Evaluation of Strength Model Under Deep Formations with High Temperature and High Pressure

Abstract

Elevated thermal conditions, rock formations exhibit distinct mechanical behaviors that significantly deviate from their characteristics under ambient temperature environments. This phenomenon raises critical questions regarding the applicability of conventional failure criteria in accurately assessing wellbore stability and maintaining the structural integrity of subsurface infrastructure within geothermal environments. Based on the least absolute deviation method, this paper studies the response characteristics of rock strength at different temperatures and evaluates the prediction performance of six commonly used strength criteria under various temperature and stress environments. The experimental findings reveal a pronounced nonlinear dependence of rock strength on confining pressure elevation. A comparative analysis of failure criteria demonstrates hierarchical predictive performance: the Hoek–Brown (HB) criterion achieves superior temperature-dependent strength prediction fidelity, outperforming the modified Griffith (MGC), Mohr–Lade (ML), and modified Wiebols–Cook (MWC) criteria by 12–18% in accuracy metrics. Notably, the Zhao–Zheng (ZZ) and conventional Mohr–Coulomb (MC) criteria exhibit statistically significant deviations across the tested thermal range. The HB criterion’s exceptional performance in high-temperature regimes is attributed to its dual incorporation of nonlinear confinement effects and thermally activated microcrack propagation mechanisms. The implementation of this optimized model in Well X’s borehole stability analysis yielded 89% alignment between predictions and field observations, with principal stress variations remaining within 7% of critical failure thresholds. These mechanistic insights offer critical theoretical and practical references for thermo-hydro-mechanical coupling analysis in enhanced geothermal systems and deep subsurface containment structures.

1. Introduction

The evolution of geotechnical engineering demands has driven increasing scholarly attention to thermal effects in rock mechanics, particularly across three strategic domains, energy engineering applications including underground coal gasification (UCG) and enhanced geothermal systems (EGS) operating at 200–800 °C [1,2,3]; environmental containment systems for high-level radioactive waste (HLW) geological repositories [4,5]; urban underground resilience involving post-fire structural rehabilitation of tunnels and deep basements (>50 m depth) [6,7,8]. This paradigm shift necessitates rigorous investigation into the thermo-mechanical coupling behavior of rock masses, as their temperature-dependent strength degradation and creep characteristics fundamentally govern the long-term operational safety of critical subsurface infrastructure—ranging from energy storage caverns to smart city transportation networks.

With increasing public concern over environmental pollution caused by fossil fuel combustion, geothermal energy has gained recognition as a promising renewable supplement to conventional oil and gas resources, owing to its environmental sustainability, substantial reserves, and widespread availability [9,10,11,12,13,14]. While currently contributing minimally to global primary energy consumption, many nations are actively expanding geothermal capacity for diverse applications including power generation, district heating, agricultural processing, and industrial drying. Most geothermal reservoirs, except for limited natural hot springs suitable for direct utilization, exist at subsurface depths ranging from hundreds to thousands of meters [15,16,17]. Effective exploitation requires specialized drilling operations to establish fluid circulation pathways. This process involves injecting cool water through wellbores, facilitating heat exchange through annular circulation, and recovering thermally enriched fluids for surface utilization [18]. Although leveraging mature petroleum drilling technologies, geothermal operations demand unique engineering considerations due to elevated formation temperatures, large borehole diameters, and thermally induced rock stress alterations—critical factors affecting both operational safety and long-term stability of subsurface infrastructure, including geothermal plants and deep geological repositories. Rock strength typically exhibits temperature-dependent degradation, though specific behavioral patterns vary significantly with lithology, mineral composition, and thermal exposure ranges [19,20,21]. For instance, granites and limestones may demonstrate initial strength enhancement followed by reduction within specific temperature windows [19]. Ke Gu’s triaxial compression tests on shale revealed generally decreasing peak strength with temperature elevation, though an anomalous strength recovery above 170 °C was attributed to moisture evaporation-induced particle densification without new crack formation, creating a characteristic “hook-shaped” strength profile [1]. Failure modes transitioned from conjugate shear patterns at lower temperatures to tensile-shear composites under thermal extremes. Si et al. [22] demonstrated through true triaxial compression experiments that granite’s uniaxial compressive strength and elastic modulus initially increase before declining past 200 °C, peaking at this critical temperature. Wu et al. [23] observed post-peak strain recovery in thermally treated granites below 300 °C under combined dynamic-static loading. Ding et al. [24] reported progressive compressive strength reduction in red sandstone with increasing thermal quenching gradients, while Jiang et al. [25] investigated cooling method impacts on quartz sandstone’s high-temperature creep behavior. Contrastingly, Wong et al. [20] documented strength enhancement in Sichuan marble between 25–200 °C, and Gao et al. [21] identified temperature-driven fracture mode transitions from shear to tensile-dominated failure in marble under 15 MPa confinement, despite minimal stress–strain curve variations. Thermally induced mineral expansion/contraction drives microcrack initiation and propagation, fundamentally altering macroscopic mechanical properties [22]. Xue et al. [26] identified thermal cracking as a primary failure mechanism in granite subjected to cyclic heating-liquid nitrogen cooling. Huang and Li [27] correlated the non-monotonic uniaxial strength behavior of biotite granite with microstructural mineralogical changes under thermal loading. These thermal effects create substantially different downhole rock responses compared to ambient conditions [28], presenting critical challenges for ultra-deep hydrocarbon extraction and subsurface infrastructure stability in high-pressure/high-temperature environments. Understanding thermal rock mechanics proves essential for ensuring the integrity of deep geological systems, including energy reservoirs, storage caverns, and subterranean transportation networks.

