# Effect of Shearing on Non-Darcian Fluid Flow Characteristics through Rough-Walled Fracture

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Study

#### 2.1. Fracture Model

#### 2.2. Numerical Method

^{3}, and the dynamic viscosity coefficient was 1.003 × 10

^{3}Pa·s (at 20 °C). The water flow in each fracture was modeled at five different shear displacements (0, 0.5, 1.0, 1.5, and 2.0 mm) and nine different injection velocities (0.001, 0.005, 0.01, 0.05, 0.1, 0.25, 0.5, 0.75, and 1.0 m/s). A total of 180 simulations were performed.

## 3. Results and Discussion

#### 3.1. Emergence of Eddies

#### 3.2. Correlation between Inlet Flow Velocity and Hydraulic Gradient

#### 3.3. Forchheimer Coefficients

^{−1}and mean aperture measured ${e}_{m}$ using L, the following equation was used to measure the variation of $\beta $:

#### 3.4. Critical Reynolds Number

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Standard profiles of fracture surfaces with different joint roughness coefficients (JRCs).

**Figure 3.**Gaussian distribution and vertical apertures for different shear displacements when JRC = 5.34.

**Figure 4.**Gaussian distribution and vertical apertures for different shear displacements when JRC = 10.02.

**Figure 5.**Gaussian distribution and vertical apertures for different shear displacements when JRC = 13.51.

**Figure 6.**Gaussian distribution and vertical apertures for different shear displacements when JRC = 16.34.

**Figure 7.**Eddy evolution resulting from different injection velocities, where JRC = 16.34 and d

_{s}= 2.0 mm. Re, Reynolds number.

**Figure 13.**Non-Darcian coefficient results and fitting surface as a function of ${\sigma}_{s}/{e}_{m}$ and JRCs.

**Figure 14.**Relationship between critical Reynolds numbers $R{e}_{c}$ and ${\sigma}_{s}/{e}_{m}$ with different JRCs.

JRC | Shear Displacement d_{s} (mm) | $\mathbf{Mean}\text{}\mathbf{Vertical}\text{}\mathbf{Aperture}\text{}{\mathit{e}}_{\mathit{m}}\text{}\left(\mathbf{mm}\right)$ | $\mathbf{Standard}\text{}\mathbf{Deviation}\text{}{\mathit{\sigma}}_{\mathit{s}}\text{}\left(\mathbf{mm}\right)$ | $\mathbf{Variation}\text{}\mathbf{Coefficient}\text{}{\mathit{\sigma}}_{\mathit{s}}/{\mathit{e}}_{\mathit{m}}$ |
---|---|---|---|---|

5.34 | 0.5 | 1.497 | 0.0665 | 0.0444 |

1.0 | 1.494 | 0.1193 | 0.0799 | |

1.5 | 1.491 | 0.1644 | 0.1103 | |

2.0 | 1.488 | 0.2040 | 0.1371 | |

10.02 | 0.5 | 1.488 | 0.0947 | 0.0636 |

1.0 | 1.477 | 0.1716 | 0.1162 | |

1.5 | 1.466 | 0.2341 | 0.1597 | |

2.0 | 1.454 | 0.2874 | 0.1977 | |

13.51 | 0.5 | 1.494 | 0.1214 | 0.0813 |

1.0 | 1.489 | 0.2211 | 0.1485 | |

1.5 | 1.483 | 0.3068 | 0.2069 | |

2.0 | 1.477 | 0.3861 | 0.2614 | |

16.34 | 0.5 | 1.490 | 0.1510 | 0.1013 |

1.0 | 1.481 | 0.2789 | 0.1883 | |

1.5 | 1.471 | 0.3876 | 0.2635 | |

2.0 | 1.461 | 0.4836 | 0.3310 |

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**MDPI and ACS Style**

Li, B.; Xu, W.; Yan, L.; Xu, J.; He, M.; Xie, W.-C.
Effect of Shearing on Non-Darcian Fluid Flow Characteristics through Rough-Walled Fracture. *Water* **2020**, *12*, 3260.
https://doi.org/10.3390/w12113260

**AMA Style**

Li B, Xu W, Yan L, Xu J, He M, Xie W-C.
Effect of Shearing on Non-Darcian Fluid Flow Characteristics through Rough-Walled Fracture. *Water*. 2020; 12(11):3260.
https://doi.org/10.3390/w12113260

**Chicago/Turabian Style**

Li, Biao, Weiya Xu, Long Yan, Jianrong Xu, Mingjie He, and Wei-Chau Xie.
2020. "Effect of Shearing on Non-Darcian Fluid Flow Characteristics through Rough-Walled Fracture" *Water* 12, no. 11: 3260.
https://doi.org/10.3390/w12113260