Phase-Field Modeling of Thermal Fracturing Mechanisms in Reservoir Rock Under High-Temperature Conditions

Abstract

Thermal stimulation represents an effective method for enhancing reservoir permeability, thereby improving geothermal energy recovery in Enhanced Geothermal Systems (EGS). The phase-field method (PFM) has been widely adopted for its proven capability in modeling the fracture behavior of brittle solids. Consequently, a coupled thermo-mechanical phase-field model (TM-PFM) was developed in COMSOL 6.2 Multiphysics to probe thermal fracturing mechanisms in reservoir rocks. The TM-PFM was validated against the analytical solutions for the temperature and stress fields under steady-state heat conduction in a thin-walled cylinder, three-point bending tests, and thermal shock tests. Subsequently, two distinct thermal fracturing modes in reservoir rock under high-temperature conditions were investigated: (i) fracture initiation driven by sharp temperature gradients during instantaneous thermal shocks, and (ii) crack propagation resulting from heterogeneous thermal expansion of constituent minerals. The proposed TM-PFM has been validated through systematic comparison between the simulation results and the corresponding experimental data, thereby demonstrating its capability to accurately simulate thermal fracturing. These findings provide mechanistic insights for optimizing geothermal energy extraction in EGS.

1. Introduction

Thermal fracturing of reservoir rock under high-temperature conditions occurs ubiquitously in subsurface engineering applications, including deep mining operations, shale gas extraction, and geothermal energy development [1,2,3,4]. Reservoir rock exhibits significant heterogeneity in mineral composition and internal structure due to physico-chemical processes during diagenesis. Two primary mechanisms govern thermal fracturing under high-temperature conditions: (i) fracture initiation driven by sharp temperature gradients during instantaneous thermal shocks, and (ii) crack propagation resulting from heterogeneous thermal expansion of constituent minerals. These mechanisms generate substantial thermal stresses within the rock matrix. When the induced stresses surpass the rock mass’s tensile strength, extensive fracture networks develop [5,6]. Thermal fractures significantly impact engineering integrity. For instance, when assessing the stability of deep rock masses, such fractures compromise structural safety and may induce catastrophic failure. Conversely, thermal stimulation strategically exploits these fractures to enhance reservoir permeability and improve energy recovery efficiency [7,8,9]. Consequently, comprehensive investigation of thermal fracture initiation and propagation mechanisms is imperative for developing evidence-based engineering protocols.

Although extensive experimental studies on rock thermal fracturing exist, most focus primarily on analyzing how heat treatment or thermal shock affects rock physico-mechanical properties, rather than elucidating underlying fracturing mechanisms. Furthermore, real-time measurement of rock properties at elevated temperatures imposes stringent instrumentation requirements. Given that thermal shock-induced fracturing constitutes a transient phenomenon occurring within millisecond-scale durations, conventional experimental techniques lack sufficient temporal resolution to capture the evolution of fracture networks [10,11]. Consequently, numerical methods have gained prominence in studying thermal fracturing mechanisms of brittle materials due to their computational efficiency, reproducibility, and cost-effectiveness. Li et al. [12] employed a non-local model to simulate ceramic thermal fracturing. Adopting a micromechanical perspective, Zhao [13] and Wu et al. [14] analyzed granite fracture initiation and propagation under high-temperature conditions via discrete element modeling in PFC2D. Tang et al. [15] implemented Weibull-distributed material properties in RFPA (Rock Failure Process Analysis) to investigate thermal shock-induced fracturing. Yan et al. [16,17] employed a FDEM for rock thermal fracturing simulations. Using extended finite element methods (XFEM), Ngo and Pellet [18] simulated large-scale rock salt cooling experiments, establishing fracture mechanics-based failure mechanisms. Furthermore, peridynamics (PD) has emerged as a promising approach for rock thermal fracturing analysis [19,20].

Recently, the PFM has emerged as a predominant approach for modeling fracture in brittle solids, owing to its robust capability in handling complex crack topologies [21,22,23,24]. Building on Griffith’s elastic fracture mechanics theory, the PFM framework was pioneered by Bourdin et al. [25] and subsequently refined through variational formulations [26,27,28]. This method regularizes sharp discontinuities through a diffuse crack representation, introducing an auxiliary phase-field variable to quantify fracture intensity—a concept akin to continuous damage mechanics. Notable applications to thermal fracturing include: Miehe et al. [29] who developed a fully TM-PFM, simulating crack propagation in heated circular plates and thermal shock failure in thin glass; Chu et al. [30] who investigated dynamic crack trajectories under thermal shock using PFM; Miao et al. [31] who analyzed temperature-induced fracture effects on thermal energy storage efficiency; Wang et al. [32] who implemented a parallelized thermo-elastic PFM algorithm in Abaqus/Explicit; and Li et al. [33] who established 3D thermo-mechanical PFM models for ceramic thermal shock fracture. While these simulations demonstrate satisfactory experimental agreement, they predominantly focus on idealized ceramic materials. Studies addressing thermal fracturing in geological materials—particularly reservoir rock with inherent mineralogical heterogeneity and differential thermal expansion properties—remain scarce. Furthermore, computational efficiency limitations persist: divergent numerical implementations within finite element frameworks yield significant variations in solution convergence rates and computational cost. Consequently, enhancing PFM’s algorithmic efficiency constitutes a critical research thrust within the computational fracture mechanics community.

