# Numerical Study of Spatial Behavior of Solute Particle Transport in Single Fracture with Variable Apertures

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Solute Transport Model: Random Walk Particle Following (RWPF) Model

^{2}/sec] to the longitudinal dispersion coefficient, $U$ [m/sec] to the mean velocity, and $t$ [sec] to time.

_{3}or P

_{4}in Figure 1). As part of the element with aperture (${e}_{i}$) and size $\Delta a\times \Delta a$, the residence time of solute particles ${t}_{i}$ is obtained by dividing the volume of the element (${V}_{i}$) by the localized flow rate (${Q}_{ij}$) as follows:

## 3. Verification of RWPF Model

^{3}] refers to the initial concentration, $L$ [m] to the length of the one-dimensional channel, $U$ [m/sec] to the velocity of solute particles moving along the direction of flow, and $D$ [m

^{2}/sec] to the longitudinal dispersion coefficient.

## 4. Solute Transport Simulation with Spatial Correlation Length and Effective Normal Stress

#### 4.1. Simulation Condition

^{2}/sec. The Monte–Carlo simulation (30 random numbers) was conducted for each value of $\lambda /L$, and ${\sigma}^{\prime}$ applied for solute transport simulation. Analysis results presented are mean values.

#### 4.2. Breakthrough Curves and Mean Residence Time

#### 4.3. Tortuosity of Solute Particles

#### 4.4. Spatial Dispersion of Solute Particles

#### 4.5. Empirical Formula for Calculating Mean Residence Time of Solutes

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Solute particles’ selection of routes at intersection $i$: black lines indicate the flow direction, and gray lines the particle routes. Random numbers exist between 0 and 1.

**Figure 2.**One-dimensional channel system applied to verify the RWPF (Random Walk Particle Following) model.

**Figure 3.**Comparison of simulation results of solute particle transport with $R$ with analytical solutions: the value of $D$ was fixed to 1.0 m

^{2}/sec.

**Figure 4.**Comparison of simulation results of solute particle transport with $D$ with analytical solutions: the value of $R$ was fixed to 3.0.

**Figure 6.**Variations in breakthrough curves of solutes based on $\lambda /L$ and ${\sigma}^{\prime}$. The red line refers to the average of breakthrough curves (bold gray lines) calculated through 30 random numbers, and the blue line to the accumulated breakthrough curves.

**Figure 8.**Travel trajectories of the solute particles for (

**a**) varying aperture distribution and (

**b**) consistent aperture distribution.

**Figure 9.**Trajectories of solute particles passing through a channel with relatively large localized flows when ${\sigma}^{\prime}$ increased at $\lambda /L$ = 0.3: (

**A**) ${\sigma}^{\prime}$ = 0.0 MPa, (

**B**) ${\sigma}^{\prime}$ = 15.0 MPa, and (

**C**) ${\sigma}^{\prime}$ = 35.0 MPa.

**Figure 12.**Relations between for ${\sigma}^{\prime}$ and ${t}_{m}/{t}_{m}^{\prime}$ with $\lambda /L$.

Effective Normal Stress $\left[\mathbf{MPa}\right]$ | Closure $\left[\mathsf{\mu}\mathbf{m}\right]$ |
---|---|

0.0 | 0.0 |

5.0 | 139.0 |

10.0 | 212.0 |

15.0 | 258.0 |

20.0 | 289.0 |

25.0 | 312.0 |

30.0 | 329.0 |

35.0 | 342.0 |

$\mathit{\lambda}/\mathit{L}$ | No Correlation | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 |
---|---|---|---|---|---|---|

a | 0.9481 | 0.8350 | 0.8213 | 0.7761 | 0.8113 | 0.9086 |

b | 0.0056 | 0.0229 | 0.0243 | 0.0316 | 0.0316 | 0.0202 |

CR | 0.7999 | 0.9592 | 0.9472 | 0.9545 | 0.9833 | 0.9818 |

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**MDPI and ACS Style**

Jeong, Y.-W.; Jeong, W.
Numerical Study of Spatial Behavior of Solute Particle Transport in Single Fracture with Variable Apertures. *Water* **2018**, *10*, 673.
https://doi.org/10.3390/w10060673

**AMA Style**

Jeong Y-W, Jeong W.
Numerical Study of Spatial Behavior of Solute Particle Transport in Single Fracture with Variable Apertures. *Water*. 2018; 10(6):673.
https://doi.org/10.3390/w10060673

**Chicago/Turabian Style**

Jeong, Yong-Wook, and Woochang Jeong.
2018. "Numerical Study of Spatial Behavior of Solute Particle Transport in Single Fracture with Variable Apertures" *Water* 10, no. 6: 673.
https://doi.org/10.3390/w10060673