# A Fractal-Based Quantitative Method for the Study of Fracture Evolution of Coal under Different Confining Pressures

^{*}

## Abstract

**:**

## 1. Introduction

_{f}of the total pore structure was calculated using the three-dimensional box-counting method. The results show that the three-dimensional fractal dimension can accurately describe the fractal characteristics of coal. The larger the D

_{f}, the greater the porosity of the coal samples [21]. However, previous CT scanning experiments have not used real-time scanning during the loading process. Therefore, to study the temporal and spatial evolution laws of internal cracks in coal, one must capture the dynamic damage and failure process of loaded coal samples in real time. More importantly, a method for accurately describing the evolution of internal cracks in coal is required.

## 2. Test Scheme and System

#### 2.1. Sample Introduction

#### 2.2. Testing Equipment

#### 2.3. Test Scheme

- (1)
- CT system initialization. The system is heated, and the ray tube is vacuumized. Then, the mechanical shaft is reset. The specimen is fixed to the specimen holder with an electronic jack, and then the holder is placed in the axial pressure loading cylinder. The assembly is installed and fixed to the CT mechanical turntable, and the pipeline is connected for exhaust extraction.
- (2)
- The CT scanning and triaxial loading parameters are set, with those such as voltage, current, exposure time, and the number of pictures set according to the imaging effect, resolution, and gray value. The triaxial loading equipment is set up with the loading parameters required for this experiment. Multiple scans can be performed before the coal sample breaks. During the scanning process, the pressure remains unchanged, and the number of scans is associated with the stress–strain curve for adjustment.
- (3)
- At the end of the experiment, the data are processed. Firstly, the mechanical parameters are exported from the three-axis software database for analysis. Secondly, the original image is corrected using the CT data reconstruction software Phoenix Datos|x2 (version 2.6.1-RTM). Finally, the image is reconstructed in three dimensions using the image processing software.
- (4)
- The above steps are repeated, carrying out triaxial tests with confining pressures of 10, 15, and 20 MPa.

## 3. Analysis of Fracture Evolution Law

#### 3.1. Qualitative Analysis of Two-Dimensional Fracture Evolution Characteristics

#### 3.2. Quantitative Analysis of Two-Dimensional Fracture Evolution Characteristics

_{b}is

^{2}space, N(s) is the minimum number of squares of length s required to cover this non-empty subset. The slope of the linear regression equation formed by plotting lgN(s) with lg(1/s) is the fractal dimension, which obeys the following linear equation:

_{1}is a constant.

_{b}of the internal cracks of the coal sample were obtained by fitting the linear relationship between the double logarithmic coordinates.

**Figure 5.**Coal sample slice binary graphics at 5 MPa from (

**a**,

**b**) Stage I, (

**c**,

**d**) Stage II, (

**e**) Stage III, and (

**f**) IV.

#### 3.3. Three-Dimensional Fracture Evolution Characteristics

#### 3.4. Quantitative Analysis of Three-Dimensional Fracture Evolution Characteristics

_{t}is reduced, indicating that an increase in confining pressure additionally and effectively inhibits the development of internal cracks in coal–rock.

^{3}, respectively.

^{3}, and the fractal dimensions were 2.1056, 2.0714, 2.0221, and 2.0016 for the samples before being subjected to 5, 10, 15, and 20 MPa, respectively. The deformation generated at this stage is plastic and cannot be restored after axial compression unloading. After the second scan, the compaction stage ended, the elastic deformation stage began, and the fracture volume and fractal dimension of the coal increased.

^{3}for the samples before being subjected to 5, 10, 15, and 20 MPa, respectively. The growth rates of the three-dimensional fractal dimension were also large, rising to 2.3277, 2.3004, 2.2578, and 2.2041, respectively. At this time, the coal gradually lost its bearing capacity and became damaged; however, because of the interaction between internal friction within the coal and confining pressure, the coal retained some strength after reaching the peak of the stress–strain curve.

## 4. Discussion

_{0}, y

_{0}). Additionally, we can infer that σ1 and σ3 have the following relationship:

