Study on the Influence of the Strain-Softening of the Surrounding Rock with Buried Depth on Gas Extraction Boreholes

: The buried depth of coal seams in China gradually increases as shallow resources decrease. The purpose of this article is to reveal the effect of buried depth on gas extraction boreholes. Firstly, we analyzed the shortcomings of the Mohr–Coulomb (M-C) constitutive model for simulating excavation problems and introduced the strain-softening (S-S) model and its advantages. Subsequently, we constructed the gas extraction models with different buried depths based on the S-S model and combined them with the evolution equations for permeability and the equations for gas migration. Then, we studied the difference between the M-C and S-S models using numerical simulations. We found that the inﬂuence of the S-S phenomenon in the surrounding rock of boreholes is more signiﬁcant as the buried depth increases—that is, the strain-softening has a signiﬁcant buried depth effect. When the buried depths were 300, 500, and 700 m, the maximum ratios of permeability to the initial permeability obtained from the Mohr–Coulomb model were 1.37, 6.88, and 97.56, respectively; the maximum ratios of permeability to the initial permeability obtained from the strain-softening model were 2.06, 291.23, and 3629.66. The differences in the increase zone of permeability from the two models also increased with the increase in the buried depth. The distribution curves of gas pressure in the surrounding rock of boreholes obtained from the S-S model were below those of the M-C constitutive model at the same buried depth, and the zones with slowly increasing gas pressure with the increase in buried depth were more obvious in the S-S model compared with the M-C constitutive model. The differences in borehole effective extraction radius also increased gradually between the S-S and M-C models as the buried depth increased. Through the analysis of the research results, we found that S-S of the surrounding rock has a lesser inﬂuence on the gas extraction boreholes, and the S-S phenomenon in the surrounding rock of boreholes can be ignored when the buried depth is relatively shallow. It is increasingly necessary to consider the S-S phenomenon in the surrounding rock of boreholes with the gradual increase in the buried depth.


Introduction
Coal mining in China has been gradually extended deeper as a result of the depletion of shallow coal resources in recent years, and some problems have gradually emerged with the increases in mining depth [1][2][3]. On the one hand, deep mines have the characteristics of high in situ stress levels, strong mining disturbances, high development of coal fractures, and the linear-to-nonlinear transformation of mechanical properties [4,5]. On the other hand, the gas pressure and contents increase as the buried depth increases in coal seams. Many mines transform from low-gas to high-gas mines or even coal and gas outburst mines after extending deeper, resulting in an increase in gas disaster accidents in mining [6,7]. The most efficacious method to control coal mine gas disaster accidents at present is to extract is quite discrepant from the stress-strain curve of coal and rock. The schematic diagram of the M-C constitutive model and the stress-strain curve of coal and rock are shown in Figure 1 (where σ is the stress, ε is the strain of coal and rock, c is the cohesion, and c p is the peak cohesion).
Processes 2023, 11, x FOR PEER REVIEW 3 of 18 within a certain area during simulation, but the matching effect on coal and rock properties in the post-peak stage is relatively poor. The principal reason for this is that the M-C constitutive model is an ideal elastic-plastic model. The strength parameters of coal and rock are constant in numerical simulation, e.g., the cohesion is the peak cohesion, which is quite discrepant from the stress-strain curve of coal and rock. The schematic diagram of the M-C constitutive model and the stress-strain curve of coal and rock are shown in Figure 1 (where σ is the stress, ε is the strain of coal and rock, c is the cohesion, and cp is the peak cohesion).
(a) (b) The S-S model was developed on the basis of the M-C constitutive model, taking into account the reduction in the strength parameters of coal and rock after the peak values. The strain-softening is more compatible with the actual situation in the field. To explore the substance of the S-S phenomenon of coal and rock, Jing et al. analyzed the strength parameters of coal and rock in the post-peak stage according to the strength test data published in 18 academic journals at home and abroad, along with master's research [28]. The study validated that the substance of the S-S phenomenon of coal and rock is that the internal friction angle remains unchanged and the cohesion decreases significantly. The full stress-strain curve of coal and rock can be simplified as a "3-line representation", and the corresponding cohesion of coal and rock is shown in Figure 2. The full stress-strain curve can be expressed as the elastic stage, softening stage, and residual stage. The cohesion corresponding to the elastic stage is the peak cohesion cp, and the cohesion corresponding to the residual stage is the residual cohesion cr. After forming boreholes, the surrounding rock of boreholes can be divided into the crushing zone, plastic zone, and elastic zone due to the redistribution of in situ stress. These three zones correspond to the residual stage, softening stage, and elastic stage in The S-S model was developed on the basis of the M-C constitutive model, taking into account the reduction in the strength parameters of coal and rock after the peak values. The strain-softening is more compatible with the actual situation in the field. To explore the substance of the S-S phenomenon of coal and rock, Jing et al. analyzed the strength parameters of coal and rock in the post-peak stage according to the strength test data published in 18 academic journals at home and abroad, along with master's research [28]. The study validated that the substance of the S-S phenomenon of coal and rock is that the internal friction angle remains unchanged and the cohesion decreases significantly. The full stress-strain curve of coal and rock can be simplified as a "3-line representation", and the corresponding cohesion of coal and rock is shown in Figure 2. The full stress-strain curve can be expressed as the elastic stage, softening stage, and residual stage. The cohesion corresponding to the elastic stage is the peak cohesion c p , and the cohesion corresponding to the residual stage is the residual cohesion c r .
