Issue of Selecting Stress Field Parameters for the Analysis of Mining Excavation Stability Using Numerical Methods in the Conditions of the LGCB Mines

Abstract

This paper concerns the issue of selecting appropriate stress field parameters for predicting the stability of headings driven under the geological and mining conditions of Polish underground copper mines. The problem is of key importance due to strict safety requirements in mine workings that serve ventilation and transport functions. Numerical analyses were carried out for four stress field variants: the stress state determined based on Bulin’s formulas (variant 1), the hydrostatic stress state (variant 2), and stress states determined from in situ measurements conducted in the Rudna mine (variant 3 and variant 4). Numerical simulations were performed for a group of four headings, supported with fully grouted rock bolts, in the geological and mining conditions of the Rudna mine. Stability assessment was performed using the finite element method (FEM). Rock mass input parameters for the modeling were obtained with RocLab 1.0, applying the Hoek–Brown classification, while numerical analyses employed the Mohr–Coulomb failure criterion. The elastic–plastic model with softening was used to describe the rock mass behaviour. Numerical calculations were conducted in the RS2 computer program in a triaxial stress state and in a plane strain state. The range of the yielded rock mass zone in the roof of the headings was assumed as the optimal measure of the headings stability. The obtained simulation results provided a basis for recommending suitable rock bolting systems to protect the stability of headings developed under various initial stress field conditions.

1. Introduction

Numerical methods are a widely used tool for solving various problems related to rock mechanics. Their main advantage is in the ability to analyze objects of almost any geometry, allowing for the application of different models to describe material behavior under load, spatial variation in rock mass properties, specific primary stress fields, dynamic loading conditions, and more. These methods also provide solutions for both two-dimensional and three-dimensional problems [1,2,3,4,5,6,7,8,9,10,11,12,13,14]. Numerical methods have significantly expanded the research capabilities related to the analysis and assessment of headings stability in underground mines. They are frequently used to solve problems associated with the driving of headings under variable or difficult geological and mining conditions [15,16,17,18,19,20,21,22,23,24,25,26]. Using numerical simulations, it is possible to predict stress concentration around mine excavations as well as cases of headings instability. This is particularly important in selecting the shape, dimensions, and location of headings, as well as the parameters of the mining systems. Numerical modeling is also applied in the design of support systems for underground excavations [27,28,29,30,31,32,33,34]. Recently, the issues of the stability of mining excavations in hard coal mines and the study of coal drawing parameters of deeply buried hard coal seams based on PFC, as well as the overburden failure characteristics and fracture evolution rule under repeated mining with multiple key strata control, have been described by some scientists [35,36,37,38,39,40,41].

To perform optimal numerical simulations, it is essential to construct a numerical model that accurately reflects the geomechanical conditions of the rock mass. Therefore, the specialist responsible for creating the numerical model must possess the necessary knowledge and data regarding the following, among other things:

−

The geological structure of the rock mass (the arrangement of rock layers);

−

The strength and strain parameters of the rock layers;

−

The dimensions, shape, and location of the underground headings within the rock mass;

−

The parameters of the stress field.

Often the greatest challenge in creating a numerical model is the selection of appropriate stress field parameters. This task becomes particularly difficult when no in situ stress measurements have been conducted in the analyzed underground mine to determine these parameters. In such cases, the experience and expertise of the specialist performing the numerical simulations play a crucial role. The specialist must select stress field parameters for the numerical model in a way that they representatively reflect the actual stress conditions present in the modeled rock mass.

The numerical simulation results of heading stability in the Rudna mine presented in this article are a continuation of the research described in the paper “Numerical methods as an aid in the selection of roof bolting systems for access excavations located at different depths in the LGCB mines” [42]. The results were obtained as part of a research program conducted at the Faculty of Geoengineering, Mining and Geology of Wroclaw University of Science and Technology, titled “Application of numerical methods for the analysis of selected natural hazards in underground mines.”

In Polish copper mines, various shapes and ranges of roof collapses are observed in underground workings. This phenomenon depends, among other factors, on the strength and strain parameters of the rocks, the size and shape of the excavations, as well as the parameters of the stress field (including the values and directions of the stress components). The research presented in this paper, along with the obtained results, may allow for a preliminary determination of the type of stress field occurring in the rock mass based on the shape and range of roof collapses in mining headings. These studies also make it possible to optimally select rock bolt support systems for mine workings, depending on the identified stress field parameters in a specific area of the mine.

