Geofluids

Volume 2019, Article ID 3587989, 15 pages

https://doi.org/10.1155/2019/3587989

## Evaluate the Probability of Failure in Rainfall-Induced Landslides Using a Fuzzy Point Estimate Method

Department of Resources Engineering, National Cheng Kung University, No. 1 University Road, Tainan City, Taiwan

Correspondence should be addressed to Hsin-Fu Yeh; moc.liamg@22heyfh

Received 26 November 2018; Accepted 26 January 2019; Published 17 April 2019

Guest Editor: Roberto Tomás

Copyright © 2019 Ya-Sin Yang and Hsin-Fu Yeh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Traditional slope stability analysis mostly adopts the limit equilibrium method, which predetermines the slope failure surface and assumes that failure occurs simultaneously at all points of the failure surface. The method is based on the balance of forces and torques. The slope stability is represented by the factor of safety. The lowest factor of safety obtained after repeated analysis indicates the most failure-prone slope surface. However, the factor of safety for only one slope failure surface is obtained when applying this method. The distribution and changes of factor of safety in the interior of the slope are not identified. In addition, the analysis of factor of safety is influenced by the uncertainty in soil mechanical parameters, whereas uncertainty is not quantified in the traditional deterministic analysis. Therefore, a probabilistic approach, which uses the probability distribution function to explain the randomness of parameters, is proposed for quantifying the uncertainty. Nonetheless, when the observation data are not sufficient for determining the probability distribution function, the fuzzy theory can be an alternative method for the analysis. The fuzzy theory is based on fuzzy sets. It expresses the ambiguity of incomplete sets of information using a membership function. Moreover, a correct judgment can be made without verbose iterations. Hence, the aim of this study is to examine the uncertainty in soil mechanical parameters. The membership functions between soil mechanical parameters, i.e., cohesion and angle of internal friction, were constructed based on the fuzzy theory. The fuzzy point estimation was used in combination with the hydrologic and mechanical coupling model on HYDRUS 2D and the Slope Cube Module. The local factor of safety at different depths of the slope was determined using the local factor of safety theory. The probability of failure at different depths was calculated through reliability analysis, which could serve as an early warning for subsequent slope failures.

#### 1. Introduction

Slope stability is affected by intrinsic and triggering factors. The intrinsic factors include soil, groundwater, vegetation, slope gradient, and lithology. The triggering factors include volcanic eruptions, earthquakes, and rainfall. A common trigger for natural slopes is rainfall [1–8]. Rainfall-induced slope failures are usually shallow, with a depth of failure not exceeding three meters, and they likely occur on slopes with a gradient of 30° to 40° [9]. Lu and Godt [10] suggested that the failure mechanism for rainfall-induced shallow failures is that, as the rainfall infiltrates into the soil, the soil matric suction declines and the pore pressure rises positively. As the soil matric suction decreases, there would be a nonlinear drop in soil shear strength. Hence, when the soil is nearly saturated, the matric suction approaches zero, resulting in slope instability and further inducing disasters such as landslides and debris flow.

Studies related to rainfall-induced slope failure can be divided into three types according to their theoretical basis: statistical-model-based [4, 11–19], contributing factor [20–24], and physical-model-based analyses [5, 25–31]. Among them, the physical-model-based analysis coupled with hydromechanical mechanism models has overcome the excessive dependence of statistical models on rainfall data. The method can describe the hydromechanical changes caused by transient rainfall in the interior of the slope, as well as the associated failure mechanism. With its higher predictive power and capability of quantifying the effect of each parameter on slope stability [32], the method is now widely used. Nevertheless, the analytical process is limited by uncertainty caused by measurement error, spatial variability, and limited information [33]. The result of slope stability analysis may deviate from reality owing to the uncertainty in model parameters [34].

Therefore, probabilistic analysis is used to quantify the uncertainty [7, 35–39]. Nawari and Liang [40] and Giasi et al. [41] suggested that an adequate number of reliable observation values are required for probabilistic analysis. Precise mean values and standard deviations are derived from the observation values to construct a reasonable probability density function [42]. In addition, Juang (in 1998) and Nawari and Liang [40] proposed that the uncertainty in parameters may be nonstochastic. Previous studies have shown that, when the data available are not sufficient for defining the probability density function, the uncertainty in rock mass parameters can be expressed effectively with the use of a fuzzy set [43, 44]. This method has been applied to some of the cases for slope stability analysis [30, 41, 45–48].

Traditional slope stability analysis adopts the limit equilibrium analysis, which discretizes the potential sliding soil mass into smaller vertical slices without considering soil deformation. It assumes that failure occurs simultaneously at all points of the failure surface. This method is based on the balance of forces and torques. The slope stability is represented by the factor of safety. Various analytical methods have been developed based on different assumptions on the balance of forces [49–52]. In recent years, the finite element method has been widely applied to slope stability analysis in order to calculate the factor of safety in slopes with high complexity (complex geometries, boundaries, and loading conditions) and to investigate the stress–strain relationship in soil [53–58]. Liu and Shao [59] introduced the finite element limit equilibrium analysis, which combines the limit equilibrium analysis and finite element analysis. It is used to examine the slope stability and evaluate the breaking load of a rigid foundation and retaining wall.