The evolution of fracture networks under external loading is governed by the solution of diffusion-type equations subject to specific boundary constraints. Given COMSOL Multiphysics’ demonstrated efficacy in handling multiphysics phenomena [34,35,36,37], this study implements a TM-PFM within this platform to simulate thermal fracture evolution. Crucially, we leverage COMSOL’s state variables to dynamically update the historical strain field—a methodological advancement that eliminates the need for auxiliary ODE/DAE interfaces required in conventional implementations. Utilizing this optimized framework, we simulate two fundamental thermal fracturing mechanisms in reservoir rock: (i) fracture initiation driven by steep thermal gradients during instantaneous thermal shocks; (ii) crack propagation induced by differential thermal expansion of heterogeneous mineral assemblages. Quantitative analysis of fracture topology evolution provides new insights into these distinct failure modes. The resulting mechanistic understanding offers actionable guidelines for optimizing thermal stimulation protocols in EGS, particularly regarding fracture network design for permeability enhancement.

2. Theoretical Background

2.1. Regularization and Variational Formulation for Brittle Fracture

An arbitrary thermoelastic continuum $\Omega \subset R^{\dim }\left(\dim \in \left\{2,3\right\}\right)$ serves as the physical domain for this investigation, with an outer boundary $\partial \Omega$ and an inner sharp crack $\Gamma$, as shown in Figure 1. Herein, $\partial {\Omega }_u$ and $\partial {\Omega }_T$ denote Dirichlet boundaries, while $\partial {\Omega }_t$ and $\partial {\Omega }_J$ represent Neumann boundaries. Within the standard PFM formulation, sharp cracks are regularized via a continuous scalar field $d\left(x,t\right)\in \left[0,1\right]$ introduction, where $d=0$ indicates the solid is intact (not damaged), and $d=1$ represents the solid is completely broken. $x$ is the position vector, and t is the time. Following variational regularization theory, the sharp crack geometry is asymptotically approximated by a diffuse fracture representation governed by the phase-field functional.

$$\underset{\mathrm{Sharp} \mathrm{crack}}{\underbrace{A={\displaystyle {\int }_{\Gamma }\mathrm{d}S}}}\approx \underset{\mathrm{Diffuse} \mathrm{crack}}{\underbrace{{\displaystyle {\int }_{{\Gamma }_d}\gamma }\left(d,\nabla d\right)\mathrm{d}\Omega =A_d}}$$

(1)

where $\gamma \left(d,\nabla d\right)$ denotes the crack surface density per unit volume, defined as [21]:

$$\gamma \left(d,\nabla d\right)={\displaystyle \frac{d^2}{2l_0}}+{\displaystyle \frac{l_0}{2}}{\left|\nabla d\right|}^2$$

(2)

where $l_0$ denotes the characteristic length scale, representing the regularization parameter in the phase-field formulation. The smaller $l_0$ means the narrower the crack. When $l_0$ tends to zero, the crack is closer to the real crack. Therefore, the crack fracture energy ${\psi }_f$ can be approximately written as:

$${\psi }_f={\displaystyle {\int }_{\Gamma }{G}_c\mathrm{d}S}\approx {\displaystyle {\int }_{\Gamma }{G}_c\left(\frac{d^2}{2l_0}+\frac{l_0}{2}{\left|\nabla d\right|}^2\right)\mathrm{d}\Omega }$$

(3)

Consistent with Griffith’s fracture theory [38], crack propagation arises from the competition between external work expenditure for new crack formation and the strain energy released during fracture advancement. Consequently, the total energy functional $\Pi \left(u,T,d,\nabla d\right)$ of the PFM is defined as [25,39]:

$$\begin{array}{c}\Pi \left(u,T,d,\nabla d\right)=\underset{{\psi }_{kin}}{\underbrace{{\displaystyle \frac{1}{2}}{\displaystyle {\int }_{\Omega }\rho \dot{u}\cdot \dot{u}\mathrm{d}}\Omega }}-\underset{{\psi }_{{\epsilon }_e}}{\underbrace{{\displaystyle {\int }_{\Omega }{\psi }_{{\epsilon }_e}\left({\epsilon }_e\right)\mathrm{d}}\Omega }}-\underset{{\psi }_f}{\underbrace{{\displaystyle {\int }_{\Gamma }{G}_c\mathrm{d}S}}}\hfill \\ +\underset{{\psi }_T}{\underbrace{{\displaystyle {\int }_{\Omega }{\psi }_T\left(T\right)\mathrm{d}}\Omega +{\displaystyle {\int }_{{\Omega }_J}\overline{J}\cdot n\mathrm{d}S}}}+\underset{W_{ext}}{\underbrace{{\displaystyle {\int }_{\Omega }b\cdot u\mathrm{d}}\Omega +{\displaystyle {\int }_{{\Omega }_t}\overline{t}\cdot u\mathrm{d}S}}}\hfill \end{array}$$

(4)

where ${\psi }_{kin}$ is the kinetic energy, ${\psi }_{{\epsilon }_e}$ represents the elastic strain energy, $G_c$ denotes the critical energy release rate, ${\psi }_T$ is the thermal energy, $W_{ext}$ is the external force work, $T$ is the temperature, $b$ is the body force, $\overline{t}$ is the traction, $u$ is the displacement, $\dot{u}={\displaystyle \frac{\partial u}{\partial t}}$, and ${\epsilon }_e$ represents the elastic strain tensor. Under thermo-mechanical coupling, materials undergo thermal expansion. The thermal elastic strain ${\epsilon }_{th}$ of the material is defined as:

$$\left\{\begin{array}{l}{\epsilon }_e={\displaystyle \frac{1}{2}}\left[\nabla u+{\left(\nabla u\right)}^{\mathrm{T}}\right]-{\epsilon }_{th}\hfill \\ {\epsilon }_{th}={\alpha }_T\left(T-T_0\right)I\hfill \end{array}\right.$$

(5)

where ${\alpha }_T$ represents thermal expansion tensor.

