# Seepage–Fractal Model of Embankment Soil and Its Application

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## Abstract

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## 1. Introduction

## 2. Methodology

#### 2.1. Hydraulic Conductivity Based on Pore Radius and Fractal Dimension

#### 2.2. Randomness of Hydraulic Conductivity of Embankment Soil

^{−5}cm/s and the fluctuation scale of the random field in any direction is fixed at 3 m. Different COV values result in changes in the dispersion degree and spatial structure of hydraulic conductivity, while the mean of hydraulic conductivity of soil is the same. In order to show this spatial distribution characteristic, different COV values were selected to generate the hydraulic conductivity random field: COV of 0.1 and 0.3 (Figure 1a) and 0.2 and 0.3 (Figure 1b). There are 1600 elements in the random field, and the element size is a 1 m square. Hydraulic conductivity is expressed in gray scale; from pure black to pure white, the scale represents hydraulic conductivity from minimum to maximum. According to the principle of minimum potential energy, groundwater mainly flows through light colored units.

## 3. Case Study

^{−6}cm/s and COV of 0.3, and then the fractal dimension of soil element was calculated according to the relationship between soil fractal dimension and hydraulic conductivity. There were three important boundary conditions: (1) the upstream slope was permeable, and the total head was 6 m; (2) the downstream slope was permeable, and the total head was 0 m; (3) without considering the influence of unsaturated zone, the bottom of the embankment was impervious.

## 4. Results and Discussion

^{−6}, 4 × 10

^{−6}], and only a few hydraulic conductivity units are distributed outside this interval. This phenomenon conforms to the objective law. In this research, the mean of hydraulic conductivity is 3.32 × 10

^{−6}cm/s, COV is 0.3, and the scale of fluctuations is 6 and 3 m, respectively. When the hydraulic conductivity random field is discretized, COV is within a certain range, and the value after discretization deviates little from the mean value. With increased COV, the value after discretization will be distributed in a larger range. In Figure 5b,c,e, the discrete points show a slight downward bending phenomenon, which means that the unit permeability may decrease slowly with increased fractal dimension.

## 5. Conclusions

- (1)
- The proposed seepage–fractal model of embankment soil is suitable for porous soil media under laminar flow. The influencing factors of hydraulic conductivity mainly include pore size, fractal dimension, and fluid viscosity coefficient, and fractal dimension is the main factor;
- (2)
- Hydraulic conductivity is inversely proportional to fractal dimension. Increased fractal dimension will reduce the connectivity of soil pores in a single direction, increase the seepage resistance of water, and reduce the hydraulic conductivity. Decreased fractal dimension will lead to consistency of seepage channels in the soil, limited seepage direction of water, decreased resistance in the seepage direction, and increased hydraulic conductivity;
- (3)
- Increased fractal dimension leads to decreased hydraulic conductivity, increased potential energy consumption through the same seepage path, and increased hydraulic gradient. When the seepage resistance increases further, the seepage path changes, and the water will bypass units with high fractal dimension and flow through units with low fractal dimension;
- (4)
- In the seepage–fractal model of embankment soil, the fractal dimension and hydraulic conductivity had significant field characteristics. Units with similar attributes formed agglomerates within which the soil interacts, and the attributes tended to be consistent. On the outside, the interaction between clusters shows obvious transition.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Gray scale diagram of hydraulic conductivity: (

**a**) COV of hydraulic conductivity (k) = 0.1 and 0.3; (

**b**) COV of k = 0.2 and 0.3.

**Figure 3.**Embankment model of Shijiu Lake: (

**a**) dimensions of section; (

**b**) mesh generation of section; (

**c**) location of section in embankment model; (

**d**) mesh generation of embankment.

**Figure 4.**Distribution of hydraulic conductivity (k) and fractal dimension (D) of sections 1−6 in the embankment: (

**a**) k in section 1; (

**b**) D in section 1; (

**c**) k in section 2; (

**d**) D in section 2; (

**e**) k in section 3; (

**f**) D in section 3; (

**g**) k in section 4; (

**h**) D in section 4; (

**i**) k in section 5; (

**j**) D in section 5; (

**k**) k in section 6; (

**l**) D in section 6.

**Figure 5.**Variation in the hydraulic conductivity with fractal dimension in different sections: (

**a**) section 1; (

**b**) section 2; (

**c**) section 3; (

**d**) section 4; (

**e**) section 5; (

**f**) section 6.

**Figure 6.**Variation in hydraulic gradient with fractal dimension in different sections: (

**a**) point 1; (

**b**) point 2; (

**c**) point 3; (

**d**) point 4.

Soil | k | COV | ${\mathit{\theta}}_{\mathit{x}}$ | ${\mathit{\theta}}_{\mathit{y}}$ |
---|---|---|---|---|

Heavy silty loam | 3.32 × 10^{−6} cm/s | 0.3 | 6 m | 3 m |

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**MDPI and ACS Style**

Zhao, X.; Yang, B.; Yuan, S.; Shen, Z.; Feng, D.
Seepage–Fractal Model of Embankment Soil and Its Application. *Fractal Fract.* **2022**, *6*, 277.
https://doi.org/10.3390/fractalfract6050277

**AMA Style**

Zhao X, Yang B, Yuan S, Shen Z, Feng D.
Seepage–Fractal Model of Embankment Soil and Its Application. *Fractal and Fractional*. 2022; 6(5):277.
https://doi.org/10.3390/fractalfract6050277

**Chicago/Turabian Style**

Zhao, Xiaoming, Binbin Yang, Shichong Yuan, Zhenzhou Shen, and Di Feng.
2022. "Seepage–Fractal Model of Embankment Soil and Its Application" *Fractal and Fractional* 6, no. 5: 277.
https://doi.org/10.3390/fractalfract6050277