Stability Grade Evaluation of Slope with Soft Rock Formation in Open-Pit Mine Based on Modified Cloud Model

Abstract

In recent years, the frequent occurrence of slope failures has brought the issue of slope problems to the forefront of widespread public concern, which significantly impedes progress toward the secure and sustainable development of open-pit mines. And, high and steep slopes of weak rock strata, being a more complex type of slope, pose a greater potential for danger. In order to ensure the reliability of the safety evaluation results of the high and steep open-pit slope containing soft rocks, an evaluation index system with quantized grade intervals was created based on the thought of an analytic hierarchy process, and the MATLAB R2021a was used to calculate the numerical characteristic values of the cloud model. Then, a standard stability cloud model based on cloud theory was established. With the opening pit mine as an example, its slope stability practice cloud image was generated and the similarity between this image and the grades of the cloud model was calculated through the ECM algorithm to effectively identify the stability and verify the scientificity and validity of the model. The results show that the similarity between the practice cloud image and the standard stability cloud image for the total evaluation of the stability of an open-pit mine is 0.021, 0.279, 0.594, and 0.106, respectively. The slope stability is at grade C, which is basically consistent with the numerical simulation and the analysis results of the traditional limit equilibrium method, verifying that the model is scientific and effective to a certain extent. The method provides substantial guidance to ensure production safety in this specific open-pit mine. It provides ideas and means for other similar complex slope stability analysis and prevention. Meanwhile, it promotes the safe and sustainable development of open-pit mines.

1. Introduction

The frequency of landslide instability accidents occurring in open-pit mines has increased due to the expanding number and scale of mining operations in recent years, which significantly impedes progress toward the secure and sustainable development of open-pit mines. According to incomplete statistics (as shown in Figure 1c), weak rock formations have caused over 500 incidents of high steep slope instability across approximately 50,000 open pit mines in China from 2014 to 2023. These accidents have resulted in more than 600 fatalities and over 1000 injuries, accompanied by direct economic losses exceeding CNY 10 billion. For example, a large-scale collapse accident occurred at an open-pit coal mine located in Alxa Left Banner, Alxa League, Inner Mongolia Autonomous Region (as shown in Figure 1a), claiming over 50 lives in February 2023. Similarly, another incident took place within Baiyin City, Gansu Province (as shown in Figure 1b) in July 2022, which led to more than ten fatalities along with numerous injuries and caused substantial adverse societal repercussions. Therefore, the evaluation of slope stability in groups of soft rock masses with steep slopes is particularly important for ensuring safety in open-pit mining and remains a challenging and prominent research topic at present.

The stability of these slopes has been investigated by numerous scholars using various methodologies. The utilization of conventional numerical simulation methods, limit equilibrium methods, and model testing methods is essential in the field. For example, Yizhou Zhuang et al. [1] utilized the strength reduction method to perform a numerical analysis on slopes containing weak layers in order to assess the stability of landslides under both natural and extreme rainstorm conditions. Dai Zhangjun et al. [2] developed an experimental model that simulated a slope with a vulnerable interlayer and subsequently conducted rainfall infiltration experiments on an expansive soil slope to examine its stability. Xu Peng et al. [3] have developed a calculation model that utilizes the limit analysis method to investigate slope stability affected by clusters of weak layers. Unconventional approaches are currently being developed. Such as, Kai Cui [4] employed the partial least squares method to establish a predictive analysis model for analyzing slope stability in the presence of weak layers. Yuan Jiahua et al. [5] combined traditional plastic theory with generalized plastic theory to apply the upper bound limit analysis method for calculating the stability coefficient of slopes that contain weak layers and so on. The assessment of slope stability has been enhanced by the advancement of scientific theory and the growing trend toward interdisciplinary integration, progressing from qualitative and quantitative methods to uncertain analytical techniques and comprehensive analytical approaches. The application of novel theories, such as fuzzy mathematics (FM), grey correlation (GRA), genetic algorithms(GAOT), and mutation theory, significantly enhances the evaluation methods employed for assessing slope stability by establishing a series of innovative approaches. For example, the enhanced grey relational analysis method was employed by Shuai-hua et al. [6] to investigate factors that affect slope stability, resulting in increased sensitivity. Based on the enhanced mutation theory, Wu Xin et al. [7] have developed a comprehensive approach to evaluate slope stability using an actual engineering case study. Jin Aibing et al. [8] developed an intelligent prediction model for slope instability using SSA-SVM with the aim of achieving accurate predictions of slope instability and so on.

However, due to the special geological structure of the soft weak layer group in open-pit mines with high and steep slopes, the overall stability of the slope is, to some extent, determined by this soft weak layer group [9], which is more sensitive to human interference or unavoidable natural factors [10]. Therefore, the study of slope stability becomes complex and comprehensive. At present, the numerical simulation method for this type of slope is not as realistic as it should be and the limit equilibrium method encounters difficulties in establishing models and complexity in calculations. Other analytical methods are generally insufficiently comprehensive enough to form a cohesive system. Furthermore, the assessment of slope engineering stability is time sensitive and dynamic as the project progresses, and the conventional calculation methods are intricate and heavily reliant on experiential knowledge. Additionally, this type of slope stability evaluation involves numerous influencing factors and complex relationships, making the process inherently unpredictable and ambiguous.

