# Particle Flow Analysis of Macroscopic and Mesoscopic Failure Process of Salt Rock under High Temperature and Triaxial Stress

Authors: Haoran Li; Ziheng Wang; Dekang Li; Yajun Zhang
Journal: Geofluids
 (2021)

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## Abstract

In order to reveal the mechanism of thermal-induced deformation and fracture development of salt rock under high temperature, the particle flow program PFC2D was used to study the triaxial compression failure process of salt rocks under different temperatures; at the same time, a combination model of Burge and Linearpbond was proposed to simulate plastic deformation and heat conduction of salt rock. Finally, the simulation results were compared with the experimental results to verify the validity of the conclusion. The simulation results show that the elastic limit points of rock gradually descend, the dilatancy points rise gradually, and the plastic deformation characteristics of salt rock become more obvious with the increase of temperature. Due to the damage of the sample, the strong chains break and disappear, increasing the proportion of the weak chains, and the high temperature intensifies the rupture of the contact between the particles in the salt rock. As the temperature increases from 50°C to 120°C, the strong chains in the rock sample decrease significantly, and the damage gradually increases; when the temperature is 150°C, the contact force decreases sharply, and the damage of salt rock is significant. According to the particle displacement cloud diagrams, it is found that the expansion direction from the middle part of the rock sample to the left and right ends is 12.08°, 9.55°, 8.2°, 6.33°, and 0°, respectively. The displacement directions of the rock sample show obvious radial expansion tendency, and the higher the temperature, the more obvious the “drum-shaped” failure phenomenon in the middle of the rock sample. During the heating process, the thermal cracks are mainly tensile cracks, and transverse cracks are gradually formed in the middle of the model. The cementation failure points at the top and bottom of the model expand in an oblique direction and form oblique cracks of about 45°. From the three different mathematical models of macroscopic and mesoscopic views, it is concluded that the effect of temperatures on salt rock is more significant after 90°C. This research is important for exploring the macroscopic and microscopic mechanics evolution of salt rock and provides a reference for determining the long-term mechanical strength of salt rock.


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## Body

## 1. Introduction

Salt rock can create an ideal environment for the storage of underground oil and gas energy, and salt cavern energy storage is generally built in the salt rock ore layer with a depth of 1500 m [1]. However, with the rapid development of shallow favourable deposits, the development of salt cavern underground energy storage is expanded from shallow (burieddepth<1500m) to deep (buried depth 1500-2500 m) and ultradeep (burieddepth>2500m), during which the construction and operation of underground storage also face more complex geological conditions, such as high in situ stress and high ground temperature. Therefore, it is very important to study the physical and mechanical properties of salt rock in a high-temperature environment to ensure the safe operation of storage.Many scholars have done a lot of experimental research on the mechanical properties of salt rock under the influence of temperature. Gao et al. [2] carried out triaxial compression tests of salt rock under the temperature of 25°C, 50°C, and 100°C to study the variation of macroscopic mechanical parameters of rock mass with temperature from a macro perspective, and they found that the peak stress and elastic modulus of salt rock decreased with the increase of temperature. Wu et al. [3] monitored the acoustic emission activities of salt rock in the process of thermal loading and explored the acoustic emission characteristics of rock at different stress stages under high temperatures. Kang et al. [4] designed the damage self-recovery test of salt rock and analyzed the damage and healing mechanism of salt rock from the perspective of crystal by using SEM technology. Under different temperatures and osmotic pressures, Yang et al. [5] carried out triaxial creep tests on calcium mirabilite salt rock and analyzed the influence of temperature on the creep process of calcium mirabilite. Because of the restriction of the test conditions, the temperature in the triaxial test of salt rock was not high (below 100°C), and the results had limitations.In addition, the laboratory tests always try to explain the temperature effect from the macroscopic perspective, which cannot reflect the mesoscopic development of salt rock in the thermodynamic coupling process; the DEM makes up this defect and is widely used in the geotechnical engineering field. Shen et al. [6] used PFC2D to simulate the dynamic failure process of shale rock under uniaxial compression and explored the evolution characteristics of microcracks in brittle shale failure under different stresses. Jiang et al. [7] used the discrete element method (DEM) to carry out triaxial simulation tests on structural sand and compared the results of numerical simulation and laboratory tests. Li et al. [8] applied the 3D-GBM model to PFC3D to explore the micromechanical properties of salt rock practical scale and verified the effectiveness of this method. Based on the DEM, Li et al. [9, 10] adopted the thermal-mechanical creep constitutive model to explore the micromechanism of the influence of temperature on the creep behaviour of salt rock. Xu [11] and Liang et al. [12] used PFC2D to establish the thermal-mechanical model and explored the fracture mechanism of granite under high temperature from a mesoscopic perspective. However, although the particle discrete element method has been widely used to analyze the damage evolution of rock mass, the research on the failure law of rock under a high-temperature environment is limited.Given the complexity of rock and earth mass mesostructure and the limitation of the research, there is no accepted mechanical theory formed to characterize the relationship between macro- and mesoevolution of rock mass. Based on the particle flow simulation technology, this paper makes up for the inadequacy of indoor high-temperature rock mechanics tests, analyzes the triaxial compression failure process of salt rock under 20°C, 50°C, 90°C, 120°C, and 150°C, and systematically studies the correlation characteristics of salt rock from mesoscopic change to macroscopic response, revealing the macro-mesoevolution mechanism of salt rock damage under high-temperature environment.


## 2. Constitutive Contact Model and Thermal Module Implementation

### 2.1. Selection of Constitutive Model

(1)
Burger model: Burger’s model is an elastoviscous rheological model, which consists of the Maxwell body in series with the Kelvin body (Figure1(a)). The main contact parameters are the stiffness Km,Kn of the elastic element and the bond coefficient Cm,Ck of the viscous element. The stiffness of the elastic element is the stiffness of contact between particles. The bond coefficient of the viscous element is the contact damping of the particles, which is related to the deformation strength. This model can well reflect the viscoplastic deformation characteristics of salt rock; however, during heating and loading, the internal thermal damage of the salt rock cannot be reflected by Burger’s model, and the built-in program of the model does not have related functions, such as crack control network, force chain, and energy(2)
Linearpbond model: the Linearpbond model consists of a bond component and a linear elastic component (Figure1(b)), which establishes an elastic contact and can better simulate the linear elastic stage of salt rock. In addition, stiffness ratio (kn/ks), bond strength (c), and effective modulus (E∗) are the main contact parameters in the Linearpbond model. The stiffness ratio is the ratio of normal stiffness to tangential stiffness (related to particle deformation). Bond strength is the contact bond strength of the particles. Effective modulus is the elastic modulus assigned to a single particle at the microscopic scale. After the contact bond is broken, the model allows the contacts between particles to produce cementation failure points, which can simulate the microcrack propagation during loading to better reflect the degradation of mechanical properties of rock and thermal expansion effects(3)
Combination constitutive model: during the triaxial compression test, the salt rock exhibits strong viscoplastic deformation characteristics, and microcracks grow significantly. In addition, the high temperature increases the ductility of the rock mass, and the thermal response of the internal cementation is strong. When the temperature reaches a certain value, the salt rock may enter a creep stage. Therefore, to better simulate the mesoscopic failure process of salt rock under high-temperature triaxial stress, the combination constitutive contact model of Burger and Linearpbond was adopted, as shown in Figure1. The constitutive models, burger and linear bond, and their parameters are quoted from the built-in database of PFC2D of Itasca commercial [13, 14]Figure 1
A combinatorial constitutive contact model of Burger and Linearpbond.


