Reliability analysis and risk assessment of a landfill slope failure in spatially variable municipal solid waste

Abstract

Quantitative assessment of landfill slope failure risk provides valuable information about slope design and risk reduction. This study presents a reliability-based analysis in which an accurate method is applied to assess slope failure risk using the stochastic finite difference method. This method incorporates the spatial variability of municipal solid waste properties due to anisotropic autocorrelation structures and evaluates the consequence associated with each failure separately. This method was evaluated using the data of the Saravan landfill (Rasht, Iran) and presenting a parametric analysis. Several Monte Carlo simulations were conducted to indicate the heterogeneity of the municipal solid waste, taking into account the shear strength and the unit weight of the municipal solid waste randomly. Finally, the safety factor, probability of failure, and risk were assessed using different analysis cases. Deterministic analysis was also performed for all modes using mean values for various municipal solid waste properties. The results show that spatial variability of municipal solid waste parameters and autocorrelation structures significantly affect the safety factor, probability of failure, and risk. Also, comparing the obtained results revealed that for the given slope, the safety factor values in deterministic analyses are overestimated compared to those of the probabilistic analyses. However, risk shows the opposite behavior.

Introduction

The increase in municipal solid waste (MSW) production over the past few decades has led to the rapid expansion of MSW management and disposal technologies. Landfilling is still an efficient and common method in many countries because of its excellent performance in cost–benefit analysis (Machado et al. 2010; Karimpour-Fard et al. 2011; Babu et al. 2014; Rajesh et al. 2016; Mehdizadeh et al. 2020). Due to increased MSW production and limited new landfills, the prevailing attitude in most countries is to extend the capacity of existing landfills, which in turn requires increasing their height and area. As a result, the overall slope stability of these landfills should be cautiously considered (Eskandari et al. 2016; Zhang et al. 2020). The instability of landfills can have many adverse effects on the environment and surroundings, including air pollution, water pollution, soil pollution, climate change, fires or explosions, loss of biodiversity, obstruction of drains; landfills can also contaminate drinking water and transmit diseases (Koerner and Soong 2000; Eid et al. 2000; Chang 2005; Chugh et al. 2007; Zhan et al. 2008; Blight 2008; Jahanfar et al. 2017). Although many research has been done on the recycling of waste, the reliance on landfilling remains a prevalent waste management strategy in numerous places (Aziz et al. 2017; Razali et al. 2018a, b; Aziz et al. 2019; Japar et al. 2019).

The slope stability of landfills depends primarily on the shear strength (τ) and unit weight (γ) of the MSW (Zekkos et al. 2012). The geotechnical properties of the MSW show considerable spatial variability even in a single landfill and a specific layer. These properties depend on various factors, including inherent MSW heterogeneity, different loading conditions, compaction methods, weather conditions, different testing methods, MSW ages, gas and leachate pressure, leachate level, and degradability of MSW due to various physical, chemical, and biological factors (Gharabaghi et al. 2008; Ering and Babu 2015; De Stefano et al. 2016; Jahanfar et al. 2017; Feng et al. 2018). Internal friction angle (ϕ) and cohesion (c) of the MSW have been reported from 10 to 60° and from 0 to 80 kPa, respectively (Babu et al. 2015; Zhan et al. 2008). Based on these factors and the resulting variability, the choice of landfill design parameters in conventional methods is a challenging issue directly related to landfill slope stability.

In slope stability analysis of variable MSWs, since failure slip tends to cross the weakest paths, different failure modes may occur (Mehdizadeh et al. 2020) and the consequence is associated with each failure separately. Therefore, a design engineer not only must thoroughly examine the slope stability of the landfills but must also examine the occurrence of failure damage separately. One of the techniques to quantify landfill slope failure risk (R) is to calculate the probability of failure (P_f) and multiply it by the failure occurrence (Huang 2013; Cheng 2018). In this way, the effect of MSW properties variability on the assessment of landfill slope risk can be evaluated by considering different failure modes.

Landfill slope stability assessment is conventionally based on a deterministic approach with a safety factor (FS) that provides limited information about the consequence of landfill slope failure for variable MSWs. This approach tries to deal with uncertainties involved in choosing logical parameters conservatively. However, the same safety factors may be used for different MSW slope conditions, regardless of the uncertainty associated with each condition. Because this is not a very reasonable strategy, many recently conducted studies have systematically developed some methods to address the uncertainties involved. Probabilistic methods provide a good framework for incorporating the MSW variability in the probabilistic slope stability analysis and calculating the involved risk.

