# Effect of Variations in Long-Duration Rainfall Intensity on Unsaturated Slope Stability

^{*}

## Abstract

**:**

## 1. Introduction

^{3}of gravel deposition in the reservoir. This increase in gravel load far exceeded the 5.61 million m

^{3}of gravel of the original design. Therefore, the present study used statistical methods to analyze the variation trends in rainfall intensity in the Zengwen reservoir catchment area to understand the effects of climate change on rainfall intensity in this area. This research was combined with slope stability analysis to investigate the changes in slope stability under the influence of climate change.

## 2. Study Area

^{2}and 7.1 km

^{2}, respectively, which make this Taiwan’s largest reservoir (Figure 1). The Zengwen reservoir catchment area is located in the Western Foothills geological area. The strata in this region are mainly from the Miocene epoch of the Tertiary Period to the Pleistocene epoch of the Quaternary Period and consist mostly of gray fine sandstone and shale. Due to the high mud content in the formation, and the relatively young geological age and poor consolidation, it is easily affected by weathering. As a result, the river terrace on both sides of the stream is well developed and there are many rainfall-triggered collapses that have occurred in this area. The terrain in this region decreases in altitude from the northeast to the southwest, with elevations ranging from 100 m to 2600 m. The geographical location of the Nan-Laio-Yeh-Hsi slope is between the Da-Dong-Shan and Ma-To-Shan rainfall stations. The overburden of the slope is composed of sand, and the bedrock is sandstone.

## 3. Materials and Methods

#### 3.1. Mann–Kendall Test

_{1}, x

_{2}, x

_{3}, …, x

_{n}is t

_{1}, t

_{2}, t

_{3}, …, t

_{n}; then, the test statistic of the Mann–Kendall test S is defined as:

_{0}and the alternative hypothesis H

_{1}are set up to assume that the time-series data does not and does show significant trends, respectively. The Z value is used to determine whether to reject H

_{0}and accept H

_{1}and is defined as:

_{(α/2)}, we accept the alternative hypothesis H

_{1}and reject the null hypothesis H

_{0}, which means that the time-series data shows significant trends. A positive S value indicates an increasing trend and vice versa. α is the significance level, and different significance levels correspond to different Z

_{(α/2)}means and statistically different thresholds for identifying significant trends. This study set α = 0.05 as the significance level for trend determination, so that Z

_{(α/2)}= 1.96. Therefore, when |Z| > Z

_{(α/2)}, the time-series data shows significant increasing or decreasing trends.

#### 3.2. Theil–Sen Estimator

_{j}and x

_{k}in the time-series and correspond to the time-points j and k (j > k).

_{t}represents the data, the value of which was estimated from the trends equation and represents rainfall intensity in this study; t represents time; and b is the intercept. Therefore, the linear trends formula can be used to predict future changes in rainfall intensity.

#### 3.3. Rainfall Data

#### 3.4. Seepage Analysis and Qualitative Analysis of Slope Stability

^{−1}), respectively; Q is the boundary flux (L

^{3}T

^{−1}); $\theta $ is the volumetric water content (-); and t is time (T). “L“, “T“, “M“ are the physical dimension of length, time, and mass, respectively. Also, “-“ means dimensionless. This equation describes the summation of changes in the flow rate and boundary flux in unit soil elements in the x-y two-dimensional plane, which is equivalent to changes in the volumetric water content in soil elements per unit time.

^{−1}T

^{−2}); a, n, and m are fitting parameters including the soil air-entry value, the gradient of the soil water retention curve, and the gradient of the curve at which time the soil enters the residual water content state, respectively; and ${k}_{s}$ is the saturated hydraulic conductivity coefficient (L T

^{−1}). From this equation, we can obtain the corresponding pore water pressure and soil water permeability of sand under different water content conditions (Figure 3).

_{r}represents the summation of shear strength of the slices, and S

_{m}represents the shear stress experienced by the damaged surface.

## 4. Results and Discussion

#### 4.1. Trends in Rainfall Intensity

_{(α/2)}= 1.645) (Table 3). This result was compared with Jung et al. [40]. Their study reported that rainfall patterns showed a phenomenon of decreased rainfall duration and increased rainfall intensity in the Zengwen watershed from 1966 to 2007. This showed an increasing trend in extreme rainfall events and their intensity in this region. The Theil–Sen estimator results in this study revealed that the slope of rainfall intensity also showed an increasing trend from 1990 to 2016, and the trend slope of the one-day rainfall events at all three rainfall stations was the largest, with an average slope of 0.1 mm/h per year (Figure 4).