Processes 2023, 11, x FOR PEER REVIEW 3 of 18 within a certain area during simulation, but the matching effect on coal and rock properties in the post-peak stage is relatively poor. The principal reason for this is that the M-C constitutive model is an ideal elastic-plastic model. The strength parameters of coal and rock are constant in numerical simulation, e.g., the cohesion is the peak cohesion, which is quite discrepant from the stress-strain curve of coal and rock. The schematic diagram of the M-C constitutive model and the stress-strain curve of coal and rock are shown in Figure 1 (where σ is the stress, ε is the strain of coal and rock, c is the cohesion, and cp is the peak cohesion).
(a) (b) The S-S model was developed on the basis of the M-C constitutive model, taking into account the reduction in the strength parameters of coal and rock after the peak values. The strain-softening is more compatible with the actual situation in the field. To explore the substance of the S-S phenomenon of coal and rock, Jing et al. analyzed the strength parameters of coal and rock in the post-peak stage according to the strength test data published in 18 academic journals at home and abroad, along with master's research [28]. The study validated that the substance of the S-S phenomenon of coal and rock is that the internal friction angle remains unchanged and the cohesion decreases significantly. The full stress-strain curve of coal and rock can be simplified as a "3-line representation", and the corresponding cohesion of coal and rock is shown in Figure 2. The full stress-strain curve can be expressed as the elastic stage, softening stage, and residual stage. The cohesion corresponding to the elastic stage is the peak cohesion cp, and the cohesion corresponding to the residual stage is the residual cohesion cr. After forming boreholes, the surrounding rock of boreholes can be divided into the crushing zone, plastic zone, and elastic zone due to the redistribution of in situ stress. These three zones correspond to the residual stage, softening stage, and elastic stage in  After forming boreholes, the surrounding rock of boreholes can be divided into the crushing zone, plastic zone, and elastic zone due to the redistribution of in situ stress. These three zones correspond to the residual stage, softening stage, and elastic stage in the stress-strain curve, respectively. Therefore, the cohesion in the different zones of the surrounding rock of boreholes is different, as shown in  Based on the results of previous research, the changes in the cohesion of coal and rock corresponding to the stress-strain curve can be formulated by the following equation [29]: where c p is the elastic stage cohesion (MPa), c s is the plastic stage cohesion (MPa), c r is the residual stage cohesion (MPa), γ p is the equivalent plastic shear strain, and γ p* is the equivalent plastic shear strain at the beginning of the plastic phase (0.01). The plastic stage cohesion can be formulated as follows: The equivalent plastic shear strain can be formulated by the following equation [30]: where ε p 1 , ε p 2 , and ε p 3 are the first, second, and third principal plastic strains, respectively.

Permeability Evolution Equations
The change in permeability with stress is principally caused by the change in fracture porosity. Research shows that there is a close relationship between the fracture porosity and volume strain. Regardless of the impact of gas adsorption and desorption on the fracture porosity, the fracture porosity after stress redistribution can be formulated as follows [31]: where φ f1 is the fracture porosity after forming boreholes, ε v is the volume strain, and φ f0 is the original fracture porosity. The relationship between permeability and fracture porosity can be formulated as follows, according to the cubic law [32]: where k is the permeability of the coal seams (m 2 ), and k 0 is the original permeability of the coal seams (m 2 ). Based on Equations (4) and (5), the permeability of the surrounding rock after forming boreholes can be formulated as follows:

Equations of Gas Migration
Coal body is a kind of double-porous medium material, and the gas migration can be divided into two steps: First, the gas in the fractures of the rocks surrounding the borehole flows into the borehole under the action of pressure difference, and then the gas of the coal matrix block is desorbed and diffuses into the fractures as source term of fractures. The mass exchange between the gas of the coal matrix and the fractures can be formulated as follows [33]: where Q s is the mass exchange rate of gas between the coal matrix and the system of fractures (kg/(m 3 ·s)), τ is the adsorption time (h), M is the molar mass of the gas (kg/mol), R is the gas constant (J/(mol·K)), T is the temperature of the coal seam (K), P m is the gas pressure of the coal matrix (MPa), and P f is the gas pressure of the fractures (MPa). The mass exchange rate of gas between the coal matrix and the system of fractures is equal to the change in the gas content of the coal matrix system with time, according to the law of conservation of mass [34]: where m m is the gas content in the coal matrix (kg/m 3 ) and t is the time (s). The gas content in the coal matrix can be formulated as follows [35]: where V L is the maximum gas adsorption capacity per unit volume of coal (kg/m 3 ), V M is the molar volume of gas (L/mol), ρ c is the density of the coal matrix block (kg/m 3 ), and φ m is the porosity of the coal matrix block. The change equation of gas pressure in the matrix with time can be calculated by using Equations (7)-(9): Gas seepage in coal fractures conforms to the conservation-of-mass equation and Darcy's law, which can be formulated using the following equations [36]: where ρ f is the fractures' gas density (kg/m 3 ), ν is the seepage velocity of the gas (m/s), and µ is the viscosity of the gas (Pa·s). Equations (10)- (12) are the gas migration control equations.

