Dynamic Evolution of Fractures in Overlying Rocks Caused by Coal Mining Based on Discrete Element Method

Abstract

Mining-induced fractures and overlying rock movement change rock layer porosity and permeability, raising water intrusion risks in the working face. This study explores fracture development in working face 31123-1 at Dongxia Coal Mine using UDEC 7.0 software and theoretical analysis. The overlying rock movement is a dynamic, spatially evolving process. As the working face advances, the water-conducting fracture zone height (WFZH) increases stepwise, and their relationship follows an S-shaped curve. Numerical simulations give a WFZH of about 112 m and a fracture–mining ratio of 14.93. Empirical formulas suggest a WFZH of 85.43 to 106.3 m and a ratio of 11.39 to 14.17. Key stratum theory calculations show that mining-induced fractures reach the 16th coarse-sandstone layer, with a WFZH of 97 to 113 m and a ratio of 12.93 to 15.07. Simulations confirm trapezoidal fractures with bottom angles of 48° and 50°, consistent with rock mechanics theories. A fractal permeability model for the mined overburden, based on the K-C equation, shows that fracture permeability positively correlates with the fractal dimension. These results verify the reliability of simulations and analyses, guiding mining and water control in this and similar working faces.

1. Introduction

After coal seam mining, when the bending tensile stress acting on the strata exceeds the corresponding ultimate tensile strength, the overlying rock will be damaged and collapse. The collapse of the overburden leads to uneven subsidence, causing the rock layers to separate from each other and generating a large number of fractures, which significantly alter the flow characteristics of the overburden, including the porosity and related permeability [1,2,3]. Among various disasters, water disasters caused by coal seam mining-induced overburden fracturing are the most destructive. Fracturing refers to the layered voids formed by uneven subsidence of continuous rock layers with different thicknesses and lithologies [4]. Fracture zones induced by coal seam mining may connect the water-bearing strata, thereby triggering a roof water inrush accident [5,6,7]. Roof water intrusion and ground subsidence are two major problems encountered in coal mining in China [8,9,10]. The overburdened rock is deformed and impaired by mining activities, and a considerable number of fractures are formed in the overburden, which can be partitioned into three zones, namely, the caved zone, the fractured zone, and the sagging zone [11,12,13]. As the number of transverse fractures in the collapsing and fracturing zones increases, the permeability of the strata also increases. Therefore, these two zones are defined as water-conducting fracture zones. During large-scale and high-intensity coal mining, due to the lack of attention to the balance between high-intensity mining methods and the self-repair capacity of the ecological environment, water resources in ecologically fragile areas have been damaged and vegetation has died, further damaging the ecological environment [14,15,16]. Consequently, studying the distribution and precise prediction of the water-conducting fracture zone height (WFZH) above coal seams is of utmost importance for the prevention of water inrush occurrences in coal mines and for ecological protection in coal mining areas.

Currently, theoretical research regarding the evolution of mining-induced fractures is relatively mature. Since the 21st century, with the rapid advancement of computer technology, numerical simulation software has witnessed rapid development, and numerical simulation has emerged as an important means for studying the development characteristics of mining-induced fractures [17]. Numerous researchers have employed numerical simulation software to investigate the development characteristics of mining-induced fractures, validating the effectiveness of numerical simulation in studying such characteristics [2,18,19]. Through numerical simulation, we can simulate the formation, propagation, and evolution of fractures in the process of coal seam mining, revealing the patterns and mechanisms of fracture development. Additionally, numerical simulation can be employed to predict the impact of fractures on coal mine safety production and geological stability, providing a scientific basis for coal mining and geological engineering.

Table 1. Some studies on mining-induced fracture development via numerical simulation in the past 2 years.

Author	Research Results	Date
Xiao et al. [20]	Three stages of “slow growth-accelerated growth-periodic increase”	2025
Zheng et al. [21]	The shape of the plastic failure zone was a typical trapezoid	2025
Li et al. [22]	The overburden rock’s displacement zone forms an “arch-beam” structure, starting from 160 m	2024
Song et al. [23]	Overburden structure on the development WFZH is studied and revealed	2024
Zhang et al. [24]	The fracture development process can be divided into three stages: extensive development of new fractures, partial compaction of fractures, and closure of numerous fractures	2024
Xu et al. [25]	The fractal dimension can be divided into four stages in time and two stages in space	2024
Zhang et al. [26]	The evolution process and development characteristics of inter-layered rock fractures have been revealed	2024

lists articles published in the past two years that have utilized numerical simulation software to study the characteristics of mining-induced fracture development, along with their research findings.

Through a conjunction of theoretical calculations and numerical simulation, a method for calculating the WFZH in the overburden in coal seam mining has been developed, revealing the failure morphology and fracture development process during thick coal seam mining [27,28]. Yang et al. [29] utilized the PFC to establish a novel numerical model for generating irregular particle roofs, and they found that as the roof thickness increases, the supporting capacity of the roof rises, and the deflection decreases. In addition, the increase in the thickness of the main roof leads to the transformation of the failure pattern from O-X to O-*, with an increasing fracture angle and the appearance of shear cracks. Ye et al. [30] established a similar simulation physical experimental system and obtained the development pattern of fractures in the overburden of a deep, inclined coal seam working face. Li et al. [31] employed both numerical simulation and similar simulations to investigate the destruction characteristics of the overburden in karst areas and discovered that the plastic zone begins to develop and accumulate on the roof. After mining, the stress and displacement of the overburden above the working face are separated into three regions, and the central region undergoes the most significant change. Ling et al. [32] studied the development features of fractures in the overburden layer by integrating similar simulation experiments, numerical simulations, and on-site observations, and they combined the fractal dimension to quantitatively divide the fractures in the overburden layer into three stages: the formation stage of fractures in the overburden layer, the expansion stage of deformation fractures in the overburden layer, and the stable development stage of fractures in the overburden layer. Xiong et al. [33,34,35] studied the characteristics and changes in fractures in the overburden of a steeply inclined coal seam. In conclusion, numerical simulation is an effective tool for studying the development characteristics of mining-induced fractures. In this study, we employed the two-dimensional discrete element software UDEC, in combination with theoretical calculations and analyses, to study the height, development morphology, and variations in mining-induced fractures in working face 31123-1 in the Dongxia Coal Mine of the Huating Coal Industry. The results of this study have instructive significance for coal seam mining in working face 31123-1 and similar situations.

