Dynamic optimization design of open-pit mine full-boundary slope considering uncertainty of rock mass strength

Abstract

Rock and soil strength profoundly influences the stability of open-pit mine slopes. Traditional slope design, based on limited drilling data, often disregards inherent uncertainties. Effectively utilizing new sample information from mining operations poses a challenge, hindering dynamic and differentiated design for the entire perimeter slope. To address this, we propose a dynamic optimization method considering rock mass strength uncertainty for the entire perimeter slope. Our approach involves designing slope angles separately in different zones, while thoroughly considering decision-makers' preferences. Furthermore, we delegate the final adjustment authority of slope angles within the safety permissible range to on-site decision-makers. Compared to traditional methods, our dynamic design method incorporates rock mass strength uncertainty into slope evaluation while also accounting for decision-makers' safety and economic preferences. Through a case study of a specific open-pit mine, our proposed dynamic design method increases the overall slope angle by approximately 2.5°, fully accommodating the influence of on-site decision-making preferences on slope design. This article introduces a new method of dynamic optimization of open-pit mine slope based on simplified observation method, which improves the flexibility of decision-making and realizes the differential design of the whole surrounding slope.

Introduction

The stability of slopes in open-pit mines is a crucial prerequisite directly affecting the normal production of the mine^1,2. Slope instability can have adverse effects on the overall profitability, safety, and environment of the mine³. Traditional methods for analyzing slope stability in open-pit mines typically employ deterministic design approaches to determine whether slopes are stable. However, these methods have drawbacks as they treat parameters of slope rock mass as constants and do not account for the uncertainty inherent in slope engineering^4,5. The uncertainty of rock mass parameters reflects various attributes of the rock-soil model that cannot be explained⁶. Due to the uncertainty in rock-soil model attributes, it is observed in engineering practice that slopes may experience instability even when safety factors are greater than 1⁷.

The stability of slopes in open-pit mines is influenced by both internal and external factors, including rock mass strength and excavation disturbance. Among these factors, rock-soil parameters are crucial for slope stability and are typically estimated using geological statistical estimation techniques based on available borehole data and sampling⁸. Sampling of the deposit through drilling at different locations followed by interpolation of sample information using geostatistical techniques is employed to understand block properties⁹. Due to the constraints of exploration costs, the number of boreholes is always limited, which restricts the accuracy of geological property estimation¹⁰. This imprecision results in unpredictable characteristics in the calculation results of slope stability, hence it is advisable to consider such uncertainties in slope design¹¹.

In open-pit mining, a major geotechnical challenge involves excavating slopes with steeper angles to achieve the lowest stripping ratio while ensuring maximum resource recovery¹². The focus of slope engineering lies in understanding when (temporally) and where (spatially) actions should be taken¹³. Therefore, safety design of slopes must be considered early in the mine planning phase¹⁴. Safety is the ultimate goal of slope design, and ensuring safety is a prerequisite for any slope design. However, given the inaccuracies in stability calculation methods and uncertainties in input parameters, it is necessary to maintain a certain redundancy in slope design. Hence, stability coefficients are commonly used as criteria for determining slope stability. However, when using stability coefficients, specific values are generally not analyzed in a targeted manner; instead, only intervals for stability coefficients are empirically determined based on specific requirements. An objective rule in open-pit mining is that for every 1° increase in slope angle, considerable economic benefits accrue to the enterprise. However, considering that safety coefficient intervals are usually set based solely on empirical experience, it is necessary to conduct targeted analyses for the specific values of stability coefficients under uncertain conditions in open-pit mining.

The credibility calculation method, by incorporating the uncertainty of input variables and evaluating the probability of system failure, provides a more rational approach to quantifying risk¹⁵. It promotes the consideration of the uncertainty of each input parameter by treating each input parameter as a random variable^16,17, making it an ideal approach for incorporating uncertainty into decision-making. In theory, there exists a unique correspondence between the reliability of slopes and stability coefficients. By calculating the reliability value of a specific slope, the stability coefficient of the slope under that condition can be obtained, enabling slope stability analysis that includes uncertainty. Due to the large mining area and long duration of mining operations in open-pit mines¹⁸, it is unreasonable to manage slopes across the entire perimeter using the same slope angle. Additionally, traditional slope management is rigid and does not allow decision-makers room for adjustment. Flexible planning methods offer a promising solution to these issues, where flexibility refers to a system's ability to respond to various uncertain factors¹⁹. Introducing the concept of flexible planning into slope morphology optimization involves considering future environmental changes during slope stability analysis. Flexible planning adapts slope designs to possible future environmental changes at minimal cost, thus mitigating potential losses caused by internal and external environmental variations.

Building upon the analysis above, this article proposes a dynamic zoning and grading theory for slopes across the entire perimeter. It fully considers the uncertainty characteristics of rock mass strength that affect slope stability. It semi-quantitatively describes the impact of valuation uncertainty on slope stability and proposes a zoning design method using different slope angles in different regions of the slope to enhance enterprise efficiency while ensuring slope stability. By introducing flexible optimization principles, the final decision-making authority for slope design is returned to on-site decision-makers. Once decision preferences are established, a set of slope zoning management plans can be promptly generated. Furthermore, throughout the production cycle, decision preferences can be adjusted multiple times to promptly respond to internal and external environmental changes, thereby enhancing the on-site applicability of slope design. Finally, the proposed flexible slope design method is applied to actual open-pit mining production to validate its effectiveness.

Construction of risk assessment system for full-boundary slope

Slope risk source identification based on fault tree analysis

The instability accidents of open-pit slopes pose a serious threat to the production safety of open-pit mining. A large amount of data indicates that slope instability accidents rank first among all mining accidents in open-pit mines. Open-pit slopes are influenced by factors such as rock mass characteristics, slope geometry, groundwater, and discontinuities²⁰. Based on field work experience, these factors can be classified into two categories: human factors and natural factors.

