Physical and Mechanical Properties and Constitutive Model of Rock Mass Under THMC Coupling: A Comprehensive Review

Abstract

Research on the multi-field coupling effects in rocks has been ongoing for several decades, encompassing studies on single physical fields as well as two-field (TH, TM, HM) and three-field (THM) couplings. However, the environmental conditions of rock masses in deep resource extraction and underground space development are highly complex. In such settings, rocks are put through thermal-hydrological-mechanical-chemical (THMC) coupling effects under peak temperatures, strong osmotic pressures, extreme stress, and chemically reactive environments. The interaction between these fields is not a simple additive process but rather a dynamic interplay where each field influences the others. This paper provides a comprehensive analysis of fragmentation evolution, deformation mechanics, mechanical constitutive models, and the construction of coupling models under multi-field interactions. Based on rock strength theory, the constitutive models for both multi-field coupling and creep behavior in rocks are developed. The research focus on multi-field coupling varies across industries, reflecting the diverse needs of sectors such as mineral resource extraction, oil and gas production, geothermal energy, water conservancy, hydropower engineering, permafrost engineering, subsurface construction, nuclear waste disposal, and deep energy storage. The coupling of intense stress, fluid flow, temperature, and chemical factors not only triggers interactions between these fields but also alters the physical and mechanical properties of the rocks themselves. Investigating the mechanical behavior of rocks under these conditions is essential for averting accidents and assuring the soundness of engineering projects. Eventually, we discuss vital challenges and future directions in multi-field coupling research, providing valuable insights for engineering applications and addressing allied issues.

1. Introduction

Over the past three decades, there has been growing curiosity about multi-field couplings within rock masses, driven by the need to accomplish assessments and optimize designs in radioactive waste disposal, geothermal energy, and deep underground mining. Multi-field coupling refers to the interaction between processes like fluid flow and stress–strain behavior, where one operation influences the commencement and progression of another process [1]. In rock mechanics, these couplings describe the interaction between stress, oozing, and thermal fields, which are integral to rock mass formation and evolution, influenced by both natural forces and human activities. Coupled processes imply that a piece of each process affects and is influenced by the others. Consequently, the behavior of rock masses in scenarios like radioactive waste storage cannot be accurately forecast by isolating these processes. While much research has focused on two-way couplings like TM and HM, understanding and predicting full THM and THMC couplings is essential, especially for repository performance [2,3,4,5,6,7,8]. This remains a significant challenge caused by the differing impermanent and spatial scales of the processes involved and the need for improved constitutive models. Coupled processes are also critical in subsurface engineering fields, including carbon sequestration, geothermal systems, energy storage, and distinctive oil and gas productivity through hydraulic stimulation. Temperature significantly impacts the behavior of geomaterials, which is essential in geotechnical and underground engineering [9]. Various applications, such as high-level nuclear waste disposal [10,11,12,13,14], heat storage [15], geothermal structures [16,17], petroleum drilling [18,19], and zones surrounding high-voltage cables [20], rely on understanding these effects. Recent studies have particularly focused on the TM behavior of geomaterials like Boom Clay [21,22,23,24], Opalinus Clay [25,26,27], Callovo-Oxfordian argillite [28], and Äspö Pillar [29,30], emphasizing the importance of evaluating their mechanical properties under long-term exposure to high temperatures. These temperature-induced changes affect mechanical parameters such as friction angle [31,32], permeability/porosity [33,34,35], elastic modulus [36,37], shear strength [38,39,40], and dilatancy [41] and lead to softening, brittle-to-ductile transitions [42,43], and creep behavior [27,28]. Wei et al. conducted Scanning Electron Microscope (SEM) tests on pre-cracked red sandstone prototypes to assess the efficacy of a THMC coupling indicator, which was developed to quantify the impact of THMC coupling fields on the stress field through rock porosity analysis. This indicator, measured in real-time using a novel permeability method for brittle fracture criteria, enables a continuous evaluation of THMC-field effects on rock stability [44]. Under external loads, natural discontinuities like cracks or joints in brittle rock can initiate and propagate rapidly, often leading to unstable fracture progression and eventual failure of the rock mass [45,46,47]. These findings are particularly relevant in the context of underground nuclear waste repositories, where elevated temperatures could alter the steadiness and long-term probity of the surrounding rock. The overall understanding of these changes provides insights into optimizing underground storage and other geotechnical applications.

As underground spaces continue to be developed and mineral resource extraction deepens, the environmental conditions in which rocks exist have become significantly more complex. Studies indicate that rocks and soils can experience significant changes in mechanical properties when exposed to elevated temperatures over various time frames. Since Prager’s foundational work on non-isothermal plastic deformation modeling [48], numerous constitutive models have emerged, incorporating these thermal impacts within computational inelasticity frameworks [49]. Many of these models extend the critical-state model [50], enabling a more comprehensive analysis of temperature effects on the short-term behavior of clays and rocks [51,52,53,54,55]. Research into the multi-field coupling of rocks began in the 1950s, initially driven by the need to understand reservoir-induced seismic activity. Early investigations revealed that the behavior of rocks under multi-field conditions could not be explained by examining individual physical fields in isolation. Instead, it became clear that these fields interact in complex ways, with changes in one field potentially triggering or intensifying effects in another. This realization marked a significant shift in the approach to studying rock mechanics under multi-field coupling conditions, emphasizing the need for a more comprehensive understanding.

In the study of multi-field coupling, two primary methods have emerged: direct coupling and sequential coupling. Direct coupling involves solving for the interactions between multiple physical fields, such as temperature, fluid flow, mechanical stress, and chemical reactions, simultaneously. This method is computationally efficient for scenarios where real-time interactions are crucial, as it avoids the need for iterative recalculations between fields. However, due to the high computational demand and the complexity of real-world systems, direct coupling is often limited to simpler models or small-scale scenarios. In contrast, sequential coupling is the more commonly used approach in multi-field research. This method involves iteratively calculating the effects of each field one after the other, feeding the results of one calculation into the next until the system reaches convergence. Although this process is more time-consuming, it allows for greater flexibility and accuracy when dealing with complex systems where interactions between fields evolve gradually over time [56,57,58,59]. Sequential coupling is particularly useful for studying the long-term behavior of rocks under prolonged exposure to combined physical forces, such as those found in underground mining operations or deep geological storage facilities. It enables researchers to model not just the immediate effects of multi-field interactions but also the cumulative impact of these forces on rock stability over time. The rock mass itself is comprised of intact rock blocks interspersed with fractures, and research into multi-field coupling primarily focuses on the evolution of damage within these fractured rock masses.

Bond et al. [60,61] synthesize findings from international modeling teams on joint novaculite and granite research. Individual team contributions detail varying approaches: McDermott et al. [62] devised hybrid mathematical and analytical solutions for modeling THMC-coupled processes with fluid flow in fragmented rock; Chittenden et al. [63] analyzed the congener significance of THMC interactions in fluid flow investigations on ruptured novaculite and granite; Pan et al. [64] employed an elastoplastic cellular automaton (EPCA) approach to simulate THMC coupling within a single crack; and Lu et al. [65] examined water–granite interactions, focusing on pressure-solution effects in flow through a single crack under confinement.

