Damage characteristics of gas extraction boreholes under different crustal stresses and their effects on gas extraction

To investigate the effects of different crustal stresses on the stability of gas extraction boreholes, several equations were deduced. Additionally, the coupled flow‐solid model of gas extraction was constructed, given the elasticity−plasticity deformation of the coal body. Using the numerical simulation and analysis software of Comsol, a study was conducted on the deformation and damage of gas extraction boreholes under different crustal stresses. The results are as follows: with vertical stress increasing, the concentration area of shear stress, deformation amount, and the area of the plastic zone around the hole increase. Varied lateral pressure coefficients led to differing distribution positions of shear stress concentration areas and plastic zones around the hole. An increase in lateral pressure coefficient shifted the distribution positions of shear stress maximum. In addition, plastic zone, and permeability ratio maximum change from the left and right sides of the borehole to the upper and lower sides, and the larger the lateral pressure coefficient is, the worse the stability of the upper and lower sides of the borehole is. Higher lateral pressure coefficients inhibited extraction, with the effect becoming more pronounced over time. Elevated vertical stress and lateral pressure coefficients hindered negative pressure propagation in the borehole.


| INTRODUCTION
China is the country with the largest coal consumption in the world and coal usage will still continue. 1In recent years, the depth of coal mining in China has become deeper and deeper, and the geological conditions have become more complex.At the same time, the conditions of crustal stress on the coal seams have also become more and more complicated, the initial gas pressure in the coal seams has become higher, and consequently, the amount of gas outflow has risen resulting in the increased possibility of hazardous accidents. 2With further depth of excavation in the mine, the conditions of crustal stress, mining, and coalbed methane (CBM) storage in coal seams become more complicated, which makes it more difficult to extract the gas.Especially, the increase of horizontal crustal stress causes the lateral pressure coefficient of the coal seam to become larger and larger, and the gas extraction boreholes are affected by the complex crustal stress, which leads to the instability and deformation of the boreholes or even the collapse of the holes. 3][9] Many scholars who are at home and abroad have carried out a great deal of research on the instability and deformation of boreholes as well as the gas flow law.Zou et al. 10 have effectively improved the stability of coal rock boreholes by utilizing surfactant compounding compositions.Hao et al. 11 found that the initial crustal stress, mining stress, and gas pressure of the coal seam are the key factors affecting the stability of the borehole through theoretical and numerical simulation methods, and proposed the mechanism of borehole zonal destabilization.Liu 12 investigated the influence of coal seam burial depth, lateral pressure coefficient, and pore pressure on the destabilization damage of extraction drill holes.They classified the stability of drill holes into three grades and proposed prevention and control technologies for extraction drill instability.Zhang et al. 13 started from the drilling mechanics model and used numerical simulation to conclude that the drilling is mainly dominated by shear damage, while the stress conditions of the coal seam, gas pressure, and construction technology all have an impact on the stability of the drilling.Based on laboratory tests and theoretical analysis of variable mass flow pressure drop, Zhang et al. 14 established a coupled mathematical model.This model takes into account the deformation instability of the borehole, gas transport from the coal seam, and the dynamic attenuation of the negative pressure along the length of the borehole.They also numerically analyzed the mechanism of the impact of the instability and collapse of extraction boreholes on gas extraction.Additionally, they put forward the detection technology for the instability and collapse characteristics of the extraction boreholes based on the test of the distribution of the negative pressure of extraction.Han et al. 15 established a mechanical model of borehole instability considering the H-B criterion and GSI number, obtained the distribution law of the plastic zone around the borehole under different lateral pressure coefficients, and classified the type of borehole instability damage.Li et al. and Zhang et al. [16][17][18] comprehensively considered the effects of effective stress, temperature, moisture, and other factors, and established a coal rock permeability model based on this.Li et al. 19 used a combination of experimental and numerical simulation to study the characteristic law of well wall deformation caused by the temperature change of surrounding rock in deep CBM wells.The numerical simulation results show that the high temperature of the surrounding rock of deep CBM wells leads to a large stress concentration range and deformation of the coal bed borehole.Zhang et al. 20 set up a 3D numerical model to simulate the deep coal seam and concluded that the crustal stresses and lateral pressure coefficients are the main controlling factors that lead to the instability and deformation of boreholes.Zhang et al. 21investigated the impact of three deformation destabilization modes on gas drainage in boreholes and analyzed the mechanism of soft coal seam destabilization.Three modes were identified, including full boreholes, collapsed boreholes, and plugged boreholes.Additionally, a solid-liquid coupling model was established to account for the dynamic pressure changes during pumping.Zhang et al. 22 and Yin et al. 23 established the flow-solid coupling equation affected by mining stress and revealed the coupling mechanism of mining stress on gas flow through field experiments and simulation experiments after considering the effect of mining stress on the permeability of the coal body.Considering the kinetic diffusion of gas in the matrix, Zhang et al. 24 proposed an effective stress-permeability model for reservoirs based on the stress-strain ontological relationship and the cubic relationship of permeability-porosity.Based on the principle of effective stress, He et al. 25 applied the ABAQUS software to carry out numerical simulation under the conditions of different compression strength, crustal stress, borehole diameter, gas pressure, and other factors and concluded that the displacement and deformation of the borehole wall will change with the vertical stress peak and position with the change of crustal stress, compressive strength of the coal body, and gas pressure of the coal seam.
There are many damage criteria that can be introduced in the study of the damage behavior of coal-rock materials. 26,27In this article, the DP criterion is mainly used to describe the damage behavior as it takes into account the effect of hydrostatic pressure and is able to better explain the yielding or damage phenomena that may also occur in rocks under hydrostatic pressure conditions.Moreover, the criterion can efficiently solve problems with optimal substructure properties, avoiding repeated calculations and reducing time complexity.In terms of using the DP criterion to model well wall stresses, Liu et al. 28 performed a semianalytical solution and finite element numerical simulation of the tunnel excavation problem under drainage conditions using the extended Drucker-Prager model.They concluded that this problem can be reduced to solving a system of ordinary differential equations, with the radial, tangential, and vertical stresses, as well as the specific volume being the basic unknowns.Rahimi et al. 29 proposed and evaluated 13 damage criteria for predicting shear failure in boreholes.And the results predicted by each damage criterion compared to actual field damage cases are derived.
Most of the above scholars' research on the fluidstructure interaction model was based on the linear theory of elasticity, and there was less research on the deformation equations of elastic-plastic coal body based on the dual pore medium model.Moreover, gas extraction borehole is affected by complex crustal stress conditions, and most of the previous studies focused on the fluid-structure interaction law of gas extraction under a single crustal stress condition, while there were fewer studies on the flow law of gas extraction with different combinations of vertical crustal stresses and lateral pressure coefficients.Therefore, this paper intends to use theoretical analysis to establish a fluid-structure interaction model of gas extraction in boreholes considering the elastic-plastic deformation of coal rock body from the dual pore medium model and use numerical simulation to investigate the destabilizing deformation of the boreholes and the flow law of the gas under the conditions of different vertical crustal stresses and lateral pressure coefficients.

