# Flow Simulation of Artificially Induced Microfractures Using Digital Rock and Lattice Boltzmann Methods

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. The 3D Digital Core

#### 2.2. Fractal Theory and Fractal Dimension

#### 2.2.1. The Concept of “Fractals”

#### 2.2.2. The Box-Counting Method and Fractal Dimension

#### 2.3. The Calculating of Permeability by LBM

_{i}(x) (at each site x, for each lattice vector e

_{i}), stands for the average movement of fluid particles. Figure 2 illustrates the numerical model of LBM. The Lattice Bhatnagar–Gross–Krook (LBGK) approximation was applied for distribution function at time t [45]:

^{−6}. According to Darcy’s law, Equation (11), the equivalent permeability of the induced fracture in Berea sandstone was calculated.

## 3. Results and Discussion

#### 3.1. The Digital Rocks of Fractures

#### 3.2. Fractal Dimension Calculating of Fractures

#### 3.3. The Permeability of Fractures

#### 3.4. The Relations of Permeability and Other Parameters

**a**) permeability and 3D box fractal dimension, (

**b**) permeability and surface roughness, (

**c**) permeability and mean fracture aperture, (

**d**) 3D box fractal dimension and porosity, and (

**e**) 3D box fractal dimension and mean fracture aperture.

#### 3.4.1. Permeability versus 3D Box Fractal Dimension

_{3d}. The equation between permeability and 3D box fractal dimension obtained is:

#### 3.4.2. Permeability versus Surface Roughness

#### 3.4.3. Permeability versus Mean Fracture Aperture

^{2}also indicates that there is a strong correlation between permeability and the square of mean fracture aperture.

#### 3.4.4. Permeability versus Minimum Fracture Aperture

#### 3.5. The Relationship of 3D Box Fractal Dimension and Other Parameters

_{3d}, was used to characterize 3D artificially induced fractures in Berea sandstone. The relationship of 3D box fractal dimension and other parameters was analyzed below.

#### 3.5.1. 3D Box Fractal Dimension versus Porosity

^{2}value is greater than 0.97). The standard errors of slope and intercept of the linear relationship lie in a small range.

#### 3.5.2. 3D Box Fractal Dimension versus Mean Fracture Aperture

^{2}value of the relationship between 3D box fractal dimension and mean fracture aperture is even larger than that of 3D box fractal dimension and porosity, which indicates that mean fracture aperture has a great impact on 3D box fractal dimension.

#### 3.5.3. 3D Box Fractal Dimension versus Surface Roughness

^{2}= 0.733).

#### 3.6. PLS Regression

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**The numerical model of the lattice Boltzmann method (LBM) (The lattice Bhatnagar–Gross–Krook (LBGK) approximation was applied).

**Figure 3.**Digital cores of Berea sandstone with induced fracture (the size of every digital core is 2.743 × 2.743 × 6.035 mm

^{3}), (

**a**) Berea 1 to (

**f**) Berea 6 are samples of all nine digital cores; their aperture and roughness are totally different.

**Figure 8.**The relationship between permeability and the square of mean fracture aperture of Berea sandstone with an induced fracture.

**Figure 9.**The relationship between permeability and minimum fracture aperture of Berea sandstone with an induced fracture.

**Figure 10.**The relationship between the 3D box fractal dimension and porosity of Berea sandstone with an induced fracture.

**Figure 12.**The relationship between 3D box fractal dimension and surface roughness of Berea sandstone with an induced fracture.

**Table 1.**The minimum aperture values, mean aperture values, and fracture porosity of all the fractures.

Core Sample | Porosity | Minimum Fracture Aperture/Pixels | Mean Fracture Aperture/Pixels | Surface Roughness/Pixels |
---|---|---|---|---|

Berea 1 | 0.111 | 7 | 24.4 | 8.962 |

Berea 2 | 0.082 | 2 | 18.4 | 8.568 |

Berea 3 | 0.068 | 0 | 15.0 | 7.865 |

Berea 4 | 0.067 | 0 | 14.7 | 7.892 |

Berea 5 | 0.093 | 2 | 20.5 | 8.424 |

Berea 6 | 0.100 | 8 | 21.9 | 8.384 |

Berea 7 | 0.117 | 7 | 25.7 | 9.141 |

Berea 8 | 0.088 | 6 | 19.3 | 7.972 |

Berea 9 | 0.106 | 8 | 23.3 | 8.743 |

r | 1 | 2 | 4 | 8 | 16 | 32 | 64 | 128 | 256 | |
---|---|---|---|---|---|---|---|---|---|---|

N(r) | Berea 1 | 243,739 | 64,473 | 14,952 | 3899 | 682 | 112 | 16 | 2 | 1 |

Berea 2 | 184,084 | 49,754 | 12,946 | 3714 | 679 | 112 | 16 | 2 | 1 | |

Berea 3 | 149,777 | 40,921 | 10,970 | 3302 | 640 | 104 | 16 | 2 | 1 | |

Berea 4 | 146,858 | 39,713 | 9936 | 2982 | 637 | 107 | 16 | 2 | 1 | |

Berea 5 | 205,178 | 54,841 | 13,687 | 3751 | 685 | 112 | 16 | 2 | 1 | |

Berea 6 | 219,441 | 58,328 | 14,339 | 3866 | 685 | 112 | 16 | 2 | 1 | |

Berea 7 | 256,973 | 67,771 | 15,537 | 4025 | 686 | 112 | 16 | 2 | 1 | |

Berea 8 | 193,450 | 51,953 | 13,339 | 3734 | 686 | 112 | 16 | 2 | 1 | |

Berea 9 | 232,925 | 61,929 | 14,801 | 3926 | 686 | 112 | 16 | 2 | 1 |

Core Sample | Berea 1 | Berea 2 | Berea 3 | Berea 4 | Berea 5 | Berea 6 | Berea 7 | Berea 8 | Berea 9 |
---|---|---|---|---|---|---|---|---|---|

3D Box Fractal Dimension | 2.356 | 2.302 | 2.259 | 2.247 | 2.323 | 2.337 | 2.367 | 2.312 | 2.349 |

Core Sample | Berea 1 | Berea 2 | Berea 3 | Berea 4 | Berea 5 | Berea 6 | Berea 7 | Berea 8 | Berea 9 |
---|---|---|---|---|---|---|---|---|---|

Permeability | 1.903 | 0.820 | 0.300 | 0.071 | 1.183 | 2.108 | 3.759 | 1.492 | 2.852 |

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**MDPI and ACS Style**

Yang, Y.; Liu, Z.; Yao, J.; Zhang, L.; Ma, J.; Hejazi, S.H.; Luquot, L.; Ngarta, T.D.
Flow Simulation of Artificially Induced Microfractures Using Digital Rock and Lattice Boltzmann Methods. *Energies* **2018**, *11*, 2145.
https://doi.org/10.3390/en11082145

**AMA Style**

Yang Y, Liu Z, Yao J, Zhang L, Ma J, Hejazi SH, Luquot L, Ngarta TD.
Flow Simulation of Artificially Induced Microfractures Using Digital Rock and Lattice Boltzmann Methods. *Energies*. 2018; 11(8):2145.
https://doi.org/10.3390/en11082145

**Chicago/Turabian Style**

Yang, Yongfei, Zhihui Liu, Jun Yao, Lei Zhang, Jingsheng Ma, S. Hossein Hejazi, Linda Luquot, and Toussaint Dono Ngarta.
2018. "Flow Simulation of Artificially Induced Microfractures Using Digital Rock and Lattice Boltzmann Methods" *Energies* 11, no. 8: 2145.
https://doi.org/10.3390/en11082145