The stability of underground headings, which serve as haulage routes for output transportation from mining fields, crew transport, and also play a vital role in mine ventilation, is crucial for long-term underground mining operations. The in situ stresses occurring in the rock mass may lead to the loss of stability of underground headings, posing a serious hazard to miners and disrupting production continuity, so this is a very important factor in designing underground excavations [43,44,45].

Determining the primary stress state in an undisturbed rock mass is the starting point for all geomechanical analyses. This is due to the fact that, during underground mining operations, any geomechanical changes occurring in the rock mass are referred to the primary stress state [46,47]. Defining this primary stress condition is the most important element in characterizing the rock mass. In general terms, it can be stated that the primary factors influencing the stress state in the lithosphere are: gravitational force, specifically, the weight of the overlying rocks in the lithological structure and the possible presence of tectonic stresses, associated with past dynamic orogenic events that have resulted in both continuous and discontinuous deformations within the rock mass. Other elements that create the primary stress state are surface topography, the course of erosion and weathering processes, and the presence of the fractures and discontinuities within the rock mass.

For a rock mass where gravity is the only acting load, the analysis can be reduced to determining the stresses acting on an elementary cubic volume extracted from the Earth’s crust at a depth H (Figure 1). The weight of the overburden acting vertically generates vertical stress described by the following formula:

$${\sigma }_z=\gamma ·H,$$

(1)

where

• σ_z—vertical stress [MPa];
• γ—unit weight of the overburden rocks [MN/m³];
• H—depth of the analyzed rock layer [m].

Figure 1. Primary stress components in undisturbed rock mass adapted from [46].

Figure 1. Primary stress components in undisturbed rock mass adapted from [46].

At a depth of H (at the assumed level), the sum of the multiplication results of the thicknesses of individual overburden layers (h_i) and their unit weights (γ_i) must be taken into account [46,48]. In this case, the vertical stress in the rock mass is determined by the following formula:

$${\sigma }_z={\sum }_{i=1}^n{\gamma }_i·h_i,$$

(2)

where

• σ_z—vertical stress [MPa];
• γ_i—unit weight of the i-th rock layer [MN/m³];
• h_i—thickness of the i-th rock layer [m].

The values of horizontal stress acting on the lateral surfaces of the elementary rock mass element can be determined using the following formula:

$${\sigma }_x={\sigma }_y={\displaystyle \frac{v}{1-v}}·{\sigma }_z,$$

(3)

where

• σ_x—horizontal stress along the x-axis [MPa];
• σ_y—horizontal stress along the y-axis [MPa];
• ν—Poisson’s ratio [-];
• σ_z—vertical stress along the z-axis [MPa].

For the horizontal stress calculated using Equation (3), the following condition is always met:

$${\sigma }_x={\sigma }_y\le {\sigma }_z,$$

(4)

If a hydrostatic stress state is assumed for the rock mass, the stress field takes the following form:

$${\sigma }_x={\sigma }_y={\sigma }_z={\sum }_{i=1}^n{\gamma }_i·h_i,$$

(5)

For certain geotechnical conditions of the rock mass, assuming that only gravitational forces act within the rock mass (i.e., no other loads are present) may lead to incorrect stress field estimations. Numerous observations indicate that the primary stress state in the Earth’s crustal rocks results from the superposition of two stress fields:

−

Gravitational stress field, related to the weight of the overlying rocks.

−

Tectonic stress field, related to tectonic processes [47,49,50].

The influence of gravitational and tectonic stresses on the primary stress field in a rock mass has led to the development of many empirical formulas for determining horizontal stress components [51]. For geostatic areas, the N.K. Bulin equation is most commonly used to calculate horizontal stress values [50]. It was developed based on stress measurements in a rock mass. It is assumed that the average horizontal normal stress σ_Bx,y increases with depth H, and is given by

$${\sigma }_{Bx,y}=2.50+0.013·H,$$

(6)

where

• σ_Bxy—horizontal stress in geostatic areas [MPa];
• H—depth at which horizontal stress is calculated [m].

