Numerical Investigation of Complex Thermal Coal-Gas Interactions in Coal-Gas Migration

School of Construction Engineering, Jiangsu Vocational Institute of Architectural Technology, Xuzhou 221116, China State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining and Technology, Xuzhou 221116, China Institute of Geotechnical Engineering, Shaanxi Provincial Key Laboratory of Loess Mechanics and Engineering, Xi’an University of Technology, Xi’an 710048, China


Introduction
Coal seam gas is an important natural source and an important part in the clean energy structure in China [1].However, most Chinese coalbed methane cannot meet the requirements of effective recovery because of the low permeability characteristic [2,3].erefore, many technological means are adopted to enhance the coal permeability and develop the gas extraction efficiency [4,5], such as, hydraulic fracturing technology [6].Besides, hot injection has been adopted to enhance the extraction of coalbed methane [7,8].
erefore, the thermal evolution characteristic evolution during the extraction of coalbed methane needs to be studied.
e thermal stimulation of gas reservoir triggers the complex interaction between coal, gas, and temperature.It changes the deformation behavior of coal, as well as the heat and gas flow.Temperature has an obvious effect on the adsorption capacity of coal seam gas, which is negatively correlated with the increase of temperature [9][10][11].At the same time, the thermal expansion coefficient is heterogeneous and anisotropic due to the different response of different components to temperature changes.Noorishad et al. [12] evaluated the coupled thermal-hydraulicmechanical characteristics of rock and investigated the effects of thermal stresses on permeability through the deformation change of the fractures.Mctigue [13] developed a fluid-saturated, porous, thermoelastic model and calculated the heating of a half space at constant temperature conditions.Harpalani and Schraufnagel [14] found that desorption induced the shrinking of the coal matrix and increased permeability through seepage tests.Zhou et al. [15] consider thermodynamically coupled water and heat flow and demonstrate the influence of thermodynamic coupling through numerical and analytical solutions.Zhu et al. [16] analyzed the influence of temperature on coal permeability based on a complex thermal-hydrological-mechanical model and found that sorption-induced volumetric strain affected the seepage behavior of coal seam significantly.Cai et al. [17] found that thermal treatment can promote methane desorption and increase the permeability through the numerical simulation.Qu et al. [18] developed a new thermal model with gas flow and matrix adsorption and suggested that the increase in temperature leads to the decrease in coal swelling and larger cleat aperture and higher coal permeability.
e interaction mechanism of heat injection has not yet fully understood for the design of engineering.It is indispensable to study the coal-gas interaction under variable temperature.In this study, a coupled thermal-hydrologicalmechanical model is established, considering the sorption characteristic and permeability evolution.en, this complex model is implemented into a finite element (FE) and solved through the numerical method.Finally, the effect of temperature on coal permeability and gas pressure is studied based on this coupled model.

Gas Seepage Characteristic in Coal
Fractures.For the coal seam reservoir, the non-Darcy flow can be expressed as where μ ⟶ is velocity vector, ρ g is density, β � 1.75/ ������� 150k g ϕ 3  , ϕ is porosity, and k g is the permeability of coal.
e above equation can be expressed in the following form: where e mass conservation equation of gas migration can be expressed as where μ ⟶ is the velocity vector and Q s is the gas source, and the mass content can be expressed as [19] m where ϕ f is the porosity, ρ ga is the gas density, ρ c is the coal density, and V sg is the absorbed content.
In the coal seams, the gas absorption volume can be expressed as e Sorption Strain ε s can be expressed as where V sg is the content of absorbed gas and α sg is the strain coefficient.e gas density can be obtained as e mass conservation equation can be expressed as

Gas Diffusion in Coal
Matrix.e gas migration process experiences three substeps: flow in fracture, gas diffusion, and sorption in matrix [20].Figure 1 shows a conceptual model for gas transport.e source term can be expressed as where Q s is the exchange from matrix to fractures, c m is the gas concentration in matrix, c f is the gas concentration, σ c is the shape factor, and τ is the sorption time.e relationship of gas concentration and gas pressure is written as where Mc is the molar mass.e diffusion equation can be written as where m e is the equilibrium gas content.en, the diffusion equation can be expressed as e permeability k g can be written as [21]

Solid Mechanics Equation.
e stress equation of coal seam can be written as 2 Advances in Civil Engineering e stress-strain equation can be written as e overall stress balance equation can be written as

Coal Permeability.
e relationship between porosity and e ective stress can be expressed as [22] Δϕ en, the porosity is expressed as By substituting the porosity, it can be rewritten as where is the initial pressure, and ϕ 0 is the initial porosity.e evolution characteristics of permeability can be expressed as 2.5.Energy Conservation.Total heat ux q T is expressed as where q T is the thermal ux and ρ s is the mass density.e thermal balance can be written as [23] z (ρC where (ρC and ρ s is the mass density.e conservation of mass yields ese equations ( 8), ( 12), ( 16), and ( 24) describe the fully coupling model of coal seam gas migration, including mechanical deformation of coal (gas desorption induces coal shrinkage and self-heat induces coal expansion), gas di usion from the matrix, gas ow, and heat transfer in fractures.e coupled relationship is illustrated in Figure 2. e THM model indicates the nonlinear response of gas migration in underground coal seams.It is hard to get an analytic solution of these equations, so they are achieved through the numerical method with Comsol Multiphysics (a powerful partial di erential equation solver) and appropriate boundary conditions.Advances in Civil Engineering simulated extraction, the numerical simulation is carried out according to the practical example of coal seam extraction.e actual size of the coal seam is 568 × 568 m 2 , and the parameters are obtained from Mora and Wattenbarger [24].