#### 4.2. Seepage Analysis

#### 4.3. Qualitative Analysis of Slope Stability

_{int}is the Factor of Safety of the slope when the rainfall begins; and FS′ is the Factor of Safety of the slope when the rainfall stops.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 4.**Rainfall intensity variation patterns from 1990 to 2016: (

**a**) 1, 2 and 3 day rainfall events at Da-Dong-Shan rainfall station; (

**b**) 5 and 7 day rainfall events at Da-Dong-Shan rainfall station; (

**c**) 1, 2 and 3 day rainfall events at Ma-To-Shan weather station; (

**d**) 5 and 7 day rainfall events at Ma-To-Shan rainfall station; (

**e**) 1, 2 and 3 day rainfall events at San-Jiao-Nan-Shan rainfall station; (

**f**) 5 and 7 day rainfall events at San-Jiao-Nan-Shan rainfall station.

**Figure 5.**Pore-water pressure change with time in 2016 (Da-Dong-Shan rainfall station, 14.5 mm/h): (

**a**) 6 h; (

**b**) 12 h; (

**c**) 18 h; (

**d**) 24 h.

**Figure 6.**Pore-water pressure change with time in 2050 (Da-Dong-Shan rainfall station, 18.8 mm/h): (

**a**) 6 h; (

**b**) 12 h; (

**c**) 18 h; (

**d**) 24 h.

**Figure 7.**Pore-water pressure change with time in 2100 (Da-Dong-Shan rainfall station, 25 mm/h): (

**a**) 6 h; (

**b**) 12 h; (

**c**) 18 h; (

**d**) 24 h.

**Figure 8.**Results of slope stability analysis for the period 2016, 2050 and 2100: (

**a**) Da-Dong-Shan rainfall station; (

**b**) Ma-To-Shan rainfall station; (

**c**) San-Jiao-Nan-Shan rainfall station.

**Figure 9.**The failure surface under different rainfall conditions (Da-Dong-Shan rainfall station): (

**a**) 2016; (

**b**) 2050; (

**c**) 2100.

Material | ${\mathit{\theta}}_{\mathit{s}}$ | a (kPa) | n | m | k_{s} (m/s) |
---|---|---|---|---|---|

Sand | 0.32 | 0.7 | 2.68 | 0.63 | $8.25\times {10}^{-5}$ |

Sandstone | 0.2 | 100 | 1.8 | 0.44 | $1.0\times {10}^{-6}$ |

Material | $\mathit{\gamma}$ (kN/m^{3}) | ${\mathit{\gamma}}_{\mathit{S}}$ (kN/m^{3}) | c’ (kN/m^{2}) | ${\mathit{\varphi}}^{\prime}$ (°) | ${\mathit{\varphi}}^{\mathit{b}}$ (°) |
---|---|---|---|---|---|

Sand | 19.2 | 21.3 | 3 | 30 | 15 |

Sandstone | 22.7 | 22.7 | 5100 | 63 | 30 |

Rainfall Duration (day) | Da-Dong-Shan | Ma-To-Shan | San-Jiao-Nan-Shan | |||
---|---|---|---|---|---|---|

Mann–Kendall Test Result | Slope Estimator | Mann–Kendall Test Result | Slope Estimator | Mann–Kendall Test Result | Slope Estimator | |

1 | 0.79 | 0.13 | 0.48 | 0.09 | 1.10 | 0.08 |

2 | 0.63 | 0.07 | 0.42 | 0.04 | 1.31 | 0.09 |

3 | 0.67 | 0.05 | 0.1 | 0.00 | 1.29 | 0.08 |

5 | 0.25 | 0.02 | −0.17 | −0.01 | 0.71 | 0.03 |

7 | 0.25 | 0.01 | −1.23 | −0.05 | 0.4 | 0.01 |

Average | - | 0.06 | - | 0.02 | - | 0.06 |

Rainfall Station | ${\mathit{I}}_{\mathit{t}}=\mathit{\beta}\times \mathit{t}+\mathit{b}$ | Rainfall Intensity in 2016 (mm/h) | Rainfall Intensity in 2050 (mm/h) | Rainfall Intensity in 2100 (mm/h) | |
---|---|---|---|---|---|

β | b | ||||

Da-Dong-Shan | 0.13 | −237.5 | 14.5 | 18.8 | 25 |

Ma-To-Shan | 0.09 | −176 | 13 | 16.2 | 20.8 |

San-Jiao-Nan-Shan | 0.08 | −141.4 | 11.4 | 13.9 | 17.7 |

**Table 5.**The decreasing time and rate of slope Factor of Safety (FS) for the period 2016, 2050 and 2100.

Rainfall Station | FS Decreasing Phenomenon | 2016 | 2050 | 2100 |
---|---|---|---|---|

Da-Dong-Shan | FS decreasing point of time (h) | 18 | 13 | 13 |

FS decreasing rate (%) | 37.7 | 41.7 | 41 | |

Ma-To-Shan | FS decreasing point of time (h) | 20 | 16 | 14 |

FS decreasing rate (%) | 35.4 | 37.8 | 42.1 | |

San-Jiao-Nan-Shan | FS decreasing point of time (h) | 21 | 19 | 14 |

FS decreasing rate (%) | 23.3 | 35.9 | 41 |

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**MDPI and ACS Style**

Yeh, H.-F.; Tsai, Y.-J.
Effect of Variations in Long-Duration Rainfall Intensity on Unsaturated Slope Stability. *Water* **2018**, *10*, 479.
https://doi.org/10.3390/w10040479

**AMA Style**

Yeh H-F, Tsai Y-J.
Effect of Variations in Long-Duration Rainfall Intensity on Unsaturated Slope Stability. *Water*. 2018; 10(4):479.
https://doi.org/10.3390/w10040479

**Chicago/Turabian Style**

Yeh, Hsin-Fu, and Yi-Jin Tsai.
2018. "Effect of Variations in Long-Duration Rainfall Intensity on Unsaturated Slope Stability" *Water* 10, no. 4: 479.
https://doi.org/10.3390/w10040479