The average horizontal stress defined by the N.K. Bulin Equation (6) is generally higher than the σ_x, σ_y values determined from Equation (3), which are based solely on gravitational forces. Stress investigations conducted in geostatic regions have shown [50] that the vertical stress component σ_z is close to the gravitational stress value σ_{z grav} and can be described as

$${\sigma }_z=\left(1.0÷1.2\right)·{\sigma }_{z grav},$$

(7)

where

• σ_z—vertical stress in geostatic areas [MPa];
• σ_{z grav}—vertical stress resulting from gravitational forces [MPa].

The application of measurement methods to determine the magnitudes and directions of stresses in rock masses makes it possible to verify assumptions about the primary stress field in the rock masses. In situ stress measurements performed in underground mines across selected continents have shown the following:

−

Vertical stress increases with depth;

−

Horizontal components of the primary stress may exceed the vertical component [47,51,52,53,54,55,56,57].

Underground stress measurements in the Rudna mine were conducted in 2012 as part of the research project: “Magnitude and directions of in situ stress in the deep part of a copper ore deposit” with an Appendix “Determination of the impact of the primary stress directions and magnitudes on the optimal geometry of mining fields” [57]. The overcoring method was used for in situ stress measurements (Figure 2). This method, classified as a relaxation technique, involves isolating a rock sample from the existing stress field and simultaneously measuring the deformations or displacements caused by the stress relief [51].

For the measurements, CSIRO HI cells were used, which were bonded in the pilot holes with a special adhesive. A single CSIRO HI measuring cell records measurements in the x, y, and z directions. This provides a sufficient amount of measurement data and enables a complete determination of the stress tensor components in a three-dimensional system from a single measurement. The obtained results made it possible to determine the values of vertical and horizontal stresses in the rock mass in the Rudna mine (

Table 1. Parameters of the stress field in the Rudna mine adapted from [57].

Parameter	Measurement Station (S) and Measurement Point (P)
	S1P2	S1P3	S2P2	S2P4	S3P2	S4P3	S4P4
σH [MPa]	24.7	30.8	22.2	11.4	16.8	33.6	26.4
αH [°]	110.0	118.0	55.0	42.0	158.0	124.0	131.0
σh [MPa]	18.0	25.3	19.0	7.3	11.8	26.8	19.6
αh [°]	20.0	28.0	145.0	132.0	68.0	34.0	41.0
σv [MPa]	22.1	27.8	18.6	9.4	20.0	22.1	23.9

The symbols used in the above table are as follows: σ_H—maximum horizontal stress, α_H—azimuth of maximum horizontal stress, σ_h—minimum horizontal stress, α_h—azimuth of minimum horizontal stress, and σ_v—vertical stress.

). Based on these data, it is also possible to calculate the principal stress values in the x, y, z coordinate system and the azimuths of their directions.

2. Prediction of the Mining Excavations Stability at the Rudna Mine

The stability prediction of the headings under the geological and mining conditions of the Rudna mine was carried out using the RS2 v. 9.030 software. The finite element method (FEM) was applied. Rock mass parameters were obtained from boreholes Jm-15/H-173 and Jm-15-460. Laboratory tests of rock samples were performed at the KGHM Cuprum Research and Development Centre—Materials Testing Laboratory. In the adopted geological profile, the immediate roof is composed of limestone dolomites characterized by high strength and deformation parameters. The sidewalls and floor consist of rocks that generally exhibit lower mechanical properties than those occurring in the roof (

Table 2. Average strength and strain parameters of rocks determined in laboratory uniaxial compression tests for boreholes Jm-15/H-173 and Jm-15-460.

Location	Rock Type	h [m]	ρ [kg/dm3]	Rc [MPa]	Rr [MPa]	Ei [GPa]	v [-]
Roof	Anhydrite I–III	5.50	2.94	108.74	5.58	40.06	0.24
	Anhydrite IV	9.30	2.94	86.93	5.94	38.30	0.24
	Calcareous dolomite I–VIII	7.20	2.74	163.13	7.81	60.18	0.25
Sidewall	Calcareous dolomite IX	1.60	2.71	95.00	10.16	27.17	0.22
	Dolomitic shale	0.60	2.69	111.49	9.06	28.89	0.23
	Quartz sandstone I	1.20	2.40	47.85	2.93	16.94	0.17
Floor	Quartz sandstone II	9.50	2.33	36.19	2.78	13.60	0.14

The symbols used in the above table are as follows: h—thickness of rock layers, ρ—volume density, R_c—rock sample strength to uniaxial compression, R_r—tensile strength of the rock sample, E_i—longitudinal modulus of elasticity, and v—Poisson’s ratio.

