#### Abstract

Hydraulic fracturing is an effective mean to prevent coal bumps in deep coal mining. The calculation of stress redistribution of coal sidewall after hydraulic fracturing can evaluate the fracturing effect and provide a reference for the hydraulic fracturing scheme design. In this paper, a theoretical model of stress redistribution of roadway sidewall after hydraulic fracturing is established with considering the effect of crushing zone. And the influences of the water injection length and the fracturing radius on the stress redistribution of roadway sidewall are investigated. The results show that the water injection section should be set at a deeper position with considering the existence of the crushing zone. The coal roadway sidewall after hydraulic fracturing is divided into five zones: the crushing zone, the shallow plastic zone, the pressure relief zone, the deep plastic zone, and the elastic zone. The stress redistribution law is obviously affected by the water injection section length and fracturing radius. With the increase of water injection length or fracturing radius, the pressure relief effect is more evident, but the axial force of rockbolt increases gradually. In order to acquire a better pressure relief effect, the matching of the water injection length and the fracturing radius should be considered crucially in the hydraulic fracturing design for preventing coal bumps.

#### 1. Introduction

Coal bump is a typical dynamic disaster with sudden and violent ejection of coal during deep coal mining, which seriously threatens the safety of people and equipment of coal mines [1–4]. High stress in the coal roadway sidewall is one of the main inducements of coal bumps [5]. Many pressure relief methods of roadway sidewall have been proposed to prevent the coal bumps in deep mining. The hydraulic fracturing method can reduce the risk of coal bump disaster effectively by transferring the high stress in coal seam to a further position away from the roadway boundary [6] and has played an important role in the coal bump prevention [7–9].

Figure 1 shows the schematic diagram of hydraulic fracturing in roadway sidewall in coal seam. A mass of coal fractures quickly form and connect to each other in hydraulic fracturing. The integrity of the coal seam in water injection section is damaged under a certain water pressure. Affected by the mechanical properties [10–12], water injection method [13, 14], and distribution of rock strata [15–17], the propagation form of coal fractures will be different [18, 19]. Coal damage caused by water pressure changes the stress distribution of the roadway sidewall [20, 21]. Through the calculation of the stress distribution, the pressure relief effect of hydraulic fracturing can be evaluated, and it provides a theoretical basis for designing an effective and reasonable hydraulic fracturing scheme. Therefore, it is necessary to analyze the stress of the surrounding rock.

**(a)**

**(b)**

**(c)**

Hydraulic fracturing can reduce the high stress of the roadway sidewall and cannot meet the energy conditions of coal bumps. Stress redistribution of surrounding rock can reflect the pressure relief effect. However, there is no unified understanding of the change pattern of stress redistribution. The stress level of roadway sidewall in a deep buried roadway is fairly high and the stress distribution is more complex [22], which increases the difficulty to determine the impact on the surrounding rock caused by hydraulic fracturing. However, the field hydraulic fracturing scheme design requires the support of the prediction of fracturing effect and estimation of stress redistribution. As a result, it is of great significance to study the stress redistribution of roadway sidewall after hydraulic fracturing.

In this paper, a theoretical model of stress redistribution in roadway sidewall after hydraulic fracturing is presented with considering the existence of the crushing zone. The effect of hydraulic fracturing in the high-stress zone of roadway sidewall was analyzed quantitatively. Finally, the influence of different length of water injection section and different fracturing radius on stress redistribution of roadway sidewall is investigated by numerical simulation.

#### 2. Theoretical Model of Stress Redistribution of Roadway Sidewall

##### 2.1. Stress Distribution of Surrounding Rock of Circle Roadway considering the Crushing Zone

Stress concentration degree and stress value is small after excavation in shallow buried roadway. It is generally considered that there are two zones of roadway sidewall: plastic zone and elastic zone. Using the elastic-plastic theory to quantitatively analyze the stress distribution of circular tunnel is a frequent issue. However, vertical stress of roadway sidewall is far greater than the uniaxial compressive strength of coal in a deep buried roadway. As shown in Figure 2, a wide range of the crushing zone in the surrounding rock of a circular roadway will be formed. The stress distribution calculated by elastic-plastic theory is suitable for the shallow buried roadway but not for the deep buried roadway. Therefore, the crushing zone should be considered in the theoretical analysis of stress redistribution in the surrounding rock of deep roadway.

