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
Abstract
:1. Introduction
2. Materials
2.1.Test Data of the Soil–Water Characteristic Curve
2.2. Slope Properties
3.Methods
3.1. SWCC Model
3.2. Bayesian Approach
3.3. Uncertainty Estimation
- (1)
- In this study, the prior distributions of the model parameters were assumed to be lognormal distributions;
- (2)
- The likelihood function 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
References
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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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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
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 StyleLiu, 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