).

A major challenge lies in the proper determination of the strength and strain parameters of the rock mass for numerical simulations. The application of the Hoek–Brown classification [58,59,60,61] enables the use of laboratory test results of rock samples to determine high-quality strength and deformation parameters of the rock mass [62,63,64]. The generalized Hoek–Brown failure criterion for fractured rock mass can be described by the following equation [61]:

$${\sigma }_1={\sigma }_3+{\sigma }_{ci}\mathsf{\bullet }{\left(m_b\mathsf{\bullet }{\displaystyle \frac{{\sigma }_3}{{\sigma }_{ci}}}+s\right)}^a,$$

(8)

where

• σ₁ and σ₃—values of maximum and minimum effective principal stress at failure;
• m_b—Hoek–Brown constant for the rock mass;
• s and a—constants dependent on the rock mass properties;
• σ_ci—uniaxial compressive strength of the intact rock sample.

When the tensile strength of the rock mass, σ_tm, is exceeded, the equation for a = 0.5, takes the following form:

$${\sigma }_{tm}={\displaystyle \frac{{\sigma }_{ci}}{2}}\mathsf{\bullet }\left(m_b-\sqrt{{m_b}^2+4s}\right),$$

(9)

For the rock layers in boreholes Jm-15/H-173 and Jm-15-460, the strength and strain parameters of the rock mass were determined (

Table 3. Rock mass parameters determined on the basis of the Hoek–Brown classification using the RocLab 1.0 program Data from [42].

Location	Rock Type	c [MPa]	φ [°]	σt [MPa]	Erm [MPa]
Roof	Anhydrite I–III	8.137	38.66	0.871	29,356.00
	Anhydrite IV	6.505	38.66	0.696	28,066.78
	Calcareous dolomite I–VIII	14.879	39.00	3.611	52,975.30
Sidewall	Calcareous dolomite IX	7.579	37.69	1.442	22,180.23
	Dolomitic shale	6.447	30.41	1.327	18,250.37
	Quartz sandstone I	3.589	40.54	0.180	10,701.33
Floor	Quartz sandstone II	2.520	39.06	0.093	7072.00

). Using the RocLab 1.0 computer program, cohesion (c), internal friction angle (φ), the uniaxial tensile strength of the rock mass (σ_t), and the rock mass modulus of elasticity (E_rm) were determined. The calculations were based on the Hoek–Brown classification [58,59,60,61].

Numerical simulations using the Coulomb–Mohr strength criterion make it possible to accurately model the shapes and ranges of roof collapses in mine workings under the geological and mining conditions of Polish copper mines. This has been confirmed by underground observations of stability loss cases in mine headings in Polish copper mines [62,63,64]. In the numerical modeling conducted using the RS2 program, the Mohr–Coulomb strength criterion was applied, according to which the rock material reaches its failure when the following condition is met:

$${\sigma }_1={\sigma }_3\mathsf{\bullet }{\displaystyle \frac{1+\sin \phi }{1-\sin \phi }}+{\displaystyle \frac{2c\mathsf{\bullet }\cos \phi }{1-\sin \phi }}$$

(10)

or

$${\sigma }_3=-{\sigma }_t$$

(11)

where

• σ₁ and σ₃—values of maximum and minimum effective principal stress at failure;
• φ—friction angle;
• c—cohesion;
• σ_t—uniaxial tension strength.

Numerical simulations were carried out using an elastic–plastic model with strain softening (for the layers forming the roof and sidewalls) and an elastic–plastic model (for the layers forming the floor). The rock mass was assumed to be isotropic and homogeneous. The RS2 software allows analyses to be performed under triaxial stress conditions and with the plane strain assumption. The strength and strain parameters of the rocks are presented in

Table 4. Strength and strain parameters of the rock mass for numerical simulations data from [42].

