# Numerical Analysis of Stress Fields and Crack Growths in the Floor Strata of Coal Seam for Longwall Mining

Authors: Lingzhi Sun; Yunyue Xie; Hongtian Xiao
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

This paper predicts the possibility of water inrush from a confined aquifer under the action of mining activities and water pressure. The study uses numerical analyses to evaluate stress redistribution and crack growth which result from coal extraction operations. Two models are presented in this study. By simplifying the distribution of the disturbed vertical stress on the coal seam and floor around a working face, a model is established to analyze the additional stresses in the floor strata induced by mining activities. And some distribution features of all the additional stress components are described. By using the superposition principle in fracture mechanics, another model is developed to analyze the crack growth in the floor strata under the action of disturbed stresses and water pressure. And the stress intensity factors at the crack tip are presented and the process of crack growth is obtained in the advancement of a working face. Because of discretizing only loading areas and crack surfaces, the present methods can obtain the accurate numerical results. Finally, some suggestions are made for preventing the water inrush from a confined aquifer.


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## Body

## 1. Introduction

In the coalfields of North China, coal seams are underlain by Ordovician limestone, which is strongly karstified and highly permeable. In most cases, the limestone aquifer contains an abundant supply of water and is confined. The activities of mining these coal seams are threatened with the confined water from the Ordovician limestone. Water inrush accidents result in heavy losses and casualties [1, 2]. With coal mining being executed in deeper sites, the pressure of confined water becomes larger and this type of water inrushes may occur more frequently.In order to avoid water inrushes and keep the mines safe, large-scale dewatering or depressurizing of the Ordovician aquifer was chosen. In practice, dewatering causes serious environmental problems such as surface subsidence and underground water pollution and reduces the amount of water available for supply in the area. To solve this dilemma, some researches were devoted to make full use of the uncontaminated karst water obtained from dewatering for water supply [3]. Another measure to prevent water inrushes is to use the floor strata between the coal seam and the aquifer as a protective rock pillar. However, once the strength of the floor strata is difficult to resist the invasion of high-pressure water, the confined water floods into working faces. In some cases, grouting into the rock layers is used to improve the strength of floor strata to resist the invasion of confined water. By taking these measures, the water in the Ordovician limestone can be preserved to some extent and the coal seams can be extracted in a relatively safe environment.As early as in 1979, the field investigations of the floor strata were executed in Fengfeng coalfield, China, where mining activities were threatened by confined aquifers [4]. After that, more field investigations of floor strata were performed in different types of mines. In Jingxing coalfield, China, it was found that the joints in floor strata were impermeable before mining, and confined water passed through the joints and flooded into the mines after mining [5]. This phenomenon may be explained by the joint growth induced by mining activities and water pressure from a confined aquifer. Additionally, the test of water injection in boreholes was used for understanding the permeability variations in floor strata induced by mining activities [6, 7]. These measured data were further used to determine the failure zones directly below the coal seam and above the confined aquifer.In the past several decades, other types of analytical methods have been employed to study mining with water pressure for preventing water inrushes. Wang et al. [8] established both physical and numerical models to gain a better understanding into fracture initiation and propagation in floor strata during a mining process. Yang et al. [9] and Li et al. [10] used finite element methods to understand the mechanisms of water inrushes. Because different coal seams have different vulnerabilities for water inrushes, Wu et al. [11] assessed the differences in vulnerability between multiple coal seams. To evaluate a major water inflow hazard in a mine, Rafiqul et al. [12] used boundary element methods to model the mining-induced stress field. Zuo et al. [13] explored the spatial morphology of karst collapse and employed the discontinuous deformation analysis method to investigate karst collapse.Although a great deal of field investigations and numerical analyses have been performed and some research results of water inrushes have been obtained, all the mechanisms of water inrushes are not known and the water inrushes are often reported. It is estimated that more than 80 percent of water inrushes may occur in the case of the floor rocks having faults or joints. This means that faults and joints in floor strata may induce water inrushes under the action of water pressure from aquifers and mining activities. Therefore, the water inrushes induced by faults or joints in longwall mining need to be further studied.The principal objectives of this paper are to understand the stress fields in floor strata induced by coal mining and to predict the joint growths in the floor strata of the coal seam threatened by confined water in the advancement of a working face. From now on, the term crack will be used to describe a joint. In this paper, firstly, two numerical methods to be used are briefly introduced. Secondly, a mechanical model for simulating coal mining is established and the additional stress fields induced by coal mining are presented. Thirdly, another mechanical model is established for analyzing the crack problems under the action of water pressure and mining activities, and the stress intensity factors (SIFs) are obtained for the cracks at different positions of floor strata. Finally, the crack growths are predicted using SIF values and fracture criteria, and some suggestions are made for preventing the water inrushes into working faces from confined aquifers in floor strata.


## 2. Analytical Methods for Elastic Fields and Crack Problems in a Layered Medium

### 2.1. Yue’s Solution Based Numerical Method for Elastic Fields in a Layered Medium

Xiao et al. [14] developed a numerical method for assessing the elastic fields in an arbitrarily depth-heterogeneous medium subject to complex loads using Yue’s solution. Yue’s solution [15] is a fundamental singular solution for the generalized Kelvin problems of a multilayered elastic solid of infinite extent subject to concentrated point body vectors. The stresses and displacements at any points of a layered medium are described as(1a)σijQ=∫Sσijk∗Q,PtkPdSP,i,j,k=x,y,z(1b)uiQ=∫Suik∗Q,PtkPdSP,i,k=x,y,zwhere σijk∗Q,P and uik∗Q,P are, respectively, stresses and displacements of Yue’s solution for the field point Q due to the unit force along the k direction at the source point P; tkP is the traction at the source point P; the integral domain S is the loading area. It should be noted that the subscript k is a dummy index.The corresponding computer program LayerSmart3D was written in FORTRAN. The techniques adopted in LayerSmart3D primarily involve the discretization of the loading area into a finite number of quadrilateral elements. Values of the loads are inputted at the node points of the discretized area. Numerical verification of LayerSmart3D indicates that numerical solutions of high accuracy can be efficiently calculated for elastic fields induced by the distributed loads in a layered medium.


### 2.2. Yue’s Solution Based DDM for Analyzing Crack Problems in a Layered Medium

Xiao and Yue [16, 17] developed a new displacement discontinuity method (DDM) for the analysis of crack problems in a layered medium of infinite extent. This approach is also based on Yue’s solution [15] and the corresponding computer program LayerDDM3D was written in FORTRAN.Assume that the crack surfaces consist ofΓ+ and Γ-. In the global coordinates x,y,z, the discontinuous displacements of the crack surfaces are defined as ΔujQΓ+=ujQΓ+-ujQΓ-. uj is a displacement along the direction j=x,y,z. The traction on the crack surface is defined as tjPΓ+. The basic equations of Yue’s solution based DDM are as follows:(2)tjPΓ++niPΓ+∫Γ+Tijk∗PΓ+,QΔukQdΓQ=0,i,j,k=x,y,zwhere niPΓ+ is the unit outward normal of the positive side of the crack surface at the source point PΓ+ and Tijk∗(PΓ+,Q) is a new kernel function which can be calculated by using the traction tij∗ of Yue’s solution.In developing the DDM for (2), the nine-node continuous and discontinuous elements are used to discretize a crack surface. Kutt’s numerical method is used to calculate the hypersingular integral in the integral equation. By solving the discretized equations of (2), the discontinuous displacements on the crack surface can be obtained. The numerical verification shows that the present results are in very good agreement with the existing ones. However, the proposed DDM is only suitable for the analysis of crack problems in layered media of infinite extent under the action of loadings on crack surfaces. Other types of crack problems in infinite media can be analyzed by using the two numerical methods together.