To address the limitations in assessing the stability of high, steep complex slopes that contain soft rock formations, in this study, the cloud theory ideology proposed by Academician Li Deyi [11], which is widely utilized in risk level assessment, stability evaluation, data prediction, and mining excavation, was introduced to construct a modified cloud model for comprehensively evaluating the stability of high and steep slopes with weak rock strata in open-pit mines. In order to ensure the reliability of the safety evaluation results of the high and steep open-pit slope containing soft rocks, an evaluation index system with quantized grade intervals was created based on the thought of an analytic hierarchy process. By applying cloud theory, the vague correlation between qualitative and quantitative data can be transformed into a precise mathematical model using cloud modeling techniques. The establishment of a comprehensive evaluation model overcomes the complexity, ambiguity, and unpredictability associated with the evaluation process.

2. Establishment of the Slope Stability Evaluation System

The soft layer group slope refers to the interlayer geological slope controlled by one or more soft structural planes, and its stability is influenced by various internal and external factors characterized by inherent uncertainty. It is essential to establish a practical, rational, and precise evaluation methodology. Therefore, after conducting an extensive literature review [12,13,14,15,16,17,18], we built the influencing factors of high steep slope stability in open mines containing soft rock strata and established an evaluation system for slope stability using the analytic hierarchy process (AHP). The primary focus of this evaluation system is to assess the stability of complex slopes with high steepness in open mines characterized by soft rock strata. The first-level index layer encompasses factors such as the slope soft strata group, the geometric properties of the slope, the engineering geology, the meteorological hydrology, and other relevant factors. These first-level indexes are further classified into a second-level index layer representing specific influencing factors related to slope stability. We constructed a main element layer consisting of five first-level evaluation indexes and fifteen second-level evaluation indexes, forming a four-level assessment scale (A: Stable; B: Understable; C: Unstable; D: Extremely-unstable) for evaluating open mine slope stability (as shown Figure 2).

The established evaluation system for open mine slope stability classifies the factors influencing stability into two levels, which consist of five categories and a total of fifteen items. Further, the standard intervals for each grade and index are established through meticulous sorting and rigorous analysis, ensuring the utmost precision in our methodology. The details are as follows:

(1)

The Slope Soft Strata Group: The soft layer group is classified based on the number of layers composed of soft rock. The distribution of weak layers is based on their impact on slope stability, which includes four distinct types: reverse slopes, oblique slopes, consequent slopes, and comprehensive slopes that exhibit varying characteristics. The characteristics of weak rock layers are evaluated and assigned scores based on their characteristics, which are categorized into four groups: general quality, low quality, very low quality, and extremely low quality [19,20].

(2)

The Geometric Properties of Slope: The slope height is categorized into intervals of 50 m, 100 m, and 150 m on steep slopes with varying heights of location. The slope gradient is divided into three categories: 20 degrees, 45 degrees, and 60 degrees. The slope angle is classified into three distinct categories: 10 degrees, 25 degrees, and 45 degrees.

(3)

The Engineering Geology: The rock and soil types are classified into four categories—excellent, good, medium, and poor—based on the integrity and strength of the rock mass. The classification of the structural plane development is based on the degree of geological deficiencies in fractures and joints, specifically categorized as 10%, 35%, and 50%. The internal friction coefficients are categorized as 0.6, 0.4, and 0.2 [20,21,22,23].

(4)

The Meteorological Hydrology: The maximum daily rainfall is classified into three levels: 30 mm, 80 mm, and 150 mm. The frequency of annual rainstorm days is classified into three categories: 5, 15, and 30 days. The seepage water from the slope is classified into three groups based on their respective flow rates of 5 m/d, 15 m/d, and 25 m/d [20,22,24].

(5)

The Other Factors: The level of excavation disturbance is classified into four levels based on the slope excavation technique used and the extent of destabilization. The degree of rock and soil weathering is classified based on the overall weathering ratio, which comprises three grades: 5%, 15%, and 30%. The vegetation coverage rate is classified based on the slope’s vegetation coverage, specifically as 40%, 25%, and 10% [23,25].

The specific evaluation criteria, derived from the established evaluation system, are outlined in

Table 1. Standards for each evaluation index.

Evaluation Index		Estimation Scale
The First Level	The Second Level	A	B	C	D
		Stable	Understable	Unstable	Extremely-Unstable
The Slope Soft Strata Group	The Weak Layer Number	<2	2	3	≥4
	The Distribution of Weak Layers (0–100)	80~100	60~80	30~60	0~30
	The Characteristics of Weak Layers (0–100)	80~100	60~80	30~60	0~30
The Geometric Properties of Slope	The Slope Height (m)	<50	50~100	100~150	>150
	The Slope Grade (°)	0~20	20~45	45~60	60~90
	The Slope Angle (°)	<10	10~25	25~45	>45
The Engineering Geology	The Rock Type (0–100)	80~100	60~80	30~60	0~30
	The Structural Plane Development (%)	<10	10~35	35~50	>50
	The Internal Friction Factor	>0.6	0.6~0.4	0.4~0.2	<0.2
The Meteorological Hydrology	The Maximum Daily Rainfall (mm)	0~30	30~80	80~150	>150
	The Annual Rainstorm Days	<5	5~15	15~25	>25
	The Slope Seepage (m/d)	<5	5~15	15~30	>30
The Other Factors	The Excavation Disturbance (0–100)	0~10	10~40	40~70	70~100
	The Degree of Rock Weathering (%)	<5	5~15	15~30	>30
	The Vegetation coverage (%)	>40	25~40	25~10	<10

.

