# Impact of Geometrical Features on Solute Transport Behavior through Rough-Walled Rock Fractures

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Generation of Rough-Walled Fracture

^{2}. The self-affinity of a fractional Brownian motion increment can be stated as:

_{1}(x,y), and the upper fracture surface z

_{2}(x,y) can be obtained by the shear displacement method from Wang et al. (1988):

## 3. Solute Transport Model in Rough Fracture

**u**, P, and µ are the fluid density, velocity vector, fluid pressure and dynamic viscosity, respectively.

**n**is the normal outward vector to the boundary.

## 4. Results and Analysis

#### 4.1. Concentration Distribution

^{−9}m

^{2}/s, and the pressure difference between the inflow boundary and outflow boundary was 0.05Pa.

#### 4.2. Breakthrough Curve

## 5. Conclusions

- (1)
- The geometric model of rough-walled fractures is successfully generated by the successive random addition method, which can guarantee consistency between the output and input values. With the increased fractal dimension and standard deviation, the fracture roughness becomes larger, and the aperture distribution becomes more scattered, which can make the streamline more tortuous, the flow distribution more uneven, and the concentration front more inhomogeneous.
- (2)
- With the growth of fractal dimensions, the average time of solute transport increases nonlinearly, and the time variance decreases linearly, respectively. The RTD curve skews more to the right, and the middle region turns to be more concentrated. The curve symmetry is enhanced, the tailing degree is weakened, and the time skewness is decreased.
- (3)
- With the increase of the standard deviation, the average time and the time variance of solute transport increase linearly and nonlinearly, respectively. The right part of RTD curves turns to be flatter with a larger range of time distribution, and the tailing degree is enhanced. Therefore, the curve symmetry is weakened while the skewness is increased.
- (4)
- Based on the curve fitting, the average time of solute transport linearly increases with the standard deviation and the fractal dimension by the power function. The time variance has a linear decreasing relationship with fractal dimension and a power function increasing relationship with standard deviation. The time skewness has a linear decreasing relationship with fractal dimension and a logarithmic increasing relationship with standard deviation.

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) The morphology of fracture surface in space; (

**b**) aperture distribution in plane; (

**c**) probability density function, where the red curve is a normal probability density function.

**Figure 8.**(

**a**) Time variance versus fractal dimension; (

**b**) time variance versus standard deviation of aperture.

**Figure 9.**(

**a**) RTD curves of different fractal dimensions; (

**b**) RTD curves of different standard deviation of aperture.

**Figure 10.**(

**a**) Skewness versus fractal dimension; (

**b**) skewness versus standard deviation of aperture.

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**MDPI and ACS Style**

Chuang, X.; Li, S.; Hu, Y.; Zhou, X. Impact of Geometrical Features on Solute Transport Behavior through Rough-Walled Rock Fractures. *Water* **2023**, *15*, 124.
https://doi.org/10.3390/w15010124

**AMA Style**

Chuang X, Li S, Hu Y, Zhou X. Impact of Geometrical Features on Solute Transport Behavior through Rough-Walled Rock Fractures. *Water*. 2023; 15(1):124.
https://doi.org/10.3390/w15010124

**Chicago/Turabian Style**

Chuang, Xihong, Sanqi Li, Yingtao Hu, and Xin Zhou. 2023. "Impact of Geometrical Features on Solute Transport Behavior through Rough-Walled Rock Fractures" *Water* 15, no. 1: 124.
https://doi.org/10.3390/w15010124