Abstract
In order to study the effect of the limit equilibrium zone energy in front of the heading face on the volume of coal and gas outburst, the concept of coalgas mixed matrix was put forward, then the width of the limit equilibrium zone was calculated, and thus the energy transformation mathematical model of the limit equilibrium zone was established through theoretical analysis and mathematical derivation. In addition, the energy density calculation formula of the limit equilibrium zone was derived and then the intensity prediction formula of coal and gas outburst was derived combining elastic energy of coal body, expansion energy of gas, crushing work and throwing work of coal body. The results showed that the energy in the limit equilibrium zone has a certain effect on the volume of coal and gas outburst and the intensity prediction formula has high accuracy in Yangou Coal Mine. These conclusions have certain guiding significance for energy calculation and intensity prediction of coal and gas outburst.
Keywords
Coal and gas outburst, limit equilibrium zone, coalgas mixed matrix, energy density, intensity prediction
1 Introduction
BP's latest 2019 World Energy Outlook report showed that coal demand of China will account for 39% of global coal demand by 2040 although the coal consumption in total energy consumption in China will fall from 60% in 2017 to 35% in 2040. Therefore, coal will continue to play an important part in energy consumption. The depth of coal mining will increase in the wake of the gradual reduction of shallow coal resources, and the possibility of coal and gas outburst (CGO) will rise [1]. Many researchers were devoted to relevant study on CGO. B.B. Hodort [2] has proposed the energy theory to explain the cause of CGO from the perspective of energy, and Liu et al. [3] analyzed that from the perspective of mechanics, Nie et al. [4] discussed the mechanism of CGO caused by vibration. Moreover, a stressseepagedamage coupling model was established to investigate the size scale of CGO [5]. Some new technologies and experiments were used for further study, such as acoustic emission and microseismic monitoring [68], numerical simulation [911], threedimensional testing system [12] and simulation experiment [1314].
There are various factors affecting CGO, Zhai et al. [15] found that the factors include geological structure, mining depth, seam thickness, the lithology of roof and floor, and the mode of operation. In addition, the widespread occurrence of tectonic coal and soluble organic matters play a crucial role in outbursts [16, 17]. Temperature is of the least importance in the outburst intensity of coal and gas among gas pressure, the thickness of coal and the temperature [18]. Human errors and the lack of safety knowledge and awareness are the main causes of CGO from the perspective of human [19].
The energy of CGO is of great importance because it determines the intensity of CGO. A method was provided to calculate gas expansion energy based on the effect of coal damage and the environmental pressure change for desorption [20]. More gas and outburst energy were demonstrated in the Gasenriched areas, which can cause an outburst easily [21]. And some researchers found the gas expansion energy is higher than the elastic energy of coal [22].
Effective prediction can reduce the probability of CGO. A new method for predicting CGO was proposed based on coal mass strength, gas pressure, and in situ stress [23]. Potential coal emissions can be predicted by measuring the initial expansion energy of coal lumps in the prone areas and the thickness of soft coal. The relationship between the initial expansion energy of released gas and the initial volume of released gas can reflect the risk of coal outburst [24]. A new comprehensive index was adopted to predict coalrock dynamic disaster hazards [25]. Jiang et al. [26] proposed a method to identify the coal seams of CGO and the evaluation indexes. And in the study of Aziz et al. [27], a mine safety management system was established to decrease the incidence of CGO. In addition, some theories and new technologies were used to predict CGO, such as BP neural network [28], Fisher criterion [29], support vector machine [30], Bayesian network [31], gray theory [32] and fault tree [33].Furthermore, some researchers demonstrated that uniform fracturing [34], hydraulic slotting technique [35, 36] and mining protective layers [3742] can effectively reduce the risk of CGO.
The researches above on CGO mainly focused on the mechanism, influencing factors, prediction and prevention measures. However, the energy of the limit equilibrium zone has been rarely concerned although some researchers have done a lot of research on the gas energy and expansion energy of CGO. This study established a mathematical model to calculate the energy density of the limit equilibrium zone and thus the total energy of it. In addition, the intensity prediction formula of CGO was derived combining elastic energy of coal body, expansion energy of gas, crushing work and throwing work of coal body, and the accuracy has been verified in Yangou Coal Mine. These results will have guiding significance for the energy calculation and intensity prediction of CGO.
The occurrence probability of CGO in the heading face is higher than other work places according to statistics. Therefore, the coal and rock mass in front of the heading face is the key point to the research of CGO. The overburden load of the goaf is transferred to the front of the heading face during the normal advancing process of the heading face. Hence, the vertical pressure of the coal rock mass in front of the heading face within a certain range is larger than the original vertical pressure. Under the action of abutment pressure, part of the coal body undergoes strength failure, and it continues to carry with the pressure after failure, and it is called the limit equilibrium zone.
The energy stored in the limit equilibrium zone is transformed from elastic energy and gas expansion energy, and there is no existing formula to calculate it directly. In order to calculate the energy of the limit equilibrium zone, the mixture of coal body and gas in this zone can be considered as a kind of even material, which is called coalgas mixed matrix. Then the energy density calculation formula of the limit equilibrium zone can be obtained, based on the energy conservation principle during the process of the elastic zone transforms into the limit equilibrium zone. The derivations are based on the following hypotheses:
a. Methane is ideal gas and desorbs in an instant;
b. Coalgas mixed matrix is an even material;
c. Coal and gas outburst process is an adiabatic process.
The pressure around the heading face will be redistributed because of the effect of roadway excavation, as shown in Figure 1. Taking the section view along the advancing direction of the heading face, the pressure distribution and the zone division can be shown clearly in Figure 2, the zones include limit equilibrium zone and elastic zone [43].
At the boundary between the limit equilibrium zone and the elastic zone, the pressures satisfy,