2.2. Governing Equation for PFM

The initial elastic strain energy density is expressed as:

$${\psi }_{{\epsilon }_e}\left({\epsilon }_e\right)={\displaystyle \frac{\lambda }{2}}{\mathrm{tr}}^2\left({\epsilon }_e\right)+\mu \mathrm{tr}\left({\epsilon }_e^2\right)$$

(6)

where $\lambda$ and $\mu$ denote the Lame constant.

$$\left\{\begin{array}{l}\lambda ={\displaystyle \frac{E\nu }{\left(1+\nu \right)\left(1-2\nu \right)}}\hfill \\ \mu ={\displaystyle \frac{E}{2\left(1+v\right)}}\hfill \end{array}\right.$$

(7)

The damage function $g\left(d\right)={\left(1-d\right)}^2+{\delta }_0$ is adopted to characterize the deterioration of material stiffness with damage. Additionally, a small dimensionless parameter ${\delta }_0$ $\left(0<{\delta }_0<<1\right)$ is introduced into the degradation function to prevent numerical singularity arising from complete loss of material stiffness when $d=1$. The elastic strain energy stored in a solid can be written as:

$${\psi }_{{\epsilon }_e}={\displaystyle {\int }_{\Omega }g\left(d\right)\left[{\psi }_{{\epsilon }_e}\left({\epsilon }_e\right)\right]\mathrm{d}}\Omega$$

(8)

This thermal component ${\psi }_T$ is defined as:

$${\psi }_T={\displaystyle {\int }_{\Omega }\left(\rho C_p\dot{T}+\nabla \cdot J-Q\right)\mathrm{d}}\Omega +{\displaystyle {\int }_{{\Omega }_J}\overline{J}\cdot n\mathrm{d}S}$$

(9)

where $\rho$ denotes the density, $C_p$ represents the specific heat capacity, and $k$ is the thermal conductivity. $J=-k\nabla T$ is obtained from Fourier’s law. Furthermore, thermal conductivity undergoes degradation proportional to material damage, ensuring zero heat flux across fully developed cracks. This is implemented through the conductivity degradation function [35,40].

$$k=g\left(d\right)k_0$$

(10)

where $k_0$ is the initial thermal conductivity.

Therefore, the Lagrange energy functional $\Pi \left(u,T,d,\nabla d\right)$ can be rewritten as:

$$\begin{array}{l}\Pi \left(u,T,d,\nabla d\right)={\displaystyle \frac{1}{2}}{\displaystyle {\int }_{\Omega }\rho \dot{u}\cdot \dot{u}\mathrm{d}}\Omega -{\displaystyle {\int }_{\Omega }g\left(d\right)\left[{\psi }_{{\epsilon }_e}\left({\epsilon }_e\right)\right]\mathrm{d}}\Omega \\ -{\displaystyle {\int }_{\Omega }{G}_c\left(\frac{d^2}{2l_0}+\frac{l_0}{2}{\left|\nabla d\right|}^2\right)\mathrm{d}\Omega }+{\displaystyle {\int }_{\Omega }b\cdot u\mathrm{d}}\Omega +{\displaystyle {\int }_{{\Omega }_t}\overline{t}\cdot u\mathrm{d}S}\\ +{\displaystyle {\int }_{\Omega }\left(\rho C_p\dot{T}+\nabla \cdot J-Q\right)\mathrm{d}}\Omega +{\displaystyle {\int }_{{\Omega }_J}\overline{J}\cdot n\mathrm{d}S}\end{array}$$

(11)

Applying the variational principle (δΠ(u, T, d, ∇d) = 0) to the total energy functional yields the following governing equations for the TM-PFM:

$$\nabla \cdot \sigma \left({\epsilon }_e,d\right)+b=\rho \ddot{u}$$

(12)

$$\rho C_p\dot{T}+\nabla \cdot J=Q$$

(13)

$${\displaystyle \frac{G_c}{l_0}}\left(d-l_0^2{\nabla }^{2}d\right)-2\left(1-d\right)H\left({\epsilon }_e\right)=0$$

(14)

where Equation (12) is the displacement field governing equation, Equation (13) is the temperature field governing equation, and Equation (14) is the phase-field governing equation. $\sigma \left({\epsilon }_e,d\right)=\partial {\psi }_{{\epsilon }_e}\left({\epsilon }_e,d\right)/\partial {\epsilon }_e$. $H\left({\epsilon }_e\right)=\underset{s\in \left[0,t\right]}{\max }\left\{{\psi }_{{\epsilon }_e}\left({\epsilon }_e\left(u,s\right)\right)\right\}$ is the historical strain field providing the driving force for PFM.

