Introduction

In underground mining operations, it is crucial to determine reasonable parameters of the stope structure. This not only helps maintain the mining area in an ideal mechanical state, ensuring that stress and strain in the surrounding rock can be evenly distributed, but also effectively avoids problems such as stress concentration, energy accumulation, and deformation damage, thereby achieving efficient ground pressure management and control1,2,3. In order to further improve the production capacity and efficiency of mines, it is often necessary to consider appropriately increasing the structural dimensions of the mining area, specifically the stope exposure area. However, the selection of the exposed area requires rigorous evaluation. If the exposed area is too small, it can limit production efficiency, increase ore loss, and affect economic benefits. Conversely, if the exposed area is too large, it may lead to increased ground pressure activity and compromise the stability of the stopes4,5. Therefore, on the premise of ensuring safe operations, determining the optimal stope through scientific calculations can optimize production conditions and effectively control ground pressure risks, which is an effective strategy to improve mining production efficiency.

The selection of the stope exposure area can be achieved through empirical methods, numerical simulation, and similar simulation experiments. Among them, the Mathews stability graph method6 is one of the most widely used empirical methods. This theory links the rock stability coefficient of the stope with the shape coefficient (hydraulic radius) to jointly evaluate the stope parameters. Since the birth of the Mathews method, the modification and development of the Mathews method are mainly related to the position and number of representative regions in the stability graph. The original Matthews graph contains three different zones, namely, the stable zone, the unstable zone, and the caving zone. Potvin7 collected more mine data in 1988, and he reduced the number of stable graph areas to 2. The graph area is separated by a transition zone and divided into two parts: stability zone and caving zone. Potvin et al.8 also improved the stability graph method to enable it to be applied in the supported stopes. Nickson9 improved Potvin’s improved stability graph method in 1992, respectively. He added more cases of supported stopes to the stability graph database. Stewart and Forsyth10 readjusted the Mathews stability graph in 1995. They redivided the graph area into four parts through three transition zones. Although they have considered the stability probability, there is no quantitative data to indicate the possibility of stability. Trueman et al.11 added many large-scale stope cases to the stability map database in 2000, forming a comprehensive stability database containing 483 cases. Mawdesley et al.12 proposed an extended Mathews stability graph based on this database in 2001. Then, Mawdesley13 validated and improved the extended Mathews stability graph method in 2004. Up to now, there are still a large number of researchers studying and improving Mathews stability graph method. Vallejos et al.14 evaluated the performance and significance of the adjustment factors used in the stability graph method through contingency matrix and performance indicator analysis to define the most representative stability boundary. These studies have improved the predictive ability and applicability of the stability graph.

In recent years, the combination of numerical simulation and empirical methods has become an important direction of stope stability analysis. As a three-dimensional numerical simulation tool based on the finite difference method, FLAC3D is widely used in mine engineering because of its advantages in simulating nonlinear behavior, stress redistribution, and plastic failure of rock mass. Zhang et al.15 used FLAC3D numerical simulation to obtain a reasonable strip width to determine the stope exposure area, and combined it with the extended Mathews stability graph method to verify its stope stability of over 85%. Heidarzadeh et al.16 integrated Monte Carlo simulation techniques with FLAC3D to evaluate the optimal ranges of geometric parameters for each stope, thereby estimating the stope exposure area. Jia et al.17 revised the Mathews stability graph to determine potential hazardous areas and used on-site monitoring and numerical simulation to jointly evaluate and warn of the stability of stopes with different exposure areas. Cui et al.18 employed FLAC3D numerical simulations to analyze the mechanical response characteristics of stopes under different exposure areas at various segment levels during the mining process. Additionally, physical modeling experiments19 have been identified as an important approach for optimizing stope exposure areas. However, these experiments are often costly and difficult to replicate. Some researchers have also integrated theoretical methods, artificial intelligence, and numerical simulations to investigate the stability of stopes20. While these methods have shown promising results in assessing stope stability, they often lack intuitiveness and convenience in obtaining stability probabilities. Moreover, the analysis frequently neglects the influence of exposure time. Most numerical simulations consider only the impact of a single stope or a single-step mining process, which may lead to overly optimistic safety assessments and thus fail to reflect the actual conditions accurately.

To address the aforementioned issues, the inclined orebody stope of a certain iron mine was taken as the research object, and an extended Mathews stability isoprobability line equation was fitted, considering the exposure time factor of the stope. The stope exposure area was analyzed and obtained. Furthermore, numerical simulations of two-step backfilling mining were conducted for stopes with different length-to-width ratios to analyze the rock mass stability characteristics. Ultimately, a rational ultimate exposure area for the mine is proposed through collaborative optimization.

