Characteristics of Stress-Displacement-Fracture Multi-Field Evolution around Gas Extraction Borehole

Abstract

To ensure the effectiveness of the gas extraction borehole, it is necessary to investigate the stress-displacement-fracture evolution of the coal around the borehole. In this study, by constructing a numerical model of a gas extraction borehole, the burial depth and side pressure coefficient are used to characterize the overall stress level of the borehole and the difference in stress distribution caused by complex stress conditions. First, the stress time-varying pattern and force chain distribution of coal around the borehole were revealed. Then, the displacement time-varying pattern and displacement distribution of coal around the borehole were elucidated. Then, the microfracture distribution of coal around the borehole, which characterizes the microfractures, was analyzed. Finally, the validity of the numerical results was verified. The results showed that, after the stress field of the coal around the borehole was adjusted, the force chain of the borehole was unevenly distributed and the stress concentration phenomenon appeared. With the increase in burial depth, the stress around the borehole gradually increased, while the range of stress concentration zone in the borehole kept increasing, and the borehole changed from unilateral instability to bilateral instability. Moreover, the displacement field around the borehole was distributed in the shape of a ”disk leaf”. With the increase in burial depth, the deformation of coal around the borehole increased. With the increase in the side pressure coefficient, the vertical and horizontal displacement also increased gradually. Furthermore, there was a certain correspondence between the development of fracture and the deformation around the coal. With the increase in burial depth, the development of fractures was gradually obvious, and the distribution characteristics were concentrated in the middle and dispersed around. This study provides a theoretical reference for the stability of gas extraction boreholes, aiming to improve the gas extraction effect.

1. Introduction

Coal occupies an important position in the energy structure of China [1,2,3]. In order to meet the national coal energy demand, the depth of coal mining has gradually increased, and the geological conditions of mining have become more complex [4,5,6,7]. Deep coal seams are characterized by high gas content, high gas pressure, high ground stress, and poor permeability [8,9,10,11], which not only increase the difficulty of coal mining but also pose a great threat to safe and efficient coal mining [12,13,14,15]. Coal seam gas pre-extraction is an important method to manage gas disasters and can effectively prevent and control gas disaster accidents [16,17,18]. Borehole is the key and difficult part of gas pre-extraction measures. After the borehole is completed and lifted, the stress in the coal around the borehole is redistributed and displacement occurs, resulting in the expansion of primary fractures and the creation of new fractures [19,20]. These are all important factors leading to borehole collapse. Therefore, to ensure the effectiveness of the gas extraction borehole, it is necessary to investigate the stress-displacement-fracture multi-field evolution of the coal around the borehole.

Many theoretical analyses, numerical simulations, and indoor experiments have been carried out on the stress in the gas extraction borehole [21,22,23,24,25]. Some scientists have used the Kozeny–Carman equation to establish a mathematical model of coal-rock-gas-solid coupling, obtained stress distribution patterns on both sides of the borehole through simulation analysis, elucidated the effective influence radius of the borehole, and documented the applicability of the coupled gas extraction and multi-physics field model in coal mines [26]. Zhang et al. [27] used theoretical analysis and numerical simulation to study the deviation principles of gas extraction boreholes in outburst coal seams. Li et al. [28] used the energy balance method and borehole process monitoring system to conduct orthogonal tests on granite to obtain the sensitivity of each borehole parameter to borehole efficiency. Zheng et al. [29] established a three-dimensional numerical simulation model of dynamic coal breaking by the borehole bit in order to determine the coal stress along the hole depth of the coal seam borehole and studied the process of coal boreholes under different coal stress conditions. Hao et al. [30] constructed a mechanical model of coal stress and borehole pipe speed and applied a borehole pipe speed test system to analyze the variation law of borehole pipe speed under different coal stress. Subsequently, Xiao et al. [31] established a limit borehole model. He et al. [32] conducted a numerical study on the stress distribution and deformation damage in boreholes based on the effective stress principle, and the location of the occurrence of vertical peak stress would be gradually moved away as the coal seam ground stress and gas pressure increased. In summary, many studies have been carried out on boreholes, and a series of research results have been achieved in terms of stress. The change in stress will lead to a change in displacement and then many fractures will be generated [33,34]. However, scholars have mainly studied the effects of parameters such as borehole diameter, borehole number, and gas pressure on borehole destabilization [35,36,37]. Many results have been obtained on the evolution of stress around boreholes. However, the evolution of displacement and fracture around the borehole, which also has important effects on gas extraction, is not clear. Moreover, the displacement and fracture changes are different for different stress states. Therefore, it is necessary to investigate the stress-displacement-fracture multi-field evolution around the borehole under different stress conditions.