The Dirichlet boundary conditions and Newman boundary conditions for the corresponding field variables are as follows:

$$\sigma \left({\epsilon }_e,d\right)\cdot n=\overline{t}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{on} \partial {\Omega }_t$$

(15)

$$J\cdot n=\overline{J}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{on} \partial {\Omega }_J$$

(16)

$$\nabla d\cdot n=0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{on} \partial \Omega \cup \Gamma$$

(17)

2.3. Energy Decomposition and Driving Force of Phase-Field Evolution

While the classical phase-field formulation (Equation (14)) does not distinguish between material failure modes, geological materials exhibit complex fracture behaviors under multi-physics coupling, including tensile, compressive, tensile-shear, and compressive-shear failure. To address this limitation, researchers have developed enhanced phase-field frameworks. Notably, Amor et al. [41,42] pioneered an energy decomposition approach that partitions the elastic strain energy density into volumetric ${\psi }_{{\epsilon }_e}^{sph}$ and deviatoric ${\psi }_{{\epsilon }_e}^{dev}$ components:

$${\epsilon }_e={\epsilon }_{sph}+{\epsilon }_{dev}, {\epsilon }_{sph}=1/3\mathrm{tr}\left({\epsilon }_e\right)I, {\epsilon }_{dev}={\epsilon }_e-{\epsilon }_{sph}$$

(18)

$${\psi }_{{\epsilon }_e}^{sph}={\displaystyle \frac{K}{2}}{\mathrm{tr}}^2\left({\epsilon }_e\right), {\psi }_{{\epsilon }_e}^{dev}=\mu \mathrm{tr}\left({\epsilon }_{dev}^2\right)$$

(19)

where $K$ is the bulk modulus, $K=\left(3\lambda +2\mu \right)/3$.

Complementarily, Miehe et al. [21] proposed a spectral decomposition approach that partitions the elastic strain energy density into the tensile and compressive part ${\psi }_{{\epsilon }_e}^{\pm }\left({\epsilon }_e\right)$ according to the spectral decomposition method.

$${\epsilon }_e^{\pm }=\sum _{i=1}^3<{\epsilon }_i{>}_{\pm }{n}_i\otimes n_i$$

(20)

$${\psi }_{{\epsilon }_e}^{\pm }\left({\epsilon }_e\right)={\displaystyle \frac{\lambda }{2}}<\mathrm{tr}\left({\epsilon }_e\right){>}_{\pm }^2+\mu \mathrm{tr}(<{\epsilon }_e{>}_{\pm }^2)$$

(21)

where ${\epsilon }_i$ denotes the principal strain, and the operator $<x{>}_{\pm }$ is defined as $<x{>}_{\pm }=\left(x+\left|x\right|\right)/2$.

Thermal fracturing in rocks under elevated temperatures predominantly manifests as tensile cracking. Although the classical PFM cannot capture complex mixed-mode failures, it demonstrates proven effectiveness in simulating tensile-dominated fracture. Consequently, this study adopts the classic PFM, specifically restricting material stiffness degradation to the tensile strain energy contribution.

$${\psi }_{{\epsilon }_e}\left({\epsilon }_e\right)=\left[{\left(1-d\right)}^2+{\delta }_0\right]{\psi }_{{\epsilon }_e}^{+}\left({\epsilon }_e\right)+{\psi }_{{\epsilon }_e}^{-}\left({\epsilon }_e\right)$$

(22)

Therefore, Equation (14) can be rewritten as:

$${\displaystyle \frac{G_c}{l_0}}\left(d-l_0^2{\nabla }^{2}d\right)-2\left(1-d\right)H_t=0$$

(23)

where $H_t=\underset{s\in \left[0,t\right]}{\max }{\left[{\psi }_{{\epsilon }_e}^{+}\left({\epsilon }_e\right)-{\psi }_c\right]}_{+}$, ${\psi }_c={\displaystyle \frac{f_t^2}{2E}}$ represents the stored energy threshold for damage evolution, and $f_t$ is the uniaxial tensile strength.

The historical strain field is dynamically updated through COMSOL Multiphysics’ state variables, eliminating the need for auxiliary ODE/DAE interfaces traditionally required to track fracture history.

Additionally, the stress tensor is rewritten as:

$$\sigma \left({\epsilon }_e,d\right)={\displaystyle \frac{\partial {\psi }_{{\epsilon }_e}\left({\epsilon }_e,d\right)}{\partial {\epsilon }_e}}=g\left(d\right){\displaystyle \frac{\partial {\psi }_{{\epsilon }_e}^{+}\left({\epsilon }_e\right)}{\partial {\epsilon }_e}}+{\displaystyle \frac{\partial {\psi }_{{\epsilon }_e}^{-}\left({\epsilon }_e\right)}{\partial {\epsilon }_e}}$$

(24)

The stiffness matrix of the material is:

$$ℂ\left({\epsilon }_e,d\right)={\displaystyle \frac{\partial \sigma \left({\epsilon }_e,d\right)}{\partial {\epsilon }_e}}=g\left(d\right){\displaystyle \frac{{\partial }^2{\psi }_{{\epsilon }_e}^{+}\left({\epsilon }_e\right)}{\partial {\epsilon }_e^2}}+{\displaystyle \frac{{\partial }^2{\psi }_{{\epsilon }_e}^{-}\left({\epsilon }_e\right)}{\partial {\epsilon }_e^2}}$$

(25)

The stiffness matrix in Equation (21) exhibits pronounced nonlinearity due to the degradation function’s dependence on the evolving phase-field variable. To overcome the limitations, the Hybrid phase field solution strategy is adopted [43,44].

3. Numerical Implementation

Within this computational framework, the strong-form partial differential equations are discretized using a Galerkin finite element formulation and implemented in COMSOL Multiphysics.