### 2.3. The Analytical Methods of Crack Problems in Layered Media

In the following, the two numerical methods mentioned above, i.e., LayerSmart3D and LayerDDM3D, and the superposition principle in fracture mechanics are utilized to analyze crack problems in a layered medium. LayerSmart3D is firstly employed to obtain the stress fields of a layered medium without a crack under the action of distributed loadings within layered media. Using the superposition principle [18], the tractions, whose values are equal to the mining-induced stress and which have opposite directions, are then applied on the crack surfaces in the layered medium without the above distributed loadings. LayerDDM3D is finally employed to obtain the discontinuous displacements of the crack surfaces under the action of the above tractions. The SIF values at the crack tip can be calculated by using the discontinuous displacements of the crack surfaces and the relationship between SIFs and discontinuous displacements.


## 2.1. Yue’s Solution Based Numerical Method for Elastic Fields in a Layered Medium

Xiao et al. [14] developed a numerical method for assessing the elastic fields in an arbitrarily depth-heterogeneous medium subject to complex loads using Yue’s solution. Yue’s solution [15] is a fundamental singular solution for the generalized Kelvin problems of a multilayered elastic solid of infinite extent subject to concentrated point body vectors. The stresses and displacements at any points of a layered medium are described as(1a)σijQ=∫Sσijk∗Q,PtkPdSP,i,j,k=x,y,z(1b)uiQ=∫Suik∗Q,PtkPdSP,i,k=x,y,zwhere σijk∗Q,P and uik∗Q,P are, respectively, stresses and displacements of Yue’s solution for the field point Q due to the unit force along the k direction at the source point P; tkP is the traction at the source point P; the integral domain S is the loading area. It should be noted that the subscript k is a dummy index.The corresponding computer program LayerSmart3D was written in FORTRAN. The techniques adopted in LayerSmart3D primarily involve the discretization of the loading area into a finite number of quadrilateral elements. Values of the loads are inputted at the node points of the discretized area. Numerical verification of LayerSmart3D indicates that numerical solutions of high accuracy can be efficiently calculated for elastic fields induced by the distributed loads in a layered medium.


## 2.2. Yue’s Solution Based DDM for Analyzing Crack Problems in a Layered Medium

Xiao and Yue [16, 17] developed a new displacement discontinuity method (DDM) for the analysis of crack problems in a layered medium of infinite extent. This approach is also based on Yue’s solution [15] and the corresponding computer program LayerDDM3D was written in FORTRAN.Assume that the crack surfaces consist ofΓ+ and Γ-. In the global coordinates x,y,z, the discontinuous displacements of the crack surfaces are defined as ΔujQΓ+=ujQΓ+-ujQΓ-. uj is a displacement along the direction j=x,y,z. The traction on the crack surface is defined as tjPΓ+. The basic equations of Yue’s solution based DDM are as follows:(2)tjPΓ++niPΓ+∫Γ+Tijk∗PΓ+,QΔukQdΓQ=0,i,j,k=x,y,zwhere niPΓ+ is the unit outward normal of the positive side of the crack surface at the source point PΓ+ and Tijk∗(PΓ+,Q) is a new kernel function which can be calculated by using the traction tij∗ of Yue’s solution.In developing the DDM for (2), the nine-node continuous and discontinuous elements are used to discretize a crack surface. Kutt’s numerical method is used to calculate the hypersingular integral in the integral equation. By solving the discretized equations of (2), the discontinuous displacements on the crack surface can be obtained. The numerical verification shows that the present results are in very good agreement with the existing ones. However, the proposed DDM is only suitable for the analysis of crack problems in layered media of infinite extent under the action of loadings on crack surfaces. Other types of crack problems in infinite media can be analyzed by using the two numerical methods together.


## 2.3. The Analytical Methods of Crack Problems in Layered Media

In the following, the two numerical methods mentioned above, i.e., LayerSmart3D and LayerDDM3D, and the superposition principle in fracture mechanics are utilized to analyze crack problems in a layered medium. LayerSmart3D is firstly employed to obtain the stress fields of a layered medium without a crack under the action of distributed loadings within layered media. Using the superposition principle [18], the tractions, whose values are equal to the mining-induced stress and which have opposite directions, are then applied on the crack surfaces in the layered medium without the above distributed loadings. LayerDDM3D is finally employed to obtain the discontinuous displacements of the crack surfaces under the action of the above tractions. The SIF values at the crack tip can be calculated by using the discontinuous displacements of the crack surfaces and the relationship between SIFs and discontinuous displacements.


## 3. The Additional Stresses of Floor Strata without Cracks Induced by Mining

### 3.1. Geological and Mining Conditions of the Working Face

Jing et al. [4] investigated the deformation and failure of floor strata in mining the coal seam threatened by confined water in No. 2 Coal Mine of Fengfeng coalfield. No. 2710 working face in this mine was chosen for field measurements. Table 1 describes the roof and floor strata of the extracted coal seam. The coal seam is about 48 m away from the Ordovician limestone in which the water pressure ranges from 1.5 to 2 MPa. The working face is located at a depth of about 145 m and the longwall panel is 90 m long and 350 m wide. The coal seam is 1.5 m thick and the dip angle is about 1-3°. For the sake of simplicity, the rock strata are treated as horizontal media in developing a mechanical model. The working face was advanced along the strike direction of the coal seam. From August 1979 to February 1980, comprehensive measurements of floor deformation and failure were performed in the advancing process of the working face. In the field measurements, large amounts of data were obtained and will be further employed to analyze the water inrush from confined aquifers here.Table 1
The roof and floor strata of the extracted coal seam in No. 2710 working face.
Strata Thickness(m) Elastic modulus (GPa) Poisson’s ratio Sandstone 14 25 0.28 Extracted coal seam 1.5 4 0.33 Sandstone 17 25 0.28 Limestone 5 40 0.28 Shale 4.2 15 0.29 Coal seam 1.8 4 0.33 Shale 20 20 0.30  Ordovician Limestone Larger than 650 45 0.30