To deeply uncover the potential causes of accidents and propose targeted prevention measures, it is necessary to systematically analyze the various factors affecting the instability of open-pit mine slopes and identify the key factors. A fault tree is a directed logic tree that describes the occurrence of events from result to cause. By conducting deductive analysis on such a tree, strategies to prevent the occurrence of results can be sought. The steps for constructing a fault tree vary depending on the purpose and required precision, but generally follow a certain procedure:

1. Determine the top event of the fault tree, i.e., identify the event to be analyzed. This paper takes "open-pit mine slope instability accident" as the research object.

2. Investigate the human and natural factors related to the top event, determine the causes of the accident, and conduct an impact analysis. Eleven possible influencing factors were identified in this study, as shown in Table 1.

3. Construct the fault tree. According to the principles of deductive analysis, starting from the top event, analyze the direct cause events step by step. Following a certain logical relationship, use logic gates to connect the top event of the fault tree with the cause events, and create a tree diagram reflecting the cause-and-effect relationship.

4. Perform qualitative analysis of the fault tree. Based on the structure of the fault tree, calculate the minimal cut sets or minimal path sets; calculate the structural importance of the basic events.

5. Perform quantitative analysis of the fault tree. Based on the occurrence probabilities of the basic events that lead to the accident, calculate the probability of the top event occurring; calculate the probability importance and critical importance of each basic event.

Table 1.

The implication of each code of fault tree.

Code	Events	Code	Events
T	Open-pit mine slope instability	X4	Fault
M1	Rock structure and properties	X5	Development of joints and fissures
M2	Slope geometry	X6	Irregular geometric shape
M3	Human factors	X7	Large individual blocks
M4	Poor rock quality	X8	Excessive height
M5	Rock mass structure	X9	Excessive slope angle
M6	Components of slope	X10	Concave slope
M7	Slope morphology	X11	Convex slope
M8	Excavation method	X12	Uneven perimeter of slope
M9	Disturbance effects	X13	Overhanging slope
M10	Structural planes	X14	Excessive horizontal bending
M11	Structural bodies	X15	Unreasonable step height
X1	Low rock strength	X16	Blasting vibration
X2	Rock weathering	X17	Earthquake
X3	Folding	X18	Water

According to the principles of fault tree analysis, this study conducts a hierarchical analysis of both human and natural factors, step by step identifying their basic cause events according to the principles of deductive analysis. Based on the analysis of slope instability accidents, a fault tree was constructed, as shown in Fig. 1.

Fig. 1.

Fault tree of slope instability accidents in open-pit mines.

The meanings of the codes in the fault tree are as shown in Table 1.

The functional structure of the fault tree is as follows:

$$\begin{array}{c}T=M_1{M}_2{M}_3=\left(M_4+M_5\right)\left(M_6+M_7\right)\left(M_8+M_9\right)\\ =\left[\left(X_1+X_2\right)+\left(M_{10}+M_{11}\right)\right]\left(M_6+M_7\right)\left(M_8+M_9\right)\\ \end{array}$$ 1

As can be seen from Fig. 1, the number of basic events in the minimal cut sets is always three, as they are derived from the combination of the M₁, M₂, and M₃ branches. Using Boolean algebra, the minimal cut sets of the fault tree are calculated to be 196. Since the number of basic events in each minimal cut set is the same, the structural importance coefficient for each minimal cut set can be calculated using Eq. (2).

$$I_{\mathrm{\Phi }\left(i\right)}=\frac{1}{k}\sum _{r=1}^k\frac{1}{m_r\left(x_i\in E_r\right)},\left(i=1,2,\dots ,n\right)$$ 2

In the equation, k represents the number of minimal cut sets; E_r denotes the r-th minimal cut set; x_i is a basic event; and m_r is the number of basic events contained in the r-th minimal cut set E_r. The calculation results are as follows:

$$\begin{array}{cc}& I_{\mathrm{\Phi }\left(1\right)}=I_{\mathrm{\Phi }\left(2\right)}=I_{\mathrm{\Phi }\left(3\right)}=I_{\mathrm{\Phi }\left(4\right)}=I_{\mathrm{\Phi }\left(5\right)}=I_{\mathrm{\Phi }\left(6\right)}=I_{\mathrm{\Phi }\left(7\right)}=0.0476,\hfill \\ \hfill & I_{\mathrm{\Phi }\left(8\right)}=I_{\mathrm{\Phi }\left(9\right)}=I_{\mathrm{\Phi }\left(10\right)}=I_{\mathrm{\Phi }\left(11\right)}=0.0833,\hfill \\ \hfill & I_{\mathrm{\Phi }\left(12\right)}=I_{\mathrm{\Phi }\left(13\right)}=I_{\mathrm{\Phi }\left(14\right)}=I_{\mathrm{\Phi }\left(15\right)}=I_{\mathrm{\Phi }\left(16\right)}=I_{\mathrm{\Phi }\left(17\right)}=I_{\mathrm{\Phi }\left(18\right)}=0.0476.\hfill \\ \hfill \end{array}$$

Sorting the structural importance of each basic event, we find that excessive height, excessive slope angle, concave slope, and convex slope have the highest structural importance coefficients. This indicates that slope morphology and shape are the primary external factors influencing slope stability. Both are closely related to the delineation of open-pit mine boundaries and directly impact the economic viability of mining operations. Therefore, a balance must be struck between slope safety and overall economic feasibility.

Determination of landslide center line of irregular boundary slope

As discussed in the previous section, the geometric shape and irregular planar boundary of slope profiles are the primary external factors leading to slope instability and failure. The irregular shape of the boundary in open-pit mines often results in serrated slope profiles, making slope design and stability control challenging. In irregular boundary open-pit mines, alternating convex and concave slope surfaces occur along the perimeter, with significant differences in slope morphology and rock characteristics along strike. Consequently, there are differences in exposure time and slope morphology at various locations within the entire mining boundary. It is evidently unreasonable to use the same slope angle for slope design across the entire boundary. Therefore, by conducting engineering geological zoning of the entire perimeter slope, we can effectively understand the geological conditions within each zone, select representative models for zone-specific studies, and provide basic parameters and evaluation criteria for slope stability analysis.