3.1. Finite Element Discretization

$$u={\displaystyle \sum _i^m{N}_i^u{u}_i}, T={\displaystyle \sum _i^m{N}_i^T{T}_i}, d={\displaystyle \sum _i^m{N}_i^d{d}_i}$$

(26)

where $N_i^u=N_i^T=N_i^d=N_i$ is the shape function of node $i$. $u_i$, $T_i$, $d_i$ are the values of the displacement field, temperature field, and the phase-field at node $i$, respectively. The corresponding gradient is given by:

$$\epsilon ={\displaystyle \sum _i^m{B}_i^u{u}_i}, \nabla T={\displaystyle \sum _i^m{B}_i^T{T}_i}, \nabla d={\displaystyle \sum _i^m{B}_i^d{d}_i}$$

(27)

where $B_i^u$, $B_i^T$, $B_i^d$ is the derivative of the corresponding shape function, respectively.

Moreover, the test functions and their derivatives inherit identical discretization properties to the trial functions.

$$\delta u={\displaystyle \sum _i^m{N}_i^u\delta u_i}, \delta T={\displaystyle \sum _i^m{N}_i^T\delta T_i}, \delta d={\displaystyle \sum _i^m{N}_i^d\delta d_i}$$

(28)

$$\delta \epsilon ={\displaystyle \sum _i^m{B}_i^u\delta u_i}, \nabla \delta T={\displaystyle \sum _i^m{B}_i^T\delta T_i}, \nabla \delta d={\displaystyle \sum _i^m{B}_i^d\delta d_i}$$

(29)

The weak form residuals of the coupled system are derived as:

$$\left\{\begin{array}{c}{R}_i^{\mathit{u}}=\underset{{\mathit{F}}_i^{u,ext}}{\underbrace{{\int }_{\Omega }\mathit{b}\cdot \delta \mathit{u}\mathrm{d}\Omega +{\int }_{{\Omega }_t}\overline{\mathit{t}}\cdot \delta \mathit{u}\mathrm{d}S}}-\underset{{\mathit{F}}_i^{u,int}}{\underbrace{{\int }_{\Omega }\mathit{\sigma }:\delta \mathit{\epsilon }\mathrm{d}\Omega }}-\underset{{\mathit{F}}_i^{u,ine}}{\underbrace{{\int }_{\Omega }\rho \ddot{\mathit{u}}\delta \mathit{u}\mathrm{d}\Omega }}\hfill \\ R_i^T=\underset{{\mathit{F}}_i^{T,ext}}{\underbrace{{\int }_{\Omega }Q\delta T\mathrm{d}\Omega +{\int }_{{\Omega }_J}\overline{\mathit{J}}\delta T\mathrm{d}S}}-\underset{{\mathit{F}}_i^{T,int}}{\underbrace{{\int }_{\Omega }\nabla T\cdot \nabla \delta T\mathrm{d}\Omega }}-\underset{{\mathit{F}}_i^{T,loc}}{\underbrace{{\int }_{\Omega }\rho C_p\dot{T}\delta T\mathrm{d}\Omega }}\hfill \\ R_i^d=-\underset{{\mathit{F}}_i^{d,int}}{\underbrace{{\int }_{\Omega }\left\{{\displaystyle \frac{G_c}{l_0}}\right(d\delta d-l_0^2\nabla d\cdot \nabla \delta d)-2(1-d\left)H\right({\mathit{\epsilon }}_e\left)\delta d\right\}\mathrm{d}\Omega }}\end{array}\right.$$

(30)

where $R_i^u$, $R_i^T$, $R_i^d$ represents the residual terms corresponding to the displacement field, the temperature field, and the phase field, respectively. $F_i^{u,ext}$, $F_i^{u,int}$, $F_i^{u,ine}$ are the external force terms. $F_i^{T,ext}$, $F_i^{T,int}$, $F_i^{u,loc}$ are the external force term, internal force term, and local internal energy increase term of the temperature field, respectively. $F_i^{d,int}$ is the internal force term of the phase field.

$$\left\{\begin{array}{l}{K}_{ij}^{uu}={\displaystyle \frac{\partial F_i^{u,int}}{\partial u_j}}={\int }_{\Omega }{\left({\mathit{B}}_i^u\right)}^{\mathrm{T}}{\mathbb{C}}_e{\mathit{B}}_j^u\mathrm{d}\Omega \\ K_{ij}^{TT}={\displaystyle \frac{\partial F_i^{T,int}}{\partial u_j}}={\int }_{\Omega }{\left({\mathit{B}}_i^T\right)}^{\mathrm{T}}k{\mathit{B}}_j^T\mathrm{d}\Omega \mathrm{d}\Omega \\ K_{ij}^{dd}={\displaystyle \frac{\partial F_i^{d,int}}{\partial u_j}}={\int }_{\Omega }\left\{N_i\right[{\displaystyle \frac{G_c}{l_0}}+2H\left({\mathit{\epsilon }}_e\right)]+{\left({\mathit{B}}_i^T\right)}^{\mathrm{T}}{G}_c{l}_0{\mathit{B}}_j^T\}\end{array}\right.$$

(31)

where $K_{ij}^{uu}$, $K_{ij}^{TT}$, $K_{ij}^{dd}$ represents the stiffness matrices of the displacement field, the temperature field, and the phase field, respectively.

3.2. COMSOL Implementation

In the numerical implementation of the TM-PFM, a segregated solution strategy is employed to enhance computational efficiency. The physics modules of choice in COMSOL Multiphysics are the Heat Transfer in Solids Module, Solid Mechanics Module, State Variables, and Helmholtz Equation. The temperature field is solved within the Heat Transfer in Solids Module, while the displacement field is computed in the Solid Mechanics Module. State variables are employed to update the historical strain field during computation. Concurrently, the phase-field variable is obtained by solving the Helmholtz equation. Temporal integration employs the implicit backward difference formula (BDF). The complete solution procedure is illustrated in Figure 2.