3. Weight Calculation of the Stability Evaluation Index

3.1. The Weight Calculation Ideas

The stability of high-steep open mine slopes with weak rock strata is influenced by a complex interaction among numerous factors, highlighting the necessity to develop a scientifically and logically weighted assessment index, and the optimization is the following basic steps (as shown in Figure 3): (1) The Establish a Judgment Matrix Group: Weight vectors are obtained by evaluating the relative importance of each index and constructing a judgment matrix using an evaluation system designed for analyzing slope stability in open pit mines with high steep slopes consisting of soft rock formations, and the construction process utilizes a 1–9 scale methodology [26] to facilitate indicator comparison. (2) The Weight Index Calculation: The characteristic value of the judgment matrix is calculated using Matlab, and then the maximum eigenvalue and its corresponding eigenvector are determined. Finally, the weight allocation for each index needs to be reckoned. (3) The Consistency Test of Calculation Results: The consistency index is determined by evaluating the maximum eigenvalue of the judgment matrix, and any judgment matrix that does not meet the consistency criteria will be adjusted accordingly [27]. (4) The Calculate Ranking of the Index Weight: The significance of each influencing factor on slope stability is determined by calculating, sorting, and analyzing the weight of the index layer in the target layer.

3.2. The Establish A Judgment Matrix Group

The evaluation index system for slope stability of high-steep slopes with soft rock stratum groups in open mines utilizes the 1–9 scale [26] method (as shown in

Table 2. 1–9 scaling method.

Number	Meaning	Number	Meaning	Number	Meaning
1	The Xi has the same effect as the Xj	3	Effect of Xi is slightly stronger than Xj	5	Effect of Xi is stronger than Xj
7	The effect of Xi is significantly stronger than Xj	9	The Xi is absolutely stronger than Xj	2, 4, 6, 8	Intermediate values of two adjacent odd scales

) to establish a judgment matrix for both the main-factor and sub-factor index layers, incorporating expert opinions from four professionals specialized in slope engineering.

Based on the evaluation system and Table 2, a relevant judgment matrix was established and the calculation method is as follows:

(1) Construct the judgment matrix for the primary index layer based on the target layer.

$$MA=\left(\begin{array}{ccccc}B1,1& B1,2& B1,3& B1,4& B1,5\\ B2,1& B2,2& B2,3& B2,4& B2.5\\ B3,1& B3,2& B3,3& B3,4& B3,5\\ B4,1& B4,2& B4,3& B4,4& B4,5\\ B5,1& B5,2& B5,3& B5,4& B5,5\end{array}\right)=\left(\begin{array}{ccccc}1& 2& 4& 4& 5\\ \frac{1}{2}& 1& 2& 2& 5\\ \frac{1}{4}& \frac{1}{2}& 1& 3& 4\\ \frac{1}{4}& \frac{1}{2}& \frac{1}{3}& 1& 3\\ \frac{1}{5}& \frac{1}{5}& \frac{1}{4}& \frac{1}{3}& 1\end{array}\right)$$

(2) Construct the judgment matrix for the subelement index layer based on the target layer.

① The Slope Soft Strata Group:

$$M_{B1}=\left(\begin{array}{ccc}{C}_{1,1}& C_{1,2}& C_{1,3}\\ C_{2,1}& C_{2,2}& C_{2,3}\\ C_{3,1}& C_{3,2}& C_{3,3}\end{array}\right)=\left(\begin{array}{ccc}1& 3& 4\\ \frac{1}{3}& 1& 2\\ \frac{1}{4}& \frac{1}{2}& 1\end{array}\right)$$

② The Geometric Properties of Slope:

$$M_{B2}=\left(\begin{array}{ccc}{C}_{4,4}& C_{4,5}& C_{4,6}\\ C_{5,4}& C_{5,5}& C_{5,6}\\ C_{6,4}& C_{6,5}& C_{6,6}\end{array}\right)=\left(\begin{array}{ccc}1& 2& 3\\ \frac{1}{2}& 1& 2\\ \frac{1}{3}& \frac{1}{2}& 1\end{array}\right)$$

③ The Engineering Geology:

$$M_{B3}=\left(\begin{array}{ccc}{C}_{7,7}& C_{7,8}& C_{7,9}\\ C_{8,7}& C_{8,8}& C_{8,9}\\ C_{9,7}& C_{9,8}& C_{9,9}\end{array}\right)=\left(\begin{array}{ccc}1& \frac{1}{4}& \frac{1}{5}\\ 4& 1& \frac{1}{2}\\ 5& 2& 1\end{array}\right)$$

④ The Meteorological Hydrology:

$$M_{B4}=\left(\begin{array}{ccc}{C}_{10,10}& C_{10,11}& C_{10,12}\\ C_{11,10}& C_{11,11}& C_{11,12}\\ C_{12,10}& C_{12,11}& C_{12,12}\end{array}\right)=\left(\begin{array}{ccc}1& \frac{1}{3}& \frac{1}{5}\\ 3& 1& \frac{1}{2}\\ 5& 2& 1\end{array}\right)$$