### 3.2. A Model of Analyzing Additional Stresses of Floor Strata Induced by Mining

#### 3.2.1. The Mining-Disturbed Area on the Interface of Coal Seam and Floor

Figure1 illustrates the disturbed area ABCD on the interface between the mined coal seam and the floor. It is assumed that the domain out of the area ABCD is not disturbed by coal mining. According to the in situ investigations, the disturbed area ABCD is 150 m long and 150 m wide. The vertical stress on the interface of the coal seam and the floor strata, which are not disturbed by mining, can be calculated using the following formula:(3)p0=γh=25×145=3625kN/m2where γ is the average unit weight of the overlying strata and h is the depth of the extracted coal seam below the ground surface.Figure 1
The disturbed area ABCD induced by coal mining.Mining activities induce the stress concentration on the coal pillars and floor around the working face. In the field investigations at No. 2710 working face, the disturbed area and stress concentration on the coal seams were obtained. Based on these data, we develop a model for the analysis of the influence of coal mining on floor strata.The vertical stresses along different cross sections on the disturbed area are assumed by using the measured data. The concentration factors of the vertical stress ahead of and behind the working face along the cross section A1-A2 are taken to be k1=3 and k2=1, respectively. And the stress concentration factors along the cross sections B1-B2 and C1-C2 are taken to be k1=3. On the floor strata near the roadways and the working face, the stresses are highly concentrated and the stress concentration factor is taken to be k3=3.5. The stress concentration factors along other cross sections can be obtained by using the methods of linear or quadratic interpolation.Figure2 illustrates the process of obtaining the additional vertical stress on the interface of coal seam and floor along the cross section A1-A2 shown in Figure 1. It can be found that the additional vertical stress in Figure 2(c) is obtained by subtracting out the weight p0 of the overburden above the mined coal seam in Figure 2(b) from the vertical pressure in Figure 2(a). Figure 3 presents the additional vertical stresses on the interface between coal seam and floor along three different cross sections. Based on the data from in situ measurements, the parameters of lengths shown in Figure 3(a) are presented as follows: a1=40m, a2=10m, a3=5m, a4=5m, a5=30m, and a6=60m. The parameters of lengths shown in Figure 3(b) are presented as follows: L=90m and b1=30m. The parameters of lengths shown in Figure 3(c) are presented as follows: L=90m, c1=30m, c2=30m, and c3=30m.Figure 2
Analytical method of additional vertical stress on the interface of coal seam and floor along the cross section A1-A2.
(a)
Vertical stress on the interface of coal seam and floor for the disturbed case (b)
Vertical stress on the interface of coal seam and floor for the undisturbed case (c)
Additional vertical stress on the interface of coal seam and floor for the disturbed caseFigure 3
Additional vertical stress on the interface of coal seam and floor along different cross sections.
(a)
Additional vertical stress on the interface of coal seam and floor along the cross section A1-A2 (b)
Additional vertical stress on the interface of coal seam and floor along the cross section B1-B2 (c)
Additional vertical stress on the interface of coal seam and floor along the cross section C1-C2


#### 3.2.2. Element Mesh of the Disturbed Area and Comparison of Numerical and Field Results

In order to analyze the elastic fields using the software LayerSmart3D, the disturbed area ABCD is discretized into the mesh with 2610 nodes and 8037 elements, shown in Figure4. The additional vertical stresses on the disturbed area ABCD (see Figure 3) are applied on the nodes of the element mesh. Figure 5 illustrates the distribution of the additional vertical stress on the interface of coal seam and floor. It can be found that the distribution of the additional vertical stress shown in Figure 5 is in good agreement with the one shown in Figure 3.Figure 4
Element mesh of the disturbed area ABCD, the projectionabcd of the calculating area at z=15m, and the plane at y=20m for calculating the stresses.Figure 5
Additional vertical stress on the disturbed area ABCD.After presenting the mechanical parameters of floor strata in Table1 and the additional vertical stress in Figure 5, the software LayerSmart3D is used to obtain the stress and displacement fields of floor strata. Table 2 lists the vertical displacements at some points in the floor strata from numerical analyses and in situ measurements. It can be found that the displacements from the two methods are much close to each other if the differences of the geological and mining conditions for two methods are considered.Table 2
Comparison of maximum vertical displacements from in situ measurements and numerical analyses along the advancing direction (unit: mm).
Positions(y,z) ahead of the face wall behind the face wall In situ measurements This study In situ measurements This study ( 0,4.9 ) 8 9.9995 5 -2.7071 ( 0,6.2 ) 8 9.7000 -2 -2.5608 ( 0,10.2 ) 7 8.9472 1 -2.1707  ( 40,2.4 ) 13 10.3224 6 8.9056 ( 40,4.7 ) 14 9.3394 13 7.0610 ( 40,7.2 ) 14 9.6763 16 8.2905


### 3.3. Variations of Additional Stresses in Floor Strata Induced by Mining

#### 3.3.1. Distribution of Additional Stresses at the Depthz=15m

As shown in Figure4, the model is symmetric with respect to the coordinate planeOxz. Thus, the rectangular area, i.e., 60 m≤x≤110m, 0 m≤y≤45m, and z=15m, is chosen to calculate the additional stresses. This area has a projectionabcd on the coordinate planeOxy shown in Figure 4. Figures 6–11 illustrate the distribution of all the stress components on the calculating area.Figure 6
Contours of the additional stressσxx at the depth z=15m.Figure 7
Contours of the additional stressσyy at the depth z=15m.Figure 8
Contours of the additional stressσzz at the depth z=15m.Figure 9
Contours of the additional stressσxy at the depth z=15m.Figure 10
Contours of the additional stressσxz at the depth z=15m.Figure 11
Contours of the additional stressσyz at the depth z=15m.Figures6–8 show the variations of three normal stresses in the floor strata. In Figure 6, the rectangular area can be divided into three areas by considering the distribution features of σxx. Below the goaf area, σxx is tensile and has small values. Below the coal seam in the front of the working face, σxx is compressive and its minimum value is about -1.6 MPa. And below the coal pillar outside the goaf area, σxx is tensile and its maximum value is about 1.6 MPa. In Figure 7, the rectangular area can also be divided into three areas. Below the goaf area, σyy approaches zero. Below the coal seam in the front of the working face, σyy is tensile and its minimum value is about 1.6 MPa. And below the coal pillar outside the goaf area, σyy is compressive and its minimum value is about 1.6 MPa. In Figure 8, the rectangular area can be divided into two areas. σzz is compressive below the goaf area and is tensile away from the goaf area.Figures9–11 show the variations of three shear stresses in the floor strata. In Figure 9, σxy has a maximum value of about -1.0 MPa at the point x,y,z=94m,46m,15m and σxy becomes small away from this point. In Figure 10, σxz is concentrated in the front of the working face and becomes small away from this zone and its maximum value is about 1.0 MPa. In Figure 11, σyz is concentrated below the coal pillar outside the goaf area and becomes small away from this zone.


#### 3.3.2. Distribution of the Additional Stresses in the Planey=20m

Figures12–17 illustrate the distribution of all the stress components in the plane y=20m for different x and z values. As shown in Figure 4, this plane is along the advancing direction of the working face and has a horizontal distance of 25 m from the coal pillar outside the goaf. For the sake of simplification, the stress variations are referred to as the ones with the depth increasing in the following analyses. From these figures, we can have the following observations:Figure 12
Variations of the additional stressσxx with depth at y=20m.Figure 13
Variations of the additional stressσyy with depth at y=20m.Figure 14
Variations of the additional stressσzz with depth at y=20m.Figure 15
Variations of the additional stressσxy with depth at y=20m.Figure 16
Variations of the additional stressσxz with depth at y=20m.Figure 17
Variations of the additional stressσyz with depth at y=20m.(i) Atx=0m, all the stresses have very small values because the position is away from the maximum loadings. In the first and second layers, the stresses vary obviously with depth. In the other layers, all the stresses are very small and vary weakly with depth.(ii)σxx has small values in the first layer and the absolute values of σxx increase in the second layer. In the first layer, σyy increases at x=100m whilst σyy decreases at the other x values. In the second layer, σyy is negative and decreases. At x=40,81m, σzz is positive and decreases. At x=90m, σzz is positive in the first layer and negative in the other remaining layers and decreases. σzz is negative and decreases at x=95m, and σzz is negative and increases at x=100m.(iii) Atx=0,40m, σxy decreases in the first, second, and third layers and increases in the remaining two layers. The variation of σxy at x=81,90,95,100m is different from the one at x=0,40m. σxz is negative at x=0,40m whilst σxz is positive at x=81,90,95,100m. In the different layers, σxz first increases and then decreases for a given z value from x=81 to 100 m. The absolute values of σxz first increase and then decrease. σyz is always positive for all the x values. For x=0m, σyz increases. Except for x=0m, in the first and second layers, σyz first increases and then decreases. And from the third layer to the fifth layer, σyz first increases and then decreases.


