# Laboratory Investigation of Flow Paths in 3D Self-Affine Fractures with Lattice Boltzmann Simulations

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Methods

#### 2.1. Generation of Fracture Rough Surfaces

#### 2.2. Lattice Boltzmann Method

_{x}/δ

_{k}defined as a characteristic lattice velocity in a cell size δ

_{x}.

## 3. Experimental Design and Setup

## 4. Results and Discussion

#### 4.1. Simulation of Water Flow through the Self-Affine Rough Fracture

_{a}= 5000, 10,000, and 60,000 ts, respectively. This is because the water flow through the fracture is at an unsteady state. Based on the simulation results, the maximum velocity and velocity distributions between the time t

_{a}= 60,000 and 70,000 ts are the same. It can be concluded that the water flow has reached a steady state at the time t

_{a}= 60,000 ts in Figure 7. The values of velocities at the steady state are used to calculate flow rates of five sections from the discharging surface and the results are compared with experimental data in the following sections.

#### 4.2. Model Validation

#### 4.3. Influence of Fractal Dimension

#### 4.4. Influence of Mismatch Wavelength

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

## Appendix A

**Table A1.**Experimental data for Figure 8.

Time (s) | Outlet 1 (mL) | Outlet 2 (mL) | Outlet 3 (mL) | Outlet 4 (mL) | Outlet 5 (mL) |
---|---|---|---|---|---|

