A Mathematical Model to Study the Coupling Effect of Deformation-Seepage-Heat Transfer on Coalbed Methane Transport and Its Simulative Application

Injection of high-temperature water or steam into low-permeability coalbed for efficient and rapid extraction of coalbed methane has been studied by our university for many years and will soon be implemented in the field.With comprehensive consideration of coupling of heat transfer, water seepage, desorption of coalbed methane, and coal-rock mass deformation, the paper establishes a more comprehensive mathematical model of the coupling effect of deformation-seepage-heat transfer on coalbed methane transport. Compared with the previous studies, this theoretical model considers the change of adsorbed and free coalbed methane at high temperature and the coalbed methane transport caused by a high-temperature gradient. Using the Tunlan Coal Mine of Shanxi Coking Coal Group to conduct the numerical simulations on the coalbed methane extraction project using heat injection technology, results show that (1) high-temperature water flowed towards the extraction hole along fractured fissures, with seepage towards the coal mass on both sides of the fissure at the same time, gradually heating the coalbed and forming an arcuate distribution of temperature from high to low for an area from the fractured fissure to the coalbed upper and lower boundaries. On the thirtieth day of heat injection, the temperature of the coalbed in the heat injection area ranged from 140°C to 260°C. (2) Under high temperatures, desorption of the coalbed gas was quick, and the adsorption gas content formed an oval funnel from the heat injection hole towards the extraction hole, centered by the fractured fissure, and migrating towards the coalbed upper and lower boundaries. Along with heat injection and extraction, the absorbed gas content rapidly decreased, and on the thirtieth day of injection, the absorbed gas content of the entire heat injection area decreased to 1.5m/t, only 7% of the original. (3) During heat injection, the coalbed gas pore pressure rapidly increased and reached 5.5MPa on the tenth day, about 4.5 times the original, and the pore pressure steadied at 3.5MPa on the thirtieth day of extraction. Such a high gas pressure gradient promoted the rapid flow and drainage of the gas.


Introduction
Understanding the controls and influences on coalbed methane transportation is an important problem in production and safety technology related to safe and efficient coalbed methane and coal mining. erefore, scholars continue to perform extensive studies from various angles. As the leader in coal production, China has also led in studies for extraction of coalbed methane and safety of coal mines. Zhao et al. [1] proposed a mathematical model of continuous media for solid and gas coupling coalbed methane transport focusing on coal and rock mass deformation. Zhao et al. [2] also included that coalbeds are typically fractured media composed of matrix blocks and fissures and took the modified effective stress law to reflect the adsorption swelling deformation of coalbed methane.
ey proposed a mathematical model of coalbed methane transport in fractured media, rather than continuous media, and performed numerical simulations on the emission of coalbed methane for drilling and extracting, along with on the coal face. Liu et al. [3] reviewed the international progress of studies on coalbed methane transport under multifield coupling effects in detail and provided many new study results. Wang et al. [4] proposed a deformation-seepage coupled model for the simulation of underground storage of carbon dioxide and improvement of coalbed methane extraction efficiency.
is model considered permeability as a function of the effective stress, but the effects on permeability from the stress caused by adsorption were inconsistent. Gu and Chalaturnyk [5] established a new model of porosity and permeability. e model took fissures and matrix blocks as equivalent continuous elastic media and conducted permeability analysis on anisotropic coal seams and deemed that gas desorption/ adsorption might cause coal matrix shrinkage/expansion. is model also considered the effect of temperature variation and coal petrologic parameters. Zhang et al. [6] considered the action of free-phase and adsorbed-phase gas on the mechanical deformation of coal and determined that their coupling might change the pore volume. e change of coal adsorption might change the coal pores, and therefore they studied the effects of the coupling of gas flow and coal deformation. Chen et al. [7] extended the singlehole elastic model and included the flow and transport of gas mixtures (binary gases such as carbon dioxide and methane). Wu et al. [8,9] further extended the model to a two-hole elastic model (dual solid media-coal matrix and fissure), used for single gas and binary gas systems. Liu and Smirnov [10,11] established a set of mathematical models regarding carbon dioxide storage, covering capillary pressure, relative permeability, porosity, coupling adsorption model, concentration, and temperature equation. e model focused on the important effects of matrix structure deformation and determined that the matrix might affect fluid flow and lead to a rapid decline in capillary pressure and relative permeability as well. Wei and Zhang [12] developed a simulation model of two-dimensional, two-phase, three-hole/dual permeability coupling fluid flow and geomechanical coalbed methane for the extraction of natural gas and water. e mobilization of mixed gas from thermal action could not be ignored, especially at high temperatures. Liang et al. and Sun et al. [13,14] studied adsorption and desorption of coalbed methane under nonisothermal conditions and proposed mathematical models of solid-fluid coupling under nonisothermal conditions and finite element numerical methods. Xia et al. [15] studied the mathematical model of spontaneous heating in coal seams, and Zhu et al. [16] established a coupling model of coal deformation, coalbed methane transport, and heat transport and studied interactions of coal and gas under variable temperatures. Li et al. [17] studied a more comprehensive mathematical model of deformation-water vapor two-phase seepage/heat transfer. Recently, a number of scholars have also carried out some innovative works on the modeling of CBM extraction considering effective stress effect and sorption induced strain [18], coal matrix block deformation [19], sorption induced swelling modification of effective stress coefficient [20], and methane solubility in water-saturated coals [21].
All the work above is the continuous and in-depth development of studies on coalbed methane, undoubtedly playing a greater role in promoting the theory of coalbed methane extraction.
First, heat injection wells and gas extraction wells are drilled in the coal seam. en, by injecting superheated steam into the heat injection well, the temperature of the coal body may rise rapidly due to the effect of heat conduction and convection heat transfer, which will promote the desorption of adsorbed gas. At the same time, from the heat injection well to the extraction well, the gas seepage pressure gradient will increase. Hence, the permeability of the coal seam is greatly improved, thereby greatly enhancing the efficiency of gas extraction. Studies on the extraction theory and technology of coalbed methane on heating have been rapidly promoted and implemented in China [22,23]. Zhao YS Team from Taiyuan University of Technology has conducted a large number of experiments with thermal-mechanical coupling of coalbed methane in recent years and have revealed the mechanisms of coalbed methane desorption, seepage, and volume expansion under high temperature, determined the effects of temperature on permeability, and found that transportation of coalbed methane over 100°C was vastly different from that at 30～50°C. e effects of temperature are noticeable in gas adsorption/ desorption processes, seepage process, and the coupling process of gas and coal seams with temperature. is paper will elaborate on this problem from the mathematical model point of view and the extraction law of coalbed methane using heat injection technology in coal seams.

