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## Thermal-Hydraulic-Mechanical (THM) Coupling Behaviour of Fractured Rock Masses

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Research Article | Open Access

Volume 2020 |Article ID 8837425 | https://doi.org/10.1155/2020/8837425

Xiao Fukun, Meng Xin, Li Lianchong, Liu Jianfeng, Liu Gang, Liu Zhijun, Xu Lei, "Thermos-Solid-Gas Coupling Dynamic Model and Numerical Simulation of Coal Containing Gas", Geofluids, vol. 2020, Article ID 8837425, 9 pages, 2020. https://doi.org/10.1155/2020/8837425

# Thermos-Solid-Gas Coupling Dynamic Model and Numerical Simulation of Coal Containing Gas

Revised11 Nov 2020
Accepted26 Nov 2020
Published22 Dec 2020

#### Abstract

Based on gas seepage characteristics and the basic thermo-solid-gas coupling theory, the porosity model and the dynamic permeability model of coal body containing gas were derived. Based on the relationship between gas pressure, principal stress and temperature, and gas seepage, the thermo-solid-gas coupling dynamic model was established. Initial values and boundary conditions for the model were determined. Numerical simulations using this model were done to predict the gas flow behavior of a gassy coal sample. By using the thermo-solid-gas coupling model, the gas pressure, temperature, and principal stress influence, the change law of the pressure field, displacement field, stress field, temperature field, and permeability were numerically simulated. Research results show the following: (1) Gas pressure and displacement from the top to the end of the model gradually reduce, and stress from the top to the end gradually increases. The average permeability of the section of the model tends to decrease with the rise of the gas pressure, and the decrease amplitude slows down from the top of the model to the bottom. (2) When the principal stress and temperature are constant, the permeability decreases first and then flattens with the gas pressure. The permeability increases with the decrease of temperature while the gas pressure and principal stress remain unchanged.

#### 1. Introduction

The aforementioned was mainly through experiments to research the effect of gas pressure, temperature, and principal stress on the permeability, and then processing and analyzing experiment data, getting the change law; there is no systematic theoretical analysis, through the analysis of theoretical model which is rarely reported. Under uniaxial and pseudotriaxial conditions, the monotonic factors are considered. The combined effects of stress, temperature, and pressure are not taken into account. Therefore, based on the theory of gas flow and coal deformation, the thermoelastic expansion, absorption gas expansion, pore pressure, and compression deformation of coal are fully considered, together with the dynamic equations of porosity and permeability which are comprehensively considered by gas pressure, temperature, and principal stress that are established, and the thermo-solid-gas coupling model were established finally. In addition, the numerical simulation model is built by using the finite element method, and the numerical solution of the model for a special loading case is given in terms of the constraint conditions and corresponding parameters. The research results may have a meaning for further enriching and improving thermo-solid-gas coupling theories for coal containing gas.

#### 2. The Basic Assumption

At a macro level, coal is a porous medium. The factors affecting the flow of gas in coal are extremely complex; therefore, in the establishment of the thermo-solid-gas coupling model of gassy coal to grasp the main factors, secondary factors were abnegated, and the relationship between the main influencing factors was found. Therefore, the following assumptions are made: (1) gassy coal is homogeneous isotropic, and the gas flow in the coal is laminar flow, obeying Darcy’s law. (2) Coal is the single-phase gas saturation, the gas adsorption and desorption were completed, and the gas follows the Langmuir equation in an instant. (3) The coal gas content is free gas and adsorbed gas. (4) The deformation of coal belongs to the small deformation.

#### 3. Theoretical Model of Porosity and Permeability of Coal

In the process of coal mining, the condition of coal is very complex, which is affected by gas pressure, stress, temperature, etc., causing the changes of internal structure of coal, which will lead to the change of porosity and permeability of coal.

##### 3.1. Theoretical Model of Gassy Coal Porosity

According to the definition of porosity of porous medium [17, 18], the expression of the gassy coal porosity is

In the formula, is the initial porosity of gassy coal, is the volumetric strain of gassy coal, is the initial frame volume of gassy coal, and is the change of frame volume of gassy coal.

