Experimental investigations on effects of gas pressure on mechanical behaviors and failure characteristic of coals

Yi Xue ,P.G.Rnjith ,Feng Go ,Zhizhen Zhng ,Songhe Wng

a State Key Laboratory of Eco-hydraulics in Northwest Arid Region,Xi’an University of Technology,Xi’an,710048,China

b Deep Earth Energy Laboratory,Department of Civil Engineering,Monash University,Building 60,Melbourne,Victoria,3800,Australia

c State Key Laboratory for Geomechanics and Deep Underground Engineering,China University of Mining and Technology,Xuzhou,221006,China

Keywords:Coal Gas pressure Acoustic emission (AE)Strain energy Fractal characteristics

ABSTRACT The mechanical behavior of coal is the key factor affecting underground coal mining and coalbed methane extraction.In this study,triaxial compression and seepage tests were carried out on coal at different gas pressures.The mechanical properties and failure process of coal were studied,as well as the acoustic emission(AE)and strain energy.The influence of gas pressure on the mechanical parameters of this coal was analyzed.Based on the conventional energy calculation formula,the pore pressure was introduced through the effective stress formula,and each energy component of coal containing gas was refined innovatively.The contribution of gas pressure to the total energy input and dissipation during loading was quantitatively described.Finally,the influence of gas pressure on coal strength was theoretically analyzed from the perspectives of Mohr-Coulomb criterion and fracture mechanics.The results show that the total absorbed energy comprises the absorbed energy in the axial pressure direction(positive)and in the confining pressure direction(negative),as well as that induced by the pore pressure(initially negative and then positive).The absorbed energy in the axial pressure direction accounts for the main proportion of the total energy absorbed by coal.The quiet period of AE in the initial stage shortens,and AE activity increases during the pre-peak stage under high gas pressure.The fractal characteristics of AE in three stages are studied using the correlation dimension.The AE process has different forms of selfsimilarity in various deformation stages.

1.Introduction

Mineral resources in shallow regions in the Earth’s crust are gradually being exhausted,and deeper resources are increasingly exploited(Xue et al.,2014;Xie et al.,2015).Geo-stress and pressure in coal seams are becoming increasingly prominent with increasing burial depth (Xie et al.,2019).The coupling of geo-stress and gas pressure elevates the severity of dynamic disasters induced by deep mining and exhibits many new characteristics.The intensity and frequency of a single dynamic disaster,such as rock burst and gas outburst,increase(Fan et al.,2017).A rock burst induces abnormal gas emission,even gas outburst.Rock burst and gas outburst affect each other,resulting in composite dynamic disasters(Pan,2016;Qi et al.,2018).The effective prevention of these disasters,which endanger the safety and efficiency of deep mining,is becoming increasingly challenging.Therefore,the mechanical behavior of coal under complex geo-stress conditions should be elucidated.

The mechanical property of coal has always been a basic research topic,which has been widely studied through uniaxial compression,tension,triaxial compression,Brazilian splitting and split Hopkinson pressure bar (SHPB) tests (Zhao et al.,2016a,b;Jiang et al.,2017,2018;Sampath et al.,2019;Kong et al.,2020;Hou et al.,2022).The physical and mechanical properties of coal mass can change considerably with the differences in gas content and pore pressure(Peng et al.,2015;Yin et al.,2015;Zhang et al.,2018).Brace et al.(1965) proposed that,in addition to the mechanical effect,other physical and chemical effects exist,which cause obvious changes in the mechanical response of rocks.Based on the strength criterion and deformation model,Lu et al.(2017)discussed the effect of internal stress and swelling deformation on the mechanical parameters of coal,and the failure mechanism of coal containing gas.From the perspective of gas desorption,Xie et al.(2017) gave the theoretical expression of stress intensity factor of mode I crack,and analyzed the parameter sensitivity of gas pressure and gas desorption rate.Ding and Yue (2018) analyzed the desorption and diffusion mechanism of gas and presented the calculation method of the expansion energy of free and adsorbed gases and the expression of the energy required for coal briquette fracturing.The influence of gas pressure on the initiation of coal briquettes outburst was analyzed from the perspective of energy.These studies are beneficial in the analysis of the mechanical behavior of such coal.However,the influence of gas pressure on the coal still needs further exploration because of the complexity of this issue.