Currently, the analysis of fracture evolution, deformation mechanisms, mechanical constitutive modeling, and the construction of multi-field coupling models are among the most challenging and active areas of research in rock mechanics. Scholars worldwide have recognized the importance of this field, and it has garnered significant attention. For example, the DECOVALEX project [66] (DEmonstration of COupled models and their VALidation against EXperiments), an international collaborative effort that began in 1992, has conducted extensive multi-scale experiments on the multi-field coupling of rocks. This project developed and validated THM cross-coupling models, which represented the cutting-edge research at the time. Later phases of the project, such as DECOVALEX–THMC (2004–2007), introduced chemical processes into the modeling framework, further advancing our understanding of how these interactions affect rock masses [67].

Several constitutive models have been established to characterize the behavior of claystone rocks [68,69,70,71,72,73,74,75]. Mánica et al. [74,75,76,77] introduced a set of leading-edge elastoplastic models within a coupled hydro-mechanical framework, integrating properties such as hardening–softening, rigidity, strength anisotropy, and time-dependency, which were validated against experimental results. The proposed model offers a promising approach to simulating key features of THMC behavior in clay rocks. It holds significant potential for applications in various fields, including the thermal fracturing analysis of underground nuclear waste disposal systems, mineral resource extraction, oil and gas, geothermal energy, and various resource and energy sectors [78]. Additionally, it applies to water conservancy, hydropower projects, alpine territory infrastructure, and subsurface construction [79,80,81,82], where multiplex physical and chemical situations can notably affect the mechanical characteristics of rock masses, potentially leading to structural collapses [83]. The THMC coupling indicator, established upon rock porousness, quantifies the influence of its interactions on the stress field by using an innovative real-time permeability measurement technique to examine how these indicator interactions affect stress and deformation within rock structures, enhancing our understanding [84]. Existing research on THMC-coupled constitutive models for rocks encompasses various aspects: the thermal field captures processes like conduction, convection, dissipation, and radiation [85,86] and influences fluid density, viscosity, and rock mechanical properties, as well as reaction rates and equilibria [87]. The hydraulic field centers on Darcy or non-Darcy flows, with factors such as porosity, volumetric strain, stability, thermal impacts [88,89], rock deformation, and mineral dissolution and precipitation [90]. The mechanical field accounts for pore pressure and cementation phase [91], with the stress domain impacting heat generation from friction, permeability alterations, and chemical transport paths [92,93]. Lastly, the chemical field models reactive particle transport, compositional changes like weathering and mineral transformations [94], contaminant infiltration, and reactive fluids from engineering activities, affecting rock permeability and conductivity via compositional and structural modifications [95]. These fluid–rock reactions occur in complex hydraulic–chemical environments [96].

In recent years, researchers have made considerable progress in both the theoretical and experimental aspects of multi-field coupling in rocks [44,97,98,99,100,101,102]. However, the complexity of these interactions means that there is still much to learn. Moving forward, it will be essential to conduct systematic analyses of previous research to identify the key achievements and challenges in this field, as shown in

Table 1. Notable multi-field coupling modeling solutions and advanced constitutive models in geological systems.

No.	Reference	Main Content/Mechanical Approach
1.	Birkholzer et al. (2022) [2]	The understanding and modeling of THMC coupling processes in geological systems
2.	Hudson et al. (2017) [4]	Coupled THMC modeling for safety assessment of geological disposal of radioactive wastes
3.	Zheng et al. (2017) [14]	Coupled THMC models for bentonite in an argillite repository for nuclear waste: Illitization and its effect on swelling stress under high temperature
4.	Fan et al. (2019) [33]	A THMC coupling model and its application in acid fracturing enhanced coalbed methane-recovery simulation
5.	Yi et al. (2021) [44]	THMC coupling fracture criterion of brittle rock
6.	Pan et al. (2016) [64]	An approach for simulating the THMC process in single novaculite fracture using EPCA
7.	Tsang et al. (2009) [66]	Introductory editorial to the special issue on the DECOVALEX–THMC project
8.	Bernier et al. (2017) [79]	Implications of safety requirements for the treatment of THMC processes in geological disposal systems for radioactive waste
9.	Fan et al. (2019) [80]	THMC couplings controlling CH4 production and CO2 sequestration in enhanced coalbed methane recovery
10.	Tao et al. (2022) [81]	Permeability and porosity in damaged salt interlayers under coupled THMC conditions
11	Zhang et al. (2015) [82]	Sequentially coupled THMC model for CO2 geological sequestration into a 2D heterogeneous saline aquifer
12.	Yi et al. (2019) [84]	A new measurement method of crack propagation rate for brittle rock under THMC coupling condition
13.	Gan et al. (2021) [94]	The use of supercritical CO2 in deep geothermal reservoirs as a working fluid: insights from coupled THMC modeling
14.	Izadi et al. (2015) [98]	The influence of THMC effects on the evolution of permeability, seismicity, and heat production in geothermal reservoirs
15.	Taron et al. (2009) [100]	Numerical simulation of THMC processes in deformable, fractured porous media
16.	Zheng et al. (2008) [101]	A coupled THMC model of the FEBEX mock-up test
17.	Zheng et al. (2010) [102]	A coupled THMC model of a heating and hydration laboratory experiment in unsaturated, compacted FEBEX bentonite
18.	Xu et al. (2021) [103]	Coupled THMC modeling on acid fracturing in carbonatite geothermal reservoirs containing a heterogeneous fracture
19.	Liu et al. (2015) [104]	Stress evolution in THMC GMZ-bentonite China-Mock-up for geological disposal of high-level radioactive waste in China
20.	Sun et al. (2018) [105]	A coupled THMC model for methane hydrate-bearing sediments using COMSOL Multiphysics
21.	Wu et al. (2017) [106]	A coupled THMC modeling application of cemented coal gangue-fly ash backfill
22.	Yin et al. (2011) [107]	Coupled THMC modeling of CO2 injection by finite element methods
23.	Jing et al. (2003) [108]	Numerical modeling for coupled THMC processes of geological media—International and Chinese experiences
24.	Shao et al. (2008) [109]	THMC coupling in geomaterials and applications
25.	Poulet et al. (2012) [110]	THMC coupling with damage mechanics using ESCRIPTRT and ABAQUS
26.	Andersson et al. (2004) [111]	THMC modeling of rock mass behavior
27.	Alexander et al. (2016) [60]	Development of approaches for modeling coupled THMC processes in single-granite-fracture experiments
28.	Chittenden et al. (2016) [63]	Bounding the uncertainty of key THMC processes during fluid flow in fractured granite
29.	Bond et al. (2016) [112]	Synthesis of coupled THMC modeling of a single novaculite fracture for DECOVALEX-2015
30.	McDermott et al. (2015) [62]	Application of hybrid numerical and analytical solutions for the simulation of coupled THMC processes during fluid flow through a fractured rock
31.	Yasuhara et al. (2011) [113]	Temporal alteration of fracture permeability in granite under hydrothermal conditions and its interpretation by coupled chemo-mechanical model
32.	Yan et al. (2020) [114]	A review of the research on physical and mechanical properties and constitutive model of rock under THMC multi-field coupling
33.	Kolditz et al. (2016) [115]	THMC processes in fractured porous media
34.	Bond et al. (2014b) [116]	Coupled THMC modeling of a single fracture in novaculite for DECOVALEX-2015
35.	Xu et al. (2016) [117]	Coupled THMC modeling on chemical stimulation in fractured geothermal reservoirs
36.	Zhou et al. (2024) [118]	A granular thermodynamic constitutive model considering THMC coupling effect for hydrate-bearing sediment
37.	Yoon et al. (2016) [119]	Assessment of THMC performances of Ca-type bentonite–graphite mixtures as buffer materials for a high-level radioactive waste repository
38.	Mortazavi et al. (2016) [120]	Coupled THMC simulation of silicate rocks with an enriched-FEM model
39.	Song et al. (2024) [121]	Fully coupled THM constitutive model for clay rocks

. By understanding the specific research characteristics and contexts of various studies, scholars can better determine the future direction of multi-field coupling research. This comprehensive approach will provide valuable insights for advancing the field and ensuring the stability and safety of underground engineering projects.