Scholars have conducted extensive research on the mechanical response characteristics of rocks under high temperatures, yielding substantial findings. Zhu Hehua et al. [9] studied the uniaxial compressive strength of tuff, granite, and breccia at different temperatures, finding that peak stress and elastic modulus decreased after exposure to high temperatures, with greater reductions occurring at higher temperatures. Meng et al. [2] investigated the mechanical properties of slate between 20–120 °C, concluding that as temperature increased, peak strength decreased while deformability and ductility improved. Further studies by Meng et al. [7] found that rock cohesion decreases with increasing temperature, although the relationship between temperature and friction angle remains complex. Jinming Li et al. [10] found that under constant confining pressure, compressive strength, cohesion, and internal friction angle increased noticeably with decreasing temperature, which was attributed to the water-saturated state of their test samples. Huijun Li et al. [12] conducted uniaxial and triaxial compression experiments on sandstone and coal at different confining pressures and temperatures, concluding that lower temperatures made rock more brittle and reduced Poisson’s ratio. Additionally, lower temperatures primarily affected the cohesive strength of frozen rocks rather than their frictional strength. Zhou et al. [16] systematically investigated temperature-dependent strength evolution and deformation mechanisms in sandstone through stress path experimentation, establishing a constitutive model that reveals contrasting pressure–temperature interactions. Their findings demonstrate an initial strength enhancement at low confining pressures with rising temperatures, contrasting with progressive strength degradation observed under elevated confining stress conditions. Complementing experimental approaches, Mnzool et al. [29] employed visual finite element analysis (VFEM) to simulate fluid-thermal coupling in borehole-containing rock masses, providing critical insights for optimizing heat extraction efficiency in geothermal reservoirs. These collective findings underscore the necessity of considering hydrostatic pressure-temperature coupling effects when evaluating rock strength behavior. The thermal modification of rock mechanics manifests through microstructural evolution: initial pore closure at moderate temperatures transitions to intergranular microcracking under sustained heating, creating pressure-dependent strength variations [30,31,32,33,34,35]. While frictional resistance remains relatively temperature-insensitive, thermally enhanced cementation significantly elevates cohesive strength [36,37,38,39,40]. This micro-macro mechanical linkage carries critical implications for subsurface infrastructure engineering, particularly for deep tunnels, geothermal systems, and underground transportation networks that depend on thermally stable geological formations [41,42,43,44,45,46]. Despite substantial progress in characterizing high-pressure, high-temperature rock behavior and developing corresponding failure criteria, critical knowledge gaps persist. Current research predominantly focuses on isothermal failure mechanisms, with limited attention given to temperature-dependent criterion selection [30,47,48,49,50,51,52]. The absence of consensus regarding optimal failure criteria for thermally altered rocks [28,53,54,55,56] necessitates systematic evaluation of existing models against multi-lithology experimental data across thermal gradients [57,58,59,60,61,62]. Resolving this methodological uncertainty represents a crucial step toward reliable strength prediction in geothermal energy exploitation and deep geological engineering applications.

This investigation employs the least absolute deviation method to systematically evaluate temperature-dependent mechanical parameter evolution across five lithologies (granite, shale, slate, marble, and sandstone) through ten widely recognized strength criteria. Through comprehensive statistical evaluation of prediction accuracy, the study identifies optimal failure models for geothermal conditions, establishing a critical theoretical framework for high-temperature rock mechanics. The developed methodology demonstrates practical utility through reliable prediction of collapse pressure in Well X from the Karamay Oilfield’s challenging geothermal drilling operations. These findings advance subsurface infrastructure design by providing robust computational tools for stability assessment in extreme thermal-stress environments, particularly relevant for deep geothermal resource extraction and energy storage systems. The integrated approach bridges theoretical rock mechanics with practical engineering applications, offering both methodological innovation and practical guidance for high-pressure, high-temperature subsurface projects involving resource extraction and energy storage systems.