The Influence of the S-S Phenomenon of the Surrounding Rock on Extraction Boreholes with Buried Depth
To investigate the buried depth effect of the S-S phenomenon on the surrounding rock of boreholes, we established numerical models of gas extraction using COMSOL. Assuming that the surrounding rock of boreholes is in a two-way isobaric state, since the extraction borehole's length is much longer than its diameter, the model can be simplified as a two-dimensional plane-strain model. The model size is 10 m × 10 m, and the borehole diameter is 115 mm. The model is shown in Figure 4. The boundaries around the model are under stress and no-flow boundary conditions, and the boundaries around the borehole are under free boundary conditions. The simulation calculation process was as follows: Firstly, we selected the M-C constitutive model and combined it with the evolution equations of permeability and the equations of gas migration to analyze the distribution of stress, permeability, and gas pressure in the rock surrounding the boreholes at different buried depths. Subsequently, the M-C constitutive model was replaced by the S-S model to study the variations of stress, permeability, and gas pressure with buried depth in the rock surrounding the boreholes. The simulated buried depths were 300, 500, and 700 m. The parameters used in the numerical calculation model are shown in Table 1. These parameters mainly refer to published papers, with some modifications [37,38].  compared with the M-C constitutive model at the same buried depth. Meanwhile, we found that the tangential stress at the borehole boundary obtained from the S-S model was smaller than that from the M-C constitutive model at the same buried depth. Our analyses show that the width of the crushing zone of the surrounding rock of boreholes increases with the increase in the buried depth, leading to the transfer of stress to greater depths. At the same buried depth, due to the S-S phenomenon of the surrounding rock near the borehole after breakage, the bearing capacity of the surrounding rock of the boreholes decreases, leading to a decrease in the tangential stress of the surrounding rock near the borehole's boundary and the transfer of the increased portion of tangential stress to greater depths. To more clearly observe the distribution of tangential stress of the surrounding rock of boreholes, the distribution curves of tangential stress from the borehole boundary to the model boundary were extracted and drawn into curves, as shown in Fig   compared with the M-C constitutive model at the same buried depth. Meanwhile, we found that the tangential stress at the borehole boundary obtained from the S-S model was smaller than that from the M-C constitutive model at the same buried depth. Our analyses show that the width of the crushing zone of the surrounding rock of boreholes increases with the increase in the buried depth, leading to the transfer of stress to greater depths. At the same buried depth, due to the S-S phenomenon of the surrounding rock near the borehole after breakage, the bearing capacity of the surrounding rock of the boreholes decreases, leading to a decrease in the tangential stress of the surrounding rock near the borehole's boundary and the transfer of the increased portion of tangential stress to greater depths. To more clearly observe the distribution of tangential stress of the surrounding rock of boreholes, the distribution curves of tangential stress from the borehole boundary to the model boundary were extracted and drawn into curves, as shown in Fig   show that the increased portions of tangential stress in the surrounding rock of boreholes gradually move deeper with the increase in buried depth under the same model conditions. The increased portions of tangential stress in the surrounding rock of boreholes obtained from the S-S model were farther from the borehole's boundary compared with the M-C constitutive model at the same buried depth. Meanwhile, we found that the tangential stress at the borehole boundary obtained from the S-S model was smaller than that from the M-C constitutive model at the same buried depth. Our analyses show that the width of the crushing zone of the surrounding rock of boreholes increases with the increase in the buried depth, leading to the transfer of stress to greater depths. At the same buried depth, due to the S-S phenomenon of the surrounding rock near the borehole after breakage, the bearing capacity of the surrounding rock of the boreholes decreases, leading to a decrease in the tangential stress of the surrounding rock near the borehole's boundary and the transfer of the increased portion of tangential stress to greater depths. To more clearly observe the distribution of tangential stress of the surrounding rock of boreholes, the distribution curves of tangential stress from the borehole boundary to the model boundary were extracted and drawn into curves, as shown in Figure 7. Figure 7a-c show that the tangential stress in the surrounding rock of boreholes is the lowest at the borehole boundary. The tangential stress increases to the maximum stress and then gradually reduces to the original stress along the coal body's depth. At the same buried depth, the peak stress values obtained from the two models were the same-12.27, 19.61, and 27.60 MPa, respectively-but the peak positions obtained from the S-S model were larger than those obtained from the M-C constitutive model. Figure 7d shows the differences in the peak position between the S-S model and the M-C constitutive model. From the figure, we can see that the differences in the peak position obtained from the two models are 31.6, 36.8, and 48.1 mm at the buried depths of 300, 500, and 700 m, respectively. The differences in the peak position are larger with the increase in the buried depth. The results indicate that the S-S phenomenon of the surrounding rock has a greater effect on the tangential stress distribution around the borehole with the increase in the buried depth.