(1)  

where is the horizontal pressure, kPa; is the vertical pressure, kPa; is the lateral pressure coefficient; K is the pressure concentration factor; is the bulk density of the overlying strata, N/m^{3}; H is the burial depth of coal seam, m.
The pressure of coal body in the limit equilibrium zone satisfies the limit equilibrium condition, and its differential equilibrium equations are:

(2)  
 

where is the shear pressure, kPa; is the internal friction angle of coal seam, °; c is the coal seam cohesion, kPa.
Combine equations (1) and (2), we can obtain

(3)  

At the point , . Then we can obtain the limit equilibrium zone width r_{0}[44].

(4) 
where m is the thickness of the coal seam, m.
2.2 Transition zone elastic energy
In the original rock pressure zone, the coal body satisfies the characteristics of elastomer, and the stored energy can be calculated according to the elastic energy calculation formula. The elastic energy in per unit volume of coal body is:

(5) 
where E is the coal seam modulus of elasticity, MPa; is the coal seam Poisson's ratio; are main pressures, MPa.
Assuming that the section of tunneling roadway is circular and its radius is r_{0}. If the tunneling distance is 1 m, the elastic zone with corresponding length will transform into limit equilibrium zone, as shown in Figure 3. The mathematical model and the orthogonal coordinate system was established as shown in Figure 4, the tunneling direction of the roadway as the positive xaxis and the vertical direction of the heading face as the positive yaxis. The shadow part is the research zone where the energy transformed from elastic zone to limit equilibrium zone, and the volume is V_{1}.

(6) 
where y is the distance from the center of the roadway, m; f_{1}(y), f_{2}(y) are the before and after transformation curve equations of the interface between the limit equilibrium zone and the elastic zone respectively.
f_{1}(y), f_{2}(y) satisfy

(7)  

The energy stored in the limit equilibrium zone that transformed from the elastic zone is E_{e}:

(8) 
2.3 Transition zone gas energy
The existing state of gas in coal body includes free state and adsorbed state. If there is no disturbance, the energy of gas is stored in coal body in the form of internal energy. The calculation formula is as follows:

(9) 
where E_{g} is the gas internal energy, J; i is the freedom degree of gas; n is the quantity of matter, mol; R is the gas constant, J/(mol·K); T is the absolute temperature, K.
Assuming that the gas in the coal body is an ideal gas and satisfies the ideal gas state equation, then

(10) 
where P is the gas pressure, MPa; V_{g} is the volume of gas, m^{3}.
In the energy transition zone,

(11) 
where q is the gas content of per ton coal, m^{3}/t.
Substituting equations (10) and (11) into equation (9) yields

(12) 
As CH_{4} is a polyatomic gas, i takes 6. Substituting it into equation (12) then

(13) 
2.4 Transition zone energy consumption
When coal body transforms from elastic state to limit equilibrium state, coal body will generate cracks, which will consume the energy of coalgas mixed matrix. Because the number and length of cracks cannot be directly measured, it is difficult to obtain a formula through theoretical derivation to calculate the energy consumption, thus the formula can be obtained only from experimental tests [45].

(14) 
where E_{b} is the energy consumed by coal body crushing, J; f is the consistent coefficient of coal body; is the coal body density, kg/m^{3}; Y_{p}_{1}is the percentage of the quality of coal samples with particle size less than 0.2 mm accounts for the total coal samples.
The displacement is assumed not to change when the coal body transforms from elastic state to limit equilibrium state, and its moving work is not considered here.
2.5 Limit equilibrium zone energy density
When elastic zone with volume of V_{1} becomes limit equilibrium zone, then the corresponding energy is stored in the limit equilibrium zone, which can be set as E_{L}.
According to the energy dynamic balance, we can obtain:

(15) 
Substituting equations (8), (13) and (14) into (15) then

(16) 
Assuming that the energy density of the limit equilibrium zone is e_{2}, then

(17) 
3 CGO intensity prediction
3.1 CGO supporting energy
CGO supporting energies contain: free gas expansion energy, adsorbed gas expansion energy, elastic energy, and limit equilibrium zone energy. Assuming that the CGO volume is V_{2}, the energies are as follows respectively.