The geological environment in which a borehole is located is complex, and there are complex spatial relationships during the borehole process. The borehole passes through different strata, with different stress states and different borehole directions, causing different stress fields around the borehole, which in turn produce differences in displacement and fracture fields. Therefore, this study establishes a numerical simulation model for gas extraction boreholes. The difference in stress distribution caused by the overall stress level and complex stress conditions in the borehole is characterized by the burial depth (h) and side pressure coefficient (λ). The study reveals the stress time-varying pattern and force chains distribution of coal around the borehole, monitors the displacement time-varying pattern and displacement distribution of coal around the borehole, analyses the microfracture distribution of coal around the borehole, and quantitatively characterizes the microfractures. This study provides a theoretical reference for the stability of the gas extraction borehole, with the aim of improving the subsequent gas extraction effect.

2. Methods

2.1. Modeling

As shown in Figure 1, a Cartesian system is established to analyze the stress field of the coal around the borehole: a linear pore elastic model and the coordinate transformation equation are used to convert the stress state distribution of the borehole in the natural spatial coordinate system to a stress model in x, y, and z directions centered on borehole to solve for the stress distribution around the borehole. The model was constructed using PFC software to build a 700 mm × 700 mm 2D model with a 30 mm diameter borehole at the center of the model, and the 4 boundaries of the model were fixed, i.e., the particles at the boundaries of the model were fixed.

The particle flow model is completed by several computational cycles, i.e., updating the contact force of the particle contact part by the force-displacement law, updating the particle and wall position by Newton’s second law, and readjusting the contact relationship between the particles, both of which act alternately to traverse the entire model ensemble until equilibrium is reached or damage occurs that cannot be kept stable.

The PFC software is used for the simulation, which has the following three advantages. First, the modeled particle unit is a sphere, the contact between objects is simpler compared with angular objects, and the simulation is computationally efficient. Then, there is no relevant size limitation in monitoring the displacement of the simulated object, and the object displacement can be monitored under various displacement conditions. Finally, since the objects consist of cemented particles, the objects can break under external action.

To ensure that the numerical simulation results are realistic and reliable, the following assumptions are made:

• The sphere or circular element is a unit of certain stiffness, the motion of the sphere and circular element involves translational and rotational transformations, which can be rotated around a certain shape center, and the shape and size of the sphere and circular element change with time.
• The contact between the spheres is regarded as point-point contact, and the contact with the boundary surface is point-edge contact.
• The contact force between the spheres or circular elements of the model is related to its stiffness, and the relative displacement between each unit is related to the initial damping coefficient.

2.2. Parameters Setting

For the PFC simulation, selecting the appropriate micromechanical parameters is crucial. The numerical model parameters were set by combining the physical mechanical parameters of No. 3 coal seam of Sangbei Coal Mine in Hancheng mining area. No. 3 coal seam of Sangbei Coal Mine is mainly thick coal seam with stable coal seam distribution and small thickness variation. Its structure is simple, with a single coal type, mainly high metamorphic poor coal, with little change in coal quality.

According to the mechanical parameters of the specimen obtained from the indoor displacement loading test, the mechanical parameters of the constructed numerical simulation model were measured in the same loading mode. The model parameters of its indoor displacement loading test are as follows: the uniaxial compression test model is 100 mm high and φ50 mm wide, and the diameter of the Brazilian disc test is 50 mm. The mechanical parameters of the displacement loading test and numerical simulation are shown in

Table 1. Basic parameters.

Mechanical Parameters	Elastic Modulus (GPa)	Poisson’s Ratio
Experimental results	1.48	0.23
Numerical results	1.55	0.21
Error (±%)	4.70	8.60

. The micromechanical parameters were adjusted to make the mechanical properties of the model in the simulation consistent with those obtained from the indoor test results. The calibrated micromechanical parameters are shown in

Table 2. Model parameters.

Particle Basic Parameters	Value	Parallel Key Parameters	Value
Particle density (kg/m3)	1450.00	Average normal strength (MPa)	4.00
Particle contact modulus (GPa)	0.80	Standard strength standard deviation (MPa)	0.10
Particle minimum radius (m)	2.00 × 10−4	Mean shear strength (MPa)	4.00
Spherical diameter ratio	1.80	Standard deviation of shear strength (MPa)	0.10
Stiffness ratio	1.00	Elasticity coefficient (GPa)	0.80
Friction coefficient	0.40	Stiffness ratio	1.00
Damping coefficient	0.70

.

2.3. Research Program

Simulation of the unloading process using an effective numerical simulation method is the key to ensuring the final effect of the simulation is realistic and reliable. The corresponding initial stress conditions need to be set before borehole, and the unloading of borehole after borehole formation causes a large change in the stress and corresponding changes in the stress, displacement, and fracture fields. In this study, the ground stress automatic release method is used. It can demonstrate the real change process of coal stress around borehole, reflect the re-equilibrium process of force distribution in coal, and reveal the unloading law of coal after borehole formation. The calculation process neither needs to calculate the release load nor superimpose the stress field. It only needs to establish the corresponding elastic model, plastic model, and plastic change assessment conditions. The rest of the process can be approximated as a linear elastic process. The process of achieving continuous drilling and continuous hole formation is very simple.