Extended Mathews stability graph method

Stability index N and shape coefficient S

The Mathews stability graph method is a stability evaluation method that integrates the rock mass stability coefficient N related to rock properties and the stope shape coefficient (hydraulic radius) S related to stope structural parameters. The two influence factors obtained are statistically judged on the graph that has been divided into predicted stability zone, potential instability zone and caving zone. The stability coefficient N and stope shape coefficient S are calculated as shown in Formula (1).

$$\begin{aligned} N & = Q^{\prime } ABC \\ S & = \frac{XY}{{2(X + Y)}} \\ \end{aligned}$$
(1)

where Q′ represents the modified Q value of the NGI tunnel quality index, assuming that the stress reduction factor (SRF) and the joint water reduction factor (Jw) are both 1. A represents the rock stress coefficient that accounts for the influence of stress on rock mass stability. B represents the orientation correction coefficient for rock mass defects, which depends on the difference between the dip angle of the major joint sets and the dip angle of the drift. C represents the exposure face orientation correction coefficient, which considers the directional influence of exposed surfaces (such as drift walls) on rock mass stability. X and Y represent the width and length of the exposed face, respectively, and the shape coefficient (hydraulic radius) S is the ratio of the exposed surface area to the circumference.

Consider the Mathews synthetic chart of isoprobability lines

Based on the Logistic model, Mawdesley utilized the percentages of stable, failure, and major failure cases from a large number of mining examples to formulate regression equations, as shown in Formula (2), and their outcome probabilities in Formula (3). Using the maximum likelihood estimation method, the database was subjected to logarithmic regression analysis to quantitatively express coefficients S, N, A, B, and C, resulting in the prediction probability Formula (4). By calculating S and N and substituting them into the probability density function model, the results were found to be consistent with the probabilities in actual mining cases, thus achieving the goal of predicting stability

$$z = \alpha + \beta_{1} x_{1} + \beta_{2} x_{2} + \beta_{3} x_{3} + \cdots + \beta_{k} x_{k}$$
(2)
$${{f(z) = }}\frac{{1}}{{{{1 + e}}^{{ - {{z}}}} }}$$
(3)
$$z = 2.9603 + 0.7928lnN - 1.4427lnS$$
(4)

where z is the predicted logarithmic probability value; α is a constant; β is the regression coefficient; f(z) is the predicted logarithmic probability value.

To more clearly determine the stability states of the stopes and the specific probability of stability, various probability equations were calculated according to Formula (3), and an extended Mathews stability isoprobability composite graph was drawn. The Mathews stability graph is divided into four regions21,22,23: the stable zone, the failure zone, the major failure zone, and the caving zone. The boundaries between these regions are the stable-failure line, the failure-major failure line, and the major failure-caving line, respectively. Among them, the stability probability of the stable-failure line is 57%, the stability probability of the failure-major failure line is 8%, and the stability probability of the major failure-caving line is 0%. According to Formula (2) and (3), the equations for the 95% stable isoprobability line, 90% stable isoprobability line, 85% stable isoprobability line, stable-failure line, and failure-major failure line were fitted. The probability lines are parallel to each other, and the results are listed in Table 1. As shown in Fig. 1, all stable equipotential lines were plotted on the stability graph.

Table 1 Equations of stable probability lines.
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Fig. 1
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The extended Mathews stability probability composite graph.

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Stability analysis of stope considering exposure time

The stability of stopes is influenced not only by rock mass properties and stope structural parameters but also by their exposure time. This time effect (e.g., rock creep, stress relaxation) is an independent factor whose mechanism differs from static factors such as rock mass quality, stress state, and geometric parameters. Suorineni24 summarized the explanations of several scholars on the influence of time factors on stope stability, and led to the dependence of stope surrounding rock stability on time obtained by referring to the experience of engineering historical data. The relationship between the exposure time of the stope and its stability can be predicted and evaluated by introducing the time factor T according to the reference through the rock mass quality Q′.

Therefore, the stability coefficient N is revised, taking into account the time factor T, and the stability index N* = Q′ABCT. The corresponding relationship between the time factor T and the rock mass quality Q′ is shown in Table 2.