The irregular boundaries directly result from variations in the working line length, which include three basic types: constant, contraction, and expansion. These variations can be combined pairwise, resulting in nine different slope types: constant-constant, constant-contraction, constant-expansion, expansion-constant, expansion-expansion, expansion–contraction, contraction-constant, contraction–expansion, and contraction-contraction, as illustrated in Fig. 2a–i.

Fig. 2.

The most dangerous area and stability analysis diagram of irregular end slope landslide.

Utilizing the strength reduction method and employing uniform strength parameters, the landslide areas and stability coefficients of each type of slope are determined through finite element simulation²¹. The stability coefficients and landslide areas of different types of slopes are depicted in Fig. 2. When there is no change in the working line length, the landslide is primarily located in the middle of the slope near the inner dumping area. However, when there is a change in the working line length, resulting in an irregular boundary, the potential landslide areas may vary with changes in the slope morphology. In most cases, irregular changes in the boundary lead to reduced slope stability. Figure 2 illustrates the central positions and sizes of landslide areas for the nine types of slope, providing a basis for zoning design of the entire perimeter slope.

It is important to note that in actual rock slopes, the materials do not satisfy the assumptions of isotropy and homogeneity, and features such as layering, joints, and fractures are inevitably present. Therefore, the numerical simulation model shown in Fig. 2 may not accurately reflect the safety conditions and potential landslide modes of the slope. The purpose of using the simplified model shown in Fig. 2 is to illustrate the variations in slope stability and the centerline of the sliding mass under irregular end-slope morphology, which is reasonable to some extent. However, when the study object is an actual open-pit rock slope, it is necessary to construct a realistic three-dimensional numerical analysis model based on the actual morphology of the slope and the development of joints and fractures. This approach will enable accurate analysis of the potential landslide modes and variations in stability coefficients.

Risk zoning of full boundary slope

Due to the reliance on geological parameters obtained from limited borehole data during the early exploration stage, the design of slope morphology cannot accurately reflect the influence of geological factors such as geological structure, lithology, weak layers, and bedding on slope stability. Therefore, the difficulty of conducting zoning and grading design is significant. Engineering geological zoning is of practical value for the study of slope stability²². By conducting engineering geological zoning of a specific area, it is possible to effectively understand the geological conditions within that area, thus providing a basis for the prevention and control of geological disasters.

In order to manage and organize mining activities and ensure the safety and efficient operation of mines, open-pit mines are typically divided into working benches, non-working benches, and end benches. Due to the different functions of each bench, there are differences in slope angle, slope height, and safety level among them. Therefore, each bench is initially roughly divided into a geological zone, and then further subdivided based on the irregular trend of the boundary, resulting in the final geological zoning.

Considering the characteristics of inclined coal seams in open-pit mines, the mine is divided into four major zones: working benches, inner dumping areas, shallow end benches, and deep end benches. The deep end bench is further subdivided into three areas based on the different slope morphologies, as shown in Fig. 3.

Fig. 3.

Slope landslide hazard zoning map of open-pit mine.

Combining the numerical simulation results shown in Fig. 2, the typical profiles 1–1, 2–2, and 3–3 are illustrated in the cross-sectional views of Fig. 3. The landslide modes of reverse-dip rock slopes are typically related to their structural characteristics and the dip of the rock layers. Possible failure modes include sliding failure, buckling failure, wedge failure, toppling failure, and combinations of circular-arc and polyline failures, among which the most common landslide mode is usually sliding failure. Therefore, Fig. 3 uses the sliding failure mode as an example to illustrate the morphology of the potentially most dangerous sliding surface. In practical applications, it is essential to select typical cross-sections based on the actual strata and geological structures to reanalyze the landslide modes and stability, thereby providing a basis for the refined zoning of the slope. For instance, the zoning map shown in Fig. 3 divides the entire slope into six zones based on slope functionality and boundary variations.

Zone I: Working benches of the mining area, characterized by relatively gentle slopes with a low likelihood of landslide occurrence.

Zone II: Shallow end benches, with slope angles close to the dip angle of the coal seam.

Zone III: Inner dumping areas, formed by the accumulation of loose discarded materials, with a focus on settlement and displacement studies.

Zones VI, V, and VI: Deep end benches, characterized by significant depth of excavation. The irregular boundary changes lead to a certain degree of reduction in slope stability, thus requiring special attention during the mining process.

Based on slope morphology, rock mechanics properties, and geological structures, the slopes are initially roughly divided into geological zones of the same morphology. Subsequently, these zones are further refined based on factors such as rock mechanics properties and geological structures to obtain the final geological zoning.

Improvement of zoning strategy considering uncertainty of rock mass parameters

Due to the relatively stable occurrence state of layered ore deposits within the same stratum, a two-dimensional geological model can be employed to represent them within the same stratum. The calculation principle is illustrated in Fig. 4.

Fig. 4.

Borehole valuation credibility expression diagram of open-pit mine.

Due to the reliance on borehole data obtained during the exploration phase, there are significant differences in the accuracy of estimation results across different areas. Each borehole can only cover a certain range, and beyond the influence radius of estimation, its impact on the estimated value of a particular segment can be ignored. As illustrated in Fig. 4, assuming there are only three boreholes within the entire exploration area, the estimation of each block is constrained by the results of these three boreholes. Suppose the influence radius of each borehole is denoted as R, then the estimation points within the entire area can be classified into four categories: no borehole control, single borehole control, dual borehole control, and triple borehole control. In theory, the accuracy of estimation increases as the estimation points are located within the influence radius of more boreholes. However, it is important to note that the accuracy of estimation from a single borehole is negatively correlated with the distance between the estimation point and the borehole. This implies that the accuracy of estimation from two boreholes located at a greater distance may not necessarily be higher than that from a single borehole located closer to the estimation point.