⑤ The Other Factors:

$$M_{B5}=\left(\begin{array}{ccc}{C}_{13,13}& C_{13,14}& C_{13,15}\\ C_{14,13}& C_{14,14}& C_{14,15}\\ C_{15,13}& C_{15,14}& C_{15,15}\end{array}\right)=\left(\begin{array}{ccc}1& 3& 4\\ \frac{1}{3}& 1& 2\\ \frac{1}{4}& \frac{1}{2}& 1\end{array}\right)$$

3.3. Calculate the Judgment Matrix and the Weight of Each Index

The eigenvector corresponding to the maximum eigenvalue of the judgment matrix was calculated using matrix operations in Matlab. After normalization, we obtained the weight vector ω_i, which served as a fundamental basis for computing indices. In order to assess the consistency of the matrix, a consistency ratio (CR) based on Equation (1) was introduced [27,28,29]:

$$CR=\frac{CI}{RI}$$

(1)

The formula represents CI as the consistency index, which was calculated using the equation CI = (λ_max − n)/(n − 1). The RI refers to the random consistency index. The calculation results of weight ω_i, consistency ratio CR and outcomes of the consistency test are presented in

Table 3. Results of the judgment matrix calculation.

Judgment Matrix (M)	Indicator Weigh (ωi)	Consistency Ratio	Consistency Check
MA	[0.4330, 0.2404, 0.1737, 0.1029, 0.0501]	0.0543	CR < 0.1
MB1	[0.6250, 0.2385, 0.1365]	0.0176	CR < 0.1
MB2	[0.5396, 0.2970, 0.1634]	0.0088	CR < 0.1
MB3	[0.0974, 0.3331, 0.5695]	0.0236	CR < 0.1
MB4	[0.1095, 0.3090, 0.5816]	0.0036	CR < 0.1
MB5	[0.6250, 0.2385, 0.1365]	0.0176	CR < 0.1

.

3.4. Calculate the Total Ranking of Index Weights

The weights of each factor in the index were calculated to assess the level of impact exerted by individual factors on this particular type of slope. ω_ACj represents the weight assigned to scheme layer C in target layer A [24]:

$${\omega }_{ACj}={\displaystyle \sum _{l=1}^m{\omega }_A{B}_L{\omega }_{BCj}}$$

(2)

This formula can be explained as follows: j = 1, 2, ... k; l = 1, 2, ... m. ω_A represents the weight coefficient of the primary indicator within the target layer, while B_L denotes the judgment matrix. ω_BCj indicates the weighting factor of secondary indicator layer C within primary indicator layer B. The weight results for each index layer C in the target layer A were calculated using Matlab according to Formula (2) and then they are sorted and organized as shown in Figure 4.

Based on our weight-based analysis, we have identified several significant factors that exert considerable influence on the stability of high steep slopes characterized by weak rock strata within open pits. These influential factors include (1) the weak layer number; (2) the slope height; (3) the distribution of weak layers; (4) the internal friction factor; (5) the slope grade; (6) the slope seepage. When compared to conventional slopes, it becomes apparent that both the existence and attributes associated with feeble stratifications play pivotal roles in determining slope stability outcomes. Therefore, assigning priority to these primary influencing elements is essential during analyses and implementation phases aimed at devising preventive measures for high steep slopes featuring compromised rock formations within open pit mining operations.

4. Construct a Comprehensive Modified Cloud Model for Slope Stability

4.1. Cloud Model Theory

The cloud model is a model proposed based on fuzzy mathematics and stochastic mathematics theory to realize the qualitative and quantitative transformation of uncertain concepts, which can transform the uncertain relationship between qualitative and quantitative into an intuitive and universal method [30]. In the number field space, the cloud model is a one-to-many mathematical mapping image. When establishing the mathematical model of the cloud model, three basic digital features, the expected value (Ex), entropy value (En), and hyperentropy value (He), are introduced and their geometric meanings are shown in Figure 5a.

In this study, we use a forward cloud model to quantitatively map qualitative information expressed through linguistic values into range and distribution rules. By incorporating randomness and fuzziness, we can effectively utilize a normal cloud generator to quantify slope stability and determine the range of qualitative indicators [31]. Subsequently, we calculated the digital characteristics of the cloud model by applying Formula (3) [32,33]:

$$\begin{array}{l}{E}_{\mathrm{x}}=\frac{C_{\max }+C_{\min }}{2}\\ E_n=\frac{C_{\max }-C_{\min }}{6}\\ H_e=k\end{array}\}$$

(3)

C_max represents the highest value within a specified range for each risk index, which is determined by its current situation. C_min represents the lowest value within a specified range for each risk index, also determined by its current situation. k indicates the circulatory display point and is assigned based on specific requirements.

According to the fundamental principles of cloud theory, the ith cloud droplet is generated by a cloud generator. By utilizing Matlab computational software and implementing Equation (4) through programming, precise coordinates [X(1, i), Y(1, i)] can be obtained for each individual cloud droplet [34,35]:

$$\begin{array}{l}X(1,i)=normrnd(E_n,\sigma ,1,1)\\ Y(1,i)=\exp \left(\frac{-{\left[X(1, i)-Ex\right]}^2}{2{\sigma }^2}\right)\\ \sigma =normrnd(E_n,H_e^2,1,1)\end{array}\}$$

(4)

where σ represents the standard deviation, E_x denotes the expected value, E_n signifies the entropy value, and H_e indicates the hyperentropy value.