### 2.2. Heat Transfer Mechanism of the Model

To study the displacement and stress changes of rock particles under the action of thermal effect, the model needs to have the ability of heat storage and heat conduction. In the particle flow program, the heat conduction between particles is realized by the heat source through the heat pipe and finally forms the network heat transfer channel. The heat pipe represents the heat transfer process, which is related to predetermined thermal parameters of the model. Thermal parameters include thermal expansion coefficient (α), thermal resistance (η), thermal conductivity (k∗), and specific heat capacity (Cv). These parameters work together to generate heat conduction between particles. These parameters are determined following Section 3.4 of this paper. The thermal effect of particles produces the following effects: thermal strain causes particle expansion, which leads to the change of contact force; at the same time, the heat transferred through the contact between particles causes the expansion of the bond; their combined effect causes rock sample deformation. In addition, when heat is transferred to the inside of the model, the force and range of contact between particles will increase, and the parallel bond between particles will be broken, or particles will be moved, resulting in partial failure of the heat pipes between particles. The number of effective heat pipes decreases, and the overall heat conduction ability of the model decreases. The above process is consistent with the actual situation [15], so the PFC2D model can better simulate the heat conduction process of rock.


### 2.3. Thermal Strain Constitutive Model

In the PFC2D, the heat source completes heat conduction through the heat pipe and finally forms a network heat transfer channel. We assume that the effect of strain on temperature can be ignored, and the continuity equation of heat conduction involving thermal-mechanical coupled quasistatic mechanics is shown in
(1)−∂qi∂xi+qv=ρCv∂T∂t,whereqi is the heat flow vector, qv is the volumetric heat source intensity or energy density, ρ is the density, Cv is the specific heat capacity, and T is the temperature.In the PFC2D, the thermal strain is obtained by calculating the expansion of the adhesion between particles, and the thermal strain of the particles under the action of temperature is calculated by
(2)ΔR=αRΔT,whereΔT is the temperature increment, α is the linear expansion coefficient of the particle, and ΔR is the change in radius under temperature.


## 2.1. Selection of Constitutive Model

(1)
Burger model: Burger’s model is an elastoviscous rheological model, which consists of the Maxwell body in series with the Kelvin body (Figure1(a)). The main contact parameters are the stiffness Km,Kn of the elastic element and the bond coefficient Cm,Ck of the viscous element. The stiffness of the elastic element is the stiffness of contact between particles. The bond coefficient of the viscous element is the contact damping of the particles, which is related to the deformation strength. This model can well reflect the viscoplastic deformation characteristics of salt rock; however, during heating and loading, the internal thermal damage of the salt rock cannot be reflected by Burger’s model, and the built-in program of the model does not have related functions, such as crack control network, force chain, and energy(2)
Linearpbond model: the Linearpbond model consists of a bond component and a linear elastic component (Figure1(b)), which establishes an elastic contact and can better simulate the linear elastic stage of salt rock. In addition, stiffness ratio (kn/ks), bond strength (c), and effective modulus (E∗) are the main contact parameters in the Linearpbond model. The stiffness ratio is the ratio of normal stiffness to tangential stiffness (related to particle deformation). Bond strength is the contact bond strength of the particles. Effective modulus is the elastic modulus assigned to a single particle at the microscopic scale. After the contact bond is broken, the model allows the contacts between particles to produce cementation failure points, which can simulate the microcrack propagation during loading to better reflect the degradation of mechanical properties of rock and thermal expansion effects(3)
Combination constitutive model: during the triaxial compression test, the salt rock exhibits strong viscoplastic deformation characteristics, and microcracks grow significantly. In addition, the high temperature increases the ductility of the rock mass, and the thermal response of the internal cementation is strong. When the temperature reaches a certain value, the salt rock may enter a creep stage. Therefore, to better simulate the mesoscopic failure process of salt rock under high-temperature triaxial stress, the combination constitutive contact model of Burger and Linearpbond was adopted, as shown in Figure1. The constitutive models, burger and linear bond, and their parameters are quoted from the built-in database of PFC2D of Itasca commercial [13, 14]Figure 1
A combinatorial constitutive contact model of Burger and Linearpbond.


## 2.2. Heat Transfer Mechanism of the Model

To study the displacement and stress changes of rock particles under the action of thermal effect, the model needs to have the ability of heat storage and heat conduction. In the particle flow program, the heat conduction between particles is realized by the heat source through the heat pipe and finally forms the network heat transfer channel. The heat pipe represents the heat transfer process, which is related to predetermined thermal parameters of the model. Thermal parameters include thermal expansion coefficient (α), thermal resistance (η), thermal conductivity (k∗), and specific heat capacity (Cv). These parameters work together to generate heat conduction between particles. These parameters are determined following Section 3.4 of this paper. The thermal effect of particles produces the following effects: thermal strain causes particle expansion, which leads to the change of contact force; at the same time, the heat transferred through the contact between particles causes the expansion of the bond; their combined effect causes rock sample deformation. In addition, when heat is transferred to the inside of the model, the force and range of contact between particles will increase, and the parallel bond between particles will be broken, or particles will be moved, resulting in partial failure of the heat pipes between particles. The number of effective heat pipes decreases, and the overall heat conduction ability of the model decreases. The above process is consistent with the actual situation [15], so the PFC2D model can better simulate the heat conduction process of rock.


## 2.3. Thermal Strain Constitutive Model

In the PFC2D, the heat source completes heat conduction through the heat pipe and finally forms a network heat transfer channel. We assume that the effect of strain on temperature can be ignored, and the continuity equation of heat conduction involving thermal-mechanical coupled quasistatic mechanics is shown in
(1)−∂qi∂xi+qv=ρCv∂T∂t,whereqi is the heat flow vector, qv is the volumetric heat source intensity or energy density, ρ is the density, Cv is the specific heat capacity, and T is the temperature.In the PFC2D, the thermal strain is obtained by calculating the expansion of the adhesion between particles, and the thermal strain of the particles under the action of temperature is calculated by
(2)ΔR=αRΔT,whereΔT is the temperature increment, α is the linear expansion coefficient of the particle, and ΔR is the change in radius under temperature.


## 3. Numerical Modeling and Parameter Calibration

### 3.1. The Establishment of the Geometric Model

Referring to the high-temperature triaxial compression test in the salt rock chamber [16], numerical simulation was carried out to perform comparative analysis and verify the conclusion drawn in numerical simulation. The two-dimensional plane stress model was established, and the size of the model is Φ100mm×200mm, which is consistent with the laboratory test. In the model, the particle size range is 0.7 mm~1.05 mm, obeying uniform probability distribution. The reasonable ratio of the sample size to the average grain diameter is greater than 30~40 [17]. The combination constitutive model is adopted to simulate the contacts between particles; the contacts between the wall and the particle adopt Linear’s model, forming a total of 17,993 contacts. The density of the rock sample is 2300 kg/m3, and the porosity is 0.03. The salt rock model is shown in Figure 2.Figure 2
Numerical model of the salt rock: (a) particle model of heat transfer; (b) different contact models between particles.
(a)(b)


### 3.2. Test Scheme of Numerical Simulation

The first step is to calibrate the parameters of the constitutive contact model based on the laboratory test results. The test temperature was set to 20°C (indoor temperature), and the axial loading speed was set to 0.05 m/s. In the quasistatic simulation, a higher local damping coefficient is usually used to remove the kinetic energy in the system effectively, making the quasistatic deformation occur at a much higher rate than the actual situation [18, 19]. Triaxial compression tests with confining pressures of 10 MPa, 20MPa, and 30 MPa were, respectively, simulated to obtain the stress-strain curves; data processing was studied by trial and error method and comparison method. Then, the results were compared with laboratory test results to confirm the reasonable parameters of the numerical simulation constitutive contact model.The second step is to determine the thermal model parameters and the new contact parameters. The triaxial compression tests were carried out at environment temperatures of 50°C, 90°C, 120°C, and 150°C while setting the triaxial confining pressure as 10 MPa and the axial loading speed as 0.05 m/s. Through parameter optimization, the stress-strain curves of salt rock were obtained. By comparing to laboratory test results, the reasonability of numerical simulation thermal model parameters was verified, and new contact parameters were determined.The third step is to conduct the triaxial high-temperature numerical simulation test of salt rock. According to the parameters determined above, the triaxial confining pressure was set to be 10 MPa, and the axial loading speed was set to be 0.05 m/s. The stress-strain curves of salt rock under the environment temperatures of 20°C, 50°C, 90°C, 120°C, and 150°C were to obtain to further explore the damage evolution mechanism and macroscopic and mesoscopic characteristics of salt rock model at high temperatures.


