# Effect of Confining Pressure on Mechanical and Energy Conversion Properties of Gas-Containing Coal under Loads

Authors: Zhizhen Zhang; Yixin Niu; Xiaoji Shang; Rui Zhou; Xingguang Liu; Feng Gao
Journal: Geofluids
 (2022)

---

## Abstract

Confining pressure is closely related to the deformation and failure characteristics of gas-containing coal and is one of the key factors for gas outburst-rockburst coupled dynamic disaster. Firstly, this paper briefly summarized the triaxial compression test results of gas-containing coal under different confining pressures, including deformation and stiffness and failure and strength. Then, based on the second developed MatDEM gas-solid coupling program, the macro- and microdeformation and failure process of coal samples under different confining pressures and constant gas pressure were simulated, and the distribution of particle stress, pore stress, and crack, as well as energy conversion characteristics, was obtained. Finally, the influence mechanism of confining pressure on deformation and failure of gas-containing coal was discussed, and the occurrence conditions of coal and gas outburst, rockburst, and their coupled dynamic disasters were distinguished. The results show that the peak strength and residual strength after failure of gas-containing coal both increase approximately linearly with the increase of confining pressure, while the elastic modulus does not change much or increases continuously, which is closely related to the number of microdefects in the coal sample. The strengthening effect of confining pressure on rock stiffness and strength mainly comes from external support and compaction. The total number of cracks decreases with the increase of confining pressure, while the number of shear cracks increases, and the inclination angle between the formed macroscopic main crack and the horizontal plane gradually decreases. The maximum cumulative elastic energy and residual elastic energy both increase with the increase of confining pressure, which leads to the increase of the impact tendency index of coal samples.


---

## Body

## 1. Introduction

There are many reasons for the occurrence of underground coal mine accidents. Among them, the instability and damage of gas-containing coal body are particularly prominent, which will cause disasters such as working face dumping, coal and gas outburst, rockburst, and coal wall peeling. According to the comprehensive hypothesis of coal and gas outburst, it is caused by the combined action of physical and mechanical properties of coal, gas pressure, and in situ stress, as seen in Figure1; that is to say, the instability and failure of gas-containing coal are one of the prerequisites for the occurrence of coal and gas outburst disasters [1–3]. The occurrence of rockburst is also recognized as closely related to the mechanical properties of coal and rock and in situ stress. Therefore, the key problem in predicting gas outburst-rockburst coupled dynamic disaster is to study the instability and failure properties of gas-containing coal.Figure 1
Schematic diagram of coal and gas outburst.The underground coal body is subjected to three-dimensional stress under natural occurrence conditions, so the confining pressure is closely related to the deformation and failure characteristics of gas-containing coal. P. G. Ranjith and Viete [4] studied the mechanical behavior of coal after adsorption of CO2 under triaxial compression and found that high confining pressure can weaken the reducing effect of CO2 on coal strength. J. Y. Wang [5] found that the confining pressure has a certain degree of influence on the elastic modulus, peak strength, and deformation characteristics of outburst briquette specimens using a gas-bearing coal triaxial servo seepage system. K. D. Liu [6] and X. Ding et al. [7] found that the effective confining pressure can increase the strength and elastic modulus of coal while decreasing its Poisson’s ratio. Confining pressure also significantly affects the permeability of coal. J. Z. Liu et al. [8] found that confining pressure can reduce the permeability of coal samples under constant axial pressure and gas pressure. K. Wang et al. [9] studied the effect of confining pressure on the mechanical and permeability properties of gas-containing coal samples, combined coal-sandstone samples, and combined coal-mudstone samples. The mechanical properties of gas-containing coal under different confining pressures are also reflected in the acoustic emission (AE) signal. Z. Y. Qiu et al. [10] and X. B. Zhang et al. [11] comprehensively discussed the effects of confining pressure on AE characteristics in coal samples. Further, R. Zhang et al. [12] and Z. Q. Jia et al. [13] explored the fractal features of the spatiotemporal evolution of AE signals. In addition, Y. Xue et al. [14], C. P. Xin et al. [15], Q. M. Li et al. [16], and C. B. Jiang et al. [17] focused on the effects of different loading and unloading paths and rates on the mechanical properties and permeability of gas-containing coal and investigated their correlation with AE signals. K. T. Kang et al. [18] and T. Teng et al. [19] applied the method of energy accumulation and dissipation to study the energy consumption characteristics of coal samples under compressive load and found that the total energy absorbed, the stored elastic strain energy, and the dissipated energy all increased with increasing confining pressure.Coal is affected by depositional environment, diagenesis, metamorphism, and tectonic action of crustal movement during its formation, resulting in its heterogeneity and anisotropy. Even at the same depth, the same coal seam, or even the same occurrence location, the heterogeneity of the coal body is very obvious, which leads to great differences in the mechanical properties of coal. Therefore, it is limited to study the effect of confining pressure on the mechanical properties of coal through laboratory tests, which is reflected in the sampling location. Besides, it is difficult to know the stress and structural changes in coal samples at the meso-scale due to the limitations of experimental techniques. These deficiencies can be partially compensated by numerical simulation methods [20]. In this paper, we first briefly summarized the experimental results of triaxial compression of gas-containing coal and then carried out discrete element simulation of coal sample compression under constant gas pressure and different confining pressure conditions. Based on this, the particle stress, gas pressure, and crack distribution of coal samples were studied, and the energy conversion behavior during deformation and failure of gas-containing coal was revealed.


## 2. A Brief Summary of Experimental Tests

Scholars have done a large number of conventional triaxial compression tests of gas-containing coal under different confining pressures, and the results obtained are similar. We have made a brief summary of this, which is divided into two aspects, namely, deformation and stiffness and failure and strength.


### 2.1. Deformation and Stiffness Characteristics

Representative stress-strain curves and variation in AE cumulative counts of coal sample under triaxial compression are shown in Figure2. The deviatoric stress gradually increases, and the axial strain and radial strain of the coal sample also increase, while the volumetric strain first decreases and then increases. It is mainly divided into four stages, namely, the compaction and elastic stage (phase I), the stable crack propagation stage (phase II), the nonstable crack propagation stage (phase III), and post-peak failure stage (phase IV). In the compaction stage, the pores and fissures inside the coal sample are gradually compacted and closed, the elastic modulus of the coal sample increases gradually, the stress-strain curve is upwardly convex, and the axial strain and radial strain of the coal sample increase, and the volume is compressed, forming an early nonlinear phase of the curve. It should be pointed out that during the triaxial compression test, some primary defects in the coal sample have been compressed and closed by the confining pressure before the axial loading, so that the compaction stage of the deviatoric stress-axial strain curve of the coal sample is not obvious. In the elastic stage (phase I), the deviatoric stress-radial strain and deviatoric stress-volumetric strain curves also show an approximate linear trend, and the stress-strain relationship conforms to Hooke’s law. At this stage, no new cracks are formed in the coal sample, the axial strain and radial strain of the coal sample continue to increase, and the volumetric strain continues to decrease, and the deformation of the coal sample is mainly elastic and reversible. As the deviatoric stress increases, the stress concentration near the primary defect causes microcracks to initiate and grow slowly, but the microcracks do not expand further when the load is kept constant (phase II). Due to the generation of microcracks, the volumetric strain increasing rate of the coal sample gradually decreased, and the axial strain rate and radial strain rate both increased slowly. The deviatoric stress-strain curve slowly deviates from the original straight line. Entering the stage of unsteady crack propagation (phase III), the deviatoric stress-strain curve exhibits obvious nonlinear changes, and the slope gradually decreases, indicating that the coal sample stiffness decreases significantly. The generation of axial cracks leads to a rapid increase in radial strain, and the increase rate is significantly higher than the axial strain rate, and the coal sample exhibits volume expansion, resulting in irreversible plastic deformation. When the loading stress exceeds the peak strength, it enters the post-peak failure stage (phase IV), and the coal sample still has a certain bearing capacity, and the three stress-strain curves (axial strain/radial strain/volumetric strain) all bend downward and appear nonlinear. The volume of the coal sample expanded rapidly, and the volumetric strain rapidly decreased to a negative value.Figure 2
Representative stress-strain curves and variation in AE cumulative counts of coal sample under triaxial compression [21].With the increase of confining pressure, the variation law of elastic modulus is not the same, which is closely related to the internal microstructure, defects, and fissures of coal samples. If there are fewer defects such as internal pores and cracks, the confining pressure has less compacting and closing effect and has less influence on the elastic modulus. On the contrary, if there are a large number of defects in the coal sample, under the action of confining pressure, the pores and cracks are gradually compressed and closed, and the ability of the coal to resist deformation and damage increases, so the elastic modulus also increases. In addition, the coal specimens exhibit a transition from brittle deformation at low confining pressure to plastic deformation at high confining pressure.


