# Study on the Fractal Characteristics of Fracture Network Evolution Induced by Mining

Authors: Tao Liang; Xiaoli Liu; Sijing Wang; Enzhi Wang; Quansheng Li
Journal: Advances in Civil Engineering
 (2018)

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## Abstract

The evolution and distribution of fracture network induced by mining is essential to determine the mechanical properties and permeability of disturbed rock mass. In this paper, the similar material model tests are employed to simulate the stress variation, cyclic breaking, and fracture formation and distribution status of the overlying strata with different loading conditions, rock properties, and mining process. The fractal dimension of mining-induced fracture network varied with mining advancing, and the evolvement laws of fracture network with mining advancing and different mining advancing footage are concerned and obtained. By establishing the relationship between the fractal dimension and the mining length in different horizontal and vertical zones, it demonstrates that the fractal dimensions in horizontal and vertical zones have a self-similar characteristic, and the distribution of the fractal dimension of the mining-induced fractures shows generally the “W”-type trend.


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## Body

## 1. Introduction

The original equilibrium state of the underground rock is disturbed by mining, and the fracture network in the overlying rock mass gradually forms and evolves with the rock stress redistribution. The mechanism of the fracturing and evolution of mining-induced cracks plays a great important role on the permeability assessment, overburden strata subsidence prediction, and the security and stability evaluation of mining rock mass [1–6]. According to Yu et al. [2], it is probably one of the most important issues in mining science to recognize and describe quantitatively the evolution of fracture network resulting from mining processes, on which many researchers have carried out fruitful research in recently years. Qian and Xu [7] proposed the two-stage development law of mining-induced cracks and revealed the “O”-shape distribution characteristics in a long-wall face of a coal mine. Li et al. [8] discussed the effect of key strata on the distribution of fractures and concluded that an elliptic paraboloid zone would be formed in the overlying strata due to the connection of the rock mass breaking induced fractures and the strata layers separation induced fractures in mining process. Li et al. [9] described the distribution characteristics of roof separation fractures in the process of fully mechanized top coal caving and derived the equivalent areas of bed-separated cracks before and after the key strata breaking and the permeability coefficients in different crack districts.The fractal geometry proposed by Mandelbrot [10] is a powerful tool to quantitatively describe complex objects and phenomenon of irregular. In recent years, increasing attention has been focused on the combined application of fractal theory and physical models, by which considerable progress has been made in quantitative description of the evolution law of mining-induced fractures [11–20].In the process of coal mining, the authors put forward underground reservoirs construction technology in mined out area, which is a new idea on coal-water coordinate mining, for comprehensive utilization of underground water resources, protection of aquifer, safe production of coal mines, and underground water reuse. This technology is meaningful for sustainable development of underground resources, including coal mine resources, especially in arid and semiarid areas, where the water resources are very short. The distribution and evolution of mining-induced fractures in the overlying and floor rock mass play an important role in determining the location and capacity of the underground reservoir and in groundwater operation and purification as well. In this study, the Daliuta coal mine in the Ordos Basin of northeastern China, where underground reservoirs were constructed, is taken as the engineering background. Fracture network evolution and distribution induced by mining is essential to determine the mechanical properties and permeability of disturbed rock mass. In this paper, the physical similarity model was made and by which the deformation and collapse of the overlying rock mass were well observed and recorded under different conditions. By analyzing the fractal dimension variation of fracture networks of different working conditions and different zones with the mining face advancing, the distribution and evolvement laws of fracture network with mining speed and in different horizontal and vertical zones are concerned, which provides a basis for studying the relationship between cracks evolution and corresponding permeability change of mining rock mass.


## 2. Experimental Procedure

### 2.1. Geological Conditions

According to the geological investigation, the representative stratigraphic distribution in the Daliuta mine field and the physical properties are presented in Table1.Table 1
Site stratigraphy and physical properties.
Strata number
Lithology
Strata thickness (m)
Initial permeability coefficient (m/d)
In situ vertical stress (MPa)
Strata density (kg/m3)
1
Aeolian sand and loess
30
0.5–5.0
2000
2
Mudstone
15
0.0002
34.86
2304
3
Medium coarse-grained sandstone
5
0.02
50.00
2550
4
Mudstone
20
0.0002
25.22
2204
5
Fine sandstone-siltstone
10
0.004
5.26
2133
6
Mudstone
15
0.0002
25.22
2204
7
Number 2 coal seam
5
0.0001
14.49
1306
8
Mudstone
15
0.0002
34.86
2304
9
Fine sandstone-siltstone
20
0.004
5.26
2133
10
Mudstone
15
0.0002
34.86
2304
11
Medium coarse-grained sandstone
20
0.02
50.00
2550
12
Mudstone
20
0.0002
25.22
2204
13
Fine sandstone-siltstone
10
0.004
5.26
2133
14
Mudstone
10
0.0002
25.22
2204
15
Medium coarse-grained sandstone
20
0.02
50.00
2550
16
Mudstone
20
0.0002
25.22
2204
17
Medium coarse-grained sandstone
20
0.004
5.26
2133
18
Number 5 coal seam
5
0.0001
19.54
1295
19
Mudstone
15
0.0002
34.86
2304
20
Medium coarse-grained sandstone
15
0.02
50.00
2550
21
Mudstone
10
0.009
34.86
2304
22
Medium coarse-grained sandstone
10
0.02
50.00
2550