### 3.3. Determination of Constitutive Contact Model Parameters

Since the cross-sectional area of the salt rock changes greatly after compression, the assumption of small strain is no longer applicable. Therefore, based on the results of the laboratory test [16], the area correction method [20] was used to process the stress-strain data (Figure 3). The principle of the area correction method is as follows.Figure 3
Stress-strain curves of different confining pressures at 20°C: (a) confining pressure 10 MPa; (b) confining pressure 20 MPa; (c) confining pressure 30 MPa.
(a)
Confining pressure 10 MPa(b)
Confining pressure 20 MPa(c)
Confining pressure 30 MPaAs the deformation changes the cross-sectional area of the rock sample, the corresponding pressure can be corrected by engineering strain; the specific expression is as follows.(3)σluc=F1A′,A′=A01−εt,whereσluc is linear corrected axial pressure, A0 is the cross-sectional area of the unpressurized sample, F1 is the axial force, and εt is the engineering strain value.In addition, according to the current research results [21–25], a set of constitutive model mesoparameters (Table 1) are determined by comparing the revised laboratory test results of salt rock so that the basic macroscopic parameters and stress-strain curves from the triaxial numerical simulation at room temperature are in good agreement with the laboratory test results (Figure 3). Comparison of stress-strain curves between laboratory test and numerical simulation shows that the correlation coefficients between the test curves and numerical curves are 0.9652, 0.9402, and 0.9601, respectively. The selected mesoparameters can reasonably reflect the macromechanical properties of the laboratory triaxial tests.Table 1
Microstructure parameters of PFC2D constitutive contact model.
Contact modelMesoscopic parameterNumerical valueLinearDensity (kg/m3)2300Linear contact modulus (Pa)3×109Stiffness ratiokn/ks1.0Coefficient of intergranular friction (μ)0.6Parallel bondBond contact modulus (Pa)3×109Normal bond strength (Pa)2.6×108Shear bond strength (Pa)2.0×108Burger’sKelvin elastic coefficient (Pa)1×109Kelvin viscosity coefficient (Pa·s)1×105Maxwell elastic coefficient (Pa)5×1011Maxwell viscosity coefficient (Pa·s)1×107Friction coefficient (μ)0.6


### 3.4. Determination of Thermal Model Parameters and New Contact Parameters

The heat conduction of the salt rock model was simulated by using the heat module of PFC2D, in which the upper and lower boundary of the salt rock model was set to be adiabatic, so the heat was transmitted from the lateral boundary of the model. With the reference data [9, 26, 27] and optimized parameters, a set of mesoscopic parameters of the thermal model were determined (Table 2). Specific heat capacity, CvJ/kgC∘, indicates the amount of heat required to raise the temperature of 1 kg of material by 1°C. The microscopic properties could be the same as the macroscopic specific heat capacity of the material. The thermal coefficient of expansion, α (1/°C), is the relative change in length of a unit length object when its temperature increases by 1°C. This microscopic property can be set by the macroscopic thermal coefficient of expansion α of the material. Upper and lower boundary thermal resistance, ηC∘/W⋅m, is the ratio of the temperature difference between the two ends of the object and the power of the heat source when heat is transmitted through the object. The value of η can be calculated by traversing the thermal network of a given granular material. Macroscopic thermal conductivity, k∗W/mC∘, refers to the heat transfer per unit time through per unit horizontal cross-sectional area when the temperature gradient is 1°C/m vertically downward.Table 2
Microstructure parameters of PFC2D thermal model.
Model parameterUnitsNumerical valueCoefficient of thermal conductivity (k∗)W/mC∘2.5Specific heat capacity (Cv)J/kgC∘1015Thermal coefficient of expansion (α)1/C∘2.8×10−6Upper and lower boundary thermal resistance (η)C∘/W⋅m1×10300In addition, as the temperature rose, the cohesion between the particles was weakened, the particles moved, and the cementation points were destroyed, resulting in the contact of the particles, the local organization, and even the overall structure greatly changed. Therefore, it is necessary to update the contact parameters of the model after the rock temperature rising to better simulate the loading process of salt rock after heating up. The contact model parameters at 20°C were used as the reference parameters, and the new contact parameters were determined through parameter optimization (Table3). Under temperatures of 50°C, 90°C, 120°C, and 150°C, the simulation results of salt rock experiments with confining pressure of 10 MPa were obtained (Figure 4). The stress-strain characteristics of the laboratory test and numerical simulation test show good agreement, which verifies the rationality of the selection of thermal parameters and new contact parameters.Table 3
New contact parameters of rock sample model at high temperature.
Model parameterNumerical valueTemperature (°C)205090120150Kelvin elastic coefficient (Pa)1×1095×1082.5×1082.2×1082.0×108Maxwell elastic coefficient (Pa)5×10114×10113.0×10112.5×10111.5×1011Maxwell viscosity coefficient (Pa·s)1×1070.8×1070.6×1070.4×1070.2×107Normal bond strength (Pa)2.6×1082.4×1082.2×1082.0×1081.8×108Shear bond strength (Pa)2.0×1081.8×1081.7×1081.54×1081.4×108Figure 4
Stress-strain curves of different temperatures under the confining pressure of 10 MPa.


## 3.1. The Establishment of the Geometric Model

Referring to the high-temperature triaxial compression test in the salt rock chamber [16], numerical simulation was carried out to perform comparative analysis and verify the conclusion drawn in numerical simulation. The two-dimensional plane stress model was established, and the size of the model is Φ100mm×200mm, which is consistent with the laboratory test. In the model, the particle size range is 0.7 mm~1.05 mm, obeying uniform probability distribution. The reasonable ratio of the sample size to the average grain diameter is greater than 30~40 [17]. The combination constitutive model is adopted to simulate the contacts between particles; the contacts between the wall and the particle adopt Linear’s model, forming a total of 17,993 contacts. The density of the rock sample is 2300 kg/m3, and the porosity is 0.03. The salt rock model is shown in Figure 2.Figure 2
Numerical model of the salt rock: (a) particle model of heat transfer; (b) different contact models between particles.
(a)(b)


## 3.2. Test Scheme of Numerical Simulation

The first step is to calibrate the parameters of the constitutive contact model based on the laboratory test results. The test temperature was set to 20°C (indoor temperature), and the axial loading speed was set to 0.05 m/s. In the quasistatic simulation, a higher local damping coefficient is usually used to remove the kinetic energy in the system effectively, making the quasistatic deformation occur at a much higher rate than the actual situation [18, 19]. Triaxial compression tests with confining pressures of 10 MPa, 20MPa, and 30 MPa were, respectively, simulated to obtain the stress-strain curves; data processing was studied by trial and error method and comparison method. Then, the results were compared with laboratory test results to confirm the reasonable parameters of the numerical simulation constitutive contact model.The second step is to determine the thermal model parameters and the new contact parameters. The triaxial compression tests were carried out at environment temperatures of 50°C, 90°C, 120°C, and 150°C while setting the triaxial confining pressure as 10 MPa and the axial loading speed as 0.05 m/s. Through parameter optimization, the stress-strain curves of salt rock were obtained. By comparing to laboratory test results, the reasonability of numerical simulation thermal model parameters was verified, and new contact parameters were determined.The third step is to conduct the triaxial high-temperature numerical simulation test of salt rock. According to the parameters determined above, the triaxial confining pressure was set to be 10 MPa, and the axial loading speed was set to be 0.05 m/s. The stress-strain curves of salt rock under the environment temperatures of 20°C, 50°C, 90°C, 120°C, and 150°C were to obtain to further explore the damage evolution mechanism and macroscopic and mesoscopic characteristics of salt rock model at high temperatures.