0 | 0 | 0 | 0 | 0 | 0 |

60 | 0 | 0 | 0 | 0 | 0 |

120 | 6.956 | 3.478 | 0 | 0 | 3.478 |

180 | 13.912 | 6.956 | 3.478 | 3.478 | 10.434 |

240 | 20.868 | 13.912 | 6.956 | 6.956 | 17.39 |

300 | 27.824 | 20.868 | 13.912 | 13.912 | 24.346 |

360 | 34.78 | 27.824 | 20.868 | 20.868 | 31.302 |

420 | 41.736 | 31.302 | 27.824 | 24.346 | 34.78 |

480 | 48.692 | 31.302 | 27.824 | 27.824 | 41.736 |

540 | 59.126 | 34.78 | 31.302 | 31.302 | 48.692 |

600 | 66.082 | 38.258 | 34.78 | 34.78 | 55.648 |

660 | 69.56 | 38.258 | 34.78 | 34.78 | 62.604 |

720 | 76.516 | 45.214 | 41.736 | 34.78 | 66.082 |

780 | 86.95 | 48.692 | 45.214 | 34.78 | 66.082 |

840 | 97.384 | 52.17 | 48.692 | 34.78 | 66.082 |

900 | 104.34 | 55.648 | 55.648 | 34.78 | 69.56 |

960 | 111.296 | 59.126 | 59.126 | 38.258 | 76.516 |

1020 | 121.73 | 62.604 | 62.604 | 38.258 | 83.472 |

1080 | 128.686 | 66.082 | 66.082 | 41.736 | 86.95 |

1140 | 132.164 | 69.56 | 69.56 | 45.214 | 93.906 |

1200 | 139.12 | 69.56 | 69.56 | 48.692 | 100.862 |

1260 | 146.076 | 73.038 | 73.038 | 55.648 | 100.862 |

1320 | 153.032 | 79.994 | 79.994 | 59.126 | 104.34 |

1380 | 163.466 | 86.95 | 86.95 | 62.604 | 111.296 |

1440 | 170.422 | 86.95 | 86.95 | 62.604 | 118.252 |

1500 | 173.9 | 93.906 | 93.906 | 62.604 | 125.208 |

1560 | 184.334 | 97.384 | 97.384 | 66.082 | 132.164 |

1620 | 194.768 | 97.384 | 97.384 | 66.082 | 135.642 |

1680 | 201.724 | 100.862 | 100.862 | 69.56 | 139.12 |

1740 | 208.68 | 104.34 | 104.34 | 69.56 | 142.598 |

1800 | 215.636 | 104.34 | 104.34 | 69.56 | 146.076 |

1860 | 222.592 | 107.818 | 107.818 | 69.56 | 153.032 |

1920 | 233.026 | 111.296 | 111.296 | 73.038 | 159.988 |

1980 | 239.982 | 118.252 | 118.252 | 79.994 | 166.944 |

2040 | 246.938 | 121.73 | 121.73 | 83.472 | 170.422 |

2100 | 253.894 | 128.686 | 128.686 | 86.95 | 173.9 |

2160 | 260.85 | 132.164 | 132.164 | 93.906 | 177.378 |

2220 | 271.284 | 132.164 | 132.164 | 97.384 | 180.856 |

2280 | 278.24 | 135.642 | 135.642 | 97.384 | 187.812 |

2340 | 285.196 | 139.12 | 139.12 | 100.862 | 194.768 |

2400 | 292.152 | 139.12 | 139.12 | 100.862 | 198.246 |

2460 | 302.586 | 142.598 | 142.598 | 104.34 | 205.202 |

2520 | 309.542 | 146.076 | 146.076 | 104.34 | 208.68 |

2580 | 316.498 | 149.554 | 149.554 | 104.34 | 212.158 |

2640 | 323.454 | 153.032 | 153.032 | 104.34 | 215.636 |

2700 | 330.41 | 156.51 | 156.51 | 107.818 | 222.592 |

2760 | 337.366 | 159.988 | 159.988 | 111.296 | 229.548 |

2820 | 347.8 | 163.466 | 163.466 | 111.296 | 233.026 |

**Table A2.**Simulation data for Figure 8.

Time (s) | Outlet 1 (mL) | Outlet 2 (mL) | Outlet 3 (mL) | Outlet 4 (mL) | Outlet 5 (mL) |
---|---|---|---|---|---|

0 | 0 | 0 | 0 | 0 | 0 |

300 | 29.184 | 16.728 | 18.723 | 17.679 | 23.382 |

600 | 58.368 | 33.456 | 37.446 | 35.358 | 46.764 |

900 | 87.552 | 50.184 | 56.169 | 53.037 | 70.146 |

1200 | 116.736 | 66.912 | 74.892 | 70.716 | 93.528 |

1800 | 175.104 | 100.368 | 112.338 | 106.074 | 140.292 |

2100 | 204.288 | 117.096 | 131.061 | 123.753 | 163.674 |

2400 | 233.472 | 133.824 | 149.784 | 141.432 | 187.056 |

2700 | 262.656 | 150.552 | 168.507 | 159.111 | 210.438 |

3000 | 291.84 | 167.28 | 187.23 | 176.79 | 233.82 |

**Table A3.**Experimental data for Figure 9.

Time (s) | Outlet 1 (mL) | Outlet 2 (mL) | Outlet 3 (mL) | Outlet 4 (mL) | Outlet 5 (mL) |
---|---|---|---|---|---|