## 3.1. Geological and Mining Conditions of the Working Face

Jing et al. [4] investigated the deformation and failure of floor strata in mining the coal seam threatened by confined water in No. 2 Coal Mine of Fengfeng coalfield. No. 2710 working face in this mine was chosen for field measurements. Table 1 describes the roof and floor strata of the extracted coal seam. The coal seam is about 48 m away from the Ordovician limestone in which the water pressure ranges from 1.5 to 2 MPa. The working face is located at a depth of about 145 m and the longwall panel is 90 m long and 350 m wide. The coal seam is 1.5 m thick and the dip angle is about 1-3°. For the sake of simplicity, the rock strata are treated as horizontal media in developing a mechanical model. The working face was advanced along the strike direction of the coal seam. From August 1979 to February 1980, comprehensive measurements of floor deformation and failure were performed in the advancing process of the working face. In the field measurements, large amounts of data were obtained and will be further employed to analyze the water inrush from confined aquifers here.Table 1
The roof and floor strata of the extracted coal seam in No. 2710 working face.
Strata Thickness(m) Elastic modulus (GPa) Poisson’s ratio Sandstone 14 25 0.28 Extracted coal seam 1.5 4 0.33 Sandstone 17 25 0.28 Limestone 5 40 0.28 Shale 4.2 15 0.29 Coal seam 1.8 4 0.33 Shale 20 20 0.30  Ordovician Limestone Larger than 650 45 0.30


## 3.2. A Model of Analyzing Additional Stresses of Floor Strata Induced by Mining

### 3.2.1. The Mining-Disturbed Area on the Interface of Coal Seam and Floor

Figure1 illustrates the disturbed area ABCD on the interface between the mined coal seam and the floor. It is assumed that the domain out of the area ABCD is not disturbed by coal mining. According to the in situ investigations, the disturbed area ABCD is 150 m long and 150 m wide. The vertical stress on the interface of the coal seam and the floor strata, which are not disturbed by mining, can be calculated using the following formula:(3)p0=γh=25×145=3625kN/m2where γ is the average unit weight of the overlying strata and h is the depth of the extracted coal seam below the ground surface.Figure 1
The disturbed area ABCD induced by coal mining.Mining activities induce the stress concentration on the coal pillars and floor around the working face. In the field investigations at No. 2710 working face, the disturbed area and stress concentration on the coal seams were obtained. Based on these data, we develop a model for the analysis of the influence of coal mining on floor strata.The vertical stresses along different cross sections on the disturbed area are assumed by using the measured data. The concentration factors of the vertical stress ahead of and behind the working face along the cross section A1-A2 are taken to be k1=3 and k2=1, respectively. And the stress concentration factors along the cross sections B1-B2 and C1-C2 are taken to be k1=3. On the floor strata near the roadways and the working face, the stresses are highly concentrated and the stress concentration factor is taken to be k3=3.5. The stress concentration factors along other cross sections can be obtained by using the methods of linear or quadratic interpolation.Figure2 illustrates the process of obtaining the additional vertical stress on the interface of coal seam and floor along the cross section A1-A2 shown in Figure 1. It can be found that the additional vertical stress in Figure 2(c) is obtained by subtracting out the weight p0 of the overburden above the mined coal seam in Figure 2(b) from the vertical pressure in Figure 2(a). Figure 3 presents the additional vertical stresses on the interface between coal seam and floor along three different cross sections. Based on the data from in situ measurements, the parameters of lengths shown in Figure 3(a) are presented as follows: a1=40m, a2=10m, a3=5m, a4=5m, a5=30m, and a6=60m. The parameters of lengths shown in Figure 3(b) are presented as follows: L=90m and b1=30m. The parameters of lengths shown in Figure 3(c) are presented as follows: L=90m, c1=30m, c2=30m, and c3=30m.Figure 2
Analytical method of additional vertical stress on the interface of coal seam and floor along the cross section A1-A2.
(a)
Vertical stress on the interface of coal seam and floor for the disturbed case (b)
Vertical stress on the interface of coal seam and floor for the undisturbed case (c)
Additional vertical stress on the interface of coal seam and floor for the disturbed caseFigure 3
Additional vertical stress on the interface of coal seam and floor along different cross sections.
(a)
Additional vertical stress on the interface of coal seam and floor along the cross section A1-A2 (b)
Additional vertical stress on the interface of coal seam and floor along the cross section B1-B2 (c)
Additional vertical stress on the interface of coal seam and floor along the cross section C1-C2


### 3.2.2. Element Mesh of the Disturbed Area and Comparison of Numerical and Field Results

In order to analyze the elastic fields using the software LayerSmart3D, the disturbed area ABCD is discretized into the mesh with 2610 nodes and 8037 elements, shown in Figure4. The additional vertical stresses on the disturbed area ABCD (see Figure 3) are applied on the nodes of the element mesh. Figure 5 illustrates the distribution of the additional vertical stress on the interface of coal seam and floor. It can be found that the distribution of the additional vertical stress shown in Figure 5 is in good agreement with the one shown in Figure 3.Figure 4
Element mesh of the disturbed area ABCD, the projectionabcd of the calculating area at z=15m, and the plane at y=20m for calculating the stresses.Figure 5
Additional vertical stress on the disturbed area ABCD.After presenting the mechanical parameters of floor strata in Table1 and the additional vertical stress in Figure 5, the software LayerSmart3D is used to obtain the stress and displacement fields of floor strata. Table 2 lists the vertical displacements at some points in the floor strata from numerical analyses and in situ measurements. It can be found that the displacements from the two methods are much close to each other if the differences of the geological and mining conditions for two methods are considered.Table 2
Comparison of maximum vertical displacements from in situ measurements and numerical analyses along the advancing direction (unit: mm).
Positions(y,z) ahead of the face wall behind the face wall In situ measurements This study In situ measurements This study ( 0,4.9 ) 8 9.9995 5 -2.7071 ( 0,6.2 ) 8 9.7000 -2 -2.5608 ( 0,10.2 ) 7 8.9472 1 -2.1707  ( 40,2.4 ) 13 10.3224 6 8.9056 ( 40,4.7 ) 14 9.3394 13 7.0610 ( 40,7.2 ) 14 9.6763 16 8.2905


