fracturing stage during the injection periods of fracturing stages 7-20, 6-20, and 1-20, respectively. During the shut-in periods of all fracturing stages, the radial stress always remains below its compressive strength.

To further quantify the radial stress under varying initial temperatures, we compare the radial stresses at the fifth, 10th, 15th, and 20th fracturing stages for various initial fluid temperatures, as depicted in Figure 25. Based on this figure, we can discern the following.

(a) Under identical initial temperatures, the radial stress diminishes as the fracturing stage increases. Through quantitative analysis, it is revealed that when the initial temperature is 5 °C, the maximum radial stresses at the fifth, 10th, 15th, and 20th fracturing stages are −39.87 MPa, −33.72 MPa, −28.92 MPa, and −24.93 MPa, respectively. From the fifth to the 20th stages, the radial stress accumulates a total reduction of 37.47%.

(b) For the equivalent fracturing stage, the radial stress rises as the initial temperature of the fracturing fluid rises. At stage 5, when initial temperatures are 5 °C, 25 °C and 45 °C, the corresponding maximum radial stresses are −39.87 MPa, −41.77 MPa and −43.91 MPa, respectively. When the initial temperature goes up from 5 °C to 45 °C, the radial stress accumulates a total increase of 10.13%.

Figure 26 demonstrates the variation patterns of cement sheath tangential stress under varying initial fracturing fluid temperatures as the fracturing stage progresses, revealing the following characteristics.

(a) Under identical initial fluid temperatures, during the first-stage injection period, the positive tangential stress initially increases and then decreases, while in subsequent injection periods, it progressively rises with extended injection time. During all shut-in periods, as casing pressure reduces to hydrostatic pressure, the tangential stress reverses direction (negative values) and gradually increases with shut-in time as temperature recovers.

(b) For the equivalent fracturing stage, the tangential stress exhibits a consistent decreasing trend with higher initial fluid temperatures.

(c) During the first-stage injection period, regardless of initial fluid temperature, the tangential stress exceeds the tensile strength for that stage. However, the tangential stress remains below the stage-specific tensile strength threshold in all subsequent fracturing stages.

To further quantify the tangential stress under varying initial temperatures, a comparison is conducted on the tangential stresses at the fifth, 10th, 15th, and 20th fracturing stages with varying initial fluid temperatures, as illustrated in Figure 27. Based on this figure, we can discern the following.

(a) Under identical initial fracturing fluid temperatures, the tangential stress decreases with the increment in the fracturing stage. Quantitative analysis reveals that when the initial fracturing fluid temperature is 5 °C, the maximum tangential stresses at the fifth, 10th, 15th, and 20th fracturing stages are −13.03 MPa, −11.85 MPa, −11.11 MPa and −10.63 MPa, respectively. From stage 5 to stage 20, the tangential stress shows a cumulative reduction of 18.42%.

(b) For the equivalent fracturing stage, the cement sheath tangential stress increases with higher initial fracturing fluid temperatures. At stage 5, when initial temperatures are 5 °C, 25 °C and 45 °C, the corresponding maximum tangential stresses measure −13.03 MPa, −13.73 MPa and −14.63 MPa, respectively. When the initial temperature climbs from 5 °C to 45 °C, the tangential stress accumulates a total increase of 3.84%.

Figure 28 illustrates the variation in the failure determination values under varying initial temperatures. The determination results relying on the modified Mohr–Coulomb criterion demonstrate the following.

(a) Under the same initial temperature, the cement sheath’s damage value shows a significant non-uniform distribution characteristic in each fracturing stage: it reaches the peak value during the first fracturing stage, and in the subsequent fracturing stages, the peak value of damage shows a monotonic increase as the fracturing stage rises.

(b) Under the same fracturing stage, the damage value maintains a continuous decrease as the initial temperature goes up.

(c) When the initial temperatures are 5 °C, 25 °C, and 45 °C, the cement sheath’s damage value exceeds the determination value in the whole injection period of the fracturing stages; however, during the shut-in periods of all fracturing stages, the damage value constantly stays beneath the determined value.

From the failure determination results, it is evident that elevating the initial temperature of the fluid can efficiently minimize the potential for cement sheath failure. Notably, when the temperature is 45 °C, the damage value experiences a substantial decrease. This is because an increment in the initial temperature of the fluid lessens the temperature difference between the internal and external walls of the cement sheath, thereby decreasing the negative effect generated by thermal stress. Although decreasing the initial temperature of the fluid does not significantly improve the integrity failure condition of the cement sheath, moderately raising the initial temperature of the fracturing fluid is conducive to optimizing the stress state of the cement sheath and decreasing its damage value.