## 3.3. Determination of Constitutive Contact Model Parameters

Since the cross-sectional area of the salt rock changes greatly after compression, the assumption of small strain is no longer applicable. Therefore, based on the results of the laboratory test [16], the area correction method [20] was used to process the stress-strain data (Figure 3). The principle of the area correction method is as follows.Figure 3
Stress-strain curves of different confining pressures at 20°C: (a) confining pressure 10 MPa; (b) confining pressure 20 MPa; (c) confining pressure 30 MPa.
(a)
Confining pressure 10 MPa(b)
Confining pressure 20 MPa(c)
Confining pressure 30 MPaAs the deformation changes the cross-sectional area of the rock sample, the corresponding pressure can be corrected by engineering strain; the specific expression is as follows.(3)σluc=F1A′,A′=A01−εt,whereσluc is linear corrected axial pressure, A0 is the cross-sectional area of the unpressurized sample, F1 is the axial force, and εt is the engineering strain value.In addition, according to the current research results [21–25], a set of constitutive model mesoparameters (Table 1) are determined by comparing the revised laboratory test results of salt rock so that the basic macroscopic parameters and stress-strain curves from the triaxial numerical simulation at room temperature are in good agreement with the laboratory test results (Figure 3). Comparison of stress-strain curves between laboratory test and numerical simulation shows that the correlation coefficients between the test curves and numerical curves are 0.9652, 0.9402, and 0.9601, respectively. The selected mesoparameters can reasonably reflect the macromechanical properties of the laboratory triaxial tests.Table 1
Microstructure parameters of PFC2D constitutive contact model.
Contact modelMesoscopic parameterNumerical valueLinearDensity (kg/m3)2300Linear contact modulus (Pa)3×109Stiffness ratiokn/ks1.0Coefficient of intergranular friction (μ)0.6Parallel bondBond contact modulus (Pa)3×109Normal bond strength (Pa)2.6×108Shear bond strength (Pa)2.0×108Burger’sKelvin elastic coefficient (Pa)1×109Kelvin viscosity coefficient (Pa·s)1×105Maxwell elastic coefficient (Pa)5×1011Maxwell viscosity coefficient (Pa·s)1×107Friction coefficient (μ)0.6


## 3.4. Determination of Thermal Model Parameters and New Contact Parameters

The heat conduction of the salt rock model was simulated by using the heat module of PFC2D, in which the upper and lower boundary of the salt rock model was set to be adiabatic, so the heat was transmitted from the lateral boundary of the model. With the reference data [9, 26, 27] and optimized parameters, a set of mesoscopic parameters of the thermal model were determined (Table 2). Specific heat capacity, CvJ/kgC∘, indicates the amount of heat required to raise the temperature of 1 kg of material by 1°C. The microscopic properties could be the same as the macroscopic specific heat capacity of the material. The thermal coefficient of expansion, α (1/°C), is the relative change in length of a unit length object when its temperature increases by 1°C. This microscopic property can be set by the macroscopic thermal coefficient of expansion α of the material. Upper and lower boundary thermal resistance, ηC∘/W⋅m, is the ratio of the temperature difference between the two ends of the object and the power of the heat source when heat is transmitted through the object. The value of η can be calculated by traversing the thermal network of a given granular material. Macroscopic thermal conductivity, k∗W/mC∘, refers to the heat transfer per unit time through per unit horizontal cross-sectional area when the temperature gradient is 1°C/m vertically downward.Table 2
Microstructure parameters of PFC2D thermal model.
Model parameterUnitsNumerical valueCoefficient of thermal conductivity (k∗)W/mC∘2.5Specific heat capacity (Cv)J/kgC∘1015Thermal coefficient of expansion (α)1/C∘2.8×10−6Upper and lower boundary thermal resistance (η)C∘/W⋅m1×10300In addition, as the temperature rose, the cohesion between the particles was weakened, the particles moved, and the cementation points were destroyed, resulting in the contact of the particles, the local organization, and even the overall structure greatly changed. Therefore, it is necessary to update the contact parameters of the model after the rock temperature rising to better simulate the loading process of salt rock after heating up. The contact model parameters at 20°C were used as the reference parameters, and the new contact parameters were determined through parameter optimization (Table3). Under temperatures of 50°C, 90°C, 120°C, and 150°C, the simulation results of salt rock experiments with confining pressure of 10 MPa were obtained (Figure 4). The stress-strain characteristics of the laboratory test and numerical simulation test show good agreement, which verifies the rationality of the selection of thermal parameters and new contact parameters.Table 3
New contact parameters of rock sample model at high temperature.
Model parameterNumerical valueTemperature (°C)205090120150Kelvin elastic coefficient (Pa)1×1095×1082.5×1082.2×1082.0×108Maxwell elastic coefficient (Pa)5×10114×10113.0×10112.5×10111.5×1011Maxwell viscosity coefficient (Pa·s)1×1070.8×1070.6×1070.4×1070.2×107Normal bond strength (Pa)2.6×1082.4×1082.2×1082.0×1081.8×108Shear bond strength (Pa)2.0×1081.8×1081.7×1081.54×1081.4×108Figure 4
Stress-strain curves of different temperatures under the confining pressure of 10 MPa.


## 4. Analysis of Numerical Calculation Results

### 4.1. Characteristic Analysis of Salt Rock Strain Curves

In this section, triaxial compression simulations were carried out on salt rock under a different confining pressure of 10 MPa and 20°C, 50°C, 90°C, 120°C, and 150°C, respectively. The transverse strain, stress-strain curves, and volume strain curves of the samples were monitored and recorded, as shown in Figures5–7.Figure 5
Stress-strain simulation curves of salt rock samples at different temperatures.Figure 6
Volume strain simulation curves of salt rock samples at different temperatures.Figure 7
Transverse strain values of salt rock model after heating termination.Generally, the deformation process of rock can be divided into a linear elastic stage, a nonlinear elastic stage, a plastic stage, and a failure stage. As shown in Figure5, the turning point from the nonlinear elastic stage to the plastic stage is defined as the elastic limit point, and it can be seen that the elastic limit points (a–e) of the rock gradually descended with the increase of temperature. The result shows that with the increase of temperature, the plastic deformation of salt rock becomes more obvious, and the stress-strain curves of salt rock tend to be flatter. Figure 6 shows that with the temperature increase from 20°C, 50°C, 90°C, 120°C, to 150°C, respectively, the dilatancy points (a–e) of salt rock continuously increase at 20°C, 50°C, 90°C, 120°C, and 150°C. The result shows that with the increase of temperature, the internal particles of the rock sample expand, which causes the volume to increase and the same sample becomes loose. Therefore, the volume strain values of the sample reaching the expansion points increase during axial loading.Figure7 shows the transverse strain of salt rock after heating, and it can be seen that with the increase of termination temperature, the transverse strain was 0.0147, 0.0170, 0.018, and 0.0182, respectively, and the corresponding increments were 0.0023, 0.001, and 0.0002; this can be explained as that the temperature causes the increase of thermal motion of particles. However, when the particles collided with the wall, particles moved in reverse, resulting in the decrease of the increments of transverse strain.


### 4.2. Evolution of Force Chain Network in Rock Mass during Loading

The force chain network represents the distribution of the interparticle contact force in the soil framework system, which can reflect the sensitivity of the particle system to the force. Figure8 shows the variation of the force chain network of the sample at different temperatures under the axial strain of 0.15. The contact force in the force chain was generalized to visually reflect the variation of the internal stress field of the sample at different temperatures.
(1)
It can be seen from Figures8(a)–8(e) that the contact force inside the salt rock gradually decreased, and the strong chains broke and disappeared with the damage of the samples; at the same time, as the temperature increased, the proportion of weak chains increased under the same strain of 0.15, showing that the high temperature intensifies the rupture of contact between particles in the salt rock(2)
The maximum contact force changed from 0.787 MPa to 0.649 MPa as the temperature changed from 20°C to 50°C. When the temperature changed from 120°C to 150°C, the maximum contact force varied from 0.6 MPa to 0.417 MPa. For the temperature of 50°C, 90°C, and 120°C, the contact force slightly changed from 0.649 MPa, 0.62 MPa, to 0.6 MPa. However, Figures8(b)–8(d) show that the force chains inside the rock sample significantly became disperse from 50°C to 120°C, indicating that the damage began to intensify gradually. The contact force decreased sharply at 150°C, indicating that the salt rock was damaged significantlyFigure 8
The force chain network evolution diagram of salt rock samples at different temperatures.
(a)
20°C, contact force(b)
50°C, contact force(c)
90°C, contact force(d)
120°C, contact force(e)
150°C, contact force