### 2.2. Similarity Theory and Similar Materials

Similar simulation study is an important means for scientific research, during which a model with artificial materials is made in the laboratory according to the similar principles and with a certain scale. With the aid of test instruments, the physical and mechanical parameters and distribution rules in the model are observed and obtained, from which the mechanical phenomenon and law of rock-pressure distribution of the prototype are inferred, and hereby to solve the practical problems in rock engineering.Similar material and prototype used in the fluid-solid coupling model test should satisfy (1) and (2) simultaneously [21, 22]:(1)Kx∂2p∂x2+Ky∂2p∂y2+Kz∂2p∂z2=S∂p∂t+∂e∂t+W,where Kx, Ky, and Kz are the permeability coefficients in x, y, and z directions, respectively; p is the water pressure; S is the water storage coefficient; e is the volumetric strain; and W is the source and sink.(2)G∇2u+λ+G∂e∂x+X+p∂α∂x+α∂p∂x=ρ∂2u∂t2,G∇2υ+λ+G∂e∂y+Y+p∂α∂y+α∂p∂y=ρ∂2υ∂t2,G∇2ω+λ+G∂e∂z+Z+p∂α∂z+α∂p∂z=ρ∂2ω∂t2,where ∇2=∂2/∂x2+∂2/∂y2+∂2/∂z2 is the Laplace operator; G=E/21+μ is the shear modulus; λ=μE/1+μ1−2μ is the lame constant; e=∂u/∂x+∂υ/∂y+∂ω/∂z is the volume strain; E is the modulus of elasticity; μ is Poisson’s ratio; u, υ, and ω are the displacements in the directions x, y, and z, respectively; X, Y, and Z are the volume forces in the directions x, y, and z, respectively; α is the effective stress coefficient; ρ is the density; p is the water pressure.Equations (1) and (2) are the basic equations of the fluid-solid coupling theory. The archetype and the model need to satisfy the above equations. Based on the similarity theory [23], the similarity coefficients of the similar model tests are derived to satisfy the following equations:(3)CGCuCl2=CλCeCl=CGCeCl=Cγ=CpCαCl=CαCpCl=CρCuCt2,(4)CkCpCl2=CSCpCt=CeCt=CW,where C is the similarity coefficient. CG=G′/G″, Cρ=ρ′/ρ″, and so on, where the lower foot mark represents the physical variables or parameters that it represents, ′ represents the prototype, and ″ represents the model.The model test should be based on the similarity principles, the hydrologic and geological conditions of the prototype, and the specific requirements of the research problems. At the same time, the similarity index and similarity ratio should be determined according to the conditions mentioned above as well as the field conditions of the model test. The similar materials were tested by adjusting the composition and proportion of similar materials and then determined through experimental verification. The main concerns of this model test are the deformation, failure, and permeability change of the overlying rock, so the main parameters of the similar model test are the bulk density, compressive strength (or tensile strength), and permeability coefficient. At the same time, because of the effect of groundwater, the softening coefficient and water absorption also need to satisfy a certain similarity with the prototype.According to the principles of the similarity model test, the geometrical size, boundary condition, load and movement conditions of the model, and physical and mechanical parameters (e.g. bulk density, strength, deformation characteristics, and water characteristics) of the similar materials are similar to that of the prototype. Based on the elastic mechanics, the dimensional analysis method combined with the fluid-solid coupling theory; the values of the similar parameters are detailed in Table2. The range of the physical and mechanical parameters of the similar materials in the model test is described in Table 3.Table 2
Similarity ratios.
Similarity coefficient
Similarity ratio
C
ω
a
1/10
C
γ, Cμ, Cε, Cφ, CR
1
C
k, Ct
10
C
L, Cδ, Cc, Cσ, Cσt, Cσc, CE
100
γ: bulk density; L: length; δ: displacement; ε: strain; E: elastic modulus; σ: stress; σt: tensile strength; σc: compressive strength; c: cohesion; φ: internal friction angle; μ: Poisson’s ratio; k: permeability coefficient; R: softening coefficient; ωa: water absorption; t: time duration.Table 3
Physical and mechanical parameters of the prototype and the model.
Number
Parameters
Prototype
Model
1
Bulk density (kN/m3)
21.0∼25.5
21.0∼25.5
2
Compressive strength (MPa)
5∼50
0.05∼0.5
3
Permeability coefficient (cm/s)
1 × 10−4∼1 × 10−3
1 × 10−5∼1 × 10−4
4
Softening coefficient
Greater than 0.75
Greater than 0.75
5
Water absorption
0.2%∼0.5%
2%∼5%Based on the materials developed in [24], after many times of proportioning tests, similar materials meeting the test requirements are made, and the similar material parameters are listed in detail in Table 4.Table 4
Similar material parameters.
Number
Simulated rock stratum
Bulk density (kg/m3)
Compressive strength (MPa)
Permeability coefficient (cm/s)
Softening coefficient
Water absorption
1
Mudstone
2300
0.35
7.2E − 5
0.88
0.02
2
Medium coarse-grained sandstone
2550
0.50
9.0E − 5
0.81
0.03
3
Mudstone
2200
0.25
1.0E − 4
0.78
0.03
4
Fine sandstone-siltstone
2100
0.05
9.0E − 5
0.75
0.03