2. Physical and Mechanical Properties of Rock Mass Under THMC Coupling

2.1. Overview of Multi-Field Coupled Rock-Deformation Studies

Natural rock masses inherently contain defects such as micro-fractures, pores, and cracks. These imperfections are critical to understanding rock behavior under stress, particularly in the context of deformation and failure. External factors, including water infiltration, temperature fluctuations, chemical interactions, and mechanical stresses, exacerbate the deterioration of rock materials over time. For instance, oil shale, which is composed of both organic matter and minerals, experiences progressive damage due to the presence of internal microcracks and micropores. When subjected to stress, these microscopic flaws evolve, leading to crack propagation and ultimately forming macroscopic fractures (Figure 1). This process of damage evolution and material deterioration under stress–chemical interaction provides insight into how fractured rock mass deforms under multi-field coupling conditions.

Yan et al. developed a THM model established upon the Discrete Element Method (DEM) to address mechanics–hydraulic coupling in fractured porous media [122,123]. Meanwhile, Fan et al. [33] proposed a THMC coupling model governed by conservation of mass and energy principles, incorporating equations for THMC fields. This model also includes permeability and porosity changes under acidic conditions, as illustrated in Figure 2. The interaction of temperature, fluid flow, stress, and chemical processes can accelerate fracture propagation and alter the physical and mechanical properties of the rock. To model this thermal hydromechanical damage (THMD)-model process, damage mechanics theory is often employed, using a damage field to quantitatively describe the deterioration of rock materials. This approach introduces damage factors to represent both the initial state of the rock and the impact of various physical fields on its subsequent deformation. The evolution of damage is influenced not only by the multi-physics of the rock but also by its mesoscopic characteristics, such as porosity and internal structure. These factors add a layer of complexity to the study of multi-field coupling, as they significantly affect how a rock mass responds to external forces. Additionally, Xu et al. [103] introduced a fully coupled THMC model to simulate acid fracturing in carbonate geothermal reservoirs with single incongruous fractures, investigating the impacts of fracture length, reservoir temperature, in situ stress, acid concentration, and injection parameters.

Zhu et al. [124] noted the significant impact of damage on the seepage, temperature, and stress fields in rock porous media, proposing a damage-evolution equation. Yang et al. [125] developed an elastoplastic damage model for hydrate-bearing sediments, incorporating the impact of hydrate cementation and compaction, reflecting volume changes and confining pressure effects. Li et al. [126] advanced a statistical model capturing strain-softening and hardening behavior in hydrate sediments by merging statistical strength and damage theories. Yan et al. [127,128] introduced models for hydrate-bearing sediments that factor in hydrate morphology, temperature, and pore pressure, providing insights into damage evolution and hydrate saturation’s influence on mechanical properties.

Further developments include Sun’s [129] optimization of the Klar A coupling model using energy dissipation theory and Wang’s [130] THMC model application for analyzing Shenhu hydrate reservoir behavior. Zhang et al. [131] created a statistical damage model, integrating a three-parameter Weibull distribution and residual strength factors, to study hydrate-bearing sediment stability during dissociation. Finally, Huang et al. [132] developed a THMC model focusing on displacement, saturation, pore pressure, and temperature changes in permafrost mining scenarios, noting that these damage models generally utilize Weibull distribution for damage-statistical relationships in sediments.

Figure 2. THMD coupling mode [133].

Figure 2. THMD coupling mode [133].

A mathematical framework was developed based on continuum mechanics to model the complex interplay between these three fields. In this model, each field—thermal, fluid, and mechanical—is affected by the others, creating a dynamic feedback loop that requires iterative methods for solving multi-field coupling equations. For the THM coupling calculations, contemplating the crack propagation, initiation, and intersection through the fracturing of joint elements was utilized to establish a 2D FDEM analysis program. This program was then applied to study the impact of fractures on the THM behavior of rocks [123]. By applying the principles of static equilibrium, the law of mass conservation, and the conservation of energy, researchers derived governing equations for the temperature–flow-stress coupling in porous media. These equations are particularly useful for modeling the behavior of rock masses in deep engineering applications such as tunneling, deep mining, and nuclear waste disposal [134]. The challenges of nuclear waste treatment, geothermal energy exploitation, and hydraulic high-pressure diversion tunnels under extreme geothermal conditions have made the multi-field coupling of rocks a major research focus in recent years. However, earlier research primarily focused on hydraulic coupling effects, leaving gaps in understanding how high-gradient temperature fields influence the mechanical characteristics and distortion behavior of surrounding rock masses (Figure 3) [135]. One pressing challenge in this area of research is improving THM models to more accurately reflect the post-peak behavior of brittle rocks, especially in terms of how strength declines after failure. Such improvements are essential for better predicting rock behavior in the long term, particularly under the extreme conditions found in deep engineering environments.

Figure 3. Schematic sketch of in situ stress on the tunnel with peak ground temperature [135].

Figure 3. Schematic sketch of in situ stress on the tunnel with peak ground temperature [135].

The importance of multi-field coupling research becomes starkly evident in the context of engineering disasters, such as slope instability in landfills. Under the combined effects of THMC processes, these instabilities can result in catastrophic consequences. For example, Figure 4 highlights how multi-field coupling effects can lead to landfill slope failure. Such disasters underscore the critical need for comprehensive research to understand and mitigate these risks. A deeper investigation into the catastrophic processes that occur in landfills due to multi-field coupling can help prevent the fugitive release of pollutants and other hazardous outcomes.

The study of rock deformation under the influence of THMC multi-field coupling is fundamentally rooted in understanding the geological conditions and environmental stresses that act on rock masses. By integrating laboratory experiments, theoretical analysis, and numerical simulations, researchers aim to uncover the mechanisms of deformation and failure, as well as the damage evolution of rocks subjected to multi-field coupling. This area of research is inherently interdisciplinary, combining principles from geology, damage mechanics, solid mechanics, fluid mechanics [99,122], and chemistry to offer a more complete understanding of rock mass behavior under complex conditions. When examining rock masses subjected to multi-field coupling, particularly in challenging environments, they are referred to as complex rock masses. Figure 5 illustrates this concept, highlighting the intricate interactions between various fields in fractured rock masses. To describe the overall deterioration of these masses, researchers study the deformation and failure mechanisms under the influence of coupled physical fields. This comprehensive approach not only helps in predicting rock behavior but also provides practical insights for applications in mining, civil engineering, and environmental management.

