# Slope Failure Risk Assessment Considering Both the Randomness of Groundwater Level and Soil Shear Strength Parameters

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{f}) and failure risk coefficient (C), one-by-one. A new slope failure risk assessment method that uses the sum of the element failure risk to calculate the overall failure risk is proposed in this paper and considers both the randomness of the groundwater level and soil shear strength parameters. The element failure probability is determined by their location information and failure situation; the element failure risk coefficient is determined by their area. It transforms the complex overall failure risk problem into a simple element failure risk problem, which simplifies the calculation process and improves the calculation efficiency greatly. The correctness is verified with the systematic analysis of a classical case. The results show that the slope failure probability and failure risk are greatly increased from 1.40% to 3.30% and 0.829 m

^{2}to 2.094 m

^{2}with rising groundwater level, respectively.

## 1. Introduction

_{f}

_{1}, C

_{1}) and deep landslide (P

_{f}

_{2}, C

_{2}). P

_{f}

_{1}is larger than P

_{f}

_{2}, but C

_{1}is smaller than C

_{2}, so which failure mode has greater impact? To fully consider the impact of P

_{f}and C, researchers proposed the concept of slope failure risk (R) [27,28]. However, whether we use the LEM or FEM to calculate the slope failure risk, the failure modes of the slope need to be identified and counted first, and there is lower computational efficiency with large sample calculations. Zhang Xiaoyan et al., using the slope safety factor and velocity field information to calculate the element failure probability (EFP), provide a new idea for the calculation of slope failure risk [29].

## 2. Methodologies

#### 2.1. Stable Seepage Field of Slope with Stochastic Groundwater Level

#### 2.2. Stochastic Field of Parameter Spatial Variability

#### 2.3. Stochastic Programming Model

## 3. Solution Strategy

- (1)
- Assuming that the stochastic groundwater level follows a truncated normal distribution, which is generated with the Monte Carlo simulation method:$$\{\begin{array}{l}{H}_{w}({T}_{w})=Random(Normal,{\mu}_{w},{\sigma}_{w},1,{N}_{w})\\ {H}_{w}^{min}\le {H}_{w}({T}_{w})\le {H}_{w}^{max}\end{array}$$
- (2)
- Assuming that the autocorrelation functions of the soil materials are exponential type, using Equations (3) and (4), the midpoint method of Cholesky decomposition for the stochastic field generation ${c}^{e}({T}_{m})$ and ${\phi}^{e}({T}_{m})$, of which ${T}_{m}=(1,\dots ,{N}_{m})$; ${N}_{m}$ is the number of random fields for the shear strength parameters.

- (3)
- The stochastic number of ${T}_{w}$ groundwater levels generated in step (1) is substituted into the stable seepage field calculation equation to obtain the pore water pressure ${p}_{1}^{e}({T}_{w})$, ${p}_{2}^{e}({T}_{w})$, ${p}_{3}^{e}({T}_{w})$; $e=(1,\dots ,{N}_{e})$; ${T}_{w}=(1,\dots ,{N}_{w})$.
- (4)
- From ${T}_{w}=1$ to ${T}_{w}={N}_{w}$ cycles, repeat ${p}_{1}^{e}({T}_{w})$, ${p}_{2}^{e}({T}_{w})$, ${p}_{3}^{e}({T}_{w})$, all the finite element nodes’ pore water pressure values are successively replaced with the stochastic programming model for the slope reliability analysis; in each cycle from ${T}_{w}=1$ to ${T}_{w}={N}_{w}$; ${c}^{e}({T}_{m})$, ${\phi}^{e}({T}_{m})$ from ${T}_{m}=1$ to ${T}_{m}={N}_{m}$ cycles, the number of ${N}_{m}$ stochastic fields are brought into Equation (5) and use the dual simplex method to obtain ${N}_{w}\times {N}_{m}$ numbers of capacity overload factors $[{k}_{\gamma}({T}_{m},{T}_{w})]$ while, at the same time, use the bisection method to obtain ${N}_{w}\times {N}_{m}$ numbers of the slope safety factor $[{k}_{m}({T}_{m},{T}_{w})]$. Figure 2 shows the specific numerical solution flow.
- (5)
- Calculation of the slope safety factor and plot the related curve.

## 4. Reliability Index for Slope

## 5. Calibration and Application

#### 5.1. Numerical Simulations

^{3}, and the permeability coefficient is $K=5\times {10}^{-7}\mathrm{m}/\mathrm{s}$, both of which are determined values. See Table 1 for the other calculation parameters [26].