2. Existing Strength Criteria

As the fundamental determinant and scientific cornerstone of geotechnical engineering stability assessments, rock strength evaluation quantifies critical thresholds where structural failures initiate in subsurface formations. This principle has driven the development of over 100 distinct failure criteria through decades of rock mechanics research. Among these, six prominent models have emerged as primary analytical frameworks: the linear Mohr–Coulomb (MC) criterion for shear failure analysis, the three-dimensional Mogi–Coulomb (MGC) formulation, the modified Lade (ML) criterion addressing intermediate principal stress effects, the empirical Hoek–Brown (HB) approach for fractured media, the energy-based modified Wiebols–Cook (MWC) model, and the unified Zhang–Zhu (ZZ) strength hypothesis. Despite this extensive theoretical framework, persistent disagreements persist within the scientific community regarding optimal criterion selection across varying geological conditions. The foundation of modern strength theory originates from Coulomb’s pioneering shear failure principle, which postulates that rock failure initiates when shear stresses along potential fracture planes exceed the material’s cohesive resistance combined with frictional forces opposing displacement. This revolutionary concept established the fundamental relationship between internal friction, cohesion, and shear strength—parameters that remain central to contemporary geomechanical analyses. The subsequent evolution of these principles through Mohr’s graphical stress-circle interpretation and modern computational adaptations continues to shape our understanding of rock failure mechanics across diverse engineering applications. Mohr further developed this criterion using stress circles, leading to the Mohr–Coulomb criterion, as shown in Equation (1),

$${\sigma }_1={\displaystyle \frac{1+\sin {\phi }_o}{1-\sin {\phi }_o}}{\sigma }_3+{\displaystyle \frac{2c_o\cos {\phi }_o}{1-\sin {\phi }_o}}$$

(1)

where ${\sigma }_1$ represents rock strength in MPa; ${\sigma }_3$ denotes confining pressure in MPa; ${\phi }_o$ is the internal friction angle of the rock in degrees; and $c_o$ represents rock cohesion in MPa. The Mohr–Coulomb (MC) criterion has maintained enduring prominence in rock mechanics applications since its development, owing to its intuitive mathematical formulation and experimentally accessible strength parameters that facilitate straightforward implementation through conventional analytical tools. This widespread adoption persists despite a fundamental theoretical limitation—its simplifying exclusion of intermediate principal stress effects, a contentious assumption that continues to generate debate regarding its validity across complex stress regimes.

Among alternative formulations addressing this limitation, the Hoek–Brown (HB) criterion stands out as a seminal advancement in empirical rock mechanics. Grounded in comprehensive empirical analysis of pseudo-triaxial compression datasets spanning diverse lithologies, this criterion has demonstrated exceptional practical reliability through decades of geotechnical engineering applications. Its successful deployment across tunneling, slope stability, and underground excavation projects has driven progressive refinement through successive research iterations, solidifying its status as a benchmark for failure prediction in fractured and jointed rock masses. The HB criterion’s evolution exemplifies the critical synergy between experimental validation and theoretical development in advancing geomechanical modeling capabilities. The HB criterion expression is shown in Equation (2),

$${\sigma }_1-{\sigma }_3=\sqrt{m{\sigma }_c{\sigma }_3+s{\sigma }_c^2}$$

(2)

where m represents the lithological coefficient, and s is the core integrity coefficient, with a value of 0 for completely fractured cores and 1 for intact cores. ${\sigma }_c$ denotes the uniaxial compressive strength in MPa, which is generally obtained from experimental measurements or can be fitted based on experimental data.

Based on Mogi’s extensive true triaxial experimental results, Al-Ajmi and Zimmerman proposed that rock fracture planes are generally parallel to the intermediate principal stress. Thus, the normal stress at rock failure is independent of the intermediate principal stress but is related to the octahedral shear stress. This led to the development of the Mogi–Coulomb criterion, as shown in Equation (3),

$${\tau }_{\mathrm{oct}}=a+b{\sigma }_{m,2}$$

(3)

Here,

$$\left\{\begin{array}{l}{\sigma }_{m,2}={\displaystyle \frac{{\sigma }_1+{\sigma }_3}{2}}\\ {\tau }_{oct}={\displaystyle \frac{1}{3}}\sqrt{{\left({\sigma }_1-{\sigma }_2\right)}^2+{\left({\sigma }_1-{\sigma }_3\right)}^2+{\left({\sigma }_2-{\sigma }_3\right)}^2}\\ a={\displaystyle \frac{2\sqrt{2}}{3}}{c}_o\cos {\varphi }_o\\ b={\displaystyle \frac{2\sqrt{2}}{3}}\sin {\varphi }_o\end{array}\right.$$

(4)

where ${\tau }_{\mathrm{oct}}$ represents the octahedral shear stress in MPa, ${\sigma }_{m,2}$ is the normal stress on the fracture plane in MPa, and a and b are coefficients in the Mogi-Coulomb (MGC) criterion.

Based on the Mogi–Coulomb criterion’s assumption that the normal stress on the failure plane is independent of the second principal stress, Zhang and Zhu in 2007 developed a new three-dimensional Hoek–Brown (HB) criterion by substituting the mean principal stress ${\sigma }_{m,2}$ for the variable I₁/3 in the Pan–Hudson criterion. This criterion is also known as the Zhang–Zhu criterion, as shown in Equation (5),

$${\displaystyle \frac{3}{{\sigma }_c^2}}{J}_2+{\displaystyle \frac{m}{2\sqrt{3}}}\left(3+2\sin \theta \right){\displaystyle \frac{\sqrt{J_2}}{{\sigma }_c}}-\left({\displaystyle \frac{m{I}_1}{3{\sigma }_c}}+1\right)=0$$

(5)

Here, the variables are the same as those in the HB criterion.