### 2.2. Failure and Strength Characteristics

The cumulative AE counts represent the overall AE activity intensity during the fracture process of the coal sample. The changes of AE counts have a good corresponding relationship with the stress-strain curve [22]. At each turning point of the stress-strain curve, the cumulative AE counts change significantly, indicating that the coal sample entered different crack propagation stages showing different crack activity intensity, resulting in changes in AE counts.In general, the transition point of the AE event to the steady increasing phase can be considered as the crack initiation stress (σci), while the acceleration point of the AE event can be considered as the crack damage stress (σcd) [23]. As shown in Figure 2, in stage I, there is almost no AE signal. When the stress exceeds σci, the tensile stress at the tip of the primary crack and the concentrated stress at the pore boundary exceed the tensile strength of the coal particles, the primary crack in the coal sample begins to expand, and new cracks are initiated, so that the cumulative AE count of the coal sample increases linearly and slowly. At the crack density level at this stage, it can be considered that the expansion of cracks is independent of each other. Under a certain stress level, the cracks in the coal sample will not continuously expand, so the cracks are stable. When the stress reaches σcd, the crack connection and penetration cause unstable expansion, which makes the cumulative AE count of the coal sample increases exponentially, and its curve rises approximately vertically near the peak stress (σp). When the stress reaches σp, a macroscopic fracture surface is formed, the bearing capacity of the coal sample decreases, and a considerable number of AE events are also recorded, and then, the increasing trend decreases slightly. It can be seen that the deformation and failure process of gas-containing coal samples is a crack evolution process of primary crack propagation, new crack generation, dense, confluence, and penetration, and finally a macroscopic fracture surface is formed. The crack initiation stress σci, crack damage stress σcd, and peak stress σp of the coal sample control the entire crack evolution process.When the gas pressure is consistent, with the increase of the confining pressure, the deformation and crack development of the coal body during the failure process are limited to a certain extent, so that both the peak strength and the residual strength of the specimen increase. Both Mohr-Coulomb and Hoek-Brown strength criteria apply to gas-containing coals. The failure characteristics of coal samples are simple, the fracture surface is relatively single, and the failure mode is mainly shear failure. The inclination angle of the fracture surface is related to the confining pressure and the gas pressure. Under the lower confining pressure, the inclination angle is larger, and the fracture surface tends to the end of the coal sample. With the increase of confining pressure, the inclination angle becomes smaller, and the fracture surface shifts from the end to the side.


## 2.1. Deformation and Stiffness Characteristics

Representative stress-strain curves and variation in AE cumulative counts of coal sample under triaxial compression are shown in Figure2. The deviatoric stress gradually increases, and the axial strain and radial strain of the coal sample also increase, while the volumetric strain first decreases and then increases. It is mainly divided into four stages, namely, the compaction and elastic stage (phase I), the stable crack propagation stage (phase II), the nonstable crack propagation stage (phase III), and post-peak failure stage (phase IV). In the compaction stage, the pores and fissures inside the coal sample are gradually compacted and closed, the elastic modulus of the coal sample increases gradually, the stress-strain curve is upwardly convex, and the axial strain and radial strain of the coal sample increase, and the volume is compressed, forming an early nonlinear phase of the curve. It should be pointed out that during the triaxial compression test, some primary defects in the coal sample have been compressed and closed by the confining pressure before the axial loading, so that the compaction stage of the deviatoric stress-axial strain curve of the coal sample is not obvious. In the elastic stage (phase I), the deviatoric stress-radial strain and deviatoric stress-volumetric strain curves also show an approximate linear trend, and the stress-strain relationship conforms to Hooke’s law. At this stage, no new cracks are formed in the coal sample, the axial strain and radial strain of the coal sample continue to increase, and the volumetric strain continues to decrease, and the deformation of the coal sample is mainly elastic and reversible. As the deviatoric stress increases, the stress concentration near the primary defect causes microcracks to initiate and grow slowly, but the microcracks do not expand further when the load is kept constant (phase II). Due to the generation of microcracks, the volumetric strain increasing rate of the coal sample gradually decreased, and the axial strain rate and radial strain rate both increased slowly. The deviatoric stress-strain curve slowly deviates from the original straight line. Entering the stage of unsteady crack propagation (phase III), the deviatoric stress-strain curve exhibits obvious nonlinear changes, and the slope gradually decreases, indicating that the coal sample stiffness decreases significantly. The generation of axial cracks leads to a rapid increase in radial strain, and the increase rate is significantly higher than the axial strain rate, and the coal sample exhibits volume expansion, resulting in irreversible plastic deformation. When the loading stress exceeds the peak strength, it enters the post-peak failure stage (phase IV), and the coal sample still has a certain bearing capacity, and the three stress-strain curves (axial strain/radial strain/volumetric strain) all bend downward and appear nonlinear. The volume of the coal sample expanded rapidly, and the volumetric strain rapidly decreased to a negative value.Figure 2
Representative stress-strain curves and variation in AE cumulative counts of coal sample under triaxial compression [21].With the increase of confining pressure, the variation law of elastic modulus is not the same, which is closely related to the internal microstructure, defects, and fissures of coal samples. If there are fewer defects such as internal pores and cracks, the confining pressure has less compacting and closing effect and has less influence on the elastic modulus. On the contrary, if there are a large number of defects in the coal sample, under the action of confining pressure, the pores and cracks are gradually compressed and closed, and the ability of the coal to resist deformation and damage increases, so the elastic modulus also increases. In addition, the coal specimens exhibit a transition from brittle deformation at low confining pressure to plastic deformation at high confining pressure.


## 2.2. Failure and Strength Characteristics

The cumulative AE counts represent the overall AE activity intensity during the fracture process of the coal sample. The changes of AE counts have a good corresponding relationship with the stress-strain curve [22]. At each turning point of the stress-strain curve, the cumulative AE counts change significantly, indicating that the coal sample entered different crack propagation stages showing different crack activity intensity, resulting in changes in AE counts.In general, the transition point of the AE event to the steady increasing phase can be considered as the crack initiation stress (σci), while the acceleration point of the AE event can be considered as the crack damage stress (σcd) [23]. As shown in Figure 2, in stage I, there is almost no AE signal. When the stress exceeds σci, the tensile stress at the tip of the primary crack and the concentrated stress at the pore boundary exceed the tensile strength of the coal particles, the primary crack in the coal sample begins to expand, and new cracks are initiated, so that the cumulative AE count of the coal sample increases linearly and slowly. At the crack density level at this stage, it can be considered that the expansion of cracks is independent of each other. Under a certain stress level, the cracks in the coal sample will not continuously expand, so the cracks are stable. When the stress reaches σcd, the crack connection and penetration cause unstable expansion, which makes the cumulative AE count of the coal sample increases exponentially, and its curve rises approximately vertically near the peak stress (σp). When the stress reaches σp, a macroscopic fracture surface is formed, the bearing capacity of the coal sample decreases, and a considerable number of AE events are also recorded, and then, the increasing trend decreases slightly. It can be seen that the deformation and failure process of gas-containing coal samples is a crack evolution process of primary crack propagation, new crack generation, dense, confluence, and penetration, and finally a macroscopic fracture surface is formed. The crack initiation stress σci, crack damage stress σcd, and peak stress σp of the coal sample control the entire crack evolution process.When the gas pressure is consistent, with the increase of the confining pressure, the deformation and crack development of the coal body during the failure process are limited to a certain extent, so that both the peak strength and the residual strength of the specimen increase. Both Mohr-Coulomb and Hoek-Brown strength criteria apply to gas-containing coals. The failure characteristics of coal samples are simple, the fracture surface is relatively single, and the failure mode is mainly shear failure. The inclination angle of the fracture surface is related to the confining pressure and the gas pressure. Under the lower confining pressure, the inclination angle is larger, and the fracture surface tends to the end of the coal sample. With the increase of confining pressure, the inclination angle becomes smaller, and the fracture surface shifts from the end to the side.