Table 2 Correspondence table of exposure time and time factor T.
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Engineering application

Project profile

The ore deposit of the mine is located in the top iron-bearing phyllite of the Yanlingguan Formation and consists of multiple band-like ore layers. These band-like ore layers extend in an NW–SE direction, with a general strike of 330°, dipping SW with dip angles ranging from 23° to 68°, and an average dip angle of 50°. The ore body is 1400 m in length and has a thickness that generally ranges from 21 to 50 m, with an average thickness of 32 m. This ore body is typical of a moderately thick, inclined deposit. The mine employs the upward high-segment pre-controlled roof backfilling mining method. The stopes are arranged perpendicular to the ore body strike. Current working faces are concentrated in the − 210 m to − 140 m levels, with each segment having a height of 12.5 m. Anchor bolts are installed on the roof of each stope for pre-controlled roof support, with a pre-controlled roof height of 3 m. During stoping operations, no crown pillars or sill pillars are retained. A ‘skip-one, mine-one’ approach is implemented: first mining odd-numbered stopes, followed by immediate backfilling, then extracting even-numbered stopes after the backfill achieves sufficient strength. High-pressure down-the-hole (DTH) drills drill parallel holes for ore breaking. Ore extraction is performed using 1 m3 electric and 2 m3 hydraulic shovel, and backfilling is carried out immediately after the stope is mined out.

The mining method requires that each segment undergo full cutting and roof support. Currently, the preparation time for single-stope mining is lengthy, and the cycle of development, mining, and backfilling is frequent, leading to high production management complexity and low stope production capacity. Post-cutting roof exposure expansion necessitates more extensive support. Additionally, anchor mesh installation and shotcreting are necessary, which significantly increase support costs and time required for support. Therefore, to develop deeper resources and achieve efficient and safe mining, it is essential to determine a reasonable limit for the exposed area in the stopes.

Simulation scheme based on the extended Mathews stability graph method

The rock mass stability coefficient N can be calculated based on the actual conditions of the mine. The Q′ value and the RMR (Rock Mass Rating) value can be converted using the formula RMR = 9lnQ′ + 4425. The Rock Mass Rating (RMR) classification was conducted based on the actual engineering geological conditions of the mine. The parameters and scores for each rock mass indicator are as follows: the uniaxial compressive strength (UCS) is 100.9 MPa, with a score of 12; the rock quality indicator (RQD) ranges from 50 to 75%, with a score of 13; a total survey length of 49 m revealed 235 joints, with an average joint spacing of 20.85 cm, resulting in a score of 10; the surface condition of discontinuities is smooth with no infill material, and the slickensides or infill thickness is less than 5 mm, yielding a score of 10; the groundwater condition is completely dry, with a score of 15; and the orientation of joints has a correction value of − 5. This RMR classification provides a comprehensive assessment of the rock mass quality, which is essential for determining the appropriate mining methods and support measures.

$${\text{RMR}} = 12 + 13 + 10 + 10 + 15 - 5 = 55$$
(5)

The Q′ value is derived from the RMR calculation:

$${{Q}}^{\prime } = {{e}}^{{{((RMR - 44)/9)}}} = {{e}}^{{{((55} - {44)/9)}}} { = 3}{{.39}}{.}$$
(6)

The stability index is calculated using Formula (1). In this formula, A accounts for the effect of high stress on rock mass stability, primarily related to the ratio of the uniaxial compressive strength of intact rock (σc) to the maximum induced stress in the working face (σi), that is, σc/σi. According to empirical data7, A is linearly related to σc/σi, with the following relationship: when σc/σi < 2, A = 0.1; when 2 < σc/σi < 10, A = 0.1125(σc/σi) − 0.125; and when σc/σi > 10A = 1. For the planned mining operation, the upper surface elevation of the ore body is approximately 150 m, the mid-elevation of the mining area is approximately − 300 m, and the excavation depth is 450 m. Through FLAC3D numerical simulation, the induced stress considering in-situ stress field and mining redistribution is 13.77 MPa. Therefore, σc/σi = 7.33, and A = 0.70.

B is determined based on field geological surveys. The major joint sets are found to be approximately 60° to the dip of the drifts. Therefore, B is assigned a value of 0.8.

C is determined based on the surface dip angle α of the stope:

$$C = 8 - 6\cos \alpha = 2$$
(7)

The stability index N is calculated as follows:

$$N = 3.39 \times 0.70 \times 0.80 \times 2 = 3.80.$$
(8)

The limit exposure area of the stope roof and floor in the same mine, under identical geological structures, hydrological conditions, and other mining backgrounds, is primarily determined by the geometric dimensions of the stope, specifically the length and width of the exposure face. Based on the calculated stability index N of 3.80, and using the extended stability graph, the shape coefficient S for the stope at 85% probability of stability is determined to be 4.87.