### 4.3. Evolution Law of Cloud Image of Displacement Field inside the Model

Figure9 shows the change of displacement field in the sample under the temperature of 20°C, 50°C, 90°C, 120°C, and 150°C, respectively, and the axial strain was 0.15.
(1)
By analyzing the displacement direction of particles, it can be known that the displacement direction of the particles in the middle of the sample showed an obvious radial expansion trend at different temperatures, and the salt rock model appeared a “drum shape” destruction phenomenon(2)
The arrows in Figures9(a)–9(e) indicate the outward expansion direction of the central particles. It can be seen that the expansion direction from the middle of the sample to the left and right ends was 12.08°, 9.55°, 8.2°, 6.3°, and 0°, respectively, with the increase of temperature, which gradually changed from incline to horizontal. Therefore, during the loading, the phenomenon of “drum shape” in the middle of the rock sample became more obvious with the increase of temperatureFigure 9
Displacement field nephograms of sample model at different temperatures.
(a)
20°C, angle:α=12.08°(b)
50°C, angle:α=9.55°(c)
90°C, angle:α=8.2°(d)
120°C, angle:α=6.3°(e)
150°C, angle:α=0°


### 4.4. Expansion Analysis of Mesofailure of Rock Mass during Heating

This section studies the initiation and evolution of thermal cracks in the salt rock during the heating process when the confining pressure of the test is 10 MPa, the initial temperature of the test is 20°C, and the heating termination temperatures are 50°C, 90°C, 120°C, 130°C, and 150°C, respectively. Figure10 shows the distribution and extension of thermal cracks inside rock samples when the temperature changed from 20°C to 50°C. A certain bonding distance is set between particles. The bonding force will be generated between particles due to the bonding action, which is called the state of cementation when the distance between two particles is smaller than this distance. Cementation failure occurred under the action of thermal stress and thermal strain. The failure criterion is as follows: the failure occurs between particles, generating cementation failure point, when the tensile stress, compressive stress, or shear stress generated between particles is greater than the cementation strength between particles.
(1)
It can be seen from Figures10(a)–10(e) that there was a temperature gradient between the boundary and the internal particles of rock mass, and the cementation failure occurred first at the edge of the sample under the combined effect of thermal stress and thermal strain. As the heat was transferred to the interior of the model, the number of cementation failures increased, and the cementation failure points near the edge of the sample gradually connected. After reaching the termination temperature, the cementation failure points became stable(2)
In the middle of the model, the cementation failure points expanded along the transverse direction. At the top and bottom of the model, the cementation failure points expanded obliquely, and the direction was about 45° from the horizontal direction(3)
Figure11 shows the curve of the number of cementation failures under different temperatures. Figure 11 (a-b1-c1-d1) is the segment point of the cementation failure curve at 50°C, and Figure 11 (a-b2-c2-d2) is the segment point of the cementation failure curve at 90°C, 120°C, 130°C, and 150°C. It can be seen that at the initial stage of temperature rising (a-b1 and a-b2), the number of internal cementation failures in the salt rock increased linearly. With the transmission of heat (b1-c1 and b2-c2), the internal temperature gradient of rock mass became smaller, and the number of cementation failures grew smoothly. However, with the increase of temperature, the thermal motion of the particles increased, the cohesion decreases, and the cementation failure points continued to increase at a slower rate. After the termination of heat transfer (c1-d1 and c2-d2), the cementation failure of rock mass no longer increased(4)
According to the statistics of rock cementation failure points during the whole process of heating (Table4), it was found that the number of cementation failure points gradually increased with the termination temperature. However, when the temperature exceeded 120°C, the number of cementation failure points decreased, which may be because the significant enhancement of thermal movement and diffusion of particles in this temperature range move particles no longer a single movement, but a granular organizational movement. At the same time, the wall became more sensitive to the movement of particles, and the confining pressure of the wall also largely prevented contact fractures caused by particle dislocationsFigure 10
Distribution and propagation of cementation failure points in the salt rock model at 50°C (blue is the tensile failure points, green is the shear failure points, and red arrow represents the microcrack propagation direction).Figure 11
Cementation failure curves of sample model at different temperatures during temperature rising.Table 4
Cementation failure number of sample model at different temperatures.
Serial numberInitial temperature (°C)Termination temperature (°C)Cementation failure numbers (piece)1#20507202#20908743#2012011484#2013010415#20150998


## 4.1. Characteristic Analysis of Salt Rock Strain Curves

In this section, triaxial compression simulations were carried out on salt rock under a different confining pressure of 10 MPa and 20°C, 50°C, 90°C, 120°C, and 150°C, respectively. The transverse strain, stress-strain curves, and volume strain curves of the samples were monitored and recorded, as shown in Figures5–7.Figure 5
Stress-strain simulation curves of salt rock samples at different temperatures.Figure 6
Volume strain simulation curves of salt rock samples at different temperatures.Figure 7
Transverse strain values of salt rock model after heating termination.Generally, the deformation process of rock can be divided into a linear elastic stage, a nonlinear elastic stage, a plastic stage, and a failure stage. As shown in Figure5, the turning point from the nonlinear elastic stage to the plastic stage is defined as the elastic limit point, and it can be seen that the elastic limit points (a–e) of the rock gradually descended with the increase of temperature. The result shows that with the increase of temperature, the plastic deformation of salt rock becomes more obvious, and the stress-strain curves of salt rock tend to be flatter. Figure 6 shows that with the temperature increase from 20°C, 50°C, 90°C, 120°C, to 150°C, respectively, the dilatancy points (a–e) of salt rock continuously increase at 20°C, 50°C, 90°C, 120°C, and 150°C. The result shows that with the increase of temperature, the internal particles of the rock sample expand, which causes the volume to increase and the same sample becomes loose. Therefore, the volume strain values of the sample reaching the expansion points increase during axial loading.Figure7 shows the transverse strain of salt rock after heating, and it can be seen that with the increase of termination temperature, the transverse strain was 0.0147, 0.0170, 0.018, and 0.0182, respectively, and the corresponding increments were 0.0023, 0.001, and 0.0002; this can be explained as that the temperature causes the increase of thermal motion of particles. However, when the particles collided with the wall, particles moved in reverse, resulting in the decrease of the increments of transverse strain.


## 4.2. Evolution of Force Chain Network in Rock Mass during Loading

The force chain network represents the distribution of the interparticle contact force in the soil framework system, which can reflect the sensitivity of the particle system to the force. Figure8 shows the variation of the force chain network of the sample at different temperatures under the axial strain of 0.15. The contact force in the force chain was generalized to visually reflect the variation of the internal stress field of the sample at different temperatures.
(1)
It can be seen from Figures8(a)–8(e) that the contact force inside the salt rock gradually decreased, and the strong chains broke and disappeared with the damage of the samples; at the same time, as the temperature increased, the proportion of weak chains increased under the same strain of 0.15, showing that the high temperature intensifies the rupture of contact between particles in the salt rock(2)
The maximum contact force changed from 0.787 MPa to 0.649 MPa as the temperature changed from 20°C to 50°C. When the temperature changed from 120°C to 150°C, the maximum contact force varied from 0.6 MPa to 0.417 MPa. For the temperature of 50°C, 90°C, and 120°C, the contact force slightly changed from 0.649 MPa, 0.62 MPa, to 0.6 MPa. However, Figures8(b)–8(d) show that the force chains inside the rock sample significantly became disperse from 50°C to 120°C, indicating that the damage began to intensify gradually. The contact force decreased sharply at 150°C, indicating that the salt rock was damaged significantlyFigure 8
The force chain network evolution diagram of salt rock samples at different temperatures.
(a)
20°C, contact force(b)
50°C, contact force(c)
90°C, contact force(d)
120°C, contact force(e)
150°C, contact force