In theory, each borehole has its own influence radius for estimation. Therefore, this article employs the Kriging method to determine the range when calculating the experimental variogram function to determine the influence radius²³. Specifically, based on the information values of each variable to be estimated (such as physical and mechanical indicators, thickness, etc.), the overall average variogram function is computed.

$$\gamma \left(h\right)=\frac{1}{2N\left(h\right)}\sum _{i=1}^{N\left(h\right)}{\left[Z\left(x_i\right)-Z\left(x_i+h\right)\right]}^2$$ 3

In the equation, $x_i$, $x_i+h$ represents the sampling points; ℎ denotes the basic calculation distance; N(h) signifies the total number of sample pairs at distance ℎ; γ(h) represents the average variogram function, while $Z\left(x_i\right)$, $Z\left(x_i+h\right)$ denote variable information values (such as stratum information, thickness, etc.).

Then, a variogram plot is generated (equivalent to the variogram plot in the Kriging method), with the distance ℎ at which the variogram exhibits a stable trend being considered as the estimation influence radius R. Since the mechanical properties of rock and soil vary not only in different spatial locations but also in different directions, and they also exhibit a certain degree of correlation²⁴, it is possible to use physical–mechanical indicators as the basis for variable information values.

Suppose the credibility of estimation at the center position of each borehole is C_r = 100%, and the credibility of estimation at the influence radius is 0%. Obviously, the credibility of estimation is negatively correlated with the distance from the estimation point to the center position of the borehole. Let's denote m as the number of boreholes controlling a certain segment v to be estimated. In this case, the accuracy of estimation at that position is influenced by multiple borehole controls. Assuming the overall credibility is C_T, we can combine the information from m boreholes using logical conjunction:

$$C_T=max\left\{C_i{W}_i\right\}$$ 4

In the equation: C_T—Total credibility of the segment to be estimated, in percentage (%); C_i—Credibility of estimation for the ith borehole, in percentage (%); m—Number of boreholes considered for estimation; W_i—Weight of the ith borehole. When all boreholes have equal weights, the overall credibility is the maximum credibility among the boreholes.

Additionally, due to the timeliness of open-pit mining construction, as the excavation progresses, the rock layers at the location of the working bench are continuously exposed. Geological re-surveying can be conducted promptly at this time. Therefore, the credibility of the rock layers controlled by the working bench can be considered as 100%. Similarly, there is a decreasing trend in credibility with increasing distance from the working bench. Hence, it is necessary to supplement the borehole data with subsequent geological conditions to obtain greater credibility, as shown in Fig. 5.

Fig. 5.

The schematic diagram of determining the overall geological reliability.

In Fig. 5, the estimation reliability at the center of the boreholes and in the newly exposed area of the working bench is 100%, and the reliability decreases with increasing distance. For ease of explanation, let's assume that the decline pattern of reliability is the same for both locations. At this point, it is only necessary to determine the function of reliability change with distance, which can then establish the overall estimation reliability of any position on the working bench, thus improving the partition evaluation method.

Slope risk assessment method analysis

Summary of reliability index calculation

Due to the multitude of factors influencing slope stability, it is understood that there is inevitably a certain degree of risk in slope engineering design. Risk, in essence, is the probability of slope failure occurring or the likelihood of adverse consequences resulting from slope failure. Throughout the entire mining cycle in open-pit mines, the final design of the optimal slope angle depends not only on the distribution of ore grade and operational costs but also on the overall characteristics of the rock mass. Therefore, it is recommended to incorporate potential failures into the final design of open-pit mines²⁵. This helps to understand in advance when and where action must be taken. Rock mass characteristic parameters are never precisely known. They always involve uncertainty. Some of this uncertainty arises due to limited data, testing errors, random data collection, and lack of knowledge. Slope reliability analysis is an effective method to address this issue. By setting fixed parameters in stability analysis as random parameters, and calculating the probability of slope failure and corresponding reliability indices, potential failures can be effectively represented.

Slope reliability design is based on the limit state of normal use of slope engineering, and establishing the limit state equation is the basis for reliability calculation. The reliability analysis method can essentially be seen as a problem of the relationship between the resisting moment (torque) R and the sliding force (torque) S of the slope. Therefore, the limit state equation of the slope can be simplified:

$$Z=g(R,S)=R-S=0$$ 5

Based on the characteristics of the slope rock strata, establish a point estimation model with rock mass strength parameters as input variables and slope safety factor as the output variable. Apply the Strength Reduction Method (SSR) to calculate the safety factor of the output slope. Upon obtaining the probability density functions of R and S, the slope stability reliability can be calculated using the following probability expression:

$$P_f=1-P_r=1-{\int }_{-\infty }^{\infty }{f}_s(s)\left[{\int }_s^{\infty }{f}_R(r)dr\right]·ds$$ 6

where $f_s(s)$ is the probability density function of S; $f_R(r)$ is the probability density function of R; $P_f$ is the probability of slope failure; and $P_r$ is the slope stability reliability.

Due to practical limitations, some random variables' probability density functions cannot be accurately obtained or their expressions are too complex to directly calculate the failure probability through integration. Therefore, reliability indices are commonly used to measure the stability of slopes. Since slope stability is influenced by various variables including unit weight, cohesion, internal friction angle, elastic modulus, and Poisson's ratio, the limit state equation can be expressed as $Z=g\left(X_1,X_2,\cdots ,X_m\right)$. Where $X_i\left(i=1,2,\dots ,n\right)$ represents the random variables affecting slope stability. The reliability index can be expressed as:

$$\beta =\frac{{\mu }_Z}{{\sigma }_Z}=\frac{{\mu }_{F_s}-1}{{\sigma }_{F_s}}$$ 7

In the equation, ${\mu }_{F_s}$ represents the mean value of the slope stability coefficient, and ${\sigma }_{F_s}$ represents the standard deviation of the stability coefficient.

The key step in calculating reliability indicators lies in determining the limit state equation or the mean and standard deviation of the stability coefficient, then used in the Eq. (7) to obtain the reliability indicators. However, since there is no direct relationship between the deterministic safety factor and the failure probability, this paper considers indirectly using the maximum and minimum target reliability indices as the upper and lower bounds for the stability coefficient. By predefining the range of stability coefficients, it is possible to assess the overall stability of the slope using reliability analysis methods. This approach allows for the incorporation of valuation uncertainty into the slope stability assessment, leading to a more accurate evaluation of stability.