### 2.3. Similarity Physical Model

The experiment is carried out in a large-scale model test bed designed by the authors, which is 8000 mm long, 1800 mm wide, and 4000 mm high, with the outer frame composed of I steel and channel steel and positive for toughened glass. There are open channels with the sealing cover on the elevation of coal seams on the back plate, used to simulate coal seam excavation, and inlet and outlet holes and flow control table on both sides of the test bed.As shown in Figure1, there are 21 similar material layers in the test bed from the bottom to the top, with a total height of 3000 mm, including the number 2 and number 5 similar coal seams with 50 mm thickness located on the elevation of 2250 mm and 500 mm, respectively. The similar pillar is 300 mm wide, by which each similar coal seam is separated in 3 mining faces with a length of 2100 mm. With different mining speeds, the coal seam in a different mining face is mined and the overburden strata movement, and the spatial distribution and evolution process of mining-induced fractures are observed and studied.Figure 1
Cross section of the physical simulation model.A real-time in situ monitoring program was carried out during and after the mining process. The items of monitoring include the stress in overburden strata and coal pillar, the displacement of overburden strata, and the water pressure in aquifer. A total of 160 sensors (e.g., displacement meter, rock-pressure gauge, and water-pressure gauge) are arranged in the model, from which the data acquisition processing system feedback to the control terminal and the real-time curve will be obtained. Meanwhile, with the aid of total station instrument, 3D laser scanner, and high-speed digital camera, the displacement of 486 monitoring points distributed on the surface of the model is obtained as well.


## 2.1. Geological Conditions

According to the geological investigation, the representative stratigraphic distribution in the Daliuta mine field and the physical properties are presented in Table1.Table 1
Site stratigraphy and physical properties.
Strata number
Lithology
Strata thickness (m)
Initial permeability coefficient (m/d)
In situ vertical stress (MPa)
Strata density (kg/m3)
1
Aeolian sand and loess
30
0.5–5.0
2000
2
Mudstone
15
0.0002
34.86
2304
3
Medium coarse-grained sandstone
5
0.02
50.00
2550
4
Mudstone
20
0.0002
25.22
2204
5
Fine sandstone-siltstone
10
0.004
5.26
2133
6
Mudstone
15
0.0002
25.22
2204
7
Number 2 coal seam
5
0.0001
14.49
1306
8
Mudstone
15
0.0002
34.86
2304
9
Fine sandstone-siltstone
20
0.004
5.26
2133
10
Mudstone
15
0.0002
34.86
2304
11
Medium coarse-grained sandstone
20
0.02
50.00
2550
12
Mudstone
20
0.0002
25.22
2204
13
Fine sandstone-siltstone
10
0.004
5.26
2133
14
Mudstone
10
0.0002
25.22
2204
15
Medium coarse-grained sandstone
20
0.02
50.00
2550
16
Mudstone
20
0.0002
25.22
2204
17
Medium coarse-grained sandstone
20
0.004
5.26
2133
18
Number 5 coal seam
5
0.0001
19.54
1295
19
Mudstone
15
0.0002
34.86
2304
20
Medium coarse-grained sandstone
15
0.02
50.00
2550
21
Mudstone
10
0.009
34.86
2304
22
Medium coarse-grained sandstone
10
0.02
50.00
2550


## 2.2. Similarity Theory and Similar Materials

Similar simulation study is an important means for scientific research, during which a model with artificial materials is made in the laboratory according to the similar principles and with a certain scale. With the aid of test instruments, the physical and mechanical parameters and distribution rules in the model are observed and obtained, from which the mechanical phenomenon and law of rock-pressure distribution of the prototype are inferred, and hereby to solve the practical problems in rock engineering.Similar material and prototype used in the fluid-solid coupling model test should satisfy (1) and (2) simultaneously [21, 22]:(1)Kx∂2p∂x2+Ky∂2p∂y2+Kz∂2p∂z2=S∂p∂t+∂e∂t+W,where Kx, Ky, and Kz are the permeability coefficients in x, y, and z directions, respectively; p is the water pressure; S is the water storage coefficient; e is the volumetric strain; and W is the source and sink.(2)G∇2u+λ+G∂e∂x+X+p∂α∂x+α∂p∂x=ρ∂2u∂t2,G∇2υ+λ+G∂e∂y+Y+p∂α∂y+α∂p∂y=ρ∂2υ∂t2,G∇2ω+λ+G∂e∂z+Z+p∂α∂z+α∂p∂z=ρ∂2ω∂t2,where ∇2=∂2/∂x2+∂2/∂y2+∂2/∂z2 is the Laplace operator; G=E/21+μ is the shear modulus; λ=μE/1+μ1−2μ is the lame constant; e=∂u/∂x+∂υ/∂y+∂ω/∂z is the volume strain; E is the modulus of elasticity; μ is Poisson’s ratio; u, υ, and ω are the displacements in the directions x, y, and z, respectively; X, Y, and Z are the volume forces in the directions x, y, and z, respectively; α is the effective stress coefficient; ρ is the density; p is the water pressure.Equations (1) and (2) are the basic equations of the fluid-solid coupling theory. The archetype and the model need to satisfy the above equations. Based on the similarity theory [23], the similarity coefficients of the similar model tests are derived to satisfy the following equations:(3)CGCuCl2=CλCeCl=CGCeCl=Cγ=CpCαCl=CαCpCl=CρCuCt2,(4)CkCpCl2=CSCpCt=CeCt=CW,where C is the similarity coefficient. CG=G′/G″, Cρ=ρ′/ρ″, and so on, where the lower foot mark represents the physical variables or parameters that it represents, ′ represents the prototype, and ″ represents the model.The model test should be based on the similarity principles, the hydrologic and geological conditions of the prototype, and the specific requirements of the research problems. At the same time, the similarity index and similarity ratio should be determined according to the conditions mentioned above as well as the field conditions of the model test. The similar materials were tested by adjusting the composition and proportion of similar materials and then determined through experimental verification. The main concerns of this model test are the deformation, failure, and permeability change of the overlying rock, so the main parameters of the similar model test are the bulk density, compressive strength (or tensile strength), and permeability coefficient. At the same time, because of the effect of groundwater, the softening coefficient and water absorption also need to satisfy a certain similarity with the prototype.According to the principles of the similarity model test, the geometrical size, boundary condition, load and movement conditions of the model, and physical and mechanical parameters (e.g. bulk density, strength, deformation characteristics, and water characteristics) of the similar materials are similar to that of the prototype. Based on the elastic mechanics, the dimensional analysis method combined with the fluid-solid coupling theory; the values of the similar parameters are detailed in Table2. The range of the physical and mechanical parameters of the similar materials in the model test is described in Table 3.Table 2
Similarity ratios.
Similarity coefficient
Similarity ratio
C
ω
a
1/10
C
γ, Cμ, Cε, Cφ, CR
1
C
k, Ct
10
C
L, Cδ, Cc, Cσ, Cσt, Cσc, CE
100
γ: bulk density; L: length; δ: displacement; ε: strain; E: elastic modulus; σ: stress; σt: tensile strength; σc: compressive strength; c: cohesion; φ: internal friction angle; μ: Poisson’s ratio; k: permeability coefficient; R: softening coefficient; ωa: water absorption; t: time duration.Table 3
Physical and mechanical parameters of the prototype and the model.
Number
Parameters
Prototype
Model
1
Bulk density (kN/m3)
21.0∼25.5
21.0∼25.5
2
Compressive strength (MPa)
5∼50
0.05∼0.5
3
Permeability coefficient (cm/s)
1 × 10−4∼1 × 10−3
1 × 10−5∼1 × 10−4
4
Softening coefficient
Greater than 0.75
Greater than 0.75
5
Water absorption
0.2%∼0.5%
2%∼5%Based on the materials developed in [24], after many times of proportioning tests, similar materials meeting the test requirements are made, and the similar material parameters are listed in detail in Table 4.Table 4
Similar material parameters.
Number
Simulated rock stratum
Bulk density (kg/m3)
Compressive strength (MPa)
Permeability coefficient (cm/s)
Softening coefficient
Water absorption
1
Mudstone
2300
0.35
7.2E − 5
0.88
0.02
2
Medium coarse-grained sandstone
2550
0.50
9.0E − 5
0.81
0.03
3
Mudstone
2200
0.25
1.0E − 4
0.78
0.03
4
Fine sandstone-siltstone
2100
0.05
9.0E − 5
0.75
0.03