Model Establishment and Analysis
e numerical results and the eld production data are compared in Figure 3. Good agreement was achieved between the results of numerical simulation and the eld production data, which proves the validity of the model.

Model Establishment.
A model is established to analyze the e ect of adsorption on the distribution of coal permeability and gas pressure.e size of the model is 0.1 m × 0.1 m, as shown in Figure 4. e upper boundary is free.And the condition of the upper boundary is the constant gas pressure 2.5 MPa and the constant temperature 333 K. e initial pressure of the coal seam is 0.1 MPa and the parameters in the calculation are listed in Table 1.

E ect of Adsorption on Gas Pressure.
ere are two di erent pressures between matrix blocks and coal fractures.Because of the high permeability in coal fractures, gas can quickly ow through the fractures.
e gas pressure in fractures is generally higher than that in the matrix blocks in the calculation model.e matrix also continues to adsorb gas, and the pressure rises gradually.As the gas is injected from the top of the model, the crack pressure reaches 2.5 MPa rstly at the upper boundary, and the matrix permeability is also increasing.Finally, pore pressure and matrix pressure reach the equilibrium state.Figure 5 shows the gas pressure distribution at di erent times.As the time goes on, the di erence of pressure becomes smaller and smaller.e initial pressure in the coal seam is 0.1 MPa. e gas pressure maximums of coal matrix are 0.3 MPa, 1.8 MPa, and 2.3 MPa at 1 h, 4 h, and 6 h, respectively.Finally, two pressures reach 2.5 MPa at about 24 hours.
Figure 6 shows the gas pressure distribution at the monitoring line.e di erence between the matrix pressure and the fracture pressure can be seen more easily from the detection line.e gas pressure at the bottom of the model is low, while the gas pressure at the entrance is high.Within the equilibrium time of 1h, 4h, 8h, 12h, and 24h, the minimum pressure of the coal fracture is 0.1 MPa, 1.42 MPa, 2.16 MPa, 2.39 MPa, and 2.5 MPa, respectively; meanwhile, the minimum pressure of the coal matrix is 0.1 MPa, 0.64 MPa, 1.66 MPa, 2.19 MPa, and 2.5 MPa.With the passage of time, the pressure of internal coal mass increases gradually.e ratios of minimum matrix pressure and the minimum fracture pressure are 45%, 76%, and 92% at 4 hours, 8 hours, and 12 hours, respectively.Advances in Civil Engineering

E ect of Adsorption on Permeability Distribution.
Based on the dual-porous medium model, the fracture pressure and the matrix pressure are calculated through di erent equations.erefore, we can further understand the dual-porous medium model by analyzing the evolution rule of permeability.Figure 7 is the variation diagram of coal permeability and gas content of coal mass at di erent time.e permeability of coal fractures decreases gradually.It is caused by the adsorption swelling of the matrix and the reduction of the crack opening in the coal mass.
Coal mass expends and the fracture aperture decreases when it absorbs gas.Meanwhile, the coal mass is limited by the boundary conditions, so coal deforms under the combined in uence of the change of gas pressure, expansion stress, and e ective stress [25].Figure 7 shows that the permeability decreases with time, and the volumetric strain caused by adsorption is the main reason.ere is an inverse relationship between the permeability and the content of gas; that is, the gas content of coal increases ceaselessly and permeability decreases gradually.
e maximum of permeability ratio decreased from 0.901 to 0.876, while the minimum of gas content increased from 0.00239 m 3 /kg to 0.0481 m 3 /kg when the time increased from 1 h to 8 h.As time goes, the permeability of coal gradually stabilizes and tends to be at a uniform state.
e changes of permeability and porosity of a point are analyzed.e point is at the center of the model, and its coordinates are (0.05 m, 0.05 m). Figure 8 is the gure of permeability, porosity, gas content, and temperature change at the monitoring point A. Figure 8 shows that the evolution of permeability and gas content at point A are consistent with Figure 7. e temperature and gas content increase with the increase of time.e permeability and porosity of coal decrease rapidly before 150 h, which is mainly caused by the large pressure di erence.Pore pressure and adsorption expansion a ect together, which causes the change of permeability and porosity.With the decrease of pressure difference in coal mass, the change speed of permeability and porosity decreases gradually and tends to be stable nally.Speci c heat capacity of gas, C g (J/kg•K) 1.625 × 10 3 [16] Initial porosity, ϕ 0 (-) 0.01 [22] Speci c heat capacity of coal, C s (J/kg•K) 1.25 × 10 3 [16] Density of coal, ρ c (kg/m 3 ) 1250 [22] Pressure coe cient, c 1 (MPa −1 ) 0.07 [16] Poisson's ratio of coal, ] (-) 0.34 [22] Sorption strain coe cient, α sg (kg/m 3 ) 0.06 [18] Initial gas permeability, k ∞0 (m 2 ) 1.09 × 10 −18 [22] Volumetric coe cient of matrix, α T (K −1 ) 2.4 × 10 −5

