# Characteristics Description of Shale Fracture Surface Morphology: A Case Study of Shale Samples from Barnett Shale

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## Abstract

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## 1. Introduction

## 2. Acquisition of Shale Fracture Surface Height Distribution

## 3. Description Parameters of Shale Fracture Surface Morphology and Their Calculation

#### 3.1. Height Distribution Pre-Processing of Shale Fracture Surface Morphology

#### 3.2. Characteristics Description of Shale Fracture Surface Morphology

#### 3.2.1. Roughness

#### 3.2.2. Joint Roughness Coefficient

_{2}was adopted to calculate the joint roughness coefficient of fracture profiles [33], while the Z

_{2}along the lateral direction was listed as follows,

_{2}and joint roughness coefficient developed by Tse and Cruden were chosen to calculate the joint roughness coefficient of the profile along the lateral direction [32], and it was formulated as,

_{2}

^{y}and the joint roughness coefficient JRC

_{3Dy}along the longitudinal direction were similar to that of the lateral direction. The joint roughness coefficient of shale fracture surface morphology is shown in Equation (6).

_{3D}of shale fracture surface morphology is shown in Table 2.

#### 3.2.3. Fractal Dimension

_{i}

_{,j}covering the fracture surface in the grid (i, j) was provided as follows,

#### 3.2.4. Tortuosity

#### 3.2.5. Dip Angle

## 4. Relationship between Roughness and Other Description Parameters of Shale Fracture Surface Morphology

#### Summary of Shale Fracture Surface Morphology Characteristics

## 5. Conclusions

- (1)
- The shale samples collected from Barnett Shale were fractured using the Brazilian test, and the information regarding shale fracture surface morphology was obtained. According to the collecting principle of shale fracture surface morphology, the calculation methods of roughness, joint roughness coefficient, fractal dimension, tortuosity, and dip angle were proposed.
- (2)
- Roughness, joint roughness coefficient, fractal dimension, tortuosity, and dip angle were all directional, and they varied greatly along different directions.
- (3)
- The parameters of roughness, joint roughness coefficient, fractal dimension, tortuosity, absolute dip angle, and overall trend dip angle were among 0.0834 mm–0.2427 mm, 2.5715–10.9368, 2.1000–2.1364, 1.0732–1.1879, 17.7498°–24.5941°, and −3.7223°–13.3042°, respectively.
- (4)
- Joint roughness coefficient, fractal dimension, tortuosity, and dip angle were all positively correlated with roughness. However, the sensitivity of each parameter to roughness varied greatly.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

ε, ε_{x}, ε_{y} | roughness of the whole shale fracture surface morphology, along the lateral and longitudinal direction, respectively, mm |

Z_{ix}, Z_{iy} | mean height of ith profile along the lateral and longitudinal direction, respectively, mm |

$\overline{{Z}_{x}}$,$\overline{{Z}_{y}}$ | mean height of the profile along the lateral and longitudinal direction, respectively, mm |

z_{i} | height of an ith point on the shale fracture surface, mm |

n | number of points on shale fracture surface |

Z_{2} | root mean square first derivative values |

JRC_{2D} | two-dimensional joint roughness coefficient |

JRC_{3D} | three-dimensional joint roughness coefficient |

L | the length of shale fracture |

x_{i} | the position of an ith point on the profile |

INT | ingetral function |

l_{x},l_{y} | the apparent length of a profile in the lateral and longitudinal direction, respectively, mm |

τ, τ_{x}, τ_{y} | tortuosity of the whole shale fracture surface morphology, along the lateral and longitudinal direction, respectively |

Δx, Δy | scanning interval along the lateral and longitudinal direction, respectively |

m, n | number of profiles along the lateral and longitudinal direction, respectively |

θ | dip angle of shale fracture surface morphology |

m | number of profiles along lateral direction |

N_{x} | number of points on a profile along the longitudinal direction |

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**Figure 1.**Fractures of shale samples. (

**a**) Shale fracture 1; (

**b**) Shale fracture 2; (

**c**) Shale fracture 3.

**Figure 3.**Shale fracture surface morphologies (the subfigures (

**a**–

**c**) are the reconstructed images of the rectangular region with 35 mm length and 21 mm width on the three shale surface morphology, and they are the zone for the calculation of characteristics description).

Shale Sample No. | ε_{x}, mm | ε_{y}, mm | ε, mm |
---|---|---|---|

1 | 0.3089 | 0.1907 | 0.2427 |

2 | 0.0306 | 0.2271 | 0.0834 |

3 | 0.0933 | 0.1080 | 0.1004 |

Shale Sample No. | JRC_{3Dx} | JRC_{3Dy} | JRC_{3D} |
---|---|---|---|

1 | 9.2163 | 12.9779 | 10.9368 |

2 | 3.9798 | 1.6616 | 2.5715 |

3 | 9.5734 | 4.2771 | 6.3989 |

Shale Sample No. | Fractal Dimension D |
---|---|

1 | 2.1364 |

2 | 2.1000 |

3 | 2.1187 |

Shale Sample No. | τ_{x} | τ_{y} | τ |
---|---|---|---|

1 | 1.1743 | 1.2015 | 1.1879 |

2 | 1.0843 | 1.0621 | 1.0732 |

3 | 1.0916 | 1.2149 | 1.1516 |

Shale Sample No. | α_{x}, ° | α_{y}, ° | α, ° |
---|---|---|---|

1 | 24.256 | 24.937 | 24.5941 |

2 | 19.406 | 16.235 | 17.7498 |

3 | 19.539 | 29.861 | 24.1548 |

Shale Sample No. | β_{x}, ° | β_{y}, ° | β, ° |
---|---|---|---|

1 | 8.531 | 20.748 | 13.3042 |

2 | 7.037 | −1.969 | −3.7223 |

3 | 5.019 | 7.432 | 6.1075 |

Shale Sample No. | 1 | 2 | 3 |
---|---|---|---|

Roughness, mm | 0.2427 | 0.0834 | 0.1004 |

Joint roughness coefficient | 10.9368 | 2.5715 | 6.3989 |

Fractal dimension | 2.1364 | 2.1000 | 2.1187 |

Tortuosity | 1.1879 | 1.0732 | 1.1516 |

Absolute dip angle, ° | 24.5941 | 17.7498 | 24.1548 |

Overall trend dip angle, ° | 13.3042 | −3.7223 | 6.1075 |

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**MDPI and ACS Style**

Qu, G.; Shi, T.; Zhang, Z.; Su, J.; Wei, H.; Guo, R.; Peng, J.; Zhao, K.
Characteristics Description of Shale Fracture Surface Morphology: A Case Study of Shale Samples from Barnett Shale. *Processes* **2022**, *10*, 401.
https://doi.org/10.3390/pr10020401

**AMA Style**

Qu G, Shi T, Zhang Z, Su J, Wei H, Guo R, Peng J, Zhao K.
Characteristics Description of Shale Fracture Surface Morphology: A Case Study of Shale Samples from Barnett Shale. *Processes*. 2022; 10(2):401.
https://doi.org/10.3390/pr10020401

**Chicago/Turabian Style**

Qu, Guanzheng, Tao Shi, Zheng Zhang, Jian Su, Haitao Wei, Rui Guo, Jiao Peng, and Kairui Zhao.
2022. "Characteristics Description of Shale Fracture Surface Morphology: A Case Study of Shale Samples from Barnett Shale" *Processes* 10, no. 2: 401.
https://doi.org/10.3390/pr10020401