Allowing for the effect of gas pressure, principal stress, and temperature on coal deformation, the increment of volumetric strain is mainly composed of the increment of thermal strains , the increment of gas pressure strain , and the increment of adsorption gas expansion strain , namely

In the formula, the increment of adsorption gas expansion strain of unit volume of coal  is

Plug formula (2) and (3) into formula (1), we get a theoretical model of gassy coal porosity:

##### 3.2. Theoretical Model of Gassy Coal Permeability

Permeability, an important index of coal seam permeability, can reflect the gas migration in coal. Therefore, study on dynamic change law of gassy coal permeability has a guiding significance to the prevention and control of gas dynamic disasters.

According to the Kozeny-Carman  equation, the expression between permeability and porosity is

In the formula, is the permeability, is the pore surface area of the coal volume per unit volume, and is the dimensionless constant.

In the process of coal stress and strain, it can be considered that the per-unit volume of coal particle surface area keeps invariant, and combining with the dynamic equation of porosity (4), there is

The theoretical model of the gassy coal permeability is

The formula (4)–(7) define the porosity and permeability equation of gassy coal under the thermo-solid-gas coupling condition, which reflects the effects of gas pressure and temperature on permeability. The influence of gas pressure and temperature on permeability is very difficult by using the finite element software COMSOL.

#### 4. The Thermo-Solid-Gas Coupling Model of Gassy Coal

##### 4.1. The Stress Equation

Under the small deformation and elastic isotropic assumption of gassy coal, the stress field should obey the equilibrium equation, thermosetting gas constitutive equations, and geometric equations. (1)Equilibrium of equationIn the formula, is a volume force.

According to the principle of effective stress of gassy coal (not considering damage) (2)Thermosetting gas constitutive equations

In the formula, is Lame coefficient; is shear modulus; is volume deformation; is thermal stress coefficient; is gas pressure coefficient; and is gas adsorption coefficient. (3)Geometric equationJoint type (9), (10), and (11) obtained stress equation of gassy coal is as follows:

##### 4.2. Seepage Equation

The paper only considers the gas flow in the coal body in one way, and to meet the Darcy’s law  (1)Gas flow equation (2)The state equation of gas flow(3)Continuous flow equation

Joint type (14), (15), and (16) obtained gas seepage equation of gassy coal:

##### 4.3. Temperature Field Equation

Coal gas adsorption, desorption, and seepage are thermal effect and are a nonisothermal process; containing gas coal body under external force deformation also produces thermal effects, so with coal and gas coupling temperature field equation of coupled stress field and seepage field effect. The temperature change of coal containing gas is mainly caused by gas adsorption and desorption, adsorption and desorption of gas equivalent to a heat source. Therefore, to solve the problem of coal containing gas temperature field is the three-dimensional unsteady heat conduction with internal heat source. The temperature field equations of coal containing gas seepage are

##### 4.4. Thermo-Solid-Gas Coupling Model

Combining the type (4), (7), (12), (17), and (18), we get the control equations of thermo-solid-gas coupling model of gassy coal. This equation reflects the interaction of porosity and permeability of gassy coal, gas flow characteristics, and dynamic changes of coal and gas. For the control equations of thermo-solid-gas coupling model of gassy coal, to the specific problems, giving the initial conditions and boundary conditions constitutes the thermo-solid-gas coupling model of gassy coal.

According to the process of gas migration in coal body, consider the following boundary conditions. (1)Initial Conditions. The initial condition refers to the time of , the original distribution of gas pressure in gassy coal (2)Boundary Conditions. Displacement boundary conditions are as follows:

Stress boundary condition:

Pressure boundary condition:

Flow boundary:

#### 5. Numerical Simulation Analysis

According to the above analysis, the mathematical model of thermo-solid-gas coupling mathematical model is a very complicated nonlinear equation group, which needs to be solved by numerical method. In this paper, using the finite element software COMSOL  by partial differential equations calculation tool PDE in the system platform for secondary development, we fill in the need of porosity, permeability, stress field, seepage field, and temperature field equation and establish the containing coal gas heat-solid-gas coupling mathematical model, suitable to determine the actual boundary conditions and initial conditions. The distribution of seepage field and deformation field of gas bearing coal body and the permeability change of gas bearing coal body under the action of different gas pressure, effective stress, and temperature is obtained by using the calculation tool, and the relationship between seepage field and deformation field and gas pressure, effective stress, and temperature is also obtained.

