Chamber roof deformation prediction and analysis of underground mining using experimental design methodologies

doi:10.21203/rs.3.rs-1630254/v1

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Chamber roof deformation prediction and analysis of underground mining using experimental design methodologies

https://doi.org/10.21203/rs.3.rs-1630254/v1

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The deformation prediction of the chamber roof is crucial in underground mining. Combined with Flac3D numerical simulation, the experimental design methodologies, including single-factor test (SFT), Plackett-Burman design (PBD), steepest ascent design (SAD), and response surface methodology (RSM), were used to evaluate the effect of multiple variables on the chamber roof deformation. Firstly, eight factors that affected the vertical displacement (D_s) of the chamber roof were selected, and the sensitive interval of every factor was obtained through SFT. Then, four factors that significantly affect the results were screened by PBD: cohesion (C_o), stope length (L_s), stope width (W_s), and internal friction angle (f_r). 29 groups of response surface schemes with 4 factors and 3 levels satisfying the Box-Behnken design (BBD) were simulated. Through the result analysis of variance (ANOVA) and sensitivity, the influence order of each factor on D_s can be determined: W_s >L_s >f_r > C_o>interaction between C_o and f_r > interaction between L_s and W_s. Finally, using the prediction model, the roof deformation of 0#N stope of a lead-zinc mine was predicted and the error was analyzed. The relative errors between the prediction value and the numerical simulation value, the measured value are 0.7% and 4.3%, respectively, which indicates that the prediction model is reasonable and has a certain reference value for mine safety.

Deformation prediction

Numerical simulation

Underground mining

Response surface methodology

Sensitivity analysis

An assessment has been made on the effect of orebody properties, orebody dip angle, and stope structure parameters on chamber roof deformation.
Four factors that had a significant impact on the chamber roof deformation were screened out through Plackett-Burman design.
A quadratic polynomial prediction model was set up through response surface methodology. The individual and interactive effects of four factors on displacement have determined significant.

The chamber deformation of the underground stope is the focus of rock mechanics workers and mine managers, and it affects the production capacity of mines and the safety of personnel directly (Abdellah et al. 2014; Brown 2012). Displacement prediction is one of the most critical challenges in mining, and it is of profound significance to mining safety. The mechanized panel slicing backfilling mining method has been widely used in China because of its high mining efficiency and exemplary safety. However, when this method is used in mine inclined orebody, pillar spalling, roof fall, and rockburst may occur in the chamber due to the unreasonable structure of the stope. As a result, it is necessary to establish a reasonable plan to predict stope deformation.

The factors affecting roof stability are very complex. The uncertainty of these factors will become stronger when they reach a certain underground depth, which seriously threatens safe mining and puts higher demands on the prediction and evaluation method (Qiu et al. 2018). Scholars have published many valuable research findings in this area in recent decades. In conclusion, the commonly used methods for stability analysis and deformation prediction of underground engineering are as follows: analytical method, empirical method, numerical modeling, on-site measurement, and model test method (Heidarzadeh et al. 2018; Zhang et al. 2021). The analytical method has good applicability to simple engineering problems. However, it has limited capacity to deal with sophisticated mining issues. Standard empirical methods (such as Barton Q classification method (Barton 2002), Mathews stability graph method (Potvin 1988), extended Mathews stability graph (Mawdesley et al. 2001; Mawdesley 2004)) also have certain limitations, such as oversimplification of irregular geometries and failure to consider the effects of pillars in the chamber, thus the functionalities of these methods are restricted. In recent years, some scholars have applied artificial intelligence technology (AI) to study the pillars strength (Mohanto and Deb 2020; Wattimena et al. 2013; Zhou et al. 2015) and the stability of the stopes (Goh et al. 2017; Qi et al. 2018; Yu et al. 2020). This method is effective when analyzing a single object, but it has limited ability to study the chamber with pillars, and the collection of training samples is complicated. In addition, in-situ measurements and indoor model experiments are time-consuming and costly. On the other hand, numerical methods can circumvent these limitations because they can adopt different failure criteria corresponding to various failure mechanisms and intricate mining problems can be solved by building accurate numerical models according to actual stope conditions (Heidarzadeh et al. 2021; Shnorhokian et al. 2015). Therefore, they have been widely employed in the research of underground excavation stability (Liu et al. 2018; Vallejos et al. 2018).

The chamber roof stability of the inclined orebody is affected by rock mass properties, in-situ stress, blasting disturbance and stope structure parameters. Among them, the stope structure parameters and rock mass properties are very essential (Heidarzadeh et al. 2018). In general, the influence of stope structure parameters and rock mass properties on the stope chamber stability with pillars is usually studied by empirical stability analyses and experiments. For instance, Milne et al. (2004) proposed that it was preferable to use the radius factor (RF) to determine the actual stope surface in the stability graph. Zhou et al. (2019) proposed a pillar risk assessment model combined with pillar stability, which quantified the impact of a single pillar failure on the danger of the whole stope. Li et al. (2020) used the model test method to simulate the pillar recovery, and they analyzed the instantaneous deformation responses and long-term settlements of the stope during the mining of the pillar. As a standard stability analysis method, numerical modeling also has been widely used to evaluate the effect of stope structure parameters and rock mass properties on the chamber roof stability. For instance, some scholars studied the effects of geometrical parameters (Heidarzadeh et al. 2019) and in-situ stress uncertainties (Saeidi et al. 2021) on the stope stability using Flac3D. Some others modified the rock stress factor in the stability graph method by combining empirical methods and numerical modeling (Jia et al. 2020), and investigated the damage behavior of rock mass using Flac3D (Yilmaz and Unlu 2013). However, the specific impacts of these factors on the chamber deformation have yet to be clarified. Moreover, while the methods discussed in this study are better than traditional research methods, they are incapable of solving the challenge of assessing multivariate systems.