Influence of the S-S Phenomenon of the Surrounding Rock of Boreholes on the Permeability Distribution with Buried Depth
The coal seam's permeability is a significant index to evaluate the gas extraction difficulty, and also an important parameter when studying extraction boreholes. The desorption of gas in coal seams is a slow process. The gas pressure and desorption amount change slightly at the initial formation stage of the rock surrounding the boreholes, so the change influence of gas pressure and desorption on the permeability of the surrounding rock of boreholes can be ignored, and only the influence of the redistribution of in situ stress on the permeability of surrounding rock of boreholes must be considered. The distribution cloud charts of the ratios of the permeability of the surrounding rock to the original permeability in the borehole with different buried depths, calculated by the M-C constitutive model and the S-S model, are shown in Figures 8 and 9, respectively. Figures 8 and 9 show that the permeability of the surrounding rock envelope of boreholes under a two-way isobaric state is distributed in a circular shape. The ratio of Figure 7a-c show that the tangential stress in the surrounding rock of boreholes is the lowest at the borehole boundary. The tangential stress increases to the maximum stress and then gradually reduces to the original stress along the coal body's depth. At the same buried depth, the peak stress values obtained from the two models were the same-12.27, 19.61, and 27.60 MPa, respectively-but the peak positions obtained from the S-S model were larger than those obtained from the M-C constitutive model. Figure 7d shows the differences in the peak position between the S-S model and the M-C constitutive model. From the figure, we can see that the differences in the peak position obtained from the two models are 31.6, 36.8, and 48.1 mm at the buried depths of 300, 500, and 700 m, respectively. The differences in the peak position are larger with the increase in the buried depth. The results indicate that the S-S phenomenon of the surrounding rock has a greater effect on the tangential stress distribution around the borehole with the increase in the buried depth.

Influence of the S-S Phenomenon of the Surrounding Rock of Boreholes on the Permeability Distribution with Buried Depth
The coal seam's permeability is a significant index to evaluate the gas extraction difficulty, and also an important parameter when studying extraction boreholes. The desorption of gas in coal seams is a slow process. The gas pressure and desorption amount change slightly at the initial formation stage of the rock surrounding the boreholes, so the change influence of gas pressure and desorption on the permeability of the surrounding rock of boreholes can be ignored, and only the influence of the redistribution of in situ stress on the permeability of surrounding rock of boreholes must be considered. The distribution cloud charts of the ratios of the permeability of the surrounding rock to the original permeability in the borehole with different buried depths, calculated by the M-C constitutive model and the S-S model, are shown in Figures 8 and 9, respectively. permeability at the borehole boundary increases with the increase in the buried depth. At the same buried depth, the permeability obtained from the S-S model was larger than that obtained from the M-C constitutive model. When the buried depths were 300, 500, and 700 m, the ratios of permeability at the borehole's boundary obtained from the M-C constitutive model were about 1.40, 7.00, and 100.00, respectively, while the ratios of permeability at the borehole's boundary obtained from the S-S model were about 2.10, 300.00, and 3700.00, respectively.  The increase in the zone of permeability of the surrounding rock of boreholes is an indicator to evaluate the effect of boreholes' permeability enhancement. The increase in permeability means that the ratio of the permeability of the surrounding rock to the original permeability is greater than 1. The increases in the zones of permeability of the surrounding rock of boreholes under different models are shown in Figures 10 and 11. permeability at the borehole boundary increases with the increase in the buried depth. At the same buried depth, the permeability obtained from the S-S model was larger than that obtained from the M-C constitutive model. When the buried depths were 300, 500, and 700 m, the ratios of permeability at the borehole's boundary obtained from the M-C constitutive model were about 1.40, 7.00, and 100.00, respectively, while the ratios of permeability at the borehole's boundary obtained from the S-S model were about 2.10, 300.00, and 3700.00, respectively.  The increase in the zone of permeability of the surrounding rock of boreholes is an indicator to evaluate the effect of boreholes' permeability enhancement. The increase in permeability means that the ratio of the permeability of the surrounding rock to the original permeability is greater than 1. The increases in the zones of permeability of the surrounding rock of boreholes under different models are shown in Figures 10 and 11.  9 show that the permeability of the surrounding rock envelope of boreholes under a two-way isobaric state is distributed in a circular shape. The ratio of permeability at the borehole's boundary is the maximum, and the ratio of permeability from the borehole's boundary to the bottom of the coal body gradually decreases-that is, the permeability is the greatest at the borehole's boundary, and it gradually decreases from the borehole's boundary to the bottom of the coal body. Under the same model, the permeability at the borehole boundary increases with the increase in the buried depth. At the same buried depth, the permeability obtained from the S-S model was larger than that obtained from the M-C constitutive model. When the buried depths were 300, 500, and 700 m, the ratios of permeability at the borehole's boundary obtained from the M-C constitutive model were about 1.40, 7.00, and 100.00, respectively, while the ratios of permeability at the borehole's boundary obtained from the S-S model were about 2.10, 300.00, and 3700.00, respectively.