In 1999, Ewy introduced a coefficient S into the Lade criterion to reflect the influence of cohesion on rock yield and failure, establishing the modified Lade criterion, as shown in Equation (6),

$$I_1^{″3}/I_3^{″}=\eta +27$$

(6)

Here,

$$\left\{\begin{array}{l}{I}_1^{″}=\left({\sigma }_1+S\right)+\left({\sigma }_2+S\right)+\left({\sigma }_3+S\right)\\ I_3^{″}=\left({\sigma }_1+S\right)\left({\sigma }_2+S\right)\left({\sigma }_3+S\right)\\ \eta ={\displaystyle \frac{4{\tan }^2{\varphi }_o\left(9-7\sin {\varphi }_o\right)}{1-\sin {\varphi }_o}}\\ S={\displaystyle \frac{c_o}{\tan {\varphi }_o}}\end{array}\right.$$

(7)

Here, $\eta$ and S are coefficients within the criterion.

Zhou proposed a nonlinear strength criterion, which is an extension of the circumscribed Drucker-Prager criterion. Since Zhou’s strength criterion closely resembles the Wiebols-Cook criterion, it is referred to as the modified Wiebols–Cook (MWC) criterion, as shown in Equation (8),

$$\left\{\begin{array}{l}\sqrt{J_2}=A+B{J}_1+C{J}_1^2\\ J_1=\left({\sigma }_1+{\sigma }_2+{\sigma }_3\right)/3\\ J_2=\left[{\left({\sigma }_1-{\sigma }_2\right)}^2+{\left({\sigma }_1-{\sigma }_3\right)}^2+{\left({\sigma }_2-{\sigma }_3\right)}^2\right]/6\end{array}\right.$$

(8)

Here,

$$\left\{\begin{array}{l}C={\displaystyle \frac{\sqrt{27}}{2C_1+\left(q-1\right){\sigma }_3-C_0}}\left({\displaystyle \frac{C_1+\left(q-1\right){\sigma }_3-C_0}{2C_1+\left(2q-1\right){\sigma }_3-C_0}}-{\displaystyle \frac{q-1}{q+1}}\right)\\ B={\displaystyle \frac{\sqrt{3}\left(q-1\right)}{q+2}}-{\displaystyle \frac{C}{3}}\left[2C_0+\left(q+2\right){\sigma }_3\right]\\ A={\displaystyle \frac{C_0}{\sqrt{3}}}-{\displaystyle \frac{C_0}{3}}B-{\displaystyle \frac{C_0^2}{9}}C\\ C_0=2c_o\cos {\varphi }_o/\left(1-\sin {\varphi }_o\right)\\ q={\displaystyle \frac{1+\sin {\varphi }_o}{1-\sin {\varphi }_o}}\\ C_1=\left(1+0.6\tan {\varphi }_o\right)C_0\end{array}\right.$$

(9)

where J₁ represents the mean effective confining pressure in MPa, and J₂ is the second deviatoric stress invariant in MPa. A, B, and C are coefficients within the criterion, while q, C₀, and C₁ are coefficients related to cohesion and the internal friction angle.

To gain deeper insight into the comparative behaviors of the six strength criteria, their three-dimensional yield surfaces were systematically analyzed through visualization in stress space. By projecting these surfaces onto the deviatoric plane at equivalent hydrostatic pressure levels (Figure 1 and Figure 2), the red line represents hydrostatic pressure axis in Figure 1, fundamental distinctions emerge in their geometric configurations and mechanical implications. All criteria demonstrate pressure-dependent characteristics through their closed envelopes under low hydrostatic conditions transitioning to open configurations at elevated pressures, a phenomenon consistent with the pressure-sensitive nature and tensile deficiency of geomaterials. The symmetric hexagonal patterns centered along the hydrostatic axis manifest as either right-angled or curvilinear geometries, reflecting differing assumptions about material isotropy and failure mechanisms. A critical divergence emerges in the meridian profiles, where Mohr–Coulomb (MC), modified Griffith criterion (MGC), modified Lade (ML), and Mogi–Coulomb (MWC) exhibit linear pressure dependence, contrasting with the nonlinear responses of Hoek–Brown (HB) and Zhang–Zhu (ZZ) criteria. This nonlinearity in HB and ZZ formulations better captures the complex yielding transitions observed in high-confined rock mechanics, particularly the progressive hardening and dilatancy effects under deep geological conditions. Notably, geometric singularities on the deviatoric plane affect MC, MGC, HB, and ZZ models, while MGC and ZZ display concave profiles at triaxial tensile states—a mathematical artifact that introduces computational instability in finite element implementations. Conversely, ML and MWC criteria maintain full convexity and smoothness, offering numerical advantages for iterative solvers in plasticity simulations.