4. Numerical Examples

4.1. Heat Conduction

A thick-wall cylinder with inner radius $a=20\mathrm{mm}$ and outer radius $b=150\mathrm{mm}$ serves as a benchmark for validating the thermo-mechanical coupling (see Figure 3). In the displacement field, the cylinder’s inner and outer walls are free boundaries. In the temperature field, the cylinder’s initial temperature is 25 °C, with the inner wall at 400 °C and the outer wall at 25 °C. A two-dimensional thick-wall cylinder is a solution to a plane strain problem.

According to previous studies [16,45], the analytical solution of the temperature field at any distance from the center of the thick-walled cylinder is given:

$$T_r={\displaystyle \frac{\ln \left(b/r\right)}{\ln \left(b/a\right)}}{T}_a+{\displaystyle \frac{\ln \left(a/r\right)}{\ln \left(a/b\right)}}{T}_b$$

(32)

The radial and tangential stresses in the thick-wall cylinder are:

$$\left\{\begin{array}{l}{\sigma }_r=-{\displaystyle \frac{E{\alpha }_T\left(T_a-T_b\right)}{2\left(1-v\right)}}\left[{\displaystyle \frac{\ln \left(b/r\right)}{\ln \left(b/a\right)}}-{\displaystyle \frac{b^2/r^2-1}{b^2/a^2-1}}\right]\hfill \\ {\sigma }_{\phi }=-{\displaystyle \frac{E{\alpha }_T\left(T_a-T_b\right)}{2\left(1-v\right)}}\left[{\displaystyle \frac{\ln \left(b/r\right)-1}{\ln \left(b/a\right)}}+{\displaystyle \frac{b^2/r^2+1}{b^2/a^2-1}}\right]\hfill \end{array}\right.$$

(33)

where $r$ represents the radial distance from any point within the cylinder to its central axis. $T_r$ donates the temperature at distance $r$. ${\sigma }_r$ and ${\sigma }_{\phi }$ are the radial and tangential stresses at distance $r$, respectively. Compressive stress is considered positive while tensile stress is taken as negative. The corresponding parameters in the simulation are as follows: $\rho =2300 {\mathrm{kg}/\mathrm{m}}^3$, $E=20 \mathrm{GPa}$, $\upsilon =0.3$, $C_p=880 \mathrm{J}/\left(\mathrm{kg}\cdot \mathrm{K}\right)$, $k=3 \mathrm{W}/\left(\mathrm{m}\cdot \mathrm{K}\right)$, and ${\alpha }_T=1.0\times {10}^{-6} /\mathrm{K}$.

Figure 4a displays the temperature distribution within the thick-walled cylinder, comparing analytical and numerical temperature solutions along the radial direction. The temperature decreases monotonically from the inner to the outer surface, and the numerical results exhibit excellent agreement with the analytical solution. Figure 4b further compares numerical and analytical solutions for radial and tangential stresses. Along the radial coordinate, the radial stress first increases and subsequently decreases while remaining compressive throughout the entire thickness; concurrently, the tangential stress transitions from initial compression to tension with increasing radius. The close correspondence between numerical and analytical solutions of stress and temperature strongly confirms the accuracy of the coupled thermo-mechanical model.

4.2. Three-Point Bending Test

A classical three-point bending test was employed to validate the PFM’s capability in simulating fracture behavior under mechanically driven loading conditions. Figure 5 represents the geometry and boundary conditions of the symmetric three-point bending beam. The parameters of the material are as follows: $G_c=42.6 \mathrm{N}/\mathrm{m}$, $l_0=5 \mathrm{mm}$, $E=37 \mathrm{GPa}$, and $\upsilon =0.2$. The geometric model is discretized by quadrilateral elements, and the mesh is locally refined in the upper region of the notch. There are a total of 39,680 quadrilateral elements. Figure 6 displays the load-CMOD curve obtained from the three-point bending beam test, showing good agreement with the corresponding results reported by Liu et al. [46]. Moreover, Figure 7 illustrates the phase-field simulated evolution of fracture propagation in the three-point bending specimen under mechanical loading. The predicted crack path demonstrates close alignment with experimental observations reported in References [46,47], confirming the model’s capability to accurately capture fracture dynamics driven by external forces.

4.3. Cryogenic Thermal Shock

Thermal shock stimulation represents an established technique for enhancing permeability in deep, low-porosity reservoirs. The process involves injecting cooling fluids (e.g., water or liquid nitrogen) to create steep thermal gradients at rock-fluid interfaces. This section employs a TM-PFM to investigate the underlying mechanisms.

We chose a classic example, the 2D brittle solid quenching test, to simulate thermal shock. First, 99% Al₂O₃ powder was melted into a 50 mm × 100 mm × 1 mm sheet. Then, the brittle solid sheet was heated to an initial temperature, which varied from 300 °C to 600 °C. Finally, the high-temperature brittle solid sheet was abruptly placed in a 20 °C water bath.