| Basic assumption
The flow of gas in a coal seam is a rather complex process.To facilitate the study, certain assumptions about the model have to be made.The specific assumptions are as follows 30 : 1.As shown in Figure 1, the coal body consists of the pore system and the fissure system is a double pore medium model.2. The gas transport in the coal follows the tandem mode of "desorption-diffusion-permeation," and the gas in the pore system and the fissure system is suitable for Fick's first law of diffusion and Darcy's law, respectively.
3. The gas in the coal bed is regarded as an ideal gas.4. The flow process of gas is an isothermal diffusion process. 5.The coal bed does not contain moisture and is regarded as a dry solid.6.The coal body is regarded as an isotropous elastic-plastic medium.7. The adsorption and desorption of gas in the coal bed follow the Langmuir Isotherm Adsorption Equation.
In the process of performing numerical simulation, the gas matrix diffusion equation in the article is determined by considering the gas as an ideal gas, to determine the calculation formula of the free gas.In this paper, the influence of vertical ground stress and lateral pressure coefficient on the borehole is mainly studied, and the temperature and moisture may affect the structure of the coal body, thus affecting the deformation of the borehole and the diffusion of gas.However, the influence of temperature and humidity on the structure of coal and gas flow is not the focus of this study.Therefore, we keep the temperature of coal constant and treat it as a dry solid.The assumption of isotropy can reduce the calculation amount of related materials and improve the calculation efficiency.The anisotropy of the coal body makes the mechanical properties of the coal body different in each direction, which makes the study of the effect of stress on the borehole more complicated, and the vertical stress and the horizontal stress cannot keep a certain direction for the transmission of force.To facilitate the calculation, the adsorbed gas in the pores is calculated according to the formula of Langmuir adsorption, and the free gas in the fissures follows the modeling through Darcy's law.

| Control equations for deformation of elastic-plastic coal bodies
The effective stress law for the coal and rock mass of the dual pore medium model is 31 : Through improving this model and introducing a coal matrix shrinkage distortion term, the effective stress law for the coal body of the dual pore medium model considering coal matrix shrinkage distortion can be obtained: GUANG and HONGBAO where, σ′ ij is effective stress; σ ij is the coal ultimate adsorbate deformation; ԑ L is the Kronecker notation; δ ij is the effective stress coefficient in fracture; β f = (1-K)/K m is the effective stress coefficient in the matrix; ) is the coal matrix bulk modulus, MPa; K s is the bulk modulus of coal skeleton, MPa; E is Young's modulus of the coal, MPa; E m is Young's modulus of the coal matrix, MPa; ν is Poisson's ratio of the coal.The coal body geometric equation is: where ԑ ij is stress tensor; u is displacement, μ i j , is displacement tensor; the first subscript is the component of the displacement u in the i direction and the second subscript is the derivative of the displacement component u i in the j direction.
The incremental form of the elastic-plastic constitutive equation for a coal rock body is given by: ( ) where, λ = νE ν ν The Drucker−Prager yield criterion was chosen to determine the plastic deformation rate in the elastoplastic constitutive equation.This criterion can be expressed as: where 2 .I′ 1 is the first invariant tensor of principal stress, J′ 2 is the effective deviator stress tensor.

| Gas diffusion equation in the coal matrix
The diffusion of gas in the coal matrix is the process of the gas flowing from the pore system to the fissure system, and the gas flows from the region of higher concentration to the region of lower concentration, which conforms to Fick's first law.So the flux expression of mass exchange between the pore and fissure systems is: where, Q s is the mass exchange rate for the gas between the pore and fissure systems, kg/(m 3 •s); D is the gas diffusion coefficient in coal seam, m 3 /s; σ s = 3π 2 /L 2 is form factor, m −2 ; L is fissure spacing, m; c m is the gas concentration in the coal matrix, kg/m 3 ; c f is the gas concentration in fissure, kg/m 3 .It can be seen from the ideal gas law: where M c is the molar mass of methane, kg/mol; R is the ideal gas constant, J/(mol•K); T is the temperature of the coal reservoir, K; P m is the gas pressure in the coal matrix, Pa; P f is the gas pressure in the fissure, Pa.
In the pore, adsorbed gas is the main form of gas storage in the coal seam and it conforms to the Langmuir adsorption equilibrium equation of state, thus: In the pore, the free gas mass is represented by the ideal gas equation of state: m m c m 2 (10)   Combined with the Langmuir adsorption equilibrium equation, the total mass of gas per unit volume of coal can be expressed as: where V L is the Langmuir's volume constant, m 3 /t; P L is the Langmuir's pressure constant, Pa; ρ a is the true density of the coal, kg/m 3 ; ρ s is the density of the gas at the standard condition, kg/m 3 ; ϕ m is the porosity of the coal matrix, %.The gas diffusion equation in the coal matrix can be obtained by combining the above equation: where V M is the Molar volume of methane at standard conditions, m 3 /mol.