The numerical model calculation process is shown in Figure 2. Based on the automatic release method of ground stress, the initial stress equilibrium state at each point on the boundary after borehole is broken, and the forces at each node on the boundary are unbalanced so that the motion of particles occurs. The motion of particles in the process is calculated to reach a new equilibrium, and the equilibrium conditions can be set to meet the relevant accuracy requirements. During the equilibrium process, the object will undergo corresponding deformation, which is expressed as the motion of the particles, to obtain the result of stress redistribution, and finally attain the stress, deformation, and fracture fields of the coal after the borehole process. The simulation of borehole process is realized by deleting the spheres in the corresponding positions, and the borehole process is decomposed into multiple steps. The steps are set according to the characteristics of the borehole process, and the particles are deleted under the corresponding conditions, and the stress, deformation, and fracture changes around the borehole under the corresponding conditions can be compared and analyzed in each borehole step.

The construction process of the borehole under different burial depths and different side pressure coefficients is studied, and the construction process of the borehole is simulated and analyzed by using PFC discrete element numerical simulation software with the field borehole construction data. Change the burial depth and side pressure coefficient for numerical simulation. The side pressure coefficient λ = 1.0 for different depths of burial depth of 200 m, 400 m, 600 m, and 800 m. When the burial depth h = 600 m, the side pressure coefficients are 0.6, 0.8, 1.2, and 1.4.

3. Results and Discussion

3.1. Stress Field Evolution around Borehole

3.1.1. Time-Varying Characteristics of Stress around Borehole

Figure 3 shows the stress variation around the borehole for burial depths of 200 m, 400 m, 600 m, and 800 m and the side pressure coefficient λ = 1.0. When the coal around the borehole produced unloading and sudden pressure drops, the magnitude of unloading gradually increased with the increase in burial depth. With the deformation of the rock, the pressure around the rock gradually restored the initial ground stress level, and the changing trend of stress in both horizontal and vertical directions was nearly the same.

As shown in Figure 4, after the formation of the borehole, the coal around the borehole will go through the force fluctuation stage and force re-equilibration process under different side pressure coefficients. When the borehole was first formed, the stress level decreased rapidly due to the unloading effect. Then, the stress gradually increased, and a new stress balance was reached. With the increase in the side pressure coefficient, the stress level in the vertical direction increased. The time required to re-equilibrate the stress in the coal around the borehole was basically the same under different side pressure coefficients. The change of stress in the horizontal direction was basically the same as that in the vertical direction. Firstly, the stress changed abruptly to unload the pressure, then the stress fluctuated and finally reached equilibrium again. The stress after equilibrium was increased compared with that before equilibrium, and the presence of a borehole will make the stress distribution of coal rock change, and the stress concentration occurred in the coal around the borehole.

3.1.2. Distribution Law of Force Chain around Borehole

Figure 5 gives the force chain development around the borehole at the early stage and after stabilization of the borehole at 200 m, 400 m, 600 m, and 800 m burial depths.

From the results of the force chain distribution (Figure 5a), when the borehole was just formed, the stress level was low, and the force chain was evenly distributed. After the stress of the coal around the borehole was stabilized, the stress distribution of the borehole was not uniform. Under the condition of 200 m burial depth, the stress concentration phenomenon appeared, and the stress concentration in the borehole did not cause failure to the borehole in the short term, but it could be presumed that the stress concentration would cause further local deformation of the coal around borehole with the passage of time. Moreover, under 400 m burial depth (Figure 5b), the overall level of stress was higher compared with the condition of 200 m. The phenomenon of stress concentration on the upper right side appeared at the early stage at h = 400 m, indicating that under high stress, the borehole was more likely to be deformed, and the deformation caused the stress concentration on the upper right side. It is basically the same with the burial depth of 200 m that the stress concentration phenomenon still appeared more obvious.

Under h = 600 m (Figure 5c), the overall stress level was further increased, and at the early stage of borehole formation, long and thin blue force chains appeared around the borehole, which was prolonged by the movement of model particles toward the center of the borehole after borehole formation. Due to the movement deformation unloading pressure, the coal stress level decreased and became thin to show a blue color. It indicated that under the burial depth of 600 m, the phenomenon of borehole wall shedding may occur, the stress connection between the borehole wall particles and the borehole wall is weak, and the borehole closure may also occur under the condition of weak disturbance. The stress concentration around the borehole appeared in a ring shape, and under h = 600m, the stress concentration was distributed in a larger radius compared with that in 200 m and 400 m. Under h = 800m (Figure 5d), the overall stress level increased again, and the stress distribution resembled a patchy shape, which was scattered and widespread. This indicated that, under the high-stress effect, a large amount of energy was accumulated inside the coal, and a small unconnected stress concentration chain existed. Under the action of external disturbance, the distributed stress concentration was connected and will cause failure in the coal and borehole. From Figure 5d, we could see that under h = 800m, the failure form of the borehole was basically the same as under h = 600m, both were affected by the high ground stress, but after the stress stabilization of the coal, the distribution of the stress chain was somewhat different from that under h = 600m, and the stress was uniformly dispersed without collapse of the borehole.