4.3.4. Well Depth Sensitivity Analysis

While the multi-stage fracturing operation was being carried out in the horizontal interval, different depths corresponded to distinct fracturing stages, with the number of cyclic fracturing operations increasing progressively toward the toe of the horizontal well. Furthermore, the thermo-mechanical conditions of the cement sheath varied with depth, resulting in depth-dependent stress evolution characteristics. To investigate the effect of well depth (fracturing stages) on the cement sheath’s integrity, the initial fracturing fluid displacement was set at 12 m³/min, the initial temperature at 25 °C, and the casing pressure at 95 MPa. The dynamic variation patterns in cement sheath stresses (radial and tangential) were calculated for varying well depths (5050 m, 4800 m, 4550 m, and 4300 m) corresponding to varying fracturing stages of five, 10, 15, and 20, respectively.

The variation patterns of the radial stress under varying well depths as the fracturing stage increased are presented in Figure 29. According to this figure, we can see the following.

(a) Under the same well depth condition, during the injection phase of the first fracturing stage, the radial stress initially shows a decreasing trend and subsequently undergoes an increase. During the injection periods of the other fracturing stages, the radial stress decreases with the increase in fracturing fluid injection time. In contrast, during all shut-in periods of fracturing, the radial stress instantaneously decreases as the casing pressure reduces to the hydrostatic pressure value. Subsequently, as the temperature of the cement sheath gradually recovers, its radial stress increases as the duration of shut-in expands.

(b) In the case of an identical number of fracturing stages, the radial stress continuously grows with the progression of well depth.

(c) As the well depth increases, the range of fracturing stages in which the radial stress surpasses its compressive strength gradually narrows. Specifically, when located at varying well depths of horizontal wells (4300 m, 4550 m, 4800 m, and 5050 m), the radial stress surpasses the compressive strength corresponding to the respective fracturing stage during the injection periods of fracturing stages 3-20, 3-15, 3-10, and 2-5, respectively. During the shut-in periods of all fracturing stages, the radial stress always remains below its compressive strength.

To systematically quantify the distribution patterns of the radial stress along the well depth, this study comparatively analyzed the evolutionary characteristics of radial stress under the fifth, 10th, 15th, and 20th fracturing stages within the well depth interval of 4300–5050 m, as depicted in Figure 30. Based on this figure, the following is evident.

(a) Under the same well depth condition, the radial stress drops as the fracturing stage increases. The results of the quantitative analysis show that, at a well depth of 4300 m, the peak radial stresses of the cement sheath during the fifth, 10th, 15th, and 20th fracturing stages are −41.77 MPa, −36.32 MPa, −31.96 MPa, and −28.23 MPa, respectively. From the fifth to the 20th fracturing stage, the cumulative reduction in the radial stress is 32.42%.

(b) Under the same fracturing stage, an increase in well depth leads to a corresponding increase in the radial stress of the cement sheath. When the count of fracturing stages is five, the peak radial stresses of the cement sheath at well depths of 4300 m, 4550 m, 4800 m, and 5050 m are −41.77 MPa, −52.07 MPa, −47.71 MPa, and −37.42 MPa, respectively. As the well depth increases from 4300 m to 5050 m, the cumulative reduction in the radial stress is 10.41%.

As the fracturing stage increases, the variation patterns of the tangential stress under varying well depths are illustrated in Figure 31. According to this figure, the following can be seen.

(a) Under the same well depth condition, during the injection phase of the first fracturing stage, the positive tangential stress shows an initial upward trend followed by a decline. During the injection periods of the other fracturing stages, the positive tangential stress demonstrates an upward tendency with the augmentation of fracturing fluid injection time. In contrast, when the casing pressure decreases to the hydrostatic pressure level during all shut-in periods of fracturing, the tangential stress reverses direction. Subsequently, with the gradual recovery of the cement sheath’s temperature, the negative tangential stress increases as the shut-in time is prolonged.

(b) Under the same fracturing stage, tangential stress presents a consistent downward tendency as the well depth progresses.

(c) During the injection phase of the first fracturing stage, the tangential stress surpasses the tensile strength corresponding to the first fracturing stage for all well depths. In the other fracturing stages, the tangential stress remains below the tensile strength applicable to each fracturing stage.

To further quantify the overall trend of the tangential stress under varying well depths during the multi-stage fracturing process, the tangential stress at varying well depths (4300 m, 4550 m, 4800 m, and 5050 m) and at the fifth, 10th, 15th, and 20th fracturing stages was compared, as depicted in Figure 32. Based on this figure, we can discern the following:

(a) Under the same well depth condition, the tangential stress exhibits a decreasing trend as the fracturing stage increases. Quantitative analysis indicates that at a well depth of 4300 m, the peak tangential stresses at the fifth, 10th, 15th, and 20th fracturing stages are −13.73 MPa, −12.81 MPa, −12.25 MPa, and −11.89 MPa, respectively. During the fracturing operations from the fifth to the 20th fracturing stage, the cumulative reduction in the tangential stress is 13.40%.