4.2. Calculate the Digital Feature Value

According to the previous study, we established the stability evaluation system specifically designed for open pit mining operations on high-steep slopes with weak rock strata (as shown in Table 1). The mathematical processing was carried out using the Matlab calculation program to standardize index grades based on the evaluation grade Table 1 for each influencing factor. Subsequently, Equation (3) was applied to calculate the modified cloud digital characteristics (Ex, En, He) corresponding to five main primary layer evaluation grades, as shown in

Table 4. Calculation of standard cloud digital feature evaluation indicators.

Primary Indicators	Stability Level	Evaluation Index		Evaluation Index (Normalized)		Ex	En	Ee
		Maximum	Minimum	Maximum	Minimum
C1	A	2	0	0.333	0.000	0.1665	0.0555	0.01
	B	3	2	0.500	0.333	0.4165	0.0278	0.01
	C	4	3	0.667	0.500	0.5835	0.0278	0.01
	D	6	4	1.000	0.667	0.8335	0.0555	0.01
C2	A	100	80	1.000	0.800	0.9000	0.0333	0.01
	B	80	60	0.800	0.600	0.7000	0.0333	0.01
	C	60	30	0.600	0.300	0.4500	0.0500	0.01
	D	30	0	0.300	0.000	0.1500	0.0500	0.01
C3	A	100	80	1.000	0.800	0.9000	0.0333	0.01
	B	80	60	0.800	0.600	0.7000	0.0333	0.01
	C	60	30	0.600	0.300	0.4500	0.0500	0.01
	D	30	0	0.300	0.000	0.1500	0.0500	0.01
C4	A	50	0	0.167	0.000	0.0835	0.0278	0.01
	B	100	50	0.333	0.167	0.2500	0.0277	0.01
	C	150	100	0.500	0.333	0.4165	0.0278	0.01
	D	300	150	1.000	0.500	0.7500	0.0833	0.01
C5	A	20	0	0.222	0.000	0.1110	0.0370	0.01
	B	45	20	0.500	0.222	0.3610	0.0463	0.01
	C	60	45	0.667	0.500	0.5835	0.0278	0.01
	D	90	60	1.000	0.667	0.8335	0.0555	0.01
C6	A	10	0	0.111	0.000	0.0555	0.0185	0.01
	B	25	10	0.278	0.111	0.1945	0.0278	0.01
	C	45	25	0.500	0.278	0.3890	0.0370	0.01
	D	90	45	1.000	0.500	0.7500	0.0833	0.01
C7	A	100	80	1.000	0.800	0.9000	0.0333	0.01
	B	80	60	0.800	0.600	0.7000	0.0333	0.01
	C	60	30	0.600	0.300	0.4500	0.0500	0.01
	D	30	0	0.300	0.000	0.1500	0.0500	0.01
C8	A	10	0	0.100	0.000	0.0500	0.0167	0.01
	B	35	10	0.350	0.100	0.2250	0.0417	0.01
	C	50	35	0.500	0.350	0.4250	0.0250	0.01
	D	100	50	1.000	0.500	0.7500	0.0833	0.01
C9	A	0.8	0.6	1.000	0.750	0.8750	0.0417	0.01
	B	0.6	0.4	0.750	0.500	0.6250	0.0417	0.01
	C	0.4	0.2	0.500	0.250	0.3750	0.0417	0.01
	D	0.2	0	0.250	0.000	0.1250	0.0417	0.01
C10	A	30	0	0.100	0.000	0.0500	0.0167	0.01
	B	80	30	0.267	0.100	0.1835	0.0278	0.01
	C	150	80	0.500	0.267	0.3835	0.0388	0.01
	D	300	150	1.000	0.500	0.7500	0.0833	0.01
C11	A	5	0	0.100	0.000	0.0500	0.0167	0.01
	B	15	5	0.300	0.100	0.2000	0.0333	0.01
	C	25	15	0.500	0.300	0.4000	0.0333	0.01
	D	50	25	1.000	0.500	0.7500	0.0833	0.01
C12	A	5	0	0.100	0.000	0.0500	0.0167	0.01
	B	15	5	0.300	0.100	0.2000	0.0333	0.01
	C	30	15	0.600	0.300	0.4500	0.0500	0.01
	D	50	30	1.000	0.600	0.8000	0.0667	0.01
C13	A	10	0	0.100	0.000	0.0500	0.0167	0.01
	B	40	10	0.400	0.100	0.2500	0.0500	0.01
	C	70	40	0.700	0.400	0.5500	0.0500	0.01
	D	100	70	1.000	0.700	0.8500	0.0500	0.01
C14	A	5	0	0.050	0.000	0.0250	0.0083	0.01
	B	15	5	0.150	0.050	0.1000	0.0167	0.01
	C	30	15	0.300	0.150	0.2250	0.0250	0.01
	D	100	30	1.000	0.300	0.6500	0.1167	0.01
C15	A	100	40	1.000	0.400	0.7000	0.1000	0.01
	B	40	25	0.400	0.250	0.3250	0.0250	0.01
	C	25	10	0.250	0.100	0.1750	0.0250	0.01
	D	10	0	0.100	0.000	0.0500	0.0167	0.01

.