## 3. Numerical Simulation Method

### 3.1. Numerical Test Platform

The discrete element method (DEM) was first proposed by Cundall and Strack [24] in 1979 to study the motion and interaction of particulate matter. The rock mass is relatively continuous macroscopically, while it is composed of a series of grains, pores, and fissures microscopically. It is difficult to solve the discrete and discontinuous problems on the microscopic level according to the conventional continuum mechanics method, but the DEM can simulate the discontinuous and nonuniform characteristics of rock mass close to the actual situation with the help of the design model of accumulation and cementation particles, which can meet the needs of various types of geotechnical engineering analysis.At present, the main international DEM commercial software includes PFC (2D), PFC (3D), and EDEM, and the open source software mainly includes Yade, ESyS-Particle, and LIGGGHTS. In China, a variety of excellent discrete element software has been designed, including 2D-Block, GDEM, SDEM, StreamDEM, and MatDEM. In the field of seepage, DEM is commonly used in water-rock interaction [25]. However, there are almost no DEM simulation analysis cases for gas-solid coupling of gas-containing coals. MatDEM software was developed by Professor Liu Chun of Nanjing University. It has the advantages of fast calculation speed and easy learning. It is a domestic software with complete independent intellectual property rights. It has been successfully applied in slope instability, hob rock breaking, and rock compaction failure [20, 26–28]. MatDEM software has the function of self-training materials. It only needs to input 6 macro-mechanical parameters to automatically obtain the material’s micromechanical parameters. Compared with general commercial discrete element software, such as PFC, which directly set the bond strength between particles, it greatly saves the time to adjust model parameters [20]. Therefore, we choose this software for secondary development to conduct gas-solid coupling research.Matdem cannot simulate the expansion and contraction process of coal mass caused by gas adsorption and desorption. In order to realize the coupling effect of adsorption expansion, pore gas pressure and stress, a permeation network is constructed by triangulation method, and pore gas pressure can be calculated. Based on the method of changing particle size in real time by adsorption expansion strain, a numerical simulation program of gas-containing gas-solid coupling was developed [20]. The specific steps of realizing the gas-solid coupling simulation process in MatDEM are as follows:
(1)
According to the particle diameter and the overall model size, a dense set of particles is generated, and the particles are given a random initial velocity, which naturally accumulates into a model structure under the action of gravity. In order to accelerate the accumulation process, a certain pressure is applied to the upper pressure plate, and the accumulation is repeated 2-6 times to obtain a rock mass structure that mimics natural deposition(2)
The aforementioned stacked particles are soft particles by default, and the preset macroscopic material properties are calculated and assigned to the microscopic particles through the macro-micro conversion formula. Equilibrium is used to release the suddenly increased elastic energy between particles. The equilibrium model adopts strong cementation equilibrium, and the cementation property between particles is not destroyed(3)
The spatial position of the particles after the material assignment does not change without loading. At this time, the particle coordinates are subjected to Delaunay triangulation to obtain the triangular matrix and the neighbor triangular matrix(4)
Calculate the pore area (2-dimensional) according to the spatial coordinates of each triangle vertex, as the gas storage space. The seepage distance between particles is the centroid distance of two adjacent triangles. The seepage channel is along the normal direction of the Delaunay edge, and the normal force is the interaction force between the two particles that constitute the Delaunay edge(5)
Fill the pore structure obtained by triangulation with a certain pressure of gas, calculate the particle expansion according to the gas pressure, calculate the pressure of the pore gas on the particles according to the particle position relationship, and assign the structure to the particles in the form of body force. Then, carry out gas pressure balance and stress balance. Among them, the gas pressure balance is based on the calculated seepage flow by the seepage parameters and the gas pressure difference at both ends of the seepage channel, and the stress balance is based on the displacement caused by the uneven force of the particles(6)
Repeat steps (4) and (5) until the model is balanced and the particle space position is determined(7)
Apply a load, perform step (4) after each stress balance and gas pressure balance, update the pore structure information, and realize the coupling process of gas pressure and particle stressThrough the above process, not only the coupling of real-time gas pressure and particle stress can be realized, but also the change of pore structure caused by gas adsorption and desorption can be taken into account, and the deformation and failure characteristics of gas-containing coal during the compression test can be obtained.


### 3.2. Numerical Simulation Scheme

Based on the developed DEM program, the numerical simulation of triaxial compression of gas-containing coal under different fixed confining pressures was carried out. The spatiotemporal evolution process of particle stress and pore pressure was obtained, the location and number of tensile cracks and shear cracks in the sample were monitored simultaneously, and the energy evolution of the particle system was also identified. The confining pressures were set to 0, 2, 4, 6, 8, and 10 MPa, respectively, and the gas pressure was constant at 1 MPa.Referring to the experimental data [29], there are 14,726 coal particles with an average radius of 3e-4m, and 27,945 pores are formed by triangulation. The size of the model after accumulation is 97 mm ×51 mm. Through the macro-micro conversion formula [26], the macro-mechanical parameters of the coal sample can be converted into the micromechanical parameters between coal particles. The macro-micro conversion formula is as shown in Equations (1)–(6). The specific macro- and microparameters are shown in Tables 1 and 2.
(1)Kn=2Ed41−2v,(2)Ks=21−5vEd41+v1−2v,(3)Xb=3Kn+Ks62KnKn+Ks·Tu·d2,(4)FS0=1−2μp6·Cu·d2,(5)μp=22+2I2+2I,(6)I=1+μi21/2+μi2,where Kn is the inter-element normal stiffness, Ks is the shear stiffness, Xb is the breaking displacement, FS0 is the shear resistance, μp is the coefficient of friction, E is the Young’s modulus, v is the Poisson’s ratio, Tu is the tensile strength, Cu is the compressive strength, and μi is the coefficient of intrinsic friction.Table 1
Macro-material parameters of coal sample.
Elastic modulusE/GPa9.534Poisson’s ratiov0.15Compressive strengthCu /MPa64Tensile strengthTu/MPa6.4Internal friction coefficientμi0.3536Densityρ/kg/m31650Table 2
Micromechanical parameters between coal particles.
Average normal stiffnessKn/MPa20.0460Average tangential stiffnessKs/MPa7.1590Average fracture forceFb/N11.8780Average shear resistanceFs/N37.6636Initial opening of poresa0/m1e-5Normal half forceF0/N1e6The limit adsorption capacityam/m3/t19.95Adsorption equilibrium constantbm/MPa-11.081The simulation results of the triaxial compression test are shown in Figure3. The simulation results before peak stress are basically consistent with the test results. In DEM, because the simulation process only relies on the input macro-parameters to build the model, the damage of the model cannot be controlled by cohesion like the continuum mechanics, so there is a certain gap between the simulation and the experimental results in the post-peak residual stage. In view of this situation, in the case of using cluster elements in contact with parallel bonds, the microscopic parameters can only be adjusted manually. After the confining pressure exceeds 30 MPa, it is close to the test results, and there is a big gap under the condition of 10 MPa [30].Figure 3
Comparison of triaxial compression simulation results and test results.