| Modeling of porosity permeability dynamics
In the case of gas-containing coal rock mining influence, the porosity and permeability are dynamically changed, and the change rule varies in the elastic and plastic stages.According to the previous research results and appropriate assumptions, the expression of the dynamic change of porosity and permeability was given.
When the gas-containing coal rock is in the elastic stage, the porosity φ e is defined as: where φ e is the porosity of the coal rock in the elastic zone; φ 0 is the initial porosity of the coal rock; ԑ v is volumetric strain; p is the gas pressure/MPa; p p p Δ = − 0 is the gas pressure change/MPa where p 0 is the initial gas pressure/MPa.When the coal rock body is in the plastic deformation stage, its porosity can be expressed as: where φ p is the porosity of the coal rock body in the plastic region; φ m is the porosity of the coal rock body when it is subjected to peak stress σ c ; σ i is the magnitude of the stress intensity in all directions of the coal rock body; and σ s is the yield stress, Therefore, the porosity of the coal rock body can be obtained as: According to other studies on the permeability model, combined with the cube law of permeability and porosity, 32 the permeability change equation in the elastic zone of coal rock body can finally be obtained as: GUANG and HONGBAO where k e is the permeability of the coal rock body in the elastic region; k 0 is the initial permeability of the coal seam.
Similarly, the equation of permeability change in the plastic zone of the coal rock body can be obtained: Therefore, the permeability change equation for the coal rock body can be obtained as:

| Equation for gas seepage in a fissure
The seepage process of free gas in the coal body satisfies the ideal gas law: where, ρ g is the density of the gas, kg/m 3 ; M g is the molar mass of the gas, kg/mol; R is the ideal gas constant, kg/(mol•K); T is the absolute temperature, K.The flow of free-state gas in a coal seam satisfies Darcy's law, so the gas seepage rate is: where m is the Klingberg coefficient, MPa; grad p is the gas pressure gradient; μ is the Kinetic viscosity coefficient of methane, Pa•s.
The flow of the gas in the coal seam satisfies the continuity equation: Combining the gas content equation, ideal gas state equation, Darcy's law, and continuity equation, the seepage equation of the gas in the fissure can be obtained as 33 : ( )

| DEFORMATION AND DAMAGE CHARACTERISTICS OF GAS EXTRACTION DRILLING UNDER DIFFERENT GROUND STRESS CONDITIONS
In this paper, the fluid-structure coupling model was imported into the PDE module of Comsol Multiphysics.
The influence that different vertical stress and lateral pressure coefficients have on the law of the gas flow around the drilling was investigated.

| Geometric model and boundary conditions
1. Geometric model and meshes.As shown in Figure 2, a rectangular coal seam with a length of 80 m, a width of 40 m, and a height of 5 m was established, and the drill hole was located in the center of the seam, with a diameter of 120 mm and a depth of 60 m.The division of the mesh will have a corresponding impact on the results.The denser the mesh, the greater the computational workload and the longer the computation period.However, our computational resources are always limited.Furthermore, as the grid becomes denser, the rounding errors caused by computer floating-point operations also increase.Therefore, in practical applications, users always seek a relatively appropriate balance between computational accuracy and computational cost, and this balance point is where the gridindependent value is achieved.Therefore, after mesh independence verification, the model was divided into 64,046 meshes using a very fine grid controlled by the physical field, and the mesh was encrypted around the drill hole to meet the requirements of computational accuracy.

Mesh-independent verification.
In order that the adopted grid cell size does not significantly affect the simulation results, gridindependent validation is performed in this article.As shown in Figure 3, before the grid number reaches 64,046, the images corresponding to different grid numbers are still differentiated, but after the grid number reaches 64,046, the images basically do not change as the grid number continues to increase.Since the computation speed becomes slower as the grid number increases, the grid number is set to 64,046 in this paper.

Boundary conditions and simulation parameters.
Seepage boundary conditions: the gas pressure of the initial coal seam was 1.2 MPa and the negative pressure for extraction was 20 KPa.The contact surface between the coal seam and the atmosphere experienced atmospheric pressure, and the model boundary conditions were set to be static.Stress boundary conditions: the bottom of the model was fixed constraint, the top of the model suffered vertical crustal stress from different depths, and both sides of the model were subjected to horizontal crustal stresses with corresponding lateral pressure coefficients.

| Deformation and damage characteristics of boreholes under different ground stresses
When the depth of the coal seam is different, the vertical crustal stress is different.When the horizontal crustal stress condition is different, the lateral pressure coefficient of the coal seam is also different.By applying different vertical stresses (5.0, 7.5, 10.0, 12.5, 15.0 MPa) and keeping the coefficient of lateral pressure as 1, and by applying different coefficients of lateral pressure (0.5, 1.0, 1.5, 2.0, 2.5) and keeping the vertical stress as 10.0 MPa, the distribution patterns of shear stress and plasticity zones of the coal seam around the borehole can be investigated under different vertical crustal stresses and lateral pressure coefficient conditions (Table 1).