As shown in Figure 6, with the increase in the side pressure coefficient, the location of stress concentration gradually changed from a horizontal direction to a vertical direction. When λ was less than 1.0, the stress concentration was mainly concentrated in the vertical direction, and the direction of the force chain was up and down. When λ was greater than 1.0, the stress concentration was mainly concentrated in the horizontal direction, and the stress on the left and right sides was concentrated in the horizontal direction of the force chain. As λ increased, the top vertical stress and horizontal stress on both sides will increase, but the growth rate of top vertical stress was faster than the growth rate of horizontal stress on both sides. Compared with the time when λ was less than 1.0, the stress concentration area was more obvious when λ was greater than 1.0, and the coal around the borehole was more deformed and more prone to instability at this time.

3.2. Displacement Field Evolution around Borehole

3.2.1. Displacement Time-Varying Characteristics around Borehole

During the borehole process, there are three stages of pressure relief deformation, deformation fluctuation, and deformation recovery in the coal around the borehole, and the stress is significantly reduced in the pressure relief deformation stage. The generation of pressure relief deformation is related to the pressure relief effect on the coal rock around the borehole after the formation of the borehole. After the stress unloading of the coal around the borehole, the high stress of the unrelieved coal around the borehole and the low stress of the relieved coal produce a stress difference, which leads to the pressure relief deformation of the coal around the borehole.

The vertical deformation of coal around the borehole under different burial depths is shown in Figure 7. When the burial depth is 200 m, the strain level of coal around the borehole was the lowest, and the spatial range exceeding the critical value of plastic deformation was relatively small. Comparing the experimental results of different burial depths, it can be seen that with the increase in burial depth, the deformation of coal around the borehole increased, and the deformation rate increased. Until the burial depth was the largest, the vertical displacement of coal around the borehole changed the most.

3.2.2. Displacement Distribution around Borehole

Figure 8 shows the strain changes around the borehole at the initial stage of the borehole and after stabilization of the borehole under the conditions of burial depths of 200 m, 400 m, 600 m, and 800 m. Under h = 200 m (Figure 8a), the strain mainly occurred in the process of borehole formation. At the end of the subsequent borehole, the strain range around the borehole was further expanded. Through simulation, it was found that there was a certain direction in the deformation process. The coal underwent a large deformation along the straight line, and the vertical direction with this straight line would appear small deformation. This was caused by the difference between the distribution of the maximum deformation direction and the minimum deformation direction at 200 m burial depth. At the same time, it could be seen that there was a π/2 deflection angle in the large and small deformation zone. Under h = 400 m (Figure 8b), the strain distribution was not uniform at the initial stage of borehole formation, and the deformation of coal around the borehole was small. After the coal deformation was stable, the deformation was large, which was different from the deformation under h = 200 m. Under high-stress action, the borehole’s deformation was more rapid and more severe after the borehole process, and the displacement field gradually changed from the initial uneven distribution to a uniform trend. The strain of coal showed random directionality at the beginning, and then with the stable deformation of coal, the deformation tended to be uniform, indicating that with the increase in in situ stress levels, the in situ stress levels became the dominant borehole deformation factor, which determined the stress distribution of coal after borehole stability.

Under h = 600 m (Figure 8c), the strain distribution of the surrounding coal was uniform when the borehole was formed. The strain developed into a flat oval with the change of time. The borehole diameter was reduced, and a slight collapse occurred in the upper part of the borehole. In the early stage of the borehole, the main deformation area of coal around the borehole showed the characteristics that the closer the distance from the center of the borehole, the more obvious the deformation was. The deformation area was distributed in layers. Moreover, after the borehole became stable, it developed into a long strip shape. In the model scale range, X-shaped deformation appeared around the borehole and the deformation was small on both sides. Under h = 800 m (Figure 8d), the displacement distribution was like that under h = 600 m. However, in the early stage of borehole formation, due to the influence of high ground stress, the deformation of coal around the borehole was severe, the overall shape was circular, and the failure form was basically the same as under h = 600 m. After the stress of coal was stable, the displacement distribution had the same long strip deformation area as that under h = 600 m. However, under h = 800 m, the 4 corner positions in the model range showed a more obvious large deformation.

Figure 9 reflects the change of displacement around the borehole caused by the formation of the borehole. After the borehole was arranged, the coal around the borehole deformed in the radial plane, and the deformation of the coal around the borehole was greatly affected by the lateral pressure coefficient. When the side pressure coefficient λ < 1.0, the overall deformation of the coal showed 2 ellipses with vertical distribution intersecting at the borehole location. When λ > 1.0, the overall deformation of the coal showed 2 ellipses with horizontal distribution intersecting at the borehole location. In addition, there was a small deformation area in the secondary deformation direction of coal around the borehole, and there was a larger deformation in the main deformation direction, and the shape of the borehole would be compressed into an ellipse. Under disturbance, the borehole was more prone to the closed-hole phenomenon. Especially when the direction of stress wave conduction was consistent with the direction of large deformation, it would lead to greater deformation of coal around the borehole, thus causing its instability.