## 2.3. Similarity Physical Model

The experiment is carried out in a large-scale model test bed designed by the authors, which is 8000 mm long, 1800 mm wide, and 4000 mm high, with the outer frame composed of I steel and channel steel and positive for toughened glass. There are open channels with the sealing cover on the elevation of coal seams on the back plate, used to simulate coal seam excavation, and inlet and outlet holes and flow control table on both sides of the test bed.As shown in Figure1, there are 21 similar material layers in the test bed from the bottom to the top, with a total height of 3000 mm, including the number 2 and number 5 similar coal seams with 50 mm thickness located on the elevation of 2250 mm and 500 mm, respectively. The similar pillar is 300 mm wide, by which each similar coal seam is separated in 3 mining faces with a length of 2100 mm. With different mining speeds, the coal seam in a different mining face is mined and the overburden strata movement, and the spatial distribution and evolution process of mining-induced fractures are observed and studied.Figure 1
Cross section of the physical simulation model.A real-time in situ monitoring program was carried out during and after the mining process. The items of monitoring include the stress in overburden strata and coal pillar, the displacement of overburden strata, and the water pressure in aquifer. A total of 160 sensors (e.g., displacement meter, rock-pressure gauge, and water-pressure gauge) are arranged in the model, from which the data acquisition processing system feedback to the control terminal and the real-time curve will be obtained. Meanwhile, with the aid of total station instrument, 3D laser scanner, and high-speed digital camera, the displacement of 486 monitoring points distributed on the surface of the model is obtained as well.


## 3. Evolution of Mining-Induced Fracture Network

### 3.1. Simulation of the Mining Process

With the aid of the physical similarity model, the deformation and collapse of the overlying rock mass can be observed directly during the mining process. Taking the mining face 5-3, for example, the immediate roof falls with a width of 300 mm and height of 50 mm at the mining length of 300 mm. From 300 mm to 800 mm of the mining length, the immediate roof falls four times at an average width of 100 mm and height of 50 mm. Then, the main roof collapses with a height of 100 mm when the mining face advances 850 mm. During the next mining process, the integral caving of the main roof occurs at the mining length of 1200 mm and 1800 mm, accompanying periodical fall of the immediate roof at an average length of 50 mm to 150 mm and several partial collapses of the main roof. With the advancement in mining process, the fractures generate and propagate in the overlying rock strata. At the end of the mining process, the fracture network evolutes within the whole mined out area with a caving height of 600 mm. The description of the mining process in detail is presented in Table5, and representative fracture network diagrams are shown in Figure 2.Table 5
Description of the mining process (mining face 5-3).
Mining length (mm)
Caving height (mm)
Caving length (mm)
Caving description
300
50
300
Immediate roof falls
500
50
200
Immediate roof falls
600
50
100
Immediate roof falls
700
50
100
Immediate roof falls
800
50
100
Immediate roof fall
850
100
800
Main roof collapses (integral)
900
50
100
Immediate roof falls
950
50
50
Immediate roof falls
1100
100
300
Main roof collapses (partial)
1150
50
50
Immediate roof falls
1200
200
1200
Main roof collapses (integral)
1250
50
50
Immediate roof falls
1350
100
100
Immediate roof falls
1400
50
50
Immediate roof falls
1450
200
250
Main roof collapses (partial)
1500
300
500
Main roof collapses (partial)
1550
50
100
Immediate roof falls
1650
100
200
Main roof collapses (partial)
1750
50
100
Immediate roof falls
1800
600
1800
Main roof collapses (integral)
1850
50
50
Immediate roof falls
1900
50
50
Immediate roof falls
1950
50
50
Immediate roof falls
2100
50
150
Immediate roof fallsFigure 2
Fracture network diagrams (mining face 5-3). Mining length: (a)850 mm; (b)1200 mm; (c)1800 mm.
(a)
(b)
(c)