## 3.1. Numerical Test Platform

The discrete element method (DEM) was first proposed by Cundall and Strack [24] in 1979 to study the motion and interaction of particulate matter. The rock mass is relatively continuous macroscopically, while it is composed of a series of grains, pores, and fissures microscopically. It is difficult to solve the discrete and discontinuous problems on the microscopic level according to the conventional continuum mechanics method, but the DEM can simulate the discontinuous and nonuniform characteristics of rock mass close to the actual situation with the help of the design model of accumulation and cementation particles, which can meet the needs of various types of geotechnical engineering analysis.At present, the main international DEM commercial software includes PFC (2D), PFC (3D), and EDEM, and the open source software mainly includes Yade, ESyS-Particle, and LIGGGHTS. In China, a variety of excellent discrete element software has been designed, including 2D-Block, GDEM, SDEM, StreamDEM, and MatDEM. In the field of seepage, DEM is commonly used in water-rock interaction [25]. However, there are almost no DEM simulation analysis cases for gas-solid coupling of gas-containing coals. MatDEM software was developed by Professor Liu Chun of Nanjing University. It has the advantages of fast calculation speed and easy learning. It is a domestic software with complete independent intellectual property rights. It has been successfully applied in slope instability, hob rock breaking, and rock compaction failure [20, 26–28]. MatDEM software has the function of self-training materials. It only needs to input 6 macro-mechanical parameters to automatically obtain the material’s micromechanical parameters. Compared with general commercial discrete element software, such as PFC, which directly set the bond strength between particles, it greatly saves the time to adjust model parameters [20]. Therefore, we choose this software for secondary development to conduct gas-solid coupling research.Matdem cannot simulate the expansion and contraction process of coal mass caused by gas adsorption and desorption. In order to realize the coupling effect of adsorption expansion, pore gas pressure and stress, a permeation network is constructed by triangulation method, and pore gas pressure can be calculated. Based on the method of changing particle size in real time by adsorption expansion strain, a numerical simulation program of gas-containing gas-solid coupling was developed [20]. The specific steps of realizing the gas-solid coupling simulation process in MatDEM are as follows:
(1)
According to the particle diameter and the overall model size, a dense set of particles is generated, and the particles are given a random initial velocity, which naturally accumulates into a model structure under the action of gravity. In order to accelerate the accumulation process, a certain pressure is applied to the upper pressure plate, and the accumulation is repeated 2-6 times to obtain a rock mass structure that mimics natural deposition(2)
The aforementioned stacked particles are soft particles by default, and the preset macroscopic material properties are calculated and assigned to the microscopic particles through the macro-micro conversion formula. Equilibrium is used to release the suddenly increased elastic energy between particles. The equilibrium model adopts strong cementation equilibrium, and the cementation property between particles is not destroyed(3)
The spatial position of the particles after the material assignment does not change without loading. At this time, the particle coordinates are subjected to Delaunay triangulation to obtain the triangular matrix and the neighbor triangular matrix(4)
Calculate the pore area (2-dimensional) according to the spatial coordinates of each triangle vertex, as the gas storage space. The seepage distance between particles is the centroid distance of two adjacent triangles. The seepage channel is along the normal direction of the Delaunay edge, and the normal force is the interaction force between the two particles that constitute the Delaunay edge(5)
Fill the pore structure obtained by triangulation with a certain pressure of gas, calculate the particle expansion according to the gas pressure, calculate the pressure of the pore gas on the particles according to the particle position relationship, and assign the structure to the particles in the form of body force. Then, carry out gas pressure balance and stress balance. Among them, the gas pressure balance is based on the calculated seepage flow by the seepage parameters and the gas pressure difference at both ends of the seepage channel, and the stress balance is based on the displacement caused by the uneven force of the particles(6)
Repeat steps (4) and (5) until the model is balanced and the particle space position is determined(7)
Apply a load, perform step (4) after each stress balance and gas pressure balance, update the pore structure information, and realize the coupling process of gas pressure and particle stressThrough the above process, not only the coupling of real-time gas pressure and particle stress can be realized, but also the change of pore structure caused by gas adsorption and desorption can be taken into account, and the deformation and failure characteristics of gas-containing coal during the compression test can be obtained.


## 3.2. Numerical Simulation Scheme

Based on the developed DEM program, the numerical simulation of triaxial compression of gas-containing coal under different fixed confining pressures was carried out. The spatiotemporal evolution process of particle stress and pore pressure was obtained, the location and number of tensile cracks and shear cracks in the sample were monitored simultaneously, and the energy evolution of the particle system was also identified. The confining pressures were set to 0, 2, 4, 6, 8, and 10 MPa, respectively, and the gas pressure was constant at 1 MPa.Referring to the experimental data [29], there are 14,726 coal particles with an average radius of 3e-4m, and 27,945 pores are formed by triangulation. The size of the model after accumulation is 97 mm ×51 mm. Through the macro-micro conversion formula [26], the macro-mechanical parameters of the coal sample can be converted into the micromechanical parameters between coal particles. The macro-micro conversion formula is as shown in Equations (1)–(6). The specific macro- and microparameters are shown in Tables 1 and 2.
(1)Kn=2Ed41−2v,(2)Ks=21−5vEd41+v1−2v,(3)Xb=3Kn+Ks62KnKn+Ks·Tu·d2,(4)FS0=1−2μp6·Cu·d2,(5)μp=22+2I2+2I,(6)I=1+μi21/2+μi2,where Kn is the inter-element normal stiffness, Ks is the shear stiffness, Xb is the breaking displacement, FS0 is the shear resistance, μp is the coefficient of friction, E is the Young’s modulus, v is the Poisson’s ratio, Tu is the tensile strength, Cu is the compressive strength, and μi is the coefficient of intrinsic friction.Table 1
Macro-material parameters of coal sample.
Elastic modulusE/GPa9.534Poisson’s ratiov0.15Compressive strengthCu /MPa64Tensile strengthTu/MPa6.4Internal friction coefficientμi0.3536Densityρ/kg/m31650Table 2
Micromechanical parameters between coal particles.
Average normal stiffnessKn/MPa20.0460Average tangential stiffnessKs/MPa7.1590Average fracture forceFb/N11.8780Average shear resistanceFs/N37.6636Initial opening of poresa0/m1e-5Normal half forceF0/N1e6The limit adsorption capacityam/m3/t19.95Adsorption equilibrium constantbm/MPa-11.081The simulation results of the triaxial compression test are shown in Figure3. The simulation results before peak stress are basically consistent with the test results. In DEM, because the simulation process only relies on the input macro-parameters to build the model, the damage of the model cannot be controlled by cohesion like the continuum mechanics, so there is a certain gap between the simulation and the experimental results in the post-peak residual stage. In view of this situation, in the case of using cluster elements in contact with parallel bonds, the microscopic parameters can only be adjusted manually. After the confining pressure exceeds 30 MPa, it is close to the test results, and there is a big gap under the condition of 10 MPa [30].Figure 3
Comparison of triaxial compression simulation results and test results.