Moreover, the coal shrunk radially, and there is a large deformation of coal around the borehole. When λ < 1.0, the maximum displacement of the borehole occurred on the upper and lower sides of the borehole. When λ > 1.0, the maximum displacement of the borehole occurred on the left and right sides of the borehole, and the deformation direction pointed to the borehole. The coal rock far away from the borehole was less affected by the borehole formation process.

In summary, the displacement field around the borehole was distributed in the shape of a “disk leaf”. The deformation of coal around the borehole had a certain direction, and it underwent a large deformation along the straight line, and the vertical direction of the straight line had a smaller deformation. Before the formation of the borehole, the model formed the distribution of the initial in situ stress field was basically consistent with the actual situation. After the borehole was formed, the stress concentration around the borehole was generated, and the deformation around the borehole was obvious.

3.3. Fracture Field Evolution around Borehole

3.3.1. Distribution Pattern of Fracture around Borehole

Figure 10 shows the distribution of fractures around the borehole with different burial depths. There was a certain correspondence between the development of fractures and the deformation around the coal. More fractures were produced in the area where large deformation occurred at the beginning of loading, and scattered fractures were produced in the high-stress area after stress stabilization. When the stress level was low, the number of fracture development was small and the distribution pattern was not obvious. As the ground stress increased, the development of fractures gradually showed a concentrated distribution in the middle and scattered distribution around. With the increase in ground stress, stress concentration occurred in more areas, and the number of distributed fractures gradually increased, thereby further promoting fracture development.

Figure 11 shows the distribution of fractures around the borehole with different side pressure coefficients. As shown in Figure 11, areas of stress concentration induced the production of large deformations, which led to the development of fractures. However, a special phenomenon existed: many fractures were still produced in the low-stress and small deformation areas perpendicular to the main deformation direction. This was due to the large deformation generated around the borehole in the high-stress direction, which drove many pores in the low-stress direction, causing the coal around the borehole to crack and form many fractures under low-stress conditions. Under the condition of λ ≠ 1, there was inhomogeneity in the development of fractures, with more fractures being produced on one side and fewer on the other side. The deformation in the high-stress direction was not completely uniform, and the corresponding inhomogeneity would also occur on the low-stress side.

3.3.2. Quantitative Characterization of Fractures around Borehole

During the borehole process, the coal around the borehole went through three stages: pressure relief deformation, deformation fluctuation, and deformation recovery. As shown in Figure 12a, in the stage of pressure relief deformation, the stress was significantly reduced due to borehole pressure relief, and the fracture development inside the coal rock mainly existed in the middle stage of deformation fluctuation. Especially under the condition of a 200 m burial depth, the number of fractures was also less due to the low ground stress. Only a few fractures were produced in the early stage, and, basically, no fractures were produced in the later stage. In the 800 m burial depth range, the overall level of fracture development was higher as the burial depth increased.

As shown in Figure 12b, when λ > 1, the fracture development was generally more obvious than that when λ < 1. In addition, the fracture development was more obvious when the side pressure coefficient was closer to one. When λ > 1, the fracture development inside the coal was more obvious. When λ = 1.2, the number of fractures was higher than that when λ = 1.4, indicating that not the greater the side pressure coefficient, the better the degree of fracture development. In addition, there was a suitable side pressure coefficient interval for fracture development. When λ < 1, the fracture with large λ was more fully developed, further verifying that there was a suitable range of fracture development that allowed the coal rock to develop fractures adequately under horizontal and vertical stresses.

3.4. Validation of Numerical Analysis Results

3.4.1. Verification of Numerical Results with Different Burial Depths

In general, the results of numerical simulations need to be validated [38,39]. The numerical simulation results are examined in combination with existing theories. The transformation equation is as follows, as shown in Equation (1). After the formation of the borehole, the ground stress is redistributed, and the change of the coal stress around the borehole [40,41] is shown in Equation (2). The transformation equation is as follows, as shown in Equation (1):

$$\left.\begin{array}{l}{\sigma }_{x0}=\left({\sigma }_1{\cos }^2\alpha +{\sigma }_2{\sin }^2\alpha \right){\cos }^2\beta +{\sigma }_s{\sin }^2\beta \hfill \\ {\sigma }_{y0}={\sigma }_1{\sin }^2\alpha +{\sigma }_2{\cos }^2\alpha \hfill \\ {\sigma }_{z0}=\left({\sigma }_1{\cos }^2\alpha +{\sigma }_2{\sin }^2\alpha \right){\sin }^2\beta +{\sigma }_s{\cos }^2\beta \hfill \\ {\tau }_{xy0}=\frac{1}{2}\left({\sigma }_2-{\sigma }_1\right)\sin (2\alpha )\cos \beta \hfill \\ {\tau }_{yz0}=\frac{1}{2}\left({\sigma }_2-{\sigma }_1\right)\sin (2\alpha )\sin \beta \hfill \\ {\tau }_{xz0}=\frac{1}{2}\left({\sigma }_1{\cos }^2\alpha +{\sigma }_2{\sin }^2\alpha -{\sigma }_s\right)\sin (2\beta )\hfill \end{array}\right\}$$