Acoustic emission (AE) can effectively reveal the internal fracturing and energy release of coal containing gas during deformation process(Lockner et al.,1991;Cai et al.,2001;Rück et al.,2017).The AE signal reflects the change in the crack initiation rate of coal at different stages of damage development in real time (Moradian et al.,2016).AE localization technology can be used to observe the spatial position,initiation rate,and expansion direction of cracks with multiple AE sensors and localization algorithms in real time.Zhang et al.(2019a) analyzed the AE characteristics at different fracturing stages of granite under true triaxial loading,and concluded that AE localization results could effectively reveal the fracture morphology.Chen et al.(2019) analyzed in detail the relationships among the fracture process,macro-crack growth,failure mode of shale-coal composite sample,and AE signal.

The energy evolution can reflect the intrinsic characteristics of coal deformation and failure(Meng et al.,2016).Energy dissipation leads to the degradation of mechanical properties and strength loss.The compression and failure of coal containing gas can be analyzed completely based on strain energy theory.Studies on rock energy evolution have primarily focused on energy constitutive and energy failure criteria (Xie et al.,2005;Gong et al.,2019;Jia et al.,2019),experimental energy evolution characteristics (Zhang and Gao,2012;Hou et al.,2019;Zhang et al.,2019b),and numerical simulation of energy evolution(Abu Al-Rub and Voyiadjis,2003;Ju et al.,2010).Xie et al.(2005) provided the criteria for strength loss and dissipated energy according to the internal relationship between energy dissipation and rock fracturing.They also analyzed the critical stress of rock element fracturing under different loading conditions.For coal containing gas,in addition to the energy input of the external load,the effect of gas pressure also produces energy input to coal.However,this kind of influence has often been overlooked in previous research,and only the influence of external load on the energy accumulation and dissipation has been considered in the analysis of coal energy evolution.If the influence of pore pressure is not considered,there will be a large deviation in the calculation of energy accumulation and dissipation of coal.

Coal exhibits distinct mechanical behavior due to the physical and chemical effects of gas.Therefore,it is of great significance to analyze the effect of gas on the coal mass using the AE technology and strain energy theory.By contrast,the influence of different gas pressures on the coal mass is relatively less explored.In this study,triaxial compression tests were conducted,and AE signal was monitored during the tests.Deformation,strength,failure,and AE characteristics of raw coal specimens were systematically analyzed,and the basic mechanical properties and deformation failure laws of coal containing gas were determined.The crack propagation of the coal was explored by adopting the AE localization method and strain energy theory.This study provides a useful reference for predicting the internal fracture mechanism and the risk formation of coal containing gas.

2.Specimen preparation and test equipment

Triaxial compression tests were carried out with the MTS815 test system,gas input system,and the PCI-2 AE test system at Sichuan University,China (Fig.1).The confining pressure was 10 MPa,the test gas was 99.9%methane,and the gas pressures were 1 MPa,2 MPa,3 MPa,and 5 MPa.The AE threshold was set to 30 dB,and 8 AE sensors were symmetrically distributed on the outer wall of the pressure chamber of the MTS system.After the coal specimen was assembled,the confining pressure was loaded in a force control mode,and the hydrostatic pressure was constantly kept at 10 MPa.The coal specimens were vacuumed to below 50 kPa,and the methane gas was filled in the pipeline.After the gas pressure was balanced and saturated,the axial stress was increased at a rate of 10 kN/min.The displacement control was adopted during the postpeak stage.The MTS815 and AE test systems simultaneously recorded the data to ensure the synchronization of stress-strain and AE results.The AE system used the time-difference localization method to retrieve the spatial position of AE activity.The basic parameters of the specimens are listed in Table 1.