In combination with the average ore zone width of 32 m and the vertical arrangement of stopes along the strike, two scenarios for the stope length are considered: 30 m and 40 m. Through calculations, it is determined that when the stope length is 30 m, the stope span is 14.42 m, resulting in an exposure area of 432.60 m2. When the stope length is 40 m, the stope span is 12.88 m, resulting in an exposure area of 515.20 m2.

Therefore, for the two scenarios where the stope length is 30 m and 40 m, the limit exposure areas of the stopes are simulated for stope span of 10 m, 12.5 m, 15 m, 17.5 m, and 20 m. The simulation schemes are detailed in Table 3 and plotted on the extended Mathews stability graph, as shown in Fig. 2. It can be observed that, overall, the two scenarios cross each other on the stability graph as the span increases. Schemes 3 and 7 closely follow the 85% stability isoprobability line, indicating that these two schemes can effectively maintain stope stability while maximizing production capacity. Therefore, these two schemes are preferred for stope design.

Table 3 Design of different stope exposure area simulation schemes.
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Fig. 2
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Stability results of stopes with different exposure areas based on the composite Mathews stability graph.

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Additionally, Barton’s permissible span analysis26 was used for verification. The relationship between the maximum unsupported span (SPAN) of the excavation, the Q value, and the excavation support ratio (ESR) is given by the following formula (9):

$$SPAN = 2 \times ESR \times Q^{0.4} .$$
(9)

The excavation support ratio (ESR) is related to the stability requirements of the excavated tunnel. Based on the technical conditions and mining methods of the mine, the studied stope can be considered a temporary tunnel. The ESR for temporary mine tunnels is typically in the range of 3 to 5. Using this range, the maximum unsupported span of the stope is calculated to be 16.30 m. The results from the permissible span analysis are consistent with those obtained using the extended Mathews stability graph method.

Analysis of stope stability considering exposure time

After considering the time factor T, the corrected stability coefficient N* and the corrected stability probability f(y)* were calculated for different exposure periods: less than 3 months, 3–5 months, 5–12 months, and more than 12 months. The results are detailed in Table 4.

Table 4 Statistics of stope stability probabilities with different exposure times.
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By comparing the stability probability f(y)* considering exposure time with the uncorrected f(y), it was found that exposure time had a significant impact on the stability of the mined-out area. After an exposure time of more than 3 months, the stability probability of the stope was consistently below 85%. As the exposure time increases, the stability of the mined-out area decreases. Particularly, for exposure times exceeding 12 months, the stability of the mined-out area is severely compromised, with the stability probability falling below the stable-failure line. This result is consistent with the findings reported by Zhang27.

Additionally, the impact of exposure time was more pronounced for stopes with inherently lower stability. The difference in stability probability between stopes not considering exposure time and those exposed for more than 12 months was more significant when the initial stability probabilities differed by 10%. In such cases, the stability probability of the less stable stope, when considering exposure times greater than 12 months, decreased by approximately 8% more. Therefore, it is essential to ensure the stability of stopes not considering exposure time, and immediate backfilling should be performed after ore extraction to minimize the impact of exposure time on stability.

Numerical model and mechanical parameters

FLAC3D is capable of simulating the mechanical behavior of three-dimensional media in engineering structures28,29,30. During the computation, the simulated medium can undergo yielding and creep, allowing for the prediction of local damage and deformation that may occur in geotechnical engineering. Therefore, FLAC3D was used to analyze the mechanical response of the stope under different excavation exposure areas.

Based on the limit exposed area of stope determined by the extended Mathews stability graph method, considering the process background of the project, it is necessary to further verify its practical engineering feasibility by combining it with the backfill process. Through the collaborative simulation of exposure area and backfill, the control effect of two-step mining on stress redistribution and plastic failure of stope can be quantitatively analyzed, and the collaborative optimization of theoretical parameters of extended Mathews stability graph method and backfill practice can be realized.

In addition, in the stope design and stability analysis of underground mines, the single-stope simulation can not capture the stress redistribution and cumulative effect in the process of multi-stope mining, which may lead to the incomplete and accurate evaluation of stope stability. In contrast, multi-stope collaborative simulation can more comprehensively evaluate the overall stability of the mine. It can not only capture the stress redistribution between multiple stopes, but also evaluate the impact of different mining sequences on global stability, and can more truly reflect the dynamic behavior and potential risks in the mining process. Therefore, concerning stope design, five stopes were established, and the mining process was divided into three steps: first, mining the middle non-adjacent stopes; second, backfilling the mined-out areas; and third, mining the remaining stopes.

