Optimization of the matching relationship between the stemming length and minimum burden in cut blasting of large-diameter long-hole stopes

Abstract

Cut blasting, in which new surfaces and relief space for subsequent blasting are created, is one of the most critical steps in the establishment of large-diameter long-hole (LDL) stopes. To reduce the damage to the chamber roof caused by stemming recoil and improve the rock breaking effect, 15 groups of small-scale model tests with minimum burdens of 3, 4, 5, 6, and 7 cm and stemming lengths of 0, 2, 4, 5, 6, and 7 cm were designed to optimize the matching relationship between the stemming length and minimum burden. First, through the model tests, values were obtained for ten evaluation indexes related to the total mass of fragments, crate size, fragment size, fragmentation energy consumption, and stemming recoil area. Then, the normal cloud combination weighting method was used to combine six subjective and objective weighting methods, and combined weights were obtained. Finally, the test schemes were optimized according to the Euclidean distance and similarity. The test results showed that the best blasting scheme involves a burden of 5 cm and a stemming length of 5 cm, followed by that involving a burden of 4 cm and a stemming length of 4 cm, and the optimal stemming length is approximately equal to the minimum burden. A field test of LDL stope cut blasting was conducted, with a stemming length of 2.2 m and a minimum burden of 2.2 m in the boreholes. The highly satisfactory field blasting effect indicates that the stemming length and minimum burden are reasonable.

Highlights

• Ten indexes were obtained and normalized for each of the eight test schemes. Six weighting methods were used to obtain the respective weights, and the normal cloud combination weighting method was used to obtain more reasonable weights.

• The optimal scheme was that involving a burden of 5 cm and a stemming length of 5 cm, followed by that involving a burden of 4 cm and a stemming length of 4 cm. The optimal stemming length was approximately equal to the minimum burden.

• An on-site cut blasting test was conducted of the N10#S stope in the Fankou Mine. The upper part of the explosive packs exhibited a stemming length of 2.2 m, and the lower part exhibited a burden of 2.2 m. The highly favorable field blasting effect proves that the above stemming length and minimum burden are reasonable.

1 Introduction

Optimizing the stemming length and minimum burden are very important for improving the blasting effect, especially in first cut blasting of an underground large-diameter long-hole (LDL) stope with one free surface. Cut blasting is the first step in LDL mining because it artificially creates new free surfaces and relief space for subsequent blasting (Wang et al. 2018). However, in the actual process of cut blasting, an unreasonable matching relationship between the stemming length and minimum burden usually leads to the destruction of the chamber roof steel mesh due to stemming recoil, and many large fragments or fine ore particles are produced after blasting. This causes potential safety hazards, affecting mining profits and increasing the workload of secondary ore fragmentation. Hence, it is imperative to enhance the blasting fragmentation efficiency and mitigate the adverse effects of stemming recoil by optimizing the matching relationship between the stemming length and minimum burden.

The initial cut blasting configuration is a crucial component of modern blasting techniques and is frequently employed in scenarios such as narrow vein mining, vertical crater retreat (VCR) methods, and similar cases. However, there is a lack of research dedicated to LDL mining, particularly regarding the optimization of the matching relationship between the stemming length and minimum burden. Stemming can improve the blasting performance and reduce the blasting costs (Zhang et al. 2021c). However, stemming for too long can increase the clogged borehole ratio, thus increasing the work cost of borehole cleaning (through the borehole). In open-pit blasting, excessive stemming also causes a rapid increase in the large boulders (Cevizci and Ozkahraman 2012). Insufficient stemming may lead to the stemming structure being flushed out and detonation gas prematurely escaping from the orifice, resulting in a poor blasting effect. In parallel, the minimum burden directly affects the effect of rock breaking. Many scholars have researched the matching relationship between the stemming length and minimum burden to optimize the blasting effect in recent years. For example, in open-pit bench blasting, the stemming length is generally 0.7~1.0 times the minimum burden or 20~30 times the borehole diameter (Chen et al. 2021). Oates and Spiteri (2021) proposed that a non-stemming method is often needed for blasting in hot holes and reactive ground, or when blasting a presplit. The blasting effect is best for larger-diameter boreholes when the stemming length is 0.7 times the burden. For smaller diameter boreholes, the stemming length is generally 20~30 times the borehole diameter. Bilgin (1991) explored the influence of the burden on the blasting effect in bench blasting and found that when the burden is almost equal to half of the bench height, the maximum volume of broken ore was achieved. Prasad et al (2017) found that the mean fragment size and maximum fragment size increased with increasing stemming length-to-burden ratio in open-pit bench blasting. There is also literature suggesting that the rock fragmentation process is completed at the moment the stemming structure is flushed out of the borehole, and the stemming length at this time is the optimal length (Chen et al. 2021; Zhao et al. 2010). Additional pertinent studies on the optimization of the matching relationship between the stemming length and burden, stemming length optimization, or burden optimization are listed in Table 1. However, the above studies are aimed at open-pit bench blasting or tunneling but do not consider cut blasting in the LDL mining method. Cut blasting, open-pit blasting, and tunneling blasting are subjected to different confinement conditions, e.g., cut blast holes typically involve only one free face, and these methods also exhibit variations in their charging structures and borehole layouts. The above is crucial, indicating that the topic of our study is cut blasting of large-diameter long-hole stopes, an area of research that has received relatively less attention. Moreover, cut blasting exhibits distinct confinement conditions and blasting schemes relative to open-pit blasting and tunneling blasting. Parameter optimization research of the LDL mining method mainly focuses on the optimization of stemming materials (Qiu et al. 2018a; Shi et al. 2023), propagation of blasting vibration (Gou et al. 2021), rock breaking mechanism (Qiu et al. 2018c; Zhang et al.2023), and charging structure optimization of lateral caving blasting (Huo et al. 2020). There is very little research on optimizing the matching relationship between the minimum burden and stemming length of cut blasting. Scholars have conducted numerical simulations to optimize the parameters of one-step raise excavation (Liu et al. 2018) and optimal design of cut blasting in high-stress rock masses (Xie et al. 2017). However, they did not analyze the rock breaking effect and stemming recoil range of the cut blasting.