## 4. Numerical Simulation Results and Analysis

### 4.1. Stress-Strain Curves

The stress-strain curves of gas-containing coal under different confining pressures are shown in Figure4. Similar to the laboratory test results, the deformation process of gas-containing coal can be divided into five stages, namely, elastic deformation, crack propagation, post-peak failure, and residual stage. However, due to the limitations of DEM, the compaction stage cannot be accurately simulated without special treatment. The crack propagation stage is reflected in a small sudden drop in the prepeak stress, indicating that the particles inside the model release stress due to the failure of cementation. The damaged coal sample continues to bear the structural pressure under the limitation of shear force and confining pressure, so the internal stress continues to rise after the sudden drop. In the post-peak failure stage, a large stress drop occurs, and the internal through cracks cause the instability of the overall structure. In the residual stage, the residual strength is closely related to the confining pressure and friction coefficient, and the bearing capacity of the overall structure is maintained only by the friction between particles.Figure 4
Stress-strain curves of gas-containing coal under different confining pressures.The elastic moduli of the specimens obtained from the numerical simulations are different from those obtained from the laboratory tests. The test found that with the increase of confining pressure, the micropores were closed, and the weak cementation was compacted, which increased the compactness of the particles inside the coal sample, and the elastic modulus increased in different amplitudes. Since the basic assumption in discrete element software (MatDEM, PFC, and Yade) is to decompose macroscopic objects into rigid spherical particles, bond springs are set between adjacent particles, and the stiffness of these springs will not change with compaction before breaking. Not only that, the axial contact between two particles does not affect the lateral strain, and the macroscopic lateral expansion strain is only achieved by the tangential displacement between the particles caused by nonaxial stress, which is also different from the actual situation. Therefore, DEM cannot simulate the significant growth of elastic modulus. F. Huang et al. [31] did not show that rock stiffness increases with the increase of confining pressure through PFC (2D) simulation. The simulation result of T. Zhang et al. [32] is that when the confining pressure increases from 2 MPa to 25 MPa, the elastic modulus of the rock only increases from 18.5GPa to 19.2GPa, with an increase of less than 4%. It is worth mentioning that Z. X. Liu et al. [33] carried out numerical simulation of coal samples with initial pores based on the SEM results of heterogeneous coal samples; that is, some particles were deleted at the position of large pores, and the simulated stiffness of coal samples was relatively significantly increased with the increasing confining pressure. This further shows that the influence of confining pressure on the elastic modulus of coal rock is largely a reflection of the compaction effect.The variation of peak strength and residual strength of gas-containing coal with confining pressure is shown in Figure5. When there is no confining pressure, the peak strength of coal sample is only 22.36 MPa, and the residual strength is 5.8 MPa, and they increase to 44.19 MPa and 21.49 MPa, respectively, when the confining pressure is 10 MPa. Both the peak strength and the residual strength increase approximately linearly with increasing confining pressure, which is consistent with the experimental results [6, 7, 9, 10, 29]. The strengthening effect of confining pressure on the strength of gas-containing coal mainly lies in the external support effect of confining pressure on the sample. The confining pressure constrains the lateral deformation, making the internal pores and cracks more compact, and it is difficult to form a failure surface, so the peak strength increases. In the post-peak stage, the confining pressure increases the normal pressure on the shear failure surface, causing the shear sliding friction to increase, and thus, the residual strength also increases.Figure 5
Variation of peak strength and residual strength of gas-containing coal with confining pressure.


### 4.2. Particle Stress and Pore Pressure

The particle stress distribution of gas-containing coal under different confining pressures is shown in Figure6. With the application of axial load, the high-stress area generated at the upper end of the sample gradually moves down, and the stress is released at the place where the microcracks are generated. After the overall instability, there is local support stress at the macroscopic penetration crack, and a large amount of elastic energy is released in other areas. Under 0 MPa confining pressure (Figure 6(a)), the failure of the coal sample starts from the upper right corner. With the increase of the load, the effective bearing area of the upper end decreases, resulting in a large stress at the end. The failure shows obvious brittleness, and the angle between the main crack and the horizontal direction is close to 90° in the middle of the crack. With the increase of the load, the high-stress areas gradually moved from top to bottom. Before the instability, the high-stress area is concentrated near the midline at the upper end and tends to concentrate at the lower right corner at the lower end. After failure, the high-stress area largely disappears and only appears near the crack, and the overall structure has almost no bearing capacity.Figure 6
The particle stress distribution of gas-containing coal under different confining pressures.
(a)
0 MPa(b)
2 MPa(c)
4 MPa(d)
6 MPa(e)
8 MPa(f)
10 MPaThe gas pressure distribution of gas-containing coal under different confining pressures is shown in Figure7. Displacement occurs due to the fracture of cementation between particles inside the coal sample, resulting in internal microcracks. Affected by the shear dilatation effect, the pore volume between particles increases, and the pore gas pressure decreases instantaneously, causing internal gas seepage. The seepage channel formed by macroscopic cracks can be clearly seen in Figure 7, which is basically the same as the stress release position due to structural failure in Figure 6. When the confining pressure is lower than 4 MPa (see Figures 7(a)–7(c)), the seepage channel is dominated by a main crack that penetrates up and down. With the increase of the confining pressure, the crack tip at the upper end moves from right to left, and the inclination angle of the crack increases gradually. The fracture form changes under the confining pressure of 6 MPa, as shown in Figure 7(d). The upper right part of the sample was partially ruptured, causing the overall instability, while the lower part remained intact. Under high confining pressures (8 MPa and 10 MPa), as shown in Figures 7(e) and 7(f), the two macroscopic cracks intersect to form a through main crack, and the intersection point is located in the middle of the sample. These phenomena are consistent with laboratory test results.Figure 7
The gas pressure distribution of gas-containing coal under different confining pressures.
(a)
0 MPa(b)
2 MPa(c)
4 MPa(d)
6 MPa(e)
8 MPa(f)
10 MPa


### 4.3. Crack Evolution

Figure8 shows the distribution of cracks in the deformation and failure process of gas-containing coal under different confining pressures. In general, the microcracks are more distributed in the upper half and the boundary of the sample. In the initial stage, due to the expansion of gas adsorption, some initial cracks are generated. As the pressure increases, local high stress occurs at the contact point between the boundary and the pressure plate, and microcracks appear. As the confining pressure increases, the inclination angle of the macroscopic main crack formed during failure gradually decreases. When there is no confining pressure, the inclination angle is close to 90°, showing obvious brittle failure, while when the confining pressure is 10 MPa, the inclination angle is about 45°, showing a certain ductile failure.Figure 8
Crack distribution during deformation and failure of gas-containing coal under different confining pressures.Shear cracks and tensile cracks can be identified by the fracture of tangential springs and normal springs between coal particles. Figure9 shows the changes in the number of shear cracks and tensile cracks of gas-containing coal under different confining pressures. In general, the total number of cracks after failure decreases with the increase of confining pressure, but at the peak stress point, the number of cracks tends to increase with the increase of confining pressure, which means that cracks grow at lower loads when the confining pressure is smaller, and the degree of rupture is greater. With the increase of the confining pressure, the strength of the rock gradually increases, and the failure strain increases, giving the microcracks a longer development time. However, the existence of the confining pressure limits the development space of the post-peak cracks, resulting in more structures remaining relatively intact. It is worth noting that the number of shear cracks after failure increases with the increase of confining pressure, indicating that the failure mode of internal shear slip gradually occupies a larger proportion.Figure 9
Changes in the number of cracks during deformation and failure of gas-containing coal under different confining pressures.
(a)
0 MPa(b)
2 MPa(c)
4 MPa(d)
6 MPa(e)
8 MPa(f)
10 MPa