Based on the actual conditions of the ore body in the mine, the average dip angle in the computational model was set to 50°, focusing on exploration lines 27 to 31. According to Saint–Venant’s principle, the area where engineering excavation has a significant impact on the stress and displacement of the surrounding rock mass is 3–5 times that of the excavation area31. Therefore, the numerical model dimensions were 600 m in length, 350 m in width, and 215 m in height, using the Mohr–Coulomb strength criterion. The initial stress field is based on the assumption of self-weight stress, with a stress of 11.82 MPa applied to the surface of the model and a horizontal stress coefficient of 0.5 applied to the side of the model accordingly. The final three-dimensional numerical model, as shown in Fig. 3, was based on the actual engineering conditions of the mine, with the side and botom boundaries of the model fixed and the top surface free. The simulation process is shown in Fig. 4. The mechanical parameters of ore and surrounding rock adopted in the numerical simulation were established through laboratory mechanical testing, as listed in Table 5.

Fig. 3
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Numerical analysis model.

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Fig. 4
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The simulation process.

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Table 5 Physical and mechanical parameters of rock mass.
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Through the comparative analysis of historical on-site measurements and numerical simulations of the mine, combined with the practical experience from other underground mine stope excavations18,32, and based on comparisons with reference case mines including similarities in geological conditions such as dipping iron orebodies, rock mass quality classifications of surrounding rock, and mineral compositions, a corresponding relationship between surrounding rock displacement and underground stope stability has been established through comprehensive consideration. According to Table 6, in the subsequent numerical simulation analysis, if the displacement of the roof and floor plates exceeds 45 mm, it is considered that the engineering structure of the rock mass has been damaged.

Table 6 The corresponding relationship between the displacement of surrounding rock and the stability of stope.
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Simulation results

During the simulation, it was found that the stability of the stope after the third step was the poorest. Therefore, the principal stresses and displacements in the surrounding rock mass after the third step of the design scheme were analyzed, as shown in Fig. 5. Except for Scheme 10, where the maximum tensile stress in the surrounding rock mass decreased with increasing stope exposure area, the tensile and compressive stresses at the ends of the stopes in other schemes increased with increasing stope exposure area. Under the same stope exposure area and span conditions, the compressive stress and its rate of change were better for stopes with a length of 40 m compared to those with a length of 30 m. The compressive stress gradient is reduced by about 15% compared to the low aspect ratio scheme with the same exposure area, indicating that appropriately increasing the stope length can improve the stress conditions. It is worth noting that in Scheme 6, where the aspect ratio of the stope is 4, the tensile stress in the surrounding rock of the stope only reaches 0.69 MPa, indicating that under these conditions, the stope is more likely to maintain its self-stability. Additionally, when the stope exposure area reached 600 m2, the maximum tensile stress around the stope approached or exceeded 80% of the tensile strength of the surrounding rock, posing a risk of tensile failure.

Fig. 5
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Principal stress distribution and displacement characteristics of surrounding rock in the stopes. (a) Principal stress distribution for different stope exposure areas; (b) Displacement distribution for different stope exposure areas.

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The displacement of the stope roof and floor increased with the stope exposure area. Under the same stope exposure area, the displacement of stopes with a length of 40 m was consistently higher than that of stopes with a length of 30 m. This is attributed to the influence of the stope shape coefficient, which results in less boundary constraint in the span direction for the longer stopes, thereby releasing more rock displacement. Furthermore, the rate of displacement increase was significantly higher for stopes with a length of 40 m, indicating that the displacement changes in the roof and floor are primarily dominated by the stope length under the same span conditions. When the stope exposure area reached 600 m2, the displacement of the stope roof in stopes with a length of 40 m approached or exceeded 80% of the maximum allowable vertical displacement of the overlying strata, indicating potential instability. In addition, Fig. 6 presents the displacement contour maps of the surrounding rock under four stope exposure area schemes. Numerical simulation results indicate that the maximum subsidence and floor heave are predominantly concentrated in the roof and floor regions of the central mined-out area. The displacement trends in the mined-out areas on both sides of the central stope, symmetrically distributed around it, exhibit convergence toward the central stope. This convergence forms interconnected displacement zones within the surrounding rock mass, which may lead to cumulative deformation of the roof and floor in the central mined-out area, ultimately triggering overall instability. Therefore, to ensure long-term safe production, and considering the stope stability influenced by exposure time, Schemes 5 and 8 to 10 are not recommended for practical application.