Table 1 Relationship between the stemming length and burden and stemming length or burden optimization equation

The optimization of matching relationships necessitates the comprehensive consideration of various criteria related to engineering activities. However, this task should also address the subjective influence of human judgment and account for the inherent uncertainty in data. In other words, the selection of the optimal matching relationship between the stemming length and minimum burden in cut blasting is a complex decision-making problem with multiple objectives, multiple factors, and multiple indexes. Moreover, the selection of the blasting scheme is affected by many uncertain factors. Therefore, it is imperative to choose an optimization method that effectively integrates the advantages of both subjective and objective weighting methods. The optimization methods for blasting schemes mainly include (1) the subjective weighting optimization method and (2) the objective weighting or combination weighting optimization method. For example, scholars have adopted the analytic hierarchy process (AHP) combined with blasting cost indexes to optimize the blasting scheme (Wang and Zhou 2011). Other scholars have adopted the AHP and the technique for order preference by similarity to ideal solution (TOPSIS) combined with economic, technical indexes, and safety indexes to optimize the crown-sill pillar blasting scheme (Shi et al. 2015). In addition, scholars have used combination weight based on game theory (CW-GT) and weighted TOPSIS improved by gray correlation (GC-WTOPSIS) methods to optimize the crown-sill pillar blasting scheme (Zhang et al. 2018). However, the AHP is a subjective technique and is greatly affected by expert knowledge or scoring experience (Yang et al. 2010). The weight selection methods adopted in these optimization models are relatively simple but cannot sufficiently reflect the uncertainty and volatility of weights. In contrast, gray correlation analysis does not consider the relative importance of each factor. In recent years, based on traditional fuzzy mathematics theory and probability statistics, Li (Li 1997; Li et al. 2009) has proposed the uncertainty conversion model between qualitative concepts and quantitative values—cloud model. With the use of the normal cloud combination weighting (NCCW) method, a variety of subjective and objective weights can be fused, and information uncertainty and fusion problems can be effectively solved. Therefore, this method is widely used in underground space suitability evaluation (Tan et al. 2021), water resources and water inrush evaluation (Peng and Deng 2020; Peng et al. 2020), flight control system fault forecasting (Dong et al. 2019), and risk assessment for energy security (Chen et al. 2022). However, there are few relevant reports on the optimization of cut blasting schemes. In this study, the cloud model was used to obtain combined weights and optimize the blasting scheme, which can address the weight uncertainty problem in cut blasting scheme optimization.

To optimize the effectiveness of cut blasting and reduce the detrimental effects of stemming recoil, with the aim of improving both the blasting mining efficiency and mining safety, 15 groups of small-scale model test blocks with burden lengths of 3, 4, 5, 6, and 7 cm and stemming lengths of 0, 2, 4, 5, 6, and 7 cm were prepared to optimize the matching relationship between the stemming length and minimum burden. First, ten indexes related to the blasting effect were selected. Then, six subjective and objective weighting methods were used to determine the weights. The NCCW method was used to obtain a comprehensive weight that can consider the uncertainty in subjective and objective weight information. Finally, the best scheme based on the matching relationship between the stemming length and burden was obtained. The optimal stemming length and burden parameters were applied in cut blasting of the LDL stope of the Fankou Lead-Zinc Mine, resulting in a commendable fragmentation performance.

2 Materials and methods

2.1 Description of the site

The Fankou Lead–Zinc Mine, situated in Guangdong Province, China, is one of the pioneering mines to employ the LDL mining method, and the layout diagram of the LDL stope is shown in Fig. 1 (Qiu et al. 2018b; Wang et al. 2018). The whole stope can be divided into a cut blasting area, an ore roof breaking blasting area, and a lateral blasting area. The first few cut blasting operations can generate more relief space and a larger free surface for subsequent blasting. After the last layer of ore in the goaf roof has been broken, 2 ~ 3 lateral blasts are conducted for the ore body that remains in the stope. A favorable cut blasting effect is significant for the subsequent mining of the stope. However, in actual blasting production, the phenomenon depicted in Fig. 2 often occurs. The stemming structure is violently flushed out of the borehole, and the chamber roof is destroyed due to stemming recoil. Furthermore, numerous fragments larger than 600 mm or even 1000 mm are created in the lower chamber, potentially posing a safety risk to later production and increasing the work of secondary fragmentation. These phenomena are often caused by an unreasonable matching relationship between the stemming length and minimum burden. To this end, we performed model tests to optimize the matching relationship between the stemming length and minimum burden in cut blasting of the LDL stope to improve the blasting effect.

Fig. 1

Mining sequence arrangement of the LDL stope and blast hole charge structure in the cut blasting area

Fig. 2

After cut blasting, the upper chamber roof steel mesh is damaged, and many large fragments are produced in the lower chamber

As shown in Fig. 1, there are generally three rounds of boreholes in the cut blasting area. The first round comprises nine boreholes, including five empty holes with a diameter of 200 mm and four blast holes with a diameter of 110 mm. The five empty holes provide compensation space for the initial blast. Two packs of emulsion explosives were continuously loaded into the blast holes, the diameter of a single package is 90 mm, and the length is 670 mm. The vertical distance from the bottom of the explosive packages to the single free surface is the minimum burden with a length of l₁ (generally, the minimum distance from the charge center to the free surface is referred to as the minimum burden; according to site blasting practices, the vertical distance from the charge bottom to the free surface is referred to as the minimum burden for the convenience of construction and measurement). The upper part of the charge is stemmed with a length of l₂, and the distance from the top of the employed stemming to the upper chamber floor is l_d. According to the equivalent charge diameter criterion (Sun et al. 2006; Zhang et al. 2022), the equivalent charge diameter is 360 mm ($D_{e} =\sqrt {n{}_{e}} \cdot d$, where D_e is the equivalent charge diameter, n_e is the number of blast holes, d is the diameter of blastholes, $\sqrt {16} \times 90=360\;{\text{mm}}$), and the length diameter ratio is 3.7 (670 × 2/360≈3.7).

