# Evaluating the Influence of Fracture Roughness and Tortuosity on Fluid Seepage Based on Fluid Seepage Experiments

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background

## 3. Materials and Methods

#### 3.1. Sample Preparation

#### 3.2. Basic Parameters and Characterization of Rough Fractures

#### 3.2.1. Roughness of the Specimens with Rough Fractures

#### 3.2.2. Tortuosity of the Specimens with Rough Fractures

#### 3.3. Procedure and Scheme of Hydraulic Experiment

#### 3.3.1. Testing Equipment and Process

#### 3.3.2. Test Scheme

## 4. Results

#### 4.1. Relationship between JRC and Seepage Flow Q

- (I)
- Under the same hydraulic pressure and normal pressure, the fluid seepage flow decreased with the increase in the JRC. The larger the JRC was, the more uneven the rough fracture was. When the fluid passed through the fracture channels, more energy was needed to overcome the seepage resistance caused by the fracture roughness, which led to a decrease in fluid seepage flow.
- (II)
- Under the condition of constant normal pressure and JRC, the seepage flow of the fluid increased with the increase in hydraulic pressure. The greater the hydraulic pressure, the greater the energy of the fluid in the fracture channel. More fluid flowed through the fracture channel.
- (III)
- In the case of constant hydraulic pressure and JRC, the seepage flow of the fluid decreased with the increase in normal stress. As shown in Figure 6, when the JRC was 11.70, and the hydraulic pressure was 0.6 MPa, the seepage flow of the fracture decreased from 2.6 × 10
^{−6}to 8 × 10^{−6}${\mathrm{m}}^{3}/\mathrm{s}$ with the increase in normal pressure.

**Figure 6.**Relationship between joint roughness coefficient and seepage flow under different normal pressure and water pressure values. (

**a**) Normal stress = 0.10 MPa, (

**b**) Normal stress = 0.15 MPa, (

**c**) Normal stress = 0.20 MPa, (

**d**) Normal stress = 0.25 MPa, (

**e**) Normal stress = 0.30 MPa, (

**f**) Normal stress = 0.40 MPa.

#### 4.2. Relation between Friction Resistance Coefficient and Reynolds Number

^{2}was between 0.96 and 0.97. The fitting parameter $a$ decreased with the increase in normal pressure, whereas the fitting parameter $b$ increased with the increase in normal pressure.

#### 4.3. Relation between Tortuous Resistance Coefficient and Reynolds Number

#### 4.4. Modification of Friction Resistance Coefficient

## 5. Discussion

#### 5.1. Frictional Resistance Coefficient Model of Validation

#### 5.2. Analysis of Head Loss

## 6. Conclusions

- (I)
- Normal pressure, hydraulic pressure, and fracture roughness affected the seepage behavior of the fluid in the rough fracture. The seepage flow of the fluid decreased with the increase in normal pressure, while increasing with the increase in hydraulic pressure. Under constant hydraulic pressure and normal pressure, the seepage flow of a fluid decreased with the increase in fracture roughness, and showed a trend of a nonlinear decrease. The greater the decreased range of fluid seepage flow, the more pronounced was the influence of normal pressure on the fluid seepage flow.
- (II)
- For rough fractures under normal pressure, the frictional resistance coefficient decreased with the increase in the Reynolds number. The roughness of the fracture affected the relationship between the frictional resistance coefficient and the Reynolds number. The relationship among the friction resistance coefficient, the Reynolds number, and the roughness was established. The calculation of the friction resistance coefficient based on the roughness coefficient and the Reynolds number provided a reliable experimental basis for analyzing the decrease in seepage flow.
- (III)
- The influence of the tortuosity of the rough fracture seepage path on fluid seepage behavior cannot be ignored. Under normal pressure conditions, the tortuous resistance coefficient of a rough fracture decreased with the increase in the Reynolds number. The tortuosity of rough fractures affected the relationship between the tortuous resistance coefficient and the Reynolds number. The relationship among the tortuosity, the frictional resistance coefficient, and the tortuous resistance coefficient was thus established. The tortuous resistance coefficient model considering fracture tortuosity can be used to analyze seepage flow loss in the local range of the seepage paths.
- (IV)
- The existing frictional resistance coefficient model was modified, considering the frictional resistance and local resistance of fluid seepage. The modified frictional resistance coefficient model, considering the tortuosity and roughness, was thus established. Experimental data were applied to the modified model and the existing model to verify its rationality. The modified model was closer to the results of the practical application and more in line with the external environment of the fracture. The total head loss of the fluid was obtained based on the modified friction coefficient model. The influence of fracture roughness and tortuosity on fluid seepage behavior can be researched in more detail and more accurately by analyzing the changes in fluid seepage flow and head loss.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