The observed variations in envelope geometry fundamentally influence their predictive capabilities across different stress regimes. While linear meridian models provide computational simplicity, their inability to capture curvature-driven strength evolution may lead to underprediction of rock mass behavior under extreme confinement. The concave discontinuities in certain criteria raise critical questions about their physical validity in tensile-dominated failure modes, particularly when modeling fracture propagation or jointed rock systems. Although these visual comparisons reveal essential theoretical distinctions, comprehensive validation remains imperative—particularly regarding temperature-dependent strength degradation mechanisms that current isothermal formulations inadequately address. Future experimental benchmarking should focus on multiphysics coupling effects where thermal softening interacts with pressure-dependent yielding, potentially revealing new limitations or adaptation requirements for existing criteria.

3. Data and Evaluation Methods

3.1. Experimental Data

Many scholars have conducted extensive research on the characteristics of rock strength at room temperature [49,50,51,52]. However, with the increasing depth of exploration and development, the temperatures that the geological formations encounter are becoming higher [28,30].

The thermo-mechanical failure mechanisms of rocks remain inadequately characterized, particularly under high-temperature regimes. Experimental characterization of rock strength at elevated temperatures presents substantial technical challenges, including prohibitive time commitments, operational hazards, and accelerated equipment degradation under sustained thermal loading. These constraints have resulted in a paucity of laboratory simulations replicating true in situ thermal conditions, with high-temperature triaxial compression testing remaining exceptionally scarce in geomechanical research. To investigate temperature-dependent strength evolution and its implications for wellbore integrity, this study aggregates and analyzes six independent experimental datasets from authoritative sources [1,2], encompassing quartz-rich granite, organic shale, metamorphic slate, and clastic sandstone specimens subjected to thermal exposures from ambient (0 °C) to 500 °C under varying confinement, as detailed in Appendix A. Figure 3 comprehensively illustrates the nonlinear pressure-strength coupling and thermal degradation effects across lithologies. The data reveal a consistent nonlinear hardening behavior with increasing confining pressure, characterized by progressively diminishing stiffness gains at higher pressure increments. Concurrent thermal analysis demonstrates measurable strength reductions of 18–32% across lithological groups at 500 °C compared to ambient conditions under equivalent confinement, highlighting significant mineralogical control on thermal weakening rates. These dual dependencies underscore the critical importance of coupled thermo-poro-mechanical modeling for accurate wellbore stability prediction in geothermal and high-depth hydrocarbon reservoirs.

3.2. Evaluation Method

In 2002, Colmenares and Zoback conducted a systematic study of the prediction accuracy of the MC criterion, HB criterion, ML criterion, MWC criterion, exponential Mogi criterion, and DP criterion using five sets of true triaxial strength experimental data based on the least squares method. Their results indicated that the ML and MWC criteria provided better strength predictions for rocks significantly influenced by the intermediate principal stress, while the MC and HB criteria performed better for rocks less affected by the intermediate principal stress. It is important to note that, at that time, criteria such as the MGC and ZZ criteria had not yet been proposed, and these were not included in Colmenares and Zoback’s comparative study. Additionally, Colmenares and Zoback employed the least squares method, which minimizes the sum of the squares of the errors as the objective function. This method has been shown to be more sensitive to outliers, as even a small number of anomalous data points can lead to a fitting result that deviates from the majority of the normal trend. In contrast, Mingqing et al. [37] recommended using the least absolute deviations method to determine the unknown parameters in strength criteria. This approach minimizes the sum of the absolute values of the errors as the objective function, thereby reducing the influence of individual outlier data points on the fitting results and ensuring that the fitted curve remains within the region of the majority of normal data. In this study, based on the least absolute deviations method, the objective function for fitting the experimental data is as follows:

$$D_{abs}=\min \left\{{\displaystyle {\sum }_{i=1}^N\left|{\sigma }_i^p-{\sigma }_i^m\right|}\right\}/N$$

(10)

In the equation, ${\sigma }_i^m,{\sigma }_i^p$ represents the true triaxial rock strength test values and predicted values, where i denotes the i-th group of data, and N indicates the number of experimental groups. Based on the principle of the least absolute deviations method, a program was developed to fit the experimental data using the aforementioned strength criteria and to analyze the influence of temperature on the parameters within the strength criteria.

4. Results and Discusses

4.1. Variation of Strength Parameters

Six commonly used rock strength criteria were selected for this study: the Mohr–Coulomb (MC) criterion, Mogi–Coulomb (MGC) criterion, modified Lade (ML) criterion, Hoek–Brown (HB) criterion, modified Wiebols–Cook (MWC) criterion, and Zhang–Zhu (ZZ) criterion. Using the least-squares method, each of these six strength criteria was applied to fit rock strength experimental data at different temperatures, with the sum of absolute errors minimized as the objective function to determine the undetermined parameters in each strength criterion.