The coal in the crushing zone still has residual strength after macroscopically broken. There is a nonlinear relationship between the peak strength of rock and confining pressure, as is the residual strength. According to Mohr–Coulomb strength theory, the strength curve can be simplified into a straight line (Figure 3), and the linear expressions of peak strength and residual strength under different confining pressures are as follows:where , , , , *ξ* is the slope of strength curve, *σ _{c}* is the compressive strength,

*c*is cohesion,

*φ*is the internal friction angle,

*c*

_{f}is the residual cohesion, and

*φ*

_{f}is the residual internal friction angle.

Based on equation (2), stress redistribution of roadway sidewall in the crushing zone is analyzed. It is assumed that both the vertical stress and lateral stress of surrounding rock before roadway excavation are *σ*_{0}, and roadway excavation radius is *R*_{a}. As shown in Figure 2, the roadway section is simplified to circular structure to investigate the stress redistribution of coal roadway sidewall at a radius of *R*.

The coal seam in the crushing zone is in the residual stress state. If a polar coordinate system is established with the center of the roadway as the origin, the relation between tangential stress *σ*_{θf} and radial stress *σ*_{Rf} in the crushing zone satisfies equation (2). The stress distribution in the crushing zone of the roadway sidewall can be obtained:

At the boundary between the crushing zone and the plastic zone, tangential stress *σ*_{θf − p} and radial stress *σ*_{Rf − p} simultaneously satisfy equations (1) and (2), and the crushing zone radius *R*_{f} and the corresponding stress can be concluded:

Tangential stress *σ*_{θp} and radial stress *σ*_{Rp} of coal in plastic zone satisfy (1), and the stress distribution in the plastic zone can be obtained by simultaneous equilibrium equation and boundary conditions:

At the boundary between the plastic zone and the elastic zone, the stress satisfies *σ*_{θp} + *σ*_{Rp} = 2*σ*_{0}. Let *R* = *R*_{P}; the boundary expression of the plastic zone is

The coal body in the elastic zone is simplified to a circular roadway with a radius of *R*_{p}. The radial stress on the boundary of the plastic zone is taken as external force acting on the sidewall of the tunnel instead of the effect of the plastic zone on the elastic zone, and stress distribution in the elastic zone can be calculated:

##### 2.2. Stress Redistribution of Roadway Sidewall after Hydraulic Fracturing

Before hydraulic fracturing, the roadway sidewall is divided into the crushing zone, the plastic zone, and the elastic zone due to coal roadway excavation, as shown in Figure 4(a). In most cases, the sealing section is located both in the crushing zone and plastic zone, and the water injection fracturing section is located in the elastic zone [23]. The coal in the water injection section considerably damages after hydraulic fracturing, and the elastic zone located in the water injection section changes into the pressure relief zone. The coal near the bottom of the borehole at the depth of the roadway is changed from elastic to plastic because of the stress redistribution. Therefore, the partition of roadway sidewall is changed into the crushing zone, shallow plastic zone, pressure relief zone, deep plastic zone, and elastic zone, as shown in Figure 4(b).

Figure 4(c) shows section A in the water injection section, which is used to study the stress distribution around the water injection hole. After hydraulic fracturing, the stress in the water injection section decreases, and the plastic zone outside the fracture zone under low stress is very small [24]. Therefore, it is supposed that the surrounding rock of the water injection hole is divided into the fracture zone and the elastic zone.

It is assumed that the range of the fracture zone is equal to the fracture length, and its radius *r*_{fw} can be obtained by [24]where *E* is the elastic modulus, *Q* is the water injection flow rate, *t* is the water injection time, *H* is the thickness of the coal seam, *p* is the water pressure, and is Poisson’s ratio.