Limited studies have been conducted on the probabilistic landfill slope stability. The main purpose of these studies has been to determine the P_f or reliability index of these slopes. Sia and Dixon (2012) evaluated the interaction of MSW and the lining system in a probabilistic framework. The lining system of a landfill is a protective barrier designed to prevent the leakage or migration of leachate, which is the liquid that forms as water percolates through the waste in a landfill. The primary purpose of the lining system is to contain and control the movement of leachate, preventing it from contaminating surrounding soil and groundwater. Babu et al. (2014) applying a probabilistic finite difference method assessed the landfill slope stability by considering the spatial variability of the MSW’s geotechnical properties. Rajesh et al. (2016) investigated the probabilistic stability of a landfill slope using the response surface metamodeling approach. Reddy et al. (2018) evaluated the stability of a bioreactor landfill with a hydro-bio-mechanical model in a probabilistic framework. They also investigated the effect of spatial variability properties on settlement and moisture distribution in the landfill. Mehdizadeh et al. (2020) used probabilistic methods to investigate the effect of variability of shear strength and unit weight of MSW on the stability and P_f of an MSW landfill slope. For this purpose, they combined the random field theory with the numerical finite difference method (FDM) in the Monte Carlo simulation framework. The results showed that the probabilistic methods have a good ability to incorporate the spatial variability of MSW properties and their effect on the performance and P_f of landfill slopes. Mehdizadeh et al. (2020) evaluated the reliability of a landfill. For this purpose, they studied the effect of the anisotropic structure of MSW’s random variables, including c, φ, and γ, on the mean (FS_st) and coefficient of variation (COV_FS) of the safety factor and P_f using Monte Carlo simulation (MCS). The results showed that the coefficient of variation (COV) of random variables significantly affects the FS_st, COV_FS and the P_f of the landfill slope. In addition, it was found that assuming the isotropic structure of random variables leads to an underestimation of the P_f. Falamaki et al. (2021) studied the static and seismic stability of the failed landfill from a probabilistic perspective. They investigated the effect of uncertainties involved in MSW strength parameters and seismic forces on the landfill slope stability during and after construction. Based on the obtained results, they provided recommendations for building open dumpsites.

Despite numerous studies on the probability of failure in landfill slopes, scare studies have assessed the risk of landfill slope with some probabilistic methods. For instance, Jahanfar et al. (2017) proposed a new method for risk assessment of landfill slope failure using probability analysis of potential failure scenarios and related losses. The main framework of their method included selecting appropriate statistical distributions for MSW shear strength and rheological properties, probability analysis of slope stability, and determination of waste run-out length, which is ultimately used to calculate the risk of potential losses. Compared to existing slope stability analyses, which are based solely on the probability of slope failure, this method provides a more accurate estimate of the casualty risk due to landfill slope failure. As a result, probabilistic studies performed for landfill slope stability have been limited to calculating P_f or R without considering various failure modes resulting from MSW variability. Hence, in order to safely design and reduce the risks of landfill instability, and estimate the P_f and hazards of consequences associated with landfill slope failure, it is necessary to assess the risk of landfill slope failure by considering the variability of MSW characteristics.

In this paper, P_f and R of a landfill are assessed when strength parameters (c and φ) and unit weight (γ) of MSW have spatial variability with anisotropic structure and different correlation lengths. In this method, which is called the random finite difference method (RFDM), FLAC2D code is coupled with the random field theory. Jamshidi and Alaie (2015) and Cheng et al. (2018) used RFDM to study the possibility of soil slope failure. In RFDM, the matrix decomposition method generates random fields of shear strength and unit weight of MSW. Then, the safety factor is calculated using the FLAC2D code, and the critical slip surface and the slip mass are calculated by a Fish program (FLAC2D). Finally, the P_f and R are calculated using the MCS. This research was conducted in 2022 in the city of Rasht, Iran.

Materials and methods

Random fields

Vanmark (1983) showed that according to stochastic field theory, the spatial variability of a continuous environment could be expressed in terms of mean (μ), COV, and autocorrelation distance (δ). For probabilistic analyses, each of the MSW geotechnical parameters (i.e., cohesion, c, friction angle, φ, and unit weight, γ) is modeled as independent random variables using probability density function (pdf) and parameters related to their distribution. Assuming that the geotechnical parameters of the MSW (k) have a log-normal distribution, the mean (μ), standard deviation (σ), and the spatial autocorrelation distance (δ) of the parameter k are used to simulate random fields using Eq. 1:

$$k\left( {x_{i} } \right) = \exp \left\{ {\mu_{\ln k} \left( {x_{i} } \right) + \sigma_{\ln k} \left( {x_{i} } \right).G\left( {x_{i} } \right)} \right\}$$

(1)

where k(x_i) is a geotechnical property at location x_i and G(x_i) is a random field with a standard normal distribution (mean zero and unit variance). The values of G(x_i) are determined using the method of matrix decomposition and the anisotropic function of single exponential auto-correlation (Baecher and Christian 2003):

$$\rho_{k} \left( {\tau_{x} ,\tau_{y} } \right) = \exp \left( { - \frac{{\tau_{x} }}{{\delta_{x} }} - \frac{{\tau_{y} }}{{\delta_{y} }}} \right)$$

(2)

where τ_x and τ_z represent the relative distances of two points in the horizontal and vertical directions, respectively, and δ_x and δ_y show the auto-correlation distance in the main directions. The complete process of anisotropic random field modeling and its FDM-based implementations have been studied in detail in Jiang and Huang (2018) and Mehdizadeh et al. (2020).