## 5. Conclusions

- (1)
- The coal samples tested under different confining pressures all showed stage characteristics. The five stages are as follows: compaction, elastic, plastic, post-peak failure, and residual. The internal pores and fissures of coal underwent closure in the initial compaction stage under loading, followed by a gradual increase in the elastic stage and a sudden increase in the failure stage. Under the confining pressure, the coal’s ability to resist failure was enhanced, and the development of fractures in coal samples was limited. The greater the confining pressure, the greater the limitation to fracture development.
- (2)
- Via two-dimensional images and three-dimensional analyses, the dynamic changes in cracks emerging in loaded coal under different confining pressures can be quantitatively analyzed. The method for accurately quantifying the evolution of internal cracks within coal samples via fractal theory has good reliability. Based on the changes in the fractal dimension and fracture volume, the two- and three-dimensional fracture evolution characteristics shown within the scanning results are consistent. For coal under different confining pressures, the fracture evolution law is highly consistent, showing a small decrease first and then a large increase. The application of confining pressure can reduce damage in coal.
- (3)
- In the triaxial compression tests, the main failure mode of the coal samples was shear failure. Using the Mohr–Coulomb yield criterion, the relationship between mechanical parameters and confining pressure can be obtained. The results of the experiment verify the theory, confirming that the confining pressure improves the mechanical properties of the coal to a large extent. In addition, greater confining pressures result in an increased compressive strength and elastic modulus of the coal.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**High-precision CT three-axis loading experimental system. (

**a**) Industrial CT system, (

**b**) CT scanning principle, (

**c**) internal structure of the CT system, and (

**d**) Phoenix Datos|x2 interface.

**Figure 3.**Two-dimensional CT images of stress–strain curves at different deformation stages under different confining pressures. (

**a**) 5 MPa; (

**b**) 10 MPa; (

**c**) 15 MPa; (

**d**) 20 MPa.

**Figure 4.**Schematic diagram of the two-dimensional fracture network and the box-counting method used for assessing the fractal dimension. (

**a**) Coal sample original slice, (

**b**–

**d**) Refinement process of grids.

**Figure 6.**Binarized graph of coal sample slices at confining pressures of (

**a**) 10, (

**b**) 15, and (

**c**) 20 MPa.

**Figure 7.**Fracture morphology and distribution during coal sample loading for confining pressures of (

**a**–

**f**) 5 MPa, (

**g**–

**l**) 10 MPa, (

**m**–

**r**) 15 MPa, and (

**s**–

**x**) 20 MPa.

Ray Tube Model | System Parameter | Maximum Power | Maximum Tube Voltage | Detail Resolution |
---|---|---|---|---|

Micron-sized ray tube | Value | 500 W | 300 kV | ≤2 μm |

Nano-scale ray tube | 15 W | 180 kV | ≤0.5 μm |

Confining Pressure (MPa) | 5 | 10 | 15 | 20 | |
---|---|---|---|---|---|

Fractal dimension | First scan | 1.3631 | 1.2817 | 1.1865 | 1.1148 |

Second scan | 1.3188 | 1.2113 | 1.0475 | 1.0217 | |

Third scan | 1.3531 | 1.2217 | 1.0868 | 1.0275 | |

Fourth scan | 1.4104 | 1.2709 | 1.1391 | 1.0807 | |

Fifth scan | 1.5237 | 1.3728 | 1.2468 | 1.1541 | |

Sixth scan | 1.5869 | 1.4842 | 1.3491 | 1.3258 |

Confining Pressure (MPa) | 5 | 10 | 15 | 20 | |
---|---|---|---|---|---|

Three- dimensional fractal dimension | First scan | 2.2046 | 2. 1770 | 2.1312 | 2.1058 |

Second scan | 2.1056 | 2.0714 | 2.0221 | 2.0016 | |

Third scan | 2.1429 | 2.1252 | 2.1093 | 2.0672 | |

Fourth scan | 2.2046 | 2.1785 | 2.1633 | 2.1162 | |

Fifth scan | 2.3277 | 2.3004 | 2.2578 | 2.2041 | |

Sixth scan | 2.4141 | 2.3982 | 2.3327 | 2.2984 |

Confining Pressure (MPa) | Peak Strength (MPa) | Peak Strain | Elastic Modulus (MPa) |
---|---|---|---|

5 | 30.79 | 0.47 | 64.13 |

10 | 38.99 | 0.51 | 76.59 |

15 | 52.93 | 0.60 | 82.08 |

20 | 80.24 | 0.78 | 92.06 |

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**MDPI and ACS Style**

Wang, A.; Wang, L.
A Fractal-Based Quantitative Method for the Study of Fracture Evolution of Coal under Different Confining Pressures. *Fractal Fract.* **2024**, *8*, 159.
https://doi.org/10.3390/fractalfract8030159

**AMA Style**

Wang A, Wang L.
A Fractal-Based Quantitative Method for the Study of Fracture Evolution of Coal under Different Confining Pressures. *Fractal and Fractional*. 2024; 8(3):159.
https://doi.org/10.3390/fractalfract8030159

**Chicago/Turabian Style**

Wang, Ancheng, and Lei Wang.
2024. "A Fractal-Based Quantitative Method for the Study of Fracture Evolution of Coal under Different Confining Pressures" *Fractal and Fractional* 8, no. 3: 159.
https://doi.org/10.3390/fractalfract8030159