The above analytical methods based on the balance of forces or on the stress field usually seek a single general slope stability index. Hence, it is almost impossible to identify the changes in pore water pressure and effective stress owing to rainfall infiltration, or the actual slope failure surface and its geometry. Therefore, Lu et al. [60] proposed the theory of local factor of safety (LFS), which can calculate the factor of safety at discrete points in the soil mass and describe the geometry and position of the potential failure surface. Previous studies have revealed that the factor of safety (probability of failure) is highly dependent on the coefficient of correlation between cohesion and angle of internal friction [61–63]. It has been shown that the two parameters are not independent of each other and that the correlation between them is mostly negative [64–67]. Jiang et al. [63] noted that, when analyzing the probability of failure, a significant deviation may occur if we assume an independent relationship between cohesion and angle of internal friction (i.e., no correlation). Aladejare and Wang [68] also pointed out that neglecting the coefficient of correlation between cohesion and angle of internal friction may result in an order-of-magnitude difference in the result of the analysis. Moreover, the factor of safety does not necessarily reflect the actual safety level. With the use of reliability analysis, considering the variability of variables and calculating the probability of failure and reliability index will provide a more valid representation of the reliability of slope stability.

Hence, the aim of this study is to examine the uncertainty in soil mechanical parameters. The membership functions for the soil mechanical parameters, i.e., cohesion and angle of internal friction, were constructed based on the fuzzy theory. The fuzzy point estimation was used in combination with the hydromechanical coupling model on HYDRUS 2D and the Slope Cube Module. The local factor of safety at different depths of the slope was determined. The probability of failure at different depths was calculated through reliability analysis, which could serve as an early warning for subsequent slope failures.

#### 2. Materials and Methods

##### 2.1. Seepage Analysis

In this study, the analytic solution of transient seepage in an unsaturated layer developed by Šimůnek et al. [69] based on the Richards equation was used as the governing equation of the two-dimensional seepage as follows: where is the volumetric water content (-), is the time (), is the pore water pressure or hydraulic head (), is the total head (), is the source or sink (), is the hydraulic conductivity function (HCF) that varies with the pore water pressure (), and is the volumetric water content that varies with the pore pressure in the soil-water retention curve (SWRC) ().

The soil water content and HCF of an unsaturated zone vary with the hydraulic head and are highly nonlinear. In this study, the relationship between soil water content and matric suction was predicted using the closed-form analytic solution proposed by van Genuchten [70] (see equation (2)). It is also referred to as the SWRC. Based on the SWRC, Mualem [71] introduced the HCF for unsaturated layers (see equation (3)). where is the saturated soil water content (), is the residual soil water content (), is the matric suction (), is the reciprocal correlation of the air-entry value (), is related to the SWRC gradient (), is the hydraulic conductivity in saturated soil (), , is the coefficient of correlation of soil porosity (), and is the equivalent degree of saturation (), shown as

##### 2.2. Principle of Effective Stress in Unsaturated Soil

We adopted the principle of effective stress proposed by Lu and Likos [72], which unified the possible physical and chemical interparticle mechanisms in soil and proposed the concept of suction stress. The effective stress based on the concept of suction stress is shown as follows [73]: where is the suction stress (), is the Born repulsive force (), is the capillary force (), is the combined van der Waals attractive force and electric double-layer force (), is the degree of saturation in the soil (), and is also the matric suction (). The matric suction, capillary force, van der Waals attractive force, and electric double-layer force balance the Born repulsive force in the soil. However, as the grain size of the soil increases, the effect of the van der Waals attractive force and electric double-layer force becomes negligible.

As each of the stress components in soil can be expressed as a function of matric suction , degree of saturation , and water content , and as the suction stress in soil is mainly controlled by the soil water content, Lu et al. [74] derived the suction stress characteristic curve (SSCC) from the soil-water characteristic curve, based on the principle of thermodynamics and by considering suction stress as the energy stored in the pedon. The following analytical solution is shown: where is the equivalent degree of saturation (-), is the residual saturation (-), is the soil water content (-), is the saturated soil water content (-), and is the residual soil water content (-). Moreover, van Genuchten [70] calculated the equivalent degree of saturation using the following closed-form equation: where and are fitting parameters correlated to the air-entry value of SWRC and the gradient, respectively. Therefore, the suction stress can be expressed in the following forms. The change in soil suction stress with water content can be illustrated by estimating the SSCC:

##### 2.3. Theory of Local Factor of Safety

The local factor of safety is based on the Mohr–Coulomb failure criterion, and is defined by the ratio between the potential Coulomb stress and the current Coulomb stress as follows: where is the potential Coulomb stress and is the current Coulomb stress. The theory is illustrated in Figure 1, in which the current state of stress in the soil is represented by the realization of Mohr’s circle. The shear stress acting on the soil when a failure occurs is obtained by translating Mohr’s circle to the Mohr–Coulomb failure envelope. When the effective stress of the soil decreases owing to the increase in water content, Mohr’s circle is translated leftward, during which its size is almost unchanged. By extending the Coulomb stress, the potential Coulomb stress at the intersection point of Mohr’s circle and the Mohr–Coulomb failure envelope (point B) is determined. The local factor of safety is obtained by the calculation of similar triangles as follows: where is the effective cohesion of the soil, is the effective angle of friction of the soil, and and are the maximum and minimum effective stresses of the soil, respectively.