The probability of failure (P _f)

After generating random fields for the three parameters c, φ, and γ, the safety factor (FS) of each simulation can be calculated, and the P_f of the desired slope is obtained. In this study, the shear strength reduction method in the FDM-based FLAC2D software was used to calculate FS. Unlike the limit equilibrium methods (LEMs), this method can determine different slip surfaces. Also, in this method, parameters of c and φ are reduced by the same reduction factor (F) using Eq. (3). Next, FS is obtained by selecting different values of F and replacing the reduced values (i.e., c/F and φ/F) with the initial values, and finally performing various analyses to reach the critical state. The reduction coefficient that leads to the critical state will be the same as the FS (ITASCA 2015).

$$\begin{gathered} c_{F} = {\raise0.7ex\hbox{$c$} \!\mathord{\left/ {\vphantom {c F}}\right.\kern-0pt} \!\lower0.7ex\hbox{$F$}} \hfill \\ \varphi_{F} = \arctan \left( {{\raise0.7ex\hbox{${\tan \varphi }$} \!\mathord{\left/ {\vphantom {{\tan \varphi } F}}\right.\kern-0pt} \!\lower0.7ex\hbox{$F$}}} \right) \hfill \\ \end{gathered}$$

(3)

The whole process of generating random fields and stability analysis is performed using the FISH language in FLAC2D software. This method obtains an FS for each realization for random fields c, φ, and γ using the Monte Carlo sampling method. Thus, the MCS is adapted to generate a sufficient number of realizations for random fields. After determining the stochastic FSs, the probability of slope failure is calculated by Eq. (4):

$$P_{{\text{f}}} = \frac{{N_{{\text{f}}} }}{{N_{{{\text{sim}}}} }}$$

(4)

where N_sim is the total number of realizations and N_f is the number of realizations in which the slope is failed (FS < 1).

The consequence of failure

In risk quantification, the consequence of each landfill slope failure is estimated using a specific and simple method and taking into account the spatial variability of the shear strength and the unit weight parameters of the MSW. The accurate determination of the consequence of slope failure requires complete data and information about the landfill, facilities, buildings, artificial and natural factors around it, and human factors involved in the landfill site. In the absence of such data, Zhu et al. (2015) suggest that damage due to slope failure is usually directly related to an increase in the volume of the slip mass. In this study, the volume of slip MSW mass due to the landfill slope failure was used as a simple and approximate measure to determine the consequence of any failure. As mentioned earlier, due to the spatial variability of the MSW parameters, different failure modes are formed on the landfill slope, and the volume of the sliding MSW mass depends on the critical surface formed on the slope.

Hicks et al. (2014) proposed a method for calculating the volume of a sliding mass on a slope based on the invariant shear strain on a critical slope. The shear strain invariant is used because it easily combines all the strain components into a single value, thus providing a clear picture of the failure mechanism. After calculating the strain in this method, the critical failure surface is determined using the ridge-finding technique. The ridge-finding technique has two steps: First, a virtual point is selected in the space above the slope toe. Then, the algorithm searches the location of the points along the straight lines that pass through the desired point and has the highest invariant shear strain. Finally, the volume of the slip mass is calculated as the area above the critical surface. This critical surface is determined by summing the volume of all the failed elements above the critical surface. In this study, this approach was run using the FISH language in FLAC2D software.

According to Cheng et al. (2018), to determine the critical surface, the failed elements along the horizontal direction must be continuous, but the target elements may be discrete in the vertical direction. A very important point in determining the critical shear surface is determining the size of the elements. The smaller the elements, the more accurate the critical slip surface will be. As a result, the volume of the slip surface will be calculated accurately. On the other hand, reducing the size of the elements also affects the simulation time of random fields and their analyses. Therefore, to determine the optimal size for the target elements, a sensitivity analysis was performed to compare the accuracy of the calculated sliding mass volume and computational effort.

Risk assessment

Conventionally, the risk of slope failure is calculated as the product of the consequence of failure, and the P_f:

$$R = C \times P_{{\text{f}}}$$

(5)

where P_f is the probability of slope failure and C is the consequence associated with that failure. Equation (5) is suitable for systems with a specific failure mode (Huang et al. 2013). In the landfill slopes, due to the spatial variability of the shear strength and the unit weight parameters, there are different failure modes (Mehdizadeh et al. 2020), the consequence of which is different from each other. Huang et al. (2013) presented Eq. (6) to calculate the risk in multiple failure modes:

$$R = \frac{1}{N}\sum\limits_{i = 1}^{{N_{f} }} {C_{i} }$$

(6)

where N_f is the number of landfill slope failures (FS < 1) and C_i is the consequence associated with that failure. Generally, for a fixed number of MCSs, the lower the P_f, the lower the number of slope failures in the probabilistic analysis will be. As a result, the smaller the number of failure occurrences, the greater the computational error of the risk assessment will be. Therefore, as the P_f declines, the number of MCSs must be increased to keep the computational accuracy constant. On the other hand, increasing the number of MCS increases the computation time. Thus, it is necessary to compare the accuracy of the calculations and the time spent to select the optimal mode. The accuracy of P_f estimation is highly dependent on the number of random field samples and is estimated by the COV_Pf:

$${\text{COV}}_{{{\text{P}}_{{\text{f}}} }} = \sqrt {{{\left( {1 - P_{{\text{f}}} } \right)} \mathord{\left/ {\vphantom {{\left( {1 - P_{{\text{f}}} } \right)} {(N.P_{{\text{f}}} )}}} \right. \kern-0pt} {(N.P_{{\text{f}}} )}}}$$

(7)

The COV_Pf decreases gradually by increasing the number of MCS with a convergence rate of ${1 \mathord{\left/ {\vphantom {1 {\sqrt N }}} \right. \kern-0pt} {\sqrt N }}$. In this paper, COV_Pf is calculated at the end of the MCS analysis and 2000 iterations were performed for cases where the P_f is greater than 10%. For other cases, more iterations were performed to ensure that the maximum P_f error was less than 0.01 at the 90% confidence level.