Fig. 6
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Vertical displacement of surrounding rock under four exposure area schemes.

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During the three-step mining process, stope instability and failure can occur in two primary forms: one is when the deformation of the stope roof exceeds the allowable range, leading to roof collapse, or when the tensile stress generated in the stope roof exceeds its ultimate tensile strength, causing tensile failure; the second is when the vertical stress on the backfill exceeds its compressive strength, resulting in plastic failure and the overall collapse of the backfill33.

Considering that the color partitioning of the yielding state figure can intuitively identify the category of regional damage patterns, avoid grid dependence, improve the robustness of results, and be associated with on-site damage conditions, the yielding state figure is used here to illustrate the plastic behavior of the region. In Fig. 7, the yielding states of Schemes 1, 5, 6, and 10 are analyzed to comprehensively assess stope stability under various exposure areas, with Scheme 1 serving as a conservative benchmark, Scheme 5 highlighting risks associated with larger exposure areas, Scheme 6 demonstrating favorable mechanical responses, and Scheme 10 representing an extreme case for stability evaluation under maximum exposure.

Fig. 7
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Yielding states in the hanging wall and footwall of stopes after backfilling for four exposure area schemes.

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According to the simulation results shown in Fig. 7, it could be observed that the distribution patterns of plastic zones during the mining process were generally similar for different stope exposure areas, concentrating at the junctions between the voids and the backfill, as well as at the end side walls. The depth of plastic failure increased with the increase in the exposure area. When the stope exposure area was less than 500 m2, sporadic red regions (shear-n, shear-p) appeared near the backfill, indicating good structural stability. As the stope exposure area increased, the range of plastic zones gradually expanded. When the stope exposure area exceeded 600 m2, significant plastic zones appeared in the stope roof and side walls, and tensile failure occurred in the roof, indicating potential instability and high safety risks during mining operations. Additionally, the connectivity between the voids and backfill in the upper part of the hanging wall and the lower part of the footwall was generally better for stopes with a length of 40 m compared to those with a length of 30 m.

Figure 8 presents the plastic failure volume after backfilling for different stope exposure areas. The plastic failure value is calculated based on the Mohr–Coulomb failure criterion implemented in FLAC3D. The software iteratively calculates the stress and strain distribution within the model, automatically identifies plastic state elements that meet the failure criteria within the region, and uses state variables to track and superimpose the plastic failure values of each tetrahedral subregion within the area.

Fig. 8
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Plastic volume for different stope exposure areas.

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Overall, the failure volume increased with the increase in stope exposure area, and the tensile failure volume was much smaller than the plastic failure volume. Additionally, considering the effect of the stope length-to-width ratio being close to 4, there was a significant improvement in the tensile failure volume for Scheme 6 and the shear failure volume for Scheme 7. Therefore, taking into account the distribution of plastic zones around the stope and the failure volume, it was recommended to adopt the stope exposure area of Scheme 6 or Scheme 7. These recommendations aligned with the stability results obtained from the extended Mathews stability graph.

Conclusions

  1. 1.

    The stability isoprobability lines were added to the Mathews stability graph to form a composite graph. The stability shape coefficient for a specific mine stope was calculated to be 3.80, and the stope shape coefficient was 4.87. It was found that the two schemes with dimensions 30 m × 15 m and 40 m × 12.5 m closely followed the 85% stability isoprobability line and were within the allowable limit span, meeting the stope stability requirements.

  2. 2.

    The longer the exposure time of the stope, the lower the stability probability of the mined-out area. To ensure safe and reliable backfilling, the stability probability should be above 85%, meaning the exposure time should not exceed 3 months. Additionally, the impact of exposure time is more significant for stopes with inherently lower stability. Therefore, it is essential to ensure the stability of stopes without considering exposure time, and timely backfilling should be performed after ore extraction to minimize the impact of exposure time on stability.

  3. 3.

    Numerical simulations showed that, under the same stope exposure area and span conditions, appropriately increasing the stope length improves the stress conditions of the stope, and the displacement changes in the roof and floor are primarily dominated by the stope length. The plastic zones during mining are concentrated at void-backfill junctions and end side walls, with failure depth increasing as the exposure area grows. Combining the results from the extended Mathews stability graph method and numerical simulations, to ensure the safety of the backfilling process, it is recommended that the limit exposure area for stopes with lengths of 30 m and 40 m be 450 m2 and 500 m2, respectively.