To calculate the digital characteristic value of a cloud model modified by a comprehensive index, the calculation Formula (5) was introduced [36]:

$$\left(\begin{array}{ccc}{E}_x^{·}& E_e^{·}& H_n^{·}\end{array}\right)=({\omega }_1,{\omega }_2\cdots {\omega }_n)=\left(\begin{array}{ccc}{E}_{x1}& E_{n1}& H_{e1}\\ E_{x2}& E_{n2}& H_{e2}\\ ⋮& ⋮& ⋮\\ E_{xk}& E_{nk}& H_{ek}\end{array}\right)$$

(5)

where ω_i is the weight vector and [E_xk, E_nk, H_ek] represents the digital characteristic values of the cloud model.

The refined cloud digital features (Ex, En, He) for each assessment level were calculated based on the data provided in Table 4. These features were then combined with the indicator weights specified in Table 3 to determine the comprehensive refined cloud digital characteristics. Furthermore, Equation (5) was used to further enhance these characteristics, and the resulting outcomes are presented in

Table 5. Comprehensive correction of the cloud digital characteristics of each indicator.

Evaluating Indicator		Evaluation Standard Grade
		A	B	C	D
		Stable	Understable	Unstable	Extremely-Unstable
Primary Indicators	B1	[0.3896, 0.0299, 0.01]	[0.4187, 0.0299, 0.01]	[0.4292, 0.0361, 0.01]	[0.5250, 0.0708, 0.01]
	B2	[0.0870, 0.0290, 0.01]	[0.2229, 0.0333, 0.01]	[0.4616, 0.0293, 0.01]	[0.7747, 0.0751, 0.01]
	B3	[0.3026, 0.0325, 0.01]	[0.4991, 0.0409, 0.01]	[0.3990, 0.0369, 0.01]	[0.6204, 0.1038, 0.01]
	B4	[0.0500, 0.0167, 0.01]	[0.1982, 0.0327, 0.01]	[0.4273, 0.0436, 0.01]	[0.4302, 0.0736, 0.01]
	B5	[0.1328, 0.0261, 0.01]	[0.2245, 0.0386, 0.01]	[0.4213, 0.0406, 0.01]	[0.6931, 0.0614, 0.01]
Overall Indicators		[0.1924, 0.0268, 0.01]	[0.3127, 0.0351, 0.01]	[0.4277, 0.0373, 0.01]	[0.6087, 0.0769, 0.01]

.

4.3. Build a Modified Standard Cloud Model

The cloud digital characteristics of the overall index, which underwent modifications and calculations as presented in Table 3, were imported into the Matlab software package. A customized program was developed using Equation (5) to generate a standardized cloud model with an adjusted comprehensive index through computational operations involving 6000 cloud droplets. This model served as the basis for evaluating slope stability on high-gradient slopes within open pit mines that encompass formations of weak rock strata groups during real-world engineering endeavors, as illustrated in Figure 5b.

4.4. Similarity Calculation Theory and Cloud Evaluation

The evaluation of the cloud model was calculated using the expectation-based cloud model (ECM) algorithm proposed by Li Hailin et al. [37], which served as a method for quantifying similarity between cloud models based on expectation curves. The fundamental principle involved evaluating the likeness between two cloud models by examining their overlapping areas. The basic formula for the calculation is as follows:

$$\begin{array}{l}ECM(C1,C2)=\frac{2S}{\sqrt{2}\pi (En1+En2)}\\ \{C1(Ex1,En1,He1),C2(Ex2,En2,He2)\}\end{array}\}$$

(6)

where (E_xi, E_ni, E_ei) represents the numerical characteristics of the cloud model. The S denotes the three-part area of overlap between the standard normal distribution and actual cloud, with S = S₁ + S₂ + S₃. The E_n1 and E_n2 refer to the areas formed by the expected curve of the normal cloud model and abscissa.

The stability assessment of slopes could then be enhanced by employing a novel approach based on cloud theory. Cloud maps were created using various engineering techniques and then compared and analyzed against a standardized model using the ECM algorithm. Subsequently, the maximum similarity criterion was employed to assess the grade of the slope stability in open pit environments characterized by high steepness and weak rock strata formations.

4.5. Revised Cloud Model Stability Evaluation Steps

According to the evaluation system and standard cloud model for soft rock layers in open-pit mines with high and steep slopes, the steps for assessing the stability of actual engineering slopes using the ECM algorithm are as follows (the flow is shown in Figure 6):

(1)

Conduct an on-site geological survey of the actual slope of the engineering project and analyze the available data using a slope stability evaluation system specifically designed for soft rock layers in open-pit mines. Subsequently, compile an engineering report.

(2)

Based on on-site investigation and data analysis, the relevant evaluation weights of the standard cloud model are adjusted and scores for evaluating high-slope slopes with soft weak rock layers in open-pit mines are scientifically and reasonably assigned. (This is usually performed by an expert group consisting of slope safety management professionals and field research personnel specializing in slopes).