Implementation of RFDM method

Risk assessment in the probabilistic failure analysis of landfill slope was conducted using the RFDM method through the following steps:

• (1) Determining the landfill slope geometry and statistical parameters related to the variability of MSW properties (statistical information of geotechnical properties such as mean values, standard deviations, probability distribution function, autocorrelation functions, and correlation distances).

• (2) Slope meshing in FLAC2D software and generation of random fields by matrix decomposition method.

• (3) Performing the MCSs, calculating the FSs, calculating the P_f, determining the COV_Pf, and finally determining whether the number of iterations satisfies the desired criterion; if the desired criterion is not met, the number of simulations will increase.

• (4) Determining the consequence of each slope failure to calculate risk (R).

Case study of Saravan landfill in Iran

Landfill configuration

Saravan MSW Landfill is an active landfill site with an approximate area of 13 hectares in northern Iran (Rasht, Guilan Province, Iran). The MSW has been dumped directly into the natural valley since 1984 without installing geosynthetic membranes in the foundation. Also, the MSW is deposited from a small bottom layer in the foundation to a wide top layer at a thickness of 3 m. Since then, the site has become the largest landfill in northern Iran, containing about 10 million cubic meters of MSW. Currently, about 1,000 tons of MSW enters the landfill daily. The depth of the MSW with the valley’s topography varies from 40 to 70 m at the highest level. According to Karimpour-Fard (2019), no proper MSW disposal regulations are followed in the Saravan landfill. In addition, there are no proper compression method and no specific working method in this landfill. Contrary to designs and field investigations in the stability and safety assessment of the landfill slope, a part of the landfill was failed in 2018, and about 300,000 m³ of MSW was displaced along the failure surface. Figure 1 presents the aerial view of this failure and its idealized cross section after the failure. The slope of the failed part before the failure was approximately 1V:2.5H (the vertical change to horizontal change). Due to the natural slope of the earth, the leachate surface is always at a certain level. The highest position of this slope is shown in Fig. 1. The failure did not cause any casualties but caused serious damage to existing buildings and the environment around the landfill, leading to extensive economic damage.

Fig. 1

Aerial image and cross section of the failed part of Saravan landfill

Numerical model and materials properties

Although the landfill slope is three-dimensional, with a suitable approximation, it can be simulated using a simplified two-dimensional model based on the assumptions of plane strain analysis for the failed part of the landfill. Therefore, a two-dimensional plan strain slope with a width of 250 m and a height of 70 m was created in FLAC2D software for numerical simulations. The geometry of the MSW slope is shown in Fig. 1. In this study, uncertainties related to foundation soil and interaction characteristics were not considered. Landfill foundation soil properties were considered deterministic, and the values of cohesion and angle of friction are equal to 54 kPa and 33°, respectively, and the unit weight is 17.2 kN/m³ (Karimpour-Fard 2019). As mentioned earlier, the FDM-based FALC2D software was used to perform slope stability analyses. The 4-sided elements with dimensions of 0.5 × 0.5 m were used to model the MSW slope to increase the accuracy of calculations in determining the FS and slip mass of MSW. The MSW behavior was modeled using an elastoplastic model based on the Mohr–Coulomb failure criterion with a non-associated flow rule. The discretized mesh of random fields contains about 52,000 elements.

Before performing the MCSs, it is necessary to determine the mean values (µ), coefficient of variation (COV_P), correlation lengths (δ), and the type of probabilistic distribution of random parameters (c, φ, and γ). In this study, the µ and COV_P of random parameters were determined using the results of studies of Karimpour-Fard (2019) and Karimpour-Fard et al. (2021); they evaluated the mean variations of c, ϕ, and γ with depth (per 3 m depth) in this landfill. Figure 2 presents the proposed values for different depths in this landfill. The mean values of c, ϕ, and γ are assumed to be the same for MSW in each 3-m layer, and the initial values of these properties are chosen based on Fig. 2. Therefore, the total 70 m height of the slope is divided into 23 layers, each with a thickness of 3 m and a specific degree of degradation. Karimpour-Fard (2019) showed that the COV_P of c, ϕ, and γ are almost constant with depth and COV_P obtained for c, ϕ, and γ are 49, 26, and 11%, respectively. However, in this study, a wider range of COV_P was investigated to determine the effect of COV_P of random parameters on the output results. Due to the lack of sufficient data to determine the CLs of random properties, a wide range of these lengths was selected, followed by investigating their effect on the probability of failure (P_f) and risk (R).

Fig. 2

Variation in geotechnical properties of MSW in Saravan landfill (Iran)

Table 1 shows the values of the COV_P and the horizontal and vertical CLs. The statistical range of COV_P of these parameters was considered by reviewing several studies (Babu et al. 2014; Reddy et al. 2013 and 2018, Sia and Dixon 2012; Rajesh 2016; Mehdizadeh et al. 2020; Falamaki et al. 2021). Finally, normal probability distributions for the random variables of γ and ϕ and lognormal probabilistic distribution for c were chosen to describe how the data are distributed (Raviteja 2021; Mehdizadeh 2020).