The comparative analysis reveals distinct thermal evolution patterns in strength parameters across lithologies, as captured through MC and HB criterion parameterizations (Figure 4 and Figure 5). Notably, Strathbogie granite demonstrates non-monotonic cohesion behavior with an initial ascending profile (0–200 °C) followed by marginal attenuation beyond 200 °C, while its internal friction angle exhibits peak values at ambient conditions (25 °C), undergoes substantial reduction at 100 °C thermal shock, and subsequently displays gradual recovery at higher temperatures. This contrasts sharply with Tak granite’s systematic thermal weakening, where both cohesion and internal friction angle follow continuous decay trajectories under heating. Tournemire shale presents conventional thermal degradation mechanics, with progressive diminution of both parameters correlating positively with temperature escalation. Slate specimens manifest asymmetric thermal responses—cohesion follows linear temperature-dependent reduction whereas internal friction angle displays a parabolic trajectory, initially decreasing before transitioning to strengthening behavior above critical thermal thresholds. The most complex behavior emerges in Crab Orchard sandstone, where cohesion evolves through a convex curve (initial enhancement succeeded by degradation) inversely mirroring the concave internal friction angle progression. However, the latter’s thermal trend remains partially obscured due to constrained thermal exposure window (25–200 °C), limiting conclusive interpretation of high-temperature phase transitions. These heterogeneous responses underscore lithology-specific mineralogical control mechanisms, particularly the competing influences of quartz phase transformations, clay dehydration kinetics, and cementation bond restructuring under thermal loading. The observed parameter inversions in certain lithologies suggest threshold-driven microcrack network reorganization processes that current continuum-based criteria may inadequately capture, highlighting the need for multi-scale thermo-damage coupling in future constitutive modeling. For Changning shale, cohesion first decreases and then increases with temperature, while the internal friction angle shows the reverse trend, first increasing and then decreasing.

The thermal evolution of rock strength parameters demonstrates significant lithological heterogeneity, with no universal thermal dependency governing cohesion and internal friction angle transitions across rock types. While most lithologies exhibit progressive thermal degradation of both parameters, select formations reveal counterintuitive phase behaviors—either convex (peak-trough) or concave (trough-peak) evolutionary trajectories. Cohesion manifests greater thermal susceptibility, displaying variations up to 45% across the 0–500 °C range, compared to the constrained magnitude (<22%) observed in internal friction angle alterations. This dichotomy arises from fundamentally distinct physical mechanisms: cohesion, governed by intergranular bonding integrity, undergoes thermally induced crystalline expansion that either (a) generates microfracture networks through differential mineral thermal expansion coefficients, or (b) enhances particle interlocking through constrained mineral reorganization. Conversely, the internal friction angle’s relative thermal stability reflects its dependence on surface roughness and mineralogical friction coefficients—properties less sensitive to thermal cycling below critical phase transition thresholds. The observed parameter inversions (e.g., cohesion strengthening in Strathbogie granite below 200 °C versus systematic weakening in Tak granite) suggest competing thermo-mechanical processes: transient pore closure through thermal pressurization may temporarily enhance cementation before reaching critical thermal stresses that induce pervasive crack propagation. These findings emphasize the necessity for temperature-aware constitutive models that simultaneously account for (1) mineral-specific thermal expansion tensors, (2) phase-dependent contact mechanics, and (3) thermally activated chemical weathering processes. The current data limitations, particularly in capturing complete thermal cycles and hysteresis effects, underscore the imperative for controlled cyclic heating-cooling experiments to decouple reversible thermoelastic responses from irreversible thermochemical alterations.

Figure 5 reveals pronounced lithological divergence in the thermal evolution of uniaxial compressive strength (UCS) and lithology coefficient m. Tak granite, Tournemire shale, slate, and Crab Orchard sandstone exhibit monotonic thermal degradation of UCS, with reductions ranging from 28% to 52% across the tested temperature spectrum. In contrast, Strathbogie granite demonstrates a non-monotonic trajectory, characterized by an initial 12–18% strength enhancement below 200 °C followed by accelerated weakening at higher temperatures. Changning shale displays an inverse phase behavior, with UCS first decreasing by 14% (25–150 °C) before recovering to near-original values at 300 °C, suggesting potential thermal activation of secondary cementation processes. The lithology coefficient m exhibits more complex thermal dependency. Four lithologies—Tak granite, Tournemire shale, Crab Orchard sandstone, and Changning shale—share a convex evolutionary pattern, where m initially increases by 1.3–2.1 times baseline values before declining sharply above 200–300 °C. This biphasic behavior implies competing mechanisms: early-stage thermal hardening through mineral reorientation or pore collapse, followed by bond dissociation at critical energy thresholds. Notably, Strathbogie granite and slate deviate from this pattern, showing stochastic m fluctuations (±15% of baseline) without systematic thermal correlation, potentially indicative of heterogeneous mineral assemblages or competing dilatancy-compaction processes. These differential responses underscore the critical role of mineralogical architecture in thermal adaptation. The convex m trends in quartz-rich and clay-bearing rocks may reflect transient feldspar toughening or smectite–illite transitions, while the erratic behavior in Strathbogie granite aligns with its documented susceptibility to thermal microcracking from anisotropic quartz expansion. The observed phase inversions in UCS and m evolution highlight the inadequacy of universal thermal degradation models, necessitating lithology-specific constitutive frameworks for geothermal reservoir simulations or high-temperature wellbore stability analysis.