(3)

The actual cloud model is established based on the scoring situation of the indicators using Matlab calculation tools according to Formulas (3) and (4). Formula (5) is used to calculate the similarity between the actual cloud model and the standard cloud model, determining the stability grade of the slope based on the criterion of maximum similarity.

5. Engineering Examples

5.1. Project Overview

The open pit mines showcase a substantial, profound, and steep slope with an approximate depth of 400 m. Encompassing an area of approximately 1.89 km², the pit descends to a depth of around 350 m, reaching its lowest point at an elevation of 1385 m above sea level. Geological exploration findings indicate that the predominant rock mass in the pit primarily consists of the RAT stratum.

According to a recent engineering geological survey and slope stability monitoring report, this study focuses on investigating the Xibang Slope. The engineering data indicate that there are two consecutive layers of soft materials present in an open pit slope, primarily composed of dolomite, shale, clay, and sandstone with unfavorable rock properties. Furthermore, cracks, joints, and other adverse geological conditions are present. The internal friction coefficient for this group of soft layers ranges from 0.12 to 0.46. Meteorological data statistics over the past fifteen years reveal that the maximum daily rainfall in the area ranged from 78 to 190 mm while the average annual number of rainstorm days varied between 13 and 28. Moreover, seepage water on the slopes varies between 1.08 and 22.87 m/day. It should be noted that rock and soil weathering is severe with vegetation coverage being less than 15% as depicted in Figure 7.

5.2. Calculate the Evaluation Index of Digital Characteristics

Four experts with specialized knowledge in slope engineering were invited to assess and assign scores to various indicators related to slope conditions and their assessment was based on the evaluation index system for appraising slope stability in high steep slopes with soft rock strata groups within open pits. Then, the obtained results were subsequently adjusted and analyzed using Formula (3) to derive the numerical characteristics for each evaluation index, as presented in

Table 6. Cloud digital characteristics of secondary evaluation indicators.

Primary Indicators	Evaluation Index		Evaluation Index (Normalized)		Ex	En	Ee
	Maximum	Minimum	Maximum	Minimum
C1	3	0	0.5	0	0.2500	0.0833	0.0050
C2	65	15	0.65	0.15	0.4000	0.0833	0.0050
C3	55	10	0.55	0.1	0.3250	0.0750	0.0050
C4	347	118	0.8233	0.3933	0.6083	0.0717	0.0050
C5	84.3	48.6	0.9367	0.54	0.7384	0.0661	0.0050
C6	65	40	0.7223	0.4445	0.5834	0.0463	0.0050
C7	65	25	0.65	0.25	0.4500	0.0667	0.0050
C8	75	45	0.75	0.45	0.6000	0.0500	0.0050
C9	0.46	0.12	0.575	0.150	0.3625	0.0708	0.0050
C10	190	78	0.63334	0.260	0.4467	0.0622	0.0050
C11	28	13	0.5600	0.2600	0.4100	0.0500	0.0050
C12	22.87	1.08	0.4574	0.0216	0.2395	0.0726	0.0050
C13	80	20	0.800	0.200	0.5000	0.1000	0.0050
C14	40	25	0.400	0.250	0.3250	0.0250	0.0050
C15	12	3	0.12	0.03	0.0750	0.0150	0.0050

.

The comprehensive cloud digital characteristics of the five first-level evaluation indexes and the overall evaluation cloud digital characteristic value of all evaluation indexes were obtained using Matlab based on the calculation results for each index weight (Table 3) and the cloud digital characteristics from slope subfactor evaluation indexes (Table 5), combined with Equation (5). These findings are presented in

Table 7. Comprehensive cloud digital characteristics of the first-level evaluation indicators.

Primary Indicators		Ex	En	Ee
Actual digital characteristics of cloud in open pit mine	The Slope Soft Strata Group (B1)	0.2960	0.0822	0.0050
	The Geometric Properties of Slope (B2)	0.6429	0.0659	0.0050
	The Engineering Geology (B3)	0.4501	0.0635	0.0050
	The Meteorological Hydrology (B4)	0.3149	0.0645	0.0050
	The Other Factors (B5)	0.4002	0.0705	0.0050
Overall Indicators		0.4434	0.0726	0.0050

.

5.3. Generate the Actual Cloud Map of the Cloud Model

The digital value of the total stability index for high steep slopes containing weak rock strata in the open pit, as presented in Table 7, was used to generate an actual cloud map (depicted in Figure 8a) consisting of 6000 cloud droplets. To enable a comprehensive evaluation and calculation using the ECM algorithm, a comprehensive cloud map was constructed based on the standard cloud map (shown in Figure 8b). The comprehensive cloud map reveals locally distributed cloud droplets within grades A, B, and D; whereas grade C exhibits concentrated cloud droplets. This preliminary determination suggests that the stability level of the open pit slope falls under grade C.

The ECM method is utilized for quantitative analysis of similarities between the actual cloud and the standard cloud. By employing Equation (6) depicted in Figure 9, Matlab computes their respective areas at all levels. Application of the maximum similarity criterion to data from Figure 5 indicates that the western slope of an open pit belongs to grade C, suggesting instability slopes.