Nadimi et al. [136] conducted triaxial creep tests to assess rock mass aging characteristics, estimating dynamic constitutive creep model parameters based on test data. Herrmann et al. [137] investigated shale creep under high confining pressure and varying temperatures, finding increased creep strain with higher temperatures, axial differential stress, and reduced confining pressure. Bhat and Bhandary [138] modified a torsional ring-shear device to examine cohesive soil’s residual-state creep, producing a predictive creep failure curve for landslide displacement. Brantut et al. [139] developed a micromechanical model simulating brittle rock creep under triaxial stress. Lastly, Cao et al. [140] introduced a nonlinear damage creep constitutive model for high-stress soft rock to explore its creep deformation, contributing valuable insights for underground engineering support design.

Figure 5. General description of fractured rock mass deformation [141]. (a) Effective confining pressure, (b) volume deformation, (c) shear deformation, (d) effective normal stress, (e) normal deformation of fracture, (f) and tangential deformation of fracture.

Figure 5. General description of fractured rock mass deformation [141]. (a) Effective confining pressure, (b) volume deformation, (c) shear deformation, (d) effective normal stress, (e) normal deformation of fracture, (f) and tangential deformation of fracture.

2.2. THMC Coupling Test of Deep Rocks

Deep rocks, subjected to high temperature, significant osmotic pressure, intense geo-stress, and complex hydro-chemical environments, experience multi-field (THMC) coupling effects that involve interactions between temperature, water flow, stress, and chemical processes. This intricate coupling process is central to current research in rock mechanics, as illustrated in Figure 6. The multi-field coupling in rocks incorporates various interrelated phenomena: the influence of temperature on mechanical properties, the stress response of fractured rock masses, fluid flow dynamics within rock structures, and the chemical reactions between water and minerals. Given this complexity, extensive research is required to understand the rock failure evolution and deformation mechanisms under these multi-field coupling conditions.

As illustrated in Figure 6, the coupling relationships among THMC fields exhibit several characteristics [14,104,105]. First, mineral dissolution and precipitation processes alter porosity and permeability in porous media, affecting fluid flow. Second, chemical reactions that absorb or release heat modify thermal field characteristics. Third, mineral dissolution impacts the mechanical state of the medium by influencing its physical parameters. The impact of the hydraulic, thermal, and mechanical fields on the chemical field is primarily observed in three aspects [106,107,142]: (1) fluid velocity in the hydraulic field influences solute transport rates, (2) temperature affects reaction rates, and (3) structural changes in porous media due to mechanical stress alter solute concentration and mass transfer rates.

When studying multi-field coupling in rocks, the definitions and priorities can differ across industries. For example, in water–force coupling, different concepts emerge depending on the focus, such as fluid–structure interaction, water–rock interaction, or simply water–force coupling. Fluid–structure interaction focuses on the mechanical interaction between solid materials and fluids, whereas water–rock interaction emphasizes the chemical reactions between water and rock minerals [94,95,96]. Water–mechanical coupling, on the other hand, primarily investigates how water and stress jointly influence the mechanical behavior, deformation, and failure patterns of rocks.

In areas like nuclear waste disposal and deep coastal mining, research on multi-field coupling goes beyond water–force interactions, focusing on the combined effects of temperature, water, stress, and chemical fields [143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158]. Testing such interactions in situ requires long-term systematic experiments that integrate stress, temperature, water flow, chemical, and damage fields. These studies aim to establish constitutive relationships for multi-field coupling and advance numerical simulation techniques to analyze the interdependencies among the various fields.

Griggs [159], for instance, conducted uniaxial and triaxial creep tests on salt rocks under different stresses, confining pressures, and temperatures. His experiments revealed the relaxation characteristics of the rocks and led to the development of creep failure criteria. Malan [160] studied the rheological behavior of deep rocks in South African gold mines at depths exceeding 2000 m. Even in hard quartzite and igneous rocks, rheological effects were significant, with steady-state creep rates surpassing 0.7 m∙h⁻¹. Similarly, Ma et al. experimentally investigated the triaxial compressive rheological properties of greenschist at the foundation of a hydropower station dam, revealing the relationship between axial and lateral strain under varying confining pressures [161].

Two primary approaches for studying crack initiation include experimental and theoretical methods. Extensive experiments have investigated cracking in pre-cracked rock samples under various load conditions. Wong and Einstein [162], along with Yang and Jing [163], performed uniaxial compression tests to explore crack initiation and propagation in rock and rock-like specimens, revealing that cracking types largely depend on geometric shape and fracture traits. Yang et al. [164] used uniaxial and triaxial compression tests to assess how temperature variations affect the deformation and fracture coalescence in pre-cracked rock samples. Studies by Feng et al. [165,166] and Lu et al. [167] on crack initiation and propagation in fractured rock under HMC coupling conditions observed that initial cracks develop and extend in the direction of uniaxial stress, with chemical effects influenced by pH and ionic concentration.

When considering the interaction of water flow and stress in fractured rock masses, particularly with the added complexity of damage evolution, the research is relatively recent. Initial studies primarily examined the permeability of rocks under stress, focusing on the coupling relationship between rock stress and water flow. These studies employed laboratory and field tests to observe how the mechanical characteristics of rocks change with permeability under stress conditions. In the elastic stage, internal micro-fractures in the rock tend to close under pressure, reducing permeability. As the rock enters the elastoplastic stage, new microcracks emerge, causing permeability to first increase gradually, then rapidly peak just before or after rock failure. In the residual flow stage, permeability varies depending on pore development within the rock. It is clear that the changes in permeability closely align with the deformation and failure processes of rocks, with circumferential and axial strain playing a key role in this evolution.

Since the 1980s, researchers have increasingly focused on understanding the mechanical behavior of underground rock masses under multi-field coupling. In particular, the need to study multi-field coupling for applications such as nuclear waste disposal and geothermal energy development has become urgent. The study of THM coupling in rocks, which involves heat, water flow, and mechanical stress, builds on prior research in TM and HM coupling models. However, THM coupling is more complex than these earlier models, requiring a systematic approach to establish reasonable coupling modes and clarify the interactions between the different fields. Gens et al. [143] conducted a large-scale in situ heating test designed to replicate conditions in a deep geological repository for high-level radioactive waste in granite. Gens focuses on THM responses within the bentonite barrier and surrounding granite, emphasizing hydration progression, heating impacts, vapor transport, and swelling pressures. Their numerical analysis effectively mirrored experimental progress, enhancing understanding of THM complexities and validating the predictive reliability of the numerical model.

These models are crucial for addressing the challenges in deep underground engineering, where the influence of these fields becomes increasingly pronounced. As scientific research continues to advance, understanding the coupling effects in these environments is becoming a critical component of engineering design and safety. Parametric analyses examine how hydraulic boundary conditions (B.C.) and post-failure behavior models influence the evolution of THM behavior. Variations in drainage conditions affect thermally induced pore pressures, impacting the onset of softening [121].