All coal masses in the test were taken from the No.8 Pingmei coal mine in China.The site has a burial depth of 610-710 m.The coal masses were processed into standard cylinder specimens with the diameter of φ=50 mm and height ofH=100 mm.The microstructure of coal is the main factor affecting its strength and permeability.Fig.2a shows the microstructure of the coal using the scanning electron microscope (SEM).The macroscopic coal type is bright coking coal,with a unit weight of 1.35 g/cm3and Protodyakonov’s coefficient of 0.15-0.5.The X-ray diffraction(XRD)test uses the DMAX-3C X-ray diffractometer produced by Japanese Science Company.The test results are shown in Fig.2b.XRD analysis shows that the coal specimens contain 77.61% carbonaceous organic matter,6.33%kaolinite,6.02% quartz,and 0.3% calcite.TheT2spectrum distribution of coal was measured using nuclear magnetic resonance(NMR),and the results are shown in Fig.2c and d.It is shown thant theT2spectrum distribution of coal specimens has three peaks,which represent the pore size distribution characteristics of different sizes.The velocity of the longitudinal ultrasonic wave is 1233 m/s.

3.Experimental study on the deformation and failure characteristics of coal

3.1.Effect of gas pressure on the deformation and strength of coal

The stress-strain curves of coal specimens under different gas pressures are displayed in Fig.3.For convenience,the strains are positive in compression.The calculated parameters of the specimens are shown in Table 2.The stress drop coefficient can be calculated according to the peak strength σpand residual strength σr,i.e.λ=(σp-σr)/σp∙Given the strong heterogeneity of coal,the average values of the mechanical parameters under each gas pressure are adopted to analyze the effect of gas pressure (Kong et al.,2022).Fig.4a and b shows the changes in the elastic modulus and peak strengths.The mechanical and physicochemical effects of gas change the mechanical properties of this material(Peng et al.,2019).These parameters show an exponential relationship with the gas pressure(Gruszkiewicz et al.,2009;He et al.,2020).Increasing gas pressure will result in a smaller change in the elastic modulus.The gas pressure acting on the structure of the coal specimen may promote the crack growth in the specimen and reduce the ability of the specimen to resist failure.The adsorbed gas facilitates the internal crack formation of the coal specimen.The peak strength decreases by 19%,and the residual strength decreases by 11%as the methane pressure is increased from 1 MPa to 5 MPa.

Table 1 Physical parameters and test conditions of coal specimens.

Table 2 Mechanical parameters of the coal specimens under different gas pressures.

Fig.1.Schematic diagram of test equipment.

3.2.Effect of gas pressure on the damage characteristics of coal

The decay of the elastic modulus is adopted as the damage variable,which can be expressed as follows (Lemaitre et al.,2000;Peng et al.,2015):

whereE′is the secant slope of the instantaneous damage point,andE0is the initial elastic modulus.

Fig.2.Coal composition and pore structure characteristics: (a) SEM image of coal;(b) Mineral content of coal;(c) T2 spectrum distribution;and (d) Pore volume distribution.

The damage variable is determined based on the stress-strain data and Eq.(1),as shown in Fig.5.Three characteristic points,namely,the initial damage,peak stress,and residual state points,are selected to further observe the evolution characteristics of the damage variable.The damage evolution is more affected by the axial strain (or axial stress).

The change in gas pressure has an impact on the damage variable,and the change in the damage variable is mostly affected by the axial strain.Given the compaction effect,the stress-strain curve in the initial stage is approximately a straight line.After the initial damage point,the coal containing gas begins to enter the plastic deformation stage,and the damage variable of the coal increases.After the peak stress point,the damage variable fluctuates with the gas pressure,however,the overall change is not significant.

4.Effect of gas pressure on the energy accumulation and dissipation of coal

4.1.Novel energy calculation formula of coal considering pore pressure

The deformation of coal containing gas is accompanied by the absorption,storage,and release of energy (Xue et al.,2022a).Assuming that the energy conversion is a pure physical process and only occurs in a closed thermodynamic system without heat exchange,the energy stored by coal containing gas is composed of elastic and dissipation energy (Fig.6),which can be expressed as follows (Xie et al.,2017):

whereUis the total absorbed energy of coal;Ueis the total elastic energy;andUdis the total dissipation energy,which reflects the thermodynamic unidirectionality and irreversibility of the deformation and failure process.

The total strain energy and elastic energy can be written as follows:

The elastic energy stored in coal without pore pressure during triaxial compression can be written as follows:

Fig.3.Stress-strain curves of coal specimens under different gas pressures: (a) 1 MPa,(b) 2 MPa,(c) 3 MPa,and (d) 5 MPa.

where(i=1,2,3) is the unloading elastic modulus.Althoughis constantly changing with stress level,it is generally regarded as a constant in the calculation.