Leveraging symmetry, we model the upper-left quadrant of the brittle plate (Figure 8). In the displacement field, the upper left is the free boundary condition, and the lower right is the displacement-constrained boundary condition. Moreover, in the temperature field, the upper left boundary is the convection boundary condition, and the lower right boundary is the thermal insulation boundary condition. The total calculation time is 100 ms, and the time step is 0.1 ms. The parameters of the material are as follows: $\rho =3980 {\mathrm{kg}/\mathrm{m}}^3$, $E=370 \mathrm{GPa}$, $\upsilon =0.3$, $G_c=42.47 \mathrm{N}/\mathrm{m}$, $l_0=0.05 \mathrm{mm}$, ${\alpha }_T=7.5\times {10}^{-6} /\mathrm{K}$, $C_p=880 \mathrm{J}/\left(\mathrm{kg}\cdot \mathrm{K}\right)$, $k=31 \mathrm{W}/\left(\mathrm{m}\cdot \mathrm{K}\right)$, and heat transfer coefficient $h=5\times {10}^5 \mathrm{W}/\left({\mathrm{m}}^2\cdot \mathrm{K}\right)$. To ensure grid independence, the element size was chosen as $h_{\mathrm{size}}=l_0/4$. In the final post-processing of the numerical results, we perform two mirroring operations on the two axes of symmetry to obtain information on the entire brittle solid sheet.

When subjected to thermal shock, high-temperature samples tend to develop a multitude of thermal cracks. Figure 9 shows that the numerical simulation results of the thermal crack pattern are in high agreement with the experimental findings, thereby substantiating the capacity of the proposed TM-PFM to accurately capture complex thermal fracture patterns. Furthermore, it is noticeable that the cracks exhibit a distinct hierarchical morphology. Based on the combined experimental and numerical findings, all cracks are categorized into three types according to their length: long, medium-long, and short. As depicted in Figure 10a, there is a gradual increase in the total number of cracks within the brittle solid sheet following thermal shock, as the initial temperature rises. Concurrently, the crack density becomes more concentrated. When comparing the numerical and experimental results, the counts of long and medium-long cracks align closely. However, the numerical results show a significantly higher number of short cracks than the experimental findings (see Figure 10b–d). This discrepancy is predominantly attributed to the difficulty in visually identifying short cracks in experiments, as they are not visible in experimental images. In contrast, numerical methods can overcome this limitation. Additionally, the boundary conditions in the experimental model may not be as ideal as those in numerical simulations, and the heterogeneity of the brittle solid sheet is not taken into account.

Figure 11 and Figure 12 illustrate the distribution of the maximum principal stress and temperature of the brittle solid sheet with varying initial temperatures at 100 ms, respectively. The maximum principal stress is concentrated at the crack tips. This is due to the significant shrinkage deformation caused by the sharp temperature drop of the outer boundary of the brittle solid sheet during thermal shock, which induces large tensile stresses. Interestingly, in the temperature field distribution (see Figure 12), the temperature exhibits discontinuity at both ends of the crack in the brittle solid sheet. This is because the crack impedes heat conduction, meaning no heat passes through the cracks.

A brittle solid sample with an initial temperature of 600 °C is chosen for an in-depth analysis of the crack propagation mechanism under thermal shock. Figure 13 and Figure 14 depict the internal temperature and maximum principal stress distribution of the 600 °C sample at different times. At 3 ms, thermal shock is in its early stage. A steep temperature gradient forms at the boundary, causing large tensile stresses there. Numerous short cracks start to form at the left and upper boundaries. Upon crack initiation, the stresses at the crack sites are relieved and subsequently redistributed throughout the sheet. Between 3 ms and 50 ms, as the cryogenic zone propagates inward, additional regions within the specimen attain the tensile strength threshold defined by the maximum principal stress.

This makes existing cracks grow or new ones form. Each time new cracks appear, stresses readjust until thermal stresses from the temperature gradient can no longer drive crack growth. Eventually, cracks compete to form different length morphologies (see Figure 15). The initial abundance of short cracks and subsequent crack initiation and propagation reflect the system’s rapid energy minimization. This aligns with Griffith fracture theory and well explains the fracture variational phase-field theory.

Thermal shock stimulation is utilized to create fracture networks in reservoir rocks, enhancing production. The heat transfer coefficient is a key parameter for evaluating the effectiveness of this process. To elucidate the influence of the heat-transfer coefficient on thermally-shock-driven fracture in brittle solids, experiments were performed on specimens initially equilibrated at 600 °C, employing heat-transfer coefficients of 50,000, 100,000, and 500,000 W/(m²·K). Figure 16 demonstrates that an increase in the heat-transfer coefficient yields a higher crack count and a corresponding reduction in crack spacing. Based on prior crack length classification, the number of long and medium-length cracks is largely unaffected by the heat transfer coefficient. However, short cracks rise sharply with increasing heat transfer coefficient, driving up the total crack count (Figure 17). This is because a higher heat transfer coefficient speeds up the boundary cooling rate and intensifies the temperature gradient. The system then releases energy rapidly through numerous short cracks to minimize total energy. In practical thermal shock stimulation, along with fluid medium temperature control, enhancing fluid properties (e.g., type and flow rate) and fluid-reservoir rock contact area can increase the heat transfer coefficient. This improves reservoir permeability and boosts geothermal energy recovery.

4.4. Thermal Fracturing Due to Rock Heterogeneity

Previous research has predominantly employed two approaches to characterize rock heterogeneity. The first assumes that rock’s physical and mechanical properties conform to a statistical distribution. The second derives the actual mineral composition via digital image processing (DIP). While these approaches yield useful results, they are not without limitations. The statistical method overlooks the real mineral composition. DIP, though precise, is confined to lab settings due to its high costs. Accordingly, we introduce the Knuth-Durstenfeld shuffle algorithm to capture rock mineral heterogeneity [5,49]. The detailed procedure of the Knuth–Durstenfeld shuffle algorithm applied in the present study is as follows: (1) Determine the total number of elements, N, in the numerical model; (2) Assign the number of elements occupied by each mineral according to its volumetric fraction. If the volume fractions of quartz, feldspar, amphibole, and clay minerals are P₁, P₂, P₃, and P₄, respectively, then the corresponding element counts are N_i = N × P_i; (3) Enumerate all elements in sequential order and endow each with the appropriate physical–mechanical parameters; (4) Apply the shuffle algorithm to reorder the parameter array, yielding a new parameter distribution. This algorithm combines the benefits of random statistical distribution with consideration of actual mineral composition. Additionally, it is well-suited for characterizing large-scale in-situ rock mass heterogeneity. A model considering the actual composition of the minerals is established, as shown in Figure 18. The physical and mechanical parameters for each mineral type are provided in

Table 1. Physical and mechanical parameters of sandstone.