##### 5.1. Geometric Model and Boundary Conditions

The built model is the actual size of gassy coal in this paper, and the model is 50 mm long, 50 mm wide, and 100 mm high. The material parameters of model are shown in Table 1. According to the test of raw coal seepage, the finite element model was set up. The upper end of the model is subjected to vertical principal stress of , the left and right end under vertical principal stress of  MPa, the front end under vertical principal stress of  MPa, and the bottom under the displacement constraints; at the same time, the model has the gravity load. The gas pressure at the upper boundary of coal has initial gas pressure of  MPa. Initial temperature of coal body is , and gas flow in coal.

 Parameter name Numerical value Modulus of elasticity Poisson ratio 0.35 Original porosity 0.0828 Initial permeability Gas density Gas dynamic viscosity coefficient Thermal conductivity 0.443 J/ Specific heat 4.35 J/ Coefficient of thermal expansion Gas compression ratio Gas specific heat 1.4
##### 5.2. Numerical Results

Figure 1 shows the displacement field of gassy coal under the gas pressure of 0.6 (MPa) and the temperature of 30 (°C). As can be seen from the figure, with the principal stress and gas pressure being applied, coal is gradually compressed along the axis, axis, and axis, and strain increment of thermoelastic expansion, absorption gas expansion, and pore-pressure compression increased cause the change of porosity and permeability.

Figure 2 shows the pressure field and velocity field, displacement field, principal stress field coupling of gassy coal under the gas pressure of 0.6 (MPa) and the temperature of 30 (°C). In Figures 2 and 3, longitudinal section shows that the change of gas pressure field and cross section represent the change of the velocity field, displacement field, and stress field, respectively. In Figure 2, red indicates that the gas pressure is relatively large, and blue shows a smaller value. As can be seen from the analysis of Figure 2, the pressure of gas is 0.6 MPa on the top ends of model. With the gas migration in coal, the gas pressure decreases gradually from top to bottom. Figure 3 shows that the gas seepage speed gradually decreases from the top to bottom of coal; the reason is that when a large amount of gas enter into the coal, the internal cracks and pores in the coal will begin to absorb, adsorbing gas will increase, and the inner wall of the coal will become thicker. The Klinkenberg effect is more obvious, namely, the gas in the coal is easier to slide, resulting in the decrease of gas seepage speed. In Figure 1, with the principal stress and pressure of gas being applied, the top of coal will become the first and the maximum to withstand stress; therefore, the maximum deformation of compressed top coal, maximum relative compressive deformation of top coal, and the displacement are maximal. With the transfer of the stress, the top to bottom of coal gradually affected by stress; thus, the displacement decreases gradually from top to bottom. Figure 2 shows the change of stress of gassy coal, the color represents the stress value, the blue color represents the area of small stress value, and red represents the area of large stress value. The deeper the blue, the smaller the stress value; the deeper the red, the larger the stress value. As seen in the figure, the stress increases gradually from top to bottom under the action of gas pressure and principal stress.

Figure 4 shows the average permeability of the section of the thermo-solid-gas coupling model of the gassy coal with the change of gas pressure. As can be seen from the figure, the average permeability of he section of the model tends to decrease with the rise of the gas pressure, and the decrease amplitude slows down from the top of the model to the bottom. As far as the reason is concerned, with the increase of gas pressure, the adsorption capacity of coal to gas is enhanced, thus reducing the porosity of coal. Porosity affects the adsorption, analysis, and permeability of coal and thus reduces the permeability of coal. With the increase of gas content, coal gas has adsorption expansion deformation, which will not produce new fine holes, but expands inward, narrowing the internal cracks, thus reducing the permeability. With the increase of gas pressure, the suction capacity of coal gradually tends to balance, so the permeability decline rate gradually decreases and tends to be flat.