2. Geologic Background and Establishment of Numerical Model

2.1. Geologic Background

The research object of this paper is working face 31123-1 in the Dongxia Coal Mine of the Huating Coal Electric Group. The Dongxia Coal Mine is situated in the southeastern part of the Huating Coalfield, and its administrative jurisdiction pertains to Donghua Town, Gansu Province. Its geographic location (106°39′14″–106°40′46″ E, 35°11′39″–35°13′16″ N) is shown in Figure 1. The boundaries are as follows: the eastern boundary is the outcrop of coal layer 6-2 and the deepest limit of the Southchuan Coal Mine in the southwest. The western boundary is the deepest limit of the Huaxin Coal Industry and Huangzhuang Coal Mine in the northwest, and it is located at a height of +950 m above sea level. The northern boundary is the boundary of mining rights. The mine extends for 2.66 km from north to south and 0.97 km from east to west, covering an area of 2.6918 km². The Dongxia Coal Mine is located on the eastern slope of the Liupan Mountains and lies on the southwestern margin or the southern end of the southwestern fold and thrust belt of the Ordos Basin (Shanxi–Gansu–Ningxia Basin). The geological strata, from bottom to top, are the Upper Triassic Yanchang Formation, the Lower Jurassic Fuxian Formation, the Middle Jurassic Yan’an Formation, the Middle Jurassic Zhiluo Formation, the Lower Cretaceous Zhidan Formation, the Neogene Gansu Formation, and the Quaternary System. The main coal seam being mined is the sixth layer, which is located in the first section of the Middle Jurassic Yan’an Formation.

In the depth direction, working face 31123-1 is located between +865 and +810 m, and it is laid out along coal seam 6-1. The thickness of coal seam 6-1 is approximately 6.5–7.5 m. The total length of the working face is 1039 m, among which the southern section measures 348 m and the northern section measures 691 m. The upper inclined part is the +940 to +875 m section of the abandoned workings of working face 37121-1, and there is no mining or excavation activity in the lower part.

2.2. Numerical Simulation Based on the Discrete Element Method (DEM)

The DEM constitutes a numerical simulation approach that is well-suited for simulating granular materials. In the investigation of mining-induced fractures, the DEM can be employed to simulate the movement and interactions of particles within coal and rock strata, thereby revealing the microscopic mechanism of fracture development.

In this study, the discrete element software UDEC 7.0 was utilized to establish a two-dimensional model for simulating the dynamic evolution characteristics of the overburden fractures as the working face advances. The simulation involved a 300-m advance of the working face, with a mining step size of 10 m, and a total of 30 excavation steps.

2.2.1. Model Construction

The coal seam extracted at working face 31223-1 in the Dongxia Coal Mine is coal seam 6-1, which has an average thickness of 7.5 m. Its mining depth is 690 m, and the overburden rock has a medium hardness. The dip angle of the coal seam is ~25°, and the overburden is composed of mudstone, medium sandstone, sandy mudstone, fine sandstone, and coarse sandstone. Based on the specific mining geological conditions of working face 31223-1 in the Dongxia Coal Mine, a two-dimensional computational model was constructed (the height of the model did not reach the surface).

(1)

Model Size. If the constructed model is overly large, it will excessively occupy the processing space of the computer and reduce the computing rate of the model, which is not conducive to the analysis of the results. If the model is too small, it will be incapable of accurately analyzing the properties and environment of the research object, resulting in a significantly larger error in the final analysis results. For our model, the size of the two-dimensional numerical calculation model generated using the “block” command in the UDEC is X × Y = 500 m × 300 m (length × height). The numerical simulation model is illustrated in Figure 2.

(2)

Grid Generation. Thin elements should be avoided in important areas of the model to minimize the difference in the grid size. Similar to the design of the model size, the degree of detail in the grid generation also has an impact on the computational speed. Regarding stress and displacement, mesh refinement means using more and smaller elements to divide the research object in the model. Generally, the finer the mesh, the higher the calculation accuracy of stress and displacement. A fine mesh can more accurately capture the stress concentration and deformation gradient changes inside the medium. In terms of the characteristics of mining-induced fracture development, mesh refinement can more accurately simulate the initiation, propagation path, and process of fractures. In a coarse mesh, fractures may propagate in a rather rough manner, failing to reflect the subtle changes during the fracture propagation process. In contrast, a refined mesh can capture more micro-mechanical information, making the simulation results of the fracture propagation closer to the actual situation. However, more elements and nodes require more memory to store the model information and intermediate calculation results. The principle of grid generation should be adhered to, that is, that the grid is refined in the key areas of the study object to obtain accurate data, and coarser grids can be assigned in areas distant from the key areas to alleviate the computational burden. The grid generation command for this simulation is Zone generate. The focus area is within the Y coordinate range of (52, 182). The final model contains 157,912 elements.

2.2.2. Allocation of Parameters and Boundary Conditions

(1)

Mechanical Parameters. When assigning mechanical properties in the UDEC, it is divided into two components, namely, the rock mass and structural surfaces. The mechanical properties of the rock mass include the bulk modulus, density, elastic modulus, tension, cohesion, and shear strength. The mechanical properties of the structural surfaces include normal stiffness, tangential stiffness, and cohesion. The physical and mechanical parameters employed in the experiment were mainly derived from physical experiments conducted on rock samples collected from the site, and field sampling involved collecting three columnar samples each of fine sandstone, coarse sandstone, coal, carbonaceous mudstone, mudstone, medium sandstone, siltstone, and sandy mudstone for rock mechanics and physical parameter experiments (such as uniaxial compression and Brazilian splitting tests).

Table 2. Simulated thermomechanical parameters of coal and rock layers.

Lithology	Density (kN·m–3)	Tension (MPa)	Bulk (GPa)	Shear (GPa)	Cohesion (MPa)	Friction (°)
coarse sandstone	2580	3.2	2.3	1.6	2.0	38
mudstone	2483	1.2	2.2	1.3	1.2	29
sand mudstone	2680	1.4	2.3	1.7	1.6	29
siltstone	2460	2.4	2.2	2.0	2.3	35
carbonaceous mudstone	2245	1.6	2.1	1.4	1.7	28
coal	1350	0.6	1.8	0.4	0.9	24
medium sandstone	2690	3.3	2.8	2.1	2.2	38
fine sandstone	2760	2.9	3.0	2.4	2.2	39

and

Table 3. Physical and mechanical parameters of structural features.

Lithology	Normal Stiffness (GPa)	Tangential Stiffness (GPa)	Cohesion (MPa)	Tension (MPa)
coarse sandstone	5.1	3.1	0.25	0.32
mudstone	3.1	2.1	0.14	0.25
sand mudstone	5.0	2.8	0.08	0.15
siltstone	4.1	2.6	0.11	0.16
carbonaceous mudstone	3.6	2.2	0.07	0.13
coal	2.3	1.4	0.04	0.09
medium sandstone	5.4	3.5	0.20	0.25
fine sandstone	5.6	3.7	0.23	0.28

presents the average values obtained from the three sets of data for each rock type. These average values were then compared with empirical formulas and rock physico-mechanical parameters in databases to ensure their accuracy. The physical and mechanical parameters of the rock mechanics are shown in Table 2, and the parameters of the structural surfaces are presented in Table 3.