Similar to the stress distribution of circular roadway, the tangential stress in the fracture zone around the water injection hole can be expressed aswhere , , *c _{w}* is cohesion after hydraulic fracturing, is the internal friction angle after hydraulic fracturing, and

*C*

_{1}is a constant.

Between the fracture zone and the elastic zone, the tangential stress distribution is continuous. Assuming that the stress in the elastic zone *σ*_{A} is the invariant after hydraulic fracturing. The tangential stress distribution in the fracture zone can be calculated:

If the spacing of water injection hole is 2*L* and the drilling radius is *r*_{aw}, the stress in the pressure relief zone can be expressed by the mean value stress of the coal between the two water injection holes:

In order to calculate conveniently, the stress at the midpoint of the fracturing zone is taken as the mean value stress in the fracturing zone; then, (11) can be simplified as follows:

In the shallow plastic zone, the stress distribution can be obtained by combined the static balance equation and (1):

It should be noted that *R*_{s} is the starting position of the pressure relief zone and *R*_{d} is the end position of the pressure relief zone. According to the stress continuity conditions at the interface between the shallow plastic zone and the pressure relief zone, the tangential stress distribution in the shallow plastic zone can be obtained:

Similarly, the tangential stress in the crushing zone after hydraulic fracturing *σ*_{θf2}, the tangential stress in deep plastic zone *σ*_{θp2}, the radius of deep plastic zone *R*_{p2}, and the tangential stress of elastic zone after pressure relief *σ*_{θe2} can be calculated, respectively, as

##### 2.3. Case Analysis

The hydraulic fracturing scheme design in the field coal bumps prevention mostly focuses on engineering experience and is weak in the design of theoretical scheme. In order to specifically analyze the stress distribution of roadway sidewall after hydraulic fracturing, a specific working condition is set to analyze the application effect of the theoretical model. Parameters of rock and coal are obtained from field tests.

It is assumed that the vertical ground stress *σ*_{0} is 25 MPa and the lateral pressure coefficient is 1.0 in this case analysis. The excavation radius of the roadway is 2.5 m, and the space between the two adjacent water injection holes is 10m. The specific mechanical parameters of the coal seam are shown in Table 1.

Vertical stress of roadway sidewall could be also expressed by tangential stress in a horizontal plane. In other words, the vertical stress is equal to the tangential stress. The vertical stress curves of roadway sidewall with considering the crushing zone and not considering the crushing zone are shown in Figure 5. The abscissa represents the distance between the coordinate point and the roadway boundary, and the ordinate represents the vertical stress of the coordinate point. The red dotted lines are demarcation lines of three zones. It can be seen from Figure 5 that the maximum vertical stress calculated without considering the crushing zone occurs at 3.0 m from the roadway boundary, while the maximum vertical stress occurs at 4.2 m when considering the crushing zone. A comparison between them shows the peak stress position moving 1.2 m depth in the roadway sidewall. This means that the water injection section in the hydraulic fracture scheme should be set in depth given the presence of the crushing zone.

The starting point *R*_{s} of the water injection section is 6 m and the end point *R*_{d} is 15 m. The vertical stress redistribution curves of roadway sidewall are shown in Figure 6. As seen from Figure 6, the stress is bimodal distribution after hydraulic fracturing. In the original high-stress concentration area, the stress is obviously reduced. A new high-stress concentration zone is formed far away from the roadway boundary after hydraulic fracturing. With the increase of water injection fracturing radius, the vertical stress in pressure relief zone decreases gradually, and the stress peak transfers to the deeper of the roadway sidewall.

The high-stress reduction ratio curve and the distance between the new peak stress point to roadway boundary are shown in Figure 7. It can be seen from the curves that the high-stress reduction ratio is linearly and positively correlated with the water injection fracturing radius. With the increase of the fracturing radius, the distance between the new peak stress point and roadway boundary increases from 21 m to 30 m, which exponentially increases with the fracturing radius.