## 4.3. Evolution Law of Cloud Image of Displacement Field inside the Model

Figure9 shows the change of displacement field in the sample under the temperature of 20°C, 50°C, 90°C, 120°C, and 150°C, respectively, and the axial strain was 0.15.
(1)
By analyzing the displacement direction of particles, it can be known that the displacement direction of the particles in the middle of the sample showed an obvious radial expansion trend at different temperatures, and the salt rock model appeared a “drum shape” destruction phenomenon(2)
The arrows in Figures9(a)–9(e) indicate the outward expansion direction of the central particles. It can be seen that the expansion direction from the middle of the sample to the left and right ends was 12.08°, 9.55°, 8.2°, 6.3°, and 0°, respectively, with the increase of temperature, which gradually changed from incline to horizontal. Therefore, during the loading, the phenomenon of “drum shape” in the middle of the rock sample became more obvious with the increase of temperatureFigure 9
Displacement field nephograms of sample model at different temperatures.
(a)
20°C, angle:α=12.08°(b)
50°C, angle:α=9.55°(c)
90°C, angle:α=8.2°(d)
120°C, angle:α=6.3°(e)
150°C, angle:α=0°


## 4.4. Expansion Analysis of Mesofailure of Rock Mass during Heating

This section studies the initiation and evolution of thermal cracks in the salt rock during the heating process when the confining pressure of the test is 10 MPa, the initial temperature of the test is 20°C, and the heating termination temperatures are 50°C, 90°C, 120°C, 130°C, and 150°C, respectively. Figure10 shows the distribution and extension of thermal cracks inside rock samples when the temperature changed from 20°C to 50°C. A certain bonding distance is set between particles. The bonding force will be generated between particles due to the bonding action, which is called the state of cementation when the distance between two particles is smaller than this distance. Cementation failure occurred under the action of thermal stress and thermal strain. The failure criterion is as follows: the failure occurs between particles, generating cementation failure point, when the tensile stress, compressive stress, or shear stress generated between particles is greater than the cementation strength between particles.
(1)
It can be seen from Figures10(a)–10(e) that there was a temperature gradient between the boundary and the internal particles of rock mass, and the cementation failure occurred first at the edge of the sample under the combined effect of thermal stress and thermal strain. As the heat was transferred to the interior of the model, the number of cementation failures increased, and the cementation failure points near the edge of the sample gradually connected. After reaching the termination temperature, the cementation failure points became stable(2)
In the middle of the model, the cementation failure points expanded along the transverse direction. At the top and bottom of the model, the cementation failure points expanded obliquely, and the direction was about 45° from the horizontal direction(3)
Figure11 shows the curve of the number of cementation failures under different temperatures. Figure 11 (a-b1-c1-d1) is the segment point of the cementation failure curve at 50°C, and Figure 11 (a-b2-c2-d2) is the segment point of the cementation failure curve at 90°C, 120°C, 130°C, and 150°C. It can be seen that at the initial stage of temperature rising (a-b1 and a-b2), the number of internal cementation failures in the salt rock increased linearly. With the transmission of heat (b1-c1 and b2-c2), the internal temperature gradient of rock mass became smaller, and the number of cementation failures grew smoothly. However, with the increase of temperature, the thermal motion of the particles increased, the cohesion decreases, and the cementation failure points continued to increase at a slower rate. After the termination of heat transfer (c1-d1 and c2-d2), the cementation failure of rock mass no longer increased(4)
According to the statistics of rock cementation failure points during the whole process of heating (Table4), it was found that the number of cementation failure points gradually increased with the termination temperature. However, when the temperature exceeded 120°C, the number of cementation failure points decreased, which may be because the significant enhancement of thermal movement and diffusion of particles in this temperature range move particles no longer a single movement, but a granular organizational movement. At the same time, the wall became more sensitive to the movement of particles, and the confining pressure of the wall also largely prevented contact fractures caused by particle dislocationsFigure 10
Distribution and propagation of cementation failure points in the salt rock model at 50°C (blue is the tensile failure points, green is the shear failure points, and red arrow represents the microcrack propagation direction).Figure 11
Cementation failure curves of sample model at different temperatures during temperature rising.Table 4
Cementation failure number of sample model at different temperatures.
Serial numberInitial temperature (°C)Termination temperature (°C)Cementation failure numbers (piece)1#20507202#20908743#2012011484#2013010415#20150998


## 5. Quantitative Analysis of Thermal Damage of Salt Rock

### 5.1. Damage Analysis Based on the Elastic Modulus

In this section, the damage value is defined by the change of elastic modulus of the sample before and after heating [28], and the damage variable DT is calculated as follows.
(4)DT=1−ETE0,where ET is elastic modulus at temperature T (MPa) and E0 is elastic modulus at 20°C (MPa).By analyzing the stress-strain curves at 20°C, 50°C, 90°C, 120°C, and 150°C, the corresponding elastic modulus of salt rock was calculated to be 4.989 GPa, 4.56 GPa, 3.72 GPa, 2.33 GPa, and 1.089 GPa, respectively.DT at different temperatures were 0, 0.0860, 0.2543, 0.533, and 0.7817, respectively, and the change curves of damage variables and elastic modulus with temperature are drawn in Figure 12.Figure 12
Damage variable curves of salt rock at different temperatures.It can be seen that the elastic modulus gradually decreased with increasing temperature, while the damage variables gradually increased, showing a negative correlation of them. Before reaching 90°C, the damage variables increased slowly from 0 to 0.2543 by 0.2543, and after 90°C, it increased by 0.5274. Between 90°C and 120°C, the damage variables changed from 0.2543 to 0.5330 by 0.2787. From 120°C to 150°C, they changed from 0.533 to 0.7817 by 0.2487. In conclusion, the damage of the salt rock model was relatively gentle before 90°C, while the influence of temperature on the salt rock was more significant after 90°C, but the higher the temperature did not cause more damage.


### 5.2. Damage Analysis Based on Cementation Accumulation Numbers

The particles are connected by the Linearpbond model and the Burger model. When the contact force is greater than the bonding strength, the contact bond will be broken and produce cementation failure points; then, the cementation failure points extend and penetrate each other to form microcracks, finally leading to the damage and deterioration of the salt rock model.In addition, the rock damage index is usually defined by the development of microcracks as the basic variable. However, in the actual test, the number of cracks is not easy to obtain [29]. Therefore, many scholars define the damage variables as the ratio of cumulative acoustic emission ringing numbers to the cumulative ringing numbers under peak stress [30, 31]; the cementation failure numbers can represent cumulative acoustic emission ringing numbers in numerical simulation of particle flow [32, 33]. Therefore, with the particle flow discrete element method, the rock damage index was calculated as follows to quantitatively analyze the damage degree of salt rock at different temperatures.
(5)DT=NTN0T,where NT is the cumulative cementation failure number at temperature T and N0Tis the cementation failure number corresponding to peak stress at temperature T.Generally, the point where the axial strain of salt rock is 0.1 is defined as the peak stress point. It can be seen from Figure13 that at different temperatures, the variation trend of the salt rock damage variable DT with the axial strain was basically the same, which can be divided into three stages. In stage I, the curve showed a slow-growth trend. In stage II, the slopes of all curves began to increase, and the damage started to be gradually intensified when the axial strain reached 0.75. In stage III, the damage variable sharply increased, and the higher the temperature, the more obvious the increment.Figure 13
Variation curves of damage variables with axial strain at different temperatures.In the early stage of loading (ε=0~0.075), the damage variable of salt rock did not change obviously at different temperatures. At the end of loading (ε=0.075~0.15), the temperature showed a great influence on the damage of salt rock. In order to visually show the influence of temperature on the internal damage of salt rock, at different temperatures, the damage variables of salt rock when the axial strain was 0.09, 0.12, and 0.15 were obtained, as shown in Table 5.Table 5
Damage variables of salt rock at different temperatures.
Temperature (°C)Strain (ε=0.09)Strain (ε=0.12)Strain (ε=0.15)200.72431.29491.6872500.91921.31131.7962900.92031.41621.99301201.101.64322.21621501.06451.78712.7097It is found that from 20°C to 50°C, the damage increments under different strains were 0.1949, 0.0164, and 0.109, respectively. From 50°C to 90°C, the damage increments were 0.0011, 0.1049, and 0.1968, respectively. From 90°C to 120°C, the damage increments were 0.1797, 0.227, and 0.229, respectively. From 120°C to 150°C, the damage increments were -0.0355, 0.1439, and 0.4935, respectively. In conclusion, under different strains, the damage increments increase significantly with the temperatures after 90°C.