Location	Rock Type	h [m]	Es [MPa]	ν [-]	σt [MPa]	φ [°]	c [MPa]	δ [°]	φres [°]	cres [MPa]
Roof	Anhydrite I–III	5.50	29,356.00	0.24	0.871	38.66	8.137	2.00	36.73	1.627
	Anhydrite IV	9.30	28,066.78	0.24	0.696	38.66	6.505	2.00	36.73	1.301
	Calcareous dolomite I–VIII	7.20	52,975.30	0.25	3.611	39.00	14.879	2.00	37.05	2.976
Sidewall	Dolomite-shale-sandstone formations	3.50	17,435.35	0.20	0.976	37.41	5.971	2.00	35.54	1.194
Floor	Quartz sandstone	9.50	7072.00	0.14	0.093	39.06	2.520	2.00	39.06	2.520

The symbols used in the above table are as follows: h—thickness of rock layers, E_s—longitudinal modulus of elasticity, v—Poisson’s ratio, σ_t—tensile strength of the rock mass, φ—internal friction angle, c—cohesion coefficient, δ—dilatancy angle, φ_res—residual internal friction angle, and c_res—residual cohesion coefficient.

.

Numerical simulations were carried out for a group of four headings, between which pillars with a width of 20.0 m were located. The excavations have an inverted trapezoidal cross-section. A sidewall inclination angle of 10° was assumed. The dimensions of the excavations in the adopted cross-sections are presented in Table 5 and Figure 3. Arnall RM-18 resin-grouted rock bolts were used for headings support. A bolting pattern of 1.5 m × 1.5 m was applied. Arnall RM-18 rock bolts are characterized by the following parameters [65]:

−

Rod diameter: 18.2 mm;

−

Rod length: 1.8 m;

−

Rod material: steel;

−

Young’s modulus: 210,000.0 MPa;

−

Ultimate load capacity: 170.0 kN;

−

Residual load capacity: 17.0 kN;

−

Initial pretension force: 30.0 kN.

Table 5. Parameters of headings.

Excavation Hight [m]	Width of Excavation Roof [m]	Width of Excavation Floor [m]	Average Width [m]	Area of the Excavation [m2]	Sidewall Inclination Angle [°]
3.5	7.0	5.8	6.4	22.4	10.0

Table 5. Parameters of headings.

Excavation Hight [m]	Width of Excavation Roof [m]	Width of Excavation Floor [m]	Average Width [m]	Area of the Excavation [m2]	Sidewall Inclination Angle [°]
3.5	7.0	5.8	6.4	22.4	10.0

Figure 3. Cross-section of heading supported by rock bolts.

Figure 3. Cross-section of heading supported by rock bolts.

To determine the stress values for numerical modeling, the R-XI shaft profile, the PN-G-05016:1997 standard [48], and in situ stress measurement results conducted at the Rudna mine in 2012 (Table 1) were used. In the calculations it was assumed that the average unit weight of rock layers in the profile of the R-XI shaft (from the ground level to a depth of 1200 m below ground level) was 0.020417 MN/m³. In the numerical modeling, four stress fields in the rock mass were assumed (Table 6):

−

Variant 1: the stress field in the rock mass was determined for a depth of 1200 m below ground level based on Bulin’s formulas (Formulas (6) and (7));

−

Variant 2: hydrostatic stress state in the rock mass for a depth of 1200 m below ground level (Formula (5));

−

Variant 3 and variant 4: stress values derived from in situ measurements at the Rudna mine in 2012 (Table 1, measurement station no. 4 and measurement point no. 4 (S4P4)).

Table 6. Primary stresses in the x, y, z coordinate system for the four stress field variants in the Rudna mine.

Modeling Variant	Horizontal Stress σx [MPa]	Horizontal Stress σy [MPa]	Vertical Stress σz [MPa]
1 (based on Bulin’s formulas)	18.10	18.10	24.50
2 (hydrostatic stress state)	24.50	24.50	24.50
3 (in situ measurements)	19.60	26.40	23.90
4 (in situ measurements)	26.40	19.60	23.90

Table 6. Primary stresses in the x, y, z coordinate system for the four stress field variants in the Rudna mine.

Modeling Variant	Horizontal Stress σx [MPa]	Horizontal Stress σy [MPa]	Vertical Stress σz [MPa]
1 (based on Bulin’s formulas)	18.10	18.10	24.50
2 (hydrostatic stress state)	24.50	24.50	24.50
3 (in situ measurements)	19.60	26.40	23.90
4 (in situ measurements)	26.40	19.60	23.90

Depending on the assumed stress field in the rock mass, the following assumptions were made in the numerical models:

• Loading variant 1 (stress state determined based on Bulin’s formulas for depth H = 1200 m b.g.l.):

  −

  Lateral edges: p_x = 18.10 MPa;

  −

  Upper and lower edges: p_z = 24.50 MPa;

  −

  Direction perpendicular to the plate: p_y = 18.10 MPa.