2.3. Theoretical Study of Multi-Field Coupling Mechanics in Rocks

In nature, rock masses are highly heterogeneous, containing inherent fractures, joints, and other defects. However, in traditional long-term engineering calculations, rock masses are often treated as homogeneous, continuous isotropic materials, a simplification based on classical material and structural mechanics [144]. This assumption significantly diverges from the reality that fractured rock masses exhibit complex mechanical behavior due to their inherent discontinuities. Traditional strength theories, which were primarily developed to explain the behavior of intact, non-damaged materials, fall short when applied to fractured or damaged rock masses. Consequently, researchers globally have dedicated efforts to developing constitutive models that accurately capture the mechanical properties of fractured and jointed rock masses.

2.3.1. Rock Strength and Failure Mechanisms

The “rock strength” refers to the inherent strength of the intact rock material, while the term “rock mass” denotes the in situ geological body, which includes the intact rock as well as discontinuities such as joints and fractures. The strength theories and failure mechanisms accurately reflect the differences in behavior between an intact rock and a larger rock mass.

Rock strength is defined as the resistance a rock mass offers against failure, which is directly tied to its stress and strain conditions. The point at which a rock loses its load-bearing capacity, referred to as rock failure, marks a special deformation state, and the stress value at this point is considered the rock’s strength. Rock failure and strength are intrinsically linked. When the rock reaches its failure threshold, its mechanical integrity collapses, leading to catastrophic results in engineering applications. Various strength theories have been developed to describe this failure process.

Rock strength theories can be broadly categorized into two groups, as shown in (Figure 7): theoretical strength theories and empirical strength theories. Theoretical strength theories derive from fundamental principles of material mechanics and elastic mechanics. These include the maximum normal stress theory, maximum shear stress theory, maximum positive strain theory, octahedral shear stress theory, the widely used Mohr–Coulomb strength theory, Griffith’s fracture theory, its subsequent modifications, and double shear strength theory [145]. Conversely, empirical strength theories are derived from experimental observations and are used to approximate real-world failure processes. One notable example is the Hoek–Brown empirical strength criterion, which is widely employed in rock mechanics due to its applicability across a variety of geological settings.

In recent years, the Hoek–Brown strength criterion, Mohr–Coulomb strength criterion, Drucker–Prager strength criterion, and Griffith strength criterion have been the most commonly utilized models for assessing rock strength and failure [146,147,148,149]. Each criterion offers insights into different failure modes, stress distributions, and fracture mechanisms, providing engineers and researchers with tools to predict rock mass behavior under various loading conditions.

2.3.2. Advancements in Fracture Mechanics

In the 1950s, Irwin made significant contributions to fracture mechanics by deriving theoretical solutions for the stress and displacement fields near the tip of a crack based on the stress function [150]. His work built upon Griffith’s energy criterion, which relates fracture propagation to the energy state of the material and provided a foundation for analyzing the direction and evolution of cracks in brittle rocks. The Griffith energy criterion explains that during the fracture process in brittle materials, such as rock, the relationship between crack evolution and propagation direction is influenced by the initial crack angle. When compressive shear stress acts on a crack, the tensile stress at the crack tip concentrates, leading to crack propagation perpendicular to the direction of the applied shear stress. This understanding led to the development of compressive-shear fracture strength theory.

Hoek et al. [147,148,149] further contributed to the field by determining the initiation angle of compression-shear cracks using the minimum stress strength criterion. They employed the Mohr–Coulomb and Drucker–Prager strength criteria to establish criteria for compression-shear failure of brittle rocks. These criteria are essential for predicting and analyzing failure in rocks subjected to complex stress states, such as those found in deep underground engineering or slope stability analysis. Tang et al. [151] expanded upon these foundational concepts through experimental studies on multi-crack materials. Their work explored the mechanisms of crack propagation in rocks with pre-existing flaws, utilizing the Rock Failure Process Analysis (RFPA^2D V1.0) software to simulate and validate the observed crack behavior. This combination of experimental and numerical methods has advanced the understanding of how cracks propagate and interact in fractured rock masses, contributing significantly to the development of models capable of predicting rock mass failure under various field conditions.

2.3.3. Current Research Directions

Recent studies in rock mechanics have focused on integrating these theoretical insights into more comprehensive models that account for the multi-field coupling effects in deep rocks. These models must consider how mechanical properties such as rock strength evolve when subjected to coupled influences, including temperature, fluid flow, and chemical reactions.

As deep engineering projects such as geothermal energy extraction, nuclear waste disposal, and deep mining demand more accurate predictions of rock behavior, understanding the deformation and mechanical failure of fractured rock masses becomes increasingly critical. The evolution of fractures, damage propagation, and interaction of multiple fields (such as stress, temperature, and fluid flow) is at the forefront of this research, as these factors are key to ensuring the stability and safety of underground constructions.

By refining the theoretical frameworks and employing advanced numerical tools, researchers are making strides toward a more complete understanding of how fractured rock masses behave under complex, coupled field conditions. This knowledge is essential not only for the safe design and operation of underground facilities but also for advancing the field of rock mechanics as a whole.

3. Constitutive Model of Rock Mass Under THMC Coupling

The constitutive model of rocks under THMC coupling describes how rocks deform and behave under the simultaneous influence of THMC fields. This coupling becomes especially relevant in deep geological formations, such as those encountered in geothermal energy extraction, deep mining, nuclear waste disposal, and oil and gas extraction.

3.1. Constitutive Model Under THMC Coupling with Governing Equations

3.1.1. Mechanical Behavior (M)

A rock’s mechanical behavior is governed by its stress–strain relationship. The constitutive equation for rock mechanics usually follows the general form of Hooke’s law in the elastic region. Still, rock behavior under long-term loads involves creep, plasticity, and damage mechanics.

The governing equation is as follows:

$${\sigma }_{ij}=D_{ijkl}{\in }_{kl}$$

(1)

where ${\sigma }_{ij}$ is the stress tensor, $D_{ijkl}$ is the elastic modulus tensor, and ${\in }_{kl}$ is the strain tensor.

However, under coupled THMC conditions, mechanical behavior is also influenced by pore pressure $p$, temperature $T$, and chemical processes.

For elastic-plastic behavior, the following equation is used:

$${\sigma }_{ij}=D_{ijkl}\left({\in }_{kl}-{{\in }^p}_{kl}\right)$$

(2)

where, ${{\in }^p}_{kl}$ is the plastic strain component in Equation (2).

The mechanical constitutive model also incorporates damage, which evolves over time:

$$D=1-{\displaystyle \frac{E_d}{E_0}}$$

(3)

where Equation (3) consists of $D$ as the damage variable, $E_d$ as the degraded modulus due to damage, and $E_0$ as the initial elastic modulus.

3.1.2. Hydraulic Behavior (H)

The hydraulic field in the rock mass is driven by the flow of water through pores and fractures, typically described by Darcy’s law. Pore pressure $p$ influences mechanical behavior (effective stress) and interacts with chemical processes such as mineral dissolution or precipitation.