_{w}= 6.8006 m), ${T}_{w}=25$ (H

_{w}= 7.4848 m), and ${T}_{w}=50$ (H

_{w}= 8.1951 m), respectively. It can be observed that the contours of the pore water pressure become steeper, and the saturated area inside the slope increases as the groundwater level rises. Figure 6 shows the key points pore water pressure. Under the same groundwater level act, the pore water pressure decreases gradually as the coordinates of the key points move to the right. At P1, the mean and standard deviation of the pore water pressure are −19.77 kPa and 2.31 kPa, respectively. At P2, the mean and standard deviation of the pore water pressure are −14.59 kPa and 1.78 kPa, respectively. At P3, the mean and standard deviation of the pore water pressure are −8.94 kPa and 1.13 kPa, respectively.

#### 5.2. Main Results of the Research

- (1)
- When ${T}_{w}=1$, ${T}_{w}=25$, and ${T}_{w}=50$, the mean of the slope safety factors with the UBM is larger than that of the LEM, but the error is small, which conforms to the features of the upper bound solution. In addition, the slope safety factors acquired with the UBM and LEM methods decrease as the groundwater level rises. The slope failure probability increases acquired with the UBM and LEM decrease as the groundwater level rises. On the basis of the upper bound theorem, the slope safety factor acquired with the UBM must be greater than the real solution. Therefore, the UBM will slightly underestimate the failure slope probability.
- (2)
- Figure 8a,b shows the PDF and CDF curves of the slope safety factors acquired with the UBM and LEM under three groundwater level acts, ${T}_{w}=1$, ${T}_{w}=25$, and ${T}_{w}=50$, respectively. It is not difficult to see that the PDF and CDF curves of the slope safety factors acquired with the UBM and LEM are very close with small errors. In addition, the PDF and CDF curves gradually move to the left as the groundwater level rises.
- (3)
- Figure 9a–c are the distribution histograms of the slope safety factors acquired with the UBM under the three groundwater level acts, ${T}_{w}=1$, ${T}_{w}=25$, and ${T}_{w}=50$, respectively. It is not difficult to see that the distribution of the slope safety factors is similar to the stochastic groundwater levels.

- (1)
- The distribution of the slope safety factors is consistent with the normal distribution. The mean of the slope safety factors tends to decrease as the groundwater level rises. The PDF and CDF curves of the slope safety factors gradually move to the left as the groundwater level rises. In addition, the standard deviation of the slope safety factor tends to decrease as the groundwater level rises. The range of the PDF curve and the trend of the CDF curve of the slope safety factors gradually narrow and steepen, respectively.
- (2)
- A polynomial fit is used to acquire the quantitative equation of the mean and standard deviation of the slope safety factors and groundwater level as follows:$${u}_{k}({T}_{w})=-0.0439{H}_{w}+1.5963$$$${\sigma}_{k}({T}_{w})=-0.0039{H}_{w}+0.1685$$
- (3)
- The quantity of 100,000 slope safety factors was acquired from 2000 stochastic fields under 50 groundwater level acts to perform the statistical analysis of all the acquired data; under 50 groundwater level acts, the mean and standard deviation of the slope safety factors are 1.2664 and 0.11, respectively.

^{2}(failure times is 997). Failure modes 5 and 6 have a large failure area that belongs to a deep landslide, and the failure area is between 89.17 and 119.38 m

^{2}(failure times is 160). Failure modes 3 and 4 are between a shallow landslide and a deep landslide, and the failure area is between 58.91 and 89.04 m

^{2}(failure times is 989). When the groundwater level is ${T}_{w}=1$, the slope has only five failure modes; when the groundwater level is ${T}_{w}=25$ and ${T}_{w}=50$, the fifth failure modes occur. The phreatic line moves up, correspondingly, and the saturated area inside the slope increases as the groundwater level rises, thus increasing the probability of the failure mode.

^{2}, 1.593 m

^{2}, and 2.325 m

^{2}, respectively. All failure modes can be acquired with the UBM. When ${T}_{w}=1$, ${T}_{w}=25$, and ${T}_{w}=50$, the slope failure risk according to the UBM with Equation (14) and the UBM Equation (22) are 0.829 m

^{2}, 1.302 m

^{2}, and 2.094 m

^{2}, respectively. It should be noted that Equation (14) has a difference calculation principle from Equation (22). All slope failure modes are required to be counted when Equation (14) is used to calculate the slope failure risk; the EFP for all elements is easy to acquire by solving Equation (5), and the element area is fixed when using Equation (22) to calculate the slope failure risk. From the calculation principle, the proposed method will simplify the calculation process and make the calculation more efficient.