• Loading variant 2 (hydrostatic stress state in the rock mass for depth H = 1200 m b.g.l.):

  −

  Lateral edges: p_x = 24.50 MPa;

  −

  Upper and lower edges: p_z = 24.50 MPa;

  −

  Direction perpendicular to the plate: p_y = 24.50 MPa.

• Loading variant 3 (stress state determined based on in situ measurements at the Rudna mine; the maximum horizontal stress component acts parallel to the direction of driving the heading):

  −

  Lateral edges: p_x = 19.60 MPa;

  −

  Upper and lower edges: p_z = 23.90 MPa;

  −

  Direction perpendicular to the plate: p_y = 26.40 MPa.

• Loading variant 4 (stress state determined based on in situ measurements at the Rudna mine; the maximum horizontal stress component acts perpendicular to the direction of driving the heading):

  −

  Lateral edges: p_x = 26.40 MPa;

  −

  Upper and lower edges: p_z = 23.90 MPa;

  −

  Direction perpendicular to the plate: p_y = 19.60 MPa.

In the numerical modeling, a flat, rectangular plate measuring 288.0 × 203.5 m was assumed. In the center of the plate, openings shaped like the analyzed mining excavations were placed. The edges of the plate were located 100 m away from the extreme points of the analyzed excavations (roof, floor, and sidewalls) on each side of the excavation. At the plate edges, fixed supports were applied in both the vertical and horizontal directions. Three-node triangular finite elements were used in the numerical models. To improve the accuracy of the numerical calculations, smaller finite elements were applied in the center of the plate (in the zone where mining excavations are performed) (Figure 4).

Past and ongoing underground observations conducted in Polish copper mines in headings where a loss of stability occurred have confirmed that the shapes and ranges of roof collapses in the rock layers are very similar to the numerically simulated shapes and ranges of yielded zones in the roofs of headings [62,63,64]. Therefore, in this paper, the range of the yielded rock mass zone was adopted as the main indicator for assessing heading stability.

3. Results of Numerical Modeling and Discussion

The numerical modeling of the stability of the group of headings located in the rock mass with varying stress fields illustrated the occurrence of different roof behavior mechanisms (the development, area, and shape of the yielded zones). The results are presented in Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12 and in Table 7 and Table 8. In some cases, these phenomena may lead to excavation instability (e.g., roof falls). The numerical simulations also showed the following:

−

The area and shape of the yielded rock mass zone around the headings vary depending on the magnitude of the stress field components and are closely related to the strength and deformation parameters of the rock layers in which the excavations are driven.

−

The area of the yielded rock mass zone (ranging from 50% to 100%) in the roof of the headings in the numerical model with variant 1 loading (stress state determined based on Bulin’s formulas) is significantly smaller than the anchorage zone (ranging from 0.46 m to 0.56 m). Values from 1.24 m to 1.34 m were observed (Figure 5 and Figure 6). The yielded zone in the roof has a characteristic shape. The smallest yielded zone occurs in the center of the excavations, while the largest area of the yielded zone is observed in the roof near the upper corners of the excavations.

−

The area of the yielded zone (ranging from 50% to 100%) in the roofs of headings in the numerical model with variant 2 loading (hydrostatic stress state) is generally larger than the anchorage zone in the roofs of headings no. 1 to no. 3 (ranging from 0.18 m to 0.28 m), where values from 1.98 m to 2.08 m were observed (Figure 7 and Figure 8). In heading no. 4, the area of the yielded zone in the roof (50% to 100%) is 1.78 m, which is only 0.02 m less than the anchorage zone.

−

The smallest size of the yielded zone (ranging from 50% to 100%) in the roofs of headings occurred in the numerical model with variant 3 loading (the maximum horizontal stress component, determined based on in situ measurements, acting parallel to the heading excavation direction). Values from 1.18 m to 1.25 m were observed (Figure 9 and Figure 10). These are significantly smaller (ranging from 0.55 m to 0.62 m) than the anchorage zone in the roofs of the headings.