The governing equation is as follows:

$$q=-k\nabla p$$

(4)

In Equation (4), $q$ is the Darcy velocity (fluid flux), $k$ represents the permeability of the rock, and $\nabla p$ is the pressure gradient.

The coupling with mechanical fields occurs through changes in pore pressure affecting the effective stress:

$${\sigma }^{\prime }=\sigma -\alpha p$$

(5)

where ${\sigma }^{\prime }$ is the effective stress, $\alpha$ is the Biot coefficient, and $p$ is the pore pressure.

The mass conservation of fluid in the porous medium is described as follows:

$${\displaystyle \frac{\partial \theta }{\partial t}}+\nabla .\mathrm{p}=\mathrm{Q}$$

(6)

where $\theta$ is the porosity, $t$ is time, and $\mathrm{Q}$ is the source or sink term (e.g., fluid injection or withdrawal).

3.1.3. Thermal Behavior (T)

Temperature changes cause thermal expansion or contraction in the rock matrix and influence other fields, such as permeability (hydraulic field) and reaction rates (chemical field). The heat conduction through rocks is governed by Fourier’s law.

Governing Equation:

$$qT=-k_T\nabla \mathrm{T}$$

(7)

In Equation (7), $qT$ is the heat flux, $k_T$ is the thermal conductivity, and $\nabla \mathrm{T}$ is the temperature gradient.

The energy conservation equation, accounting for heat conduction and generation, is as follows:

$$\rho c{\displaystyle \frac{\partial T}{\partial t}}=\nabla .\left(k_T\nabla \mathrm{T}\right)+Q_T$$

(8)

where Equation (8) contains $\rho$ as the rock density, $c$ as the specific heat capacity, $T$ as the temperature, and $Q_T$ as the heat source term (e.g., due to radioactive decay or chemical reactions).

Thermal expansion is integrated into the mechanical strain equation:

$$ϵT={\alpha }_T\left(T-T_0\right)$$

(9)

where $ϵT$ is the thermal strain, ${\alpha }_T$ is the coefficient of thermal expansion, and $T_0$ is the reference temperature in Equation (9).

3.1.4. Chemical Behavior (C)

Chemical processes such as mineral dissolution, precipitation, or other geochemical reactions can modify the rock’s porosity and permeability, affecting its hydraulic and mechanical behavior. The reactive transport model is often used to describe chemical species movement and reaction.

Governing Equation (Advection-Diffusion-Reaction Equation):

$${\displaystyle \frac{\partial c}{\partial t}}+\nabla .\left(-D{\nabla }_c+\upsilon c\right)=R$$

(10)

In Equation (10), $c$ is the concentration of a chemical species, $D$ is the diffusion coefficient, $\upsilon$ is the fluid velocity, and $R$ is the reaction rate term.

Chemical reactions can alter the rock matrix, affecting the stress–strain relationship and permeability. The dissolution and precipitation processes alter the pore structure, which can be described as follows:

$${\displaystyle \frac{\partial \varnothing }{\partial t}}=-{\displaystyle \frac{1}{\rho s}}R$$

(11)

In Equation (11), $\varnothing$ is the porosity, $\rho s$ is the density of the solid, and $R$ is the rate of the chemical reaction.

3.1.5. Coupling Effects in THMC

The THMC model requires coupling these fields to understand rock behavior holistically. The mechanical field is influenced by temperature changes (thermal expansion), pore pressure (effective stress), and chemical changes (alteration of rock matrix). The governing equations for these fields are interdependent, and solving them simultaneously is necessary for accurate predictions.

Full THMC-Coupled Governing Equations

• Mechanical equilibrium equation:

  $$\nabla .\sigma +\rho g=0$$

  (12)

  where $\sigma$ is the stress tensor, $\rho$ is density, and $g$ is the gravitational acceleration in Equation (12).

• Heat conduction and convection equation:

$$\rho c{\displaystyle \frac{\partial T}{\partial t}}=\nabla .\left(k_T\nabla T\right)+QT-\rho c\upsilon .\nabla T$$

(13)

• Darcy’s Law for fluid flow:

  $$q=-k\nabla \mathrm{p}, \nabla .q=S_s{\displaystyle \frac{\partial p}{\partial t}}$$

  (14)

  where $S_s$ is the specific storage in Equation (14).

• Reactive transport equation:

$${\displaystyle \frac{\partial T}{\partial t}}+\nabla .\left(-D\nabla c+\upsilon c\right)=R$$

(15)

The constitutive model of rocks under THMC coupling integrates the mechanical, hydraulic, thermal, and chemical processes that occur in deep rock formations. The governing equations for each of these fields are coupled, reflecting the interdependence of stress, temperature, fluid flow, and chemical reactions. This holistic approach allows for better predictions of rock behavior in complex geological environments such as those encountered in geothermal energy extraction, nuclear waste disposal, or deep mining. Solving these equations requires numerical simulations, as analytical solutions are not feasible due to the nonlinear and coupled nature of the problem.

3.2. Multi-Field Coupled Constitutive Model of Rocks

3.2.1. Multi-Field Coupled Constitutive Model

The flow–stress coupling constitutive models enable theoretical and quantitative predictions of rock behavior by integrating mathematical and mechanical principles, verified through experimental and numerical methods. These models, categorized into six types—the equivalent continuum model, dual-medium model, fracture network model, continuous damage mechanics model, fracture mechanics model, and statistical model— highlight different facets of flow–stress interactions [11]. The first three, based on viscoelastic-plastic behavior, focus on water flow’s impact on rock structure, while the latter three capture the complex coupling effects of flow and stress on rock damage and internal structure changes due to damage evolution.

THM Coupling Models

Research on THM coupling in rocks has advanced significantly. This multi-field approach is essential for understanding the complex interactions within rock formations under varying environmental conditions. THM coupling is particularly relevant in contexts such as geothermal energy extraction, underground mining and geothermal energy [152], carbon sequestration, and nuclear waste disposal, where rock stability and permeability are critical. Formulated governing equations for THM coupling use principles of mass conservation, momentum balance, and energy conservation, detailing processes at each coupling stage [156,157,168]. Key interactions within the THM coupling framework include (1) temperature-induced stress and strain, (2) rock thermal property changes due to internal heat dissipation, (3) variations in porosity and fracture width, (4) water pressure effects on stress, (5) fluid property shifts with temperature, and (6) fluid thermal convection effects, as shown in Figure 8.

To understand the thermal responses of claystone rocks under thermal loading, it is essential to adopt a THM framework [153,154,155,156,157,158,169]. Within this framework, temperature changes significantly impact hydraulic behavior by altering both fluid viscosity and pore pressure through thermal pressurization. This temperature-induced pressure increase affects the permeability and flow characteristics of pore water, integrating TH interactions. Simultaneously, TM coupling arises as temperature shifts create thermally induced strains, which affect rock stability and stress distribution.

HM interactions further enhance the complexity of the THM behavior; hydraulic changes, such as variations in pore water pressure, directly influence the mechanical behavior by altering effective stresses in the rock matrix, while strain variations impact pore pressures and adjust intrinsic permeability [76,77,170]. Although hydraulic processes can theoretically influence thermal behavior through heat convection, in low-permeability claystone, such thermal-hydraulic effects remain minor due to limited fluid movement [72]. This interdependence of THM processes forms a multi-directional interaction network crucial for predicting rock behavior under thermal loads, particularly in low-permeability geomechanical contexts.