E ect of Temperature on Permeability Distribution.
Figure 9 gives the temperature change law of the model.e temperature at the entrance is relatively high, and the temperature in the coal seam keeps rising.e temperature at point A increases from 303.8 K at 1 h to 308.6 K at 8 h, nally reaching 322.7 K at 40 h.e deformation characteristics of coal seam are given in Figure 10.After the coalbed is heated up, the adsorption increases and the volume expands.e bottom, left, and right boundaries of coal seam are restricted, so there is no normal displacement in this boundaries.When the gas pressure rises on the upper boundary of the coal mass, the coal mass absorbs gas constantly, which leads to expansive deformation of coal mass and expanding outwards of the upper boundary.ese characteristics also show that the volumetric strain induced by gas adsorption plays a key role.If the e ect of gas adsorption is not considered, the e ective stress will decrease, and the permeability of coal may increase.If the in uence of gas adsorption is ignored, pore pressure may be underestimated.
e distributions of gas pressure and permeability under di erent temperatures are given in Figures 11 and 12.It indicates that temperature has a greater impact on the gas migration in the coal seam.e in uence of temperature is reduced as the coal seam gas pressure approaches the setting pressure.High temperature can promote the desorption of gas and then change the fracture aperture and porosity of coal mass.e in uence of temperature on the porosity of coal   6 in Civil Engineering seam is a complicated process.e temperature evolution is obtained under the coupling e ect of multiple factors, such as restriction on the boundary condition and adsorptive expansion deformation of coal mass.At the initial stage, the gas pressure and porosity of coal mass di er greatly at di erent temperatures.Gas pressure and permeability change constantly with the increase of time, and nally attain states of equilibrium.e gas pressure of point A is 1.05 MPa, 1.27 MPa, and 1.41 MPa at 1h when the initial boundary temperature is 333 K, 353 K, and 373 K, respectively.
In order to quantify the in uence of gas di usion behavior, the fracture pressure distribution without Klinkenberg e ect and di erent adsorption times was also plotted as shown in Figure 13.e adsorption time signi cantly a ects the distribution of fracture pressure.When the adsorption time is large, the gas pressure of coal seam will be underestimated.e in uence of di usion e ect is the same as that of Klinkenberg e ect at the same order of magnitude.In the sorption stage, the total volume expansion e ect and Klinkenberg e ect of coal increase the permeability.When the Klinkenberg e ect is not considered, the adsorption pressure will be underestimated.

Conclusions
A new model about heat transfer was developed to study the dynamic problem of gas adsorption in coal seam under   variable temperature conditions.In the model, the coal mass was regarded as a dual-porous medium, and each point of the coal mass has two di erent pressures, i.e., pore pressure and matrix pressure.e relationship between adsorption equilibrium and permeability evolution was analyzed through the numerical method.e main results are as follows: e gas pressure in fractures is generally higher than that in the matrix blocks in the calculation model.e matrix also continues to adsorb gas, and the pressure rises gradually.Finally, pore pressure and matrix pressure reach the equilibrium state.
e adsorption of coal increases with the increase of gas pressure, and the gas content increases continuously.e porosity of coal mass decreases with the increase of time, while the temperature and gas content increase with the increase of time.
e higher temperature induces the faster increase of the gas pressure.Temperature has a greater impact on the gas migration.As the coal seam gas pressure approaches the setting pressure, the in uence of temperature is reduced.Advances in Civil Engineering

3. 1 .Figure 1 :
Figure 1: A conceptual model for gas storage and transport in the coal seams.(a) Natural fracture networks.(b) Stage 1: desorption from internal surfaces.(c) Stage 2: ow/di usion in the matrix.(d) Stage 3: ow in the natural fracture network.

Figure 2 :
Figure 2: Couplings in the gas migration process.

Figure 3 :
Figure 3: Comparison between numerical result and eld test.

[ 18 ]Figure 5 :
Figure 5: Gas pressure change law of the model.

Figure 6 :
Figure 6: Gas pressure distribution along the monitoring line.

Figure 8 :
Figure 8: Permeability distribution of the monitoring point A.

Figure 7 :
Figure 7: Permeability distribution along the monitoring line.

Figure 11 :
Figure 11: Gas pressure distribution with di erent temperatures.

Figure 9 :Figure 10 :
Figure 9: Temperature change law of the model.

Figure 12 :Y
Figure 12: Permeability distribution with di erent temperatures.

Figure 13 :
Figure 13: Gas pressure distribution under di erent cases.

Table 1 :
Parameters of the model.