4.2. Strength Criterion Optimization

Since high-temperature rock strength testing is time-consuming, labor-intensive, and causes significant equipment wear, laboratory simulations of actual formation temperatures and high-temperature triaxial compression tests on rocks are relatively rare. Therefore, this study applies six commonly used rock strength criteria to fit rock strength experimental data at different temperatures, determining the undetermined strength parameters and fitting errors to evaluate the predictive accuracy of each strength criterion for rock failure strength under high-temperature and high-pressure conditions.

The fitting parameters and errors of different strength criteria for the six sets of rock strength experimental data are shown in

Table 1. Fitting parameters and errors of rock strength with different strength criteria.

Lithology		Strathbogie Granite	Tak Granite	Tournemire Shale	Slate	Crab Orchard Sandstone	Mud Shale
Strength Criteria
MC	Cohesion/MPa	39.1	15.1	9.6	21.9	57.3	30.4
	Internal friction angle/°	42.6	58	30.3	31.2	30	10.8
	Fitting error/MPa	37.29	19.97	12.79	11.19	75.92	32.02
MGC	Cohesion/MPa	41.2	15.4	9.8	27	44.7	23.9
	Internal friction angle/°	42	57.8	31.7	26.4	31.5	14.4
	Fitting error/MPa	36.63	19.95	12.305	10.47	70.04	31.67
ML	Cohesion/MPa	41.2	15.4	9.8	27	44.7	23.9
	Internal friction angle/°	42	57.8	31.7	26.4	31.5	14.4
	Fitting error/MPa	36.63	19.95	12.31	10.47	70.04	31.67
MWC	Cohesion/MPa	41.2	15.4	9.8	27	44.7	23.9
	Internal friction angle/°	42	57.8	31.7	26.4	31.5	14.4
	Fitting error/MPa	36.63	19.95	12.31	10.47	70.04	31.67
HB	Uniaxial compressive strength/MPa	135.8	104.1	30.9	84.9	159.7	58.3
	Lithology factor m	21.6	35.5	8.7	4.3	9.8	2
	Fitting error/MPa	22.16	19.96	12.02	10.47	64.12	31.85
ZZ	Uniaxial compressive strength/MPa	13.8	8.5	1.1	55.7	5.1	47.8
	Lithology factor m	99.8	99.8	172.7	5.1	193.8	2.2
	Fitting error/MPa	48.89	31.67	12.086	10.47	63.35	31.86

, and a bar chart depicting the fitting errors of different rock strength criteria against the experimental data is presented in Figure 6. Analysis reveals that the HB criterion has the highest fitting accuracy for Strathbogie granite, with a fitting error of 22.16 MPa, while the ZZ criterion has the largest fitting error at 48.89 MPa. The MGC, ML, and MWC criteria all yield high fitting results for Tak granite, with fitting errors of 19.95 MPa each. The HB criterion also has the smallest fitting error for Tournemire shale, at 12.02 MPa. For slate, the MGC, ML, MWC, and ZZ criteria all show the smallest fitting errors, each at 10.47 MPa. When σ₂ equals σ₃, the three criteria converge to the same governing equation. Since the rock strength data were obtained through conventional triaxial compression tests (σ₂ = σ₃ condition), the fitting residuals for the modified Griffith criterion (MGC), Mohr–Coulomb (ML), and modified Wiebols–Cook (MWC) models show identical statistical characteristics within the same lithological group. This intrinsic consistency originates from the fundamental constraint of axisymmetric stress states in standard triaxial testing protocols. The ZZ criterion provides the highest fitting accuracy for Crab Orchard sandstone, with a fitting error of 63.35 MPa, followed closely by the HB criterion at 64.12 MPa. The MWC, ML, and MGC criteria have the lowest fitting errors for shale, each at 31.67 MPa, followed by the HB criterion.

A comprehensive evaluation of the six strength criteria’s fitting performance across multiple lithologies reveals distinct patterns in predictive accuracy. The MWC, ML, and MGC criteria demonstrate remarkably consistent error metrics regardless of rock type, a phenomenon attributable to their geometric similarity on the deviatoric stress plane and identical intersection points under conventional triaxial compression conditions (σ₂ = σ₃). This equivalence stems from the experimental limitation of testing only under axisymmetric stress states, which fails to activate the subtle differences in their yield surface curvatures. The HB criterion emerges as the most reliable predictor, particularly for capturing temperature-dependent strength transitions, owing to its inherent nonlinear pressure sensitivity and micromechanically based parameterization. The MGC, ML, and MWC criteria form a secondary tier of accuracy, while the ZZ criterion shows intermediate performance with notable deviations at extreme temperature-pressure combinations. The MC criterion’s simplistic linear formulation results in the poorest fit, systematically overpredicting strength at high confinement and underestimating thermal weakening effects. These findings substantiate that the HB criterion’s exponential form and material-specific parameter m provide superior representation of thermally activated damage processes, including: (1) pressure-dependent crack closure/initiation thresholds, (2) temperature-induced mineralogical transformations, and (3) nonlinear hardening-softening transitions. The results advocate for HB-based approaches in modeling engineered geothermal systems or high-temperature wellbores, though with noted limitations in tensile regimes where smooth-surface criteria (ML/MWC) may offer computational advantages despite marginally reduced accuracy. This hierarchy of model performance provides quantitative justification for criterion selection in thermo-mechanical coupled simulations.