(1)

where σ₁ is the maximum horizontal ground stress, MPa; σ₂ is the minimum horizontal ground stress, MPa; σ_s is the overburden pressure, MPa; α is the azimuth of borehole axis projection in the horizontal plane, °; and β is the borehole inclination angle, °.

$$\left.\begin{array}{l}{\sigma }_{\theta 1}={\sigma }_{x0}+{\sigma }_{y0}-2\left({\sigma }_{x0}-{\sigma }_{y0}\right)\cos (2\theta )-4{\tau }_{xy0}\sin (2\theta )\\ {\sigma }_{z1}={\sigma }_{z0}-v\left[2\left({\sigma }_{x0}-{\sigma }_{y0}\right)\cos (2\theta )+4{\tau }_{xy0}\sin (2\theta )\right]\\ {\tau }_{\theta z1}=2\left(-{\tau }_{xz0}\sin \theta +{\tau }_{yz0}\cos \theta \right)\\ {\tau }_{r\theta 1}={\tau }_{rz1}=0\end{array}\right\}$$

(2)

σ_r1 is radial effective stress, MPa; σ_θ1 is circumferential effective stress, MPa; σ_z1 is vertical effective stress, MPa; τ_θz1, τ_rθ1, τ_rz1 are tangential shear stress, MPa; v is Poisson’s ratio; and θ is the azimuth of a point in the circumference of the borehole measured clockwise from the x-axis of the borehole coordinate system.

The stress distribution of the coal around the borehole after the formation of the borehole can be obtained by substituting Equation (1) into Equation (2), as shown in Equation (3).

$$\left.\begin{array}{c}\begin{array}{l}{\sigma }_{\theta 1}={\sigma }_1\left(2\sin (2\alpha )\cos \beta \sin (2\theta )+\left(1-2{\cos }^2\alpha \right){\cos }^2\beta \cos (2\theta )+2{\sin }^2\alpha (1+\cos (2\theta )\right)\\ +{\sigma }_2\left({\sin }^2\alpha {\cos }^2\beta +{\cos }^2\alpha +\left({\sin }^2\alpha {\cos }^2\beta +{\cos }^2\alpha \right)2\cos (2\theta )-2\sin (2\alpha )\cos \beta \sin (2\theta )\right)+{\sigma }_S\left({\sin }^2\beta 2\cos (2\theta )+{\sin }^2\beta \right)\end{array}\\ \begin{array}{l}{\sigma }_{z1}={\sigma }_1\left({\sin }^2\beta {\cos }^2\alpha -2v{\cos }^2\alpha {\cos }^2\beta \cos (2\theta )-2v{\sin }^2\alpha \cos (2\theta )-2v\sin (2\alpha )\cos \beta \sin (2\theta )\right)\\ +{\sigma }_2\left({\sin }^2\beta {\sin }^2\alpha +2v{\sin }^2\alpha {\cos }^2\beta \cos (2\theta )+2v{\cos }^2\alpha \cos (2\theta )+2v\sin (2\alpha )\cos \beta \sin (2\theta )\right)+{\sigma }_S\left({\cos }^2\beta +2v{\sin }^2\beta \cos (2\theta )\right)\end{array}\\ \left.{\tau }_{\theta z1}={\sigma }_2\left(\sin (2\alpha )\sin \beta \cos \theta -{\sin }^2\alpha \sin (2\beta )\sin \theta \right)-{\sigma }_1\left(\sin (2\alpha )\sin \beta \cos \theta +{\cos }^2\alpha \sin (2\beta )\sin \theta \right)+{\sigma }_5\sin (2\beta )\sin \theta \right)\\ {\tau }_{r\theta 1}={\tau }_{rz1}=0\end{array}\right\}$$

(3)

According to the analysis of the stress state of the borehole and engineering [42,43], the following two stress states of the borehole are more common: σ_z1 ≥ σ_θ1 ≥ σ_r1 and σ_θ1 ≥ σ_z1 ≥ σ_r1.

From the plane strain assumption, we can obtain the change of the main stress in each direction and transform the equation of the change of three stress directions in the coal around the borehole by using Equation (3), as shown in Equation (4).

$$\left.\begin{array}{l}{\sigma }_1=\frac{1}{2}\left({\sigma }_{\theta 1}+{\sigma }_{z1}-2P_0\right)+\sqrt{{\tau }_{\theta z1}+\frac{1}{4}{\left({\sigma }_{\theta 1}-{\sigma }_{z1}\right)}^2}\\ {\sigma }_3={\sigma }_{r1}\\ {\sigma }_2=\frac{1}{2}\left({\sigma }_{\theta 1}+{\sigma }_{z1}-2P_0\right)-\sqrt{{\tau }_{\theta z1}+\frac{1}{4}{\left({\sigma }_{\theta 1}-{\sigma }_{z1}\right)}^2}\end{array}\right\}$$

(4)

Equation (5) shows the calculation of the parameters.