Eqs.(2)-(5) are the conventional formulas for calculating the energy accumulation and dissipation of coal.For the coal containing gas,as a typical dual-porous medium,gas pressure causes the deformation of coal,and produces certain energy input to the coal.However,in the previous research,this effect is usually ignored,and only the influence of the external load is considered.During compression,the input and dissipation of energy are caused by three factors:axial pressure,confining pressure and pore pressure.The expression of the effective stress principle is introduced,i.e.=σij-αp,where α is the Biot’s coefficient,andpis the gas pressure.For the convenience of calculation,α is set to 1 in the analysis of the test data.The gas pressurepand confining pressure σ3are fixed in the test.

Considering the effect of pore pressure and the additivity of work done by the external force,Eqs.(3) and (4) are correspondingly written as follows:

where εvis the volumetric strain,and εv=ε1+ε2+ε3∙The total absorbed strain energy of coal containing gasUconsists of theabsorbed energy in the axial pressure direction (U1) and in the confining pressure direction (U3) and that induced by the pore pressure (Up) The total elastic energy of coal containing gas,(Ue)consists of the elastic energy in the axial pressure direction ()and in the confining pressure direction () and that induced by the pore pressure ()∙

Fig.4.(a) Elastic modulus and (b) peak strength of the coal specimens under different gas pressures.

The elastic energy of the coal mass during the unidirectional compression can be written based on Eq.(6) as follows:

The difference in the elastic energy obtained by Eqs.(5)and(7)is due to the effect of pore pressure considered on the basis of the conventional energy calculation formula.The pore pressure also leads to the coal deformation,and accounts for a certain proportion in the total energy accumulation of coal.

4.2.Energy accumulation and dissipation characteristics of coal

Fig.7 shows the energy evolution of the coal specimens.In the initial stage,the energy (total absorbed,elastic,and dissipation energy) evolution curves nonlinearly grow with increasing axial stress(or axial strain).Coal has a certain bearing capacity before the peak point.Thus,most of the absorbed energy is stored in the coal,and the dissipation energy only accounts for a small part.In the plastic yield stage,the total absorbed,elastic,and dissipation energy increases with the axial stress.With the rapid stress drop after the coal failure,the total absorbed energy appears to present a rebound phenomenon of “first rapid drop and then gradual increase”,as shown by the specimen CS-1-1 in Fig.7a.The total absorbed energy continues to increase after the peak point.

Fig.5.Damage variable-axial strain curves of the coal specimens under different gas pressures.

Fig.6.Calculation of strains and elastic and dissipation energy of coal containing gas under triaxial compression from the stress-strain curves.

4.3.Absorbed energy component characteristics of coal

Fig.8 shows the evolution of the total absorbed energy and each absorbed energy component,which are calculated by the new energy formulas considering the pore pressure(defined in Section 4.1).During the external force loading,the absorbed energy of coal containing gas is primarily obtained from the action of axial pressure.Given that the circumferential deformation of coal is mainly induced by the axial compression,the absorbed energy evolution in the confining pressure direction is similar to that in the axial pressure direction.The absorbed energy in the direction of confining pressure is negative,because the direction of circumferential deformation is opposite to that of confining pressure.As the coal specimen first compresses and then expands,the absorbed energy induced by pore pressure first does negative work and then positive work.The volume expansion of coal is significant after the peak point,and the absorbed energy induced by the pore pressure is also accelerated considerably after the post-peak stage.The absorbed energy in the axial pressure direction decreases under high gas pressure.Besides,the energy induced by the pore pressure increases evidently.When the gas pressure is 5 MPa,the elastic energy induced by the gas pressure at the peak stress accounts for 7%of the total elastic energy.Therefore,when the pore pressure is high,the energy accumulation caused by the pore pressure cannot be ignored.