The most effective methods to analyze the influence of stope structure characteristics and rock mass properties on stopes roof stability is probabilistic method and numerical modeling (Idris et al. 2011). Among the several probabilistic approaches used in stope stability concerns, experimental design methodologies like response surface methodology (RSM) have been widely employed to analyze the stability of underground excavations (Lü and Low 2011; Mollon et al. 2009; Pytel and Świtoń 2014). RSM can assess individual and interactive effects of multiple influencing factors on the stability evaluation index and establish a prediction and analysis model for the stope roof stability. The influence of each factor on stope stability and the interaction between each factor can be directly evaluated through the generated mathematical model. The current research aims to give a complete assessment of the influence of critical factors on the chamber roof stability of inclined orebody, and the implementation steps of the experimental design methodologies are as follows:

1. The results of single-factor test (SFT) are used to select the suitable sensitive interval for each factor.
2. Plackett-Burman design (PBD) is used to screen out the significant factors.
3. RSM combined with Flac3D is employed to assess the individual and interactive effects of the above significant factors.
4. A polynomial regression model of chamber roof displacement prediction is set up.

The prediction model, which can reflect the relationship between the vertical displacement (D_s) of the chamber roof and various factors, is then applied in the deformation prediction of underground stopes and achieved good results.

2.1 Site Description

After more than 50 years of mining, a lead-zinc mine in Guangdong Province, China, has a large number of ore bodies with a dip angle between 35° and 55° currently, and the ore grade is high, a mechanized panel slicing backfilling mining method was adjusted to be used to mine such inclined orebody at present. The thickness of the inclined orebody in this mine is generally medium-thick, so the stopes have changed from the vertical direction to along the direction of the orebody. The width of one slicing is typically similar to the stope thickness, up to 15 m-22 m, to control dilution and loss. The length of the stope becomes longer, up to 40m approximately, to improve the mining efficiency (Zhang et al. 2021). Therefore, the exposed area of the chamber roof is comparatively large. A pillar is left in the chamber to improve mining safety, and the stope layout diagram is shown in Fig. 1 (taking the stope chamber of the 55° orebody as an example).

The entire zone of the stopes can be divided into the cutting groove area and lateral blasting area, and the lateral blasting area is blasted after the cutting groove area. During on-site blasting mining, one high layer (7 m-9 m) will be mined once, and the orebody will be recovered at the same time as the ore pillar. The occurrence state of the inclined orebody stope is complex, and the influence of various factors on the chamber stability is not apparent totally. Hence it is urgent to establish a reasonable and practical stope stability prediction and evaluation method.

2.2 Numerical modeling

Selected the 0#N stope as the research object, and Flac3D was used to analyze the chamber roof stability with a pillar comprehensively. The Fig. 2 shows the built model of a 55° dip angle. The foot wall and hanging wall of the orebody are all wall rock. According to the principle of Saint-Venant, the surrounding area that is significantly affected is 3–5 times the size of the excavation (Shi et al. 2020), and the size of the entire model is X×Y×Z = 100 m×100 m×70 m. Valid and stable numerical results were obtained by testing with different mesh sizes. A finer mesh is created around the stope chamber and pillar for higher accuracy, whereas the mesh becomes coarser for the wall rock.

As shown in Fig. 3, we set four monitoring points: 1#, 2#, 3#, and 4#. The monitoring points utilized in this paper are also these four monitoring points, and the chamber displacement and the plastic zone were observed using these points.

Two typical rock samples were selected for the physical-mechanical parameters tests, and the parameters of rock mass required for numerical simulation were listed in Table 1. Where ρ is the density, K is the bulk modulus, G is the shear modulus.

Table 1 : Physical-mechanical parameters of rock mass
Rock type	ρ (kg/m³)	K (GPa)	C_o (MPa)	f_r (°)	G (GPa)	T (MPa)
orebody	2500	9	5.3	35	4.5	1.2
Wall rock	2710	8	3.32	41	4.4	1.5

The rock mass is already subject to in-situ stress before an underground excavation. These stresses are due to the weight of the overburdened, tectonic activity or are related to the local topography. The underground stability will be affected by in-situ stress. The stope studied in this paper is generally concentrated in the deep underground, and the actual measured gravity stress load is very close to the vertical component of in-situ stress, which can be determined according to the occurrence depth of the orebody. The ratio of the maximum horizontal components to the vertical component is between 1.16 and 1.49. The direction of the maximum horizontal components is parallel to the strike of the orebody (Wang et al. 2017). Assuming that the rock mass is uniform and continuous, the initial stress fields can be calculated by the following formula:

σ_z = γH = ρgH	(1)
σ_y = 0.000045H² + 0.0121H + 0.49	(2)
σ_x = 0.000035H² + 0.0117H + 0.28	(3)
Where γ is the bulk density of the rock, H is the depth of the rock mass.