−

The largest size of the yielded zone (ranging from 50% to 100%) in the roofs of headings occurred in the numerical model with variant 4 loading (the maximum horizontal stress component, determined based on in situ measurements, acting perpendicular to the direction of heading excavation). Observed values ranged from 1.97 m to 2.18 m (Figure 11 and Figure 12). These significantly exceed the anchorage zone in the heading roofs by 0.17 m to 0.38 m.

−

The maximum size of the yielded zone (ranging from 50% to 100%) in the sidewalls of the headings (Table 8), depending on the values of the stress field parameters, reached the following values:

▪

From 2.86 m to 3.22 m (variant 1 loading);

▪

From 2.83 m to 3.03 m (variant 2 loading);

▪

From 2.83 m to 3.24 m (variant 3 loading);

▪

From 2.83 m to 3.11 m (variant 4 loading).

Figure 5. Yielded zones, headings no. 1 and no. 2, variant 1 loading (stress field determined based on Bulin’s equations).

Figure 5. Yielded zones, headings no. 1 and no. 2, variant 1 loading (stress field determined based on Bulin’s equations).

Figure 6. Yielded zones, headings no. 3 and no. 4, variant 1 loading (stress field determined based on Bulin’s equations).

Figure 6. Yielded zones, headings no. 3 and no. 4, variant 1 loading (stress field determined based on Bulin’s equations).

Figure 7. Yielded zones, headings no. 1 and no. 2, variant 2 loading (hydrostatic stress state in the rock mass).

Figure 7. Yielded zones, headings no. 1 and no. 2, variant 2 loading (hydrostatic stress state in the rock mass).

Figure 8. Yielded zones, headings no. 3 and no. 4, variant 2 loading (hydrostatic stress state in the rock mass).

Figure 8. Yielded zones, headings no. 3 and no. 4, variant 2 loading (hydrostatic stress state in the rock mass).

Figure 9. Yielded zones, headings no. 1 and no. 2, variant 3 loading (stress state determined based on in situ measurements, headings driven parallel to the maximum horizontal stress component).

Figure 9. Yielded zones, headings no. 1 and no. 2, variant 3 loading (stress state determined based on in situ measurements, headings driven parallel to the maximum horizontal stress component).

Figure 10. Yielded zones, headings no. 3 and no. 4, variant 3 loading (stress state determined based on in situ measurements, headings driven parallel to the maximum horizontal stress component).

Figure 10. Yielded zones, headings no. 3 and no. 4, variant 3 loading (stress state determined based on in situ measurements, headings driven parallel to the maximum horizontal stress component).

Figure 11. Yielded zone, headings no. 1 and no. 2, variant 4 loading (stress state determined based on in situ measurements, headings driven perpendicular to the maximum horizontal stress component).

Figure 11. Yielded zone, headings no. 1 and no. 2, variant 4 loading (stress state determined based on in situ measurements, headings driven perpendicular to the maximum horizontal stress component).

Figure 12. Yielded zone, headings no. 3 and no. 4, variant 4 loading (stress state determined based on in situ measurements, headings driven perpendicular to the maximum horizontal stress component).

Figure 12. Yielded zone, headings no. 3 and no. 4, variant 4 loading (stress state determined based on in situ measurements, headings driven perpendicular to the maximum horizontal stress component).

Table 7. The size of the yielded zones in the roof of the analyzed headings (yielded zones ranged from 50% to 100%).

Heading	Size of Yielded Zones in the Roof [m]
	Loading Variant 1	Loading Variant 2	Loading Variant 3	Loading Variant 4
1	1.24	1.99	1.24	2.15
2	1.34	1.98	1.21	2.17
3	1.30	2.08	1.18	2.18
4	1.30	1.78	1.25	1.97

Table 7. The size of the yielded zones in the roof of the analyzed headings (yielded zones ranged from 50% to 100%).

Heading	Size of Yielded Zones in the Roof [m]
	Loading Variant 1	Loading Variant 2	Loading Variant 3	Loading Variant 4
1	1.24	1.99	1.24	2.15
2	1.34	1.98	1.21	2.17
3	1.30	2.08	1.18	2.18
4	1.30	1.78	1.25	1.97

Table 8. The size of the yielded zones in the sidewalls of the analyzed headings (yielded zones ranged from 50% to 100%).