The increase in the zone of permeability of the surrounding rock of boreholes is an indicator to evaluate the effect of boreholes' permeability enhancement. The increase in permeability means that the ratio of the permeability of the surrounding rock to the original permeability is greater than 1. The increases in the zones of permeability of the surrounding rock of boreholes under different models are shown in Figures 10 and 11.  Processes 2023, 11, x FOR PEER REVIEW 10 of 18   Figures 10 and 11 show that the zone of permeability gradually increases with the increase in the buried depth under the same model. The increase in the zone of permeability obtained from the S-S model was greater than that obtained from the M-C constitutive model at the same buried depth. To compare and analyze the distribution of the ratio of permeability and the variation in the increase in the zone of permeability with the buried depth for the different models more clearly, the distribution curves of the ratio of permeability and the increase in the zone of permeability for the M-C constitutive model and the S-S model at different buried depths were drawn into the same graphs, as shown in Figure 12.   Figures 10 and 11 show that the zone of permeability gradually increases with the increase in the buried depth under the same model. The increase in the zone of permeability obtained from the S-S model was greater than that obtained from the M-C constitutive model at the same buried depth. To compare and analyze the distribution of the ratio of permeability and the variation in the increase in the zone of permeability with the buried depth for the different models more clearly, the distribution curves of the ratio of permeability and the increase in the zone of permeability for the M-C constitutive model and the S-S model at different buried depths were drawn into the same graphs, as shown in Figure 12.  Figures 10 and 11 show that the zone of permeability gradually increases with the increase in the buried depth under the same model. The increase in the zone of permeability obtained from the S-S model was greater than that obtained from the M-C constitutive model at the same buried depth. To compare and analyze the distribution of the ratio of permeability and the variation in the increase in the zone of permeability with the buried depth for the different models more clearly, the distribution curves of the ratio of permeability and the increase in the zone of permeability for the M-C constitutive model and the S-S model at different buried depths were drawn into the same graphs, as shown in Figure 12.   Figures 10 and 11 show that the zone of permeability gradually increases with increase in the buried depth under the same model. The increase in the zone of perm bility obtained from the S-S model was greater than that obtained from the M-C const tive model at the same buried depth. To compare and analyze the distribution of the ra of permeability and the variation in the increase in the zone of permeability with the b ied depth for the different models more clearly, the distribution curves of the ratio of p meability and the increase in the zone of permeability for the M-C constitutive model a the S-S model at different buried depths were drawn into the same graphs, as shown Figure 12.  From the comparative an ysis, we found that the ratio of permeability increased significantly with the increase the buried depth under both models, indicating that the buried depth has a signific influence on the permeability of the surrounding rock of boreholes. At the same bur depth, the ratios of permeability obtained from the S-S model were larger than those tained from the M-C constitutive model, and the S-S model showed a greater increase the ratios of permeability with the increase in the buried depth compared to the M-C c stitutive model. This indicates that the deeper the buried depth, the more prominent influence of strain-softening on the surrounding rock of boreholes. Figure 12d shows the variation curves of the increase in the zone of permeability w buried depth. From the figure, we can see that the increase in the width of the zone permeability S (the width from the borehole's boundary to the permeability ratio of actly 1) is positively correlated with the buried depth H under the two models, which consistent with the power function relationship. The increase in the width of the zone permeability obtained from the M-C model was consistent with the buried depth: 0.2288H 0.9100 . The increase in the width of the zone of permeability obtained from the model was also consistent with the buried depth: S = 1.2436H 0.7131 . Meanwhile, we can a see from the figure that the differences in the increase in the width of the zone of perm ability obtained from the S-S model and the M-C constitutive model increase with increase in the burial depth. The differences in the increase in the width of the zone permeability obtained from the S-S model and the M-C constitutive model were 31.3, 3 and 43.9 mm for the buried depths of 300, 500, and 700 m, respectively.

Influence of the S-S Phenomenon of the Surrounding Rock of Boreholes on the Distributio of Gas Pressure with Buried Depth
Coal seams' gas pressure is one of the important parameters of gas occurring in c mines, and it is also one of the important indicators to evaluate whether the coal seam poses a potential hazard. Therefore, coal mines usually use the residual pressure of ga  From the comparative analysis, we found that the ratio of permeability increased significantly with the increase in the buried depth under both models, indicating that the buried depth has a significant influence on the permeability of the surrounding rock of boreholes. At the same buried depth, the ratios of permeability obtained from the S-S model were larger than those obtained from the M-C constitutive model, and the S-S model showed a greater increase in the ratios of permeability with the increase in the buried depth compared to the M-C constitutive model. This indicates that the deeper the buried depth, the more prominent the influence of strain-softening on the surrounding rock of boreholes. Figure 12d shows the variation curves of the increase in the zone of permeability with buried depth. From the figure, we can see that the increase in the width of the zone of permeability S (the width from the borehole's boundary to the permeability ratio of exactly 1) is positively correlated with the buried depth H under the two models, which is consistent with the power function relationship. The increase in the width of the zone of permeability obtained from the M-C model was consistent with the buried depth: S = 0.2288H 0.9100 . The increase in the width of the zone of permeability obtained from the S-S model was also consistent with the buried depth: S = 1.2436H 0.7131 . Meanwhile, we can also see from the figure that the differences in the increase in the width of the zone of permeability obtained from the S-S model and the M-C constitutive model increase with the increase in the burial depth. The differences in the increase in the width of the zone of permeability obtained from the S-S model and the M-C constitutive model were 31.3, 39.6, and 43.9 mm for the buried depths of 300, 500, and 700 m, respectively.