(2)

Boundary Conditions. The Mohr–Coulomb plasticity model was employed, with 100 m of rock retained on both sides to eliminate the boundary effect. Based on the burial depth at the top of the model, an evenly distributed load of 9 MPa was applied at the top boundary, and a stress gradient of 2.5 MPa/100 m was set. Due to coal mining, the boundary conditions of the rock layers in the unique equilibrium state were altered, and the significance and the path of the strain at each point in the stress field of the rock mass, as well as the stress ratio in the abnormal criterion, would change [36], so a new equilibrium state could be attained, within which the rock mass could be deformed, be crushed, and shift. The sideways pressure coefficient derived for the different burial depths was set as 1.0 [14] (Figure 3). The sides of the model were regarded as rolling support boundaries, and the bottom was set as a fixed boundary (Figure 4).

3. Theoretical Analysis of the Development and Morphology of the Mining Fractures

3.1. Methods for Calculating Fractal Dimension

In this paper, the acquisition of fracture images is based on two-dimensional slices along the strike direction of the fracture network obtained from the UDEC numerical simulation results. Therefore, the fractal dimension of the fracture network diagram can be calculated using the box-counting method, which can accurately describe the morphological parameters of fracture development, such as fracture size, number, and density, and is a commonly used method for studying underground rock mass mining fractures, as shown below (for two-dimensional graphics, 0 < D < 2):

$$D=\underset{\epsilon \to 0}{\lim }-{\displaystyle \frac{lgN\left(\epsilon \right)}{lg\epsilon }}$$

(1)

In this formula, D is the fractal dimension, N(ε) is the number of grid cells needed to cover the entire fracture network, and ε is the side length of the grid cells used to cover the fracture network graph.

Fraclab, a fractal analysis tool in the Matlab toolbox, is used to process the images for fractal analysis. The process includes three steps: preliminary image processing, binary image processing, and fractal calculation. The images of the overburden movement characteristics and fracture development features of the coal seam at different mining distances are exported from UDEC numerical simulation software. Then, the main research area of the images is selected and cropped, and the interference elements in the simulation images of different mining stages are removed. After that, the images are binarized using Matlab software 2022b. Finally, the numerical matrix processed by the Fraclab toolbox is imported, and the fractal dimension of the images is calculated using the box-counting dimension method.

3.2. Experienced Formula for Calculating the Development of Mining Fractures

In working face 31123-1, coal seam 6-1 has an average thickness of 7.5 m, the mining method is top coal caving, and the machine mining height is 2.7 m. The roof strata are mudstone and sandstone with uniaxial compressive strengths of 20.1–66.8 MPa, the roof structure consists of medium-hard to hard rock strata. The dip angle of the coal seam ranges from 22° to 43°.

According to the formula for calculating the WFZH (2) in the Design Code for Ground Vertical Well Gas Extraction under Mining-Induced Seismic Effects [37], the empirical formulas under the condition of the medium-hard roof type are as follows:

$$H={\displaystyle \frac{100M}{0.23M+6.10}}\pm 10.42$$

(2)

where H is the WFZH, and M is the thickness of the mined coal seam, measuring 7.5 m.

Based on the calculation using Equation (2), the height of the water-conducting fracture zone is 85.43 to 106.30 m, with a fracture-mining ratio of 11.39 to 14.17.

3.3. Theoretical Calculation of Fracture Development Height During Mining

Under normal circumstances, hard and thick strata experience relatively small amounts of deformation due to their properties and structures, while soft and thin strata experience relatively large amounts of deformation due to their inherent weakness [38,39]. The interfaces between rock layers are weak structural joints with negligible tensile strength compared to that of the rock layers. Hence, the WFZH can be determined by analyzing the structural and mechanical characteristics of the coal-bearing rock layers.

Table 4. Physical and mechanical parameters of the roof of working face 31123-1.

ID	Lithology	Thickness (m)	Density (kN·m–3)	Tension (GPa)	Elasticity (GPa)
25	siltstone	19.0	24.60	2.5	6.3
24	mudstone	3.0	24.83	1.2	4.6
23	siltstone	7.0	24.60	2.5	6.3
22	mudstone	15.0	24.83	1.2	4.6
21	fine sandstone	2.0	27.60	3.4	33.2
20	sand mudstone	22.0	26.80	0.8	5.2
19	mudstone	30.0	24.83	1.2	4.6
18	sand mudstone	20.0	26.80	0.8	5.2
17	fine sandstone	17.0	27.60	3.4	33.2
16	coarse sandstone	16.0	25.80	2.6	24.6
15	siltstone	18.0	24.60	2.5	6.3
14	coarse sandstone	8.0	25.80	2.6	24.6
13	mudstone	11.0	24.83	1.2	8.6
12	siltstone	3.5	24.60	2.5	6.3
11	medium sandstone	5.0	26.90	3.3	28.1
10	fine sandstone	9.0	27.60	3.4	33.2
9	siltstone	4.0	24.60	2.5	6.3
8	fine sandstone	6.5	27.60	3.4	33.2
7	medium sandstone	6.0	26.90	3.3	28.1
6	carbonaceous mudstone	3.0	22.45	1.6	4.3
5	fine sandstone	7.0	27.60	3.4	33.2
4	siltstone	6.5	24.60	2.5	6.3
3	medium sandstone	6.0	26.90	3.3	28.1
2	carbonaceous mudstone	1.5	22.45	1.6	4.3
1	sand mudstone	2.0	26.80	0.8	5.2

presents the physical and mechanical parameters of the overlying rock strata above working face 31123-1. The strata include claystone, siltstone, sandy claystone, carbonaceous mudstone, coarse sandstone, medium sandstone, and fine sandstone, which have rock swelling coefficients of 1.02, 1.06, 1.05, 1.03, 1.07, 1.08, and 1.09, respectively [40]. The rock swelling coefficient is the ratio of the volume of the rock after fracturing to that before fracturing. The higher the rock strength, the poorer the rounding of the crushed stones, the more prominent the edges, and the larger the voids. The lower the rock strength, the better the rounding of the crushed stones, the less prominent the edges, and the smaller the voids [41]. This can be expressed as follows:

$$K_p={\displaystyle \frac{V_h}{V_o}}$$

(3)

where K_p is the coefficient of rock swelling; V_h is the volume of the rock in a loose state after fracturing; and Vo is the volume of the rock in its original, intact state (before fracturing).