$f$ | the friction resistance coefficient |

$\mathrm{Re}$ | the Reynolds number |

$e$ | the rough fracture aperture |

$\xi $ | the height of the unevenness of rough fracture |

$\lambda $ | the tortuous resistance coefficient |

${f}_{0}$ | the modified frictional resistance coefficient |

$w$ | the fracture width |

$\mu $ | the viscosity of the fluid |

$Q$ | the seepage flow of the fluid |

$\Delta P$ | the pressure difference between the inlet and outlet of the fracture |

${D}_{h}$ | the hydraulic diameter, equal to two times that of the fracture aperture $e$ |

$\rho $ | the fluid density |

$\nu $ | the average velocity of the fluid |

$\Delta {P}_{f}$ | the frictional head loss |

$\Delta {P}_{j}$ | the regional head loss |

$L$ | the fracture length |

${L}_{a}$ | the effective length of the rough fracture seepage path |

${L}_{0}$ | the equivalent length of the rough fracture, ${L}_{0}={L}_{a}-L$ |

$\mathrm{JRC}$ | the joint roughness coefficient |

$\tau $ | the fracture tortuosity |

## Appendix A

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**Figure 7.**Relationship between the Reynolds number and the frictional resistance coefficient under different normal stresses and different JRC values. (

**a**) Normal stress = 0.10 MPa, (

**b**) Normal stress = 0.15 MPa, (

**c**) Normal stress = 0.20 MPa, (

**d**) Normal stress = 0.25 MPa, (

**e**) Normal stress = 0.30 MPa, (

**f**) Normal stress = 0.40 MPa.

**Figure 9.**Relation between Reynolds number and tortuous resistance coefficient at different tortuosity under different normal pressures. (

**a**) Normal stress = 0.10 MPa, (

**b**) Normal stress = 0.15 MPa, (

**c**) Normal stress = 0.20 MPa, (

**d**) Normal stress = 0.25 MPa, (

**e**) Normal stress = 0.30 MPa, (

**f**) Normal stress = 0.40 MPa.

**Figure 10.**Comparison of the parallel-plate model, Lomize model, Nazridoust model, Zhang model, and the proposed model under different JRC values. (

**a**) Normal pressure = 0.10 MPa, (

**b**) Normal pressure = 0.40 MPa.

Number | Model | Expression | Seepage State |
---|---|---|---|

1 | Parallel-plate [18] | $f=\frac{96}{Re}$ | Laminar flow |

2 | Zhang and Nemcik [6] | $f=\frac{96}{Re}\left[1+9.57115\times {10}^{-4}{\left(\xi /e\right)}^{1.1172}\right]$ | $Re$ < 10 |

3 | Nazridoust K, Ahmadi G, Smith DH [21] | $f=\frac{123}{Re}\left(1+0.12R{e}^{0.687}\right)$ | $Re\le 10$ |

4 | Lomize [23] | $f=\frac{96}{Re}\left[1+6{\left(\xi /e\right)}^{1.5}\right]$ | Laminar flow |

Density (kg/cm^{3}) | Compressive Strength (MPa) | Tensile Strength (MPa) | Elasticity Modulus (MPa) | Poisson’s Ratio |
---|---|---|---|---|

2.32 | 32.12 | 2.85 | 8.9 | 0.31 |

**Table 3.**Test scheme of fluid seepage in rough fractures under different normal pressures and water pressures.

Normal Pressure (MPa) | Hydraulic Pressure (MPa) | Normal Pressure (MPa) | Hydraulic Pressure (MPa) | Normal Pressure (MPa) | Hydraulic Pressure (MPa) |
---|---|---|---|---|---|

0.1 | 0.2 | 0.15 | 0.2 | 0.2 | 0.2 |

0.3 | 0.3 | 0.3 | |||

0.4 | 0.4 | 0.4 | |||

0.5 | 0.5 | 0.5 | |||

0.6 | 0.6 | 0.6 | |||

0.25 | 0.2 | 0.3 | 0.2 | 0.4 | 0.2 |

0.3 | 0.3 | 0.3 | |||

0.4 | 0.4 | 0.4 | |||

0.5 | 0.5 | 0.5 | |||

0.6 | 0.6 | 0.6 |

**Table 4.**Rate of decline in seepage flow under different water pressure and normal pressure conditions.

Hydraulic Pressure (MPa) | Normal Pressure (MPa) | |||||
---|---|---|---|---|---|---|