$$\left.\begin{array}{l}{\sigma }_{\theta 1}={\sigma }_{x0}+{\sigma }_{y0}-2\left({\sigma }_{x0}-{\sigma }_{y0}\right)\cos (2\theta )-4{\tau }_{xy0}\sin (2\theta )\hfill \\ {\sigma }_{z1}={\sigma }_{z0}-v\left[2\left({\sigma }_{x0}-{\sigma }_{y0}\right)\cos (2\theta )+4{\tau }_{xy0}\sin (2\theta )\right]\hfill \\ {\sigma }_{r1}=P_0\hfill \\ {\tau }_{\theta z1}=2\left(-{\tau }_{xz0}\sin (2\theta )+{\tau }_{yz0}\cos \theta \right)\hfill \end{array}\right\}$$

(5)

Combining Equation (5), it can be obtained that σ_θ1, σ_z1, σ_r1, τ_θz1 in which σ_r1 is a constant, τ_θz1 is smaller in magnitude than σ_θ1, σ_z1, and σ_θ1, σ_z1 is the main control factor of σ₁, σ₂, where σ_θ1, σ_z1 is a function of θ, so the orientation of the point of maximum and minimum stress value can be obtained as the following Equation (6):

$${\theta }_1=\frac{1}{2}\arctan \left(\frac{2{\tau }_{xy0}}{{\sigma }_{x0}-{\sigma }_{y0}}\right);\hspace{1em}{\theta }_2={\theta }_1+\pi /2$$

(6)

Combining Equation (6), the relationship between θ and the burial depth can be obtained, and the value of θ decreases with the increase in burial depth, which is consistent with the numerical simulation results (Figure 13). It indicates that with the increase in burial depth, the instability mode of coal around the borehole gradually changes from oblique to large deformation instability of the borehole wall to circumferential to large deformation instability of the borehole wall.

To justify the reasonableness of the numerical simulation results, the linear elastic infinitely infinite plate inner circle model (Figure 14a) is introduced to obtain the fracture distribution law around the borehole with different burial depths [44,45]. The stress distribution around the borehole can be expressed as Equation (7).

$$\left\{\begin{array}{l}{\sigma }_r=\frac{kP}{2}\left(1-\frac{a^2}{r^2}\right)-\frac{kP}{2}\left(1-4\frac{a^2}{r^2}+3\frac{a^4}{r^4}\right)\cos 2\theta \\ {\sigma }_{\theta }=\frac{kP}{2}\left(1+\frac{a^2}{r^2}\right)+\frac{kP}{2}\left(1+3\frac{a^4}{r^4}\right)\cos 2\theta \\ {\tau }_{r\theta }=\frac{kP}{2}\left(1+2\frac{a^2}{r^2}-3\frac{a^4}{r^4}\right)\sin 2\theta \end{array}\right.$$

(7)

σ_r is residual stress, MPa; σ_θ is the tangential stress of the rock, MPa; τ_rθ is the shear stress of the rock, MPa; P is in situ stress, MPa; k is the parameter characterizing different depths; a is the radius of the circle, m.

The formation of fractures should satisfy the normal stress or tangential strength over the normal strength of interparticle bonding σ_cn and shear stress over τ_cs, which can be obtained as the distribution range of fractures around the borehole, as shown in Equation (8).

$$\left\{{\sigma }_r>{\sigma }_{cn}\right\}\cup \left\{{\sigma }_{\theta }>{\sigma }_{cn}\right\}\cup \left\{{\tau }_{r\theta }>{\tau }_{cs}\right\}$$

(8)

The distribution range of fractures around the borehole is analyzed with σ_θ > σ_cn as an example. It is assumed that σ_θ is equal to σ_cn at the boundary of the damage zone. Therefore, the following Equation (9) can be deduced:

$$\frac{kP}{2}\left(1+\frac{a^2}{r^2}\right)+\frac{kP}{2}\left(1+3\frac{a^4}{r^4}\right)\cos 2\theta ={\sigma }_{cn}$$

(9)

According to Equation (9), the damaged area around the borehole with increasing burial depths can be obtained with the side pressure coefficient λ = 1.0, and its schematic diagram is shown in Figure 14b. According to the results of theoretical analysis, it can be seen that under λ = 1.0, the fracture mainly shows circular distribution under different burial depths, and the fracture circle keeps increasing with the increase in burial depth (Figure 15).

3.4.2. Verification of Numerical Results with Different Side Pressure Coefficients

To demonstrate the reasonableness of the numerical results under different side pressure coefficients, the linear elastic infinite plate inner circle model (Figure 16a) is introduced to obtain the fracture distribution law around the borehole under different side pressure coefficients. The stress distribution around the borehole can be expressed as Equation (10).

$$\left\{\begin{array}{l}{\sigma }_r=\frac{P}{2}(1+\lambda )\left(1-\frac{a^2}{r^2}\right)-\frac{P}{2}(1-\lambda )\left(1-4\frac{a^2}{r^2}+3\frac{a^4}{r^4}\right)\cos 2\theta \\ {\sigma }_{\theta }=\frac{P}{2}(1+\lambda )\left(1+\frac{a^2}{r^2}\right)+\frac{P}{2}(1-\lambda )\left(1+3\frac{a^4}{r^4}\right)\cos 2\theta \\ {\tau }_{r\theta }=\frac{P}{2}(1-\lambda )\left(1+2\frac{a^2}{r^2}-3\frac{a^4}{r^4}\right)\sin 2\theta \end{array}\right.$$

(10)

where σ_r is residual stress, MPa; σ_θ is the tangential stress of rock, MPa; τ_rθ is the shear stress of rock, MPa; P is in situ stress, MPa; a is the radius of circle, m; λ is the side pressure coefficient.