3.3.1. Determination of the Locations of the Critical Rock Layers

Based on the theory of critical layers, numerous recent studies have been carried out on the identification and division of critical layers, and a novel method for identifying critical layers in geological formations has been put forward [42,43,44].

According to the principle of combined beam action, the load on the first hard layer is as follows:

$${(q_n)}_1={\displaystyle \frac{E_1{h}_1^3\left({\gamma }_1{h}_1+{\gamma }_2{h}_2+\cdots +{\gamma }_n{h}_n\right) }{E_1{h}_1^3+E_2{h}_2^3+\cdots E_n{h}_n^3}}$$

(4)

where ${(q_n)}_1$ is the equivalent load that the first rock layer bears when the influence of the nth layer on the first layer is considered; E₁, E₂, …, E_n are the elastic moduli of the coal seam overburden layers; h₁, h₂, …, h_n are the thicknesses of the coal seam overburden layers; and ${\gamma }_1$, ${\gamma }_2$, …, ${\gamma }_n$ are the specific weights of the coal seam overburden layers.

When ${(q_{n+1})}_1$ < ${(q_n)}_1$ is satisfied, the first layer to the nth layer in the coal seam overburden layers is judged to be a composite rock layer, the nth layer is separated from the (n + 1)th layer, and the first layer is the critical rock layer.

3.3.2. Determination of the Height of the Free Space Below the Rock Layer

According to the masonry beam theory, the break distance of the ith hard rock layer can be obtained, as follows:

$$l_i=h_i\sqrt{{\displaystyle \frac{2{\sigma }_i}{q_i}}}$$

(5)

where l_i is the critical distance from the ith layer of the key stratum to the break; h_i is the thickness of the ith layer of the key stratum; σ_i is the ultimate tensile strength of the ith layer of the key stratum; and q is the load applied to the key stratum.

For weak rock formations, such as shale, sandy shale, and carbonaceous shale, the horizontal tensile strain is used as the evaluation criterion for whether a failure occurs. When the maximum horizontal tensile strain in the weak rock formation exceeds 1–2 mm/m, plastic failure occurs. The maximum horizontal tensile deformation value ε_max of the weak rock formation is as follows:

$${\epsilon }_{max}={\displaystyle \frac{3q{l}^2}{8E{h}^2}}$$

(6)

where l is the distance of the failure in the weak rock layer; E is the elasticity modulus; and h is the thickness of the rock layer.

According to the theory of the mechanics of materials, the strain can be analyzed according to the deflection, and the formula for calculating the maximum deflection of the rock layer is as follows:

$$w_{max}={\displaystyle \frac{5q{l}^4}{384EI}}$$

(7)

where I is the moment of inertia, and I = lh³/12.

Considering the fracture swelling property of the rock mass, the formula for calculating the height of the free space (HFS) below the rock mass is as follows [45]:

$$F_n=M_c-{\displaystyle \sum }_{j=1}^{n-1}{h}_j\left(k_j-1\right)$$

(8)

where F_n is the HFS at the lower part of the nth rock layer, k_j is the swelling coefficient of the jth rock layer, and M_c is the thickness of the mined coal seam.

For hard rock, when the working face moves forward a distance L greater than l_i and there is a free space in the lower part of the rock layer, the hard rock will break; otherwise, it will not break.

For weak rock, when the HFS F_n in the lower part of the weak rock is greater than the maximum deflection value of the weak rock wmax, the weak rock will undergo plastic failure, i.e.,

$$M_c-{\displaystyle \sum }_{j=1}^{n-1}{h}_j\left(k_j-1\right)>{\displaystyle \frac{5q{l}^4}{384EI}}$$

(9)

Therefore, the WFZH must meet the following two conditions: the ith hard rock layer breaks and the maximum deflection value of the overlying soft rock layer from the ith layer is less than the HFS of the below. Otherwise, the fracture development stops, and it is expressed as Equation (10).

$$\left\{\begin{array}{l}L>L_i\\ M_c-{\displaystyle {\displaystyle \sum }_{j=1}^{n-1}}{h}_j\left(k_j-1\right)>{\displaystyle \frac{5q{l}^4}{384EI}}\end{array}\right.$$

(10)

3.4. The Theory of the Development of Mining-Induced Fractures

In previous studies, it has been found that when the hard rock or critical layer has not reached the fracture limit value, shear fracturing occurs, and the overall fracture development morphology exhibits a trapezoid shape [46]. In this study, the formation of the trapezoid angle in the development process of the mining-induced fractures was analyzed by taking working face 31123-1 in the Dongxia Coal Mine as an example. In our model, the top board is treated as a simply supported one-span, superstatically determinate rectangular section beam subjected to a uniformly distributed load q, and the self-weight of the beam is ρg. This model is a two-dimensional stress problem with a thickness of 1 (Figure 5a). The inertial force in the model is a constant, and for the sake of convenience, the inertial force is replaced by a surface force (Figure 5b).

The initial equilibrium results of the numerical simulation (Figure 6) offer valuable insights. In the stress nephogram, a visually uniform color distribution is observed, and a quantitative analysis reveals minimal variations in stress values across different positions. This compelling evidence strongly indicates that the stress field exhibits remarkable uniformity within the numerical simulation framework (Figure 6b), aligning seamlessly with the stress uniformity assumption posited by the mechanical model. Turning our attention to the displacement nephogram, it presents a clear and systematic pattern. The displacement magnitude progressively diminishes from the upper part to the lower part, and notably, reaches zero at the bottom. This distinct distribution pattern serves as a robust verification of the accuracy of the predefined boundary conditions.

$$\mathrm{Let} {\sigma }_x={\sigma }_x^{\prime }-f_{x}x, {\sigma }_y={\sigma }_y^{\prime }-f_{y}y, \mathrm{and} {\tau }_{xy}={\tau }_{xy}^{\prime }$$

(11)

The stress σ_y is mainly caused by the external loads q and self-weight, both of which are constant and do not vary with x; therefore, it is assumed that σ_y does not vary with x and is only a function of y, i.e., σ_y = f(y). Using the semi-inverse method, we can obtain ${\sigma }_y={\displaystyle \frac{{\partial }^2\phi }{\partial x^2}}$ from the equation. We assume that the stress function φ is as follows:

$$\mathrm{\phi }={\displaystyle \frac{1}{2}}{x}^{2}f\left(y\right)+x{f}_1\left(y\right)+f_2\left(y\right)$$

(12)

Substituting Equation (12) into the consistency equation (Equation (12)) yields:

$${\displaystyle \frac{{\partial }^4\phi }{\partial x^4}}+2{\displaystyle \frac{{\partial }^4\phi }{\partial x^2{y}^2}}+{\displaystyle \frac{{\partial }^4\phi }{\partial y^2}}=0$$

