# Uncertainty of the Soil–Water Characteristic Curve and Its Effects on Slope Seepage and Stability Analysis under Conditions of Rainfall Using the Markov Chain Monte Carlo Method

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials

#### 2.1.Test Data of the Soil–Water Characteristic Curve

^{3}. These soil properties were measured for the granite residual soil using standard soil test methods (State Standard of the People’s Republic of China).

^{3}, 1.50 g/cm

^{3}, 1.55 g/cm

^{3}, 1.60 g/cm

^{3}, 1.65 g/cm

^{3}. The particles (which were also sieved with sieves of 0.5 mm and 1 mm aperture size, respectively) were made into soil specimens with a dry density of 1.60 g/cm

^{3}. The SWCCs were measured by means of a pressure plate extractor (GEO-Experts) that realized the soil specimen drying or wetting process by increasing or decreasing the matrix suction, which was based on the axis translation technique. Soil specimens were prepared using a cutting ring of 70 mm in diameter and 19 mm in height. The soil specimens were remolded. The SWCCs were measured following the drying path in the suction range of 0–400 kPa. Seven soil samples in total were used in the study. Among them, 49 suction-degree of saturation pairs were obtained, and are plotted in Figure 2. The details of fitting SWCC using optimal model parameters in Figure 2 are presented in Section 4.1.

#### 2.2. Slope Properties

^{3}, the unit weight at 18.5 KN/m

^{3}, the saturated permeability coefficient at 2.5 × 10

^{−6}m/s, the saturated volumetric water content at 0.488, the effective cohesion at 18 kPa, and the angle of shearing resistance at 30°.

## 3.Methods

#### 3.1. SWCC Model

#### 3.2. Bayesian Approach

#### 3.3. Uncertainty Estimation

- (1)
- In this study, the prior distributions $p\left(\zeta \right)$ of the model parameters $\zeta [\alpha ,n]$ were assumed to be lognormal distributions;
- (2)
- The likelihood function $p\left(D|\zeta \right)$ was determined by measured data of the SWCC;
- (3)
- The posterior PDF was used as the target distribution function, and the samples were generated from posterior PDF using DRAM;
- (4)
- The samples at the beginning stage were discarded, and the remaining samples used to represent the posterior distribution;
- (5)
- The confidence intervals of the SWCC and its model parameters corresponding to different percentiles (PCT) were obtained.

#### 3.4. Water Flow through Unsaturated Soil

#### 3.5. Calculation Theory of the Safety Factor

## 4. Results and Discussion

#### 4.1. The Results of Uncertainty Estimation

#### 4.2. Influence of the SWCC with Different Confidence on Seepage Analysis

#### 4.2.1. Pore-Water Pressure Profiles with Different Percentiles

#### 4.2.2. Pore-Water Pressure Profiles after Same Rainfall Duration

#### 4.2.3. Gradient of the Pore-Water Pressure

#### 4.3. Influence of the SWCC with Different Confidence on Stability Analysis

## 5. Conclusions

- (1)
- Assuming the Van Genuchten model as a fitting model of the SWCC, and taking the model parameters of Van Genuchten as random variables, the posterior distribution of the model parameters of Van Genuchten could be effectively sampled by the DRAM algorithm. The proposed Bayesian approach could be effectively applied to estimate a SWCC with a given level of confidence for the specific soil.
- (2)
- Based on the confidence interval of the SWCC, a new approach for evaluating the effect of uncertainty of the SWCC on seepage and stability analysis of an unsaturated soil slope under rainfall was proposed. It was found that, under the same conditions of saturated permeability coefficient and rainfall, the pore-water pressure profile was dependent on the confidence interval and was significantly influenced by the change of PCT. For the same conditions of rainfall, the higher the PCT value, the greater the reduction of negative pore-water pressures from the initial condition; and the deeper and faster the wetting front will advance due to the greater water storage capacity of the soil. Hence, a SWCC with a higher PCT leads to a lower safety factor for the slope.
- (3)
- The uncertainty of the SWCC was important for stability analysis under rainfall. We want to study the reliability analysis for the slope using the Bayesian approach and MCMC method in future research.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 2.**Fitted soil–water characteristic curve (SWCC) using optimal model parameters and laboratory data.

**Figure 8.**Pore-water pressure profiles at section O–O in slopes with different percentiles of SWCC. (

**a**) PCT = 2.5; (

**b**) PCT = 5; (

**c**) PCT = 12.5; (

**d**) PCT = 25; (

**e**) PCT = 50; (

**f**) PCT = 75; (

**g**) PCT = 87.5; (

**h**) PCT = 95; (

**i**) PCT = 97.5.

$\widehat{\mathit{a}}$ | ${\mathit{\sigma}}_{\mathit{a}}$ | $\widehat{\mathit{n}}$ | ${\mathit{\sigma}}_{\mathit{n}}$ | ${\widehat{\mathit{\sigma}}}_{\mathit{\epsilon}}^{2}$ |
---|---|---|---|---|

0.0166 (kPa^{−1}) | 8 × 10^{−7} | 1.2964 | 1.0 × 10^{−4} | 0.0026 |

Percentiles (PCT) | 2.5 | 5 | 12.5 | 25 | 50/Mean | 75 | 87.5 | 95 | 97.5 |
---|---|---|---|---|---|---|---|---|---|

a (kPa^{−1}) | 0.0361 | 0.0321 | 0.0271 | 0.0226 | 0.0190 | 0.0138 | 0.0117 | 0.0103 | 0.0093 |

n | 1.4472 | 1.4167 | 1.3759 | 1.3347 | 1.2962 | 1.2471 | 1.2235 | 1.2030 | 1.1919 |

Remark | LB of 95%CI | LB of 90%CI | LB of 75%CI | LB of 50%CI | Mean curve | UB of 50%CI | UB of 75%CI | UB of 90%CI | UB of 95%CI |

Percentiles (PCT) | 2.5 | 5 | 12.5 | 25 | 50 | 75 | 87.5 | 95 | 97.5 |

Safety factor | 1.790 | 1.782 | 1.744 | 1.679 | 1.562 | 1.329 | 1.239 | 1.187 | 1.162 |

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**MDPI and ACS Style**

Liu, W.; Luo, X.; Huang, F.; Fu, M. Uncertainty of the Soil–Water Characteristic Curve and Its Effects on Slope Seepage and Stability Analysis under Conditions of Rainfall Using the Markov Chain Monte Carlo Method. *Water* **2017**, *9*, 758.
https://doi.org/10.3390/w9100758

**AMA Style**

Liu W, Luo X, Huang F, Fu M. Uncertainty of the Soil–Water Characteristic Curve and Its Effects on Slope Seepage and Stability Analysis under Conditions of Rainfall Using the Markov Chain Monte Carlo Method. *Water*. 2017; 9(10):758.
https://doi.org/10.3390/w9100758

**Chicago/Turabian Style**

Liu, Weiping, Xiaoyan Luo, Faming Huang, and Mingfu Fu. 2017. "Uncertainty of the Soil–Water Characteristic Curve and Its Effects on Slope Seepage and Stability Analysis under Conditions of Rainfall Using the Markov Chain Monte Carlo Method" *Water* 9, no. 10: 758.
https://doi.org/10.3390/w9100758