The above method is known as the First-Order Second-Moment (FOSM) method, which is widely used due to its simplicity and ability to quickly solve probability integrals. However, the FOSM method also has certain limitations. For nonlinear limit state functions or random variables with non-normal distributions and correlations, the accuracy of the FOSM method is poor. Specifically, for complex nonlinear limit state functions, the FOSM method assumes that the linear combination of random variables can be approximated as a normal distribution, which may not hold in some practical situations. However, in the scenario described in this article, the FOSM method is well-suited for effectively assessing slope stability. Therefore, this article uses the FOSM method as the basis for slope reliability calculations.

Determination of target reliability considering estimation uncertainty

In the process of using reliability analysis methods to evaluate slope stability, an important step is to determine the target reliability. It represents the reliability used as the design basis and, from a risk perspective, indicates the level of risk that the design allows or accepts. The slope influences the stripping ratio to some extent (for example, the ratio of the tonnage or volume of overburden to be removed to the tonnage or volume of ore to be extracted), thereby affecting the profitability of mining.

The target reliability of slope engineering is determined by the probability of failure and the consequences of failure, reflecting the risk attitude of decision-makers. It requires a balance between subjective judgment, engineering characteristics, and importance, considering the trade-off between the risks undertaken and the potential economic benefits. Therefore, higher reliability in slope engineering is not always better. Increasing reliability from 99% to 99.9% may lead to exponentially higher costs. Clearly, as the probability of slope failure decreases and reliability increases, direct costs of the project will rise while the costs associated with instability (including economic losses from landslides and landslide mitigation expenses) will decrease, and vice versa.

As an important document in slope reliability analysis, the target reliability directly determines the accuracy of the stability evaluation. Since reliability analysis methods can be considered a form of risk analysis, the calculation of target reliability should comply with the ALARP (As Low As Reasonably Practicable) principle. This principle divides risks into three zones through two risk boundaries (upper acceptable limit and lower acceptable limit): the unacceptable zone (high risk), the tolerable zone (medium risk), and the broadly acceptable zone (low risk), as shown in Fig. 6.

Fig. 6.

Determination of target reliability based on ALARP principle.

If the risk level determined by the risk assessment falls into the unacceptable risk zone, this risk cannot be accepted under any circumstances, except for special cases. If the risk level falls into the broadly acceptable risk zone, no safety improvement measures are required due to the very low risk level. If the risk level falls into the ALARP zone, the consequences of implementing various risk reduction measures need to be considered, and a cost–benefit analysis should be conducted to determine whether the risk can be accepted. If the implementation of additional safety measures does not significantly reduce the system risk level, then the risk can be considered unacceptable.

In the method proposed in this paper, the decision-making power is returned to the on-site decision-makers, who can determine the specific target reliability value based on the current production status. However, adjustments are only permitted within the medium risk zone, thus requiring the definition of risk boundaries first. Based on existing open-pit mine slope engineering experience, the overall slope stability factor is generally controlled between 1.2 and 1.3, setting the upper and lower acceptable limits to correspond to Fs = 1.2 and Fs = 1.3, respectively, with the corresponding reliability index range being 4.2 to 4.3.

During the decision-making preference adjustment process, the impact of geological uncertainty on the target reliability value is considered. Therefore, this article proposes incorporating geological uncertainty into the calculation of the target reliability value. However, in this process, the influence of geological uncertainty on the target reliability value has not been considered. Therefore, this article proposes to incorporate geological uncertainty into the calculation of the target reliability value. By categorizing geological uncertainty into five levels, targeted adjustment coefficients K are assigned to the target reliability value, ultimately considering geological uncertainty in slope stability assessment. The specific classification of adjustment coefficients K is detailed in Table 2.

Table 2.

Open-pit mine slope geological uncertainty classification and corresponding adjustment coefficient table.

Uncertainty Level	Range	Description	Adjustment Coefficient K
Extremely low	0–5%	Adequate detailed geological investigation and exploration data support. Geological conditions are very clear, with sufficient previous engineering experience	0.8
Low	6%-15%	There is detailed geological investigation and exploration data, but there are some minor uncertainties. Moderate previous engineering experience	0.9
Moderate	16%-30%	There is some geological investigation and exploration data, but there are still significant uncertainties. Limited previous engineering experience	1.0
High	31%-50%	Lack of detailed geological investigation and exploration data, significant uncertainties in geological conditions. Limited previous engineering experience	1.2
Relatively high	51%-75%	Geological conditions are relatively unclear, and geological investigation and exploration data are not detailed. Very little previous engineering experience	1.4
Extremely high	76%-100%	Geological conditions are very unclear, with almost no detailed geological investigation and exploration data. The project area may be new or have almost no previous engineering experience	1.5

It is important to note that Table 2 is based on limited data, and its applicability to other open-pit mines needs further validation. To address the practical conditions of most open-pit mines, the following measures can be taken to make necessary adjustments to Table 2:

1. Use high-quality data collection methods to gather actual data on geological uncertainty, and perform model validation and calibration.

2. Collect opinions from multiple experts and use fuzzy mathematics methods to handle subjective judgments, reducing the bias of individual expert opinions.

3. Develop high-precision, low-error data processing methods to minimize human influence and improve the scientific accuracy and reliability of the classification.

By implementing these measures, geological uncertainty and other model and human uncertainties can be effectively managed, enhancing the accuracy and reliability of slope stability analysis.

The proposing of the dynamic design method for full boundary slope

Improvement of target reliability index calculation method

In the process of determining the target reliability, adjustments were made to the target reliability values by assigning certain adjustment coefficients. However, due to the inherent uncertainty which is unpredictable in occurrence and magnitude, relying solely on fixed target reliability values for slope stability evaluation exhibits the following drawbacks:

1. Under the influence of uncertainties, the evaluation fails to fully and accurately reflect the actual slope stability conditions in response to changes in uncertain factors.

2. Most target reliability values are based on system safety considerations, which may compromise the economic efficiency of the system.

Therefore, this article proposes a flexible optimization design approach based on this premise, introducing subjective human factors to allow for adjustments to the target reliability values to seek more flexible and adaptable slope design solutions. By adjusting the target reliability values, the article aims to integrate safety and economy, returning the adjustment authority to on-site decision-makers.