## 3.2.1. The Mining-Disturbed Area on the Interface of Coal Seam and Floor

Figure1 illustrates the disturbed area ABCD on the interface between the mined coal seam and the floor. It is assumed that the domain out of the area ABCD is not disturbed by coal mining. According to the in situ investigations, the disturbed area ABCD is 150 m long and 150 m wide. The vertical stress on the interface of the coal seam and the floor strata, which are not disturbed by mining, can be calculated using the following formula:(3)p0=γh=25×145=3625kN/m2where γ is the average unit weight of the overlying strata and h is the depth of the extracted coal seam below the ground surface.Figure 1
The disturbed area ABCD induced by coal mining.Mining activities induce the stress concentration on the coal pillars and floor around the working face. In the field investigations at No. 2710 working face, the disturbed area and stress concentration on the coal seams were obtained. Based on these data, we develop a model for the analysis of the influence of coal mining on floor strata.The vertical stresses along different cross sections on the disturbed area are assumed by using the measured data. The concentration factors of the vertical stress ahead of and behind the working face along the cross section A1-A2 are taken to be k1=3 and k2=1, respectively. And the stress concentration factors along the cross sections B1-B2 and C1-C2 are taken to be k1=3. On the floor strata near the roadways and the working face, the stresses are highly concentrated and the stress concentration factor is taken to be k3=3.5. The stress concentration factors along other cross sections can be obtained by using the methods of linear or quadratic interpolation.Figure2 illustrates the process of obtaining the additional vertical stress on the interface of coal seam and floor along the cross section A1-A2 shown in Figure 1. It can be found that the additional vertical stress in Figure 2(c) is obtained by subtracting out the weight p0 of the overburden above the mined coal seam in Figure 2(b) from the vertical pressure in Figure 2(a). Figure 3 presents the additional vertical stresses on the interface between coal seam and floor along three different cross sections. Based on the data from in situ measurements, the parameters of lengths shown in Figure 3(a) are presented as follows: a1=40m, a2=10m, a3=5m, a4=5m, a5=30m, and a6=60m. The parameters of lengths shown in Figure 3(b) are presented as follows: L=90m and b1=30m. The parameters of lengths shown in Figure 3(c) are presented as follows: L=90m, c1=30m, c2=30m, and c3=30m.Figure 2
Analytical method of additional vertical stress on the interface of coal seam and floor along the cross section A1-A2.
(a)
Vertical stress on the interface of coal seam and floor for the disturbed case (b)
Vertical stress on the interface of coal seam and floor for the undisturbed case (c)
Additional vertical stress on the interface of coal seam and floor for the disturbed caseFigure 3
Additional vertical stress on the interface of coal seam and floor along different cross sections.
(a)
Additional vertical stress on the interface of coal seam and floor along the cross section A1-A2 (b)
Additional vertical stress on the interface of coal seam and floor along the cross section B1-B2 (c)
Additional vertical stress on the interface of coal seam and floor along the cross section C1-C2


## 3.2.2. Element Mesh of the Disturbed Area and Comparison of Numerical and Field Results

In order to analyze the elastic fields using the software LayerSmart3D, the disturbed area ABCD is discretized into the mesh with 2610 nodes and 8037 elements, shown in Figure4. The additional vertical stresses on the disturbed area ABCD (see Figure 3) are applied on the nodes of the element mesh. Figure 5 illustrates the distribution of the additional vertical stress on the interface of coal seam and floor. It can be found that the distribution of the additional vertical stress shown in Figure 5 is in good agreement with the one shown in Figure 3.Figure 4
Element mesh of the disturbed area ABCD, the projectionabcd of the calculating area at z=15m, and the plane at y=20m for calculating the stresses.Figure 5
Additional vertical stress on the disturbed area ABCD.After presenting the mechanical parameters of floor strata in Table1 and the additional vertical stress in Figure 5, the software LayerSmart3D is used to obtain the stress and displacement fields of floor strata. Table 2 lists the vertical displacements at some points in the floor strata from numerical analyses and in situ measurements. It can be found that the displacements from the two methods are much close to each other if the differences of the geological and mining conditions for two methods are considered.Table 2
Comparison of maximum vertical displacements from in situ measurements and numerical analyses along the advancing direction (unit: mm).
Positions(y,z) ahead of the face wall behind the face wall In situ measurements This study In situ measurements This study ( 0,4.9 ) 8 9.9995 5 -2.7071 ( 0,6.2 ) 8 9.7000 -2 -2.5608 ( 0,10.2 ) 7 8.9472 1 -2.1707  ( 40,2.4 ) 13 10.3224 6 8.9056 ( 40,4.7 ) 14 9.3394 13 7.0610 ( 40,7.2 ) 14 9.6763 16 8.2905


## 3.3. Variations of Additional Stresses in Floor Strata Induced by Mining

### 3.3.1. Distribution of Additional Stresses at the Depthz=15m

As shown in Figure4, the model is symmetric with respect to the coordinate planeOxz. Thus, the rectangular area, i.e., 60 m≤x≤110m, 0 m≤y≤45m, and z=15m, is chosen to calculate the additional stresses. This area has a projectionabcd on the coordinate planeOxy shown in Figure 4. Figures 6–11 illustrate the distribution of all the stress components on the calculating area.Figure 6
Contours of the additional stressσxx at the depth z=15m.Figure 7
Contours of the additional stressσyy at the depth z=15m.Figure 8
Contours of the additional stressσzz at the depth z=15m.Figure 9
Contours of the additional stressσxy at the depth z=15m.Figure 10
Contours of the additional stressσxz at the depth z=15m.Figure 11
Contours of the additional stressσyz at the depth z=15m.Figures6–8 show the variations of three normal stresses in the floor strata. In Figure 6, the rectangular area can be divided into three areas by considering the distribution features of σxx. Below the goaf area, σxx is tensile and has small values. Below the coal seam in the front of the working face, σxx is compressive and its minimum value is about -1.6 MPa. And below the coal pillar outside the goaf area, σxx is tensile and its maximum value is about 1.6 MPa. In Figure 7, the rectangular area can also be divided into three areas. Below the goaf area, σyy approaches zero. Below the coal seam in the front of the working face, σyy is tensile and its minimum value is about 1.6 MPa. And below the coal pillar outside the goaf area, σyy is compressive and its minimum value is about 1.6 MPa. In Figure 8, the rectangular area can be divided into two areas. σzz is compressive below the goaf area and is tensile away from the goaf area.Figures9–11 show the variations of three shear stresses in the floor strata. In Figure 9, σxy has a maximum value of about -1.0 MPa at the point x,y,z=94m,46m,15m and σxy becomes small away from this point. In Figure 10, σxz is concentrated in the front of the working face and becomes small away from this zone and its maximum value is about 1.0 MPa. In Figure 11, σyz is concentrated below the coal pillar outside the goaf area and becomes small away from this zone.


### 3.3.2. Distribution of the Additional Stresses in the Planey=20m

Figures12–17 illustrate the distribution of all the stress components in the plane y=20m for different x and z values. As shown in Figure 4, this plane is along the advancing direction of the working face and has a horizontal distance of 25 m from the coal pillar outside the goaf. For the sake of simplification, the stress variations are referred to as the ones with the depth increasing in the following analyses. From these figures, we can have the following observations:Figure 12
Variations of the additional stressσxx with depth at y=20m.Figure 13
Variations of the additional stressσyy with depth at y=20m.Figure 14
Variations of the additional stressσzz with depth at y=20m.Figure 15
Variations of the additional stressσxy with depth at y=20m.Figure 16
Variations of the additional stressσxz with depth at y=20m.Figure 17
Variations of the additional stressσyz with depth at y=20m.(i) Atx=0m, all the stresses have very small values because the position is away from the maximum loadings. In the first and second layers, the stresses vary obviously with depth. In the other layers, all the stresses are very small and vary weakly with depth.(ii)σxx has small values in the first layer and the absolute values of σxx increase in the second layer. In the first layer, σyy increases at x=100m whilst σyy decreases at the other x values. In the second layer, σyy is negative and decreases. At x=40,81m, σzz is positive and decreases. At x=90m, σzz is positive in the first layer and negative in the other remaining layers and decreases. σzz is negative and decreases at x=95m, and σzz is negative and increases at x=100m.(iii) Atx=0,40m, σxy decreases in the first, second, and third layers and increases in the remaining two layers. The variation of σxy at x=81,90,95,100m is different from the one at x=0,40m. σxz is negative at x=0,40m whilst σxz is positive at x=81,90,95,100m. In the different layers, σxz first increases and then decreases for a given z value from x=81 to 100 m. The absolute values of σxz first increase and then decrease. σyz is always positive for all the x values. For x=0m, σyz increases. Except for x=0m, in the first and second layers, σyz first increases and then decreases. And from the third layer to the fifth layer, σyz first increases and then decreases.


