# Stability of Unsaturated Soil Slope Considering Stratigraphic Uncertainty

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Enhanced CMC Model for Simulating Stratigraphic Uncertainty

_{LR}and K

_{RL}, respectively), and the larger one of K

_{LR}and K

_{RL}is taken as the estimated K. As shown in Figure 1, when estimating the K value from left to right, the likelihood of the observed scenario can be expressed as [20]:

_{LR}and K

_{RL}. If K

_{LR}≥ K

_{RL}, the generating sequence is from left to right. If K

_{LR}< K

_{RL}, the generating sequence is from right to left. As shown in Figure 2, when the generating sequence is from left to right, the probability of state ${S}_{k}$ at cell $(t,j)$ can be calculated by

#### 2.2. Mapping the Simulated Strata into the FEM Model of Slope

#### 2.3. Coupled Pore Fluid Flow and Stress Analysis

**=**${k}_{0}$). While when the uncertainty of the strata is considered, the values of ${k}^{\left(\eta \right)}$ are related to the coordinates of the center point of meshed elements. Therefore, the seepage matrix $\mathbf{H}$ can be expressed as

#### 2.4. Algorithm Development for Automatically Calculating the Safety Factor of Unsaturated Soil Slope

- (1)
- Collect the necessary data for slope stability modeling including the borehole data of the stratum, the parameters of various types of soil, the slope geometric parameters, the boundary conditions of slope and other information contained in the site.
- (2)
- Discretize the stratigraphic profile into cells of appropriate size. Estimate the VTPM and HTPM based on the borehole data. Determine the simulation sequence according to the estimated values of K
_{LR}and K_{RL}, and then use the enhanced CMC model introduced in Section 2.1 to generate the stratum. - (3)
- Establish the initial FEM model of slope in ABAQUS software based on the information collected in Step 1 including the geometry and the boundary conditions of slope.
- (4)
- Map the simulated stratum into the FEM model of slope through python script and generate the input file named “Slope.inp”.
- (5)
- Calculate the initial stress field under the action of gravity by submitting the input file “Slope.inp” to analysis in ABAQUS. The initial stress field is saved in the “Slope.odb” file.
- (6)
- Incorporate the initial stress field into the FEM model of slope and modify the finite element model by editing the “Slope.inp” file. The input file of the modified model is named “Slope-New.inp”.
- (7)
- Submit the “Slope-New.inp” file to the ABAQUS solver through python script. The safety factor of slope is calculated using the strength reduction method, and the results are stored in the file named “Slope-New.odb”.
- (8)
- Extract the calculation results such as safety factor and groundwater table of slope from the “Slope-New.odb” file using python script.
- (9)
- Perform Monte Carlo simulation. Repeat steps 1–7 until the required N times are reached.
- (10)
- Conduct statistical analysis based on the results containing the slope safety factor and the groundwater table.

## 3. Case Study

#### 3.1. Borehole Data

#### 3.2. Simulation and Evaluation of Stratigraphic Uncertainty for Different Borehole Schemes

^{2}is used to divide the stratigraphic profile in this paper. After the first-order Markovian property of the stratum is confirmed by Li et al. [49] using the method of hypothesis test, the enhanced CMC model can be used for stratigraphic uncertainty simulation.

#### 3.2.1. The “Real” Stratum

#### 3.2.2. Borehole Schemes

_{LR}is always greater than or equal to K

_{RL}. Thus, K

_{LR}is taken as the estimated K value and the simulation sequence is from left to right. After estimating the VTPM, K and HTPM, the enhanced CMC model can be used to simulate the stratum of a specific borehole scheme.

#### 3.2.3. Stratigraphic Uncertainty for Different Borehole Schemes

#### 3.3. Stability Analysis of Unsaturated Soil Slope Considering Stratigraphic Uncertainty

#### 3.3.1. FEM Model of Slope

^{2}. The left and right sides of the slope are fixed in the horizontal direction. The bottom boundary is fixed in both the horizontal and vertical directions. The matric suction of the area 3 m down from the surface is set to be 60 kPa [28]. The groundwater tables at the left and right sides of the slope are 15 m and 9 m (0 at the bottom), respectively, and other boundaries of the slope are impermeable. Three virtual groundwater table observation boreholes (BW1, BW2 and BW3) are arranged at the top, middle and toe of the slope to observe the groundwater table in the vicinity of the slope. The parameters of the unsaturated soil are shown in Table 2. Here, the van Genuchten model with two parameters, $a$ and $n$, are used for describing the SWCC curve of unsaturated soil. The SWCC curves of the three types of soil are shown in Figure 11.