4.4.Effect of gas pressure on the energy evolution of coal at stress characteristic points

Three characteristic points (damage,volume expansion,and peak stress points) are selected to analyze the influence of gas pressure on the energy evolution of coal.The first point is the damage stress point,at which obvious AE signal appears for the first time.The volume expansion point is the extreme point of volume compression of coal specimen,that is,the turning point from expansion to compression state.The strain energy of the coal specimens at the three characteristic points is shown in Fig.9.The total absorbed and elastic energy at each characteristic point generally decreases with the gas pressure.When the gas pressure increases from 1 MPa to 5 MPa,the average value of the total absorbed energy at the damage stress point declines from 0.081 MJ/m3to 0.031 MJ/m3.

Fig.7.Evolution characteristics of the total absorbed,elastic,and dissipative energy of the coal specimens under different gas pressures:(a)CS-1-1 under gas pressure of 1 MPa;(b)CS-2-1 under gas pressure of 2 MPa;(c) CS-3-2 under gas pressure of 3 MPa;and (d) CS-5-2 under gas pressure of 5 MPa.

Fig.8.Evolution characteristics of each component of the absorbed energy of the coal specimens under different gas pressures:(a)Energy input in the axial pressure direction;(b)Energy input in the confining pressure direction;(c) Energy input induced by the pore pressure;and (d) Total energy input.

4.5.Evolution characteristic of each energy component of coal

Fig.10 illustrates the complete stress-strain curves and energy evolution characteristics of coal containing gas (specimen CS-3-1 with the gas pressure of 3 MPa).The total dissipation energy is very low at first stage,and the strain energy ratio curves are changing.Then,at the elastic deformation stage,the absorbed energy in the axial pressure direction is positive,whereas the absorbed energy in the confining pressure direction and that induced by the pore pressure are negative.The absorbed energy in the axial pressure direction has the highest value,followed by the absorbed energy in the confining pressure direction and that induced by the pore pressure.At the stable micro-crack growth stage,the elastic strain energy still occupies an absolute position in the total absorbed energy.At this stage,the total dissipation energy also increases because of the propagation of micro-cracks in coal containing gas.Finally,in the post-peak failure stage,the macro-fracture surface appears.The total absorbed energy decreases after the peak point because of the stress drop.With the rapid release of the stored energy,the elastic energy decreases rapidly,and the dissipation energy increases rapidly.

5.Effect of gas pressure on the AE characteristics of coal

5.1.AE characteristics of coal

The evolution of the internal microstructure of coal containing gas can be inferred by analyzing the AE signal.The AE count curve is shown in Fig.11.The change in the AE count of coal specimens under different gas pressures shows similar rules.Compared with the evolution of deformation,the change in the cumulative AE count curve consists of the initial silent,slow rise,rapid rise,violent activity,and stable activity stages.Fig.12 shows the AE localization points of coal containing gas at different stress levels.Firstly,a few randomly distributed points appear in the middle of the coal specimen.The AE localization points of coal specimens are gradually concentrated with loading,forming macro-cracks after the coal failure.The macro-fractures change from axial splitting fractures to multiple shear fractures or X-shaped shear fractures.

To analyze the variation in the AE activity at different stages,10%of the peak deviatoric stress is used as the increment.The AE count in each increment is shown in Fig.13a.A positive abscissa value indicates the pre-peak stage and a negative abscissa value indicates the post-peak stage.With increasing deviatoric stress ratio to a certain extent,a stable development stage of the AE count appears.When the deviatoric stress ratio exceeds 70%,the AE count increases rapidly,reaching the maximum value in the peak stage.With increasing gas pressure,the initial silence stage of AE count is shortened and AE activity rises before the peak point.Under gas pressure of 1 MPa,the AE count in each increment is very small before 50% of the peak deviatoric stress is reached.The AE signals can be monitored obviously after 20%of the peak deviatoric stress under gas pressure of 5 MPa.

The peak AE count ratio is defined as the ratio of the AE count between 90%and 100%of the peak deviatoric stress to the total AE count.Fig.13b shows the fitting curve of the peak AE count ratio.The rise in gas pressure induces the exponential decrease in the peak AE count ratio.This phenomenon shows that the intensity of the coal failure and the energy release during the peak stage gradually decline.

5.2.Fractal characteristics of AE signals

Fractal dimension is the basic quantity of fractal theory,and the correlation dimension is one of the most commonly used fractal parameters(Kusunose et al.,1991;Gao et al.,2005).The AE signal of coal is a sequence set which changes with time.The fractal characteristics of AE are studied using the correlation dimension in this section.