### 3.2. Calculation of Fractal Dimension

In order to quantitatively analyse the expansion of fractures in mining rock masses, real-time video recording is performed in the process of the model test. Through the observation section in front of the physical model, the whole process of fracture generation, expansion, evolution, and strata failure during mining is well recorded. With the help of digital processing technology, the fracture network diagrams at different mining lengths are digitized (as shown in Figure3). In this study, the calculation of fractal dimension is done by the method of “counting the boxes.” Each diagram is extended to a square area with a length of integer power times of 2. The fracture network with a unit length size of box is covered, and the number of boxes needed to cover the network is counted. The size of the box is enlarged by 2 times, the calculation of the number of boxes required to cover the fracture network is repeated, and so on (as shown in Figure 4). For each fracture network, a set of data corresponding to box length r and the number of boxes N is obtained in this way. If the fracture network has a fractal character, the size of the box r and the number of boxes N(r) will follow the relationship as shown in the following equation:(5)Nr=r−Df,where Df is the fractal dimension. By drawing the relationship between Nr and 1/r in the double logarithmic coordinates, a straight line will be obtained, of which the slope is equal to Df. Taking the mining face 5-1 at the mining lengths of 1200 mm, 1500 mm, 1800 mm, and 2100 mm, for example, N and r of the covering boxes for calculating the fractal dimension are listed in Tables 6–9. By drawing the relationship between Nr and 1/r in the double logarithmic coordinates, the slope of the straight line equal to Df is obtained (as shown in Figure 5).Figure 3
Fracture network evolution (mining face 5-1). Mining length: (a) 300 mm; (b) 450 mm; (c) 600 mm; (d) 800 mm; (e) 1000 mm; (f) 1100 mm; (g) 1200 mm; (h) 1250 mm; (i) 1350 mm; (j) 1500 mm; (k) 1700 mm; (l) 1800 mm; (m) 1900 mm; (n) 1950 mm; (o) 2050 mm; (p) 2100 mm.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
(m)
(n)
(o)
(p)Figure 4
Schematic diagram for the “counting the boxes” method. Mining face 5-1 at a mining length of (a) 1200 mm; (b) 1500 mm; (c) 1800 mm; (d) 2100 mm.
(a)
(b)
(c)
(d)Table 6
List for calculatingDf (mining face 5-1 at a mining length of 1200 mm).
r
1
2
4
8
16
32
64
128
N
24509
7071
2130
711
249
92
34
14
ln (1/r)
0
−0.69
−1.39
−2.08
−2.77
−3.47
−4.16
−4.85
lnN
10.11
8.86
7.66
6.57
5.52
4.52
3.53
2.64Table 7
List for calculatingDf (mining face 5-1 at a mining length of 1500 mm).
r
1
2
4
8
16
32
64
128
N
38677
11458
3750
1350
510
180
63
21
ln (1/r)
0
−0.69
−1.39
−2.08
−2.77
−3.47
−4.16
−4.85
lnN
10.56
9.35
8.23
7.21
6.23
5.19
4.14
3.04Table 8
List for calculatingDf (mining face 5-1 at a mining length of 1800 mm).
r
1
2
4
8
16
32
64
128
N
49959
14383
4579
1599
578
190
66
22
ln (1/r)
0
−0.69
−1.39
−2.08
−2.77
−3.47
−4.16
−4.85
lnN
10.82
9.57
8.43
7.38
6.36
5.25
4.19
3.09Table 9
List for calculatingDf (mining face 5-1 at a mining length of 2100 mm).
r
1
2
4
8
16
32
64
128
N
59485
18261
6243
2320
910
316
104
34
ln (1/r)
0
−0.69
−1.39
−2.08
−2.77
−3.47
−4.16
−4.85
lnN
10.99
9.81
8.74
7.75
6.81
5.76
4.64
3.53Figure 5
Schematic diagram for calculatingDf. Mining face 5-1 at a mining length of (a) 1200 mm; (b) 1500 mm; (c) 1800 mm; (d) 2100 mm.
(a)
(b)
(c)
(d)From (5) and the fractal dimension calculation process, we can see that for each fracture network, the fractal dimension reflects the density of fractures in the unit area, and the increase in the fractal dimension indicates the expanding of fractures and the increasing of the fractures density. By analyzing the fractal dimension variation of fracture diagrams of different working conditions and different zones at different mining lengths, we can quantitatively analyse the fracture expansion and evolution laws.


### 3.3. Fractal Dimension Evolution of Fracture Networks with Mining Process

The evolution of the fractal dimension of fracture networks with mining length is shown in Figure6, by which the following features can be obtained:(1)
The curve can be separated in several stage-type folding lines, containing increasing segment and descending segment.(2)
The descending point corresponds exactly to the mining length, at which the main roof collapses (as shown in Table5).(3)
Every point in the linear increasing segment corresponds to the mining length, at which the immediate roof falls (as shown in Table5).(4)
The increasing segment can be fitted to a straight line, which means the regression relation between the fractal dimensionD and the mining length L can be given by the following equation:(6)D=a∗L+b,where a and b are the parameters determined by the hydrogeological conditions, rock mechanical parameters, mining method, and so on.Figure 6
Evolution of the fractal dimension of fracture network with mining length.


