# Probabilistic Slope Seepage Analysis under Rainfall Considering Spatial Variability of Hydraulic Conductivity and Method Comparison

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Governing Equations for Rainfall Infiltration Analysis

#### 2.2. First-Order Stochastic Moment Analysis

#### 2.2.1. Two-Dimensional Correlation Structure of Saturated Hydraulic Conductivity

#### 2.2.2. First-Order Formulation

#### 2.3. Monte Carlo Simulation Verification

#### 2.4. Setup of Simulations

## 3. Results and Discussion

#### 3.1. Effect of Spatial Variability of Saturated Hydraulic Conductivity on Pressure Heads

#### 3.1.1. Deterministic Analysis

#### 3.1.2. Pressure Head Variance

#### 3.1.3. Upper and Lower Bounds of Pressure Head Profile

#### 3.1.4. Pressure Head Response within Heterogeneous Slope

#### 3.2. Monte Carlo Simulation Verification

#### 3.2.1. Impacts of the Grid Density

^{2}. Therefore, the minimum number of realizations needed for Case 8 (a fine grid with 50 × 40 cells of random fields) to reach convergence was 10,000.

^{2}: both are larger and more fluctuating than the corresponding result for the fine grid case. In addition, the mean pressure head still maintains a slight fluctuation when the number of realizations reaches a quite high value, 20,000, indicating that the Monte Carlo simulation with a coarse grid size requires a greater number of realizations to reach convergence than the simulation with a fine grid size. This is for to two reasons: firstly, each realization of a coarser grid contains a lesser amount of random data; it therefore needs more realizations to make the results of the Monte Carlo experience the same amount of random data as in the fine grid case to reach convergence. Secondly, a finite element numerical simulation with a coarse grid size reduces the accuracy of the pressure head prediction, making the mean and variance of the pressure head more fluctuating and uncertain.

#### 3.2.2. Comparison of First-Order Analysis and Monte Carlo Simulation

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Moisture retention and hydraulic conductivity curves: (

**a**) is the hydraulic conductivity curve and (

**b**) is the moisture retention curve.

**Figure 4.**Pressure head variance at steady state for various correlation lengths of $\mathrm{ln}\left({K}_{s}\right)$: (

**a**) ρ = 0.1 m; (

**b**) ρ = 0.5 m; (

**c**) ρ = 1 m; (

**d**) ρ = 10 m and (

**e**) ρ = 100 m.

**Figure 5.**Bounds of pressure head along section 1-1′: (

**a**) above the phreatic surface; (

**b**) below the phreatic surface.

**Figure 6.**Pressure head response along section 1-1′ within heterogeneous slope: (

**a**) ρ = 0.1 m; (

**b**) ρ = 0.5 m; (

**c**) ρ = 1 m; (

**d**) ρ = 10 m and (

**e**) ρ = 100 m.

**Figure 7.**Convergence of the mean and variance of the pressure head of Point F (slope top) obtained by Monte Carlo simulation using two different grid densities: (

**a**,

**b**) are for Case 8 with the fine grid (50 × 40 cells) and (

**c**,

**d**) are for Case 9 with the coarse grid (25 × 20 cells).

**Figure 8.**Comparison of variance of pressure head along section 1-1′ obtained by first-order analysis using two different grid densities.

**Figure 9.**Comparison of mean pressure head obtained by different methods: (

**a**) ${\sigma}_{\mathrm{ln}{K}_{s}}^{2}$ = 0.7 and (

**b**) ${\sigma}_{\mathrm{ln}{K}_{s}}^{2}$=0.4.

**Figure 10.**Comparison of variance of pressure head obtained by different methods: (

**a**) ${\sigma}_{\mathrm{ln}{K}_{s}}^{2}$ = 0.7 and (

**b**) ${\sigma}_{\mathrm{ln}{K}_{s}}^{2}$ = 0.4.

Parameters | Values |
---|---|

Mean of saturated hydraulic conductivity, ${\mu}_{{K}_{s}}$ | $1.0\text{}\mathrm{m}/\mathrm{d}$ |

Coefficient in MVG model, $\alpha $ | $0.4\text{}{\mathrm{m}}^{-1}$ |

Exponent in MVG model, $n$ | $2$ |

Exponent in MVG model, $m$ | $0.5$ |

Residual volumetric water content, ${\theta}_{r}$ | $0.07$ |

Saturated volumetric water content, ${\theta}_{s}$ | $0.4$ |

Vertical infiltration flux, $q$ | $0.01\text{}\mathrm{m}/\mathrm{d}$ |

Variance of $\mathrm{ln}\left({K}_{s}\right)$, ${\sigma}_{\mathrm{ln}{K}_{s}}^{2}$ | $0.4$, $0.7$ |

Correlation length of $\mathrm{ln}\left({K}_{s}\right)$, $\lambda $ | $0.1$, $0.5$, $1$, $10$, $100\text{}\mathrm{m}$ |

Case No. | ${\mathit{\sigma}}_{\mathbf{ln}{\mathit{K}}_{\mathit{s}}}^{2}$ | $\mathit{\lambda}$$\text{}\left(\mathit{m}\right)$ | Grid | Method |
---|---|---|---|---|

1 | 0.7 | 0.1 | 50 × 40 cells | First-order analysis |

2 | 0.7 | 0.5 | 50 × 40 cells | First-order analysis |

3 | 0.7 | 1 | 50 × 40 cells | First-order analysis |

4 | 0.7 | 10 | 50 × 40 cells | First-order analysis |

5 | 0.7 | 100 | 50 × 40 cells | First-order analysis |

6 | 0.7 | 10 | 25 × 20 cells | First-order analysis |

7 | 0.4 | 10 | 50 × 40 cells | First-order analysis |

8 | 0.7 | 10 | 50 × 40 cells | Monte Carlo simulation |

9 | 0.7 | 10 | 25 × 20 cells | Monte Carlo simulation |

10 | 0.4 | 10 | 50 × 40 cells | Monte Carlo simulation |

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**MDPI and ACS Style**

Zou, H.; Cai, J.-S.; Yan, E.-C.; Tang, R.-X.; Jia, L.; Song, K.
Probabilistic Slope Seepage Analysis under Rainfall Considering Spatial Variability of Hydraulic Conductivity and Method Comparison. *Water* **2023**, *15*, 810.
https://doi.org/10.3390/w15040810

**AMA Style**

Zou H, Cai J-S, Yan E-C, Tang R-X, Jia L, Song K.
Probabilistic Slope Seepage Analysis under Rainfall Considering Spatial Variability of Hydraulic Conductivity and Method Comparison. *Water*. 2023; 15(4):810.
https://doi.org/10.3390/w15040810

**Chicago/Turabian Style**

Zou, Hao, Jing-Sen Cai, E-Chuan Yan, Rui-Xuan Tang, Lin Jia, and Kun Song.
2023. "Probabilistic Slope Seepage Analysis under Rainfall Considering Spatial Variability of Hydraulic Conductivity and Method Comparison" *Water* 15, no. 4: 810.
https://doi.org/10.3390/w15040810