5.4. Verification and Analysis

To ensure the accuracy and scientific validity of our research methodology, as well as to verify notable stability concerns and potential safety risks within the mine area’s slope, we identified a representative high-risk section (section A) on its western side based on findings from geological surveys and real-time slope monitoring data. Subsequently, a FLAC^3D 6.0 numerical simulation analysis was employed to investigate its failure mode while determining the slope stability factor (Fs) using the limit equilibrium method.

Based on the prevailing engineering conditions, a numerical simulation model was established using FLAC^3D software to simulate a typical A section of the west slope that contains a weak strata group. By conducting a comprehensive analysis of the numerical simulations, displacement and stress distribution patterns related to slope deformation were obtained and illustrated in Figure 10a. According to the numerical simulations, significant horizontal and overall displacements were observed in both quaternary rocks and heavily weathered RAT rock formations. These findings are supported by displacement maps illustrating movements along the x-axis as well as comprehensive displacement maps. The vector representation demonstrates that these geological units primarily experience settlement-induced deformations with a predominant tendency toward horizontal movement within the horizon plane. Additionally, examination of shear stress distribution reveals intensified stresses at the foundation of steeply eroded RAT rocks on slopes, suggesting prevalent shear stress conditions in those areas. Considering localized occurrences of weak strata alongside gently inclined rock formations coupled with outcomes derived from stress–strain analyses conducted through numerical simulations, it is postulated that “settlement-sliding” along vulnerable strata constitutes a plausible mechanism for deformation and failure (Figure 10b).

After conducting numerical simulations, a comprehensive analysis was carried out on section A of the western slope. The Morgenstern–Price method [38] was utilized to determine the slope stability factor (Fs). Taking into account factors such as faults, fracture zones, tectonic movements, geological investigations, and preliminary on-site surveys, two scenarios were considered: weak interlayer penetration and nonpenetration. Consequently, the stability factor for section A of the mine boundary slope was calculated (

Table 8. The stability coefficient is calculated using the limit balance principle.

Section	Computing Method	Side Slope Scale	Slope Height	Slope Angle	Stability Coefficient (Fs)	Remark
A section	Extreme equilibrium method	Part	200 m	40°	1.198	The weak layer is not connected
					1.003	Weak layer connectivity
		Entirety	340 m	31°	1.046	The weak layer is not connected
					0.943	Weak layer connectivity
	Numerical Simulation (Intensity reduction)	Not Distinguished	350 m	29°–42°	1.001	No remarks

). The data in Table 8 indicate that, regardless of whether there is weak interlayer penetration or not in section A, the final slope stability factor fails to meet engineering safety requirements (greater than 1.05). Both numerical simulation results and limit equilibrium method calculations support each other’s findings. Section A has been chosen as representative due to its typicality; however, it poses significant safety risks regarding slope instability. Through numerical simulation and limit equilibrium analysis, it has been verified that the open pit slope exhibits a high potential risk for instability.

After conducting a thorough investigation into the slope, we observed numerous instances of local sliding, collapse, and cracking phenomena on the western side slope of the open pit mine. The presence of multiple cracks, fissures, and localized unstable slopes on this side poses a significant risk for extensive landslides. In conclusion, it can be inferred that the stability of the open pit mine’s slope is compromised. The analysis results obtained from our established model are scientifically sound and have considerable guiding significance in assessing the risks associated with complex soft layer slopes.

5.5. Comparison of Each Evaluation Method

The numerical simulation analysis method, the limit equilibrium calculation method, and the modified cloud model evaluation method all provide valuable insights into slope stability; however, there are significant disparities among these approaches. Constructing a numerical model using the numerical simulation approach presents challenges due to weak layers that cause discrepancies between simulated stress and strain results and real-world conditions. The limit equilibrium approach faces difficulties in determining an appropriate calculation section and failure mode, resulting in complex computations. Both methods oversimplify weak layers while failing to conduct a comprehensive assessment of slope stability.

But the refined cloud model evaluation method accurately incorporates various factors such as the distribution, number of layers, and properties of the weak layer group; it also considers geological, natural, and engineering disturbance factors. This comprehensive approach fully encompasses slope analysis while providing a relatively simple calculation process that can be dynamically adjusted based on actual engineering conditions. By addressing the ambiguity and randomness inherent in other evaluation methods for weak layer groups, this method offers relatively precise conclusions with high credibility.

6. Conclusions

• The stability evaluation system of slopes is established based on the analytic hierarchy process (AHP) by constructing a comprehensive correction cloud model for high-steep slopes containing weak rock strata. This model visually represents the actual situation of engineering slope stability through quantified grade intervals and provides recognition results for each index on slope stability grade through evaluation grades, standard cloud maps, actual cloud maps, comprehensive cloud maps, and cloud digital characteristics.
• The conclusions derived from numerical simulation and the limit equilibrium method regarding slope stability evaluation are consistent with those obtained through the comprehensive evaluation method of the modified cloud model, thereby validating the guiding role, scientific nature, and rationality of the modified cloud model in engineering practice. The method provides substantial guidance to ensure production safety in this specific open-pit mine.
• Through the analysis of weight values, key factors that influence the stability of slopes containing weak rock strata groups in high-steep open pits include the weak layer number, the slope height, the distribution of weak layers, the internal friction factor, the slope grade, and the slope seepage. In engineering construction, it is imperative to carefully consider these factors and implement effective treatment measures to ensure slope safety and stability.