The boundary conditions of this numerical simulation adopt displacement constraints. As illustrated in Fig. 2, the four sides constrain the horizontal displacement, and the lower surface constrains the vertical displacement, the upper surface applies gravity stress load, and the four sides apply tectonic stress loads.

2.3 Experimental design

There are four basic steps in the experiment. The initial stage is to consider as many factors that affect the chamber roof stability as possible and use SFT to obtain the sensitive interval of each factor. The second step is to use PBD to screen out the factors that significantly impact the results and reduce the number of subsequent experiments. Next, the third step is to use the steepest ascent design (SAD) to determine the center point of the response surface, which provides the basis for the next stage of RSM. In the end, the fourth step is to create a response surface scheme which satisfying the Box-Behnken design (BBD). Thus, we can determine the relationship between each influencing factor and the chamber roof displacement.

2.3.1 single-factor test design (SFT)

This paper uses the Mohr-Coulomb constitutive model for Flac3D numerical simulation. The main parameters related to ore strength are: density (ρ), bulk modulus (K), cohesion (C_o), internal friction angle (f_r), shear modulus (G) and tension strength (T). In slope engineering, shear failure dominates, and the factors that have a significant effect on the safety are C_o and G, while for underground stopes, C_o, f_r and T are the main parameters affecting ore strength and stope safety (Chen et al. 2017; Shi et al. 2011), so other parameters remain unchanged and only change these three factors when we consider the effect of ore strength on stope stability.

In addition, we selected orebody dip angle (α) and four factors closely related to the stope structure parameters: pillar height (H_p), pillar width (W_p), stope length (L_s), and stope width (W_s). Therefore, a total of 8 factors that have a more significant effect on the chamber roof were considered in SFT. In actual production, the height required for rock drilling in the chamber and the working height of the workers determined the pillar height, so H_p is generally 3.0-4.2 m. As mentioned earlier, W_s can reach 15–22 m to improve mining efficiency, α of the inclined orebody ranges from 35° to 55°, the three factors mentioned above are greatly restricted by the site conditions, so their sensitive range is also set within this range. The following fixed α, H_p, and W_s at 55°, 4.2 m, 15 m, respectively. SFT was performed on W_p, L_s, C_o, f_r, and T, and determined the sensitivity ranges of these factors. As illustrated in Table 2, fixed four factors, and the other factor is changed for the experiment. The influence factors (X) and 10%X were used to determine the error to prevent the random character of the statistical process in the test process. In other words, the simulation is carried out on the three levels of 90%X, X, and 110%X. The simulation data for each group are calculated by averaging the deformation results of 1#, 2#, 3#, 4# monitoring points.

Table 2 : SFT design
Factor	C_o (MPa)	f_r (°)	T (MPa)	W_p (m)	L_s (m)
1	2	20	1	2	25
2	5	30	3	3	35
3	8	40	5	4	45
4	11	50	7	5	55
5	14	60	9	6	65

First, change C_o (initial W_p =3 m, L_s =45 m, f_r =40°, T = 3 MPa) from 2 MPa to 14 MPa by uniformly increasing the strength by 3 MPa, and gather the displacement date of 1#, 2#, 3#, 4# points by the levels C_o-0.3 MPa, C_o, C_o+0.3 MPa. Next, initial W_p =3 m, L_s =45 m, C_o=5 MPa, T = 3 MPa and monitored the vertical displacement of the roof by changing f_r (the range of f_r is 20° to 60°, and the simulation was performed once every 10° increments), the displacement of each monitoring point was collected by the levels f_r and f_r±1°. After that, change T (initial W_p =3 m, L_s =45 m, C_o=5 MPa, f_r =40°) from 1 MPa to 9 MPa by uniformly increasing the strength 2 MPa, and gather the displacement date of 1#, 2#,3#, 4# points by the levels T-0.2 MPa, T, T + 0.2 MPa. And then, initial L_s =45 m, C_o=5 MPa, f_r =40°, T = 3 MPa, and monitored the vertical displacement of the roof by changing W_p (the range of W_p is 2 m to 6 m, and the simulation is performed once every 1 m increment), the displacement of each monitoring point was collected by the levels W_p and W_p±1 m. Finally, change L_s(initial W_p =3 m, C_o=5 MPa, f_r =40°, T = 3 MPa) from 25 m to 65 m by uniformly increasing the 10 m, and gather the displacement date of 1#, 2#,3#, 4# points by the levels L_s-10 m, L_s, L_s+10 m.