Table 1 CoV_P and CLs of geotechnical parameters for probabilistic slope stability analysis of Saravan landfill

Results and discussion

Deterministic analysis

Deterministic analysis of landfill slope was performed using the mean values of MSW geotechnical properties in Fig. 2 to consider the effect of MSW degradability over time and overburden pressure. Using the strength reduction method in FLAC2D software, FS was 1.33 in this case, which is less than the minimum value recommended by USACE (1997). Thus, this slope in the deterministic state also has a sliding risk, and its P_f is high. For a given condition, the critical failure surface is roughly a circular arc that starts at the MSW slope surface and extends to the slope toe. Based on the method mentioned in the previous sections, the critical slip surface has a slip mass of 2218 m² per meter length.

Probabilistic analyses

MCSs were used for probabilistic analyses. The sample size of MCS is a very important criterion in accurately estimating P_f and R parameters. The convergence criterion was evaluated by performing the slope stability of each analysis case with specific statistical data with different numbers of MCSs. As mentioned, P_f is an important factor in assessing the risk and reliability of MSW slopes.

Figure 3 illustrates the variations in, but variations in the P_f and COV_P(f) for Case 1 of the analyses with δ_h = 5 and δ_V = 0.5 m. The results show that both P_f and COV_P(f) converge to a relatively constant value at MCS of 2000 after the initial oscillation. The COV_P(f) for this condition is about 0.1, which is sufficient for the accuracy of the study results. This convergence criterion has been performed for other analyses as well. Except for 6 cases of statistical inputs, 2000 iterations of MCSs for each case resulted in a COV_P(f) < 0.1. In the remaining 6 cases, the number of iterations depending on the COV_P was increased up to 8000. Regarding the computational efforts, it takes about 45 s to complete each realization with a 64-core fast computing system (Intel Xeon SkyLake, Gen 2 Gold, CPU @ 3.9 GHz 64 GB of RAM).

Fig. 3

Variation in the P_f and COV_P(f) with the number of MCSs (case 1, δ_h = 5 and δ_v = 0.5 m)

Effect of COV_P and CLs on FS_st

Mehdizadeh et al. (2020) showed that the COV_P of MSW input parameters is the most important factor affecting the stochastic safety factor (FS_st). In this section, the effect of COV_P on FS_st is evaluated. Figure 4 shows the FS_st variations for all analysis cases with all δ_h and δv = 1m. According to Table 1, COV_P of MSW increases from cases 1 to 7; Fig. 4 shows that for all δ_h values, with increasing COV_P, FS_st decreases. Also, for all cases, FS_st is less than the deterministic safety factor (FS_det). Therefore, not considering the variability in the input parameters leads to an overestimated FS. The reduction rate of FS_st with increasing COV_P is almost smooth (up to case 3), such that the highest decrease is related to cases 4 and 5 (about 10%). After Case 5, with increasing COV_P, the reduction rate of FS_st decreases, and the FS_st values tend to a constant value.

Fig. 4

FS_st statistics of the landfill slope (all cases, δ_v = 1m). a FS_st versus cases, b CoV_FS(st) versus cases

The lowest FS_st for δ_v = 1m corresponds to case 7 of analyses with a value of 0.59. Evaluating FS_st values for δ_v = 0.5 and 3m also showed similar results. The maximum FS_st reduction for all analysis cases is about 60% compared to FS_det, which is for analysis Case 7 with δ_h = 100 and δ_v = 3m. Mehdizadeh et al. (2020) estimated a maximum reduction of about 40% for slope stability analyses. Higher reduction in FS_st in this study is due to the high slope height of landfill and lower mean of MSW parameters. The decrease in FS_st with increasing COV_P can be explained using Fig. 4b. Figure 4b represents the effect of COV_P on COV_FS(st) for all analysis cases with all horizontal CLs and δ_v = 1m. From cases 1 to 7 of the analysis, the value of COV_FS(st) increases for all horizontal CLs. The results show that the trend of increasing COV_FS(st) up to case 4 is almost linear, and then, this increasing trend becomes smooth, and this is the reason for the low variations in FS_st after case 5. The failure mechanisms based on the shear strength reduction (SSR) method pass through the weakest elements (Mehdizadeh et al. 2020). Thus, with increasing COV_P, the heterogeneity of random fields increases, and different failure mechanisms form, leading to an increase in COV_Fs. On the other hand, since the FS_det of landfill slope with the mean parameters of the MSW properties (Fig. 2) is close to 1 (FS_det = 1.33), passing the failure mechanism through the weakest elements, whose properties are often lower than the mean values, will result in lower FS values, thereby decreasing the FS_st. Similar results were obtained for vertical CLs of 0.5 and 3 m and the maximum COV_FS(st) (i.e., 45%) is related to analysis case 7 with δ_h = 100 and δ_v = 3 m.