2.2 Model test design

In this study, the matching relationship between the stemming length and minimum burden in cut blasting was studied by the model test method. The length of the cut blasting area must not exceed 20 m, and the height must not exceed 15 m. However, it is unnecessary and unrealistic to prepare a test model of the actual size, so a similar simulation test is conducted by considering the similarity factor using similar materials. Fifteen sets of test blocks were prepared with a rectangular steel die. The size of the test blocks is length × width × height = 400 × 400 × 300 mm, and a 300-mm long, 8-mm diameter borehole was centrally positioned in each specimen, as shown in Fig. 3. The primary objective of this study was to investigate the blasting effectiveness under different combinations of stemming lengths and minimum burdens. Therefore, uniform electronic detonators were used in the experiments, and the model size and mechanical parameters were consistent. The initial configuration of the cut blasting hole in the test specimens is a blast with one free surface, and cut blasting generates more relief space and a larger free surface for subsequent mining operations. In this test, a digital electronic detonator with a diameter of approximately 8 mm was used. The charge length is approximately 30 mm, and the length diameter ratio is also 3.7. Therefore, the model test charges share the same length diameter ratio as the charges used in the field, ensuring geometric similarity between the model and actual field conditions. According to the similarity criterion, the similarity ratio is a quantitative indicator to measure the similarity between actual engineering and test models, and the geometric similarity ratio K_L is determined at 45 (360/8 = 45). In addition to geometric parameters, matching similarity related to the matching relationship between explosives and rock is a crucial parameter. Its significance lies in the fact that when the model and field conditions meet the matching similarity criteria, this ensures that the experimental blasting process and results are similar to those in the field (Yang 1987). Typically, the matching similarity can be calculated as follows:

$$\frac{{K_{D}^{2} K_{\rho B} }}{{K_{c1}^{2} K_{\rho 0} }}$$

(1)

where K_D is the explosive detonation velocity similarity ratio; K_ρB is the explosive density similarity ratio; K_c1 is the rock P wave velocity similarity ratio; and K_ρ0 is the rock density similarity ratio. By substituting the model test and field blasting parameters into the above equation, we can obtain [(6000/5000)² × (1.13/1.35)] / [(3590/3400)² × (2151/2500)] = 1.4/1.1, and the calculated result close to 1, indicating that the matching similarity is essentially satisfied.

Fig. 3

Experimental setup for blasting; a picture of the test block; b sketch maps of the specimens

Several groups of cubic blocks with length × width × height = 75 × 75 × 75 mm were poured involving the same mixture to obtain the physical and mechanical parameters of the test blocks. After 28 days of curing, we measured the mechanical properties of the samples. Table 2 lists the basic mechanical parameters of the lead–zinc ore and test blocks. In terms of the mechanical property similarity of materials, both the strain similarity ratio K_ε and Poisson's ratio similarity ratio K_µ are equal to 1. According to mechanical similarity theory (Yan et al. 2023; Zhu et al. 2023), the geometric similarity ratio K_L, density similarity ratio K_ρ0, and strength similarity ratio K_σ should meet the following criteria: K_σ = K_p0 K_L. After calculation, we find that K_σ is 52. This suggests that the compressive strength of the similar material should reach approximately 103.5 ÷ 52 ≈ 2.0 MPa. However, in the blasting experiments, it was observed that using specimens with a uniaxial compressive strength of 2.0 MPa did not produce normal blast craters and fragments (often resulting in complete fracturing of the specimen). Additionally, specimens with such a low strength were highly susceptible to breakage during curing and transportation. Subsequent experiments revealed that specimens with a uniaxial compressive strength of at least 20 MPa were needed to generate proper fragments and blast craters without complete fracture. Therefore, we ultimately used a mixture of rod-mill tailings (particle size: 1 ~ 5 mm) from the Fankou Mine and high-strength #42.5 ordinary Portland cement in a 1:1 ratio to manufacture the specimens, achieving a uniaxial compressive strength of 20.5 MPa. Given that the production of fragments and craters during blasting is a critical evaluation criterion in this study, it is considered reasonable to increase the specimen strength appropriately.

Table 2 Physico-mechanical parameters of the materials

The aim of this study is the optimization of the stemming length and minimum burden in cut blasting to avoid stemming recoil from impacting the chamber roof and achieve a better rock-breaking effect. Therefore, the test device shown in Fig. 3b was developed. Two bricks with a height of 6.5 cm support the test block to simulate the lower chamber with a height of 3 m. A piece of white paper with scale marks was laid atop the test block to monitor the recoil range generated by blasting. The digital electronic detonators were installed in reverse. There was only a 3-cm long explosive at the end of the detonator. Therefore, we regarded the vertical distance from the charge bottom to the free surface as the minimum burden l₁, and the other parts of the detonator were regarded as part of the burden. After blasting, the fragments were collected, screened, and analyzed. Fifteen groups of model blasting tests with stemming lengths of 0, 2, 4, 5, 6, and 7 cm and minimum burdens of 3, 4, 5, 6, and 7 cm were conducted, as summarized in Table 3.

Table 3 Parameters of the fifteen experiments

2.3 Determination of the evaluation indexes

As shown in Table 4, ten evaluation indexes U1-U10 were set for each test scheme. These intuitive indexes include the total mass of fragments, mean radius, and crater area. The fragments size index includes the maximum fragment size et al. The fragmentation energy consumption index and stemming recoil range index were also included. Among the ten indexes, U1 to U5 and U10 are mainly related to geometric similarity, U6 to U9 are mainly related to matching similarity.

Table 4 Ten evaluation indexes

(1) The crater radius and area were measured using the measurement method shown in Fig. 4. The average radius R₁-R₈ of the blasting crater along eight directions was adopted to obtain the mean crater radius R, and the crater area can be obtained through contour measuring.

Fig. 4

Measurement of the crater size: Crater radius and crater area

(2) The reasonable fragment size refers to fragments of a moderate size, and the particle size refers to particles that are larger than fine ore but smaller than large blasting fragments. Generally, in actual mining production, a fragment size less than 30 mm is defined as fine ore because a fragment size below 30 mm is difficult to move, which can easily cause ore loss. A fragment larger than 500 mm is defined as a large fragment because further fragmentation is needed.

(3) A portion of the energy released by the explosive is used to break and fragment the rock, which is denoted as the fragmentation energy consumption E_p (Shi et al. 2023; Zhang et al. 2023). The fragmentation energy consumption is proportional to the newly increased surface area of rock (Hamdi et al. 2001), and it can be calculated by Eq. (2) (Liu and Katsabanis 1997; Lei et al. 2021).

$$E_{p} = \frac{{2K_{ic}^{2} A_{x} }}{{E_{d} }} = \frac{{2K_{ic}^{2} (A - A_{y} )}}{{E_{d} }}$$

(2)

where A is the surface area of all broken blocks, A_y is the original surface area of the block, A_x is the new surface area in the fragmentation process, and K_ic is the fracture toughness coefficient of the rock.