(13)

$${\nabla }^4\varnothing ={\displaystyle \frac{1}{2}}{\displaystyle \frac{d^{4}f\left(y\right)}{d{y}^4}}{x}^2+{\displaystyle \frac{d^4{f}_1\left(y\right)}{d{y}^4}}x+{\displaystyle \frac{d^4{f}_2\left(y\right)}{d{y}^4}}+2{\displaystyle \frac{d^{2}f\left(y\right)}{d{y}^2}}=0$$

(14)

where ∇² is the Laplace differential operator. According to Equation (9), the forms of f(y), f₁(y), and f₂(y) can be obtained, as follows:

$$\left\{\begin{array}{l}f\left(y\right)=A{y}^3+B{y}^2+Cy+D\\ f_1\left(y\right)=E{y}^3+F{y}^2+Gy\\ f_2\left(y\right)=-{\displaystyle \frac{A}{10}}{y}^5-{\displaystyle \frac{B}{6}}{y}^4+H{y}^3+K{y}^2\end{array}\right.$$

(15)

The stress function is $\mathrm{\phi }={\displaystyle \frac{1}{2}}{x}^2\left(A{y}^3+B{y}^2+Cy+D\right)+x\left(E{y}^3+F{y}^2+Gy\right)$

$$-{\displaystyle \frac{A}{10}}{y}^5-{\displaystyle \frac{B}{6}}{y}^4+H{y}^3+K{y}^2$$

(16)

Based on the stress function and the displacement and bending moment boundary conditions of a beam with both ends fixed, and in combination with the physical and geometric equations in elasticity, the stress components are obtained, as follows:

$$\left\{\begin{array}{l}{\sigma }_x^{\prime }={\displaystyle \frac{{\partial }^2\phi }{\partial y^2}}={\displaystyle \frac{1}{2}}{x}^2\left(6Ay+2B\right)+x\left(6Ey+2F\right)-2A{y}^3-2B{y}^2+6Hy+2K\\ {\sigma }_y^{\prime }={\displaystyle \frac{{\partial }^2\phi }{\partial x^2}}=A{y}^3+B{y}^2+Cy+D\\ {\tau }_{xy}^{\prime }=-{\displaystyle \frac{{\partial }^2\phi }{\partial x\partial y}}-x\left(3A{y}^2+2By+C\right)-\left(3E{y}^2+2Fy+G\right)\end{array}\right.$$

(17)

Because the external load is symmetrically distributed, the stress components are also symmetrically distributed. That is, σ_x and σ_y are even functions of x, and τ_xy is an odd function of x.

The calculated stress components σ_x, σ_y, and τ_xy are as follows:

$$\left\{\begin{array}{l}{\sigma }_x={\sigma }_x^{\prime }=\left(q+\rho gh\right)\left(-{\displaystyle \frac{6}{h^3}}{x}^{2}y+{\displaystyle \frac{4}{h^3}}{y}^3+{\displaystyle \frac{2}{h^3}}{l}^{2}y+\mu {\displaystyle \frac{6}{4h}}y\right)-{\displaystyle \frac{\mu q}{2}}\\ {\sigma }_y={\sigma }_y^{\prime }-\rho gy=\left(q+\rho gh\right)\left(-{\displaystyle \frac{2}{h^3}}{y}^3+{\displaystyle \frac{3}{2h}}y\right)-{\displaystyle \frac{q}{2}}-\rho gy\\ {\tau }_{xy}={\tau }_{xy}^{\prime }={\displaystyle \frac{3\left(q+\rho gh\right)}{h}}\left({\displaystyle \frac{2}{h^2}}{y}^2-{\displaystyle \frac{1}{2}}\right)x\end{array}\right.$$

(18)

The variations in each component in the vertical direction of the building height (y-direction) are approximately illustrated in Figure 7.

If the area of the inclined section is dA, then the area of face ef is dAcosα, and the area of face af is dAsinα.

Thus,

$$\left\{\begin{array}{c}\sum F_x={\sigma }_{\alpha }dA+\left({\tau }_{xy}dAcos\alpha \right)sin\alpha -\left({\sigma }_{x}dAcos\alpha \right)cos\alpha \\ +\left({\tau }_{yx}dAsin\alpha \right)cos\alpha -\left({\sigma }_{y}dAsin\alpha \right)sin\alpha =0\\ \sum F_y={\tau }_{\alpha }dA-\left({\tau }_{xy}dAcos\alpha \right)cos\alpha -\left({\sigma }_{x}dAcos\alpha \right)sin\alpha \\ +\left({\tau }_{yx}dAsina\right)sin\alpha +({\sigma }_{y}dAsin\alpha )cos\alpha =0\end{array}\right.$$

(19)

As a result,

$$\left\{\begin{array}{l}{\sigma }_{\alpha }={\displaystyle \frac{{\sigma }_x+{\sigma }_y}{2}}+{\displaystyle \frac{{\sigma }_x-{\sigma }_y}{2}}cos2\alpha -{\tau }_{xy}sin2\alpha \\ {\tau }_{\alpha }={\displaystyle \frac{{\sigma }_x-{\sigma }_y}{2}}sin2\alpha +{\tau }_{xy}cos2\alpha \end{array}\right.$$

(20)

$${\left({\displaystyle \frac{d{\sigma }_{\alpha }}{d\alpha }}\right)}_{\alpha ={\alpha }_0}=-2\left[\left({\displaystyle \frac{{\sigma }_x-{\sigma }_y}{2}}\right)sin2{\alpha }_0+{\tau }_{xy}cos2{\alpha }_0\right]=0$$

(21)