### 3.4. Evolution of Fractal Dimension with Different Mining Footage

In order to analyse the influence of mining speed on the damage of overlying strata and the evolution of fracture propagation, mining faces 5-1 and 5-3 are mined with different mining advance footage. The relationship between the fractal dimension and mining length with different mining footage is shown in Figure7, by which we can see that the mining face 5-1 (with a faster mining speed) has a larger fractal dimension than the mining face 5-3. Both the mining faces have the similar stage-type of the fractal dimension curve. Nevertheless, the 5-3 mining face has more points in increasing segment of the fractal dimension curve, which means having the shorter weighting interval in the immediate roof and a more developed fracture network in the caving zone, which can also be concluded from the fracture network at the end of the mining process (as shown in Figure 8).Figure 7
Relationship between the fractal dimension and mining length with different mining footage.Figure 8
Fracture networks at the end of the mining process. Mining face: (a) 5-1; (b) 5-3.
(a)
(b)


### 3.5. Evolution of Fractal Dimension in Different Caving Zones

With the advancing of mining face, how the fractures generate and develop in different horizontal and vertical zones is an aspect deserving to pay more attention on in this study. Based on this concern, 7 horizontal zones and 6 vertical zones are separated in mining face 5-1 (as shown in Tables10 and 11 and Figure 9); the fractal dimension is calculated, respectively, and the fractal dimension curves are shown in Figures 10–13. The key findings are as follows:(1)
Figure10 describes the evolution of the fractal dimension in horizontal zones with the mining face advancing. The evolution curves indicate that the fractal dimension in horizontal zones has a similar trend with the fractal dimension on the whole, that is, gradually increasing in general and reduces when the main roof collapses. The main reason may be that the collapse of the main roof gives rise to the closure of fractures in the caving zone.(2)
Figures10 and 11 show that the fractal dimension at different mining progresses (such as 90 m, 120 m, 150 m, and 190 m) decreases with the increase of mining footage. On the whole, at the end of the mining process, the horizontal zone H3 and zone H6 have a relatively low fractal dimension, and the zones H1, H4, and H7, on the contrary, have a relatively high fractal dimension. The distribution of the fractal dimension of the fractures shows generally the “W”-type trend.(3)
Through the analysis of the fractal characteristics of the fracture networks of the working face and combining the district partition of the fractured rock mass after mining [15], it is found that the fractal dimension of the fractured rock mass near the cutting area decreases gradually with the increase of mining footage, which indicates that the fractures in the area are gradually compressed and closed, which is called the re-compressed area. The fractal dimension of fractured rock mass near the mining face area increases, which indicates that the fractures are gradually developed and expanded with the increase of mining footage, which is called the separation zone. With the increase of mining footage, the separation area away from the mining face is gradually transformed into a re-compressed area and the re-compressed area is gradually transformed into a stable area so that the fractal dimension of the horizontal zones of the fractured rock mass in the goaf is in dynamic changing.(4)
Figure12 shows the similar developing pattern of fractal dimension in vertical zones with that in horizontal zones. At the same time, it may be concluded by Figure 13 that, in general, the fractal dimensions reduce with the increase of the caving height; that is, the caving zone has a more developed fracture network than the fractured zone. Meanwhile, however, it should be pointed out that there is an exception sometimes in the transition zone between the fractured zone and sinking zone, due to the formation of the strata-separated fractures, which contribute much to the increase of the fractal dimension.Table 10
Horizontal zone description.
Zone number
H1
H2
H3
H4
H5
H6
H7
Mining length(mm)
0–300
300–600
600–900
900–1200
1200–1500
1500–1800
1800–2100Table 11
Vertical zone description.
Zone number
Zone height(mm)
V1
0–100
V2
100–200
V3
200–300
V4
300–400
V5
400–500
V6
500–600Figure 9
Schematic diagram for the fracture zones.Figure 10
Evolution of the fractal dimension of crack network in different horizontal zones.Figure 11
3D colour map surface of evolution of the fractal dimension of crack network with mining length in different horizontal zones.Figure 12
Evolution of the fractal dimension of crack network in different vertical zones.Figure 13
Ratio of the fractal dimension of crack network in different vertical zones.


## 3.1. Simulation of the Mining Process

With the aid of the physical similarity model, the deformation and collapse of the overlying rock mass can be observed directly during the mining process. Taking the mining face 5-3, for example, the immediate roof falls with a width of 300 mm and height of 50 mm at the mining length of 300 mm. From 300 mm to 800 mm of the mining length, the immediate roof falls four times at an average width of 100 mm and height of 50 mm. Then, the main roof collapses with a height of 100 mm when the mining face advances 850 mm. During the next mining process, the integral caving of the main roof occurs at the mining length of 1200 mm and 1800 mm, accompanying periodical fall of the immediate roof at an average length of 50 mm to 150 mm and several partial collapses of the main roof. With the advancement in mining process, the fractures generate and propagate in the overlying rock strata. At the end of the mining process, the fracture network evolutes within the whole mined out area with a caving height of 600 mm. The description of the mining process in detail is presented in Table5, and representative fracture network diagrams are shown in Figure 2.Table 5
Description of the mining process (mining face 5-3).
Mining length (mm)
Caving height (mm)
Caving length (mm)
Caving description
300
50
300
Immediate roof falls
500
50
200
Immediate roof falls
600
50
100
Immediate roof falls
700
50
100
Immediate roof falls
800
50
100
Immediate roof fall
850
100
800
Main roof collapses (integral)
900
50
100
Immediate roof falls
950
50
50
Immediate roof falls
1100
100
300
Main roof collapses (partial)
1150
50
50
Immediate roof falls
1200
200
1200
Main roof collapses (integral)
1250
50
50
Immediate roof falls
1350
100
100
Immediate roof falls
1400
50
50
Immediate roof falls
1450
200
250
Main roof collapses (partial)
1500
300
500
Main roof collapses (partial)
1550
50
100
Immediate roof falls
1650
100
200
Main roof collapses (partial)
1750
50
100
Immediate roof falls
1800
600
1800
Main roof collapses (integral)
1850
50
50
Immediate roof falls
1900
50
50
Immediate roof falls
1950
50
50
Immediate roof falls
2100
50
150
Immediate roof fallsFigure 2
Fracture network diagrams (mining face 5-3). Mining length: (a)850 mm; (b)1200 mm; (c)1800 mm.
(a)
(b)
(c)