(18) 
where W_{1} is the free gas expansion energy, J; V_{g} is the free gas content, m^{3}; a is the proportion of free gas participating in expansion work, J; T_{g}is the absolute temperature after gas expansion, K; p_{0} is the initial gas pressure, MPa; p_{1} is the pressure after gas expansion, MPa; V_{m}is the molar volume of gas in standard state, m^{3}/mol; s is the variable process index; is the coal body density, kg/m^{3}.

(19) 
where b is the percentage of free gas in total gas.
Substituting equation (19) into (18), we can get

(20) 
(2) Adsorbed gas desorption expansion energy

(21) 
where W_{2} is the desorption expansion energy of adsorbed gas, J; V_{a} is the gas volume in adsorbed gas participating in work, m^{3}; T_{0} is the gas initial absolute temperature, K.

(22) 
where d is the percentage of adsorbed gas participating in expansion work.
Substituting equation (22) into (21), then we can get

(23) 
(3) Elastic energy
When the CGO volume is V_{2}, the corresponding elastic energy is E_{e}.

(24) 
(4) Limit equilibrium zone energy
When the CGO volume is V_{2}, the limit equilibrium zone is V_{1} and the corresponding limit equilibrium zone energy is E_{L}.

(25) 
3.2. CGO Energy consumption
The main energy consumption of CGO contain the crushing work and throwing work of coal body, as well as the energy consumption of friction and noise. In view of the less energy consumption of friction and noise, this paper will not consider it for the convenience of calculation.
(1) Crushing Work

(26) 
where W_{3} is the crushing work corresponding to CGO volume V_{2}, J.
(2) Throwing Work
The throwing work of the coal body is the corresponding kinetic energy when the broken coal is thrown out,

(27) 
where W_{4} is the throwing work corresponding to CGO volume V_{2}, J; v is the throwing speed of broken coal, m/s.
3.3. CGO intensity prediction
According to the law of conservation of energy, we can obtain

(28) 
That is

(29) 
Deform equation (29), we can set

(30) 
where B is the CGO intensity, t.
4. Model validation
4.1. Variables and parameters
The variables and parameters of CGO in Yangou coal mines are shown in Table 1 by querying the relevant data and references [45].
4.2. Result analysis
The energy of CGO is the key factor to determine the outburst intensity. In this paper, the energy in the limit equilibrium zone was considered. Through theoretical derivation, a method for accurately calculating the energy of CGO was proposed. Then, according to the actual situation of Yangou coal mine, the method was verified. The specific analysis process is as follows.
According to equations (20), (23), (24), (25), (26), (27), the parameters containing the width and energy of limit equilibrium zone, elastic energy, free gas expansion energy, adsorbed gas expansion energy, crushing work and throwing work were calculated, thus the CGO intensity of Yangou Coal Mine can be obtained. And the each energy of CGO is shown respectively in Tabal 2.
Variable  Parameter  Value  Unit 
E_{L}  Energy of the limit equilibrium zone  8.326×104  J 
E_{e}  Elastic energy  1.217×102  J 
W_{1}  Expansion energy of free gas  6.019×106  J 
W_{2}  Expansion energy of adsorbed gas  5.472×105  J 
W_{3}  Crushing work  5.771×106  J 
W_{4}  Throwing work  8.789×105  J 
From Table 2, the limit equilibrium zone energy is 8.326×104 J, which accounts for 1.25% of the total energy, and the CGO volume corresponding to this energy is about 25 t. Moreover, the CGO prediction intensity is approximately 1953 t, and the error is 2.35% (lower than 5%) compared to the actual intensity 2000 t. Therefore, the accuracy of the mathematical model is reasonable in Yangou Coal Mine.
5. Conclusions
In this paper, we studied the effect of the limit equilibrium zone energy on the volume of CGO. The concept of coalgas mixed matrix was put forward and the width of the limit equilibrium was calculated. Based on energy conservation law, a series of calculation and derivation were carried out to obtain the limit equilibrium zone energy density formula. Then the CGO intensity prediction model was established and verified. The results showed that in the total CGO volume of Yangou Coal Mine, 25 t was contributed by the limit equilibrium zone energy in front of the heading face. In addition, the accuracy of CGO intensity prediction was very high.
Acknowledgement
This work was supported by the National Natural Science Foundation of China (grant number 51374121) and the Liaoning Distinguished Professor Funding Project (grant number 551710007007).
References
Published on 16/09/20
Submitted on 08/09/20
Licence: CC BYNCSA license
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