Bekele et al. [168] studied the deformation of fractured rock mass under the coupling action of three fields in nuclear waste treatment and proposed the following problems: Figure 9 shows a schematic diagram of the three-field coupling of a nuclear waste repository.

3.2.2. Evolution of Permeability and Chemical Reactions

Flow–Chemical Coupling Models

Recent theoretical models aim to capture the evolution of permeability in fractured rocks under flow–chemical and flow–stress–chemical coupling. Dreybrodt and Buhmann [171] propose a thin-film theoretical model that examines calcite dissolution and precipitation under various chemical and flow conditions. Similarly, Taron et al. [100] extended this approach by creating a partial differential equation system to simulate water flow–chemical interactions in single rock fractures, showing how inhomogeneities in rock surface chemistry and flow paths can affect the dissolution–precipitation processes. These models emphasize the complexity of coupling processes in which water chemistry, driven by ion concentrations, influences rock integrity and porosity evolution.

Pressure Dissolution and Permeability Evolution

Yasuhara [172] introduced a pressure-dissolution model for rock particles, examining how fracture permeability evolves under stress through analogies with spherical-particle dissolution. His work reveals that as rock particles dissolve under pressure, the permeability within fractures changes dynamically. Meanwhile, Yeh and Tripathi [173] present a rock chemical reaction transport model, treating fractured rock as a heterogeneous, anisotropic porous medium. This model accounts for solute migration, precipitation, redox reactions, adsorption, ion exchange, and acid–base interactions, although it lacks the ability to represent tubular fracture evolution and pore structure changes during dissolution and precipitation.

3.2.3. Influence of Multi-Field Coupling on Rock Damage

The mechanical damage that occurs in rocks under the influence of multi-field coupling cannot be neglected. Initial rock damage often takes the form of fractures. As these fractures evolve under the coupled influences of temperature, water flow, and stress, they can lead to significant structural changes and eventual failure. In establishing mathematical models of multi-field coupling, it is crucial to account for the impact of mesoscopic damage on coupling effects. For instance, rock stress impacts permeability and thermal conductivity, and the heat and fluid within them significantly influence the mechanical properties of rocks.

Challenges and Future Directions

To construct a comprehensive mathematical model for multi-field coupling in rock mechanics, several assumptions must be made: (1) rock material heterogeneity must be acknowledged, with mesoscopic damage distributions derived from digital image analysis; (2) heat conduction within the rock must follow Fourier’s law; (3) the water flow process should be based on Biot’s consolidation theory and the modified Terzaghi effective stress principle; and (4) mesoscopic units of the rock are elastic and brittle. Various strength criteria, such as the Mohr–Coulomb criterion and maximum tensile strain criterion, along with elastic damage theory, are applied to describe the mechanical behavior of rocks.

In coupled THM interactions within rock masses, damage plays a crucial role. Initial fracture distribution influences mechanical properties, while fracture evolution under coupled fields leads to progressive damage until failure. Modeling THM coupling must thus account for mesoscopic damage, impacting stress, permeability, thermal conductivity, and mechanical properties. Notably, post-failure, thermal conductivity and permeability shift significantly, necessitating an evolution equation to describe interactions between the temperature and water flow fields during rock failure. These simplifications can limit the models’ applicability in complex field scenarios, where real rock structures may display unpredictable permeability, porosity, and mechanical responses. Advanced models that combine multiple scales (micro, meso, and macro) are needed to accurately simulate the complex evolution of fractured rock properties under multi-field influences. Overall, studies in THMC coupling provide critical insights into the behavior of rocks under diverse field conditions, highlighting the intricate interactions between THMC processes. By integrating laboratory experiments, mathematical modeling, and numerical simulations, researchers continue to refine these models to improve predictive accuracy, aiding engineering applications where rock stability is paramount.

Multi-Field Coupling for Geomechanical Applications

Researchers applied fractal theory to construct models of fracture structural planes. Their approach calculated the average fracture opening of a single fracture under high-pressure water, using a mathematical model to characterize permeability changes under the coupled effects of water flow, stress, and chemistry. This model illustrated how water flow channels form during rock dissolution under pressure. They developed a comprehensive coupling model of dissolution–solid–water flow–mass transfer, integrating three-dimensional solid deformation with water flow and diffusion to reveal interactions and transformations in rock variables. They developed a rock model that considered a rock matrix, fractures, and spherical pores, with evenly distributed pores, deriving an effective porosity expression and establishing a mesoscopic statistical water flow model. This model, which treats permeability during crack evolution as a nonlinear dynamic process, considers the combined effects of crack development and water flow.

The time-dependent coupling of water flow and stress fields in fractured rock masses has valuable implications for engineering stability. Some researchers also use equivalent continuum models to study the rheological effects of fluid–structure interactions in rocks, deriving rheological models that are analyzed with finite element programs.

Research into the THMC coupling of rocks has advanced considerably, yet challenges remain due to the complexity of interactions between rock mechanical properties, permeability, and thermal–chemical processes, both at micro and macro scales [67]. Progress has relied on integrating microscopic research, physical experimentation, mathematical modeling, and numerical simulation to better understand and quantify these interactions. Mathematical models for multi-field coupling, including thermodynamic, fluid flow, mechanical, and chemical transport equations, have been developed, accompanied by sophisticated numerical simulation techniques to simulate THMC processes.

Numerical Simulations and Constitutive Models

Studies suggest that effective THMC research requires a multiscale approach, incorporating models of microscopic and mesoscopic pore structures to accurately capture coupling dynamics across scales. Yan et al. [114] developed a constitutive model for THMC coupling in porous media, providing a basis for numerical simulation in rock mechanics. Similarly, a model addressing the interactions among temperature, pore pressure, mechanics, and water chemistry within shale formations enhances the framework for understanding shale’s response to multi-field influences [118].

3.2.4. Creep Constitutive Model Study of Deep Rocks

The rheological behavior of rocks, particularly under multi-field coupling, is a critical area of study in rock mechanics. Scholars are particularly focused on uncovering the mechanical effects of these multi-field interactions and developing rheological models that accurately describe the behavior of rocks under such conditions [161]. Only through detailed rheological experiments can constitutive models be developed that reflect the long-term deformation characteristics of rocks, thereby guiding practical engineering applications.

Water Content and Rock Creep

One of the primary factors influencing the creep properties of rocks is their water content [174]. Water affects several aspects of rock behavior, including elastic deformation, the duration of creep, and the magnitude of creep deformation [175]. Typically, researchers investigate the effect of water content by comparing the creep properties of dry rock specimens and water-saturated specimens. This method allows for a qualitative analysis of how water presence alters rock creep characteristics. However, since rock water content varies between these two extremes, it becomes crucial to study the full spectrum of moisture content and its impact on the mechanical properties of rock. A comprehensive understanding of these effects can lead to more accurate creep models that account for the variability of water content in natural and engineered settings.