Mineral	Quartz	Feldspar	Amphibole	Clay Minerals
Content (%)	40.14	30.75	15.16	13.95
$E$ (GPa)	12	8	10	8
$\nu$	0.16	0.19	0.23	0.22
$G_c$ (N/m)	2.56	1.19	1.15	0.76
$\rho$ (kg/m3)	2650	2570	2650	2410
$k$ (W/(m·K))	7.69	2.31	3.00	2.15
$C_p$ (J/(kg·K))	700	630	800	700

.

To explore the thermal damage mechanism induced by mineral heterogeneity during high-temperature heat treatment of sandstone specimens, a sandstone sample heated to 600 °C was chosen. Figure 19 and Figure 20 illustrate the changes in temperature and stress fields inside the sandstone during heat treatment, as well as the temperature distribution of sandstone samples at different cross-sections when heated to 600 °C. During heating, heat gradually propagates inward from the exterior. However, due to sandstone’s heterogeneity, the thermophysical parameters of different mineral crystals vary, causing the temperature contour lines to exhibit noticeable tortuosity (Figure 19a). Once thermal cracks form, they impede heat conduction, leading to significant temperature differences across the cracks and temperature discontinuity. The overall temperature on different cross-sections roughly follows a parabolic trend, but fluctuations occur in the temperature curve due to differences in thermophysical parameters. The temperature near the boundary is higher, while the middle part has the lowest temperature. The temperature at cracks on the intercept changes significantly, such as at points (38 mm, 10 mm) and (25.2 mm, 30 mm). Due to the heterogeneity of mineral thermal expansion coefficients, non-uniform thermal expansion occurs within the sandstone at high temperatures. This results in huge thermal stresses at mineral boundaries (Figure 19b,c). When thermal stress exceeds the strength of mineral crystals, numerous thermal cracks emerge (Figure 19d).

Figure 21 presents the distribution of internal stress, elastic modulus, thermal conductivity, and thermal cracks in specimens subjected to different temperatures. As the treatment temperature rises, thermal stress inside the sandstone shows a gradual increase. However, due to the varying thermal expansion coefficients of mineral crystals, high thermal stress is generated at the mineral boundaries, which in turn causes thermal damage to sandstone (Figure 21a,b). Figure 21e shows the thermal damage distribution inside sandstone at different temperatures. It can be found that from when temperature is below 400 °C, there are almost no thermal cracks generated inside the sandstone; When temperature reaches 400 °C, due to the thermal stress exceeding the tensile strength of some mineral crystals, thermal cracks begin to appear in the sandstone, resulting in a rapid decrease in the elastic modulus and thermal conductivity of the sandstone, as shown in Figure 21c,d. When the temperature exceeds 400 °C, the cracks generated by heat treatment further propagate and connect, causing irreversible damage.

Figure 22 compares the thermal cracks from the TM-PFM with the FEM results for sandstone specimens at different heat treatment temperatures. They show good consistency, proving the coupled model’s accuracy in simulating thermal fractures from non-uniform mineral crystal expansion. For quantitative analysis of sandstone damage at different heat treatment temperatures, the average elastic modulus of sandstone specimens after various heat treatments was obtained through post-processing (Figure 23). The elastic modulus trend matches experimental results. Few thermal damage units exist in sandstone specimens when the heat treatment temperature is below 400 °C. However, when it exceeds 400 °C, numerous thermal cracks form.

5. Conclusions

In this study, we established a thermodynamically consistent TM-PFM within COMSOL Multiphysics to simulate two distinct thermal fracturing modes in reservoir rocks under high-temperature conditions. Key findings include:

• The developed coupled thermo-mechanical phase-field model (TM-PFM) demonstrates a pronounced capability for accurately capturing the complete evolution of thermal fractures in reservoir rocks.
• During initial thermal shock, the substantial temperature difference induces a significant temperature gradient at the contact boundary within an extremely short duration. This rapid thermal loading generates immense thermal stresses. To relieve these stresses and minimize the system’s total energy, numerous short cracks initiate along the convective boundary. As the cryogenic region propagates inwards, these cracks undergo competition, evolving into cracks with diverse lengths and morphologies.
• Under thermal shock conditions, crack density progressively increases with higher heat transfer coefficients, while crack spacing decreases correspondingly. This phenomenon stems from the fact that a larger heat transfer coefficient accelerates the boundary cooling rate, exacerbating the temperature gradient and consequently the thermal stress magnitude.
• Elevated temperatures cause reservoir sandstone to exhibit differential thermal expansion due to mineral heterogeneity, inducing significant thermal stresses. Thermal cracks initiate once these stresses exceed the mineral strength. For the reservoir sandstone studied, 400 °C represents the threshold temperature for thermal crack initiation. Below this temperature, cracking is minimal, however, exceeding 400 °C triggers a marked increase in crack density.