#### 3.3.2. Stability of Unsaturated Soil Slope for Different Borehole Schemes

#### 3.3.3. Relationship between the Standard Deviation of $FS$ (or $AGT$) and the Average Information Entropy

#### 3.3.4. Relationship between $FS$ and $AGT$ for Different Schemes

## 4. Conclusions

- (1)
- Information entropy can well quantify the overall and local uncertainty of strata. There is a linear relationship between the standard deviation of the $FS$ (or $AGT$) and the average information entropy. Since the calculation of information entropy is simple and fast, the variation of $FS$ and $AGT$ can be estimated by information entropy in practical engineering.
- (2)
- When the number of boreholes is 11, the mean values of $FS$ and $AGT$ are close to the “real” values, which proves that the proposed algorithm can accurately calculate the $FS$ and $AGT$. The statistics of $FS$ and $AGT$ will not monotonically increase or decrease with the increase in the borehole number. But the trend is that the mean values of $FS$ and $AGT$ gradually approach and eventually converge to the real values and the standard deviations of $FS$ and $AGT$ decrease. Therefore, when there are more known boreholes, the mean value of $FS$ (or $AGT$) is more likely to be close to the real value, and the standard deviation of $FS$ (or $AGT$) is more likely to be small. Although some additional boreholes may be “bad” data and increase the deviation between the estimated values of $FS$ and $AGT$ and the true values, the overall trend is still that the more boreholes are drilled, the more accurate the estimation is. Increasing boreholes is beneficial in practical engineering.
- (3)
- The sample points with $AGT$ and $FS$ as abscissa and ordinate, respectively, are centralized and distributed in a downward inclined band, and some scattered sample points are distributed below the band, especially for the borehole schemes with more than six boreholes. The widths of the bands are related to the uncertainty of the stratum. The $FS$ and the $AGT$ are negatively correlated considering stratigraphic uncertainty. In practical engineering, the $FS$ of slope can be roughly estimated by observing the groundwater table according to this negative correlation property. However, the estimated results are not deterministic, have probabilistic properties and have a certain level of credibility.
- (4)
- The influence of stratigraphic uncertainty on the stability of unsaturated soil slopes is investigated in this paper. In the future, we will consider both the stratigraphic uncertainty and uncertainty of soil parameters and study their influence on the stability of unsaturated soil slopes.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 5.**Borehole locations and distribution of soil types in the boreholes (after Li et al. [49]). (

**a**) Relative location of the boreholes; (

**b**) Distribution of soil types in the boreholes.

Borehole Scheme | Number of Boreholes | B1 | B2 | B3 | B4 | B5 | B6 | B7 | B8 | B9 | B10 | B11 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Scheme 1 | 3 | √ | √ | √ | ||||||||

Scheme 2 | 4 | √ | √ | √ | √ | |||||||

Scheme 3 | 4 | √ | √ | √ | √ | |||||||

Scheme 4 | 5 | √ | √ | √ | √ | √ | ||||||

Scheme 5 | 5 | √ | √ | √ | √ | √ | ||||||

Scheme 6 | 5 | √ | √ | √ | √ | √ | ||||||

Scheme 7 | 6 | √ | √ | √ | √ | √ | √ | |||||

Scheme 8 | 7 | √ | √ | √ | √ | √ | √ | √ | ||||

Scheme 9 | 7 | √ | √ | √ | √ | √ | √ | √ | ||||

Scheme 10 | 8 | √ | √ | √ | √ | √ | √ | √ | √ | |||

Scheme 11 | 9 | √ | √ | √ | √ | √ | √ | √ | √ | √ | ||

Scheme 12 | 9 | √ | √ | √ | √ | √ | √ | √ | √ | √ | ||

Scheme 13 | 9 | √ | √ | √ | √ | √ | √ | √ | √ | √ | ||

Scheme 14 | 10 | √ | √ | √ | √ | √ | √ | √ | √ | √ | √ | |

Scheme 15 | 10 | √ | √ | √ | √ | √ | √ | √ | √ | √ | √ | |

Scheme 16 | 10 | √ | √ | √ | √ | √ | √ | √ | √ | √ | √ | |

Scheme 17 | 11 | √ | √ | √ | √ | √ | √ | √ | √ | √ | √ | √ |

Soil Type | Unit Weight, $\gamma $ (kN/m ^{3}) | Elastic Modulus, E (MPa) | Poisson’s Ratio, $\mathit{\nu}$ | Effective Cohesion, $\mathit{c}$ (kPa) | Effective Friction Angle, $\mathit{\phi}$ (Degree) | Fully Saturated Hydraulic Conductivity, k (m/s) | $\mathit{a}$ (kPa) | $\mathit{n}$ |
---|---|---|---|---|---|---|---|---|

Clay | 20 | 30 | 0.3 | 18 | 25 | 5 × 10^{−5} | 100 | 1.5 |

Sand | 20 | 50 | 0.3 | 2 | 33 | 5 × 10^{−4} | 10 | 2.5 |

Silt | 20 | 30 | 0.3 | 6 | 28 | 1 × 10^{−4} | 20 | 2.0 |

Borehole Schemes | Borehole Number | Intercept A | Slope B | Pearson’s R | Residual Sum of Squares |
---|---|---|---|---|---|

1 | 3 | 2.050 | −0.059 | −0.596 | 0.700 |

2 | 4 | 2.182 | −0.069 | −0.717 | 0.471 |

4 | 5 | 2.146 | −0.070 | −0.604 | 0.346 |

7 | 6 | 2.223 | −0.075 | −0.669 | 0.282 |

8 | 7 | 2.256 | −0.077 | −0.410 | 0.251 |

10 | 8 | 2.162 | −0.070 | −0.351 | 0.204 |

11 | 9 | 1.801 | −0.044 | −0.161 | 0.202 |

14 | 10 | 1.835 | −0.046 | −0.217 | 0.150 |

17 | 11 | 2.068 | −0.063 | −0.231 | 0.148 |

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## Share and Cite

**MDPI and ACS Style**

Cao, W.; Wan, Z.; Li, W.
Stability of Unsaturated Soil Slope Considering Stratigraphic Uncertainty. *Sustainability* **2023**, *15*, 10717.
https://doi.org/10.3390/su151310717

**AMA Style**

Cao W, Wan Z, Li W.
Stability of Unsaturated Soil Slope Considering Stratigraphic Uncertainty. *Sustainability*. 2023; 15(13):10717.
https://doi.org/10.3390/su151310717

**Chicago/Turabian Style**

Cao, Wei, Zheng Wan, and Wenjing Li.
2023. "Stability of Unsaturated Soil Slope Considering Stratigraphic Uncertainty" *Sustainability* 15, no. 13: 10717.
https://doi.org/10.3390/su151310717