2.3.2 Plackett-Burman Design (PBD)

It is necessary to screen all the factors and identify the significant factors to reduce the number of tests after obtaining the sensitive interval of each factors. PBD, which has been used in this paper, can also be called screening experimental design, mainly used to screen important factors, reduce workload and save costs (Vanaja and Rani 2007; Zhao et al. 2010). This design does not take into consider the interaction between factors but only determines the factors that significantly impact the results to screen and optimize the experimental factors. Here, we created a PBD with 12 tests, and 2 levels were taken for the 8 factors of the design. The selected factor's high level is set to be 1.0–2.0 times the low level, and other test conditions such as in-site stress and model mesh sizes are fixed.

2.3.3 Steepest ascent design (SAD)

After founding the significant factors through the PBD, it is necessary to change the significant factors towards the maximum direction of the response value at the same time to approach the maximum response interval and use SAD to obtain the center point of the response surface, so that an effective response surface equation can be built. The factors that have no significant effect on roof displacement are fixed in SAD. In contrast, the significant influencing factors take the change direction of the test results as the raised path so that the optimal value area can be approached as quickly as possible (Zhou et al. 2011).

2.3.4 Box-Behnken Design (BBD)

RSM is used to evaluate the effects of selected parameters associated with orebody properties parameters, orebody dip angle, and stope structure parameters on the obtained chamber roof displacement. It can be used to design experiments, optimize mathematical modeling, and get the optimal level value of each factor (Mason et al. 2003). The fundamental thought of RSM is to obtain a large amount of data through tests, estimate the implicit statistical relationship between the input variables and the response factor. Box-Behnken Design (BBD) is a commonly used RSM, which can be used to evaluate the nonlinear relationship between indicators and factors. Compared with the RSM, although the orthogonal method is simpler to calculate, it cannot find the exact optimal combination and the optimal value of the response variable in a given area. The best combination obtained by the orthogonal experiment can only be limited to the level determined by oneself before rather than the best combination necessarily. The RSM considers the random error of the experiment, and the obtained prediction model is continuous, which can optimize and evaluate the level and interaction of each factor.

Two main steps are taken when using RSM: create a mathematical equation to identify the input parameters and interactions that have a substantial impact on the result; optimize each critical parameter to obtain the desired response value (Ahmadi et al. 2014). A polynomial regression model with quadratic coefficients as formula (4) can be obtained by fitting the experimental data:

D_p = β₀+∑β_ix_i+∑β_ijx_ix_j+∑β_iix_i² + ε

(4)

Where D_p stands for the predicted displacement, β₀, β_i, β_ij, β_ii are constant coefficient, linear coefficients, interaction coefficients, and quadratic coefficients respectively, ε is considered as the random error. The significant input variables are denoted by x_i and x_j, and they can be converted by formula (5):

X_i=(x_i-x₀)/∆x

(5)

Where X_i is coded of the input variable, x₀ represents the actual value of the input variable at the center point, and Δx is the steps between independent variables. The entire experimental design process is shown in Fig. 4.

3.1 SFT results

The difference between the vertical displacement of the four monitoring points 1#, 2#, 3#, and 4# on the chamber roof is almost nonexistent, so the average value of the vertical displacement Ds is taken to replace the displacement of those four points. The fluctuation of the vertical displacement of each scheme at 10%X is investigated using error bars, and Fig. 5 depicts the results.

Figure 5(a) illustrates a negative connection between C_o and D_s. The impact process can be broken down into 3 stages. The shear failure region (C_o<5 MPa) is the first stage, which can be produced by low ore strength or external stresses loads, the vertical displacement of the roof is close to or even more than 30 mm, and the possibility of failure is very high (Chen et al. 2016), as shown in Fig. 6(a) (C_o=2 MPa), the entire pillar is in shear failure, the roof and both sides of the chamber are subjected to a large shear force, the pillar undergoing compression deformation under the external loads, and the side walls also tend to collapse to the interior of the chamber. Stage Ⅱ is a sensitive area (5 MPa < C_o<11 MPa), the vertical displacement decreases significantly with C_o increasing. Stage III is the stable area (C_o༞11 MPa), in which the vertical displacement does not change dramatically with the increase of C_o, the error bars show the variations in displacement.

A declining and nonlinear relation between f_r and D_s is displayed in Fig. 5(b), and it can be separated into 3 stages. Stage I is the failure area (f_r <35°), which may be caused by the low ore strength and occurs on the pillar and both sides of the chamber, the entire pillar and the chamber roof are all in shear failure, as shown in Fig. 6(b) (f_r =20°). As seen by the error bars, Stage II is a sensitive area (35°<f_r<55°) that fluctuates with variations in f_r. Stage III is the stable area (f_r >55°), in which the vertical displacement does not change significantly with the increase of f_r.

As illustrated in Fig. 5(c), D_s satisfy the nonlinear declining relation for different T, 3 stages exist: Stage I is the failure area (T < 3 MPa), which the low ore strength may cause. Stage Ⅱ is a sensitive area (3 MPa < T < 5.5 MPa), the error bars show that the change in D_s is noticeable in this location. Stage III is the stable area (T༞5.5 MPa), in which the vertical displacement does not change significantly with the increase of T. Figure 5(d) illustrates a negative connection between W_p and D_s, 3 stages exist: Stage I is the failure area (W_p<3 m), which may be caused by the too narrow pillar and the large load in the stud. Stage Ⅱ is a sensitive area (3 m < W_p<4.5 m). Stage III is the stable area (W_p >4.5 m), in which the vertical displacement does not change substantially with the increase of W_p. Figure 5(e) illustrates a positive connection between L_s and D_s, 3 stages exist: Stage I is the stable area (L_s<35 m), in which the vertical displacement is small, and the stope safety is good. Stage Ⅱ is a sensitive area (35 m < L_s<50 m). Stage III is the failure area (L_s >50 m), which the too-long stope may cause. In this paper, the mathematical function fitting (Table 3) is used to study the nonlinear relationship between D_s and C_o, f_r, T, W_p, L_s, and it can be seen from the table below that the fitting coefficients are all above 0.9, and the fitting degree is reasonable.

