# Experimental Investigation on Water Seepage through Transparent Synthetic Rough-Walled Fractures

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Parameters Affecting Fracture Flow

_{R}, to the projected area, A

_{0}(smooth reference surface), which can be expressed as:

_{f}and the fractional part of the fractal dimension, D*. The following theoretical relationship between R and D* has been proposed by Mandelbrot [35]:

_{f}, minus the value two of the Euclidian dimension of a smooth surface [36]. Fractal dimension as the representative of the geometric variation of fractures can also be expressed as:

^{−3}) is the density difference between the infiltrated water and air, e (L) is the fracture aperture, g (LT

^{−2}) is the gravity acceleration, β is the fracture inclination, σ (MT

^{−2}) is the water surface tension, and γ is the contact angle.

^{−2}T) is the fluid viscosity and u (LT

^{−1}) is the fluid velocity.

^{2}) of the preferential pathways along the single fracture can be related to the fractal dimension D

_{f}[38] and the mismatch length λ [39]. Eker and Akin [38] indicate that the relation between ${D}_{T}$, $\lambda ,$ and $K$ can be described as:

#### 2.2. Synthetic Fracture Designing

_{f}), as detailed in Table 1.

#### 2.3. Experimental Setup

#### 2.4. Flow Rate Prediction

## 3. Results and Discussion

#### 3.1. Flow Rate along the Fracture Outlets

#### 3.2. Preferential Flow Paths in Dry Fractures

#### 3.3. Effect of Fractal Dimension and Mismatch Length on Preferential Flow Path

#### 3.4. Prediction of Total Flow Rate

#### 3.5. Prediction of Flow Rate Time Series

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Realization of the synthetic fracture (

**a**) 3D printed fracture mold; (

**b**) synthetic fracture bottom surface. The length of the pen in figure is 165 mm.

**Figure 4.**Outlet volume (mL) vs. time (min) at different test sections for Fracture-1, with an inclination of (

**a**) 45°; (

**b**) 55°; (

**c**) 65°.

**Figure 5.**Outlet volume (mL) vs. time (min) at different test sections for (

**a**) Fracture-2 with a fracture inclination of 45°; (

**b**) Fracture-4 with a fracture inclination of 45°.

**Figure 6.**Comparison between the discharge rates along the outlet sections Q2 and Q3, obtained from experiments conducted on concrete and resin fracture surfaces.

**Figure 7.**Flow movement through Fracture-1 for an inclination angle of 45°, under dry initial conditions.

**Figure 8.**Shape of the first flow path in Fracture-1 under dry initial condition at varying the inclination angle; (

**a**) β = 55°; (

**b**) β = 65°.

**Figure 9.**Preferential flow paths in the fractures with different fractal dimensions and mismatch lengths for inclination angle of α = 45°, 10 min after the start of the infiltration processes in the fracture: (

**a**) Fracture-1; (

**b**) Fracture-2; (

**c**) Fracture-3; (

**d**) Fracture-4.

**Figure 10.**Flow paths along the aperture over cross sections: (

**a**) A–A; (

**b**) B–B; (

**c**) C–C; (

**d**) D–D. Red arrows indicate the flow paths with the maximum flow rate.

**Figure 11.**Details of coefficients of the flow rate time series in the Wavelet algorithm. The d

_{1}d

_{2}, d

_{3}and d

_{4}are the Wavelet coefficients.

**Figure 12.**Prediction of flow rate time series: (

**a**) comparison between observed and predicted flow rate over time; (

**b**) scatter plot of predicted flow rate versus observed flow rate.

Fracture Name | Physical Size (mm^{2}) | Fractal Dimension D _{f} | Mismatch Length (mm) | Standard Deviation (mm) | Mean Aperture (mm) |
---|---|---|---|---|---|

Fracture-1 | 200 × 200 | 2.2 | 30 | 2 | 2.04 |

Fracture-2 | 2.4 | 30 | 2 | 3.07 | |

Fracture-3 | 2.3 | 10 | 2 | 0.94 | |

Fracture-4 | 2.2 | 10 | 3 | 0.70 |

**Table 2.**The mean number of flow paths and intermediate channels for different fractures and inclinations.

Fracture | Flow and Intermediate Flow | α = 45° | α = 55° | α = 65° |
---|---|---|---|---|

1 | flow path | 1 | 1 | 1 |

1 | Intermediate channel | 3 | 3 | 4 |

2 | flow path | 1 | 2 | 2 |

2 | Intermediate channel | 2 | 2 | 3 |

3 | flow path | 2 | 2 | 2 |

3 | Intermediate channel | 3 | 3 | 4 |

4 | flow path | 3 | 4 | 4 |

4 | Intermediate channel | 4 | 3 | 6 |

t | F-1 α = 45° | F-1 α = 55° | F-1 α = 65° | F-2 α = 45° | F-2 α = 55° | F-2 α = 65° | F-3 α = 45° | F-3 α = 55° | F-3 α = 65° | F-4 α = 45° | F-4 α = 55° | F-4 α = 65° |
---|---|---|---|---|---|---|---|---|---|---|---|---|

10 | 3 | 3 | 4 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 5 |

20 | 3 | 4 | 4 | 3 | 4 | 4 | 4 | 5 | 5 | 4 | 5 | 5 |

30 | 3 | 4 | 5 | 4 | 5 | 5 | 5 | 5 | 6 | 5 | 5 | 5 |

40 | 4 | 5 | 6 | 5 | 6 | 6 | 5 | 6 | 6 | 6 | 6 | 6 |

50 | 6 | 6 | 7 | - | - | - | - | - | - | 5 | 6 | 6 |

60 | 6 | 6 | 6 | - | - | - | - | - | - | - | - | - |

Parameters | Minimum Value | Maximum Value |
---|---|---|

${e}^{{D}_{f}}$ | 8.96 | 10.94 |

$\lambda $ | 10 | 30 |

$\mathrm{sin}\beta $ | 0.707 | 0.906 |

Linear Model Number | Logical Condition | Linear Relation |
---|---|---|

$\mathrm{LM}1$ | $\mathrm{If}{e}^{{D}_{f}}\le 9.43,\mathrm{then}$ | $V=14.25t+12.31$ |

$\mathrm{LM}2$ | $\mathrm{If}9.43{e}^{{D}_{f}}10.42,\mathrm{then}$ | $V=14.97t+10.7$0 |

$\mathrm{LM}3$ | $\mathrm{If}10.42\le {e}^{{D}_{f}},\mathrm{then}$ | $V=15.09t+16.66$ |

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**MDPI and ACS Style**

Ranjbar, A.; Cherubini, C.; Pastore, N.
Experimental Investigation on Water Seepage through Transparent Synthetic Rough-Walled Fractures. *Water* **2022**, *14*, 3199.
https://doi.org/10.3390/w14203199

**AMA Style**

Ranjbar A, Cherubini C, Pastore N.
Experimental Investigation on Water Seepage through Transparent Synthetic Rough-Walled Fractures. *Water*. 2022; 14(20):3199.
https://doi.org/10.3390/w14203199

**Chicago/Turabian Style**

Ranjbar, Ali, Claudia Cherubini, and Nicola Pastore.
2022. "Experimental Investigation on Water Seepage through Transparent Synthetic Rough-Walled Fractures" *Water* 14, no. 20: 3199.
https://doi.org/10.3390/w14203199