### 4.4. Energy Conversion

During the triaxial compression process of gas-containing coal, the energy accumulates and dissipates continuously. Coal samples absorb and store energy in the form of elastic strain energy. When the load reaches the peak value, the elastic strain energy stored in the coal sample is instantly released and transformed into dissipated energy, which is the driving force of coal sample failure [34]. Confining pressure and gas pressure have a greater influence on the energy conversion characteristics of coal samples.In order to ensure energy conservation and realize energy dissipation, mechanical energy is converted into heat under the action of damping force, friction force, and fracture. The discrete element system can accurately calculate various types of mechanical energy and heat, and the sum of mechanical energy and heat is always constant.Mechanical energy in discrete element systems includes kinetic energy, elastic potential energy, and gravitational potential energy, and the three can be converted to each other [26].
(1)
The elastic potential energy is the sum of the strain energies of the normal and tangential springs between particles:(7)Ee=0.5KnXn2+0.5KsXs2,where Kn, Ks, Xn, and Xs are the normal stiffness, tangential stiffness, normal displacement, and tangential displacement between particles, respectively.
(2)
Gravitational potential energy is the energy possessed by particles due to gravity:(8)Eg=mgh,where m is the mass of the particle, g is the acceleration of gravity, and h is the height of the particle from the reference plane.
(3)
Kinetic energy is the energy possessed by particles due to motion:(9)Ek=0.5mv2,where v represents the particle velocity.The heat generated in this model can be divided into damping heat, frictional heat, and fracture heat according to the different acting forces, which correspond to damping dissipation, frictional force, and cementation fracture [26].
(1)
Damping Heat: When elastic waves propagate between particles, some mechanical energy can be converted into thermal energy through friction and scattering. Therefore, in discrete element systems, damping is introduced to weaken the elastic wave and dissipate the kinetic energy. The damping force acting on the particles is(10)Fd=−ηv,where η represents the damping coefficient. Since the time step determined in the simulation is quite small, if the velocity of the particle remains constant over a time step, the expression for the damping heat calculation is
(11)Qd=−Fddx,where dx represents the displacement of the particles within the current time step.
(2)
Fracture Heat: In the real world, after the fracture of cementation between particles, the elastic potential energy of an object is partially or completely converted into thermal energy through the damping vibration of the elastic structure, and this process cannot be realized in DEM. Therefore, it is assumed that the spring representing the elastic structure stops vibrating immediately when it breaks, and its elastic potential energy is directly converted into heat.When the interparticle cement breaks in the tensile state, the elastic potential energy of the normal and tangential springs is instantly reduced to 0, and the reduced elastic energy is the sum of the elastic potential energy of the normal and tangential springs:(12)Ee,t=0.5KnXn2+0.5KsXs2.When the interparticle cement breaks in the compressed state, the normal force and the corresponding elastic potential energy do not change. The tangential force is reduced fromFsmax to Fsmax, and the elastic energy consumed can be expressed as:
(13)Ee,c=0.5Fsmax2−Fsmax′2/Ks.The sum of the elastic potential energy dissipated when fracture occurs between particles is the heat of fracture, namely(14)Qb=Ee,t,Fn>0Ee,c,Fn≤0.(3)
Frictional Heat: When the tangential force between the two particles is greater than the maximum static friction force, the particles enter a relative sliding state, and the product of the average sliding friction force and the sliding distance is the frictional heat generated by the sliding:(15)Qf=0.5Fs1+Fs2⋅dS,where Fs1 and Fs2 are the sliding friction forces that start and end within the current time step, respectively, and dS represents the sliding distance.Based on the above theory, the elastic energy and dissipation energy in the process of deformation and failure of gas-containing coal are calculated. Since kinetic energy is a transient quantity, it is eventually converted into frictional heat and damping heat by damping vibration. Frictional heat and fracture heat are cumulative, so the dissipation energy in the system is the sum of damping heat, friction heat, and fracture heat.Figure10 shows the change in elastic energy of gas-containing coal under different confining pressures when the gas pressure is 1 MPa. Due to the expansion stress caused by gas adsorption, the elastic energy is not zero when not loaded, and the elastic energy gradually increases as the load increases. Define the peak value of the elastic energy curve as the peak elastic energy, which reflects the energy absorption capacity of the coal sample. The stable value of post-peak elastic energy is defined as residual elastic energy, which reflects the residual energy absorption capacity after the destruction of coal samples. Both the peak elastic energy and the residual elastic energy increase approximately linearly with the increase of confining pressure, as shown in Figure 11. Among them, the sample destruction is incomplete when the confining pressure is 6 MPa, and the lower half retains a relatively complete structure as shown in Figure 8, so the residual elastic energy is relatively large.Figure 10
Changes in the elastic energy of gas-containing coal under different confining pressures.Figure 11
Changes of the peak elastic energy and the residual elastic energy with confining pressure.The changes of dissipated energy and stress are almost synchronous. Figure12 shows the variation of dissipated energy of coal samples under different confining pressures. It increases linearly before failure, indicating that irreversible energy dissipation still exists in the elastic stage, which may result from the increase of a small number of microcracks (see Figure 8). The dissipated energy increases sharply near the peak point and then tends to be stable. Different from the elastic energy, the maximum value of dissipated energy under different confining pressures seems irregular.Figure 12
Changes in the dissipated energy of gas-containing coal under different confining pressures.


### 4.5. Burst Tendency

In the national standard of the People’s Republic of China “Determination Method of Coal Burst Tendency Index GB/T 25217.2-2010,” the impact energy index can be used to evaluate the burst tendency of coal, which is defined as the ratio of the deformation energy accumulated before peak to the deformation energy dissipated after peak in the stress-strain curve.(16)KE=AeAx,where Ae is the deformation energy accumulated before the peak and Ax is the deformation energy dissipated after the peak. If the value of impact energy index is greater than 5, it is a strong burst tendency. If the value ranges from 1.5 to 5, it is a weak burst tendency. If the value is less than 1.5, it is no burst tendency.Figure13 shows the change of coal burst tendency with confining pressure. The maximum cumulative elastic energy before peak point increases linearly with the confining pressure, while the rate of increase in post-peak dissipative energy decreases gradually, resulting in a gradual increase in the impact energy index. This is the opposite of the effect of gas pressure on coal burst tendency. X. Liu et al. [35] conducted uniaxial compression tests on gas-containing coal under different gas pressures, and evaluated the burst tendency according to four different indices, including dynamic failure time, elastic energy index, impact energy index, and uniaxial compressive strength, and the results showed that as the gas pressure increased, the burst tendency of gas-containing coals would be weak to none. The discrete element simulation results conducted by Z.Z. Zhang et al. [20], and the energy evolution analysis by Y. Xue et al. [36] also confirms this.Figure 13
Changes in coal burst tendency under different confining pressures.


## 4.1. Stress-Strain Curves

The stress-strain curves of gas-containing coal under different confining pressures are shown in Figure4. Similar to the laboratory test results, the deformation process of gas-containing coal can be divided into five stages, namely, elastic deformation, crack propagation, post-peak failure, and residual stage. However, due to the limitations of DEM, the compaction stage cannot be accurately simulated without special treatment. The crack propagation stage is reflected in a small sudden drop in the prepeak stress, indicating that the particles inside the model release stress due to the failure of cementation. The damaged coal sample continues to bear the structural pressure under the limitation of shear force and confining pressure, so the internal stress continues to rise after the sudden drop. In the post-peak failure stage, a large stress drop occurs, and the internal through cracks cause the instability of the overall structure. In the residual stage, the residual strength is closely related to the confining pressure and friction coefficient, and the bearing capacity of the overall structure is maintained only by the friction between particles.Figure 4
Stress-strain curves of gas-containing coal under different confining pressures.The elastic moduli of the specimens obtained from the numerical simulations are different from those obtained from the laboratory tests. The test found that with the increase of confining pressure, the micropores were closed, and the weak cementation was compacted, which increased the compactness of the particles inside the coal sample, and the elastic modulus increased in different amplitudes. Since the basic assumption in discrete element software (MatDEM, PFC, and Yade) is to decompose macroscopic objects into rigid spherical particles, bond springs are set between adjacent particles, and the stiffness of these springs will not change with compaction before breaking. Not only that, the axial contact between two particles does not affect the lateral strain, and the macroscopic lateral expansion strain is only achieved by the tangential displacement between the particles caused by nonaxial stress, which is also different from the actual situation. Therefore, DEM cannot simulate the significant growth of elastic modulus. F. Huang et al. [31] did not show that rock stiffness increases with the increase of confining pressure through PFC (2D) simulation. The simulation result of T. Zhang et al. [32] is that when the confining pressure increases from 2 MPa to 25 MPa, the elastic modulus of the rock only increases from 18.5GPa to 19.2GPa, with an increase of less than 4%. It is worth mentioning that Z. X. Liu et al. [33] carried out numerical simulation of coal samples with initial pores based on the SEM results of heterogeneous coal samples; that is, some particles were deleted at the position of large pores, and the simulated stiffness of coal samples was relatively significantly increased with the increasing confining pressure. This further shows that the influence of confining pressure on the elastic modulus of coal rock is largely a reflection of the compaction effect.The variation of peak strength and residual strength of gas-containing coal with confining pressure is shown in Figure5. When there is no confining pressure, the peak strength of coal sample is only 22.36 MPa, and the residual strength is 5.8 MPa, and they increase to 44.19 MPa and 21.49 MPa, respectively, when the confining pressure is 10 MPa. Both the peak strength and the residual strength increase approximately linearly with increasing confining pressure, which is consistent with the experimental results [6, 7, 9, 10, 29]. The strengthening effect of confining pressure on the strength of gas-containing coal mainly lies in the external support effect of confining pressure on the sample. The confining pressure constrains the lateral deformation, making the internal pores and cracks more compact, and it is difficult to form a failure surface, so the peak strength increases. In the post-peak stage, the confining pressure increases the normal pressure on the shear failure surface, causing the shear sliding friction to increase, and thus, the residual strength also increases.Figure 5
Variation of peak strength and residual strength of gas-containing coal with confining pressure.