Table 3

The fitting results between single-factor and the vertical displacement
Relationship	Polynomial model equations	Correlation coefficient, R²
D_s - C_o	D_s=0.0829C_o²-1.9495C_o + 35.446	0.9731
D_s - f_r	D_s=0.0047f_r²-0.6083f_r + 43.806	0.9933
D_s - T	D_s=0.0499T²-0.6915T + 28.703	0.9805
D_s – W_p	D_s= -0.5783W_p + 28.709	0.9948
D_s - L_s	D_s= -0.0032L_s² + 0.4382L_s + 14.397	0.9275

3.2 PBD simulation results

12 groups of specific tests were implemented according to the scheme of PBD, and the vertical displacements of the 1–4# monitoring points D_s were averaged and listed in the following Table 4. The significance of the factors was analyzed.

Table 4

Simulation parameters and results of PBD
Simulation order	H_p (m)	W_p (m)	C_o (MPa)	L_s (m)	W_s (m)	f_r (°)	α (°)	T (MPa)	D_s (mm)
1	4.2	4.5	11	35	22	55	30	5.5	25.54
2	3	4.5	11	35	22	35	30	3	31.25
3	3	3	5	35	15	35	30	3	24.24
4	3	4.5	5	35	15	55	55	5.5	18.21
5	4.2	3	11	35	15	35	55	5.5	22.49
6	4.2	3	5	35	22	55	55	3	34.46
7	3	3	5	50	22	55	30	5.5	36.02
8	3	4.5	11	50	15	55	55	3	20.51
9	4.2	4.5	5	50	22	35	55	3	41.09
10	3	3	11	50	22	35	55	5.5	36.58
11	4.2	3	11	50	15	55	30	3	21.06
12	4.2	4.5	5	50	15	35	30	5.5	23.55

The findings in Table 4 showed a broad range of D_s from 18.21 mm to 41.09 mm in the 12 tests, and this variety revealed that optimizing influencing factors were essential for enhancing chamber roof stability. The above test results are analyzed, the estimated effect, regression coefficient, and corresponding t and p values are given in Table 5. The calculated standard deviation of the calculation is 1.16029. The sum of prediction errors squares is 96.9312, pred-R² is 99.59%, and adj-R² is 97.56%, which indicates that better prediction results are obtained, t-values and p-values were used to determine the significance of each variable.

Table 5

Estimated effect, regression coefficient and corresponding t and p values for D_s in eight variable PBD
Variables	Effect	Coefficient	Standard error	t-value	p-value
H_p/(m)	0.064	0.032	0.3349	0.10	0.933^x
W_p/(m)	-2.616	-1.308	0.3349	-3.91	0.060^x
C_o/(MPa)	-3.189	-1.594	0.3349	-4.76	0.041^z
L_s/(m)	3.603	1.801	0.3349	5.38	0.033^y
W_s/(m)	12.314	6.157	0.3349	18.38	0.003^y
f_r/(°)	-3.729	-1.864	0.3349	-5.57	0.031^z
α/(°)	1.780	0.890	0.3349	2.66	0.117^x
T/(MPa)	-1.539	-0.769	0.3349	-2.30	0.148^x

Table 5 Estimated effect, regression coefficient and corresponding t and p values for D_s in eight variable PBD

The variable with a confidence level above 95% (p-value < 0.05) is considered the significant parameter. Analysis of the regression coefficients and the p-values of eight factors (Table 5) show that C_o, L_s, W_s, f_r were the significant parameter, at the same time, H_p, W_p, α, T were considered non-significant, they were fixed as 3.6 m, 4.5 m, 55°, 1.2 MPa, respectively in the following path of SAD and BBD tests.

3.3 SAD simulation results

By analyzing the PBD results of Table 5, it is found that L_s and W_s positively affect the D_s. In contrast, C_o, f_r have a negative effect on the D_s, the path of steepest ascent was moved along the direction in which C_o, f_r decreased and L_s, W_s increased. Five tests were designed as presented in Table 6.