## 3.3.1. Distribution of Additional Stresses at the Depthz=15m

As shown in Figure4, the model is symmetric with respect to the coordinate planeOxz. Thus, the rectangular area, i.e., 60 m≤x≤110m, 0 m≤y≤45m, and z=15m, is chosen to calculate the additional stresses. This area has a projectionabcd on the coordinate planeOxy shown in Figure 4. Figures 6–11 illustrate the distribution of all the stress components on the calculating area.Figure 6
Contours of the additional stressσxx at the depth z=15m.Figure 7
Contours of the additional stressσyy at the depth z=15m.Figure 8
Contours of the additional stressσzz at the depth z=15m.Figure 9
Contours of the additional stressσxy at the depth z=15m.Figure 10
Contours of the additional stressσxz at the depth z=15m.Figure 11
Contours of the additional stressσyz at the depth z=15m.Figures6–8 show the variations of three normal stresses in the floor strata. In Figure 6, the rectangular area can be divided into three areas by considering the distribution features of σxx. Below the goaf area, σxx is tensile and has small values. Below the coal seam in the front of the working face, σxx is compressive and its minimum value is about -1.6 MPa. And below the coal pillar outside the goaf area, σxx is tensile and its maximum value is about 1.6 MPa. In Figure 7, the rectangular area can also be divided into three areas. Below the goaf area, σyy approaches zero. Below the coal seam in the front of the working face, σyy is tensile and its minimum value is about 1.6 MPa. And below the coal pillar outside the goaf area, σyy is compressive and its minimum value is about 1.6 MPa. In Figure 8, the rectangular area can be divided into two areas. σzz is compressive below the goaf area and is tensile away from the goaf area.Figures9–11 show the variations of three shear stresses in the floor strata. In Figure 9, σxy has a maximum value of about -1.0 MPa at the point x,y,z=94m,46m,15m and σxy becomes small away from this point. In Figure 10, σxz is concentrated in the front of the working face and becomes small away from this zone and its maximum value is about 1.0 MPa. In Figure 11, σyz is concentrated below the coal pillar outside the goaf area and becomes small away from this zone.


## 3.3.2. Distribution of the Additional Stresses in the Planey=20m

Figures12–17 illustrate the distribution of all the stress components in the plane y=20m for different x and z values. As shown in Figure 4, this plane is along the advancing direction of the working face and has a horizontal distance of 25 m from the coal pillar outside the goaf. For the sake of simplification, the stress variations are referred to as the ones with the depth increasing in the following analyses. From these figures, we can have the following observations:Figure 12
Variations of the additional stressσxx with depth at y=20m.Figure 13
Variations of the additional stressσyy with depth at y=20m.Figure 14
Variations of the additional stressσzz with depth at y=20m.Figure 15
Variations of the additional stressσxy with depth at y=20m.Figure 16
Variations of the additional stressσxz with depth at y=20m.Figure 17
Variations of the additional stressσyz with depth at y=20m.(i) Atx=0m, all the stresses have very small values because the position is away from the maximum loadings. In the first and second layers, the stresses vary obviously with depth. In the other layers, all the stresses are very small and vary weakly with depth.(ii)σxx has small values in the first layer and the absolute values of σxx increase in the second layer. In the first layer, σyy increases at x=100m whilst σyy decreases at the other x values. In the second layer, σyy is negative and decreases. At x=40,81m, σzz is positive and decreases. At x=90m, σzz is positive in the first layer and negative in the other remaining layers and decreases. σzz is negative and decreases at x=95m, and σzz is negative and increases at x=100m.(iii) Atx=0,40m, σxy decreases in the first, second, and third layers and increases in the remaining two layers. The variation of σxy at x=81,90,95,100m is different from the one at x=0,40m. σxz is negative at x=0,40m whilst σxz is positive at x=81,90,95,100m. In the different layers, σxz first increases and then decreases for a given z value from x=81 to 100 m. The absolute values of σxz first increase and then decrease. σyz is always positive for all the x values. For x=0m, σyz increases. Except for x=0m, in the first and second layers, σyz first increases and then decreases. And from the third layer to the fifth layer, σyz first increases and then decreases.


## 4. A Model of Analyzing Crack Problems in Floor Strata

As shown in Figure18, a rectangular crack is located in floor strata and is subject to mining-disturbed stresses and water pressure. A global coordinate systemOxyz, which is completely the same as the one shown in Figure 1, is attached to the model to be analyzed and a local coordinate system O′x′y′z′ is attached on the crack surface. The rectangular crack is 2 m wide (AB and CD sides) and 4 m long (AD and BC sides). Assume that the crack surfaces are parallel to the advancing direction of the working face and the crack center is located at the point x,y,z=d1,d2,46m.Figure 18
Model of analyzing crack problems in floor strata under the action of distributed stresses and water pressure.Using the superposition principle in fracture mechanics [18], the disturbed stresses induced by mining activities can be transformed to the tractions on the crack surfaces. The disturbed stresses should be the summation of the horizontal normal stress σh induced by rock weights and the additional stresses σyy,σxy,σzy induced by mining activities. Additionally, the crack surfaces are subject to the water pressure pw from confined aquifer. Thus, the crack surfaces are subjected to the following tractions:(4a)fx=-σxy(4b)fy=-σyy+σh+pw(4c)fz=-σyzwhere pw is taken to be 2 MPa.In (4b), σh is calculated by the following formula:(5)σh=ν1-νγhwhere ν and γ are, respectively, the average values of Poisson’s ratio and unit weight of rock strata and h is the depth of the crack center below the ground surface. Take γ=25kN/m2, ν=0.28, and h=192m. Thus, σh≈1.8667MPa.The crack surface is discretized into 435 nodes and 98 nine-node elements with 60 nine-node isoparametric elements, 34 discontinuous elements of type I, and four discontinuous elements of type II. The discontinuous displacements of the crack surfaces can be obtained using the element mesh, the tractionsfx,fy,fz on the crack surface, and the computer code LayerDDM3D.


## 5. SIFs of Cracks under the Action of Water Pressure and Disturbed Stresses

### 5.1. General

The stress intensity factors (SIF) values at the crack tip can be calculated by using the discontinuous displacements of crack surfaces. When the crack is not located at the interfaces of multilayered solids, the crack tip field singularities and angular distributions are the same as those in a homogeneous medium and the material parameters exert an influence on the magnitudes of the SIFs. Based on the relationship of displacements and the SIFs, the formulae of the SIFs along the crack side BC are as follows:(6a)KI=E41-ν2π2rΔuy′r,θ=±π,φ=0,(6b)KII=E41-ν2π2rΔuz′r,θ=±π,φ=0,(6c)KIII=E41+νπ2rΔux′r,θ=±π,φ=0,where (r,θ,φ) is the spherical coordinates located at the crack front and θ is an angle in the coordinate plane normal to the crack line of crack front.The following two cases are analyzed: Case 1: a rectangular crack is located in the coordinate planeOxz, i.e., d2=0m; Case 2: a rectangular crack is located in the plane y=40m, i.e., d2=40m.