0 | 0 | 0 | 0 | 0 | 0 |

60 | 6.956 | 0 | 3.478 | 0 | 0 |

120 | 13.912 | 3.478 | 10.434 | 3.478 | 3.478 |

180 | 20.868 | 10.434 | 17.39 | 10.434 | 6.956 |

240 | 27.824 | 17.39 | 24.346 | 17.39 | 13.912 |

300 | 34.78 | 24.346 | 31.302 | 24.346 | 20.868 |

360 | 41.736 | 27.824 | 31.302 | 27.824 | 27.824 |

420 | 48.692 | 27.824 | 34.78 | 31.302 | 31.302 |

480 | 55.648 | 27.824 | 41.736 | 34.78 | 34.78 |

540 | 62.604 | 27.824 | 52.17 | 38.258 | 38.258 |

600 | 69.56 | 27.824 | 59.126 | 38.258 | 38.258 |

660 | 73.038 | 31.302 | 62.604 | 38.258 | 41.736 |

720 | 79.994 | 34.78 | 69.56 | 38.258 | 48.692 |

780 | 90.428 | 38.258 | 73.038 | 38.258 | 55.648 |

840 | 97.384 | 45.214 | 79.994 | 41.736 | 59.126 |

900 | 107.818 | 52.17 | 86.95 | 45.214 | 62.604 |

960 | 114.774 | 59.126 | 93.906 | 52.17 | 66.082 |

1020 | 121.73 | 62.604 | 97.384 | 55.648 | 66.082 |

1080 | 128.686 | 66.082 | 100.862 | 59.126 | 69.56 |

1140 | 135.642 | 66.082 | 100.862 | 62.604 | 73.038 |

1200 | 142.598 | 69.56 | 104.34 | 66.082 | 79.994 |

1260 | 146.076 | 69.56 | 104.34 | 66.082 | 83.472 |

1320 | 153.032 | 73.038 | 111.296 | 66.082 | 90.428 |

1380 | 159.988 | 76.516 | 118.252 | 66.082 | 93.906 |

1440 | 166.944 | 83.472 | 125.208 | 69.56 | 97.384 |

1500 | 173.9 | 90.428 | 132.164 | 69.56 | 104.34 |

1560 | 180.856 | 97.384 | 139.12 | 73.038 | 104.34 |

1620 | 187.812 | 104.34 | 142.598 | 76.516 | 104.34 |

1680 | 194.768 | 107.818 | 142.598 | 79.994 | 107.818 |

1740 | 201.724 | 107.818 | 146.076 | 83.472 | 114.774 |

1800 | 208.68 | 111.296 | 153.032 | 83.472 | 121.73 |

1860 | 215.636 | 118.252 | 156.51 | 86.95 | 128.686 |

1920 | 222.592 | 121.73 | 159.988 | 93.906 | 132.164 |

1980 | 229.548 | 125.208 | 166.944 | 97.384 | 139.12 |

2040 | 236.504 | 132.164 | 170.422 | 97.384 | 139.12 |

2100 | 243.46 | 135.642 | 173.9 | 100.862 | 146.076 |

2160 | 250.416 | 139.12 | 177.378 | 104.34 | 149.554 |

2220 | 257.372 | 139.12 | 180.856 | 104.34 | 153.032 |

2280 | 267.806 | 142.598 | 184.334 | 107.818 | 159.988 |

2340 | 274.762 | 146.076 | 187.812 | 107.818 | 163.466 |

2400 | 281.718 | 149.554 | 194.768 | 107.818 | 166.944 |

2460 | 288.674 | 156.51 | 201.724 | 111.296 | 173.9 |

2520 | 295.63 | 163.466 | 205.202 | 114.774 | 173.9 |

2580 | 302.586 | 166.944 | 205.202 | 114.774 | 177.378 |

2640 | 309.542 | 173.9 | 212.158 | 118.252 | 180.856 |

2700 | 316.498 | 177.378 | 215.636 | 125.208 | 187.812 |

2760 | 323.454 | 177.378 | 215.636 | 128.686 | 194.768 |

2820 | 330.41 | 180.856 | 222.592 | 132.164 | 198.246 |

2880 | 340.844 | 184.334 | 226.07 | 132.164 | 205.202 |

**Table A4.**Simulation data for Figure 9.

Time (s) | Outlet 1 (mL) | Outlet 2 (mL) | Outlet 3 (mL) | Outlet 4 (mL) | Outlet 5 (mL) |
---|---|---|---|---|---|

0 | 0 | 0 | 0 | 0 | 0 |

300 | 29.844 | 17.109 | 20.244 | 17.298 | 24.048 |

600 | 59.688 | 34.218 | 40.488 | 34.596 | 48.096 |

900 | 89.532 | 51.327 | 60.732 | 51.894 | 72.144 |

1200 | 119.376 | 68.436 | 80.976 | 69.192 | 96.192 |

1800 | 179.064 | 102.654 | 121.464 | 103.788 | 144.288 |

2100 | 208.908 | 119.763 | 141.708 | 121.086 | 168.336 |

2400 | 238.752 | 136.872 | 161.952 | 138.384 | 192.384 |

2700 | 268.596 | 153.981 | 182.196 | 155.682 | 216.432 |

3000 | 298.44 | 171.09 | 202.44 | 172.98 | 240.48 |

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**Figure 1.**Velocity vectors in an LBM cell of the D3Q15 model (Galindo-Torres et al., 2013 [59]).

**Figure 6.**The self-affine rough fracture model. (

**a**) Injection side of the model; (

**b**) Discharging side of the model.

**Figure 7.**Velocity distributions for different times. (

**a**) t

_{a}= 5000 ts (

**b**) t

_{a}= 10,000 ts (

**c**) t

_{a}= 60,000 ts (

**d**) t

_{a}= 70,000 ts.

**Figure 8.**The outlet volume of the fracture inclination 65° from experimental and simulation results.

**Figure 9.**The outlet volume of the fracture inclination 75° from experimental and simulation results.

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**MDPI and ACS Style**

Li, J.; Cherubini, C.; Galindo Torres, S.A.; Li, Z.; Pastore, N.; Li, L.
Laboratory Investigation of Flow Paths in 3D Self-Affine Fractures with Lattice Boltzmann Simulations. *Energies* **2018**, *11*, 168.
https://doi.org/10.3390/en11010168

**AMA Style**

Li J, Cherubini C, Galindo Torres SA, Li Z, Pastore N, Li L.
Laboratory Investigation of Flow Paths in 3D Self-Affine Fractures with Lattice Boltzmann Simulations. *Energies*. 2018; 11(1):168.
https://doi.org/10.3390/en11010168

**Chicago/Turabian Style**

Li, Jiawei, Claudia Cherubini, Sergio Andres Galindo Torres, Zi Li, Nicola Pastore, and Ling Li.
2018. "Laboratory Investigation of Flow Paths in 3D Self-Affine Fractures with Lattice Boltzmann Simulations" *Energies* 11, no. 1: 168.
https://doi.org/10.3390/en11010168