4. Results and Discussion

4.1. Development Characteristics of Mining-Induced Fractures

Figure 8 depicts the dynamic evolution of the fractures formed along the working face in the strike direction during the first 300 m of underground mining in the UDEC simulation. Based on the numerical simulation results, it is inferred that the overburden failure process is a dynamic and temporal evolving one. When the working face is advanced to 30 m after coal mining, the overburden and the basic roof begin to exhibit signs of sagging (Figure 8a). When the working face moves forward to 40 m, the fine sandstone and carbon mudstone start to sag, and a tensile fracture forms at a height of 3.5 m above the carbon mudstone roof, with a failure height of 3.5 m (Figure 8b). When the working face moves forward to 50 m, the fine sandstone above the carbon mudstone completely collapses, with a failure height of 9.5 m (Figure 8c). At this time, the first layer of fine sandstone is in contact with the bottom of the coal seam. When the working face moves forward to 60 m, the fine sandstone above the medium sandstone completely collapses, and a larger tensile fracture forms in the medium sandstone above (Figure 8d). When the working face moves forward to 70 m, the subcritical fine sandstone is initially damaged, with a failure height of 23 m (Figure 8e), and a tensile fracture forms in the fine sandstone and the carbon mudstone above. At this point, the density of the tensile fractures is significantly greater than that in the early stage of mining, and a hinge-bearing structure has formed at both ends of the rock mass (Figure 8i). The sub-key stratum regulates the distance between the rock mass and the critical layer, resulting in a reduction in pressure. When the working face moves forward to 80 m, the failure height extends to the bottom of the critical fine sandstone layer, but the critical fine sandstone layer is not yet damaged, and only a few tensile fractures have developed. At this time, the overlying rock mass undergoes the first cycle of failure, and the failure pattern is trapezoid-shaped (Figure 8f). When the working face moves forward to 90 m, the fine sandstone in the primary key stratum is initially fractured, and the overlying rock layer undergoes a second cycle of failure, attaining a failure height of 51.5 m (Figure 8g). When the working face moves forward to 100 m, the direct roof continues to be damaged, the failure morphology is trapezoid-shaped, and the height of the trapezoidal belt constantly increases (Figure 8h). When the working face moves forward to 110 m, the failure height reaches 64.5 m, and a considerable number of bed separations emerge above (Figure 8i). When the working face moves forward to 160 m, the failure height reaches the maximum value of 112 m. Subsequently, as the working face moves ahead, the height of the trapezoidal belt stops increasing, and the lateral extent of the trapezoidal belt expands (Figure 8j).

The variation in the free space beneath the rock strata is bound to give rise to distinct characteristics of rock movement. In the collapsing zone (Figure 8b), after a minor amount of displacement, the rock strata fracture for the first time. However, the fractured rock strata do not descend due to the restricted space (Figure 8c). After a small amount of counterclockwise rotation, the broken rock blocks interconnect and cease rotating, and subsequently, a voussoir structure is formed (Figure 8i), which possesses a certain bearing capacity and can greatly reduce the load on the overlying rock strata. When the roof undergoes failure due to shear damage, it slides along the failure plane, generating impact loads on the working face. As shown in Figure 8k–n, when the working face advances to within a range from 200 to 300 m, upward bending subsidence commences above, and the subsidence distance gradually increases as the working face moves forward. The transverse fractures penetrate the rock strata, while the displacement fractures continuously close, resulting in the upward transmission of the destruction of the overlying rock. The basic roof collapses and sinks under the influence of the fracturing and sinking of the upper rock strata, and the fractured rock strata at both ends form a voussoir structure (Figure 8i). When the height of the destruction reaches a maximum value of 112 m (Figure 8j), the displacement fracture at the bottom of the rock strata begins to close. When the working face advances to 240 m, the displacement fracture is nearly completely closed, and non-penetrating transverse fractures develop at both ends of the working face, while penetrating transverse fractures develop in the middle. The overall development pattern of the mining fractures is approximately an isosceles trapezoid. The lower-left angle is 48°, and the lower-right angle is 50° (Figure 8n).

After coal seam mining, a pressure equilibrium arch forms in the overlying rock layers. The weight of the rock layers outside the arch is transferred to the arch feet through this equilibrium arch, while the rock layers within the arch bend due to their own gravity. In the caved zone, the deterioration of rock layers undergoes four clear stages: bed layer separation, initial rupture, collapse, and cyclic rupture (see Figure 8b,e,f) [47].

Discontinuous fractures are the fractures that occur between the rock layers in the overburden, as a result of uneven subsidence induced by tensile stress during mining operations [48]. These fractures are mainly located in the fracture zone and can act as storage pathways for water sources. The development of discontinuous horizontal fractures is closely related to the physical and mechanical attributes of the rock layers, changes in the mining stress field, and the distribution of the primary fractures. Shear fractures formed as a result of the vertical shear stress on the rock layers or tensile fractures caused by the bending of the overburdened rock layers under stress are called transverse breakage fractures. The development of interlayer fractures is more pronounced at the ends of the cut and the other end of the working face than in the middle, and the width of the upper interlayer fractures is significantly larger than that of the lower interlayer fractures. The vertical breakage fractures developed at both ends constitute the main water conduits that lead to water-related hazards, especially the through-cutting fractures (Figure 8g).

Some scholars have studied the relationship between surface subsidence and burial depth through numerical simulation methods [49,50,51]. When the mining depth is shallow, surface subsidence becomes more pronounced, characterized by a localized subsidence zone and significant deformation. The limited thickness of the overlying rock strata facilitates the transmission of rock displacement to the surface, leading to the formation of a subsidence basin. Additionally, the subsidence rate is accelerated, which may result in abrupt surface collapses. In contrast, when the mining depth is greater, surface subsidence is relatively mitigated, exhibiting a broader subsidence area but with more gradual deformation. The increased thickness of the overlying rock layers absorbs and dissipates the rock displacement during its propagation, thereby reducing the magnitude of surface subsidence. The subsidence rate is slower, typically presenting as a progressive deformation process. Overall, surface subsidence demonstrates a negative correlation with mining depth: as the mining depth increases, the magnitude of subsidence diminishes, while the affected area expands. Conversely, as the mining depth decreases, surface subsidence becomes more pronounced and concentrated.

4.2. Fractal Evolution Law of Mining-Induced Fractures

Combining Equation (1), the fracture network models for various mining distances were analyzed, and the fractal dimensions of these networks were calculated. The relationship between the fracture characteristics at different stages of mining-induced fractures and their corresponding fractal dimensions is summarized in

Table 5. Relationship between the fracture stage and the fractal dimension of the mining-induced fractures.

Excavation Steps	Fractal Dimension	Correlation Coefficient	Breaking Characteristics
70 m	1.168	0.9806	sub-key stratum break
90 m	1.180	0.9957	primary key stratum break
120 m	1.286	0.9941	upward extension
160 m	1.284	0.9973	reach WFFZmax
200 m	1.277	0.9846	fracture compaction
240 m	1.276	0.9854	tend to stabilize
280 m	1.278	0.9785	tend to stabilize
300 m	1.295	0.9863	tend to stabilize

. Binary maps and fracture fractal values for selected mining stages are illustrated in Figure 9.

According to the fracture network topology maps shown in Figure 9, the fractal dimension of the overburden fracture network in the working face calculated by the phase correlation method is always higher than 0.97, indicating that the evolution of the complex rock fractures under mining action has highly regular fractal characteristics. From the binary maps of fracture development, it can be seen that the overburden layer in front of the working face is affected by its own weight and mining disturbance, and as the working face advances, it continues to collapse, sink, horizontally shift, and expand fractures, eventually reaching a new equilibrium state. The fractures in the overburden layer have dynamic development characteristics, and the range of fracture expansion gradually expands upwards from the mined-out area. These expanding fractures can be divided into lateral shear fractures and vertical fault fractures. From the fractal dimension value maps of fracture development, it can be seen that the fractal dimension of the overburden fractures dynamically changes with the advance of the working face. Initially, during the advance of the working face from 0 to 70 m, mining disturbance causes the fractures to rapidly develop, increasing the fractal dimension to 1.116. Subsequently, when the working face advances from 70 to 160 m, both the sub-key stratum and the primary key stratum are damaged, and the fractal dimension increases to 1.284. Finally, when the working face advances from 160 to 300 m, the overburden layer reaches the fully mined-out state, and the generation and compression of new and old fractures stabilize, leading to a stable fluctuation in the fractal dimension.