Heading	Size of Yielded Zones in the Sidewalls [m]
	Loading Variant 1	Loading Variant 2	Loading Variant 3	Loading Variant 4
1 (left sidewall)	2.98	2.83	2.83	2.83
1 (right sidewall)	3.05	2.98	3.24	2.84
2 (left sidewall)	3.00	3.00	2.95	2.89
2 (right sidewall)	3.02	3.03	2.97	2.88
3 (left sidewall)	2.86	2.97	3.18	3.02
3 (right sidewall)	3.05	2.98	2.85	3.11
4 (left sidewall)	2.94	3.00	3.18	2.99
4 (right sidewall)	3.22	2.83	2.89	3.02
Average value	3.02	2.95	3.01	2.95

Table 8. The size of the yielded zones in the sidewalls of the analyzed headings (yielded zones ranged from 50% to 100%).

Heading	Size of Yielded Zones in the Sidewalls [m]
	Loading Variant 1	Loading Variant 2	Loading Variant 3	Loading Variant 4
1 (left sidewall)	2.98	2.83	2.83	2.83
1 (right sidewall)	3.05	2.98	3.24	2.84
2 (left sidewall)	3.00	3.00	2.95	2.89
2 (right sidewall)	3.02	3.03	2.97	2.88
3 (left sidewall)	2.86	2.97	3.18	3.02
3 (right sidewall)	3.05	2.98	2.85	3.11
4 (left sidewall)	2.94	3.00	3.18	2.99
4 (right sidewall)	3.22	2.83	2.89	3.02
Average value	3.02	2.95	3.01	2.95

However, it should be noted that the average range of the yielded zone (from 50% to 100%) in the sidewalls of the headings across all loading variants is similar and ranges from 2.95 m (variant 2 and variant 4 loading) to 3.02 m (variant 1 loading).

4. Conclusions

The research presented in this paper is part of a broader research program aimed at developing, based on the results of numerous numerical simulations, innovative criteria for the preliminary determination of the type of stress field occurring in a rock mass, inferred from the observed shape and range of roof collapses in mine workings in Polish copper mines. These studies are also intended to contribute, in the future, to the modification of the currently applied criteria for the selection of rock bolt support systems in Polish copper mines, depending on the identified stress field parameters in specific areas of the mine.

The conducted numerical modeling allowed for the optimal selection of rock bolt support for headings driven under the conditions of the Rudna mine and in the rock mass with a variable stress field. For safety reasons, it was assumed that the anchored zone of the rock mass in the roof must exceed the maximum range of the yielded zone (with 50–100% yield). Based on this principle, the following rock bolt support was selected:

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For headings driven in a stress field determined using Bulin’s formulas (for a depth of 1200 m below ground level), RM-18 resin-grouted bolts with a length of 1.6 m and a bolt pattern of 1.5 × 1.5 m were selected.

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For headings driven in a rock mass characterized by a hydrostatic stress state (at a depth of 1200 m below ground level), RM-18 resin-grouted bolts with a length of 2.2 m and a bolt pattern of 1.5 × 1.5 m were selected.

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For headings driven parallel to the maximum horizontal stress component (based on in situ measurements conducted in the Rudna mine in 2012), RM-18 resin-grouted bolts with a length of 1.6 m and a bolt pattern of 1.5 × 1.5 m were selected.

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For headings driven perpendicular to the maximum horizontal stress component (based on in situ measurements conducted in the Rudna mine in 2012), RM-18 resin-grouted bolts with a length of 2.6 m and a bolt pattern of 1.5 × 1.5 m were selected.

In cases where a significant area of the yielded zones (50–100%) is present in the sidewalls of the headings, effective reinforcement can be achieved using bolts not shorter than 1.6 m, installed in a 1.5 × 1.5 m pattern. Additionally, the bottom row of bolts should be placed approximately 1.8 m above the floor. If justified, sidewalls may also be reinforced using so-called deep anchoring (bolts longer than 2.6 m), e.g., cable bolts.

The magnitude of the stress field components can be a crucial factor for the stability of headings in the LGCB mines. Problems with stability issues may arise when the yielded zone in the roof of a heading extends beyond the anchored zone. The results of the numerical simulations using an elastic–plastic rock mass model with strain softening are most consistent with the observed cases of headings instability in Polish copper mines.