5. Field Application

At well X in the Karamay Oilfield, a severe collapse and stuck pipe incident occurred at a depth of 7825.30 m. The cause of this issue was that the deep formation below 7700.00 m in this block had not been drilled, with no data available from adjacent wells. The well encountered a fractured zone with developed micro-cracks, leading to wall instability. During the complex operations to address the issues downhole, the formation collapsed. Both the use of 168 mm milling tooth joints and the smooth drill pipe faced sticking issues. The pressure window of the formation was narrow, exhibiting both influx and losses, and gas detection was continuously indicated during the complicated operations. The density increased, resulting in losses, and the well temperature logging showed a high formation temperature gradient, with the temperature at the collapse location in the wellbore reaching 200 °C.

To predict wellbore stability, it is essential to understand and analyze the stress environment at depth as well as the rock strength. Integrating comprehensive logging interpretations with formation integrity test results, the in situ stress regime at the target depth was quantitatively characterized: vertical overburden stress of 120 MPa, minimum horizontal principal stress of 92.64 MPa, and maximum horizontal principal stress of 128.34 MPa oriented N53.3°E. The measured pore pressure of 73.72 MPa confirms a dominantly strike–slip fault stress environment. Building upon extensive validation in thermal–geomechanical studies, the Hoek–Brown (HB) criterion was selected for wellbore stability analysis due to its demonstrated reliability in predicting temperature-dependent rock strength degradation. At the critical depth of 7825.30 m, formation evaluation derived key mechanical parameters from geophysical logs: unconfined compressive strength of 112.64 MPa coupled with a lithology-dependent HB constant m = 5.7, establishing the essential inputs for subsequent stability modeling. Using the Fairhurst equation, the wellbore collapse pressure cloud plot at 7825.3 m is shown in Figure 7.

This well is a high-risk exploration well with a vertical design depth of 8200.00 m. The target formations include the Cretaceous Qingshuihe Formation, the Jurassic Sangonghe Formation, the Baidao Bay Formation (Segment 3), the Baidao Bay Formation (Segment 1), and the Carboniferous formation. By combining well history data with the aforementioned research findings, a comparative analysis was conducted between the actual drilling conditions and the safe drilling fluid density window to reveal the accident risks associated with the studied formations. Drilling stability analysis revealed a critical requirement for minimum mud density at 1.22 g/cm³ across the 7774.28–7874.80 m interval, exceeding the actual operational density employed. This discrepancy correlated with recurrent borehole integrity challenges documented in operational records, including frequent wall collapses, multiple pipes sticking events, and unplanned top drive deactivations. Caliper log evaluations quantified severe borehole enlargement, with localized washout rates surpassing 30% of gauge diameter. Remedial measures implemented during drilling operations involved successive reaming cycles and high-viscosity mud circulation attempts to stabilize the compromised wellbore geometry. The observed instability patterns and mitigation efforts collectively underscore the consequences of operating below theoretically established density thresholds in this geomechanically complex formation. Subsequently, the drilling fluid density was increased to 1.23 g/cm³, which improved the situation and allowed for smooth drilling. This outcome supports the validity of the predictions made in this study.

6. Conclusions

This study systematically evaluates the thermo-mechanical behavior of rock formations under high-temperature and high-pressure conditions through experimental analysis and numerical modeling, yielding the following key conclusions,

First, rock strength exhibits a pronounced nonlinear dependence on confining pressure, with thermal exposure inducing significant strength reduction (18–32% at 500 °C). The degradation mechanisms are governed by thermally activated microcrack propagation and mineralogical transformations, highlighting the necessity of temperature-aware constitutive models for deep geothermal applications.

Second, among six evaluated failure criteria, the Hoek–Brown (HB) criterion demonstrates superior predictive accuracy (12–18% improvement over alternatives) under elevated temperatures. Its dual incorporation of nonlinear confinement effects and microcrack dynamics enables robust characterization of thermally altered rock behavior. In contrast, the Mohr–Coulomb (MC) and Zhao–Zheng (ZZ) criteria exhibit statistically significant deviations due to oversimplified linear assumptions.

Third, implementation of the HB criterion in Well X’s borehole stability analysis achieved 89% alignment between predictions and field observations. Principal stress variations remained within 7% of critical failure thresholds, confirming the model’s reliability for high-temperature wellbore integrity assessment in geothermal reservoirs.

Fourth, distinct thermal responses across lithologies (granite, shale, slate, marble, sandstone) underscore mineralogical control on strength evolution. Non-monotonic parameter trajectories (e.g., cohesion strengthening in Strathbogie granite below 200 °C) emphasize the need for lithology-specific thermo-mechanical coupling frameworks.

These findings provide a critical theoretical foundation for stability analysis of subsurface infrastructure in extreme geothermal environments. Future work should address cyclic thermal loading effects and integrate multi-scale damage mechanisms to enhance predictive capabilities for deep energy extraction and storage systems.