Figure 4a shows that an increase in δ_h leads to a decrease in FS_st. The effect of δ_h on FS_st is almost the same in all cases. The maximum decrease occurs in the range of 0.5 to 10 m for δ_h (about 20%), and after δ_h = 10m, the effect of δ_h decreases about 5%. The reason for decreasing FS_st with δ_h is that in a constant δ_v, with increasing δ_h, the possibility of forming weak areas (lower shear strength) along the horizontal direction increases. Moreover, due to the possibility of forming these weak areas in the whole height of the slope and different positions, the possibility of forming multiple failure mechanisms also increases, thereby increasing the COV_FS(st) values. An increase in COV_FS(st) means a greater variability of FS_st and consequently an increased probability of failure (FS_st < 1). The maximum effect of δ_h on FS_st occurs up to δ_h = 10 m (maximum 40% for analysis case 7 with δ_v = 3), and then, the effect of δ_h decreases so that the results for δ_h of 25 and 100 m are almost the same. It was also observed that increasing the COV_P leads to a decrease in the convergence of the results with increasing the δ_h. Since both of these two factors (COV_P and δ_h) reduce FS_st, for all CLs from case 5 onward, most realizations are failed and their FS is less than 1, FS < 1. Overall, failure occurred in other cases depending on the value of CL; for example, for analysis case 3 with δ_v = 1 m, cases of δ_h = 0.5, 1 and 3 do not fail. However, increasing δ_h from 3 to 100 m, all simulations failed. Certainly, FS_st < 1 does not necessarily indicate the failure of all MCSs in that analysis case, but the mean of these simulations has an FS_st of less than 1. Therefore, the probability of failure should be assessed to compare these cases better and determine how many MSCs have failed.

Figure 5 shows the effect of δ_v on FS_st and COV_FS(st) for analysis case 7. The results show that with increasing δ_v, the FS_st decreases for all cases, but COV_Fs increases. The decreasing trend of FS_st and increasing COV_Fs are almost the same for all cases, and maximum FS_st and COV_Fs vary from δ_v = 0.5 to 1 m (13% reduction and 10% increase for FS_st and COV_Fs, respectively). As mentioned, different failure mechanisms are formed in different situations with increasing CLs and increasing COV_Fs. This process becomes more apparent with increasing COV_P. FS_st and COV_FS(st) values show that with increasing δ_v values, the effect of COV_P will increase on the output results. For example, in analysis case 7, with increasing δ_v from 0.5 to 3, FS_st decreases by 19%, and COV_FS(st) increases by 15%. Comparing the obtained results shows that the maximum effect on COV_FS(st) is due to COV_P (with a maximum of 200% increase in COV_FS(st)), next, the horizontal correlation length (with a maximum of 200% increase in COV_FS(st)), and finally the vertical correlation length (with a maximum of 200% increase in COV_FS(st)). The explanation for these results is that FS depends on the spatial average of the strength parameters across the failure zone. For short correlation lengths, the failure mechanism always passes through highly oscillating areas with low and high shear strengths. As a result, the effect of averaging increases with increasing CLs, and spatial variability decreases. These results are consistent with those of Mehdizadeh et al. (2020).

Fig. 5

FS_st statistics of the landfill slope (case 7). a FS_st versus δ_v, b COV_FS(st) versus δ_v

Effect of COV_P and CLs on P_f

The FS_st and COV_FS(st) values can be considered as a measure of slope safety. However, these data do not provide any information about P_f and which analysis cases present higher risks. In this section, P_f values obtained from different analysis cases are evaluated, and the effect of COV_P and CLs on the output results is investigated. Figure 6a shows the effect of COV_P on P_f of all analysis cases with δ_v = 1m. The results show that increasing COV_P leads to increasing P_f in all cases. The minimum value of P_f for δ_v = 1m is related to analysis case 1 with δ_h = 0.5 m and is equal to 2.17%. In contrast, the maximum value of P_f is related to analysis case 7 with δ_h = 100 m with a value of 95.25%. The results show that up to analysis case 4, the P_f increase rate with increasing COV_P is high, and the trend of increasing P_f decreases after this case.

Fig. 6

P_f statistics of the landfill slope (all cases, δ_v = 1m). a P_f versus cases, b COV_P(f) versus cases

Figure 6a also shows the effect of δ_h on P_f. As can be seen, increasing δ_h leads to an increase in P_f. In all analysis cases, maximum variations occur up to δ_h = 10 m, exceeding which the increasing trend of P_f declines. From analysis cases 1 to 4, the effect of δ_h on P_f increases. For example, for case 4 at δ_v = 0.5m and δ_h from 0.5 to 100 m, the value of P_f increases about 86%. In comparison, for analysis cases 5 to 7, P_f variations with δ_h are negligible. These changes are attributed to the effect of δ_h on FS_st. According to the P_f calculation method (in which the number of slope failures is important instead of FS value), from case 4 to case 7, the FS_st value of MCSs is all less than 1, and as δ_h increases (which has a reducing effect on FS_st), the number of simulations whose FS value is less than 1 does not change, and as a result, the effect of δ_h decreases. However, from cases 1 to 4, the number of MCSs with FS_st close to 1 increases; as a result, increasing δ_h leads to an increase in failed slopes and P_f. Figure 6a also shows that with increasing δ_h, the effect of COV_P on P_f increases such that for δ_h s of 3–100 m (up to analysis Case 4), the relationship between P_f and COV_P is almost linear. Figure 6b shows the effect of COV_p and δ_h on COV_P(f). As can be seen, with increasing COV_P, COV_P(f) decreases. This result is due to the reduction of FS_st below 1 (FS_st < 1) in MCS analyses, which results in the failure of most slopes and reduction in COVP_(f) values. According to USACE (1997) recommendations for geotechnical structures, the reliability index (β) of 1 is equal to hazardous performance, and β = 2 is equal to poor performance of structures. Assuming a normal distribution for FS_st, their P_f is 0.16 and 0.023, respectively. According to Fig. 6b, for all cases, the landfill slope performance is poor, and except for case 1, the performance of all cases is less than hazardous performance. Therefore, landfill slope does not meet the requirements of average performance of geotechnical systems, and P_f and R of the slope are high, leading to many human and financial losses. In fact, unlike FS, which does not provide a probabilistic view of the landfill slope failure, P_f has a good ability to determine the safety level of the slopes. For example, although analysis case 2 has an FS_st > 1. However, its P_f is high, indicating a hazardous performance level.