Influence of the S-S Phenomenon of the Surrounding Rock of Boreholes on the Distribution of Gas Pressure with Buried Depth
Coal seams' gas pressure is one of the important parameters of gas occurring in coal mines, and it is also one of the important indicators to evaluate whether the coal seam gas poses a potential hazard. Therefore, coal mines usually use the residual pressure of gas to evaluate the effectiveness of borehole extraction. The cloud charts of the distribution of gas pressure in the surrounding rock of boreholes with different buried depths for 30 days of extraction, obtained from the M-C constitutive model and the S-S model, are shown in Figures 13 and 14, respectively.    Figures 13 and 14 show that the gas pressure in the surrounding rock envelope of boreholes under a two-way isobaric state is distributed in a circular shape. The gas pressure is the lowest at the borehole's boundary, and it gradually increases to the original gas pressure from the borehole's boundary to the bottom of the coal body. Under the same model, the zone of low gas pressure near the borehole increases with the increase in the buried depth (red zone in the figures). At the same buried depth, the zone of low gas pressure in the S-S model was larger than that in the M-C constitutive model. This indicates that the S-S phenomenon of the surrounding rock of boreholes also has an influence on the distribution law of gas pressure. In practical field applications, the effective extraction radius of the borehole is commonly used to evaluate the gas extraction effect of boreholes, and the effective extraction radius is also an important parameter to define the spacing of boreholes. The method of relative gas pressure is commonly used to determine the effective extraction radius of boreholes for coal seam gas below 0.74 MPa-that is, the gas pressure drops to 51% as the standard extraction boundary [39]. The distance from the standard extraction boundary to the borehole center is the effective extraction radius of the borehole. In this study, the standard extraction boundary for the borehole is that the gas pressure is below 0.245 MPa. The standard extraction zones of boreholes at different buried depths obtained from the M-C constitutive model and the S-S model are shown in Figures 15 and 16, respectively.    Figures 13 and 14 show that the gas pressure in the surrounding rock envelope of boreholes under a two-way isobaric state is distributed in a circular shape. The gas pressure is the lowest at the borehole's boundary, and it gradually increases to the original gas pressure from the borehole's boundary to the bottom of the coal body. Under the same model, the zone of low gas pressure near the borehole increases with the increase in the buried depth (red zone in the figures). At the same buried depth, the zone of low gas pressure in the S-S model was larger than that in the M-C constitutive model. This indicates that the S-S phenomenon of the surrounding rock of boreholes also has an influence on the distribution law of gas pressure. In practical field applications, the effective extraction radius of the borehole is commonly used to evaluate the gas extraction effect of boreholes, and the effective extraction radius is also an important parameter to define the spacing of boreholes. The method of relative gas pressure is commonly used to determine the effective extraction radius of boreholes for coal seam gas below 0.74 MPa-that is, the gas pressure drops to 51% as the standard extraction boundary [39]. The distance from the standard extraction boundary to the borehole center is the effective extraction radius of the borehole. In this study, the standard extraction boundary for the borehole is that the gas pressure is below 0.245 MPa. The standard extraction zones of boreholes at different buried depths obtained from the M-C constitutive model and the S-S model are shown in Figures 15 and 16, respectively.  Figures 13 and 14 show that the gas pressure in the surrounding rock envelope of boreholes under a two-way isobaric state is distributed in a circular shape. The gas pressure is the lowest at the borehole's boundary, and it gradually increases to the original gas pressure from the borehole's boundary to the bottom of the coal body. Under the same model, the zone of low gas pressure near the borehole increases with the increase in the buried depth (red zone in the figures). At the same buried depth, the zone of low gas pressure in the S-S model was larger than that in the M-C constitutive model. This indicates that the S-S phenomenon of the surrounding rock of boreholes also has an influence on the distribution law of gas pressure. In practical field applications, the effective extraction radius of the borehole is commonly used to evaluate the gas extraction effect of boreholes, and the effective extraction radius is also an important parameter to define the spacing of boreholes. The method of relative gas pressure is commonly used to determine the effective extraction radius of boreholes for coal seam gas below 0.74 MPa-that is, the gas pressure drops to 51% as the standard extraction boundary [39]. The distance from the standard extraction boundary to the borehole center is the effective extraction radius of the borehole. In this study, the standard extraction boundary for the borehole is that the gas pressure is below 0.245 MPa. The standard extraction zones of boreholes at different buried depths obtained from the M-C constitutive model and the S-S model are shown in Figures 15 and 16, respectively.  Processes 2023, 11, x FOR PEER REVIEW 13 of 18  From the above figures, we can observe that the standard extraction zone of boreholes increases gradually with the increase in the buried depth under the same model. At the same buried depth, the standard extraction zones of boreholes obtained from the S-S model were larger than those obtained from the M-C constitutive model. To compare and analyze the distribution of gas pressure in the surrounding rock of boreholes and the variation law of the effective extraction radius of boreholes with buried depth under different models, the distribution curves of gas pressure and the effective extraction radius curves of the M-C constitutive model and the S-S model at different buried depths were drawn into the same graphs, as shown in Figure 17.  From the above figures, we can observe that the standard extraction zone of boreholes increases gradually with the increase in the buried depth under the same model. At the same buried depth, the standard extraction zones of boreholes obtained from the S-S model were larger than those obtained from the M-C constitutive model. To compare and analyze the distribution of gas pressure in the surrounding rock of boreholes and the variation law of the effective extraction radius of boreholes with buried depth under different models, the distribution curves of gas pressure and the effective extraction radius curves of the M-C constitutive model and the S-S model at different buried depths were drawn into the same graphs, as shown in Figure 17. From the above figures, we can observe that the standard extraction zone of boreholes increases gradually with the increase in the buried depth under the same model. At the same buried depth, the standard extraction zones of boreholes obtained from the S-S model were larger than those obtained from the M-C constitutive model. To compare and analyze the distribution of gas pressure in the surrounding rock of boreholes and the variation law of the effective extraction radius of boreholes with buried depth under different models, the distribution curves of gas pressure and the effective extraction radius curves of the M-C constitutive model and the S-S model at different buried depths were drawn into the same graphs, as shown in Figure 17.  