## 4.2. Particle Stress and Pore Pressure

The particle stress distribution of gas-containing coal under different confining pressures is shown in Figure6. With the application of axial load, the high-stress area generated at the upper end of the sample gradually moves down, and the stress is released at the place where the microcracks are generated. After the overall instability, there is local support stress at the macroscopic penetration crack, and a large amount of elastic energy is released in other areas. Under 0 MPa confining pressure (Figure 6(a)), the failure of the coal sample starts from the upper right corner. With the increase of the load, the effective bearing area of the upper end decreases, resulting in a large stress at the end. The failure shows obvious brittleness, and the angle between the main crack and the horizontal direction is close to 90° in the middle of the crack. With the increase of the load, the high-stress areas gradually moved from top to bottom. Before the instability, the high-stress area is concentrated near the midline at the upper end and tends to concentrate at the lower right corner at the lower end. After failure, the high-stress area largely disappears and only appears near the crack, and the overall structure has almost no bearing capacity.Figure 6
The particle stress distribution of gas-containing coal under different confining pressures.
(a)
0 MPa(b)
2 MPa(c)
4 MPa(d)
6 MPa(e)
8 MPa(f)
10 MPaThe gas pressure distribution of gas-containing coal under different confining pressures is shown in Figure7. Displacement occurs due to the fracture of cementation between particles inside the coal sample, resulting in internal microcracks. Affected by the shear dilatation effect, the pore volume between particles increases, and the pore gas pressure decreases instantaneously, causing internal gas seepage. The seepage channel formed by macroscopic cracks can be clearly seen in Figure 7, which is basically the same as the stress release position due to structural failure in Figure 6. When the confining pressure is lower than 4 MPa (see Figures 7(a)–7(c)), the seepage channel is dominated by a main crack that penetrates up and down. With the increase of the confining pressure, the crack tip at the upper end moves from right to left, and the inclination angle of the crack increases gradually. The fracture form changes under the confining pressure of 6 MPa, as shown in Figure 7(d). The upper right part of the sample was partially ruptured, causing the overall instability, while the lower part remained intact. Under high confining pressures (8 MPa and 10 MPa), as shown in Figures 7(e) and 7(f), the two macroscopic cracks intersect to form a through main crack, and the intersection point is located in the middle of the sample. These phenomena are consistent with laboratory test results.Figure 7
The gas pressure distribution of gas-containing coal under different confining pressures.
(a)
0 MPa(b)
2 MPa(c)
4 MPa(d)
6 MPa(e)
8 MPa(f)
10 MPa


## 4.3. Crack Evolution

Figure8 shows the distribution of cracks in the deformation and failure process of gas-containing coal under different confining pressures. In general, the microcracks are more distributed in the upper half and the boundary of the sample. In the initial stage, due to the expansion of gas adsorption, some initial cracks are generated. As the pressure increases, local high stress occurs at the contact point between the boundary and the pressure plate, and microcracks appear. As the confining pressure increases, the inclination angle of the macroscopic main crack formed during failure gradually decreases. When there is no confining pressure, the inclination angle is close to 90°, showing obvious brittle failure, while when the confining pressure is 10 MPa, the inclination angle is about 45°, showing a certain ductile failure.Figure 8
Crack distribution during deformation and failure of gas-containing coal under different confining pressures.Shear cracks and tensile cracks can be identified by the fracture of tangential springs and normal springs between coal particles. Figure9 shows the changes in the number of shear cracks and tensile cracks of gas-containing coal under different confining pressures. In general, the total number of cracks after failure decreases with the increase of confining pressure, but at the peak stress point, the number of cracks tends to increase with the increase of confining pressure, which means that cracks grow at lower loads when the confining pressure is smaller, and the degree of rupture is greater. With the increase of the confining pressure, the strength of the rock gradually increases, and the failure strain increases, giving the microcracks a longer development time. However, the existence of the confining pressure limits the development space of the post-peak cracks, resulting in more structures remaining relatively intact. It is worth noting that the number of shear cracks after failure increases with the increase of confining pressure, indicating that the failure mode of internal shear slip gradually occupies a larger proportion.Figure 9
Changes in the number of cracks during deformation and failure of gas-containing coal under different confining pressures.
(a)
0 MPa(b)
2 MPa(c)
4 MPa(d)
6 MPa(e)
8 MPa(f)
10 MPa


## 4.4. Energy Conversion

During the triaxial compression process of gas-containing coal, the energy accumulates and dissipates continuously. Coal samples absorb and store energy in the form of elastic strain energy. When the load reaches the peak value, the elastic strain energy stored in the coal sample is instantly released and transformed into dissipated energy, which is the driving force of coal sample failure [34]. Confining pressure and gas pressure have a greater influence on the energy conversion characteristics of coal samples.In order to ensure energy conservation and realize energy dissipation, mechanical energy is converted into heat under the action of damping force, friction force, and fracture. The discrete element system can accurately calculate various types of mechanical energy and heat, and the sum of mechanical energy and heat is always constant.Mechanical energy in discrete element systems includes kinetic energy, elastic potential energy, and gravitational potential energy, and the three can be converted to each other [26].
(1)
The elastic potential energy is the sum of the strain energies of the normal and tangential springs between particles:(7)Ee=0.5KnXn2+0.5KsXs2,where Kn, Ks, Xn, and Xs are the normal stiffness, tangential stiffness, normal displacement, and tangential displacement between particles, respectively.
(2)
Gravitational potential energy is the energy possessed by particles due to gravity:(8)Eg=mgh,where m is the mass of the particle, g is the acceleration of gravity, and h is the height of the particle from the reference plane.
(3)
Kinetic energy is the energy possessed by particles due to motion:(9)Ek=0.5mv2,where v represents the particle velocity.The heat generated in this model can be divided into damping heat, frictional heat, and fracture heat according to the different acting forces, which correspond to damping dissipation, frictional force, and cementation fracture [26].
(1)
Damping Heat: When elastic waves propagate between particles, some mechanical energy can be converted into thermal energy through friction and scattering. Therefore, in discrete element systems, damping is introduced to weaken the elastic wave and dissipate the kinetic energy. The damping force acting on the particles is(10)Fd=−ηv,where η represents the damping coefficient. Since the time step determined in the simulation is quite small, if the velocity of the particle remains constant over a time step, the expression for the damping heat calculation is
(11)Qd=−Fddx,where dx represents the displacement of the particles within the current time step.
(2)
Fracture Heat: In the real world, after the fracture of cementation between particles, the elastic potential energy of an object is partially or completely converted into thermal energy through the damping vibration of the elastic structure, and this process cannot be realized in DEM. Therefore, it is assumed that the spring representing the elastic structure stops vibrating immediately when it breaks, and its elastic potential energy is directly converted into heat.When the interparticle cement breaks in the tensile state, the elastic potential energy of the normal and tangential springs is instantly reduced to 0, and the reduced elastic energy is the sum of the elastic potential energy of the normal and tangential springs:(12)Ee,t=0.5KnXn2+0.5KsXs2.When the interparticle cement breaks in the compressed state, the normal force and the corresponding elastic potential energy do not change. The tangential force is reduced fromFsmax to Fsmax, and the elastic energy consumed can be expressed as:
(13)Ee,c=0.5Fsmax2−Fsmax′2/Ks.The sum of the elastic potential energy dissipated when fracture occurs between particles is the heat of fracture, namely(14)Qb=Ee,t,Fn>0Ee,c,Fn≤0.(3)
Frictional Heat: When the tangential force between the two particles is greater than the maximum static friction force, the particles enter a relative sliding state, and the product of the average sliding friction force and the sliding distance is the frictional heat generated by the sliding:(15)Qf=0.5Fs1+Fs2⋅dS,where Fs1 and Fs2 are the sliding friction forces that start and end within the current time step, respectively, and dS represents the sliding distance.Based on the above theory, the elastic energy and dissipation energy in the process of deformation and failure of gas-containing coal are calculated. Since kinetic energy is a transient quantity, it is eventually converted into frictional heat and damping heat by damping vibration. Frictional heat and fracture heat are cumulative, so the dissipation energy in the system is the sum of damping heat, friction heat, and fracture heat.Figure10 shows the change in elastic energy of gas-containing coal under different confining pressures when the gas pressure is 1 MPa. Due to the expansion stress caused by gas adsorption, the elastic energy is not zero when not loaded, and the elastic energy gradually increases as the load increases. Define the peak value of the elastic energy curve as the peak elastic energy, which reflects the energy absorption capacity of the coal sample. The stable value of post-peak elastic energy is defined as residual elastic energy, which reflects the residual energy absorption capacity after the destruction of coal samples. Both the peak elastic energy and the residual elastic energy increase approximately linearly with the increase of confining pressure, as shown in Figure 11. Among them, the sample destruction is incomplete when the confining pressure is 6 MPa, and the lower half retains a relatively complete structure as shown in Figure 8, so the residual elastic energy is relatively large.Figure 10
Changes in the elastic energy of gas-containing coal under different confining pressures.Figure 11
Changes of the peak elastic energy and the residual elastic energy with confining pressure.The changes of dissipated energy and stress are almost synchronous. Figure12 shows the variation of dissipated energy of coal samples under different confining pressures. It increases linearly before failure, indicating that irreversible energy dissipation still exists in the elastic stage, which may result from the increase of a small number of microcracks (see Figure 8). The dissipated energy increases sharply near the peak point and then tends to be stable. Different from the elastic energy, the maximum value of dissipated energy under different confining pressures seems irregular.Figure 12
Changes in the dissipated energy of gas-containing coal under different confining pressures.