Table 6

Simulation parameters and results of SAD
Simulation order	C_o (MPa)	L_s (m)	W_s (m)	f_r (°)	D_s (mm)
1	5	50	22	35	42.91
2	6.5	46	20	40	34.18
3	8	43	19	45	29.31
4	9.5	39	17	50	24.63
5	11	35	15	55	20.98

It is noted that the D_s of the 5 SAD tests has been decreasing from Table 6 above. The analysis shows that increasing the C_o, f_r, and decreasing the L_s, W_s of the stope will lead to smaller D_s and better chamber safety. However, mining efficiency will be significantly reduced when the stope length and stope width are reduced to a certain extent. The rock drilling, charging, mining, and other works will also be significantly hindered. So the goal of optimization is to increase the stope structure parameters as much as possible under the premise of safety. Therefore, the displacement failure criterion of 30mm (Chen et al. 2016) is taken as the “top of the ascent” in SAD. The highest response was 29.31 mm with C_o (8 MPa), L_s (43 m), W_s (19 m), f_r (45°) at test 3, and it was decided that this position was close to the ideal point, therefore it was picked for further optimization.

3.4 BBD results

A 3-level, 4-factor BBD model that requires 29 simulations was employed to assess the individual or combined effects between the four significant factors and the response variable D_s. The four factors, namely C_o, L_s, W_s, and f_r, were considered independent input variables. We can predict the D_s using comprehensive factors of the sensitive areas (C_o (5–11 MPa), L_s (35–50 m), W_s (15–22 m), f_r (35°-55°)) and center point (C_o (8 MPa), L_s (43 m), W_s (19 m), f_r (45°)). As shown in Table 7, the simulation order 1 to 24 are factorial points, the order 25 to 29 correspond to the zero factor points, the design matrix of tested variables, and the results are represented, where D_s is the average displacement of the monitoring points (1#, 2#, 3#, 4#) obtained by using Flac3D, and the predictive values of average displacement D_p are calculated by the BBD analysis.

Table 7

Experimental design and results of BBD
Simulation order	Co (MPa)	L_s (m)	W_s (m)	f_r (°)	D_s (mm)	D_p (mm)	Prediction error ε
1	5(-1)	35(-1)	19(0)	45(0)	27.98	28.05	0.26%
2	11(+ 1)	35(-1)	19(0)	45(0)	26.46	26.50	0.17%
3	5(-1)	50(+ 1)	19(0)	45(0)	32.27	32.03	0.74%
4	11(+ 1)	50(+ 1)	19(0)	45(0)	30.52	30.25	0.88%
5	8(0)	43(0)	15(-1)	35(-1)	24.41	24.50	0.39%
6	8(0)	43(0)	22(+ 1)	35(-1)	36.95	36.84	0.29%
7	8(0)	43(0)	15(-1)	55(+ 1)	22.77	22.71	0.24%
8	8(0)	43(0)	22(+ 1)	55(+ 1)	34.63	34.32	0.89%
9	5(-1)	43(0)	19(0)	35(-1)	33.06	33.02	0.13%
10	11(+ 1)	43(0)	19(0)	35(-1)	29.87	30.03	0.55%
11	5(-1)	43(0)	19(0)	55(+ 1)	29.28	29.49	0.71%
12	11(+ 1)	43(0)	19(0)	55(+ 1)	28.71	29.13	1.45%
13	8(0)	35(-1)	15(-1)	45(0)	21.48	21.21	1.27%
14	8(0)	50(+ 1)	15(-1)	45(0)	23.41	23.67	1.13%
15	8(0)	35(-1)	22(+ 1)	45(0)	31.88	31.87	0.02%
16	8(0)	50(+ 1)	22(+ 1)	45(0)	36.04	36.80	2.12%
17	5(-1)	43(0)	15(-1)	45(0)	24.16	24.21	0.23%
18	11(+ 1)	43(0)	15(-1)	45(0)	22.81	22.72	0.39%
19	5(-1)	43(0)	22(+ 1)	45(0)	36.42	36.35	0.19%
20	11(+ 1)	43(0)	22(+ 1)	45(0)	34.80	34.53	0.78%
21	8(0)	35(-1)	19(0)	35(-1)	28.30	28.42	0.42%
22	8(0)	50(+ 1)	19(0)	35(-1)	32.69	32.47	0.67%
23	8(0)	35(-1)	19(0)	55(+ 1)	26.36	26.41	0.18%
24	8(0)	50(+ 1)	19(0)	55(+ 1)	30.39	30.08	1.01%
25	8(0)	43(0)	19(0)	45(0)	29.53	29.33	0.69%
26	8(0)	43(0)	19(0)	45(0)	29.36	29.33	0.09%
27	8(0)	43(0)	19(0)	45(0)	29.49	29.33	0.54%
28	8(0)	43(0)	19(0)	45(0)	29.11	29.33	0.77%
29	8(0)	43(0)	19(0)	45(0)	29.18	29.33	0.51%

The adequacy and the significance level for the regression model were checked using ANOVA. The regression relationship between D_s and the independent variables A(C_o), B(L_s), C(W_s), D(f_r) is significant if p-Value ≤ 0.05 of the model item, and when p-Value ≤ 0.01, it means that the regression relationship is highly significant. A more minor “Lack of fit” represent a more significant p-value, which means the model is more significant with linear and quadratic model terms. The closer the coefficient of determination (R²) is to 1, the better the correlation of the model. When R² is greater than 80%, the correlation of the regression equation is thought to be good.