The rock fragment size is classified by a grading sieve, and y(d_i) is the cumulative mass percentage of each grade of rock fragments. All rock fragments are treated as cubes, and the total surface area A of the rock fragments can be calculated by Eq. (3).

$$A = \sum\limits_{i = 1}^{n} {A_{i} } = \sum\limits_{i = 1}^{n} {\frac{{T\left[ {y(d_{i} ) - y(d_{i - 1} )} \right]6\overline{d}_{i}^{2} }}{{\rho_{rf} \overline{d}_{i}^{3} }}} = \sum\limits_{i = 1}^{n} {\frac{{12T\left[ {y(d_{i} ) - y(d_{i - 1} )} \right]}}{{\rho_{rf} (d_{i - 1} + d_{i} )}}}$$

(3)

where ρ_rf is the density of the rock fragments, and T is the total mass of each grade of rock fragments.

By combining Eq. (2) and Eq. (3), the rock fragmentation energy based on the new surface area of the rock fragments is:

$$E_{p} = \frac{{2K_{ic}^{2} }}{{E_{d} }}\left[ {\frac{12T}{{\rho_{rf} }}\sum\limits_{i = 1}^{n} {\frac{{\left[ {y(d_{i} ) - y(d_{i - 1} )} \right]}}{{(d_{i - 1} + d_{i} )}} - A_{y} } } \right]$$

(4)

Therefore, the fragmentation energy density e can be calculated from the ratio of the fragmentation energy E_p to the total volume V of the fragments. A higher e value indicates that the rock is more notably fragmented. The utilization ratio of fragmentation energy η can be calculated by the ratio of the fragmentation energy consumption to the total chemical energy of explosives (Zhang et al. 2023).

$$\eta = \frac{{E_{p} }}{SQ}$$

(5)

where S is the explosive heat, and Q is the charge quantity.

2.4 Normalization of the evaluation indexes

These ten evaluation indexes for matching relationship optimization are related to the total mass of fragments, crater size, fragment size, fragmentation energy consumption, and stemming recoil area. According to the nature of these evaluation indexes, they can be categorized as positive, reverse and moderate indexes. Positive indexes, also known as benefit-oriented or production-oriented indexes, are preferable when the values are higher. Reverse indexes, also referred to as cost indexes, are more desirable when the values are lower. Moderate indexes are favorable when the values fall within a specific data range (Wu et al. 2021; Yu et al. 2009). According to the classification criteria mentioned above, it is evident that U1 to U3 and U6 to U9 are positive indexes, U4 and U10 are reverse indexes, and U5 is a moderate index. Due to the different measurement units of each index and to eliminate any dimensional effects among the evaluation indexes, it is necessary to normalize the evaluation indexes when comprehensively assessing the matching relationships. The original data of the positive, reverse and moderate indexes can be processed as follows:

$${\text{Positive}}\;{\text{Indexes}}:r_{ij} = \frac{{x_{ij} }}{{x_{i\max } + x_{i\min } }},\;x_{ij} \ge 0$$

(6)

$${\text{Reverse}}\;{\text{Indexes}}:\;r_{ij} = 1 - \frac{{x_{ij} }}{{x_{i\max } + x_{i\min } }},\;x_{ij} \ge 0$$

(7)

$${\text{Moderate}}\;{\text{Indexes}}:\;r_{ij} = \left\{ {\begin{array}{*{20}l} {1 - \frac{{x_{imid} - x_{ij} }}{{\max (x_{imid} - x_{ij} )}},} \hfill & {x_{ij} < x_{imid} } \hfill \\ {1,} \hfill & {x_{imid} \le x_{ij} \le x_{imid}^{\prime } } \hfill \\ {1 - \frac{{x_{ij} - x_{imid}^{\prime } }}{{\max (x_{ij} - x_{imid}^{\prime } )}},} \hfill & {x_{imid}^{\prime } < x_{ij} } \hfill \\ \end{array} } \right.$$

(8)

where r_ij denotes the normalized final results and x_ima and x_imin denote the maximum and minimum values of the i-th index, respectively, with the interval of [x_imid, x´_imid] representing the optimal range of the i-th index.

3 Model test results and analysis

3.1 Total mass of fragments, mean radius and area of the blasting crater

After blasting the test block, the fragments were collected. As shown in Fig. 5, 11 of the 15 test blocks generated fragments, while no fragments were produced in tests 4, 5, 12, and 15. The 11 groups of test blocks formed fragments of various sizes, including very fine particles, while some test blocks also produced larger fragments, and the maximum fragment size in some tests reached 52 mm.

Fig. 5

Fragment distribution of 11 specimens (fragments are not produced in tests 4, 5, 12, and 15) (the handwritten numbers in each test block are arranged in chronological order, independent of the test order)

Figure 6a shows the total weight of the fragments produced by each test block. Craters were not produced in tests 4, 5, 12, and 15, and the mass of the fragments is 0. In the remaining 11 groups of tests, fragments were produced, and the total weight of the fragments ranged from 21.5 g in test 1 to 294.31 g in test 14. Figure 6b shows the mean radius and area of the blasting crater. The mean radius ranged from 2.46 cm in test 3 to 9.21 cm in test 14, and the crater area ranged from 33.24 cm² in test 1 to 270.27 cm² in test 14.

Fig. 6

Total mass of the fragments, mean radius of the blasting crater, and crater area. a Total weight of each test block in the 15 test schemes (among them, the total weight in tests 4, 5, 12, and 15 is 0). b Crater radius and crater area in the 11 tests

Figure 6a shows that the total weight, crater mean radius, and crater area in tests 6 to 15 (except tests 12 and 15) with stemming are significantly larger than those in tests 1 to 5 (except tests 4 and 5) without stemming. Therefore, the data for tests 1, 2, and 3 are omitted in the following statistical analysis, and only the remaining eight groups of test data are analyzed later.

3.2 Fragment size

Figure 7a, b show the maximum fragment size, mean fragment size, proportion of fragments with a reasonable size, and fractal dimension of the fragments generated by eight groups of test blocks. The maximum fragment size ranges from 30 mm in test 9 to 52 mm in test 11, the mean fragment size ranges from 17.45 mm in test 9 to 29.51 mm in test 14, the proportion of the fragments with a reasonable size ranges from 17.9% in test 8 to 69.4% in test 9, and the fractal dimension ranges from 1.6 in test 11 to 2.62 in test 13.

Fig. 7

Maximum fragment size, average fragment size, proportion of fragments with a reasonable size, and fractal dimension. a Maximum fragment size, an average block size in the eight tests. b Proportion of fragments with a reasonable size and the fractal dimension in the eight tests

3.3 Fragmentation energy consumption

The fragmentation energy consumption density and the utilization rate of fragmentation energy of the eight groups of test blocks calculated by Eqs. (2) to (5) are shown in Fig. 8. The fragmentation energy consumption density ranged from 615.4 kJ/m³ in test 6 to 5020.2 kJ/m³ in test 13, and the utilization rate of fragmentation energy ranged from 0.58% in test 8 to 11.75% in test 13.

