# Temporal Mixing Behavior of Conservative Solute Transport through 2D Self-Affine Fractures

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Fracture Generation

#### 2.2. Computational Fluid Dynamics (CFD) Simulations of the Flow Field and Solute Transport in Single Rough Fractures

#### 2.3. Mixing: Scalar Dissipation Rate (SDR)

## 3. Results and Discussion

#### 3.1. Model Setup

^{3}and $\mu $ = 1.002 × 10

^{−3}Pa∙s) was used to saturate the void space in the fractures. The typical conservative solute transport (e.g., ${\mathrm{Cl}}^{-}$ in water) and the corresponding ${D}_{m}=2.03\times {10}^{-9}{\mathrm{m}}^{2}/\mathrm{s}$ were assumed depending on the reference of [47]. The matrix of the fracture was assumed impermeable and the rough fracture walls were considered as non-slip boundaries. As background flow, the steady-state flow was induced by a given pressure drop over the entire fracture. The solved flow field serves as the input for the transient solute transport model. The flow field and transient solute transport models based on Equations (10)–(16) were implemented in the COMSOL Multiphysics package version 5.2 (COMSOL Inc., Burlington, MA, USA) using the Galerkin finite-element method [48]. In order to ensure numerical stability and accuracy, the fracture domain was discretized into ~152,000 triangular elements. The number of triangular elements was determined by the mesh independence analysis. Under the same pressure gradient ($-\nabla p=185\mathrm{Pa}/\mathrm{m}$), the steady-state flow rate changes about 0.95% (from 9.550 $\times {10}^{-4}{\mathrm{m}}^{3}/\mathrm{s}$ to 9.641 $\times {10}^{-4}{\mathrm{m}}^{3}/\mathrm{s}$) as the number of triangular elements increases by about 104% (from 152,000 to 310,000). This indicates that 152,000 triangular elements are sufficient to provide stable and accurate numerical results.

#### 3.2. Influence of the Roughness of Fracture Walls on the Temporal Behavior of the Global SDR

#### 3.3. Validity of Predicting Global SDR from the Longitudinal SDR

## 4. Summary and Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

$b\left(x\right)$ | The local aperture of the fracture |

$\overline{b}$ | The distance normal to the mean plane |

$c$ | The solute concentration |

$\overline{c}\left(x,t\right)$ | The mean longitudinal concentration projected in the transverse direction |

${d}_{0}$ | The shear displacement distance along the horizontal direction |

${D}_{m}$ | The molecular diffusion coefficient |

$H$ | Hurst exponent |

${h}_{x}$ | The horizontal distance |

L | The length of the whole fracture |

$\Delta L$ | The width of injected solute |

${m}_{0}$ | The mass of injected solute |

${M}^{2}\left(t\right)$ | The concentration second moment |

n | The normal direction to the outlet boundary |

$Pe$ | Peclet number |

$p$ | The fluid pressure |

${S}_{1}\left(x\right)$ | The top fracture wall |

${S}_{2}\left(x\right)$ | The bottom fracture wall |

$t$ | Time |

$\mathit{u}$ | The velocity vector |

W | The width of fracture in the out of plane direction |

${x}_{L}^{*}$ | The initial injection location of the solute mass |

$Z\left(x\right)$ | A function of independent variable $x$ |

$\lambda $ | Scaling factor |

$\mu $ | The dynamic viscosity |

$\rho $ | The density of fluid |

${\sigma}^{2}\left(\lambda \right)$ | The variance |

${\sigma}_{\lambda {h}_{x}}^{2}$ | The variance of increments with the distances $\lambda {h}_{x}$ |

${\sigma}_{{h}_{x}}^{2}$ | The variance of increments with the distances $\lambda $ |

${\sigma}_{b}$ | The standard deviation of the aperture |

${\tau}_{D}$ | The CHARACTERISTIC diffusion time |

${\tau}_{a}$ | The characteristic advection time |

$\chi \left(t\right)$ | The scalar dissipation rate |

${\chi}_{0}\left(t\right)$ | The analytical 1D SDR solution |

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**Figure 1.**Dolomite rock fracture surface 3D profile and distribution of Hurst exponent for the different 2D profiles along the longitudinal direction.

**Figure 2.**Self-affine fracture walls with the different Hurst exponents. (

**a**) The self-affine fracture walls with H = 0.6, H = 0.7, and H = 0.8, respectively. (

**b**) The zoom-in self-affine fracture wall with H = 0.6 between x = 80 mm and x = 90 mm.

**Figure 3.**Reconstruction of aperture field in the self-affine rough fracture with H = 0.7. (

**a**) Reconstruction of constant-aperture rough fracture. (

**b**) Reconstruction of variable-aperture rough fracture. (

**c**) Gaussian distribution of aperture field for the variable-aperture rough fracture.

**Figure 4.**The flow fields in variable-aperture and constant-aperture fractures with H = 0.6 for Pe = 1000.

**Figure 5.**The solute transport in variable-aperture and constant-aperture fractures with H = 0.6 for Pe = 1000.

**Figure 6.**Scalar dissipation rate estimated in the constant-aperture fracture (

**a**–

**c**) and variable-aperture fracture (

**d**–

**f**) with H = 0.6, 0.7, and 0.8 for Pe = 10, 100, and 1000, respectively.

**Figure 7.**The global SDR estimated from the full concentration field and the longitudinal SDR estimated from the average of the concentration over the transverse cross-section in the constant-aperture fractures with the Hurst exponent H = 0.6 (

**a**), H = 0.7 (

**b**), and H = 0.8 (

**c**) for the Pe = 10, 100, and 1000.

**Figure 8.**(

**a**,

**c**,

**e**) represent the global concentration second moment estimated from the full concentration field and the longitudinal concentration second moment estimated from the average of the concentration over the transverse cross-section in the variable-aperture fractures with H = 0.6, H = 0.7, and H = 0.8, respectively. (

**b**,

**d**,

**f**) represent the global SDR estimated from the full concentration field and the longitudinal SDR estimated from the average of the concentration over the transverse cross-section in the variable-aperture fractures with the Hurst exponent H = 0.6, H = 0.7, and H = 0.8 for the Pe = 10, 100, and 1000, respectively.

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## Share and Cite

**MDPI and ACS Style**

Dou, Z.; Sleep, B.; Mondal, P.; Guo, Q.; Wang, J.; Zhou, Z.
Temporal Mixing Behavior of Conservative Solute Transport through 2D Self-Affine Fractures. *Processes* **2018**, *6*, 158.
https://doi.org/10.3390/pr6090158

**AMA Style**

Dou Z, Sleep B, Mondal P, Guo Q, Wang J, Zhou Z.
Temporal Mixing Behavior of Conservative Solute Transport through 2D Self-Affine Fractures. *Processes*. 2018; 6(9):158.
https://doi.org/10.3390/pr6090158

**Chicago/Turabian Style**

Dou, Zhi, Brent Sleep, Pulin Mondal, Qiaona Guo, Jingou Wang, and Zhifang Zhou.
2018. "Temporal Mixing Behavior of Conservative Solute Transport through 2D Self-Affine Fractures" *Processes* 6, no. 9: 158.
https://doi.org/10.3390/pr6090158