By introducing the Integrated Risk Index (IRI), safety and economy are considered comprehensively. Safety can be represented by the reliability of the slope. The stability of the slope can be quantified using reliability indicators such as the Factor of Safety (FOS) or Probability of Failure (POF), which can be obtained through various slope stability analysis methods. Economy can be represented by considering the cost of slope improvement measures, including mining equipment, labor, and environmental management.

The Integrated Risk Index (IRI) is defined as the weighted ratio of safety and economy:

$$IRI=\frac{w_{S}SI}{w_{E}EI}$$ 8

where SI and EI represent safety and economic indicators, respectively; w_S and w_E are the weighting factors for safety and economy, indicating their relative importance in the composite index; IRI is a positive number, with IRI > 1 indicating a higher emphasis on safety, IRI < 1 indicating a higher emphasis on economy, and IRI = 1 indicating equal weight. The determination of weighting factors can be obtained through discussions with relevant stakeholders and professionals, or using multi-objective optimization methods. This can accommodate the specific requirements of different projects, regions, or ores. The improved target reliability value β_u can be calculated using the following equation.

$${\beta }_U={\beta }_0·\eta ={\beta }_0·K·IRI$$ 9

where: β_u is the improved target reliability value in flexible optimization design; β₀ is the target reliability value obtained from historical experience; η is the comprehensive adjustment coefficient; K is the valuation adjustment coefficient, as detailed in Table 2. IRI is the integrated risk index.

The relative importance of safety and economy is determined in collaboration with on-site decision-makers. This can be obtained through questionnaire surveys, expert interviews, or engineering experience. Based on the weights and adjustment parameters provided by on-site decision-makers, the stability of the slope is evaluated in real-time, and information regarding safety and economy is provided. Ultimately, decision-making authority is returned to the site. Once the relative importance of safety and economy, as well as the preferences for adjustment coefficients, are determined, a comprehensive slope zoning assessment plan can be generated promptly. This guides on-site production, balancing safety and economy while considering uncertain factors.

Flexible design method based on the observational method

In the field of open-pit mining, the observational method is commonly used to modify slope designs. This method involves predicting a comprehensive network of potential events and consequences and deciding on subsequent project adjustments before excavation begins. The key to the observational method is the advance prediction and planning of all possible events and outcomes, along with the development of corresponding measures. Its advantage lies in the ability to comprehensively and systematically study all potential conditions faced by open-pit mine slopes. However, its complexity and high cost limit its application under certain conditions. For example, the factors affecting slope stability generally have uncertain characteristics, leading to unforeseeable situations. For instance, geological exploration may indicate stable coal and rock conditions within a mining area, but during actual production, some areas may show drastically deteriorated coal and rock conditions, which were not considered before open-pit mining. Therefore, it may be impossible to provide corresponding solutions based on the observational method.

To address the shortcomings of the observational method, this paper proposes a simplified flexible slope design method. This method involves designing slope angles for different zones, fully considering human decision-making preferences, and evaluating slope stability in real-time based on weights and adjustment parameters provided by on-site decision-makers. It also provides information on safety and economic aspects. The ultimate decision-making power is returned to the site. Once the relative importance of safety and economics, as well as adjustment coefficients preferences, are determined, a comprehensive slope zoning evaluation scheme can be promptly generated to guide on-site production. This method considers uncertain factors while balancing safety and economic aspects. Compared with the observational method, the proposed method has the following characteristics:

1. It does not require advance prediction of a complete and complex network of potential events and consequences before excavation. Instead, it considers key parameters for slope zoning design. During the mining process, slope designs are dynamically adjusted based on actual on-site data, reducing the initial planning workload and improving prediction accuracy.

2. In slope design adjustments, the method fully considers the characteristics of new data and the potential changes in on-site decision-making preferences, returning decision-making power to on-site decision-makers.

Compared with the observational method, the proposed method can consider changes in safety and economic decisions, flexibly adjusting slope design schemes and providing effective solutions for slope design in most cases.

Case study

Mining conditions and slope risk zoning of an open-pit mine

The geological formation of a certain open-pit mine exhibits a monocline structure dipping to the northwest, with coal seam dips ranging from 8° to 20°. The strike and dip orientations show little variation, and there are no faults disrupting the strata. The primary coal seam being mined is the thick coal seam of the B_m group, which is a typical inclined coal seam for open-pit mining. From top to bottom, the strata consist of the Quaternary (Q_3-4^p1+a1), Tuyulu Formation (K₁^tg), Upper Yaha Formation (J_2-3^shb), Lower Yaha Formation with iron-bearing sandstone (J_2-3^sha), Lower Yaha Formation sandstone-mudstone (J₂), and coal seams.

According to the initial design, the open-pit mine begins trenching at the coal seam outcrop to the northeast of the primary mining area and progresses southeastward along the strike. The final slope angle for the mining area is designed to be 30°, while the outer dumping area slope angle is 25°, and the inner dumping area slope angle is 20°.

However, due to the large extent of the primary mining area, inevitable differences exist in the geological conditions of various regions. It is evidently unreasonable to control the slopes of the entire area with the same final slope angle. Numerous studies have shown that for open-pit mines, each degree increase in slope angle brings considerable economic benefits. Therefore, it is necessary to divide the slopes of the entire mining area into zones and then take targeted management measures.

Taking the primary mining area of this open-pit mine as the research object, and considering the development process, the end dumps are selected as the focus of the study. Since the slope angle of the superficial end dump is essentially equal to the dip angle of the coal seam, there is no risk of instability, and thus this area is not further divided. Based on this, the deep end dumps are selected as the focus of the study, and considering the irregular changes in surface boundaries, the deep end dumps are further divided into five zones for research, as shown in Fig. 7.

Fig. 7.

Stability partition of end slope in the first mining area.

Based on the identified most critical slip surface locations in Fig. 3, typical profiles were selected for each area. By employing a two-dimensional slope reliability analysis method, reliability indicators can be obtained while significantly saving computational costs.