From the above figures, we can observe that the standard extraction zone of b holes increases gradually with the increase in the buried depth under the same mode the same buried depth, the standard extraction zones of boreholes obtained from the model were larger than those obtained from the M-C constitutive model. To compare analyze the distribution of gas pressure in the surrounding rock of boreholes and the iation law of the effective extraction radius of boreholes with buried depth under diffe models, the distribution curves of gas pressure and the effective extraction radius cu of the M-C constitutive model and the S-S model at different buried depths were dr into the same graphs, as shown in Figure 17.  and 700 m, respectively. From the figures, we can recognize that the curves of gas press obtained from the S-S model are below those obtained from the M-C constitutive mo at the same buried depths, indicating that the gas pressure obtained from the S-S mo is lower than that obtained from the M-C constitutive model at the same position from borehole boundary. The difference between the curves obtained from the two model greater in a certain range with the increase in the buried depth, indicating that the in ence of the S-S phenomenon on the distribution of gas pressure increases gradually w the increase in the buried depth. We can also see that a slowly increasing zone of pressure will appear near the borehole's boundary with the increase in the buried dep in which the value of gas pressure and its rate of increase are relatively low. The slo increasing zone of gas pressure under the S-S model is more obvious than that under M-C constitutive model. From the above analysis, we hypothesize that the reason for results is that the permeability of the surrounding rock near the borehole's boundary nificantly increases with the increase in the buried depth, which leads to easier commu cation between this part and the outside, and the gas can flow into the boreholes m easily. At the same buried depth, the permeability and its amplitude obtained from th S model are larger than those obtained from the M-C constitutive model with the incre in the buried depth, which is reflected in the effect on the gas pressure distribution, as slowly increasing zone of gas pressure under the S-S model is larger than that under M-C constitutive model. Figure 17d shows the variation curves of the effective extraction radius of gas bo holes with the buried depth. From the figure, we can observe that the effective extract radius r of boreholes is positively correlated with the buried depth H under the two m els, which is consistent with the exponential function relationship. The effective extract radius of gas boreholes obtained from the M-C constitutive model is consistent with buried depth: r = 91.7277e 0.0010H . The effective extraction radius of gas boreholes obtai from the S-S model is also consistent with the buried depth: r = 83.0673e 0.0017H . We can a see from the figure that the differences in the effective extraction radius of gas boreho obtained from the S-S model and the M-C constitutive model increase with the incre in the buried depth. When the buried depths are 300, 500, and 700 m, the effective ext tion radius of gas boreholes from the S-S model is 135. 4, 195.3, and 263.6 mm, respectiv and the effective extraction radius of gas boreholes from the M-C constitutive mode 123.6, 148.2, and 182.8 mm, respectively. The differences in the effective extraction rad obtained from the S-S model and the M-C at these depths are 11.8, 47.1, and 80.8 m respectively.
According to the study of the distribution of stress, permeability, and gas pressur the surrounding rock of boreholes at different buried depths, we can recognize that th Figure 17a-c show the comparison curves of gas pressure in the surrounding rock of boreholes for the M-C constitutive model and the S-S model at buried depths of 300, 500, and 700 m, respectively. From the figures, we can recognize that the curves of gas pressure obtained from the S-S model are below those obtained from the M-C constitutive model at the same buried depths, indicating that the gas pressure obtained from the S-S model is lower than that obtained from the M-C constitutive model at the same position from the borehole boundary. The difference between the curves obtained from the two models is greater in a certain range with the increase in the buried depth, indicating that the influence of the S-S phenomenon on the distribution of gas pressure increases gradually with the increase in the buried depth. We can also see that a slowly increasing zone of gas pressure will appear near the borehole's boundary with the increase in the buried depth, in which the value of gas pressure and its rate of increase are relatively low. The slowly increasing zone of gas pressure under the S-S model is more obvious than that under the M-C constitutive model. From the above analysis, we hypothesize that the reason for the results is that the permeability of the surrounding rock near the borehole's boundary significantly increases with the increase in the buried depth, which leads to easier communication between this part and the outside, and the gas can flow into the boreholes more easily. At the same buried depth, the permeability and its amplitude obtained from the S-S model are larger than those obtained from the M-C constitutive model with the increase in the buried depth, which is reflected in the effect on the gas pressure distribution, as the slowly increasing zone of gas pressure under the S-S model is larger than that under the M-C constitutive model. Figure 17d shows the variation curves of the effective extraction radius of gas boreholes with the buried depth. From the figure, we can observe that the effective extraction radius r of boreholes is positively correlated with the buried depth H under the two models, which is consistent with the exponential function relationship. The effective extraction radius of gas boreholes obtained from the M-C constitutive model is consistent with the buried depth: r = 91.7277e 0.0010H . The effective extraction radius of gas boreholes obtained from the S-S model is also consistent with the buried depth: r = 83.0673e 0.0017H . We can also see from the figure that the differences in the effective extraction radius of gas boreholes obtained from the S-S model and the M-C constitutive model increase with the increase in the buried depth. When the buried depths are 300, 500, and 700 m, the effective extraction radius of gas boreholes from the S-S model is 135.4, 195.3, and 263.6 mm, respectively, and the effective extraction radius of gas boreholes from the M-C constitutive model is 123.6, 148.2, and 182.8 mm, respectively. The differences in the effective extraction radius obtained from the S-S model and the M-C at these depths are 11.8, 47.1, and 80.8 mm, respectively.