The vertical stress redistribution with different lengths of the water injection section can be obtained by changing *R*_{d}. As shown in Figure 8, the longer length of the water injection section is, the larger range of the pressure relief zone is formed. The high stress concentration area occurs again in a deeper location of the roadway sidewall. With the increase of the water injection section length, the position of the new stress peak point moves more and more deeper.

With different lengths of water injection section, the high-stress reduction ratio and the distance between the new peak stress point and the roadway boundary are shown in Figure 9. It can be seen that the distance between the new peak stress point and the roadway boundary is linearly and positively correlated with the length of the water injection section. The stress reduction ratio has a power function relationship with the water injection section length. As water injection section length increases, the high-stress reduction is not obvious, but the distance between peak stress and roadway boundary increases significantly.

#### 3. Numerical Study on Hydraulic Fracturing in Roadway Sidewall

##### 3.1. Simulation Program

In order to study the main control factors of hydraulic fracturing effect at the engineering scale. Flac3D software is used to simulate the sidewall hydraulic fracturing of a rectangle roadway at 1000 m buried depth. The numerical calculation model is established and shown in Figure 10.

The size of the numerical model is *X* × *Y* × *Z* = 100 m × 6 m × 50 m. The stress boundary condition is adopted at the upper part and the vertical stress is 25 MPa. The bottom boundary of the model is fixed in all directions and the displacement of out-of-plane is fixed in the *y* direction. The displacement in *x* direction is limited on the left and right sides of the model. The size of the rectangle roadway is *H* × *W* = 5 m × 5 m. The Mohr–Coulomb model is used to simulate the constitutive model of the rock strata. The corresponding parameters of roof and floor strata are listed in Table 2.

Pile units are selected to simulate rockbolt in roadway supporting [25]. The length of the rockbolt is *L* = 3 m and the anchored length is 3 m too. The space between the rockbolts is 1 m × 1 m, the yield axial force of rockbolt is 266 kN, and the elastic modulus is 210 GPa. The bulk modulus and cohesion of the anchoring agent are 20 MPa and 4.5 MPa, respectively.

The hydraulic fracturing effect is realized by weakening the mechanical parameters of coal in this numerical simulation. The surrounding area centered on the borehole is set as a weakening area of water injection fracturing. The drilling diameter of the hole in coal seam is 0.1 m, sealing section is 10 m, and water injection section is 15 m. The parameters of coal before water injection and after fracturing are listed in Table 3.

To study the influences of the water injection section length and fracturing radius on the pressure relief effect of hydraulic fracturing, two groups under different conditions were established in this numerical simulation. In the first group, the sealing section length is set at 10 m, the water injection section length is fixed at 16 m, and the fracturing radius is set at 2 m, 2.5 m, 3 m, and 3.5 m, respectively. The vertical stress of the roadway sidewall is monitored to analyze the influence of fracturing radius on the effect of hydraulic fracturing. In the second group, the sealing section length is fixed at 10 m, and the water injection section length is set as 5 m, 10 m, 16 m, 20 m, and 25 m, respectively. To eliminate the influences of rock lithology and water injection mode, the mechanical parameters of surrounding rock and water injection pattern are set as identical for each simulation scheme in this paper.

##### 3.2. Simulation Results

Figure 11 shows the vertical stress redistributions of the roadway sidewall before and after the hydraulic fracturing. In the numerical simulation results, the stress at the original stress peak point of roadway sidewall decreases after hydraulic fracturing. A new stress peak appears inward the water injection section and a bimodal stress distribution is formed, which is consistent with the theoretical results.

Before the hydraulic fracturing, the range of 0–6 m away from the roadway boundary is the crushing zone, the range of 6–10 m is the plastic zone, and the range of over 10 m is the elastic zone. There is an obvious stress concentration phenomenon in Figure 11. After the hydraulic fracturing, the range of 10–25 m is the pressure relief zone and the stress is significantly reduced. The elastic zone boundary moves to the deeper location of the roadway sidewall and the stress concentration phenomenon occurs again at the interface between the pressure relief zone and the elastic zone. The simulation result agrees well with the proposed theoretical model.