Table 8 shows the results, the “F-value” of the model was 316.28 and the “p-value”< 0.0001, indicating that the model was highly significant, the value of 1.46 showed that the “Lack of fit” was not the significant compared to the pure error, so the model was suitable for making a prediction. Linear terms of A(C_o), B(L_s), C(W_s), D(f_r), quadratic terms of A²、B²、C²、D² and interactive terms of AD、BC were significant for D_S, and the other terms are insignificant. A larger F-value indicates that the factor is more critical in the model. From this, the sensitivity of the four factors is W_s >L_s >f_r > C_o.

Table 8

ANOVA for quadratic Model
Source	Sum of squares	Degree of freedom	Mean square	F-Value	p-Value	Significant level
model	506.51	14	36.18	316.28	< 0.0001	significant
A(C_o)	7.88	1	7.88	68.85	< 0.0001	significant
B(L_s)	39.80	1	39.80	347.95	< 0.0001	significant
C(W_s)	421.85	1	421.85	3687.81	< 0.0001	significant
D(f_r)	13.37	1	13.37	116.90	< 0.0001	significant
AB	0.0135	1	0.0135	0.1179	0.7364	—
AC	0.0263	1	0.0263	0.2296	0.6392	—
AD	1.73	1	1.73	15.09	0.0017	significant
BC	1.55	1	1.55	13.52	0.0025	significant
BD	0.0353	1	0.0353	0.3089	0.5872	—
CD	0.1354	1	0.1354	1.18	0.2950	—
A²	1.46	1	1.46	12.75	0.0031	significant
B²	1.43	1	1.43	12.51	0.0033	significant
C²	1.62	1	1.62	14.16	0.0021	significant
D²	2.41	1	2.41	21.08	0.0004	significant
Residual	1.60	14	0.1144	—	—	—
Lack of fit	1.46	10	0.1462	4.19	0.0901	not significant
Pure error	0.1397	4	0.0349	—	—	—
Total	508.11	28	—	—	—	—

Based on the data acquired from the Flac3D, a statistical analysis via RSM was used to construct the best-fit regression model for D_s as follows:

For the programming regression equation:

D_s=28.35-0.8233A + 1.85B + 5.95C-1.07D-0.058AB-0.0805AC + 0.6569AD + 0.6166BC-0.0938BD-0.1827CD + 0.4741A²-0.4723B²+0.5129C²+0.6097D²

(6)

At the same time, the actual regression equation obtained from the programming regression equation according to the actual testing data is as follows:

D_s=25.96701-1.85129C_o + 0.602135L_s-0.552446W_s-0.681461f_r-0.002577C_oL_s-0.007666C_oW_s + 0.021896C_of_r + 0.023491L_sW_s-0.001251L_sf_r-0.005221W_sf_r + 0.052681 C_o²-0.008397 L_s² +0.041873 W_s²+0.006097 f_r²

(7)

The coefficient of determination R² and the adj-R² were calculated as 0.9968, 0.9937, respectively, indicating that the testing and the predicted values are in perfect accordance and that RSM may be represented by a quadratic function. Table 7 lists the predicted value D_p of the chamber roof displacement obtained by using Eq. (7), and the prediction error is also counted in Table 7 to validate the prediction model, the prediction error ε can be calculated by Eq. (8) as follows:

ε=\(\frac{\left|{\text{D}}_{\text{p}}\text{-}{\text{D}}_{\text{s}}\right|}{{\text{D}}_{\text{s}}}\text{×100%}\)

(8)

The model has favorable reliability because the statistics in Table 7 shows that the ε of the 16th group is 2.12%, and the rest errors are less than 2%, the prediction error is tiny, and the model prediction accuracy is satisfactory.

3.5 Comprehensive study of BBD

It can be seen from the above analysis that D_s is not only related to the single factor of A(C_o), B(L_s), C(W_s), D(f_r), but also to the interaction between the four factors. The order of influence of each factor on stope displacement D_s can be determined: W_s >L_s >f_r > C_o> interaction between C_o and f_r > interaction between L_s and W_s.

The three-dimensional response surface (Fig. 7) can be plotted using the regression equation by selecting any two parameters from the four factors above, and the correlation of each variable on D_s can be reflected by the horizontal plane of the three-dimensional surface.

Figure 7 Response surface figures for D_s by showing variable interactions of (a) C_o and L_s, (b) C_o and W_s, (c) f_r and C_o, (d) W_s and L_s, (e) L_s and f_r, (f) W_s and f_r

The effect of C_o and L_s on D_s is given in Fig. 7(a). D_s increases when C_o decrease and L_s increases; the growing trend indicates that L_s has a more significant effect on D_s than C_o. As shown in Fig. 7(b), D_s increases when C_o decrease and W_s increases, the curvature of the graph indicates that W_s has a more significant effect on D_s than C_o. The adverse effects of f_r and C_o on D_s are depicted in Fig. 7(c), which shows that D_s increases when f_r, and C_o decrease and the effect of f_r is almost equivalent C_o, there is a zero factor point (when C_o(0) = 8 MPa, L_S(0) = 43 m, W_S(0) = 19 m, f_r(0) = 45°), which has a mutation phenomenon, that is, the D_s increases significantly after the zero factor point. As indicated in Fig. 7(d), W_s and L_s positively affect D_s, and the influence of W_s on D_s is more significant from the response surface increasing trend. The adverse effects of f_r and positive effects of L_s on D_s are depicted in Fig. 7(e), the effect of L_s and f_r on D_s is similar to the growing trend of the response surface. Figure 7(f) shows the influence of W_s and f_r on D_s, displacement rises with an increase in W_s and a decrease in f_r, W_s has a more significant effect on D_s than f_r.