The formation of fractures should satisfy the normal stress or tangential strength over the normal strength of interparticle bonding σ_cn and shear stress over τ_cs, which can be obtained as the distribution range of fractures around the borehole, as shown in Equation (11).

$$\left\{{\sigma }_r>{\sigma }_{cn}\right\}\cup \left\{{\sigma }_{\theta }>{\sigma }_{cn}\right\}\cup \left\{{\tau }_{r\theta }>{\tau }_{cs}\right\}$$

(11)

By analyzing the distribution range of fractures around the σ_θ > σ_cn borehole and assuming that σ_θ is equal to σ_cn at the boundary of the damaged area, Equation (12) can be derived.

$$\frac{P}{2}(1+\lambda )\left(1+\frac{a^2}{r^2}\right)+\frac{P}{2}(1-\lambda )\left(1+3\frac{a^4}{r^4}\right)\cos 2\theta ={\sigma }_{cn}$$

(12)

When θ = 0, σ_θ = (3 − λ)P, θ = π/2, σ_θ = (3λ − 1)P.

The damaged area around the borehole with different side pressure coefficients is schematically shown in Figure 16b.

Combined with Figure 16b and Figure 17, when λ > 1.0, the stress in the horizontal direction is larger than that in the vertical direction. From the theoretical derivation, the fracture range in the horizontal direction is larger than that in the vertical direction, which can be approximated as an ellipse with a long axis in the horizontal direction. When λ < 1.0, the stress in the horizontal direction is smaller than that in the vertical direction. From the theoretical derivation, the distribution range of fractures in the horizontal direction is smaller than that in the vertical direction. It indicates that the fractures and the long axis are approximated as an ellipse in the vertical direction.

The numerical calculation results are basically consistent with the theoretical results, so the experimental results are reasonable. When λ < 1.0, the fractures generally expand in a vertical direction but also expand in a horizontal direction. The damaged area can be regarded as an ellipse with a long axis in the vertical direction. When λ > 1.0, the fracture is mainly concentrated in the horizontal direction, and the fracture area can be regarded as a long-axis ellipse in the horizontal direction.

In this paper, numerical simulations of boreholes under different stress states were carried out to explore the evolution of the stress-displacement-fracture three-field, and theoretical verification was carried out. However, it would be better if samples or actual field observations are available. Therefore, we will carry out mechanical experiments with hole-containing specimens and borehole field tests in the future, and combine the two with numerical simulations and theoretical analyses in order to obtain better results for the study of gas extraction boreholes.

4. Conclusions

In this study, the evolution of stress, deformation, and fracture fields of coal around the borehole were studied by using numerical simulations, and the effects of the burial depth and side pressure coefficient on the multi-field evolution characteristics of coal around the borehole were analyzed. Our main conclusions are as follows:

(1)

The evolution characteristics of the stress field around the borehole were clarified. When the borehole was just formed, the stress level was low and the force chain was uniformly distributed. After the adjustment of the stress field of coal around the borehole, the force chains of the borehole were unevenly distributed and the stress concentration phenomenon appeared. With the increase in burial depth, the stress around the borehole gradually increased, while the range of the stress concentration zone in the borehole kept increasing, the borehole changed from unilateral instability to instability on both sides, and the stability of the borehole decreased with the increase in the burial depth.

(2)

The evolution characteristics of the displacement field around the borehole were clarified. The deformation of coal around the borehole had a certain directionality, and the deformation of coal would be larger along the straight direction, and smaller deformation would occur in the vertical direction. With the increase in burial depth, the deformation of coal around the borehole increased; with the increase of side pressure coefficient, the stress of coal around the borehole increased gradually in both vertical and horizontal directions, and the vertical displacement and horizontal displacement also increased gradually. When λ < 1.0, the horizontal displacement was smaller than the vertical displacement; while when λ > 1.0, the vertical displacement was smaller than the horizontal displacement.

(3)

The evolution characteristics of the fracture field around the borehole were clarified. There was a certain correspondence between the development of fractures and the deformation around coal. When the burial depth was small, the fracture development was not obvious, the fracture expansion rate was low, and the fracture field as a whole had a circular distribution. With the increase in burial depth, the fracture development was gradually obvious, and the distribution characteristics were concentrated in the middle and dispersed around. When λ < 1.0, the fracture mainly expanded along the horizontal direction, while when λ > 1.0, the fracture was mainly concentrated in the vertical direction.