### 5.2. Case 1: The Crack Is Located in the Coordinate PlaneOxz

In this case, the shear stressesσxy and σzy in the coordinate planeOxz are equal to zero because of the symmetry of the model. As a result, KII and KIII are equal to zero. Figure 19 illustrates the variations of KI for the crack at different positions along the advancing direction of the working face. For the crack only subject to water pressure, the maximum value of KI is 0.1134 MPa·m1/2. However, for the crack subject to water pressure and disturbed stresses, the maximum value of KI is 0.1351 MPa·m1/2 at d1=0m. With d1 increasing, KI increases, arrives at maximum values at d1=40,50m, and then decreases. For d1=80m and x′≥0.5m, KI is equal to zero and KI along the whole BC side is equal to zero at d1≥90m. These findings mean that the crack below the coal seam is closed and the crack below the goaf is opened under the action of disturbed stresses and water pressure. When the crack is located at some horizontal distance of 40-50m after the coal wall of the working face, there are the maximum opening displacements. In this case, the crack becomes easy to open and induces the uplifting of the confined water.Figure 19
Variations ofKI values at the crack side (BC) for Case 1.


### 5.3. Case 2: The Crack Is Located in the Planey=40m, i.e., d2=40m

In this case, the crack is close to the coal pillars outside the working face and has a horizontal distance of 5 m from the coal pillar. As discussed in Section3, the shear stresses induced by mining exist at this position and as a result the crack has three types of fracture modes. Figures 20–22 show the variations of KI, KII, and KIII along the crack front BC, respectively.Figure 20
Variations ofKI values at the crack side (BC) for Case 2.Figure 21
Variations ofKII values at the crack side (BC) for Case 2.Figure 22
Variations ofKIII values at the crack side (BC) for Case 2.In Figure20, the KI values for d1=0,40m are very close to the ones of the crack only subject to water pressure whilst the KI values for d1=30m are larger than the ones of the crack only subject to water pressure. As the d1 values increase (d1≥30m), the KI values decrease. When d1≥90m, the KI values are equal to zero. This means that the crack directly below the goaf is opened and the crack directly below the coal seam is closed under the action of water pressure and disturbed stresses. By comparing the values shown in Figures 19 and 20, it can be found that the KI values for d2=40m are smaller than the ones for d2=0m at a given d1 value.In Figure21, the KII values are, respectively, negative and positive for d1=0-90m and d1=110-150m. For d1=0-90m, as the d1 values increase, the KII values first decrease, arrive at the minimum values at d1=50m, and then increase. For d1=110-150m, as the d1 values increase, the KII values first increase, arrive at the maximum values at d1=130m, and then decrease. It is believed that the KII values tend to zero when the absolute values of d1 tend to an infinite value.In Figure22, the KIII values for d1=0m are positive. As the d1 values increase, the KIII values decrease along the most part of the crack side BC, become negative values, and arrive at the minimum values at d1=90m. Then, the KIII values increase along the most part of the crack side BC. It is believed that the KIII values tend to zero when the absolute values of d1 tend to an infinite value.


## 5.1. General

The stress intensity factors (SIF) values at the crack tip can be calculated by using the discontinuous displacements of crack surfaces. When the crack is not located at the interfaces of multilayered solids, the crack tip field singularities and angular distributions are the same as those in a homogeneous medium and the material parameters exert an influence on the magnitudes of the SIFs. Based on the relationship of displacements and the SIFs, the formulae of the SIFs along the crack side BC are as follows:(6a)KI=E41-ν2π2rΔuy′r,θ=±π,φ=0,(6b)KII=E41-ν2π2rΔuz′r,θ=±π,φ=0,(6c)KIII=E41+νπ2rΔux′r,θ=±π,φ=0,where (r,θ,φ) is the spherical coordinates located at the crack front and θ is an angle in the coordinate plane normal to the crack line of crack front.The following two cases are analyzed: Case 1: a rectangular crack is located in the coordinate planeOxz, i.e., d2=0m; Case 2: a rectangular crack is located in the plane y=40m, i.e., d2=40m.


## 5.2. Case 1: The Crack Is Located in the Coordinate PlaneOxz

In this case, the shear stressesσxy and σzy in the coordinate planeOxz are equal to zero because of the symmetry of the model. As a result, KII and KIII are equal to zero. Figure 19 illustrates the variations of KI for the crack at different positions along the advancing direction of the working face. For the crack only subject to water pressure, the maximum value of KI is 0.1134 MPa·m1/2. However, for the crack subject to water pressure and disturbed stresses, the maximum value of KI is 0.1351 MPa·m1/2 at d1=0m. With d1 increasing, KI increases, arrives at maximum values at d1=40,50m, and then decreases. For d1=80m and x′≥0.5m, KI is equal to zero and KI along the whole BC side is equal to zero at d1≥90m. These findings mean that the crack below the coal seam is closed and the crack below the goaf is opened under the action of disturbed stresses and water pressure. When the crack is located at some horizontal distance of 40-50m after the coal wall of the working face, there are the maximum opening displacements. In this case, the crack becomes easy to open and induces the uplifting of the confined water.Figure 19
Variations ofKI values at the crack side (BC) for Case 1.


## 5.3. Case 2: The Crack Is Located in the Planey=40m, i.e., d2=40m

In this case, the crack is close to the coal pillars outside the working face and has a horizontal distance of 5 m from the coal pillar. As discussed in Section3, the shear stresses induced by mining exist at this position and as a result the crack has three types of fracture modes. Figures 20–22 show the variations of KI, KII, and KIII along the crack front BC, respectively.Figure 20
Variations ofKI values at the crack side (BC) for Case 2.Figure 21
Variations ofKII values at the crack side (BC) for Case 2.Figure 22
Variations ofKIII values at the crack side (BC) for Case 2.In Figure20, the KI values for d1=0,40m are very close to the ones of the crack only subject to water pressure whilst the KI values for d1=30m are larger than the ones of the crack only subject to water pressure. As the d1 values increase (d1≥30m), the KI values decrease. When d1≥90m, the KI values are equal to zero. This means that the crack directly below the goaf is opened and the crack directly below the coal seam is closed under the action of water pressure and disturbed stresses. By comparing the values shown in Figures 19 and 20, it can be found that the KI values for d2=40m are smaller than the ones for d2=0m at a given d1 value.In Figure21, the KII values are, respectively, negative and positive for d1=0-90m and d1=110-150m. For d1=0-90m, as the d1 values increase, the KII values first decrease, arrive at the minimum values at d1=50m, and then increase. For d1=110-150m, as the d1 values increase, the KII values first increase, arrive at the maximum values at d1=130m, and then decrease. It is believed that the KII values tend to zero when the absolute values of d1 tend to an infinite value.In Figure22, the KIII values for d1=0m are positive. As the d1 values increase, the KIII values decrease along the most part of the crack side BC, become negative values, and arrive at the minimum values at d1=90m. Then, the KIII values increase along the most part of the crack side BC. It is believed that the KIII values tend to zero when the absolute values of d1 tend to an infinite value.