### 5.3. Damage Analysis Based on Crack Density Parameters

In the particle flow program, the rock model was established based on the assumptions of nonintersecting cracks, uniform crack distribution, and simple crack shape, which is the hypothetical model without cracks interaction (NIA). Based on NIA, Budiansky and O’Connell deduced the calculation model of rock elastic modulus and crack density parameters [34, 35], as follows.
(6)ρ=452−v0161−v0210−3v0E0E−1,where E0 is the elastic modulus of undamaged rock (MPa), V0 is Poisson’s ratio of undamaged rock, E is the elastic modulus of damaged rock (MPa), and ρ is the crack density parameters.Referring to the elastic modulus and Poisson’s ratio of undamaged salt rock at room temperature, this paper explored the damage degree of salt rock at 20°C, 50°C, 90°C, 120°C, and 150°C. The elastic modulus and Poisson’s ratio of undamaged salt rock are 7.157 MPa and 0.187, respectively [16]. Formula (3) is used to calculate the crack density of salt rock at different temperatures (see Table 6) and draw the curve of crack density with temperature, as shown in Figure 14.Table 6
Crack density parameters of salt rock at different temperatures.
Temperature (°C)Elastic modulus (MPa)Crack density parameters204.9890.2433504.560.3188903.720.51721202.331.15971501.0893.1192Figure 14
Crack density parameter curve of salt rock at different temperatures.When there is no thermal damage, the crack density parameter of rock is set to 0. It can be seen from Figure14 that the crack density parameters increased slowly from 0.2433 to 0.5172, totally by 0.2739 before 90°C. From 90°C to 120°C, the value changed from 0.5172 to 1.1597 by 0.6425. From 120°C to 150°C, it changed from 1.1597 to 3.1192 by 1.9595. It can be seen that the number of cracks in the unit volume sharply increased after 90°C, and the temperature had a significant impact on the damage of salt rock at 150°C.


### 5.4. Comparative Analysis of Calculation Models

The elastic modulus, cumulative cementation numbers, and derived crack density parameters are closely related to the damage deterioration of rock mechanical properties.(1)
The elastic modulus and crack density showed a consistent variation with temperature, indicating that the temperature has a significant effect on the damage of salt rock after 90°C(2)
According to the calculation of cementation accumulation numbers, it was found that the damage of salt rock gradually increased after the axial strain reaching 0.075. Under different strains, the increments were 0.196, 0.1213, and 0.3018 from 20°C to 90°C, while the increments were 0.1442, 0.3079, and 0.7225, respectively, from 90°C to 150°C. After 90°C, the damage variables increased significantly with the change of temperatures; with the increase of loading, the damage variables increased significantly at 150°C. In addition, the crack density parameter also represents the damage of salt rock from the perspective of microcracks. The crack density parameter increased by 0.2739 from 20°C to 90°C, while it increased by 2.602 from 90°C to 150°C. After 90°C, the number of cracks per unit volume increased sharply. It can be seen from Figure13 that the temperature has a significant effect on the damage of salt rock at 150°C. The damage analysis of microcracks in salt rock from two different calculation models reflects the damage of salt rock under the action of temperature(3)
Model ①: the calculation model is established from the perspective of macroscopic elastic modulus. Model ②: the calculation model is established from the perspective of mesoscopic cementation failure points. Model ③: the microcrack density parameter is established by using the macroscopic modulus, and the microphysical quantity characterizing the material is connected with the macroscopic modulus. From three different mathematical models, it is concluded that the temperature has a significant effect on the damage of salt rock after 90°C


## 5.1. Damage Analysis Based on the Elastic Modulus

In this section, the damage value is defined by the change of elastic modulus of the sample before and after heating [28], and the damage variable DT is calculated as follows.
(4)DT=1−ETE0,where ET is elastic modulus at temperature T (MPa) and E0 is elastic modulus at 20°C (MPa).By analyzing the stress-strain curves at 20°C, 50°C, 90°C, 120°C, and 150°C, the corresponding elastic modulus of salt rock was calculated to be 4.989 GPa, 4.56 GPa, 3.72 GPa, 2.33 GPa, and 1.089 GPa, respectively.DT at different temperatures were 0, 0.0860, 0.2543, 0.533, and 0.7817, respectively, and the change curves of damage variables and elastic modulus with temperature are drawn in Figure 12.Figure 12
Damage variable curves of salt rock at different temperatures.It can be seen that the elastic modulus gradually decreased with increasing temperature, while the damage variables gradually increased, showing a negative correlation of them. Before reaching 90°C, the damage variables increased slowly from 0 to 0.2543 by 0.2543, and after 90°C, it increased by 0.5274. Between 90°C and 120°C, the damage variables changed from 0.2543 to 0.5330 by 0.2787. From 120°C to 150°C, they changed from 0.533 to 0.7817 by 0.2487. In conclusion, the damage of the salt rock model was relatively gentle before 90°C, while the influence of temperature on the salt rock was more significant after 90°C, but the higher the temperature did not cause more damage.


## 5.2. Damage Analysis Based on Cementation Accumulation Numbers

The particles are connected by the Linearpbond model and the Burger model. When the contact force is greater than the bonding strength, the contact bond will be broken and produce cementation failure points; then, the cementation failure points extend and penetrate each other to form microcracks, finally leading to the damage and deterioration of the salt rock model.In addition, the rock damage index is usually defined by the development of microcracks as the basic variable. However, in the actual test, the number of cracks is not easy to obtain [29]. Therefore, many scholars define the damage variables as the ratio of cumulative acoustic emission ringing numbers to the cumulative ringing numbers under peak stress [30, 31]; the cementation failure numbers can represent cumulative acoustic emission ringing numbers in numerical simulation of particle flow [32, 33]. Therefore, with the particle flow discrete element method, the rock damage index was calculated as follows to quantitatively analyze the damage degree of salt rock at different temperatures.
(5)DT=NTN0T,where NT is the cumulative cementation failure number at temperature T and N0Tis the cementation failure number corresponding to peak stress at temperature T.Generally, the point where the axial strain of salt rock is 0.1 is defined as the peak stress point. It can be seen from Figure13 that at different temperatures, the variation trend of the salt rock damage variable DT with the axial strain was basically the same, which can be divided into three stages. In stage I, the curve showed a slow-growth trend. In stage II, the slopes of all curves began to increase, and the damage started to be gradually intensified when the axial strain reached 0.75. In stage III, the damage variable sharply increased, and the higher the temperature, the more obvious the increment.Figure 13
Variation curves of damage variables with axial strain at different temperatures.In the early stage of loading (ε=0~0.075), the damage variable of salt rock did not change obviously at different temperatures. At the end of loading (ε=0.075~0.15), the temperature showed a great influence on the damage of salt rock. In order to visually show the influence of temperature on the internal damage of salt rock, at different temperatures, the damage variables of salt rock when the axial strain was 0.09, 0.12, and 0.15 were obtained, as shown in Table 5.Table 5
Damage variables of salt rock at different temperatures.
Temperature (°C)Strain (ε=0.09)Strain (ε=0.12)Strain (ε=0.15)200.72431.29491.6872500.91921.31131.7962900.92031.41621.99301201.101.64322.21621501.06451.78712.7097It is found that from 20°C to 50°C, the damage increments under different strains were 0.1949, 0.0164, and 0.109, respectively. From 50°C to 90°C, the damage increments were 0.0011, 0.1049, and 0.1968, respectively. From 90°C to 120°C, the damage increments were 0.1797, 0.227, and 0.229, respectively. From 120°C to 150°C, the damage increments were -0.0355, 0.1439, and 0.4935, respectively. In conclusion, under different strains, the damage increments increase significantly with the temperatures after 90°C.