## 3.2. Calculation of Fractal Dimension

In order to quantitatively analyse the expansion of fractures in mining rock masses, real-time video recording is performed in the process of the model test. Through the observation section in front of the physical model, the whole process of fracture generation, expansion, evolution, and strata failure during mining is well recorded. With the help of digital processing technology, the fracture network diagrams at different mining lengths are digitized (as shown in Figure3). In this study, the calculation of fractal dimension is done by the method of “counting the boxes.” Each diagram is extended to a square area with a length of integer power times of 2. The fracture network with a unit length size of box is covered, and the number of boxes needed to cover the network is counted. The size of the box is enlarged by 2 times, the calculation of the number of boxes required to cover the fracture network is repeated, and so on (as shown in Figure 4). For each fracture network, a set of data corresponding to box length r and the number of boxes N is obtained in this way. If the fracture network has a fractal character, the size of the box r and the number of boxes N(r) will follow the relationship as shown in the following equation:(5)Nr=r−Df,where Df is the fractal dimension. By drawing the relationship between Nr and 1/r in the double logarithmic coordinates, a straight line will be obtained, of which the slope is equal to Df. Taking the mining face 5-1 at the mining lengths of 1200 mm, 1500 mm, 1800 mm, and 2100 mm, for example, N and r of the covering boxes for calculating the fractal dimension are listed in Tables 6–9. By drawing the relationship between Nr and 1/r in the double logarithmic coordinates, the slope of the straight line equal to Df is obtained (as shown in Figure 5).Figure 3
Fracture network evolution (mining face 5-1). Mining length: (a) 300 mm; (b) 450 mm; (c) 600 mm; (d) 800 mm; (e) 1000 mm; (f) 1100 mm; (g) 1200 mm; (h) 1250 mm; (i) 1350 mm; (j) 1500 mm; (k) 1700 mm; (l) 1800 mm; (m) 1900 mm; (n) 1950 mm; (o) 2050 mm; (p) 2100 mm.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
(m)
(n)
(o)
(p)Figure 4
Schematic diagram for the “counting the boxes” method. Mining face 5-1 at a mining length of (a) 1200 mm; (b) 1500 mm; (c) 1800 mm; (d) 2100 mm.
(a)
(b)
(c)
(d)Table 6
List for calculatingDf (mining face 5-1 at a mining length of 1200 mm).
r
1
2
4
8
16
32
64
128
N
24509
7071
2130
711
249
92
34
14
ln (1/r)
0
−0.69
−1.39
−2.08
−2.77
−3.47
−4.16
−4.85
lnN
10.11
8.86
7.66
6.57
5.52
4.52
3.53
2.64Table 7
List for calculatingDf (mining face 5-1 at a mining length of 1500 mm).
r
1
2
4
8
16
32
64
128
N
38677
11458
3750
1350
510
180
63
21
ln (1/r)
0
−0.69
−1.39
−2.08
−2.77
−3.47
−4.16
−4.85
lnN
10.56
9.35
8.23
7.21
6.23
5.19
4.14
3.04Table 8
List for calculatingDf (mining face 5-1 at a mining length of 1800 mm).
r
1
2
4
8
16
32
64
128
N
49959
14383
4579
1599
578
190
66
22
ln (1/r)
0
−0.69
−1.39
−2.08
−2.77
−3.47
−4.16
−4.85
lnN
10.82
9.57
8.43
7.38
6.36
5.25
4.19
3.09Table 9
List for calculatingDf (mining face 5-1 at a mining length of 2100 mm).
r
1
2
4
8
16
32
64
128
N
59485
18261
6243
2320
910
316
104
34
ln (1/r)
0
−0.69
−1.39
−2.08
−2.77
−3.47
−4.16
−4.85
lnN
10.99
9.81
8.74
7.75
6.81
5.76
4.64
3.53Figure 5
Schematic diagram for calculatingDf. Mining face 5-1 at a mining length of (a) 1200 mm; (b) 1500 mm; (c) 1800 mm; (d) 2100 mm.
(a)
(b)
(c)
(d)From (5) and the fractal dimension calculation process, we can see that for each fracture network, the fractal dimension reflects the density of fractures in the unit area, and the increase in the fractal dimension indicates the expanding of fractures and the increasing of the fractures density. By analyzing the fractal dimension variation of fracture diagrams of different working conditions and different zones at different mining lengths, we can quantitatively analyse the fracture expansion and evolution laws.


## 3.3. Fractal Dimension Evolution of Fracture Networks with Mining Process

The evolution of the fractal dimension of fracture networks with mining length is shown in Figure6, by which the following features can be obtained:(1)
The curve can be separated in several stage-type folding lines, containing increasing segment and descending segment.(2)
The descending point corresponds exactly to the mining length, at which the main roof collapses (as shown in Table5).(3)
Every point in the linear increasing segment corresponds to the mining length, at which the immediate roof falls (as shown in Table5).(4)
The increasing segment can be fitted to a straight line, which means the regression relation between the fractal dimensionD and the mining length L can be given by the following equation:(6)D=a∗L+b,where a and b are the parameters determined by the hydrogeological conditions, rock mechanical parameters, mining method, and so on.Figure 6
Evolution of the fractal dimension of fracture network with mining length.