## 6. Analysis of Crack Growths Induced by Mining

### 6.1. Case 1: The Crack Is Located in the Coordinate PlaneOxz

As discussed in Section5.2, the crack has only mode I deformation type in this case. The crack growth can be analyzed by using the fracture criterion KI=KIC. Chong et al. [19] investigated the fracture toughness KIC of layered rocks by using three-point-bend specimens and presented the KIC values of oil shale. Using an analogical method, the KIC value of the shale in Table 1 is estimated to be 0.74 MPa·m1/2. As shown in Figure 19, the maximum KImax values are about 0.2335 MPa·m1/2 and KImax<KIC. This means that this type of crack does not grow and the confined water does not uplift under the action of water pressure and disturbed stresses.


### 6.2. Case 2: The Crack Is Located in the Planey=40m, i.e., d2=40m

As discussed in Section5.3, the mode I, II, and III deformations of the crack are coupled together. Growth of a crack in elastic solids subject to complex stress states can be assessed using the so-called S-criterion proposed by Sih [20]. TheS-criterion was developed on the basis of the strain energy density concept. It states that local instability is assumed to occur when the local minimum energy factor Smin reaches a critical value Scr. Using the fracture criterion and the SIF values in Section 5.3, the crack growth for Case 2 will be discussed in the following.The strain energy density factorS of a three-dimensional crack can be defined as [21](7)Sθ=a11θKI2+2a12KIKII+a22θKII2+a33θKIII2where(8)a11θ=116ππμ3-4μ-cos⁡θ1+cos⁡θa12θ=18πμsin⁡θcos⁡θ-1+2νa22θ=116πμ41-ν1-cos⁡θ+3cos⁡θ-11+cos⁡θa33=14πμwhere μ is the shear modulus of elasticity, ν is Poisson’s ratio, and θ is an angle in the coordinate plane normal to the crack line of crack front.Using the SIF values presented in Section5.3 and (7), the local minimum energy factor Smin can be calculated. Figure 23 illustrates the variations of Smin values with the distance d1. At d1=0,90,110,130,150m, the Smin values are smaller than the ones at d1=30,50,60,70m. From d1=0 to 90 m, Smin first increases, arrives at the maximum value at d1=50m, and then decreases. From d1=90 to 150 m, Smin first decreases, arrives at the minimum value at d1=110m, and then increases. It can be estimated that Smin tends to zero as the floor strata are far from the coal wall of the working face.Figure 23
Variations ofSmin values at the crack side (BC) for Case 2.The following conclusions can be drawn:(i) Ford1≤90m, Smin is the largest at d1=50m whilst for d1>90m, Smin is the largest at d1=110m. This means that Smin has two largest values before and after the coal pillar of the working face. Whether the crack grows or not depends on the Scr magnitude of the shale.(ii) The maximum values ofSmin at d1=50m are larger than the ones at d1=110m. The crack growth becomes easy when the crack is located below the goaf. It has been found that the KI values for Case 1 are the largest at d1=40,50m. This means that the growth of any cracks parallel to the coordinate planeOxz is easy at some horizontal distance of 40-50m behind the coal wall of the working face.


## 6.1. Case 1: The Crack Is Located in the Coordinate PlaneOxz

As discussed in Section5.2, the crack has only mode I deformation type in this case. The crack growth can be analyzed by using the fracture criterion KI=KIC. Chong et al. [19] investigated the fracture toughness KIC of layered rocks by using three-point-bend specimens and presented the KIC values of oil shale. Using an analogical method, the KIC value of the shale in Table 1 is estimated to be 0.74 MPa·m1/2. As shown in Figure 19, the maximum KImax values are about 0.2335 MPa·m1/2 and KImax<KIC. This means that this type of crack does not grow and the confined water does not uplift under the action of water pressure and disturbed stresses.


## 6.2. Case 2: The Crack Is Located in the Planey=40m, i.e., d2=40m

As discussed in Section5.3, the mode I, II, and III deformations of the crack are coupled together. Growth of a crack in elastic solids subject to complex stress states can be assessed using the so-called S-criterion proposed by Sih [20]. TheS-criterion was developed on the basis of the strain energy density concept. It states that local instability is assumed to occur when the local minimum energy factor Smin reaches a critical value Scr. Using the fracture criterion and the SIF values in Section 5.3, the crack growth for Case 2 will be discussed in the following.The strain energy density factorS of a three-dimensional crack can be defined as [21](7)Sθ=a11θKI2+2a12KIKII+a22θKII2+a33θKIII2where(8)a11θ=116ππμ3-4μ-cos⁡θ1+cos⁡θa12θ=18πμsin⁡θcos⁡θ-1+2νa22θ=116πμ41-ν1-cos⁡θ+3cos⁡θ-11+cos⁡θa33=14πμwhere μ is the shear modulus of elasticity, ν is Poisson’s ratio, and θ is an angle in the coordinate plane normal to the crack line of crack front.Using the SIF values presented in Section5.3 and (7), the local minimum energy factor Smin can be calculated. Figure 23 illustrates the variations of Smin values with the distance d1. At d1=0,90,110,130,150m, the Smin values are smaller than the ones at d1=30,50,60,70m. From d1=0 to 90 m, Smin first increases, arrives at the maximum value at d1=50m, and then decreases. From d1=90 to 150 m, Smin first decreases, arrives at the minimum value at d1=110m, and then increases. It can be estimated that Smin tends to zero as the floor strata are far from the coal wall of the working face.Figure 23
Variations ofSmin values at the crack side (BC) for Case 2.The following conclusions can be drawn:(i) Ford1≤90m, Smin is the largest at d1=50m whilst for d1>90m, Smin is the largest at d1=110m. This means that Smin has two largest values before and after the coal pillar of the working face. Whether the crack grows or not depends on the Scr magnitude of the shale.(ii) The maximum values ofSmin at d1=50m are larger than the ones at d1=110m. The crack growth becomes easy when the crack is located below the goaf. It has been found that the KI values for Case 1 are the largest at d1=40,50m. This means that the growth of any cracks parallel to the coordinate planeOxz is easy at some horizontal distance of 40-50m behind the coal wall of the working face.


## 7. Conclusions

Numerical methods and mechanical models have been proposed for analyzing the stress fields and crack growths in floor strata for longwall mining. The additional stresses in the floor strata induced by mining are calculated and the crack problems subject to mining-induced stresses and water pressure are investigated. The research results presented in this paper can be summarized as follows:(1) The additional stresses of the floor strata for longwall mining are described in detail and the distribution features of all the stress components with the depth below the floor and the distance from the working face are obtained.(2) By analyzing the crack problems under the action of water pressure and disturbed stresses, the SIF values of the crack at different positions of floor strata are obtained. When the crack is located in the coordinate planeOxz, only the mode I deformation of the crack exists. However, when the crack is not located in the coordinate planeOxz, the mode I, II, and III deformations of the crack are coupled together.(3) At some horizontal distance of 40-50m behind the working face, the vertical crack in the coordinate planeOxz has the largest values of KI and the vertical crack, which is not located in the coordinate planeOxz, has the largest values of Smin. At these positions, the crack growths become easier. Therefore, the field measures should be taken in the advancement of the working face to prevent the water inrushes into the goaf behind working faces.It is the authors’ belief that the analytical methods presented in this paper can be a powerful numerical tool, which can apply to other types of cracks in the floor strata of the coal seam threatened by confined aquifers. Some of the related problems are currently under investigation by the authors for further understanding the complex mechanisms of water inrushes.



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*Source: 4012451-2018-08-15.xml*