The vertical stress redistribution of roadway sidewall after hydraulic fracturing in different lengths of water injection section is shown in Figure 12. Stress curves show that the longer the injection section has the smaller stress of the overall roadway sidewall after hydraulic fracturing. The injection section length is positively related to the degree of stress reduction, and the new peak stress point occurs deeper with the increase in water injection length. The law of numerical simulation results conforms to the theoretical results in Figure 8.

Especially, the new stress peak value decreases with the increase of the water injection section length in the numerical simulation. This is not consistent with the theoretical model. That is because the theoretical model is established under plain strain condition and does not consider the stress distribution in the axial direction of the roadway. In the direction of the depth of roadway sidewall, the stress variation trend obtained by the simulation is basically consistent with that calculated from the theoretical model. It shows that the theoretical results are still reasonable.

The vertical stress redistribution of roadway sidewall after hydraulic fracturing with different water injection fracturing radius is shown in Figure 13. The original peak point of vertical stress of roadway sidewall decreases significantly with the increase of the water injection fracturing radius. In contrast, new peak point increases gradually. When the fracturing radius is 3 m, the first peak stress is significantly reduced, while the second peak stress is not very large. It is implied that there is an optimal fracturing radius in hydraulic fracturing design of the coal seam in roadway sidewall.

Axial force curves of rockbolt under the condition of different water injection section lengths are shown in Figure 14. As the length of the water injection section increases, the axial force of rockbolt gradually increases. For each 5 m increase in the length of the water injection section, the axial force of rockbolt increases by about 2 kN. The axial force of rockbolt under the different water injection fracturing radius is shown in Figure 15. Comparing Figures 7 and 15, when the water injection radius increases from 2 m to 5 m, the high-stress reduction ratio increases by 35% and the axial force of rockbolt increases by 16 kN. It indicates that the greater the drilling length or fracturing radius of the water injection section, the greater the axial force of the rockbolt. However, the too long water injection section or too great fracturing radius will lead to the instability of the coal seam. Therefore, in order to achieve a better pressure relief effect, the matching of length and fracturing radius should be considered crucially in the hydraulic fracturing design for preventing coal bumps.

#### 4. Conclusions

Pressure relief caused by hydraulic fracturing in coal roadway sidewall is one of the most effective means to prevent coal bumps. In this paper, stress redistribution of roadway sidewall after hydraulic fracturing in deep coal mining was studied, which could provide a guidance for the hydraulic fracturing scheme design. The main conclusions are shown below.(1)With considering the crushing zone, a theoretical model of stress redistribution of roadway sidewall is built. After hydraulic fracturing, the stress redistribution curve presents a bimodal curve. The roadway sidewall could be divided into five zones: the crushing zone, the shallow plastic zone, the pressure relief zone, the deep plastic zone, and the elastic zone. At the end of the pressure relief zone, a new plastic zone in roadway sidewall is created.(2)Through the case analysis, it can be found that the water injection section length has an important influence on the distance between the new peak stress position and roadway boundary. The longer the water injection length, the greater the distance. In contrast, the water injection fracturing radius has a greater impact on the peak stress reduction ratio more than the water injection length. The greater the fracturing radius, the greater the stress reduction ratio.(3)The numerical simulation results of the stress distribution law are basically consistent with the results calculated by the theoretical model. It is noted that both the water injection length and the fracturing radius have significant effects on the axial force of rockbolt. Too long water injection length or excessive fracturing radius will lead to the larger axial force of rockbolt, which dramatically affects the stability of the coal sidewall.

#### Data Availability

The data used to support the findings of this study are included within the article.

#### Conflicts of Interest

The authors declare that they have no any commercial or associative interest that represents conflicts of interest in connection with the work submitted.

#### Acknowledgments

This work was financially supported by the Major Program of Shandong Provincial Natural Science Foundation (no. ZR2019ZD13), National Science Foundation of China (nos. 52074167 and 52104137), Shandong Provincial Natural Science Foundation (no. ZR2020QE122), and China Postdoctoral Foundation (no. 2019M660024).