To study the variation of permeability in the thermo-solid-gas coupling, the dynamic model of gassy coal, the numerical value of different gas pressure, different principal stress, and different temperatures was calculated at the outlet end of the model, the numerical solution is introduced into MATLAB, the permeability curve of gassy coal is obtained by programming calculation, and the variation law of permeability is obtained. In Figure 5, when the temperature and the main stress are kept constant, the permeability of the gas pressure increases with the increase of the gas pressure at 0.3-0.9 MPa; and when the gas pressure is increased to 0.9 MPa, the trend of the decline of permeability is gentle. This is an increase of gas pressure, inevitably accompanied by an increase in gas flow, when a large number of gas penetration into the inside of the coal body, the existing cracks and pores in the coal body, will begin adsorption. When a large amount of gas is adsorbed, under the restraint of the three principal stresses, the coal body will not expand outward but will develop internally. Due to internal expansion effect, the original fractures and pores in the coal are compressed, or the gas molecules occupy; originally, supplied gas flow channel was diverted, resulting in gas more difficult from the coal permeability, and hence, the permeability decreases. The original offer of gas seepage channel was diverted, causing gas harder to penetrate from the coal body, so the permeability decreased. On the other hand, as the gas pressure and flow rate increase, the inner wall of the coal becomes thicker. The Klinkenberg effect is more obvious, namely, the gas in the coal is easier to slide; this leads to a decrease in permeability. With the increase of gas pressure, the suction capacity of coal gradually tends to be balanced. Swelling deformation of coal framework is also becoming smaller and smaller; therefore, the rate of permeability decline gradually decreases and tends to be flat. When the gas pressure and principal stress remain constant, the permeability decreases with the increase of temperature. This is because the viscosity coefficient of the gas increases with the increase of temperature, making it more difficult for the gas to flow in the coke. Expansion occurs inside the coal, and the pores inside the coal are compressed, which narrows the gas passage and thus reduces the permeability of the coal. When the gas temperature and pressure are fixed, the coal permeability reduces rapidly firstly and slowly where after with the principal stress increasing. With the increase of the principal stress, the deformation of coal body is gradually increasing, the fracture pore is compacted, the seepage channel becomes smaller, and so the permeability decreases. When the principal stress is higher, the coal body is gradually compressed, the compression effect is reduced, and the seepage channel is stable, so the permeability change tends to be gentle.

#### 6. Conclusion

(1)Based on gas seepage characteristics, effects of gas pressure, principal stress, and temperature on gas coal are considered; the porosity model and the dynamic permeability model of coal body containing gas were derived. Furthermore, taking into account the basic theory of thermo-solid-gas coupling, the thermo-solid-gas coupling dynamic model was established(2)By using the thermo-solid-gas coupling model, the gas pressure, temperature, and the principal stress influence, the change law of the pressure field, displacement field, and other physical field and permeability were numerically simulated. Research results show that gas pressure and displacement from the top to the end of the model gradually reduce, and stress from the top to the end gradually increases. Average permeability of section of the model tend to decrease with the rise of the gas pressure, and the decrease amplitude slows down from the top of the model to the bottom(3)Applying the COMSOL Multiphysics finite element software to numerically solve the thermo-solid-gas coupling mode of gassy coal permeability, the effects of gas pressure, temperature, and the principal stress on relative permeability of gassy coal are analyzed. The results show that when temperature and principal stress are fixed, the permeability reduces rapidly firstly and slowly whereafter with gas pressure increasing. When gas pressure and principal stress are kept constant, the permeability decreases with the increase of temperature. If gas temperature and pressure are fixed, the relative permeability of coal reduces rapidly firstly and slowly whereafter with the principal stress increasing; the results have an important significance of the prevention and control of gas dynamic disasters and improving gas drainage efficiency

#### Data Availability

The data used to support the findings of this study are included within the article.

#### Conflicts of Interest

The authors declare no conflict of interest.

#### Acknowledgments

The research was supported by the National Natural Science Foundation of China (51774121), Joint Guidance Project of Natural Science Foundation of Heilongjiang Province (LH2019E087), the National Nature Science of China Foundation (51674107), and Innovative Scientific Research Project for Postgraduates of Heilongjiang University of Science and Technology (YJSCX2020-103HKD).

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