0.10 | 0.15 | 0.20 | 0.25 | 0.30 | 0.40 | |

0.20 | 0.88 | 0.89 | 0.90 | 0.91 | 0.94 | 0.95 |

0.30 | 0.78 | 0.81 | 0.86 | 0.90 | 0.93 | 0.95 |

0.40 | 0.76 | 0.80 | 0.84 | 0.90 | 0.93 | 0.94 |

0.50 | 0.76 | 0.77 | 0.82 | 0.88 | 0.93 | 0.93 |

0.60 | 0.73 | 0.77 | 0.81 | 0.85 | 0.91 | 0.92 |

**Table 5.**Fitting parameters $a$, $b$ and fitting coefficient R

^{2}of the JRC and $\mathrm{Re}\cdot f/96$ curve.

Normal Stress (MPa) | $\mathbf{Parameter}\text{}\mathit{a}\text{}\mathbf{(}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{13}}\mathbf{)}$ | $\mathbf{Parameter}\text{}\mathit{b}$ | Fitting Coefficient R^{2} |
---|---|---|---|

0.10 | 450,000 | 7.49 | 0.96 |

0.15 | 16,900 | 8.83 | 0.96 |

0.20 | 3390 | 9.03 | 0.96 |

0.25 | 548 | 10.36 | 0.96 |

0.30 | 14 | 10.77 | 0.97 |

0.40 | 1.43 | 11.41 | 0.96 |

**Table 6.**Changes in the Reynolds number, friction resistance coefficient, and tortuous resistance coefficient in rough fracture.

Joint Roughness Coefficient JRC | $\mathbf{Tortuosity}\text{}\mathit{\tau}$ | Normal Stress = 0.10 MPa | Normal Stress = 0.20 MPa | Normal Stress = 0.30 MPa | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Re | $\mathit{f}$ | $\mathit{\lambda}$ | Re | $\mathit{f}$ | $\mathit{\lambda}$ | Re | $\mathit{f}$ | $\mathit{\lambda}$ | ||

11.69 | 1.06 | 148.16 | 3.50 | 0.13 | 54.04 | 1.10 | 0.07 | 31.87 | 0.28 | 0.02 |

13.20 | 1.07 | 322.18 | 3.79 | 0.17 | 75.63 | 1.95 | 0.14 | 38.92 | 0.36 | 0.03 |

14.35 | 1.09 | 342.73 | 4.11 | 0.39 | 138.26 | 3.54 | 0.34 | 86.94 | 2.13 | 0.18 |

14.85 | 1.10 | 355.14 | 8.35 | 0.88 | 191.32 | 7.82 | 0.63 | 93.27 | 4.74 | 0.60 |

17.12 | 1.13 | 387.15 | 22.22 | 2.39 | 201.69 | 11.11 | 1.90 | 145.21 | 10.02 | 0.98 |

17.22 | 1.14 | 434.25 | 41.64 | 4.55 | 249.75 | 28.08 | 2.97 | 171.19 | 16.99 | 2.15 |

18.96 | 1.16 | 557.56 | 66.04 | 6.56 | 251.98 | 38.87 | 4.40 | 233.77 | 26.26 | 2.97 |

19.64 | 1.18 | 695.94 | 75.28 | 9.02 | 483.54 | 59.76 | 4.65 | 254.13 | 48.89 | 6.16 |

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**MDPI and ACS Style**

Wang, S.; Xu, Y.; Zhang, Y.; Yu, Q.; Wang, L.
Evaluating the Influence of Fracture Roughness and Tortuosity on Fluid Seepage Based on Fluid Seepage Experiments. *Appl. Sci.* **2022**, *12*, 7661.
https://doi.org/10.3390/app12157661

**AMA Style**

Wang S, Xu Y, Zhang Y, Yu Q, Wang L.
Evaluating the Influence of Fracture Roughness and Tortuosity on Fluid Seepage Based on Fluid Seepage Experiments. *Applied Sciences*. 2022; 12(15):7661.
https://doi.org/10.3390/app12157661

**Chicago/Turabian Style**

Wang, Shuai, Ying Xu, Yanbo Zhang, Qinglei Yu, and Ling Wang.
2022. "Evaluating the Influence of Fracture Roughness and Tortuosity on Fluid Seepage Based on Fluid Seepage Experiments" *Applied Sciences* 12, no. 15: 7661.
https://doi.org/10.3390/app12157661