In the practical mine production process, obtaining optimal points from the RSM should not be the only goal. Although the most intuitive judgment for promoting the security of the chamber roof is the smallest vertical displacement, we need to increase the structural parameters as much as possible based on security to improve mining efficiency and optimize the benefits of production.

The roof deformation of the 0#N stope was predicted using Eq. (7) obtained from the study. For the 0# N stope, the orebody dip angle is 55°, the main parameters of ore properties are T is 1.2 MPa, Co is 5.3 MPa, and fr is 35°, Wp is 4.5 m, Hp is 4.5 m, Ls is 40 m, and Ws is 18 m. The stope deformation under this structural parameter was calculated using the prediction model and verified by numerical simulation. The stope contours of the vertical displacement and the plastic zone distribution are given in Fig. 8.

The roof displacement calculated by the prediction model is 30.18 mm, and the displacement obtained by the numerical simulation is 30.39 mm, they only differ by 0.21 mm, the error between them calculated by the formula (8) is only 0.7%, indicating that the prediction model satisfies the high accuracy.

To further validate the accuracy of the model, a monitoring point on the roof center was set on the right side of the pillar in 0#N stope (as shown in Fig. 9), the measured displacement of the point is 28.89 mm, and the error is 4.3% compared to the prediction value. Field support may explain the more meaningful error in 0#N stope, and arranging multiple monitoring points to take the average value is also conducive to reducing the error.

It can be observed in Fig. 9 that the stope chamber is stable, and the safety of the chamber is guaranteed after the bolt and mesh support. Therefore, the research conclusions in this paper are consistent with the actual situation at the chamber site, and the prediction model has definite guiding significance for the stope safety. It can be seen from the contour of the plastic zone distribution that the pillar is in a shear failure state, and the pillar on the site has a small amount of ore spalling, which is also consistent with the simulation results.

Combined with Flac3D, the experimental design methodologies, including SFT, PBD, SAD, and RSM, were used to evaluate the individual and interactive effects of 8 factors related to chamber roof deformation of an underground mine in China. These 8 factors are related to orebody properties, orebody dip angle, and stope structure parameters.

(1) The sensitivity interval of multiple factors was obtained through a mix of SFT and numerical simulation. Reliable curve fitting and correlation coefficients were calculated by fitting multiple functions, and the fitting coefficients are all above 0.9.

(2) Through PBD, the difference significance parameters of each factor were obtained, and four factors that had a significant impact on the results were screened out: C_o, L_s, W_s, and f_r. The center point of the response surface was determined by SAD: C_o(8 MPa), L_s(43 m), W_s(19 m), f_r (45°).

(3) The numerical simulation was organized using a 3-level, 4-factor BBD model that required 29 simulations, linear terms of A(C_o), B(L_s), C(W_s), D(f_r), quadratic terms of A²、B²、C²、D²and interactive terms of AD、BC were significant according to ANOVA and the sensitivity analysis. The polynomial equation accurately described the mathematical relationship between the vertical displacement (D_s) and these significant factors. The order of influence of each factor on D_s can be determined: W_s>L_s>f_r> C_o> interaction between C_o and f_r> interaction between L_sand W_s.

(4) The prediction model was applied to predict the roof deformation of 0#N stope in a lead-zinc mine. The deformation relative errors between the prediction model and the numerical simulation, the field measured data are 0.7%, 4.3%, respectively, which indicates that the prediction model is effective and has some reference value for mining safety.

Conflicts of interest/ Competing interests:

The authors declare no conflict of interest.

Funding:

National Natural Science Foundation Project of China (Xianyang Qiu, 52004329)

National Natural Science Foundation Project of China (Xiuzhi Shi, 51874350)

CRediT authorship contribution statement

Zongguo Zhang: Investigation, Software, Writing - original draft. Xianyang Qiu: Project administration, Software, Writing - review & editing. Xiuzhi Shi:, Supervision. Writing - review & editing.

Acknowledgments

This study is supported by the National Natural Science Foundation Project of China (Grant Nos. 52004329 and 51874350);

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Editorial decision: Minor revisions
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Chamber roof deformation prediction and analysis of underground mining using experimental design methodologies

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Version 1

Abstract

Figures

Highlights

1 Introduction

2 Experimental Procedures

2.1 Site Description

2.2 Numerical modeling

2.3 Experimental design

2.3.1 single-factor test design (SFT)

2.3.2 Plackett-Burman Design (PBD)

2.3.3 Steepest ascent design (SAD)

2.3.4 Box-Behnken Design (BBD)

3 Numerical Results And Discussions

3.1 SFT results

3.2 PBD simulation results

3.3 SAD simulation results

3.4 BBD results

3.5 Comprehensive study of BBD

4 Engineering Applications

5 Conclusions

Declarations

References

Status:

Version 1