In many underground projects, such as tunneling or mining, the groundwater often contains chemical ions that can influence rock properties. Acidic and alkaline solutions, as well as cyclical wetting and drying processes, further complicate the rheological behavior of rocks. Automatic rheological servo testing machines are used to analyze the effects of water on creep deformation and permeability over time to study these influences, providing insight into the long-term stability of underground rock formations.

Temperature and Time-Dependent Deformation

As deep underground engineering projects continue to develop, especially in geothermal energy extraction and nuclear waste storage, the interaction between temperature and time-dependent deformation requires closer examination. For instance, Nadimi et al. [136] studied the triaxial compressive creep and back analysis using 3D DEM of the time-dependent behavior of Siah Bisheh on rocks around the cavern. To model the primary and secondary creep regions, a power-constitutive creep model was designed for in situ measurements and creep tests. This research led to the development of a new modeling strategy based on DEM and Voronoi polyhedral to simulate the deformation and microstructural damage behavior of rock salt [141]. Similarly, Herrmann et al. [137] conducted a study on time-dependent long-term creep properties and the influence of temperate, confining pressure, and empirical relationships between stress-inducing creep strain and time.

Key Rheological Models

A major focus in rock rheology research is the establishment of constitutive models based on experimental data. These models represent the stress–strain–time relationship of rocks, enabling accurate predictions of how rocks will behave under prolonged stress [176]. Several key types of rheological constitutive models are currently used, each with its strengths and limitations.

I.

Empirical constitutive models

These models are derived from experimental data and establish functional relationships between stress, strain, and time. Common forms include power function, logarithmic, exponential, and polynomial relations. Rock creep typically progresses through three stages: (1) deceleration creep, where the strain rate decreases over time; (2) steady-state creep, where the strain rate stabilizes; and (3) accelerated creep, where the strain rate increases rapidly, often leading to failure. Although empirical models are widely used, they tend to provide only a simplified view of the creep process and are often limited by their dependency on experimental data.

II.

Component combination models

These models are built by combining basic mechanical elements such as the Hookean spring (elastic), Newtonian dashpot (viscous), and Saint-Venant element (plastic). Through creep and relaxation tests, the elastic, viscous, and plastic characteristics of the rock are evaluated to establish a rheological constitutive model. Popular models in this category include the Burgers model, Nishihara model, Bingham model, Maxwell model, and Kelvin model [176]. These combination models provide a more nuanced understanding of the stress–strain–time relationship by accounting for different mechanical behaviors. Researchers are focusing on integrating nonlinear components into rock rheology models to capture the nonlinear characteristics of rock rheology. This modification is crucial for accurately describing the full creep process of rocks, from the initial deceleration stage to the eventual failure [177].

III.

Rheological models in the form of integrals

These models use integral formulations to describe rock creep behavior. They derive constitutive equations based on genetic rheology theory, taking into account previous stresses and deformations. Integral models are particularly effective in representing the rock’s behavior under variable stress conditions. By incorporating both creep and relaxation equations, these models are able to describe the full range of rock behavior, including deceleration, steady-state, and accelerated creep [178].

IV.

Damage rheological models

In multi-field coupling conditions, the rheological behavior of rocks is often influenced by damage mechanics, particularly in scenarios where rock deformation leads to the development of fractures or microcracks. Damage rheology models aim to describe the evolution of damage and its effects on rock properties over time. These models are particularly useful for predicting long-term rock behavior in underground settings, where cumulative damage can significantly alter rock properties.

Advances in rheological testing equipment and the integration of nonlinear components into traditional models have significantly improved our ability to describe the rheological properties of rocks. By continuing to combine experimental research, theoretical modeling, and numerical simulation, scholars are working towards developing multi-field coupled rheological models that can accurately guide engineering practices in deep underground environments. These efforts are critical for ensuring the safety and stability of geothermal projects, nuclear waste repositories, and other infrastructures that rely on the integrity of deep rock formations.

4. Conclusions and Prospects

This paper provides a comprehensive review of the research progress on fracture evolution, deformation mechanisms, and the development of mechanical constitutive models for fractured rock masses under the THMC coupling effects. It highlights the critical challenges in rock mechanics research under multi-field coupling and identifies key areas for future research and exploration. The following points outline the primary directions and focal areas of future work in this field:

I.

Mathematical models and analytical methods for multi-scale coupling:

Future research in multi-field coupling must account for the behavior of rock masses at different scales, from the microscopic to the macroscopic. To solve real-world engineering challenges, it is essential to develop mathematical models and analytical methods that consider the complex interactions across these scales. A deeper understanding of scale-dependent coupling mechanisms can be achieved through advanced simulations that replicate the physical field environments of rocks at various scales. This multi-scale perspective will help engineers develop more reliable and accurate models for predicting rock behavior in various practical applications, such as deep mining and underground construction.

II.

Comprehensive study of the fully coupled THMC processes:

While the THMC coupling of rock mechanics has gained significant attention, it is still in the early stages of development. Most current studies focus on waste disposal applications, such as the THMC coupling in landfills and nuclear waste repositories. However, much remains unexplored regarding the coupling mechanisms of THMC in other critical fields, such as underground space development and deep mineral resource extraction. Of particular interest are the chemical effects in deep rock formations, such as those impacted by seawater in coastal mining or offshore oil extraction. These environments present new challenges where chemical interactions cannot be ignored. Therefore, advancing the study of the THMC multi-field coupling in these conditions will be crucial for future research.

III.

Intelligent systems for multi-field coupling:

The increasing complexity of deep mining and resource extraction necessitates the development of intelligent mining strategies that can handle the dynamic interplay between temperature, water flow, stress, and chemical fields in rock masses. To address this, researchers need to focus on the creation of intelligent theories and control systems. For instance, strategies involving stress-dominated energy control, high-temperature management, and water resource optimization are essential for the efficient utilization of deep mineral deposits. Furthermore, by leveraging advanced sensing technologies and machine learning, a more precise understanding of the multi-field environment can be achieved. This would enable the development of intelligent systems capable of identifying potential hazards and improving the safety and efficiency of resource extraction in complex underground environments.

IV.

Multidisciplinary and interdisciplinary integration:

The complexity of multi-field coupling in rocks goes beyond the interaction of temperature, water flow, stress, and chemical processes. Rock formations in real engineering environments are complex systems that cannot be accurately modeled using simple approaches. Therefore, advancing the field requires a multidisciplinary approach that integrates knowledge from fields such as computer science, environmental science, geophysics, fluid mechanics, and chemical science. This interdisciplinary approach will allow for a more comprehensive analysis of rock behavior under multi-field coupling, yielding models that better reflect the complexities of real-world engineering scenarios. By integrating the latest scientific advances and technologies, future research can more accurately represent the complex multi-field interactions in deep rock masses.

In conclusion, the study of multi-field coupling in rocks is at a critical juncture. While significant progress has been made in understanding the individual effects of temperature, water flow, stress, and chemistry on rock behavior, the challenge remains to develop integrated models that can effectively describe their combined effects. Future research should focus on refining mathematical models, expanding the scope of applications, embracing intelligent technologies, and fostering interdisciplinary collaboration. Only through such comprehensive efforts can we fully understand and manage the complex behavior of rocks in deep underground environments, ensuring the success of projects like geothermal energy extraction, nuclear waste disposal, and deep mining operations.