| Deformation and damage characteristics of boreholes under different vertical ground stresses
The cloud diagram of shear stress distribution under different vertical stresses is shown in Figure 4.The coal seam is in compression at the bottom of the ground.At the extraction borehole, the shear stress first increases rapidly to the peak stress, and then gradually decreases and stabilizes, generating a uniformly distributed circular stress concentration area around the borehole.The peak shear stresses around the hole under different vertical stresses are shown in Table 2.When the lateral pressure coefficient is unchanged and the vertical crustal stress increases, the peak shear stress around the hole increases, the distribution area of the shear stress concentration area increases, and the stability of the hole becomes worse.
Plastic zone distributions of different vertical crustal stresses are shown in Figure 5.As we can see, the black circle is the relative position before borehole destabilization and deformation.When the coal seam is in compression, different degrees of deformation will occur at various locations in the coal seam, and the perimeter of the drill hole is affected by large shear stresses, resulting in irreversible plastic deformation.
The plastic zone area of different vertical crustal stresses is shown in Table 3.When the lateral pressure coefficient of the coal seam remains unchanged and the vertical crustal stress increases, the overall inward contraction of the borehole increases.As the borehole diameter gradually decreases, the area of the plastic zone around the borehole and the degree of deformation gradually increases, leading to worse stability of the borehole and a weaker ability to resist deformation and damage.

| Deformation and failure characteristics of boreholes under different lateral pressure coefficients
The distribution of shear stress under different lateral pressure coefficients is shown in Figure 6.When the vertical crustal stress of the coal seam is unchanged and the lateral pressure coefficient changes, the distribution of shear stress around the borehole will also change accordingly.When the lateral pressure coefficient is 0.5, the shear stress in the coal rock body above and below the borehole increases gradually with distance from the borehole.However, in the coal rock body on the left and right sides of the borehole, the shear stress increases rapidly, reaching a peak value of 7.8 MPa, before gradually decreasing and stabilizing.The concentration area of shear stress takes the shape of a fan on the left and right sides of the borehole.On the other hand, when the lateral pressure coefficient is 1.0, the shear stress around the hole increases rapidly to the peak value of 8.6 MPa.Subsequently, it gradually decreases and stabilizes, with the shear stress concentration area uniformly distributed in a circular shape around the hole.When the lateral pressure coefficient exceeds 1, the shear stress in the coal rock body above and below the borehole exhibits a rapid increase to a peak value as the distance from the borehole increases.The peak values of shear stress corresponding to lateral pressure coefficients of 1.5, 2.0, and 2.5 are 11.0, 13.8, and 16.8 MPa, respectively.After reaching the peak, the shear stress gradually decreases and tends to stabilize.In contrast, the The cloud diagram of plastic zone distribution under different lateral pressure coefficients is shown in Figure 7, and the plastic zone area under different lateral pressure coefficients is shown in Table 4.When the vertical stress is unchanged and the lateral pressure coefficient changes, the distribution pattern of the plastic zone also changes.When the lateral pressure coefficient is less than 1, the plastic zone is distributed in the shape of a fan on the left and right sides of the borehole, and the left and right sides of the borehole are more prone to deformation damage.When the lateral pressure coefficient is equal to 1, the plastic zone is uniformly distributed in the circumference of the borehole in a round shape, and the area of the plastic zone is further increased compared with that of the lateral pressure coefficient of 0.5.When the lateral pressure coefficient exceeds 1, the plastic zone is distributed in an elliptical and butterfly-wing shape on the top and bottom of the borehole, making the top and bottom more susceptible to deformation and damage as the lateral pressure coefficient changes.With the increase of the lateral pressure coefficient, the area of the plastic zone becomes larger.When the vertical stress remains unchanged and the lateral pressure coefficient increases, the ability of the top and bottom sides of the borehole to resist destabilization decreases.
Above all, vertical crustal stress and lateral pressure coefficient can largely affect the stability of the borehole.In particular, the borehole's stability becomes worse under the high vertical crustal stress conditions.The stability of the upper and lower sides of the borehole deteriorates under high lateral pressure coefficient conditions.

| THE INFLUENCE OF GROUND STRESS ON GAS FLOW IN BOREHOLES 4.1 | The influence of vertical ground stress on gas flow in boreholes
When coal seams are at different depths, they are subjected to different vertical crustal stresses, which leads to a change in the law of gas flow during borehole extraction.By applying vertical crustal stresses of 5.0, 7.5, 10.0, 12.5, and 15.0 MPa to the model respectively, keeping the lateral pressure coefficient as 1, the influence laws of different vertical crustal stress conditions on the permeability of coal seams, gas pressure, gas flow rate, and effective extraction radius can be investigated.