Acquisition of probability distribution of rock mass strength parameters

Under the concept of flexible optimization design, using reliability analysis methods to evaluate slope stability is considered an ideal approach. Considering that the calculation of reliability indicators is the core content of slope reliability analysis, it is necessary to first obtain the probability distribution of rock mass strength parameters, and then calculate the mean and standard deviation of the stability coefficient (Gong et al., 2015; Jiang et al., 2017).

The rock mass parameters that affect slope stability mainly include cohesion, internal friction angle, and unit weight. Therefore, seven exploration lines are selected throughout the entire mining area, as shown in Fig. 8.

Fig. 8.

An open-pit mine exploration line and drilling layout diagram.

16 boreholes distributed along 7 exploration lines are used to obtain the basic data, however, some of the boreholes are located in the shallow part of the coal seam, unable to fully expose all strata; the distribution of the Quaternary system is extremely uneven, so its strength parameters were not statistically analyzed. In slope engineering stability analysis, the values ​​of rock mass parameters significantly impact the analysis results. When further utilizing reliability for slope stability analysis, the optimal probability density or distribution function model of rock mass parameters directly affects the accuracy of the reliability results.

Inferring the probability distribution of the population from a sample of random variables is a general problem in parameter estimation. Initially, assuming that rock and soil parameters follow certain classic probability distributions (such as normal distribution, lognormal distribution, Weibull distribution, beta distribution, extreme value distribution, etc.), the parameter values are estimated. Then, a goodness-of-fit test is conducted at a given level of acceptance to determine whether the statistical distribution of the actual samples conforms to the assumed distribution. Studies have shown that most rock and soil parameters follow a normal probability distribution, based on which the parameter values are estimated. The probability distribution parameters of rock formations exposed in various strata are listed in Table 3.

Table 3.

Statistics of stratum exposure and probability distribution parameters of rock mass.

Strata	Code	Total exposure count	Rock mass strength parameters	Mean	SD
Tuyulu Formation	K1tg	16	Bulk Density γ (g/cm3)	2.12	0.0808
			Cohesion C (MPa)	0.96	0.4893
			Angle of Internal Friction φ (°)	36.75	4.6146
Upper Subgroup Rock Strata	J2-3shb	10	Bulk Density γ (g/cm3)	2.16	0.1036
			Cohesion C (MPa)	0.92	0.3741
			Angle of Internal Friction φ (°)	39.06	3.3173
Lower Subgroup Iron-bearing Sandstone Strata	J2-3sha	13	Bulk Density γ (g/cm3)	2.19	0.1320
			Cohesion C (MPa)	1.21	0.5662
			Angle of Internal Friction φ (°)	39.43	2.6934
Lower Subgroup Sandy Mudstone Strata	J2	14	Bulk Density γ (g/cm3)	2.20	0.1562
			Cohesion C (MPa)	1.85	1.4524
			Angle of Internal Friction φ (°)	38.01	4.8096
Coal Seam	Bm	16	Bulk Density γ (g/cm3)	1.21	0.0596
			Cohesion C (MPa)	0.8	0.2345
			Angle of Internal Friction φ (°)	39.86	2.3776
Xishan Formation Rock Strata (Coal Seam Floor)	J2x	16	Bulk Density γ (g/cm3)	2.27	0.1281
			Cohesion C (MPa)	1.92	0.9050
			Angle of Internal Friction φ (°)	36.04	5.1808

Target reliability improvement considering estimation uncertainty

As mentioned earlier, each borehole can be considered to have a certain influence distance. Only when the size of the borehole valuation influence range is determined, can the credibility of geological valuation be analyzed. Since coal seams are typical sedimentary rocks, it can be assumed that the physical and mechanical parameters of rock and soil at different positions of the same stratum are correlated. Therefore, five parameters, namely density, tensile strength, compressive strength, cohesion, and friction angle, are selected as the parameters for calculating the overall average variation function. By organizing the distances between the 16 boreholes in the valuation area, the average variation function is calculated using Eq. (5), and scatter data of distance-average variation functions are constructed. The borehole influence radius is obtained through polynomial fitting function relationships using the variation chart, as shown in Fig. 9.

Fig. 9.

Determination of valuation influence radius based on variation graph.

From Fig. 8, it can be observed that when the distance reaches 2507.47 m, the average variation function essentially reaches its maximum value. Therefore, the drilling influence distance is approximately 2507.47 m, and the credibility of valuation beyond this range decreases to 0. By reverse-engineering the data, the relationship between credibility and distance can be calculated as $C_r=1.06·{10}^{-10}{x}^3-3.26·{10}^{-7}{x}^2-2.49·{10}^{-4}x+1.01$.

As credibility and geological valuation credibility are complementary, once the distance between a certain valuation point and the drill hole is obtained, the target reliability index can be improved using the table. This allows for targeted control of the stability of slopes in each zone while considering valuation uncertainty. However, the magnitude of the adjustment coefficient still needs to be determined based on the actual opinions of on-site decision-makers, which will be the focus of the next section.

Table 4 depicts the credibility and adjustment coefficients for each typical profile. As shown in Fig. 7, the credibility of the typical profile locations is influenced by multiple exploration lines, assuming that the credibility along the entire exploration line is 100%. According to Eq. (4), when the relative weight values of each exploration line are equal, the overall credibility of the valuation is the maximum value among them. Therefore, it is only necessary to determine the distance between the profile center and the nearest exploration line. This enables the determination of the credibility value on the profile using the credibility-distance function relationship, thereby establishing the valuation adjustment coefficient K.

Table 4.

Table of typical profile estimation credibility and adjustment coefficient in each partition.

Typical profile no	Distance to nearest exploration line (m)	Credibility of valuation (%)	Adjustment coefficient K
I	246.37	93.41	0.9
II	375.23	87.99	0.9
III	251.69	93.19	0.9
IV	50.28	100	0.8
V	207.36	84.89	1.0

In the course of mining activities, as layers of the open-pit mine are gradually exposed, additional sample information helps to improve the credibility of valuation. However, since this article assumes that mining has not yet commenced, this aspect has been disregarded in this section. In actual production processes, adjustments can be made to the adjustment coefficient K based on the revealed geological layers to further enhance the slope angle.