## 3.4. Evolution of Fractal Dimension with Different Mining Footage

In order to analyse the influence of mining speed on the damage of overlying strata and the evolution of fracture propagation, mining faces 5-1 and 5-3 are mined with different mining advance footage. The relationship between the fractal dimension and mining length with different mining footage is shown in Figure7, by which we can see that the mining face 5-1 (with a faster mining speed) has a larger fractal dimension than the mining face 5-3. Both the mining faces have the similar stage-type of the fractal dimension curve. Nevertheless, the 5-3 mining face has more points in increasing segment of the fractal dimension curve, which means having the shorter weighting interval in the immediate roof and a more developed fracture network in the caving zone, which can also be concluded from the fracture network at the end of the mining process (as shown in Figure 8).Figure 7
Relationship between the fractal dimension and mining length with different mining footage.Figure 8
Fracture networks at the end of the mining process. Mining face: (a) 5-1; (b) 5-3.
(a)
(b)


## 3.5. Evolution of Fractal Dimension in Different Caving Zones

With the advancing of mining face, how the fractures generate and develop in different horizontal and vertical zones is an aspect deserving to pay more attention on in this study. Based on this concern, 7 horizontal zones and 6 vertical zones are separated in mining face 5-1 (as shown in Tables10 and 11 and Figure 9); the fractal dimension is calculated, respectively, and the fractal dimension curves are shown in Figures 10–13. The key findings are as follows:(1)
Figure10 describes the evolution of the fractal dimension in horizontal zones with the mining face advancing. The evolution curves indicate that the fractal dimension in horizontal zones has a similar trend with the fractal dimension on the whole, that is, gradually increasing in general and reduces when the main roof collapses. The main reason may be that the collapse of the main roof gives rise to the closure of fractures in the caving zone.(2)
Figures10 and 11 show that the fractal dimension at different mining progresses (such as 90 m, 120 m, 150 m, and 190 m) decreases with the increase of mining footage. On the whole, at the end of the mining process, the horizontal zone H3 and zone H6 have a relatively low fractal dimension, and the zones H1, H4, and H7, on the contrary, have a relatively high fractal dimension. The distribution of the fractal dimension of the fractures shows generally the “W”-type trend.(3)
Through the analysis of the fractal characteristics of the fracture networks of the working face and combining the district partition of the fractured rock mass after mining [15], it is found that the fractal dimension of the fractured rock mass near the cutting area decreases gradually with the increase of mining footage, which indicates that the fractures in the area are gradually compressed and closed, which is called the re-compressed area. The fractal dimension of fractured rock mass near the mining face area increases, which indicates that the fractures are gradually developed and expanded with the increase of mining footage, which is called the separation zone. With the increase of mining footage, the separation area away from the mining face is gradually transformed into a re-compressed area and the re-compressed area is gradually transformed into a stable area so that the fractal dimension of the horizontal zones of the fractured rock mass in the goaf is in dynamic changing.(4)
Figure12 shows the similar developing pattern of fractal dimension in vertical zones with that in horizontal zones. At the same time, it may be concluded by Figure 13 that, in general, the fractal dimensions reduce with the increase of the caving height; that is, the caving zone has a more developed fracture network than the fractured zone. Meanwhile, however, it should be pointed out that there is an exception sometimes in the transition zone between the fractured zone and sinking zone, due to the formation of the strata-separated fractures, which contribute much to the increase of the fractal dimension.Table 10
Horizontal zone description.
Zone number
H1
H2
H3
H4
H5
H6
H7
Mining length(mm)
0–300
300–600
600–900
900–1200
1200–1500
1500–1800
1800–2100Table 11
Vertical zone description.
Zone number
Zone height(mm)
V1
0–100
V2
100–200
V3
200–300
V4
300–400
V5
400–500
V6
500–600Figure 9
Schematic diagram for the fracture zones.Figure 10
Evolution of the fractal dimension of crack network in different horizontal zones.Figure 11
3D colour map surface of evolution of the fractal dimension of crack network with mining length in different horizontal zones.Figure 12
Evolution of the fractal dimension of crack network in different vertical zones.Figure 13
Ratio of the fractal dimension of crack network in different vertical zones.


## 4. Conclusions

The fractal dimension is a very good tool to describe the evolution of mining-induced fractures network and its evolution. In this paper, the fractal characteristics of fractures network induced by mining are studied in detail, and the following conclusions can be drawn:(1)
The evolution curves of the fractal dimension with mining length can be separated into several stage-type folding lines, containing increasing segment and descending segment. The descending point corresponds exactly to the mining length, at which the main roof collapses. The increasing segment can be described by linear relationship.(2)
The mining face with a faster mining speed has a larger fractal dimension. Nevertheless, the opposite of that has a shorter weighting interval in immediate roof and a more developed fracture networks in the caving zone.(3)
The fractal dimensions in horizontal and vertical zones have a self-similar characteristic, that is, gradually increasing in general and reduces when the main roof collapses, similar to the features on the whole.(4)
At the end of the mining process, the spatial distribution of mining-induced cracks may be saddle shaped.(5)
Generally, the fractal dimension reduces with the increase of the caving height, while in the transition zone between the fractured zone and sinking zone, on the contrary, the fractal dimensions increase because of the separation of strata layers.



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*Source: 9589364-2018-06-20.xml*