(b) Under the same number of fracturing stages, an increase in well depth results in a corresponding rise in the tangential stress. When the count of fracturing stages is five, the peak tangential stress at well depths of 4300 m, 4550 m, 4800 m, and 5050 m is −13.73 MPa, −14.39 MPa, −15.06 MPa, and −16.56 MPa, respectively. As the well depth increases from 4300 m to 5050 m, the cumulative increase in the tangential stress is 20.61%.

Figure 33 illustrates the variation in the failure determination values under varying well depths. The determination results relying on the modified Mohr–Coulomb criterion indicate that:

(a) Under the same well depth condition, the cement sheath’s damage value shows a significant non-uniform distribution characteristic in each fracturing stage: it reaches the peak value during the first fracturing stage, and the peak damage value during the successive fracturing stages rises in a monotonous manner as the fracturing stage goes up.

(b) Under the same fracturing stage, at the fifth fracturing stage, the damage value steadily rises as the well depth increases.

(c) When the depth of the horizontal well is 4300m, 4550 m, 4800 m and 5050 m, respectively, the cement sheath’s damage value exceeds the determination value in all injection periods of the fracturing stages; however, during the shut-in periods of all fracturing stages, the cement sheath’s damage value always remains below the determination value.

From the Mohr–Coulomb determination results, it is evident that the nearer to the horizontal well heel, the more severe the failure of the cement sheath becomes, with the peak damage value being the highest at a well depth of 5050 m. This situation arises because, as the well depth increases, the period during which heat exchange takes place between the fracturing fluid and the wellbore is extended, significantly reducing the disturbance effect on the surrounding temperature field and thereby weakening its coupled influence on the cement sheath stress changes. Therefore, a lower horizontal well depth is advantageous for safeguarding the cement sheath’s integrity.

5. Conclusions

This study analyzes transient variations in cement sheath stresses (radial and tangential) and their influence on cement integrity under temperature–pressure coupling and alternating conditions during multi-stage fracturing. A numerical simulation method has been established to assess the integrity of the cement sheath during the entire multi-stage fracturing procedure. The integrity of the cement sheath has been analyzed with respect to varying casing pressures, fracturing fluid displacements, initial fracturing fluid temperatures, and well depths. Compared with previous studies that mainly focused on cement sheath integrity under individual temperature/pressure variations, production or gas-storage conditions, vertical well sections, or heterogeneous in situ stress conditions, this study couples cyclic temperature–pressure loads during multi-stage hydraulic fracturing with stage-dependent degradation of cement sheath mechanical properties, thereby revealing the dynamic evolution of radial and tangential stresses and providing a more applicable integrity-evaluation framework for shale gas horizontal wells. The following conclusions are obtained.

(1) A numerical model for evaluating the cement sheath’s integrity has been established throughout the present research endeavour. The results of the simulation show that the transient coupled temperature–pressure action and cyclic loading in shale gas reservoirs lead to periodic evolution characteristics of the cement sheath stress field. Specifically, this manifests as follows: (a) radial and tangential stresses display cyclic variations as the fracturing stage increases; (b) the cement sheath’s mechanical properties progressively deteriorate, with its compressive and tensile strengths continuously decreasing as the fracturing stage increases. The above coupling effects bring about a diminution in the load-bearing capacity of the cement sheath, a significant cumulative damage effect, and the failure risk increases progressively as the fracturing stage rises.

(2) During the 20-stage fracturing operations, the evolution process of the stress field of the cement sheath demonstrates distinct dynamic properties: (a) within a single fracturing cycle, the injection period leads to a decline in radial stress and a rise in tangential stress; (b) when the injection period commences for each fracturing stage, the entry of low-temperature fracturing fluid into the well causes a sudden change in well temperature, making this stage a high-risk period for cement sheath failure. The cement sheath’s damage value shows a significant non-uniform distribution characteristic in each fracturing stage: it reaches the maximum value during the first fracturing stage, and the peak damage value in the following fracturing stages shows a monotonically increasing trend as the fracturing stage increases.

(3) Sensitivity analyses were conducted for casing pressure, fracturing fluid displacement, fracturing fluid initial temperature, and horizontal well depth. The results indicate that reducing the casing pressure can effectively alleviate the radial and tangential stresses in the cement sheath; decreasing the fracturing fluid displacement can improve the stress distribution of the cement sheath; increasing the initial fracturing fluid temperature increases the radial stress and suppresses the tangential stress; and although horizontal well depth is typically a fixed design parameter and is therefore difficult to adjust, the observed influence patterns suggest that a smaller horizontal well depth helps weaken the coupled temperature–pressure effect and reduce the damage value of the cement sheath. Optimizing these parameters can mitigate stress-induced damage to the cement sheath, thereby enhancing its structural integrity and ensuring smooth fracturing operations.