Determination of slope flexible optimization design scheme

The core of slope flexible optimization design lies in balancing safety and economy, which is controlled by on-site decision-makers. This article manipulates the preferences of decision-makers by controlling the value of the Integrated Risk Index (IRI). Since determining the IRI is a dynamic process that adjusts with changes in internal and external environments, it is impractical to determine the IRI values for the entire mining process in the first mining area.

Therefore, this article simplifies the process of determining the IRI index. In the long run, due to technological advancements, the safety of slopes will be further ensured. Hence, decision-makers may prioritize the economy of slopes. According to Eq. (8), the value of the IRI index is always less than 1 and will further decrease over time. Therefore, the IRI index for the five mining stages is set as 0.98, 0.96, 0.94, 0.92, and 0.90 respectively. Combining with Table 4, the comprehensive adjustment coefficients η are calculated as 0.88, 0.86, 0.85, 0.74, and 0.90. The target reliability index values β₀ for each zone are determined to be 4.2. Consequently, based on Eq. (9), the flexible target reliability index values for each zone are calculated as 3.696, 3.612, 3.570, 3.108, and 3.780 respectively.

Using the five typical cross-sections as references, the two-dimensional slope reliability analysis method is employed to calculate the slope reliability for each cross-section with slope angles of 25°, 30°, 35°, 40°, and 45°, as shown in Fig. 10.

Fig. 10.

The reliability analysis results of slope in each stage and the determination of optimal slope angle.

Figure 10 illustrates the variation of slope reliability indicators with slope angle, and in conjunction with the flexible target reliability indicators for each zone, the optimal slope angle for each area can be determined. The optimal slope angles for Zones I, II, III, IV, and V are approximately 31.7°, 32.4°, 31.3°, 32.9°, and 31.3°, respectively, representing an increase of 1.7°, 2.4°, 1.3°, 2.9°, and 1.3° compared to the original design. This value is determined considering the uncertainty of the valuation, thus reducing the probability of slope instability to some extent. However, since the estimation of the IRI indicator is somewhat idealized, adjustments need to be made during the production process based on actual conditions to obtain slope design schemes that better meet the needs of decision-makers, further enhancing the safety and economy of slope design.

Discussion

This article aims to address the lack of specificity and flexibility in current open-pit slope designs. Through fault tree analysis, the main risk factors for slope instability were identified, with a focus on irregular variations in slope planar shapes. Using three-dimensional numerical simulation methods, the most critical slip surface locations for nine different types of slopes were determined, providing a basis for selecting typical slope profiles across the entire mining area.

Based on the commonly used observational method in open-pit mining design, improvements have been made to the target reliability index, which is a fundamental document for slope reliability analysis, considering the uncertainty in geological valuation. Furthermore, a flexible design method for slope zoning has been proposed that aggregates the decision preferences of on-site decision-makers. Compared to traditional rigid design methods, the proposed approach takes into account the uncertainty in geological valuation and allows for timely adjustments to the slope design scheme based on on-site decision preferences, thereby enhancing the flexibility of slope optimization design. However, since the proposed method can be considered a simplified version of the observational method, it may not be as comprehensive and reliable as the observational method under extremely complex or uncertain geological conditions. In summary, the following shortcomings still exist in this study:

1. The division of five zones for the deep-end slope is not detailed enough, requiring further efforts to improve the method's applicability.

2. Only considering estimation uncertainty as a representative factor, the article did not consider the improvement strategy for zone division under multiple uncertain coupling conditions. The next step will focus on how to aggregate multiple uncertain factors.

3. When improving target reliability considering estimation uncertainty, the article used a coefficient adjustment table, and the effectiveness of interval division needs to be verified. Additionally, future research needs to explore the functional relationship between uncertainty and adjustment coefficients to make more effective adjustments.

4. The core parameter of flexible optimization design is the integrated risk index, but due to current research methods limitations, it is difficult to make a reasonable prediction about the determination of the integrated risk index.

5. The proposed dynamic analysis method relies on on-site decision-making, which can be influenced by the experience and expertise of the decision-makers. Furthermore, due to the simplification of the initial design, it may not fully account for all possible events. Therefore, in certain situations, it may be necessary to combine it with a more comprehensive observational method to ensure the safety and stability of slope engineering.

Conclusion

Based on considering the uncertainty characteristics of rock mass strength affecting slope stability, this article proposes improvements to the zonal evaluation method for open-pit mine slopes. By integrating flexible optimization principles, a comprehensive boundary slope dynamic optimization design method is introduced. Using a specific open-pit mine as a production case study, the effectiveness of the proposed method is validated. The main conclusions of the article are summarized as follows:

1. By utilizing fault tree analysis, the key factors controlling slope stability were determined to be slope height, slope angle, concave slopes, and convex slopes.

2. Based on the functionality of slopes and irregular variations in boundaries, the process of dividing the entire mining area into slope zones was described.

3. Combining principles of geological estimation, a method for determining the overall geological credibility of open-pit mining was derived.

4. Considering estimation uncertainty, the magnitude of adjustment coefficients assigned to target reliability values was determined.

5. By introducing the integrated risk index, a flexible optimization design method for slopes that fully considers on-site decision preferences was proposed.

6. The proposed method was applied in practical engineering, focusing on the deep-end slope of a specific open-pit mine. The slope was divided into 5 zones, and under the set decision preferences, the optimal slope angles for each zone were determined to be approximately 31.7°, 32.4°, 31.3°, 32.9°, and 31.3° respectively, representing increases of 1.7°, 2.4°, 1.3°, 2.9°, and 1.3° compared to the original design.

Author contributions

Shuai Wang wrote the main manuscript text; Bo Cao organize and analyze the data; Runcai Bai put forward the whole idea of the article; Guangwei Liu prepared the figures; all authors developed the algorithms and reviewed the manuscript.

Data availability

The data used to support the findings of this study are available from the corresponding author upon request.

Competing interests

The authors declare no competing interests.