Fig. 8

Fragmentation energy consumption density and utilization rate of fragmentation energy in the eight tests

3.4 Stemming recoil area

Figure 9 shows the stemming recoil area of the eight groups of tests. As shown in the figure below, the recoil area under the different stemming lengths and burdens varies, and the recoil area ranges from 3.41 cm² in test 14 to 70.7 cm² in test 8. The recoil range in tests 13 and 14 with larger stemming lengths is minimal, all within 4 cm², while the recoil range in tests 6, 7, 8, and 11 with smaller stemming lengths is relatively large, exceeding 35 cm². The figure also reveals that the longer the stemming length is under the same burden, the smaller the recoil area.

Fig. 9

Stemming recoil area in the 8 groups of tests

3.5 3.5 Summary of the test results

Based on the above analysis, we preliminarily selected eight out of the 15 schemes. We summarized the experimental results for the 10 evaluation indexes of each scheme, and the outcomes are listed in Table 5.

Table 5 Summary of the experimental results for the 10 evaluation indexes of the eight schemes

4 Test scheme optimization

Selecting the optimal matching relationship between the stemming length and minimum burden in cut blasting is a complex decision-making problem with multiple objectives. For example, the crater area in test 6 is large, but the reasonable fragment size proportion is not high, and the recoil range is extensive. Test 9 exhibits the highest reasonable fragment size proportion, but its fragmentation energy utilization rate is not high. The total mass in test 14 is the largest, but the proportion of fragments with a reasonable size is relatively low. Therefore, all ten evaluation indexes of each blasting scheme must be considered during optimization, and the weight of the indexes must be reasonably determined. Scheme optimization cannot be based only on the subjective choice of personal preference, consideration of engineering activities or objective data information. Combining the advantages of subjective and objective optimization methods is necessary. In this regard, the cloud model optimization method is the preferred method.

4.1 Cloud model

The cloud model is an uncertain transformation model that can be used to transform quantitative and qualitative concepts (Li et al. 2009). As shown in Fig. 10, it can be described by three numerical features, i.e., expectation (E_x), entropy (E_n), and super-entropy (H_e), where E_x is the expectation of cloud droplets in domain space distribution theory. E_n is entropy, a measure of qualitative concept uncertainty. The smaller the E_n, the easier it is to describe quantitatively. Super-entropy H_e reflects the dispersion degree of cloud droplets. That is, H_e reflects the uncertainty of E_n (Wang et al. 2014, 2021).

Fig. 10

Three parameters of the normal cloud model (Wang et al. 2003)

The basic idea of the comprehensive evaluation method for matching relationship optimization between the stemming length and minimum burden based on the normal cloud combination weighting method is as follows: first, various subjective and objective weighting methods are used to obtain weights, and a cloud generation algorithm is used to generate cloud model combination weights from the multiple weights. Then, information fusion of the cloud model combination weights and the decision matrix of the evaluation scheme is performed to generate a comprehensive decision, and a comprehensive cloud model of each test scheme is obtained. Finally, the Euclidean distance similarity (Liao et al. 2015) between the comprehensive normal cloud model of each test scheme and the positive ideal scheme is calculated to obtain the ranking result of each test. The combined weight of the cloud model combines the advantages of subjective and objective weights, yielding a more reasonable evaluation result. The main steps of cloud model evaluation are as follows (Martin et al. 2017):

Step 1: To evaluate the blasting matching relationship optimization results involving l test schemes and y indexes, a decision matrix of the evaluation indexes is established. The decision matrix of the test scheme M_i (i = 1,2,…,l) of the indexes N_j (j = 1,2,…,y) is converted into a standardized matrix $V = \left( {\nu_{ij} } \right)_{l \times y}$, and the idealized scheme $M^{ + }$ of the decision scheme is denoted as $V^{ + } = \left( {\mathop {\max }\limits_{i} \nu_{i1} ,\mathop {\max }\limits_{i} \nu_{i2} , \ldots ,\mathop {\max }\limits_{i} \nu_{iy} } \right)$.

Step 2: A total of a₁ subjective weighting methods (such as the analytic hierarchy process (AHP) and fuzzy comprehensive evaluation (FCE) technique) and a₂ objective weighting methods (such as the entropy method, CRITIC, coefficient of variation method (CVM), and Information quantity (IQ)method) are adopted, and index weights are obtained as $W =\left( {W_{1} ,W_{2} , \ldots ,W_{a} , \ldots ,W_{A} } \right)$, (A = a₁ + a₂); among them, $W_{a} = \left( {\omega_{a1} ,\omega_{a2} , \ldots ,\omega_{ay} } \right)$, (for a = 1, 2, …, A). Then, the A weight samples can yield the following index weight matrix:

$$W =\left[ {\begin{array}{*{20}c} {\omega_{11} } & \cdots & {\omega_{1y} } \\ {\omega_{a1} } & \cdots & {\omega_{ay} } \\ {\omega_{A1} } & \cdots & {\omega_{Ay} } \\ \end{array} } \right]$$

(9)

Step 3: With the use of the cloud generator algorithm to generate the NCCW $W_{j}^{\prime } = \left( {E_{xj} ,E_{nj} ,H_{ej} } \right)$ of each index, where E_xj, E_nj, and E_ej are the expectation, entropy, and super-entropy of the j-th cloud combination weight, respectively, (j = 1,2,…,y), the primary process is as follows (Zhou et al. 2019):

$$E_{xj} = \frac{1}{A}\sum\limits_{a = 1}^{A} {\omega_{aj} }$$

(10)

$$E_{nj} = \sqrt {\frac{\pi }{2}} \frac{1}{A}\sum\limits_{a = 1}^{A} {\left| {\omega_{aj} - E_{xj} } \right|}$$

(11)

$$H_{ej} = \sqrt {\left| {\frac{1}{A - 1}\sum\limits_{a = 1}^{A} {\left( {\omega_{aj} - E_{xj} } \right)}^{2} { - }E_{nj}^{2} } \right|}$$

(12)