^{2}is 675, the frequency of EFR between 0.00062 and 0.00124 m

^{2}is 36, the frequency of EFR between 0.00124 and 0.00186 m

^{2}is 58, the frequency of EFR between 0.00186 and 0.00248 m

^{2}is 69, the frequency of EFR between 0.00248 and 0.00310 m

^{2}is 140, the frequency of EFR between 0.00310 and 0.00372 m

^{2}is 11, and the slope failure risk is 0.829 m

^{2}. When ${T}_{w}=25$, the frequency of EFR between 0 and 0.00082 m

^{2}is 670, the frequency of EFR between 0.00082 and 0.00164 m

^{2}is 32, the frequency of EFR between 0.00164 and 0.00246 m

^{2}is 54, the frequency of EFR between 0.00246 and 0.00328 m

^{2}is 56, the frequency of EFR between 0.00328 and 0.00410 m

^{2}is 165, the frequency of EFR between 0.00410 and 0.00492 m

^{2}is 12, and the slope failure risk is 1.302 m

^{2}. When ${T}_{w}=50$, the frequency of EFR between 0 and 0.00150 m

^{2}is 666, the frequency of EFR between 0.00150 and 0.00300 m

^{2}is 41, the frequency of EFR between 0.00300 and 0.00450 m

^{2}is 48, the frequency of EFR between 0.00450 and 0.00600 m

^{2}is 62, the frequency of EFR between 0.00600 and 0.00750 m

^{2}is 158, and the frequency of EFR between 0.00750 and 0.00900 m

^{2}is 14. Under all potential groundwater level acts, the frequency of EFR between 0 and 0.04820 m

^{2}is 674, the frequency of EFR between 0.04820 and 0.09640 m

^{2}is 30, and the frequency of EFR between 0.09640 and 0.14460 m

^{2}is 57. The frequency of EFR between 0.014460 and 0.19280 m

^{2}is 59, the frequency of EFR between 0.19280 and 0.24100 m

^{2}is 157, the frequency of EFR between 0.24100 and 0.28892 m

^{2}is 12, and the slope failure risk is 2.094 m

^{2}. The element failure risk comprehensively reflects the contribution of the EFP and the element failure risk coefficient, which can make a quantitative judgment on the slope failure risk of each part. The slope failure risk is 1.332 m

^{2}, obtained by the sum of all element failure risks. It is observed that the slope failure risk assessment method proposed in this paper can avoid the screening and statistical work of failure modes compared with the failure risk calculation results in Table 3.

^{2}to 2.094 m

^{2}with the gradual increase in the groundwater level. Through cubic polynomial fitting, the EFR of the slope versus the groundwater level acquired (as shown in Figure 20) is as follows:

## 6. Conclusions

- (1)
- When the randomness of the groundwater level and soil shear strength parameters are considered comprehensively, the traditional LEM will ignore multiple failure modes and may miscalculate the slope failure risk. However, all failure modes can be acquired with the UBM for seeking the minimum value of the KAVF. Thus, the result is more consistent with the real situation. In addition, the traditional LEM only judges the slope stability by the safety factor, which only reflects the degree of the IPF. The EFP is used to calculate the EFR of the slope, which cannot only reflect the degree of the IFP but, also, the slope failure risk can be accurately acquired. It should be noted that this calculation method can greatly reduce the calculation cost.
- (2)
- The IFP and EFR of the slope are increasing from 1.40% to 3.30% and 0.829 m
^{2}to 2.094 m^{2}with the rise of the groundwater level, respectively. Based on the EFP, the proposed method can accurately obtain the EFR of the slope under each groundwater level act by using the element’s location information and failure situation. This will provide engineers with realistic reference values for the slope reinforcement design to achieve sustainable development. - (3)
- Groundwater level and earthquakes are two important causes of slope instability and failure. However, this study does not consider the impact of earthquakes on slope reliability. Therefore, relevant studies on seismic slope stability will be carried out in the future. In addition, according to the upper bound theory, the upper bound solution is inevitably greater than the true solution. Therefore, the failure probability will be underestimated when using the UBM for slope reliability analysis. To solve this problem, there is a necessity to study the slope reliability calculation method on the basis of the lower bound theory in future research work. The solution of slope failure probability with the UBM and LBM can be obtained at the same time, so the interval range of the real failure probability can be accurately judged, and the reliability index of the slope can be quantified more accurately.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 7.**Stochastic fields of the slope shear strength parameters: (

**a**) c

^{r}(500); (

**b**) φ

^{r}(500).

**Figure 8.**Distribution characteristics of the slope safety factors: (

**a**) PDF curve of the safety factor; (

**b**) CDF curve of the safety factor.

**Figure 9.**Distribution histograms of the slope safety factors: (

**a**) T

_{w}= 1; (

**b**) T

_{w}= 25; (

**c**) T

_{w}= 50.

**Figure 13.**PDF and CDF curves of the slope safety factors under all potential groundwater level acts.

**Figure 14.**Schematic diagram results of the slope failure modes: (

**a**) Failure mode 1; (

**b**) Failure mode 2; (

**c**) Failure mode 3; (

**d**) Failure mode 4; (

**e**) Failure mode 5; (

**f**) Failure mode 6.

**Figure 15.**EFP of the slope under T

_{w}th groundwater level acts: (

**a**) T

_{w}= 1; (

**b**) T

_{w}= 25; (

**c**) T

_{w}= 50.