The time series of AE is{x(ti)} (i=1,2,…,n)∙The phase space is reconstructed according to the time delay method.The time series is extended to the phase type distribution ofm-dimensional phase space,i.e.{x(ti),x(ti+τ),x(ti+2τ),…,x(ti+(m-1)τ)},where τ=kΔtis the delay time,and Δtis the sampling period.In order to ensure that the phase space can contain the features of the original state space attractor,the embedding dimension is at least twice the dimension of the attractor,i.e.m≥2d+1,wheremis the embedding dimension,anddis the space dimension of the original state space attractor.

The Gerchberg-Papoulis (G-P) algorithm is used to analyze the AE time series(Grassberger and Procaccia,1984;Kong et al.,2019).Selecting any one ofN0vectors in the embedded space as the reference vector An,the distance from otherN0-1 vectors to it can be calculated:

Repeating this process for An(i=1,2,…,N0),the correlation integral function is obtained:

Fig.10.Evolution of each energy component of coal specimen CS-3-1 under gas pressure of 3 MPa: (a) Total strain energy and strain energy in the axial pressure direction;(b) Strain energy in the confining pressure direction and that induced by the pore pressure;and (c) Ratios of the elastic and dissipation energy to the total strain energy.

whereHis the Heaviside function,which can be written as

The correlation integral can be obtained by the following formula:

For differentrvalues,if these points satisfy the above formulas and have a certain linear relationship,it is shown that the data have fractal characteristics (Li et al.,2019).The slope of the regression line of these points is the correlation dimensionD.

Because the initial compression stage is not obvious under the condition of confining pressure,the whole deformation and failure process is divided into three stages,i.e.initial elastic stage (I),plastic stage (II) and post-peak failure stage (III).To study AE as a relatively unified process with the evolution of the internal material structure,the AE time series in each stage is extracted for analysis,and the results are shown in Fig.14 and Table 3(whereR2is the coefficient of determination).The curves of the three deformation stages show linear characteristics in a certain range.Therefore,the AE process has obvious fractal characteristics.The fractal characteristics of AE have a certain scale range,beyond which the fractal characteristics are not well defined.The correlation dimension of each deformation stage is different,but it has certain regularity.The change trend of the AE correlation dimension shows that the AE process has different self-similarity in different deformation stages.The correlation dimension of the AE process can be used as a characteristic parameter to describe the mechanical characteristics of materials.

Table 3 Goodness of fit and correlation dimension for various stages of coal deformation and failure process.

The relationship between AE correlation dimension and gas pressure is shown in Fig.15.The decrease of correlation dimension is often accompanied by the occurrence of main fractures,which means the transformation of damage from disordered to a macroscopically ordered failure(Gao et al.,2005).In the post-peak stage,a large number of micro-and macro-damages with different sizes and random distribution occur in the coal structure.As a measure of disorder,the correlation dimension reflects well the statistical evolution law of these micro-damages.The increase in correlation dimension means increased damage.Therefore,the reduction in the correlation dimension can be used as a concise and valuable criterion to predict the occurrence of main fracture,and it is easy to apply to engineering.It can be further refined in the next study,and more detailed stages can be divided according to the stress-strain curve to analyze the fractal characteristics of AE.Besides,the correlation dimension of AE and gas pressure show a positive correlation in each stage.It may be that the high gas pressure results in a decrease in micro-porosity and micro-fracture strength of coal,and more obvious destruction of coal,causing the increase in AE correlation dimension.

6.Discussion

6.1.Effect of gas pressure on the strength based on strength criterion

The complex interaction mechanism between coal and gas affects the strength characteristics.Mohr-Coulomb criterion is suitable for studying the strength characteristics of coal(Perera and Sampath,2020).

When the stresses meet the following condition,the shear yield failure occurs:

where τ′and σ′are the effective shear stress and effective normal stress,respectively;andcand φ represent the cohesion and internal friction angle,respectively.