In order to further refine the analysis of the development of mining fractures, the fracture map covering a working face advance of 300 m was divided into 1500 subregions (30 rows × 50 columns). To further refine the analysis of the development of mining-induced fractures, the fracture map covering a 300-m advance of the working face was divided into 1500 sub-regions (30 rows × 50 columns). The fractal dimension of each sub-region was calculated using Matlab software, and a contour map was plotted, as shown in Figure 10.

Due to the generally uneven and rough surface of the collapsed rock mass in the mined-out area, the fractal dimension of the rock mass surface can be approximately taken as D = 2.5. In the field, the minimum scale of the collapsed rock mass in the mined-out area is generally δ_min ≤ 1 × 10⁻⁵ m, therefore, in engineering, the fractal permeability model of the mined-out rock mass based on the K-C equation can be simplified as follows [52]:

$$K={\displaystyle \frac{1}{5k_s^2}}{\displaystyle \frac{{\left({\rho }_0-\rho \right)}^3{M}^2}{{\rho }_0{\rho }^4}}{\displaystyle \frac{{\left(1.5-D\right)}^2}{D^2}}{\displaystyle \frac{1}{{\delta }_{max}^4}}$$

(22)

ρ is the density of the broken rock mass (kg/m³), ${\rho }_0$ is the apparent density of the collapsed rock mass (kg/m³), k is the shape coefficient of the rock surface, M is the total mass of the collapsed rock mass.

Equation (22) is the K-C equation for mining-induced overburden with fractal characteristics. This equation demonstrates that permeability is a function of both the fractal dimension of the fracture network in the mining-induced overburden and macroscopic physical property parameters. In practical applications, the apparent density of the collapsed rock mass can be measured directly, while the bulk density can be derived using the swell coefficient, as shown by the following relationship:

$${\displaystyle \frac{\rho }{{\rho }_0}}={\displaystyle \frac{1}{k_p}}$$

(23)

In the formula, k_P represents the rock swelling coefficient. The mass can be determined using the product of density and volume; the maximum characteristic size can be obtained through theoretical calculations, numerical simulations, and physical modeling; and the fractal dimension can be calculated using the box-counting method via image processing techniques.

Based on Equation (22) and Figure 10, the permeability of the fracture network exhibits a positive correlation with its fractal dimension. Specifically, as the fractal dimension increases, the complexity of the fracture network rises, leading to enhanced permeability. This relationship can be attributed to the fact that a more complex fracture network provides a greater number of fluid pathways, thereby facilitating easier fluid transport. It is evident that the fractal dimension of overburden fractures in different regions shows distinct trends as mining progresses. Investigating the relationship between fractal dimension and permeability enables more accurate predictions of fluid flow dynamics and the optimization of resource extraction efficiency. Near the working face cut, the fractal dimension remains relatively stable at approximately 1.35, corresponding to the highest permeability. In contrast, near the shear zone, the fractal dimension gradually decreases from 1.35 to around 1.1. Although the overall fracture morphology remains unchanged, as the working face advances, the central region of the mined-out area becomes increasingly compacted, resulting in a continuous decrease in both fractal dimension and permeability.

4.3. Development Height of Mining-Induced Fractures

According to the calculations based on critical layer theory, the 5th, 8th, 10th, and 17th overlying rock layers are identified as critical layers, all of which are located within the sandstone layer. This is attributed to the fact that the sandstone layer has a considerable thickness and is relatively hard. Among them, the 10th layer of sandstone is the primary key stratum, while the 5th layer is a sub-key stratum. The HFS of the lower in the sub-key stratum in the 5th layer of sandstone is 6.49 m, and the HFS of the lower in the primary key stratum in the 10th layer of sandstone is 4.46 m. The HFS of the lower in the 15th layer of siltstone is 2.26 m, and its maximum deflection is 2.34 m. The HFS of the lower in the 16th layer of coarse sandstone is 1.18 m, and its maximum deflection is 0.62 m. The HFS of the lower is greater than the maximum deflection, satisfying Equation (9). Hence, the development of mining-induced fractures terminates at the 16th layer of coarse sandstone.

Based on the numerical simulation of the excavation process, it was found that the overlying rock layer fills the mined-out area directly via roof collapse. Owing to the crushability and bearing capacity of the collapsed rock mass, this breakage will lead to the formation of a hinge-bearing structure. The basic roof undergoes cyclic failure, and the overlying rock layer above the basic roof is prone to overall collapse and subsidence under the influence of the breakage of the basic roof and the load imposed by the loose surface layer. When the failure height has not reached the maximum value, the three zones are not formed. Before the maximum failure height is attained, the WFZH continues to be transmitted upward in the form of hinging and layer separation, and the shear fractures close when the maximum failure height is reached.

A curve was fitted between the excavation steps of the working face and the WFZH (Figure 9). As illustrated in Figure 9, the connection between the excavation steps of the working face and the WFZH conforms to the Sigmoid curve equation, which is an S-shaped curve:

$$y=A_2-{\displaystyle \frac{A_2-A_1}{1+e^{\frac{x-x_0}{B}}}}$$

(24)

where y is the WFZH (m); x is the distance of the workface advancement (m); A₁, A₂, x₀, and B are constants; and R² = 0.993.

As illustrated in Figure 11, as the working face moves forward, the WFZH undergoes a stepwise increase, and each stage contains a point of sudden increase. This is attributed to the fact that after the basic roof undergoes periodic breakage, the rock layers overlying it also become damaged, causing the entire structure to be prone to fracturing and subsidence, ultimately giving rise to the stepwise increase phenomenon. Among the points of sudden increase, point 2 represents the breaking of the sub-key stratum, which results in the destruction of the rock layers overlying the sub-key stratum. Point 3 represents the breaking of the primary key stratum, which causes the rock layers overlying the primary key stratum to collapse and subside as a whole, until reaching the 17th layer of fine sandstone in the roof.