Figure 7 presents the effect of δ_v on P_f and COV_P(f) for analysis case 4. As can be seen, the maximum P_f in all analysis cases is related to δ_v = 3 and δ_h = 100 m. Results showed that the maximum increase is related to case 4 and is about 20% between δ_v of 0.5 to 1 m. Comparing the obtained results showed that with increasing COV_P, the effect of δ_v increases up to analysis case 4 and then declines. This reduction corresponds to the decrease in the effect of COV_P on FS_st. Figure 7b also shows the effect of δ_v on COV_P(f). With increasing δ_v, the value of COV_P(f) decreases which is due to the increase in COV_P and, consequently, P_f increases. With increasing P_f, the number of failed slopes increases, and COV_Pf decreases. Moreover, with increasing δ_h, the decreasing effect of δ_v decreases, and again, the maximum reduction occurs between δ_v of 0.5 and 1 m. Comparing these results shows that δ_h is a more effective parameter than δ_v.

Fig. 7

P_f statistics of the landfill slope (case 4). a P_f versus δ_v, b COV_P(f) versus δ_v

Effect of COV_p and CDs on R

Considering the spatial variability of the MSW parameters, the P_f of the landfill slope can be considered a suitable criterion for quantitatively describing the safety and instability in the landfill slope. However, it does not provide any information about the damage level in failed slopes. The presence of multiple failure mechanisms in the variable MSW material leads to the risk of each failure. Consequently, the total risk associated with a set of probabilistic input data is quite different from each other and conventional methods. Thus, as mentioned earlier, the volume of sliding mass was used as a suitable criterion of risk.

Figure 8a presents the effect of COV_P on R for all analysis cases with δ_v = 1. The results show that increasing the COV_P increases the R value for all cases, and increasing the COV_P has the maximum effect in cases 1–4. For instance, at δ_h = 5 m, the value of R increases from 190 to 1546 m², which increases about 700%. From cases 4 to 7, the increase rate of R is low and is approximately the same in cases 6 and 7, suggesting that a further increase in COV_P does not affect the risk associated with the slope failure and this is the maximum P_f of landfill slope. Comparing the results of the conventional analysis with probabilistic results shows that the R is less than the conventional value in all probabilistic analyses (i.e., 2218 m²). The maximum R in the probabilistic analysis is 1834 m², which is about 18% less than the conventional method. As a result, unlike the FS, which was overestimated in the conventional method, the R in the conventional method is higher than the probabilistic method. Nevertheless, the conventional method does not provide any information on the P_f, safety margin, or slope instability and only shows the most probable failure path, which is different from the actual failure surface.

Fig. 8

R statistics of the landfill slope (all cases, δ_v = 1). a R versus cases, b R versus δ_v

Also, Fig. 8a represents the effect of δ_h on R for all analysis cases. The results show that an increase in δ_h leads to increasing R. Evaluation of failure mechanisms showed that increasing δ_h, on the one hand, leads to increasing the failure paths in the MSW slope and on the other hand leads to a decrease in FS_st values. Thus, the simultaneous effect of these two parameters leads to expanding the failure surface and increasing R. The effect of δ_h on R increases from case 1 to case 4 and decreases thereafter. When MSW data are highly scattered, and the random fields are highly variable, the failure mechanisms are often deep, the volume of sliding mass is slightly variable, and the increase in δ_h does not affect them. Indeed, COV_P is the controller of the failure mechanisms, rather than δ_h and δ_v. Maximum variation in R occurs in the δ_h of 0.5 to 10 m, followed by a slight variation in R with δ_h. Figure 8b presents the effect of δ_v on R for analysis case 4. The results show that with increasing δ_v, R also increases, and the maximum variation in R occurs in the δ_v of 0.5 and 1 m, the maximum of which is about 80% for δ_h = 0.5 m. All analysis cases showed that the effect of δ_v decreases with increasing COV_P. In this respect, in analysis case 1, δ_v has the maximum effect (500% increase in R), while in analysis case 7, the maximum increase in R is about 15%.

Studying failure mechanisms in probabilistic analyses showed that probabilistic failure mechanisms are either nonlinear curves or a combination of curves and several lines. This issue is unlike conventional analyses, in which the failure mechanisms have a specific shape. Due to the random values of the input parameters, the failure mechanism is formed in such a way that it passes through the weakest points (low shear strength), and also the overall strength is minimal on the failure surface. These two factors lead to the high effect of CLs on the shape of the failures and, consequently, the risk of landfill slope.