According to the study of the distribution of stress, permeability, and gas pressure in the surrounding rock of boreholes at different buried depths, we can recognize that the S-S phenomenon of the surrounding rock has an effect on the stress, permeability, and gas pressure of the surrounding rock of the boreholes. The S-S phenomenon of the surrounding rock of boreholes has a significant effect on the buried depth. The greater the buried depth, the more significant the effect of the S-S phenomenon. When the buried depth is shallow, the differences in the permeability, gas pressure, and effective extraction radius of the surrounding rock of boreholes obtained from the M-C constitutive model and the S-S model are not large, so ignoring the influence of the S-S phenomenon of the surrounding rock of boreholes produces little error in the research results. When the buried depth is deep, the permeability, gas pressure, and effective extraction radius obtained from the M-C constitutive model and the S-S model differ greatly, and ignoring the effect of the S-S phenomenon in the surrounding rock of the boreholes will have a significant effect on the research results. Therefore, the influence of the S-S phenomenon of the surrounding rock of boreholes cannot be ignored when studying gas extraction in deep coal seams.

Discussion
China's shallow coal resources will be mined out after long-term large-scale miningespecially the main coal consumption areas in the central and eastern regions of the North-China-type coalfield. The coal seam depth of the principal mines has reached as deep as 800~1000 m, and there are more than 50 mines with a mining depth of over 1000 m, while the maximum coal seam depth has reached 1501 m [40,41]. More and more mines will transition to deep mining in the future. Usually, the coal body has a low strength and is unable to resist the high stress caused by the redistribution of stress after forming boreholes, resulting in damage to the surrounding rock near boreholes. Additionally, the damaged coal body has a reduced strength due to the S-S phenomenon, which has an impact on the distribution of the stress, permeability, and gas pressure. From the above results, we can recognize that the permeability, the increase in the width of the zone of permeability, the distribution of gas pressure, and the effective extraction radius of gas boreholes obtained from the S-S model will all differ compared with the M-C constitutive model. Moreover, the error of the results obtained from two models increases gradually with the increase in the buried depth. Therefore, the S-S phenomenon of the surrounding rock of boreholes should be considered when studying the gas extraction of deep coal mines. Meanwhile, with the development of technology in recent years, some new technologies for gas extraction in coal mines -such as hydraulic flushing and mechanical reaming-have emerged. These technologies use a high-pressure water jet or metal mechanical arms to break the coal body and then use water flow to flush out the coal, expanding the original borehole diameter to achieve the purpose of increasing the gas extraction efficiency. The hydraulic flushing and mechanical reaming technologies cause greater disturbance and damage to the coal body compared with the original borehole. Consideration of the S-S phenomenon of the surrounding rock of boreholes is even more necessary when studying the use of these technologies in deep coal mines.

Conclusions
(1) We established the models of gas extraction at different buried depths according to the S-S model, which is more realistic in the field, and combined it with the evolution equations of permeability and the equations of gas migration based on the analysis of the shortcomings of the M-C constitutive model for simulating the excavation problems. Then, the results were compared with the M-C constitutive model. (2) The permeability of the surrounding rock of boreholes is sensitive to the buried depth and the strain-softening. The maximum permeability ratios obtained from the M-C constitutive model and the S-S model were 1.37 and 2.06, 6.88 and 291.23, and 97.56 and 3629.66 for buried depths of 300, 500, and 700 m, respectively. When the buried depths were 300, 500, and 700 m, the increases in the width of the zone of permeability obtained from the strain-softening model at the above depths were 31.3, 39.6, and 43.9 mm larger than those obtained from the Mohr-Coulomb constitutive model, respectively.
(3) The influence of the S-S phenomenon of the surrounding rock on the distribution of the gas pressure and the effective extraction radius of gas boreholes increases with the increase in the buried depth. At the same buried depth, the gas pressure curves obtained from the S-S model were below those obtained from M-C constitutive model. A slowly increasing zone of gas pressure will appear near the borehole's boundary with the increase in the buried depth, and the slowly increasing zone of gas pressure obtained from the S-S model is more obvious than that obtained from the M-C constitutive model. The differences in the effective extraction radius of gas boreholes obtained from the S-S model and the M-C constitutive model also increase gradually with the increase in the buried depth: which are 11.8, 47.1, and 80.8 mm for buried depths of 300, 500, and 700 m, respectively. (4) The results of this research show that the influence of the S-S phenomenon of the surrounding rock of boreholes is more significant with the increase in the buried depth; that is, the strain-softening has an obvious buried depth effect. When the buried depth is shallow (no more than 300 m under the coal seam parameters in this paper), the permeability, gas pressure, and effective extraction radius of the borehole obtained from the M-C constitutive model and the S-S model are not significantly different, and the S-S phenomenon of the surrounding rock of the borehole can be ignored. It is increasingly necessary to consider the S-S phenomenon of the surrounding rock of boreholes with the increase in the buried depth when studying gas extraction boreholes.

Data Availability Statement:
The data are available from the corresponding author upon reasonable request.