# Salt Cavern Thermal Damage Evolution Investigation Based on a Hybrid Continuum-Discrete Coupled Modeling

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## Abstract

**:**

## 1. Introduction

#### 1.1. Problem Statement

#### 1.2. Micromechanism of Thermal Damage

#### 1.3. Development of Hybrid Modeling

## 2. Numerical Method

#### 2.1. Model Description

#### 2.2. Coupling Mechanism of FLAC-PFC

## 3. Thermal Progressive Damage Evolution

#### 3.1. Thermal Effect at Three Observed Locations

#### 3.2. Dynamic Response to Depressurization and Progressive Damage Characteristics

_{1}and l

_{2}is given by:

^{−3}m

^{2}/m

^{3}, as it is the value of the fracture surface area per unit volume. The absolute index ${c}_{d}$ is the fracture mass density, which is defined as the fracture surface area per unit volume. The normalized concentration degree of crack index ${c}_{d}$ ranging from 0 to 1 is defined as the absolute index normalized by the maximum to minimum value among all the cracks. Figure 11 shows the comparison of three different locations of the surrounding wall of the salt cavity, which are: (a) location 1—at the knee point of the cavity shoulder, i.e., the convexity closest to the rim; (b) location 2—right part off the knee point, deep in the rock salt formation; (c) location 3—$11.8\mathrm{m}$ away from the cavity surrounding the wall edge, the most farthest from the centerline of the cavity. The microcrack forms and distributes into the entire specimen in location 1. The crack distribution is more concentrated compared with the other two locations far away from the convexity. The microcracks become less intensive in the region that is not close to the inner cavity with no convexity–concavity. The damage failure exhibits operation confinement and shape dependence. Figure 12 indicates the trend of the crack number using the normalized relative index ${c}_{d}$. It shows that most of the cracks are with ${c}_{d}=0.2$. To investigate the heterogeneities of thermal cracking in rock salt, Figure 13 illustrates the rose diagrams of tensile and shear cracks at location 1, location 2, and location 3, respectively. The radial length of each bin indicates the number of shear or tensile cracks oriented within the corresponding angles. It shows that tensile cracks tend to initiate in the horizontal orientation at location 1 which is close to the cavity surrounding the wall. While the orientation of shear cracks in all three selected locations is uniformly distributed.

#### 3.3. Energy Tracking

_{K}is the kinetic energy of particles under thermal-mechanical force and B

_{K}is the kinetic energy of particles under mechanical force only.

_{L}is the sliding energy of particles under thermal-mechanical force and B

_{L}is the sliding energy of particles under mechanical force only.

_{S}is the strain energy of particles under thermal-mechanical force and B

_{S}is the strain energy of particles under mechanical force only.

## 4. Results and Discussion

#### 4.1. Influence of Confining Pressure

#### 4.2. Effect of Particle Microproperties

#### 4.3. Effect of Thermal Properties

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**A field case in Jintan, sonar results of Cavern L in the years 2009 and 2015, respectively [8]. (

**a**) Sonar monitoring results of the cavity; (

**b**) Displacement difference before and after cavity collapse.

**Figure 2.**Hybrid continuum-discrete model for a salt cavern subjected to gas-cycling loading. (

**a**) Salt cavern surrounding rock continuum part; (

**b**) Selected continuous-discrete coupling part; (

**c**) The size of the discrete part of the ball ranges from 0.04 to 0.06 m.

**Figure 6.**Gas temperature and pressure variations during five years of gas-cycling loading operation (Li et al., 2021) [8].

**Figure 7.**Three selected locations for observation of microcrack development in PFC (

**a**); and the corresponding temperature (

**b**) after sharp pressure drop due to gas withdrawal at 3.14 years of operation.

**Figure 8.**Comparison of crack development at three locations for coupled thermal-mechanical effect and mechanical effect only.

**Figure 9.**Temporal and spatial evolution of crack increments: A—Location 1; B—Location 3; C—Location 2; D—The ratio of tensile cracks to total cracks at Location 1; E—The ratio of tensile cracks to total cracks at Location 3; F—The ratio of tensile cracks to total cracks at Location 2.

**Figure 10.**Comparison of crack development at three locations before and after the sharp pressure drop.

**Figure 11.**Concentration degree of distribution of microcracks at (

**a**) Location 1; (

**b**) Location 2; (

**c**) Location 3.

**Figure 16.**Influence of confining pressure on (

**a**) crack increments; (

**b**) failure mode; Yellow represents the intact particles; Blue represents the detached particles induced by tension force; Red represents the detached particles induced by shear force.

**Figure 17.**Temporal and spatial evolution of microcracking at operation pressure of 8 MPa: (

**a**) crack increment; (

**b**) damage propagation process with retain time.

**Figure 19.**Effect of interfacial friction coefficient on (

**a**) stress–strain curves; (

**b**) microcrack increments.

**Figure 20.**Microcrack increments with variations of (

**a**) thermal conductivity; (

**b**) particle-specific heat; and (

**c**) thermal expansion coefficient.

Parameters | Units | Values |
---|---|---|

Young’s modulus | GPa | 30 |

Poisson’s ratio | / | 0.3 |

Density | kg/m^{3} | 2160 |

Friction angle, φ | degrees | 45 |

Tensile strength | MPa | 4 |

Cohesion strength | MPa | 4 |

Thermal conductivity | W/m·$\xb0\mathrm{C}$ | 6.5 |

Specific heat | J/kg·$\xb0\mathrm{C}$ | 880 |

Linear thermal expansion coefficient | $\xb0\mathrm{C}$^{−1} | 5 × 10^{−5} |

Parameters | Units | Values |
---|---|---|

Particle density | kg/m^{3} | 2160 |

Coefficient of interparticle friction | / | 0.3/0.4/0.5/0.6 |

Normal-to-shear stiffness ratio | / | 1.0/1.2/1.4/1.6 |

Thermal conductivity | W/m·$K$ | 6.5/7.5/8.5/9.5 |

specific heat | J/kg·$\xb0\mathrm{C}$ | 1000/2000/3000/4000 |

Thermal expansion coefficient | $1/\xb0\mathrm{C}$ | 1 × 10^{−5}/0.7 × 10^{−5}/0.3 × 10^{−5}/1 × 10^{−6} |

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**MDPI and ACS Style**

Feng, K.; Li, W.; Nan, X.; Yang, G.
Salt Cavern Thermal Damage Evolution Investigation Based on a Hybrid Continuum-Discrete Coupled Modeling. *Sustainability* **2023**, *15*, 8718.
https://doi.org/10.3390/su15118718

**AMA Style**

Feng K, Li W, Nan X, Yang G.
Salt Cavern Thermal Damage Evolution Investigation Based on a Hybrid Continuum-Discrete Coupled Modeling. *Sustainability*. 2023; 15(11):8718.
https://doi.org/10.3390/su15118718

**Chicago/Turabian Style**

Feng, Kai, Wenjing Li, Xing Nan, and Guangzhi Yang.
2023. "Salt Cavern Thermal Damage Evolution Investigation Based on a Hybrid Continuum-Discrete Coupled Modeling" *Sustainability* 15, no. 11: 8718.
https://doi.org/10.3390/su15118718