| The evolution law of porosity and permeability under different vertical ground stresses
The porosity and permeability ratios under different vertical crustal stresses are shown in Figure 8, and the specific data are shown in Table 5.Because the lateral pressure coefficient is 1, and the shear stress concentration zone and plastic zone are uniformly distributed in a circle around the hole with high symmetry, the data from 180 days of pumping on the horizontal line running through the center of the borehole on the right side of the model were selected for analysis.The distribution of porosity and permeability ratio is essentially identical, and the ratio of porosity and permeability is the largest at the extraction borehole, then decreases gradually from the borehole outwardly and finally tends to stabilize.In the plastic area around the hole, when the lateral pressure coefficient is constant and the vertical stress increases, the degree and range of deformation in the plastic area increase.The number of secondary pores and cracks also increases, leading to an increase in porosity and permeability ratio with the increase of vertical stress.Meanwhile, in the elasticity area far away from the extraction borehole, when the lateral pressure coefficient is constant and the vertical stress increases, the pore and the cracks in the seam are compressed to a greater extent.Consequently, gas diffusion and seepage channels decrease, and the porosity and permeability ratio of the seam decreases, with a decrease of the pore and permeability ratio.The ratio of porosity and permeability of the coal seam decreases with an increase in vertical crustal stress.degree of deformation and the distribution range of the plastic zone around the extraction boreholes increase.This leads to an increase in the permeability of the plastic zone, and the law that the gas pressure is lower under high vertical crustal stresses is presented only in the plastic zone.In the elastic area outside the plastic zone, the permeability of the coal seam under high vertical crustal stress is lower, which leads to more difficult gas extraction and shows the law of higher gas pressure under high vertical crustal stress.The three sets of curves from top to bottom in Figure 10 are the gas pressure curves at 1, 90, and 180 days of extraction, and the overall gas pressure curve under high vertical crustal stress is located above the gas pressure curve under low vertical crustal stress.Because the area of the plastic zone is very different from the area of the elastic zone, the gas pressure in the plastic zone has less influence on the overall gas pressure of the coal seam.In the overall scope of the coal seam, with the increase of the vertical crustal stress, the gas pressure of the coal seam decreases and the effect of gas extraction deteriorates, thus reducing the extraction efficiency.From 1 to 90 to 180 days, it can be seen that with the extension of the extraction time, the spacing of the gas pressure curves under each vertical crustal stress condition increases, and the difference between the gas pressures increases.Therefore, this phenomenon shows that, with the extension of the extraction time, the disadvantage of gas extraction under the high vertical crustal stress is more significant.
The gas flow rate at each monitoring point under different vertical stresses is shown in Figure 11 and the arrangement diagrams of different monitoring points are shown in Figure 12.The coordinates of monitoring point 1 are (0.11, 0, 0) with a distance of 5 cm from the borehole, which is located in the plastic zone around the borehole, and the coordinates of monitoring point 2 are (1.06,0, 0) with a distance of 1 m from the borehole, which is located in the elastic zone.The coordinate of monitoring point 3 is (0, 0, 0.11) 5 cm from the borehole, located in the plastic zone on the upper side of the borehole, and the coordinate of monitoring point 4 is (0, 0, 1.06) 1 m from the borehole, located in the elastic zone on the upper side of the borehole.The maximum value of the gas flow rate at monitoring point 1 appeared in the early stage of extraction, and with the extension of extraction time, the gas flow rate gradually decreased.The maximum value of the gas flow rate at monitoring point 2 appeared in the middle stage of extraction, and with the extension of extraction time, the gas flow rate first increased to the maximum value and then gradually decreased.The size of the gas flow rate is the slope of the gas pressure curve.When the extraction time is the same, the closer the coal body is to the drill hole, the larger the gas flow rate is, and the less time it takes to reach the maximum value of the gas flow rate.This is because gas extraction is a gradual transfer process.Initially, during extraction, the gas in the coal rock body around the borehole is the first to be extracted.Subsequently, a pressure difference between this area and the gas in the peripheral coal rock body is generated.As a result, the gas in the peripheral coal rock body begins to be transported to the borehole.The further the distance from the borehole, the later the gas in the coal rock body  is extracted.The farther away from the drill hole, the later the gas is extracted.Therefore, the farther away from the drill hole, the longer it takes for the gas flow rate to reach the maximum value in the coal rock body.
Monitoring point 1 is situated within the plastic zone.As the vertical stress on the coal seam intensifies, the deformation degree of the plastic zone surrounding the borehole also escalates.Consequently, there is an increase in the number of secondary pores and cracks, leading to an enhanced permeability of the coal seam.This, in turn, causes the gas flow rate at monitoring point 1 to rise with the increasing vertical stress.Monitoring point 2 is located in the elastic zone.When the vertical crustal stress on the coal seam increases, the degree of extrusion in the elastic zone of the coal seam increases.On the contrary, the number of pore cracks decreases, and the permeability of the coal seam decreases.All these result in the decrease of gas flow rate at monitoring point 2 with the increase of vertical crustal stress.

| The effect of lateral pressure coefficient on gas flow in boreholes
Under the influence of complex crustal stress conditions, the horizontal crustal stress applied to the coal seam increases significantly, which leads to the increase of lateral pressure coefficient applied to the coal seam.Keeping the vertical crustal stress at 10.0 MPa, the influence of different lateral pressure coefficients on the permeability, gas pressure, gas flow rate, and effective extraction radius of the coal seam is investigated by applying lateral pressure coefficients of 0.5, 1.0, 1.5, 2.0, and 2.5 to the model, respectively.