Eq.(13) can be transformed into the expression of principal stresses:

According to the effective stress formula σ′=σ-αp,the influence of pore pressure on the coal strength can be analyzed.The effect of gas on the internal friction angle of coal is small,and thus it can be assumed that the internal friction angle of coal does not change (He et al.,1996).

Fig.11.AE characteristics of the coal specimens under different gas pressures:(a)CS-1-1 under gas pressure of 1 MPa;(b)CS-2-1 under gas pressure of 2 MPa;(c)CS-3-1 under gas pressure of 3 MPa;and (d) CS-5-1 under gas pressure of 5 MPa.

Combined with the effective stress equation,Eq.(14) can be expressed as

The effective compressive strength is related to the elastic modulus and surface energy and it can be expressed as (Roy and Gouda,1973):

The ratio of the strength of coal with and without gas can be written as

whereEpandE0are the Young’s moduli of coal with and without gas,respectively;and λpand λ0are the surface energy of coal with and without gas,respectively.

The cohesion of the coal under gas pressure can be expressed as

Change in the solid surface energy caused by adsorption can be expressed as (Boyd and Livingston,1942):

whereVis the molar volume of gas,Seis the surface area of coal,Tis the temperature,Ris the universal gas constant,andVsis the adsorbed gas volume.

The increment of the internal friction angle can be written as

The change in the cohesion of coal containing gas is mainly composed by two parts besides that induced by the initial internal friction angle,i.e.those induced by the initial cohesion and confining pressure.The first part is the influence of adsorbed gas,which is mainly affected by the gas adsorption capacity and gas pressure of the matrix.The other part is the influence of free gas induced by the pore pressure in the form of effective stress.

The influence of gas on the strength can be observed from the Mohr stress circle(Fig.16).The free gas acts on the coal in the form of gas pressure,which induces the Mohr circle of coal to move horizontally to the left.The higher the gas pressure is,the more likely the coal will reach the strength limit and be destroyed.Besides,the cohesion of coal also receives the combined effect of free and adsorbed gases.It also leads to a parallel downward movement of stress envelope,which also increases the risk of coal failure.

Fig.12.Failure behavior of the coal specimens under different gas pressures:(a)CS-1-1 under gas pressure of 1 MPa;(b)CS-2-1 under gas pressure of 2 MPa;(c)CS-3-1 under gas pressure of 3 MPa;and (d) CS-5-1 under gas pressure of 5 MPa.

Fig.13.AE count distribution and AE count ratio of coal specimens under different gas pressures: (a) AE count in each loading range;and (b) Peak AE count ratio.

Fig.14.AE correlation dimension of coal under different gas pressures: (a) CS-1-1 under gas pressure of 1 MPa;(b) CS-1-2 under gas pressure of 1 MPa;(c) CS-2-1 under gas pressure of 2 MPa;(d)CS-2-2 under gas pressure of 2 MPa;(e)CS-3-1 under gas pressure of 3 MPa;(f)CS-3-2 under gas pressure of 3 MPa;(g)CS-5-1 under gas pressure of 5 MPa;and (h) CS-5-2 under gas pressure of 5 MPa.

Fig.15.Relationship between AE correlation dimension and gas pressure: (a) AE correlation dimension;and (b) Mean value of correlation dimension.

Fig.16.Influence of gas on the coal strength based on Mohr circle (after Perera and Sampath,2020).

6.2.Influence of gas pressure on the coal strength based on fracture criterion

Adsorption and desorption of gas usually follow Langmuir’s law of adsorption,and gas diffusion is determined by the difference in gas pressure (or gas concentration) between the matrix and fracture,as shown in Fig.17 (Guo et al.,2014;Xue et al.,2022b).Adsorption/desorption is a prolonged process (Liu et al.,2015,2022).Based on the experiments,Van Bergen et al.(2009) found that the adsorption-induced deformation of coal was not stable until 35 h and 25 h after adsorption of CO2and CH4,respectively.