Step 4: For y NCCW $W_{1}^{\prime } \left( {E_{x1} ,E_{n1} ,H_{e1} } \right)$, $W_{2}^{\prime } \left( {E_{x2} ,E_{n2} ,H_{e2} } \right)$,…, $W_{j}^{\prime } \left( {E_{xj} ,E_{nj} ,H_{ej} } \right)$,…, $W_{y} \left( {E_{xy} ,E_{ny} ,H_{ey} } \right)$ and constant λ_j (j = 1,2,…,y), the Eqs. (13)-(15) can be used to calculate the comprehensive cloud model with normal cloud weighting of l test schemes $C_{i} \left( {E_{x} ,E_{n} ,H_{e} } \right)$ and the ideal scheme $C^{ + } \left( {E_{x} ,E_{n} ,H_{e} } \right)$ (i = 1,2,…,l).

$$E_{x} = \sum\limits_{j = 1}^{y} {\lambda_{j} E_{xj} }$$

(13)

$$E_{n} = \sqrt {\frac{1}{y}\sum\limits_{j = 1}^{y} {\lambda_{j} E_{nj}^{2} } }$$

(14)

$$H_{e} = \sqrt {\frac{1}{y}\sum\limits_{j = 1}^{y} {\lambda_{j} H_{ej}^{2} } }$$

(15)

Step 5: Eqs. (16)-(17) can be used to calculate the Euclidean distance similarity between scheme M_i and positive ideal scheme $M^{ + }$. The Euclidean similarity represents the distance between any two points in m-dimensional space. It can be considered that larger the distance between two points is, the lower the similarity between the feature vectors. In contrast, if the distance is small or even zero, the similarity of the feature vectors is highly significant.

In this method, the comprehensive cloud models $C_{i} \left( {E_{x} ,E_{n} ,H_{e} } \right)$ and $C^{ + } \left( {E_{x} ,E_{n} ,H_{e} } \right)$ of schemes Mi and M⁺ are regarded as two points $u\left( {E_{xu} ,E_{nu} ,H_{eu} } \right)$ and $v\left( {E_{xv} ,E_{nv} ,H_{ev} } \right)$, respectively, in the three-dimensional coordinate system. The expectation E_x, entropy E_n, and super-entropy H_e are mapped to three-dimensional space. The distance dis(u, v) between the two points is calculated. The exponential function is used as the standardization function to convert the Euclidean distance into the similarity value. The value of f [dis (u,v)] can be defined within the range of [0,1] by Eq. (16). When the function value is close to 1, this indicates that the distance between this scheme and the ideal scheme is the smallest, the similarity is the highest, and this scheme performs the best. When the function value approaches 0, the lowest similarity occurs, and this scheme performs the worst.

$$f\left[ {dis\left( {u,v} \right)} \right] = e^{ - dis(u,v)}$$

(16)

$$dis\left( {u,v} \right) =\sqrt {\left( {E_{xu} - E_{xv} } \right)^{2} + \left( {E_{nu} - E_{nv} } \right)^{2} + \left( {H_{eu} - H_{ev} } \right)^{2} }$$

(17)

4.2 Optimization results and analysis

Ten evaluation indexes U1-U10 for each of the eight blasting schemes were obtained by collecting test data. The data are normalized, and then the optimal value of each index is taken as the ideal scheme. Table 6 shows the results.

Table 6 Standardized index values of the eight groups of test schemes

Two subjective weighting methods, including AHP and FCE, and four objective weighting methods, including Entropy, CRITIC, CVM, and IQ methods, were used to weight ten indicators. Table 7 provides the results. Table 7 indicates that the weights of the indexes obtained by the six subjective and objective weighting methods are very different, indicating the uncertainty in the index weights. Therefore, it is not recommended to use only a single weighting method.

Table 7 Subjective and objective weights of each index

The NCCW method was used to obtain more reasonable weights in this study. In this method, the cloud generation algorithm is used to combine the six kinds of subjective and objective weights, and the normal cloud combination weight $W^{\prime } = \left( {W_{1}^{\prime } ,W_{2}^{\prime } , \cdots ,W_{10}^{\prime } } \right)$ of each index can be calculated according to Table 7 and Eqs. (9)-(12),

W₁′=(0.1014,0.0280,0.0123) ,W₂′=(0.0857,0.0209,0.0065), W₃′=(0.0601,0.0263,0.0146), W₄′=(0.0507,0.0382,0.0217), W₅′=(0.1651,0.0600,0.0235), W₆′=(0.0995,0.0481,0.0319), W₇′=(0.0571,0.0567,0.0188), W₈′=(0.1199,0.0638,0.0268), W₉′=(0.1652,0.0864,0.0253), W₁₀′=(0.0954,0.0166,0.0087).

The NCCW method can suitably capture the uncertainty and volatility of the weights. It regards the weight as a group of cloud droplet groups, and the weight is represented by a normal distribution form comprising the different cloud droplet groups. The NCCW and the evaluation index value of each scheme were comprehensively integrated according to Eqs. (13)-(15), and the comprehensive normal cloud models of the eight schemes and the ideal scheme were obtained as follows: C₁(0.3549,0.0268,0.0119),C₂(0.3651,0.0287,0.0123), C₃(0.1539,0.0199,0.0085), C₄(0.5254,0.0335,0.0145),

C₅(0.3986,0.0287,0.0126),C₆(0.3233,0.0248,0.0103), C₇(0.7717,0.0444,0.0160), C₈(0.4234,0.0267,0.0110), C₊(0.8534,0.0455,0.0185).

The comprehensive normal cloud models of the above eight schemes and the ideal scheme were plotted in a comprehensive cloud model diagram, as shown in Fig. 11. The ordinate of the graph is the certainty degree, and the abscissa is the comprehensive evaluation index. The comprehensive evaluation index of the ideal scheme is plotted on the far right.

Fig. 11

Comprehensive cloud model of the eight test schemes and the ideal scheme

Since the comprehensive cloud models of the above schemes differ and cannot be directly compared, the Euclidean distance and similarity between each scheme and the ideal scheme were calculated by Eqs. (16)-(17) for comparison. The closer the similarity value is to 1, the closer the scheme to the ideal solution and the better the scheme. Table 8 indicates that the best scheme involves a burden of 5 cm and a stemming length of 5 cm, followed by that involving a burden of 4 cm and a stemming length of 4 cm and that involving a burden of 6 cm and a stemming length of 6 cm. The optimal stemming length is equal to the minimum burden.

Table 8 Similarity and ranking between each scheme and the ideal scheme

5 Field application

In Sects. 3 and 4, the matching relationship between the stemming length and burden in cut blasting of LDL stopes was studied, and it was determined that the ideal stemming length is equal to the burden. The research conclusions of this study were applied to the field, and a cut blasting test was conducted of the N10#S stope, S4#N stope, N1# stope, and other stopes of the Fankou Lead–Zinc Mine. In this Section, N10#S stope is selected as an example.