| The evolution law of porosity and permeability under different lateral pressure coefficients
When the lateral pressure coefficient changes, the shape and distribution location of the shear stress and plastic zone around the borehole changes, leading to changes in the permeability characteristics of different locations in the borehole.So it is necessary to comprehensively consider the changes in porosity and permeability with the lateral pressure coefficient at various locations around the borehole.Taking into account the symmetry of the model and the boundary conditions, the data from the right side of the borehole and the upper side of the pumping of the 180 days were selected for analysis.
The change curves of porosity and permeability ratio with lateral pressure coefficient at the right and upper sides of the borehole are shown in Figures 13 and 14, and the specific data are shown in Table 6.The distribution of porosity and permeability is essentially identical, and the ratio of porosity and permeability is the largest at the extraction borehole, which gradually decreases from the borehole outward and finally tends to stabilize.Within the plastic zone of the coal rock body on the right side of the borehole, as the lateral pressure coefficient gradually increases while the vertical stress remains constant, the area and deformation degree of the plastic zone decrease.Simultaneously, the number of secondary pore fissures also decreases, resulting in a decline in the permeability ratio of this specific area with the increasing lateral pressure coefficient.On the other hand, in the elastic zone of the coal rock body on the right side of the borehole, the permeability ratio of the coal seam decreases with the increase of the lateral pressure coefficient.In the plastic zone of the coal rock body on the upper side of the borehole, with the increase of lateral pressure coefficient, the area and deformation degree of the plastic zone increase, and the number of secondary pores and cracks increases, which results in the permeability ratio of this zone increasing with the increase of lateral pressure coefficient.While in the elastic zone of the coal rock body on the upper side of the borehole, the permeability ratio of the coal seam decreases with an increase in the lateral pressure coefficient.In summary, when the lateral pressure coefficient is less than 1, the maximum permeability ratio value is mainly distributed on the left and right sides of the borehole.When the lateral pressure coefficient is equal to 1, the maximum permeability ratio value is uniformly distributed around the borehole in a circle.When the lateral pressure coefficient is greater than 1, the maximum permeability ratio value is mainly distributed on the top and bottom sides of the borehole.

| The gas flow law under different lateral pressure coefficients
The cloud diagram and curve of gas pressure distribution under different lateral pressure coefficients are shown in Figures 14 and 15.When the vertical crustal stress is unchanged, with the increase of lateral pressure coefficient, the deformation degree and range of the plastic zone around the borehole increases, increasing the permeability of the coal seam in this region.Moreover, only a very small range of the plastic zone shows the law that the gas pressure is lower under the high lateral pressure coefficient, and the shape of the distribution of the low gas pressure zone is consistent with the distribution of the plastic zone.The shape of the low gas pressure zone is consistent with that of the plastic zone.When the lateral pressure coefficient is less than 1, the low gas pressure zones are distributed on the left and right sides of the borehole in a fan shape.When the lateral pressure coefficient is equal to 1, the low gas pressure zones are uniformly distributed around the borehole in a circle.When the lateral pressure coefficient is greater than 1, the low gas pressure zones are distributed on the top and bottom of the borehole in the form of an ellipse and butterfly wing, and the larger the lateral pressure coefficient is, the larger the area of the low gas pressure zones.In the majority of the elastic zones, excluding the plastic zone, the degree of compression of the pore and crack structure intensifies as the lateral pressure   coefficient rises.Consequently, the permeability of the coal seam diminishes, and the decrease in gas pressure follows suit.This observation demonstrates the correlation between higher gas pressure and a higher lateral pressure coefficient.
In Figure 16, the three sets of curves from top to bottom are the gas pressure at 1, 90, and 180 days of extraction, and the gas pressure curve under a high lateral pressure coefficient is located above the gas pressure curve under a low lateral pressure coefficient.The gas pressure in the plastic zone has less influence on the overall gas pressure of the coal seam because the area of the plastic zone is very different from the area of the elastic zone.In the overall range of the coal seam, with the increase of lateral pressure coefficient, the gas pressure of the coal seam decreases, the effect of gas extraction deteriorates, and the extraction efficiency decreases.From 1 to 90 to 180 days, it can be seen that with the prolongation of the extraction time, the spacing of the gas pressure curves under the conditions of various lateral pressure coefficients increases, and the difference between the gas pressures increases.This indicates that, with the prolongation of the extraction time, the disadvantage of gas extraction under the high lateral pressure coefficients becomes more significant.The gas flow rate at each monitoring point with different lateral pressure coefficients is shown in Figure 17, and the arrangement of the monitoring points is the same as that in Figure 12.The gas flow rate at the monitoring points in the two plastic zones increases rapidly to the peak in the early stage of extraction and then decreases gradually.Conversely, the gas flow rate at the monitoring points in the two elastic zones reaches the peak in the middle stage of extraction and then decreases gradually.Similarly, the closer to the drill hole, the larger the peak gas flow rate and the shorter the time it takes to reach the peak.On the other hand, the farther away from the drill hole, the smaller the peak gas flow rate and the longer the time it takes to reach the peak.
At monitoring point 1 (in the plastic zone on the right side of the borehole), the gas flow rate decreases with the increase of lateral pressure coefficient; at monitoring point 3 (in the plastic zone on the upper side of the borehole), the gas flow rate increases with the increase of lateral pressure coefficient.In the plastic zone on the left and right sides of the borehole, as the lateral pressure coefficient increases while the vertical crustal stress remains constant, the area of the plastic zone and the degree of deformation decrease.Additionally, the number of secondary pores and cracks generated also decreases, leading to a reduction in the permeability of the coal seam.Consequently, the gas flow rate in this area decreases with the increasing lateral pressure coefficient.In the plastic zone on the upper and lower sides of the borehole, with the increase of the lateral pressure coefficient, the area of the plastic zone and the degree of deformation increases.Additionally, the number of secondary pores and cracks generated increases, resulting in a reduction of the permeability of the coal seam.As the lateral pressure coefficient increases, the area and deformation degree of the plastic zone increase, and the number of secondary pores and cracks also increases.This leads to an increase in the permeability of the coal seam, and the gas flow rate increases with the increase of the lateral pressure coefficient in this area.Whether in the elastic zone (monitoring points 2 and 4) on the right side or the upper side of the borehole, the gas flow rate decreases with the increase of the lateral pressure coefficient, and the gas flow rate in the elastic zone at the same distance from the borehole is mostly identical.This is due to the fact that in the elastic zone, as the lateral pressure coefficient increases, the degree of extrusion of the coal seam also increases, resulting in a decrease in the number of holes and cracks.This, in turn, leads to a reduction in the permeability of the coal seam, ultimately causing a decrease in the gas flow rate.