A micro-body containing pore and fracture is taken from the coal to analyze the influence mechanism of gas pressure,as shown in Fig.18a(Xie et al.,2017;Ding and Yue,2018).At the beginning of the test,after the gas is absorbed by coal specimens,the gas pressure in the matrix is equal to that in the fracture.During test,under the action of axial and confining pressures,the matrix and pore are compressed.Due to the large elastic modulus of the matrix,the deformation of the pore is larger than that of the matrix.According to the gas state equation,after the pore is compressed,the pore pressure will increase and there will be a gas pressure difference between the matrix and the pore.This pressure difference will promote the matrix to absorb more gas.This kind of adsorption cannot reach the equilibrium state quickly.In the short time of the test process,the gas pressure difference caused by the deformation of coal always exists,as shown in Fig.18.

Fig.17.Dual-porous medium characteristics and gas migration in coal (after Liu et al.,2015).

The stress intensity factorKIat the tip of a type-I crack can be derived from the theory of fracture mechanics.According to the stress state of coal,the geometric model is simplified as a circular pore with two micro-cracks.The expansion of micro-cracks is mainly caused by the effective stresses (σ1-αpand σ2-αp) and pore pressure difference Δp∙

Based on the superposition characteristics of stresses,the stress intensity factor can be decomposed into different parts according to the stresses(as shown in Fig.18b):

The first part,i.e.,mainly considers the action of external load.Under the action of horizontal and vertical in situ stresses,this part can be expressed as

whereG1andG3are the geometric factors.

In the fracturing process,the gas in the wellbore will also exert mechanical load on the wellbore wall,which can be expressed as

whereGpis the geometric factor.

After the gas enters the micro-crack,it will also exert an effect on both sides of the micro-crack and induce the crack to expand along the crack tip.This part can be expressed as

When the critical stress intensity factor exceeds the fracture toughnessKIC,the cracks begin to expand:

The total stress intensity factor can be expressed as

Fig.18.Stress intensity factor of coal element with pore and fracture.

where γ and θ are the constants related to the pore diameterRand fracture lengthl.

It is found from Eq.(26) that there is a positive correlation betweenKIand σ1,σ2or Δp.Increasing the external stress σ1,σ2or Δpcan increase the stress intensity factor and increase the risk of failure.When the gas pressure increases,the pressure difference Δpwill also increase due to pore compression.mainly considers the role of free gas (pore pressure).After the matrix adsorbs gas,the adsorbed gas will cause the strength (including compressive strength and fracture toughness) of the matrix itself to decrease.The decrease in fracture toughness will increase the risk of coal failure.

7.Conclusions

(1) The pore pressure is introduced through the effective stress formula,and each energy component of coal containing gas is refined innovatively.The strain energy of coal containing gas varies with the gas pressure specifically.During the triaxial compression,the absorbed energy of coal includes positive energy in the axial pressure direction and negative energy in the confining pressure direction,whereas the energy induced by the pore pressure is initially negative and then becomes positive.The absorbed energy in the axial pressure direction occupies the main component of the total absorbed energy of coal.The absorbed and elastic energy of coal containing gas at the initial damage,volume expansion,and peak stress points decreases with increasing gas pressure.

(2) With increasing gas pressure,the shorter the quiet period of AE in the initial loading stage,the more active the AE activity during the pre-peak stage.Under gas pressure of 1 MPa,the AE count in each load increment is very small before 50%of peak deviatoric stress.Under gas pressure of 5 MPa,a considerable number of AE signals can be monitored after 20% of peak deviatoric stress.The peak AE count ratio decreases exponentially with increasing gas pressure.The fractal characteristics of AE in three stages (initial elastic stage,plastic stage and post-peak failure stage) are studied using the correlation dimension.The AE process has different forms of self-similarity in various deformation stages.

(3) The influence of gas pressure on coal strength is theoretically analyzed from two aspects: Mohr-Coulomb criterion and fracture criterion.The joint action of free and adsorbed gases weakens the internal friction angle,and the gas pressure changes the effective principal stress.According to the Mohr-Coulomb criterion,this change of effective stress increases the risk of coal failure.Based on the theory of fracture mechanics,the stress intensity factors of micro-element including pore and fracture are obtained.The change rate of gas pressure is an important factor causing coal failure,and it is related to the change of gas desorption,gas emission and effective stress.Once the change rate of gas pressure exceeds the threshold,it will lead to the fracture of coal.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This study is sponsored by the National Natural Science Foundation of China (Grant No.12002270),and the China Postdoctoral Science Foundation(Grant Nos.2021T140553 and 2021M692600).