The Sh214a ore body within the design range of the N10#S stope is mainly hosted in the D₃t^a stratum, and the stability of the ore body is moderate. The ore body contains pyrite (sh214b) with a high sulfur content and waste rock. The stope elevation ranges from − 565.0 to − 537.0 m. The upper chamber is located in the second sublevel of Sh-550 m, and the lower chamber is located in the fourth section of Sh-600 m. The width of the stope is 8 m, the length is approximately 35 m, and the pattern parameters of the boreholes are 1.8 m ~ 2.0 m × 2.2 m. There are three rounds of boreholes in the cut area (as shown in Fig. 12a), L1-L16 are 16 blast holes (charging holes) with a diameter of 110 mm, and K0-K4 are five empty holes with a diameter of 200 mm. Figure 12c shows the stope layout and the cross-section of the second-row of boreholes in the stope. Two packs of emulsion explosives with a length of 1.34 m (670 mm × 2) are continuously loaded into the blast holes. The best test scheme involves a burden of 5 cm and a stemming length of 5 cm. According to a geometric similarity coefficient of 45, both the actual optimal stemming length and minimum burden are 2.2 m.

Fig. 12

Stope layout and effect after cut blasting. a There are three rounds of boreholes in the cut area on-site, and round I includes five empty holes with a diameter of 200 mm. b Upper chamber roof after blasting. c Cross-section of the second-row of boreholes in the stope. d Ore shoveled by a scraper. e Blasting pile formed in the lower chamber

Field observations were obtained after stope cut blasting. Figure 12b shows the upper chamber roof steel mesh facing the blast hole in the cut blasting area. The steel mesh was not damaged after blasting. Stemming and nonel-tubes were found around the blast hole on the upper chamber floor, indicating that the explosion energy exactly pushed the stemming structure out of the blast hole, which significantly reduced the clogged borehole ratio. Stemming did not damage the upper chamber roof, so this stemming length is reasonable. Figure 12d, e show the blasting pile formed in the lower chamber after blasting and the ore shoveled by the scraper after a period of removal, respectively. The photo reveals that there are no apparent large fragments, and the ore fragments are relatively uniform, indicating that an excellent stope blasting effect was achieved. In addition, during our field experiments, we conducted various blasting tests with different matching relationships, such as a burden of 2.2 m and a stemming length of 1.2 m, a burden of 2.2 m and a stemming length of 1.5 m, and a burden of 1.5 m and a stemming length of 2.2 m. We observed that none of these alternative blasting schemes performed as effectively as the combination of a stemming length of 2.2 m and a burden of 2.2 m. This finding strongly suggests that the blasting scheme involving a stemming length of 2.2 m and a burden of 2.2 m is indeed reasonable and optimal.

6 Discussion

6.1 Theoretical calculation of the matching relationship between the stemming length and minimum burden

The purpose of stemming is to fully use the explosive energy for rock breaking and obtain a better blasting effect, i.e., to optimally utilize the stemming effect and ensure that the rock fragmentation process is completed before flushing the stemming structure out of the blast hole. The stemming length at this time is the optimal length.

Figure 1 shows a diagram of the stemming. The diameter of the stemming, which is d₁, is the same as the diameter d of the blast hole. The mass and length of the stemming are m, and l₂, respectively. The pressure of the initial explosion gas in the blast hole is P₀, and the pressure after moving across distance x is P_x. The assumed mass and shape of the stemming do not change during movement in the blast hole. The pressure of explosive gas decreases with stemming movement, which can be calculated as follows:

$$P_{x} = \left( {\frac{{V_{0} }}{{V_{x} }}} \right)^{\gamma } P_{0}$$

(18)

where, γ is the adiabatic coefficient and is typically chosen as 3; V₀ denotes the volume obtained after subtracting the stemming from the initial volume in the process of compression and fragmentation; and V_x is the volume after removing the stemming when the displacement of the stemming is x, V₀ and V_x can be calculated as follows:

$$V_{0} = \frac{1}{4}\pi D^{2} l^{\prime }$$

(19)

$$V_{x} = \frac{1}{4}\pi D^{2} l^{\prime } + \frac{1}{4}\pi d^{2} x$$

(20)

where l´is the blast hole length between the explosive and the stemming during the formation of the crushing zone; D is the diameter of the crushing zone, and d is the diameter of the blast hole. Substituting Eqs. (19) and (20) into Eq. (18) yields:

$$P_{x} = \left( {\frac{1}{{1 + d^{2} x/(D^{2} l^{\prime } )}}} \right)^{\gamma } P_{0}$$

(21)

The stemming initially driven under the action of the explosive gas. At this time, the stemming is affected by the thrust of the explosive gas, its gravity G, and the sliding resistance R with the blast hole wall. The resultant force F is:

$$F = ma = \frac{\pi }{4}d^{2} l_{2} \rho_{1} a = \frac{{1}}{{4}}\pi d^{2} P_{x} - G - R$$

(22)

When the stemming moves in the blast hole (x < l_d), R can be obtained as:

$$R =\pi dl_{2} \lambda fp_{x} = \frac{{\pi dl_{2} fp_{x} \mu_{d} }}{{1 - \mu_{d} }}$$

(23)

where λ is the lateral pressure coefficient, λ = μ_d/(1-μ_d), and μ_d is the dynamic Poisson's ratio. f is the sliding friction coefficient between the stemming and the blast hole wall, which is set to 0.01 in this study.

When the stemming is about to be flushed out of the blast hole (x ≥ l_d), R can be expressed as:

$$R =\pi d\left[ {l_{2} - \left( {x - l_{d} } \right)} \right]\lambda fP_{x} = \frac{{\pi d\left[ {l_{2} - \left( {x - l_{d} } \right)} \right]fP_{x} \mu_{d} }}{{1 - \mu_{d} }}$$

(24)

Substituting Eqs. (21) and (23) into Eq. (22), the acceleration a₁ of the stemming moving in the blast hole can be obtained as:

$$a_{1} = \frac{{d - 4l_{2} \lambda f}}{{dl_{2} \rho_{1} }}\left( {\frac{1}{{1 + d^{2} x/D^{2} l^{\prime } }}} \right)^{\gamma } P_{0} - g$$

(25)