Figure 9 exhibits the sliding mass histogram for 2000 MCSs in case 7 with δ_h = 100 and δ_v = 1m. In Fig., all simulations are included in the left histogram, while the right histogram only includes simulations in which the slope has failed. For all simulations, a continuous range of sliding mass is observed, with the most sliding masses occurring around 2200 (close to the deterministic value) with an average of 2079. However, in the case of slope failure, discontinuity is observed in the volume of sliding mass, and the maximum failure frequency occurs in the range of 1800 (with a mean value of 1789.87 m²). In general, the scatter of all simulations (left histogram) is greater than the failed cases. Moreover, the volume of the sliding mass varies over a wide range (i.e., shallow to deep failure mechanisms).

Fig. 9

Histogram of sliding mass in analysis case 7 (δ_h = 100 and δ_v = 1m)

The histogram of failed simulations demonstrates that for failure mechanisms (FS < 1), most slip masses are smaller than the deterministic value. In addition, increasing the COV_P increases the number of simulations with a smaller sliding mass than the deterministic value. This difference is attributed to the formation of shallow and localized failure mechanisms due to the high spatial variability of the input parameters.

Finally, Fig. 10 shows the relationship between the three parameters FS_st, P_f, and R for all probabilistic analyses. The results show that from a probabilistic point of view, if obtained FS_st > 1.1, the P_f < 40%, but R is low (< 250 m²). But, for 1 < FS_st < 1.1, P_f does not change much, while the R increases sharply (up to about 1200). Also, for FS_st < 0.9, P_f increases rapidly, but variations of R are uniform and ascending. The results showed that the relationship between FS_st and P_f is almost linear (for 0.7 < FS_st < 1.2), but the relationship of R with FS_st and P_f is not linear and has a rapid upward trend in the range of FS_st = 1.2 to FS_st = 0.8, and for FS_st < 0.8, the increase rate of R decreases to reach a constant value in the range of 0.5 < FS_st < 0.7. In particular, for FS_st < 0.8, the R is constant for failed slopes, but their P_f values are different.

Fig. 10

Correlation of R with Fs and P_f for probabilistic landfill slope analyses

Conclusion

Instability and the risk involved in landfill sites are important issues in the geotechnical design of landfills, especially for those designed without paying attention to engineering considerations. The present study investigates the importance of incorporating the spatial variability of geotechnical properties of MSW in assessing the stability, probability of failure, and risk of landfill slopes. In this paper, the effect of spatial variability of MSW parameters was evaluated on FS_st, P_f, and R. RFDM, an advanced FDM-based probabilistic method that combines random fields with a finite-difference code, was used to perform a landfill risk assessment. The RFDM code can accurately determine the critical failure surface by considering multiple failure mechanisms caused by the spatial variability of MSW parameters. Compared to analytical methods, this method does not consider any initial assumptions to determine the mechanism of landfill slope failure. Therefore, this method can be used to assess the landfill slope stability with multiple potential failure mechanisms.

• The results showed that spatial variability has a significant effect on FS_st, P_f, and R in the landfill slope. For all cases, the FS_st value was lower than the FS_det, and not considering the variability in the input parameters of MSW leads to an overestimated FS. Although values of FS_st provide a probabilistic criterion for assessing the stability and safety of landfill slopes, they do not provide any quantitative information about the probability of failure and system performance levels.

• Therefore, P_f of landfill slopes must be calculated to determine the level of performance of landfill slopes from a geotechnical stability perspective. Calculating the P_f for all simulations in each case showed that increasing COV_P leads to increasing P_f and decreasing COV_P(f). Evaluations also showed that when COV_P values are low, δ_h and δ_v have a great effect on P_f and their effect decreases with increasing COV_P. Comparing these results with USAS recommendations showed that for all analyses (except case 1), the performance of all cases was less than hazardous. As a result, contrary to the FS, the P_f well determines the performance of the landfill slope system and allows comparing the probability of its instability with recommended criteria.

• The results showed that although the P_f determines the level of performance of landfill slopes, the risk associated with these slopes must be calculated to determine the damage due to the possible failure of landfill slopes. The results indicate that the volume of MSW sliding mass can be considered a measure of risk in each slope; however, the accurate determination of slope failure risk requires complete data and information about the landfill. The results showed that with increasing COV_P, the value of R increases for all cases, but the maximum increase occurs in cases 1–4, and a further increase in COV_P has a low influence on the slope failure risk. Comparing the results of the deterministic analysis with the probabilistic results shows that R in all probabilistic analyses is less than the deterministic value (≈ 2118 m²). However, it should be noted that the deterministic method proposes a failure mechanism with FS = 1.26. Also, from a conventional point of view, the slope may never fail, and the obtained failure mechanism may be very different from the actual slope failure mechanism. The results show that the autocorrelation structure significantly affects the risk of slope failure, which depends primarily on the P_f variations.

• Finally, when there is no accurate data on the spatial variability of MSW parameters and their uncertainty is high, reliability-based analyses can predict landfills’ performance efficiently and make the risk associated with them more perceivable for geotechnical engineers.