| THE INFLUENCE OF DIFFERENT GROUND STRESS CONDITIONS ON THE DISTRIBUTION OF NEGATIVE PRESSURE DURING BOREHOLE EXTRACTION
Previous studies 34 have indicated that the instability and deformation of the borehole can greatly impact the distribution of negative pressure within it.Furthermore, different types of instability and deformation can have varying effects on the negative pressure, which subsequently affects the gas flow in the coal seam.Therefore, investigating the distribution of negative pressure in the borehole under different vertical crustal stresses and lateral pressure coefficients is of significant importance in guiding efforts to enhance gas extraction efficiency.
Gas flow in the borehole will produce flow resistance, mainly along the resistance loss, wall inflow loss, and local resistance loss of three parts.The entire borehole is subdivided into countless tiny micro-element bodies; the borehole micro-element diagram is shown in Figure 18.
The on-way resistance loss is generated by the gas overcoming the frictional resistance in the flow and is given by: where λ d is the on-way resistance coefficient; l is the depth of the whole borehole, m; d is the hole diameter, m; v is the mean flow rate of certain fracture surfaces, m/s.The on-way resistance coefficient needs to be determined based on the flow state of the gas in the borehole, and the Reynolds number is generally used to determine the fluid flow state: When the gas is in laminar conditions in the borehole, the on-way resistance coefficient is: When the gas is in a turbulent condition state in the borehole, according to the empirical equation, the onway resistance coefficient is: ε is the roughness of the drilled hole wall.This paper takes the value of 5 mm.
Any infinitesimal L t whose gas flow rates Q t and Q t-1 between the two end faces on the entire borehole is chosen, while the inflow of gas from the section is q t .The value is derived from the integral: where, L is the length of the infinitesimal section L t , m; l is the perimeter of the borehole, m; and p f  is the gas pressure gradient of the section, Pa/m.And the front and back end face flow rates are: Then the mean flow rate of the infinitesimal segment L t is: Combining with the equation gives the along-travel resistance loss of the infinitesimal segment L t as: Where，ρ is the density of gas; L is the length of borehole; d is the radius of borehole; λ is coefficient of along-travel resistance loss.The wall inflow loss is obtained by correcting the on-way resistance coefficient 35 as: ( ) The local resistance loss caused by the change in the deformed section of the borehole instability is: where ξ is the local resistance coefficient, d′ is the equivalent diameter of the borehole after its deformation.In summary, it can be seen that the total resistance loss of the infinitesimal segment L t is the sum of the three: ( ) Finally, the negative pressure attenuation is obtained by integrating over the entire length of the borehole.

| The distribution law of negative pressure under different vertical ground stresses
The data of negative pressure distribution inside the borehole at 180 days of extraction is selected for analysis; the negative pressure at the fixed borehole is 20,000 Pa and the negative pressure distribution inside the borehole under different vertical crustal stresses is shown in Figure 19, and the data are shown in Table 7.When the lateral pressure coefficient is unchanged and the vertical crustal stress increases, the amount of negative pressure loss in the extraction borehole increases.Furthermore, the spacing of the negative pressure distribution curves under each vertical crustal stress condition decreases.This indicates that as the coal seam is buried deeper, the increase in the amount of negative pressure loss becomes smaller when the same vertical crustal stress is increased, and the degree of influence on the negative pressure loss becomes weaker.
With the increase of vertical crustal stress, the area of plastic zone around the hole and the degree of instability and deformation of the hole increase, and the hole diameter of the hole decreases.These lead to the increase of local resistance in each section of the axial direction of the hole, and the obstruction of the propagation of the negative pressure of pumping inside the hole increases.
Therefore, it presents the law that the amount of loss inside the borehole is larger and the value of negative pressure is smaller under the vertical high crustal stress.

| Under different lateral pressure coefficients, the distribution law of negative pressure
The negative pressure at the fixed porthole is 20,000 Pa and the negative pressure distribution in the borehole under different lateral pressure coefficients is shown in Figure 20, and the data are shown in Table 8.When the vertical crustal stress remains constant and the lateral pressure coefficient increases, the amount of negative pressure loss in the extraction borehole increases.Additionally, the spacing between the negative pressure distribution curves decreases under various lateral pressure coefficients.This indicates that as the lateral pressure coefficient becomes larger, the increase in the amount of negative pressure loss becomes smaller, even with the same horizontal crustal stress.Furthermore, the degree of influence on the negative pressure loss weakens.
With the increase of lateral pressure coefficient, the area of the plastic zone around the hole increases, the distribution shape and location change and the degree of compression of the borehole increases.These lead to a decrease in borehole diameter and the local resistance of each section in the axial direction of the borehole increases with the increase of lateral pressure coefficient, which in turn plays a greater impediment to the propagation of the negative pressure of the pumping in the borehole.Therefore, it presents the law that the amount of negative pressure loss in the borehole is larger and the negative pressure value is smaller under high lateral pressure coefficient.