**Figure 17.**EFR of the slope under T

_{w}th groundwater level acts: (

**a**) T

_{w}= 1; (

**b**) T

_{w}= 25; (

**c**) T

_{w}= 50.

Shear Parameter | Mean | Correlation of Variation | Distribution Type | Fluctuation Range | Correlation Coefficient |
---|---|---|---|---|---|

c(kPa) | 10 | 0.3 | Lognormal | L_{h} = 40 mL _{v} = 4 m | ρ_{c,φ} = −0.5 |

φ(°) | 30 | 0.2 | Lognormal |

Groundwater Level | Method | Mean | Standard Deviation | Failure Probability (%) |
---|---|---|---|---|

T_{w} = 1 | UBM | 1.2956 | 0.1420 | 1.40 |

LEM | 1.2923 | 0.1488 | 1.90 | |

T_{w} = 25 | UBM | 1.2678 | 0.1394 | 2.10 |

LEM | 1.2622 | 0.1461 | 2.70 | |

T_{w} = 50 | UBM | 1.2357 | 0.1365 | 3.30 |

LEM | 1.2310 | 0.1432 | 4.65 |

Failure Mode | Failure Area (m^{2}) | Failure Times | Failure Probability (%) | Failure Risk (m^{2}) |
---|---|---|---|---|

Mode 1 | 28.68–43.70 | 246 | 0.246 | 0.097 |

Mode 2 | 43.81–58.91 | 751 | 0.751 | 0.384 |

Mode 3 | 58.91–73.96 | 768 | 0.768 | 0.508 |

Mode 4 | 74.21–89.04 | 221 | 0.221 | 0.180 |

Mode 5 | 89.17–103.70 | 88 | 0.088 | 0.081 |

Mode 6 | 106.71–119.38 | 72 | 0.072 | 0.081 |

Sum | / | 2146 | 2.146 | 1.332 |

Groundwater Level | Failure Mode | Failure Times | Failure Probability (%) | Failure Risk (m^{2}) |
---|---|---|---|---|

T_{w} = 1 | Mode 1 | 3 | 0.15 | 0.057 |

Mode 2 | 13 | 0.65 | 0.334 | |

Mode 3 | 10 | 0.50 | 0.342 | |

Mode 4 | 1 | 0.05 | 0.038 | |

Mode 6 | 1 | 0.05 | 0.057 | |

T_{w} = 25 | Mode 1 | 5 | 0.25 | 0.096 |

Mode 2 | 14 | 0.70 | 0.366 | |

Mode 3 | 16 | 0.80 | 0.528 | |

Mode 4 | 5 | 0.25 | 0.210 | |

Mode 5 | 1 | 0.05 | 0.045 | |

Mode 6 | 1 | 0.05 | 0.056 | |

T_{w} = 50 | Mode 1 | 8 | 0.40 | 0.160 |

Mode 2 | 19 | 0.95 | 0.490 | |

Mode 3 | 24 | 1.20 | 0.793 | |

Mode 4 | 11 | 0.55 | 0.445 | |

Mode 5 | 2 | 0.10 | 0.091 | |

Mode 6 | 2 | 0.10 | 0.115 |

Method | LEM with Equation (12) | UBM with Equation (14) | UBM with Equation (22) | |
---|---|---|---|---|

Groundwater | ||||

T_{w} = 1 | 1.121 | 0.829 | 0.829 | |

T_{w} = 25 | 1.593 | 1.302 | 1.302 | |

T_{w} = 50 | 2.325 | 2.094 | 2.094 |

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## Share and Cite

**MDPI and ACS Style**

Peng, P.; Li, Z.; Zhang, X.; Liu, W.; Sui, S.; Xu, H.
Slope Failure Risk Assessment Considering Both the Randomness of Groundwater Level and Soil Shear Strength Parameters. *Sustainability* **2023**, *15*, 7464.
https://doi.org/10.3390/su15097464

**AMA Style**

Peng P, Li Z, Zhang X, Liu W, Sui S, Xu H.
Slope Failure Risk Assessment Considering Both the Randomness of Groundwater Level and Soil Shear Strength Parameters. *Sustainability*. 2023; 15(9):7464.
https://doi.org/10.3390/su15097464

**Chicago/Turabian Style**

Peng, Pu, Ze Li, Xiaoyan Zhang, Wenlian Liu, Sugang Sui, and Hanhua Xu.
2023. "Slope Failure Risk Assessment Considering Both the Randomness of Groundwater Level and Soil Shear Strength Parameters" *Sustainability* 15, no. 9: 7464.
https://doi.org/10.3390/su15097464