Similarly, by substituting Eqs. (21) and (24) into Eq. (22), the acceleration a₂ of the stemming ejected from the blast hole (x ≥ l_d) can be obtained as:

$$a_{2} = \frac{{d - 4\left( {l_{2} + l_{d} - x} \right)\lambda f}}{{dl_{2} \rho_{1} }}\left( {\frac{1}{{1 + d^{2} x/D^{2} l^{\prime } }}} \right)^{\gamma } P_{0} - g$$

(26)

The stemming length l₂ is much smaller than l_d, and x is l₂ + l_d when the stemming is ejected from the blast hole, so a₂ can be considered to reach 0. Due to the complexity of the acceleration calculation, it is not convenient for engineering applications. Therefore, simplification is needed. Because d²x/D²l´approaches 0, g is smaller than the previous term. As a result, Eq. (25) can be simplified as:

$$a_{1} = \frac{{d - 4l_{2} \lambda f}}{{dl_{2} \rho_{1} }}P_{0}$$

(27)

The explosion gas pressure can be estimated through the following Equation (Lu et al. 2012; Yilmaz and Unlu 2013): $P_{0} = \frac{{\rho_{1} C_{p1}^{2} }}{8}$, where ρ₁ is the density of the stemming and C_p1 is the P wave velocity of the stemming. The time needed for the stemming to be ejected from the blast hole is non-significant relative to the time needed for the stemming to move inside the blast hole. Therefore, the total time t₁ for the movement of the stemming is satisfied:

$$t_{1} = \sqrt {\frac{{2l_{d} }}{{a_{1} }}}$$

(28)

According to explosion mechanics and stress wave theory, the time of rock mass fragmentation is (Wang et al. 2016; Zhao et al. 2010):

$$t_{d} = \frac{{l + 2l_{1} }}{{C_{p} }}{ + }\frac{{0.5l + l_{1} }}{{C_{R} }}$$

(29)

where, C_p is the propagation velocity of P waves in the rock, and C_R is the surface Rayleigh wave velocity.

The optimum blast hole stemming must satisfy t₁ = t_d. Adopting first cut blasting of the N10#S stope of the Fankou Lead–Zinc Mine as an example, the optimum stemming length is calculated according to the above equations. The decoupling coefficient of the charge in this stope is 1.22, the density of the stemming is 1800 kg/m³, the rock P wave velocity is 3400 m/s, and the explosive density is 1350 kg/m³. When the stemming length l₂ is 1.2, 1.5, 1.8, 2.0, 2.3, and 2.5 m, the corresponding minimum burden l₁ can be calculated as 1.17, 1.52, 1.86, 2.07, 2.40 and 2.61 m, respectively according to the above equations. The matching relationship between these parameters is l₁ = (0.97 ~ 1.04) l₂, similar to the model test results. The optimal stemming length is approximately equal to the minimum burden.

6.2 Optimal scheme obtained by the different weighting methods

To further assess the rationality and effectiveness of the comprehensive evaluation method of cloud combination weighting, another six representative comprehensive evaluation methods were selected to weight the eight blasting schemes, as listed in Table 9. The results showed that the optimal and suboptimal schemes obtained by the seven comprehensive evaluation methods are schemes 7 and 4, respectively. The ranking of the remaining six blasting schemes is differs, but the evaluation results are also very similar. A comprehensive analysis was conducted combined with the actual situation of cut blasting in the Fankou Mine. The schemes involving a stemming length of 5 cm and a burden of 5 cm and a stemming length of 4 cm and a burden of 4 cm were selected as the best solutions. In the ranking based on the other evaluation methods, the ranking results obtained by the first four comprehensive evaluation methods are inconsistent. The main reason is that the use of the AHP weight is easily affected by subjective factors. The latter two evaluation methods entail the use of objective weights calculated from different angles, in which the uncertainty in the weights themselves is easily ignored, and they are prone to information loss. The cloud combination weighting method proposed in this study can be used to fuse a variety of subjective and objective weights, maximize subjective and objective weight information and maintain the uncertainty in weight information. It is superior to other comprehensive evaluation methods.

Table 9 Ranking results of different weighting methods

7 Conclusions

The LDL mining method has been widely used in China due to its high mining efficiency. In on-site cut blasting, it was found that stemming recoil is readily generated given an unreasonable matching relationship between the stemming length and the minimum burden, causing damage to the upper chamber roof steel mesh and a poor rock breaking effect. In this study, 15 groups of model tests were designed to optimize the cut blasting effect by optimizing the matching relationship between the stemming length and the minimum burden.

(1) Fifteen groups of model blasting tests with minimum burden lengths of 3, 4, 5, 6, and 7 cm and stemming lengths of 0, 2, 4, 5, 6, and 7 cm were designed. In the 15 groups of tests, fragments were not produced in four groups of tests (a burden of 6 cm and a stemming length of 0 cm, a burden of 7 cm and a stemming length of 0 cm, a burden of 7 cm and a stemming length of 4 cm, and a burden of 7 cm and a stemming length of 7 cm). The total fragment mass, crater radius, and crater area in the tests involving a stemming length of 4 cm under the same minimum burden are significantly greater than those in the tests with no stemming. Therefore, the three tests (a burden of 3 cm and a stemming length of 0 cm, a burden of 4 cm and a stemming length of 0 cm, and a burden of 5 cm and a stemming length of 0 cm) are removed from the 11 groups of tests, and the remaining eight groups of schemes are optimized.

(2) Values of ten indexes for each of the eight test schemes were obtained and normalized. A total of six weighting methods, including two subjective weighting methods (AHP and FCE) and four objective weighting methods (entropy, CRITIC, CVM, and IQ), were used to obtain their respective weights, and the normal cloud combination weighting method was used to obtain more reasonable weights. The comprehensive normal cloud models of the eight schemes were then calculated. Furthermore, the optimal scheme was finally identified as involving a burden of 5 cm and a stemming length of 5 cm, followed by that with a burden of 4 cm and a stemming length of 4 cm and that with a burden of 6 cm and a stemming length of 6 cm. The optimal stemming length was approximately equal to the minimum burden.

(3) An on-site cut blasting test was conducted of the N10#S stope in the Fankou Lead–Zinc Mine. Two packs of emulsion explosives are loaded into the blast holes. The upper part of the explosive packs exhibits a stemming length of 2.2 m, and the lower part exhibits a burden of 2.2 m. After cut blasting, it was found that the stemming structure was just flushed out of the blast holes, and the upper chamber roof was not damaged. No apparent large fragments occurred in the lower chamber, and the ore fragments were relatively uniform. Therefore, a favorable rock fragmentation effect could be achieved under this stemming length and minimum burden.