| CONCLUSION
In light of the above work, the main conclusions of this paper are as follows: 1. Based on the dual-pore and dual-fracture structure characteristics of coal, an elastic-plastic deformation equation for coal rock mass is established using the D-P strength criterion.The diffusion equation of gas in the matrix, the dynamic change equation of permeability, and the gas seepage equation in the fracture are derived separately.A gas extraction flow-solid coupling model considering the elastic-plastic deformation of the coal body is constructed.2. Increasing vertical stress, when the lateral pressure coefficient remains constant, to a larger stress concentration around the borehole, deformation, and plastic area.In the plastic zone around the borehole, the porosity-to-permeability ratio increases with vertical stress, while the opposite occurs in the elastic zone.High vertical stress inhibits gas extraction and this effect becomes more pronounced over time.3. When the lateral pressure coefficient is less than 1, the zone of concentrated circumferential stress and plastic area exhibit a fan-shaped distribution on the left and right sides of the borehole.When the lateral pressure coefficient is equal to 1, they exhibit a circular and uniform distribution around the borehole.When the lateral pressure coefficient is greater than 1, they exhibit an elliptical and butterfly-shaped distribution on the upper and lower sides of the borehole.Furthermore, the distribution shape of the low gas pressure zone is consistent with that of the plastic area.The larger the lateral pressure coefficient, the poorer the stability on the upper and lower sides of the borehole.4. Both increased vertical stress and lateral pressure coefficient hinder gas extraction and impede negative pressure propagation in the borehole.High vertical stress and lateral pressure coefficient decrease borehole diameter, increasing local resistance in different sections along the axial direction and hindering negative pressure propagation, leading to increased loss over time.

F I G U R E 4 8 F
Shear stress distribution under different vertical crustal stresses.(A) 5.0 MPa.(B) 7.5 MPa.(C) 10.0 MPa.(D) 12.5 MPa.(E) 15.0 MPa.shear stress in the coal rock body on the right and left sides of the borehole increases slowly as the distance from the borehole increases.Additionally, the shear stress concentration area is uniformly distributed around the borehole in a circular pattern.In the coal rock body on the left and right sides of the borehole, the shear stress increases slowly with the increase of distance from the borehole.The shear stress concentration area is elliptical and butterfly-shaped, distributed on the upper and lower sides of the borehole.The larger the lateral pressure coefficient is, the larger the distribution range of the T A B L E 2 Peak perforation shear stresses under different vertical crustal stresses.I G U R E 5 Distribution of plastic zones under different vertical crustal stresses.(A) Vertical crustal stress 5.0 MPa.(B) Vertical crustal stress 7.5 MPa.(C) Vertical crustal stress 10.0 MPa.(D) Vertical crustal stress 12.5 MPa.(E) Vertical crustal stress 15.0 MPa.shear stress concentration area.When the vertical stress remains constant and the lateral pressure coefficient increases, the peak of shear stress increases.The distribution of shear stress concentration zones is shifted from the left and right sides of the borehole to the top and bottom sides of the borehole, which leads to the instability of the top and bottom sides of the borehole under the high lateral pressure coefficient.

T A B L E 3
Plastic zone sizes of different vertical crustal stresses.

1 . 2 |
The gas flow law under different vertical ground stresses When the coal seams are at different depths, they are subject to different vertical stresses; the coal seam has different deformation and instability of the drill holes, and the permeability of the coal seams changes, which in turn affects the flow pattern of the gas.The cloud and curve diagrams of gas pressure under different vertical crustal stresses are shown in Figures 9 and 10.When the lateral pressure coefficient is unchanged and the vertical crustal stress increases, the

F I G U R E 7
Distribution of the plastic zone for different lateral pressure coefficients.(A) Lateral pressure coefficient 0.5.(B) Lateral pressure coefficient 1.0.(C) Lateral pressure coefficient 1.5.(D) Lateral pressure coefficient 2.0.(E) Lateral pressure coefficient 2.5.

F I G U R E 8
Porosity and permeability curves at different vertical crustal stresses.T A B L E 4 Plastic zone sizes for different lateral pressure coefficients.

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I G U R E 10 Gas pressure curves under different vertical crustal stresses.F I G U R E 11 Gas velocity of flow at each monitoring point under different vertical crustal stresses.(A) Monitoring point 1 gas flow rate.(B) Monitoring point 2 gas flow rate.

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I G U R E 12 Layout of monitoring sites.F I G U R E 13 Porosity and permeability variation curves with lateral pressure coefficient on the right-hand side of the borehole.GUANG and HONGBAO | 2607

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I G U R E 14 Variation curve of porosity and permeability on the upper side of the borehole with lateral pressure coefficient.(A) is change of porosity on the upper side of the borehole with lateral pressure coefficient.(B) is change of permeability on the upper side of the borehole with lateral pressure coefficient.T A B L E 6 Permeability ratios of right and upper boreholes under different side pressure coefficients.

F I G U R E 15
Gas pressure cloud map under different side pressure coefficients.(A) Lateral pressure coefficient 0.5.(B) Lateral pressure coefficient 1.0.(C) Lateral pressure coefficient 1.5.(D) Lateral pressure coefficient 2.0.(E) Lateral pressure coefficient 2.5.

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I G U R E 16 Gas pressure curves under different side pressure coefficients.

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I G U R E 17 Gas flow velocity at each monitoring point under different lateral pressure coefficients.(A) Monitoring point 1 gas flow rate.(B) Monitoring point 2 gas flow rate.(C) Monitoring point 3 gas flow rate.(D) Monitoring point 4 gas flow rate.

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I G U R E 18 Micro diagram of gas flow in a borehole.

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I G U R E 19 Distribution of negative pressure in borehole under different in situ stresses.T A B L E 7 Negative pressure and loss of negative pressure in boreholes under different in situ stresses.I G U R E 20 Distribution of negative pressure in the borehole with different lateral pressure coefficients.
Main parameters of the simulation.
T A B L E 1 Negative pressure and loss of negative pressure in boreholes under different lateral pressure coefficients.
T A B L E 8