## 4.5. Burst Tendency

In the national standard of the People’s Republic of China “Determination Method of Coal Burst Tendency Index GB/T 25217.2-2010,” the impact energy index can be used to evaluate the burst tendency of coal, which is defined as the ratio of the deformation energy accumulated before peak to the deformation energy dissipated after peak in the stress-strain curve.(16)KE=AeAx,where Ae is the deformation energy accumulated before the peak and Ax is the deformation energy dissipated after the peak. If the value of impact energy index is greater than 5, it is a strong burst tendency. If the value ranges from 1.5 to 5, it is a weak burst tendency. If the value is less than 1.5, it is no burst tendency.Figure13 shows the change of coal burst tendency with confining pressure. The maximum cumulative elastic energy before peak point increases linearly with the confining pressure, while the rate of increase in post-peak dissipative energy decreases gradually, resulting in a gradual increase in the impact energy index. This is the opposite of the effect of gas pressure on coal burst tendency. X. Liu et al. [35] conducted uniaxial compression tests on gas-containing coal under different gas pressures, and evaluated the burst tendency according to four different indices, including dynamic failure time, elastic energy index, impact energy index, and uniaxial compressive strength, and the results showed that as the gas pressure increased, the burst tendency of gas-containing coals would be weak to none. The discrete element simulation results conducted by Z.Z. Zhang et al. [20], and the energy evolution analysis by Y. Xue et al. [36] also confirms this.Figure 13
Changes in coal burst tendency under different confining pressures.


## 5. Mechanism of Confining Pressure Effect

Gas-containing coal is a typical heterogeneous multiphase medium composed of coal matrix, pores, fissures, free gas and adsorbed gas. A large number of studies have shown that the deformation and failure of gas-containing coal is jointly driven by confining pressure, gas pressure, and adsorption expansion stress, as shown in Figure14.Figure 14
Deformation and failure mechanism of coal driven by confining pressure, free gas and adsorbed gas.The influence of gas on coal strength mainly includes mechanical and nonmechanical effects. The free gas in the internal pores and cracks not only expands the volume of coal and reduces its density, but also exerts a force opposite to the confining pressure, reducing the effective confining pressure of the coal body, and promoting the expansion of primary and new cracks, thereby accelerating the instability and destruction of coal. This is the mechanical effect. The nonmechanical effect can contain two aspects. On the one hand, the adsorption of gas by the coal body reduces the surface tension of micropores and fractures, resulting in a decrease in the attractive force between coal molecules, and the confinement ability of the coal matrix to coal molecules is also weakened accordingly, which leads to the adsorption expansion deformation of the coal matrix. Therefore, additional expansion stress is generated in the coal body, resulting in an additional tensile stress zone at the tip of the fracture [37]. On the other hand, the adsorbed gas will chemically react with the functional groups on the coal surface, that is, the phenomenon of chemical adsorption. These chemical reactions will change the macromolecular structure of the coal, which may reduce the strength of the coal body. These effects will reduce the mechanical parameters such as peak strength, elastic modulus, cohesion, and internal friction angle macroscopically. However, the confining pressure has the opposite effect on the coal strength and stiffness. The confining pressure acts as an external uniform support for the coal sample, making the pores and fissures more compact, the spacing between coal particles decreases, and the cohesion between particles increases. Macroscopically, the ability of coal samples to resist deformation is enhanced, and this strengthening effect increases with the increase of confining pressure.From the perspective of engineering scale, assuming that the gas pressure is constant, when the confining pressure is low, the coal body would be destroyed in a short time, and less elastic deformation energy of the coal would be released, while the gas expansion energy is relatively large, which means that the possible disaster is coal and gas outburst. As the confining pressure increases, the coal strength becomes higher, and the magnitude of the elastic energy released by the coal during failure is comparable to the gas expansion energy, which indicates that the type of disaster may be gas outburst-rockburst coupled dynamic disaster. When the confining pressure is high, the elastic energy released by coal may be much larger than the expansion energy released by gas, and the corresponding disaster type may be rockburst.


## 6. Conclusions

The triaxial compression test results of gas-containing coal under different confining pressures were briefly summarized, and then, the specific steps of realizing gas-solid coupling simulation for gas-containing coal in MatDEM software were introduced. Based on this, the stress-strain curves, pore stress distribution, crack evolution, and energy conversion characteristics of gas-containing coal were studied, and the influence of confining pressure on the deformation and the failure of gas-containing coal and its mechanism were discussed. The main conclusions are as follows:(1)
Based on MatDEM software, the gas-solid coupling simulation method for porous media was developed, which realized the coupling effect of adsorption expansion, gas pressure, and load, and determined the specific steps of gas-containing coal triaxial compression simulation(2)
The influence of confining pressure on the elastic modulus of coal is largely a reflection of the compaction effect. With the increase of confining pressure, the variation law of elastic modulus is not the same, which is closely related to the internal microstructure, defects, and fissures of coal samples. If there are fewer defects, the confining pressure has less compacting and closing effect and has less influence on the elastic modulus. On the contrary, if there are a large number of defects in the coal sample, under the action of confining pressure, the pores and cracks are gradually compressed and closed, and the ability of the coal to resist deformation and damage increases, so the elastic modulus also increases. In addition, the coal specimens exhibit a transition from brittle deformation at low confining pressure to plastic deformation at high confining pressure(3)
Both the peak strength and the residual strength increase approximately linearly with increasing confining pressure. The strengthening effect of confining pressure on the strength of gas-containing coal mainly lies in the external support effect on the sample. The confining pressure constrains the lateral deformation, making the internal pores and cracks more compact, and it is difficult to form a failure surface, so the peak strength increases. In the post-peak stage, the confining pressure increases the normal pressure on the shear failure surface, causing the shear sliding friction to increase, and thus, the residual strength also increases(4)
With the application of axial load, the high-stress area generated at the upper end of the sample gradually moves down, and the stress is released at the place where the microcracks are generated. Affected by the shear dilatation effect, the pore volume between particles increases, and the pore gas pressure decreases instantaneously, causing internal gas seepage. The seepage channel composed of macro-cracks is basically consistent with the location of stress release caused by structural failure and instability(5)
Due to the expansion of gas adsorption, some initial cracks are generated. As the confining pressure increases, the inclination angle of the macroscopic main crack formed during failure gradually decreases. The total number of cracks after failure decreases with the increase of confining pressure, but the number of shear cracks increases, indicating that the failure mode of internal shear slip gradually occupies a larger proportion(6)
Due to the expansion stress caused by gas adsorption, the elastic energy is not zero when not loaded, and the elastic energy gradually increases as the load increases. The maximum cumulative elastic energy and residual elastic energy increase with the increase of confining pressure, resulting in the increase of burst tendency index of coal samples



---

*Source: 3074156-2022-07-15.xml*