## 5.3. Damage Analysis Based on Crack Density Parameters

In the particle flow program, the rock model was established based on the assumptions of nonintersecting cracks, uniform crack distribution, and simple crack shape, which is the hypothetical model without cracks interaction (NIA). Based on NIA, Budiansky and O’Connell deduced the calculation model of rock elastic modulus and crack density parameters [34, 35], as follows.
(6)ρ=452−v0161−v0210−3v0E0E−1,where E0 is the elastic modulus of undamaged rock (MPa), V0 is Poisson’s ratio of undamaged rock, E is the elastic modulus of damaged rock (MPa), and ρ is the crack density parameters.Referring to the elastic modulus and Poisson’s ratio of undamaged salt rock at room temperature, this paper explored the damage degree of salt rock at 20°C, 50°C, 90°C, 120°C, and 150°C. The elastic modulus and Poisson’s ratio of undamaged salt rock are 7.157 MPa and 0.187, respectively [16]. Formula (3) is used to calculate the crack density of salt rock at different temperatures (see Table 6) and draw the curve of crack density with temperature, as shown in Figure 14.Table 6
Crack density parameters of salt rock at different temperatures.
Temperature (°C)Elastic modulus (MPa)Crack density parameters204.9890.2433504.560.3188903.720.51721202.331.15971501.0893.1192Figure 14
Crack density parameter curve of salt rock at different temperatures.When there is no thermal damage, the crack density parameter of rock is set to 0. It can be seen from Figure14 that the crack density parameters increased slowly from 0.2433 to 0.5172, totally by 0.2739 before 90°C. From 90°C to 120°C, the value changed from 0.5172 to 1.1597 by 0.6425. From 120°C to 150°C, it changed from 1.1597 to 3.1192 by 1.9595. It can be seen that the number of cracks in the unit volume sharply increased after 90°C, and the temperature had a significant impact on the damage of salt rock at 150°C.


## 5.4. Comparative Analysis of Calculation Models

The elastic modulus, cumulative cementation numbers, and derived crack density parameters are closely related to the damage deterioration of rock mechanical properties.(1)
The elastic modulus and crack density showed a consistent variation with temperature, indicating that the temperature has a significant effect on the damage of salt rock after 90°C(2)
According to the calculation of cementation accumulation numbers, it was found that the damage of salt rock gradually increased after the axial strain reaching 0.075. Under different strains, the increments were 0.196, 0.1213, and 0.3018 from 20°C to 90°C, while the increments were 0.1442, 0.3079, and 0.7225, respectively, from 90°C to 150°C. After 90°C, the damage variables increased significantly with the change of temperatures; with the increase of loading, the damage variables increased significantly at 150°C. In addition, the crack density parameter also represents the damage of salt rock from the perspective of microcracks. The crack density parameter increased by 0.2739 from 20°C to 90°C, while it increased by 2.602 from 90°C to 150°C. After 90°C, the number of cracks per unit volume increased sharply. It can be seen from Figure13 that the temperature has a significant effect on the damage of salt rock at 150°C. The damage analysis of microcracks in salt rock from two different calculation models reflects the damage of salt rock under the action of temperature(3)
Model ①: the calculation model is established from the perspective of macroscopic elastic modulus. Model ②: the calculation model is established from the perspective of mesoscopic cementation failure points. Model ③: the microcrack density parameter is established by using the macroscopic modulus, and the microphysical quantity characterizing the material is connected with the macroscopic modulus. From three different mathematical models, it is concluded that the temperature has a significant effect on the damage of salt rock after 90°C


## 6. Discussions

The physical and mechanical properties of the salt rock are affected by temperature, and the mechanism is complicated. On the one hand, the temperature is related to the thermal motion of the crystal particles. The higher the temperature, the weaker the binding force between particles, resulting in the enhanced plasticity of the rock and the decreased strength. The cohesion between rock particles is composed of true cohesion and apparent cohesion; true cohesion represents the bonding strength between particles, while apparent cohesion represents the frictional occlusal effect between particles [36]. In the low-temperature environment, the occlusal effect plays a dominant role, so the particle displacement is small, the elastic tensile deformation between the particles is not obvious, and the bonding strength is not weakened. In the high temperature, the friction and occlusion strength between particles are basically destroyed, and true cohesion plays a major role. With the increase of temperature, the bonding strength between particles is gradually destroyed, and the plastic deformation characteristics of rock become obvious. On the other hand, salt rock has damage repair function, including damage microcrack healing based on diffusion, microcrack filling based on NaCl crystal recrystallization, and the healing and bonding of NaCl crystal particles in the fracture zone [4]. The energy recovery, healing speed, and crystallization development of salt rock are closely related to the ambient temperature. The simultaneous occurrence of damage and repair of salt rock is also the reason for the complexity of the research topic. Through numerical simulation, it is found that there exists a postpeak hardening stage during triaxial compression of salt rock at high temperatures; the higher the ambient temperature, the more obvious this stage is. Due to the restriction of test instruments and high-temperature conditions, the curves of this stage generally cannot be obtained in laboratory rock mechanics tests. In this stage, the salt rock still has a higher load-bearing capacity, the stress increment is small, and the strain keeps increasing. In addition, it is found in the numerical simulation that the number of cementation failures in the salt rock model decreases when the temperature exceeds 120°C. Meanwhile, the damaged salt rock has short-term self-healing characteristics under temperature conditions in laboratory tests. To sum up, the formation mechanism and physical significance of the above problems need to be further studied.


## 7. Conclusions

In this paper, through particle flow simulation analysis technology, the triaxial compression failure process and strength characteristics of salt rock under a high-temperature environment were systematically studied, and the following conclusions can be drawn.(1)
With the increase of temperature, the elastic limit points (a–e) of rock gradually decrease, which indicates that the plastic deformation characteristics of salt rock become more obvious. In addition, as the termination temperature gradually increases, the transverse strain values are 0.0147, 0.0170, 0.018, and 0.0182, respectively, and the increments of transverse strain are 0.0023, 0.001, and 0.0002, which maybe because of the decrease of the increment of transverse strain caused by the limitation of the wall and confining pressure.(2)
With the increase of temperature, the strong chains break and disappear, and the proportion of weak chains increase, which shows that the high temperature intensifies the rupture of contact between particles in the salt rock. The strength chains inside the rock sample decrease significantly from 50°C to 120°C, indicating that the damage begins to intensify gradually. The contact force decreases sharply at 150°C, indicating that the salt rock is damaged significantly at a temperature of 150°C during the loading process(3)
By analyzing the displacement direction of particles in the cloud map of the displacement field, it can be known that the displacement direction of the particles in the middle of the sample shows an obvious radial expansion trend at different temperatures, and the salt rock model shows a “drum shape” destruction phenomenon. With the increase of temperature, the expansion direction from the middle of the sample to the left and right ends is 12.08°, 9.55°, 8.2°, 6.33°, and 0°, respectively, and the phenomenon of “drum shape” in the middle of rock sample becomes more obvious(4)
During the heating process, the salt rock model is mainly controlled by tensile stress, so the thermal cracks in the sample are mainly tensile cracks. In the middle of the sample model, the microcracks roughly expand along the transverse direction. At the top and bottom of the model, the microcracks expand diagonally, and the direction of expansion is about 45° in the horizontal direction. In addition, the number of microcracks in the salt rock decreases when the temperature exceeds 120°C, which is different from the general rock, indicating that the damage self-healing phenomenon seems to occur in this temperature range(5)
The damage variables of salt rock are analyzed from the macro and micro perspectives; it is concluded that the damage of salt rock is more significant after 90°C. In addition, it is found that the damage of salt rock is significantly aggravated at 150°C in the mathematical model of cumulative cementation numbers and crack density parameter, which is also consistent with the evolution law of the force chain network in salt rock



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*Source: 8238002-2021-11-15.xml*