In Stage 1, the immediate roof and the direct roof have not reached the limit of the breaking distance, and the overlying rock layers remain undamaged. In Stage 2, the immediate roof and the direct roof reach the limit of the breaking distance and collapse, filling the void left by the mining, and the damage to the overlying rock gradually spreads upward. In Stage 3, the sub-key stratum breaks, and the shear fractures rapidly propagate upward during the excavation of the mining face. As a result, the secondary critical layer is prone to being damaged and forming a cantilever beam structure instead of a stable hinge-bearing structure. In Stage 4, as the upward breakage distance of the basic roof increases, when the cantilever beam structure becomes unstable, the working face generates a low amount of pressure, resulting in simultaneous subsidence of the overlying rock and sub-critical layer, triggering the damage of the primary key stratum. The damage of the primary key stratum then leads to periodic damage of the sub-key stratum, which imposes a significant load on the working face. The overlying rock collapses as a whole, and the development of mining fractures surges significantly. In Stage 5, the basic roof continues to break upward to the 16th layer of coarse sandstone in the roof. However, the 17th layer of fine sandstone has a considerable thickness and high degree of hardness, preventing it from breaking. The development of the mining fractures ends at the coarse sandstone, and the WFZH reaches the maximum value and stabilizes. In summary, it can be categorized into three stages: the slow growth stage, the rapid growth stage, and the stable equilibrium stage.

The WFZH in working face 31123-1, calculated using the formula for water-conducting fractures, from equation (2), ranges from 85.43 to 106.3 m, and the fracture–mining ratio ranges from 11.39 to 14.17. The WFZH, calculated using the key stratum theory, ranges from 97 to 113 m and terminates at the 16th layer of coarse sandstone. The corresponding fracture–mining ratio ranges from 12.93 to 15.07. The development height of the fractured zone, calculated via numerical simulation, is approximately 112 m (Figure 8j), and the fracture–mining ratio is 14.93. The three calculated results are highly similar.

4.4. Development Morphology of Mining-Induced Fractures

According to the numerical simulation results and previous research findings, the development pattern of mining-induced fractures is trapezoid-shaped [26,53,54]. After the first breakthrough at the subordinate layer, the failure pattern of the overlying rock layer assumes the shape of a trapezoid. As the working face advances, both the height and lateral extent of the trapezoidal belt increase. When the working face advances to a certain distance, that is, when the development of the mining fractures reaches the maximum value, the height of the trapezoidal belt stops increasing, but the lateral extent continues to increase.

The results calculated using Equation (18) are ${\sigma }_x$ = 14.14 MPa, ${\sigma }_y$ = –17.03 MPa, and ${\tau }_{xy}$ = –13.15 MPa.

The results calculated using Equation (21) are $tan2{\alpha }_0={\displaystyle \frac{-2{\tau }_{xy}}{{\sigma }_x-{\sigma }_y}}$, and among them $\left({\displaystyle \frac{{\sigma }_x-{\sigma }_y}{2}}\right)sin2{\alpha }_0+{\tau }_{xy}cos2{\alpha }_0={\tau }_{{\alpha }_0}$.

Consequently, ${\tau }_{{\alpha }_0}$ = 0. The shear stress at the section where the maximum positive stress occurs is zero, and the plane where the maximum positive stress occurs is the principal plane. Hence, the maximum positive stress is the principal stress.

Since ${\sigma }_x$ < ${\sigma }_y$, the larger angle corresponds to the principal stress, and the calculation reveals that α = 48°.

That is, the base angle of the trapezoid-shaped fractures caused by the mining is 48°. This is consistent with the numerical simulation results, verifying the accuracy of the theoretical calculations and numerical simulation.

5. Conclusions

In this paper, we investigated the development characteristics of mining-induced fractures in working face 31123-1 in the Dongxia Coal Mine via numerical simulation and theoretical analysis. Here are the key findings of this study, in summary:

(1)

The spatiotemporal dynamics characterize the evolution of the overburden failure movement. During coal mining, as the working face advances, the overburden and the immediate roof begin to collapse. After the sub-key stratum fractures, the overburden layer undergoes the first periodic failure, and the failure pattern is trapezoid shaped. As the distance between the working face and the coal face increases, the height of the step-shaped band also increases. The overburden layer gradually collapses from bottom to top, and a large number of shear fractures develop before the height of the mining-induced fractures reaches the maximum value. The shear fractures are more developed at the ends of the cut and the other end of the working face than in the middle, and the width of the shear fractures above is significantly greater than that of the fractures below. Both ends develop through-cutting inter-bedding fractures, while only a small number of non-through-cutting fractures develop in the middle. When the height of the mining-induced fractures reaches the maximum value, the shear fractures gradually close and eventually close completely. The through-cutting inter-bedding fractures at both ends are the main channels for connecting the aquifers and causing water-related hazards.

(2)

Initially, during the advance of the working face from 0 to 70 m, mining disturbances cause rapid fracture development, resulting in an increase in the fractal dimension to 1.116. Subsequently, as the working face advances from 70 to 160 m, both the sub-key stratum and the primary key stratum are damaged, leading to a decrease in the fractal dimension to 1.284. Finally, when the working face advances from 160 to 300 m, the overburden layer reaches a fully mined-out state, and the generation and compression of new and old fractures stabilize, resulting in a stable fluctuation in the fractal dimension. Combining fractal theory, a fractal permeability model of the overburden affected by mining was established based on the K-C equation, and it was found that the permeability of the mining-induced fractures is positively correlated with their fractal dimension. Specifically, as the fractal dimension increases, the complexity of the fracturing network increases, thereby enhancing permeability. This relationship can be attributed to the fact that the more complex fracture network provides more fluid channels, thereby facilitating easier fluid transport. The fractal dimension of the overburden fractures in different regions shows different trends as mining proceeds.

(3)

According to the numerical simulation results, the correlation between the WFZH and the excavation steps of the working face is an S-shaped curve. The WFZH exhibits a stepwise increase, with a sudden increase at each stage. This is because the rock layer overlying the basic roof breaks periodically after the basic roof is fractured, making the entire rock mass more prone to fracture and subsidence, and ultimately, leads to a stepwise increase phenomenon. The numerical simulation results indicate that the WFZH is approximately 112 m, and the fracture–mining ratio is 14.93. The WFZH, calculated using its formula, ranges from 85.43 to 106.3 m, and the fracture–mining ratio ranges from 11.39 to 14.17. The WFZH, calculated based on the key stratum theory, extends to the 16th layer of the coarse sandstone in the roof. The calculated WFZH ranges from 97 to 113 m, and the fracture–mining ratio ranges from 12.93 to 15.07. The three calculation results are similar.

(4)

The numerical simulation results reveal that the development shape of the mining fractures is a trapezoid, with a lower-left angle of approximately 48° and a lower-right bottom angle of about 50°. This is similar to the theoretical calculation result of 48°, verifying the accuracy and reliability of the theoretical calculations and numerical simulation results.