Establishment of the Mathematical Model of Solid-Fluid-Heat Coupling Effect on Coalbed Methane Extraction Using Heat Injection Technology
2.1. Basic Assumptions. Extraction of coalbed methane using heat injection not only involves the adsorption and desorption of coal and coalbed methane, heat convection, conduction and transmission, changes in the solid stress field, and several other processes but also involves changes in the permeability coefficient, adsorption constant, and other physical parameters related to the coal mass. To make the model better reflect these essential laws, the following basic assumptions are proposed: (1) With the increase of temperature, some adsorbed coalbed methane is instantaneously desorbed and converted to free coalbed methane. (2) e coalbed methane, water, and the solid matrix of coal mass can instantly reach local thermal equilibrium. (3) e coalbed methane in coal seams is reserved in free and adsorbed states, in which coalbed methane in a free state can be regarded as an ideal gas, with the content of 2 Mathematical Problems in Engineering (1) e content of the absorbed coalbed methane, along with the temperature variation law, is subordinated to the Langmuir adsorption equation: According to the experimental results of [24], the adsorption coefficients a and b obey the exponential change law affected by temperature: e total content of the coalbed methane in the coal mass can be determined by (4) e volumetric strain caused by adsorbed coalbed methane of the coal mass conforms to the exponential law [23]: (5) e seepage law of coalbed methane and water in coal seams conforms to the linear Darcy's law at the microsegment pressure gradient [23]: e entire interval conforms to the following formula [23]: K ij � K(Θ, p); that is, the permeability coefficient K is a function of stress and pore pressure [25]: (6) e pressure of water and coalbed methane is always consistent throughout the entire seepage field; that is, the effect of surface tension is not considered at the gas-liquid interface, and the pressure of coalbed methane is the same as that of water; therefore, (7) Pores and fissures within the coal mass are saturated by coalbed methane and water; therefore, (8) e coal and rock mass in the elastic deformation stage obeys the generalized Hooke's law: (9) Under the action of coalbed methane, the effective stress state of the coal mass follows the modified Biot effective stress law [26]: e effective stress coefficient a of Biot can be expressed as (10) e volume deformation of coal and rock mass consists of two parts, the deformation of the solid matrix of coal and rock mass, and the deformation of pores and fissures: It is assumed that (1 − n)α s ≪ nα p and the volume deformation of the coal and rock mass is equal to that of the pores. (11) e density of the water is no longer a constant but a function of pressure and temperature ρ w � ρ w (p w , T w ) and is expressed as follows [25]:

Coalbed Methane Seepage Equation.
Study on mass conservation of any representative elementary volume (REV) can refer to the following equation: e component form is written as follows: Darcy's law (7) and coalbed methane content equation (4) can be substituted into formula (16), providing

Mathematical Problems in Engineering
Right � zC zt , including z zt z zt e following can be obtained with the integration of formulas (16)∼(21): ese equations are the general transport and seepage equations of coalbed methane considering solid deformation, temperature action, adsorption parameter variations, and several other factors. e following formulas can be obtained from partial derivative of time from formula (3): za/zt and zb/zt in formula (23) can then be substituted into formula (22): Equation (24) is the coalbed methane seepage equation considering deformation, saturation, seepage pressure, temperature action, and several other factors.

Water Seepage Control Equation.
e mass conservation of water in any representative elementary volume can be expressed by the following formula: With respect to the saturation of coalbed methane and water in the pores, then According to assumption (11), water density is a function of temperature and water pressure; therefore, Assumption (5) regarding Darcy's law for water seepage can then be substituted into formula (25), providing is formula is the water seepage equation considering the deformation of coal and rock mass. As seen from the right side of the equation, the distribution of seepage pressure is affected by the solid matrix deformation, saturation, fluid pressure changing with time, and several other factors.

Temperature Field Control Equation considering Gas-Water Convective Heat Transfer and Heat Conduction of Coal and Rock
Mass. During the heat injection and extraction of coalbed methane, the fluid undergoes heat transfer by convection, and the coal and rock mass matrix undergoes heat transfer by heat conduction, with different heat transfer characteristics between the two materials, so the heat transfer equations need to be established, respectively, for analysis.
Heat conduction equation of a solid matrix is as follows: Fluid convective heat transfer equation is as follows: Formulas for specific heat, heat conductivity, and density of the mixed fluid are shown as follows: Combined with assumption (6), formula (31) can be substituted into formula (30): According to assumption (2), coal, rock mass, and fluids are always in a state of thermal equilibrium, and formulas (29) and (32) can be superposed: In the formulas above, (ρc) t is defined as the equivalent heat capacity coefficient of two-phase fluid in coal and rock mass.

Coal and Rock Mass Deformation Equation.
According to the elasticity theory, the static equilibrium equation of coal and rock mass is According to assumptions (8) and (9), in view of the pore pressure action of coalbed methane, deformation caused by coalbed methane desorption, and the deformation of coal seams caused by temperature variation, the stress equilibrium equation expressed by displacement is Equation (34) is the equation of deformational movement of coal and rock mass expressed by displacement, with the consideration of pore pressure, temperature, and adsorption and desorption of coalbed methane.

Mathematical Model of Heat-Fluid-Solid Coupling Effect on Coalbed Methane Extraction Using Heat Injection
Technology. Combined with the coalbed methane seepage equation considering the temperature action, the water seepage equation considering rock deformation, the temperature field control equation considering gas-water convective heat transfer and heat conduction of coal and rock mass, and related coupling equations, the solid-fluid-heat coupling control equation for coalbed methane extraction using heat injection technology can then be established and expressed as Mathematical Problems in Engineering 5 e above equations, containing eight unknowns and eight variables and supplemented by initial and boundary value conditions, form the complete mathematical model of coalbed methane extraction using heat injection technology. e model is extremely nonlinear and unable to obtain its analytical solutions directly but can have approximate solutions determined by numerical methods.

Mathematical Model Analysis and Discussion
Liang et al. [13] considered that coal temperature might be reduced by 5°C or so during gas drainage in "the mathematical model and numerical solution of gas flow under unequal temperature," which cannot be ignored during the gas seepage process. e model reflected the effect of temperature on the gas seepage process, with certain advanced nature. Zhao [23] established a model of coal gas interaction under variable temperatures, took the coal gas content under high temperature as an important action term of the seepage equation, and conducted detailed reasoning on the evolution of the coal matrix pores affected by temperature and solid deformation. ey then proposed a mathematical model of solid-fluid-heat coupling effect on gas flow and carried out numerical simulation of coal gas interaction under variable temperatures, while he did not consider the action on gas seepage from the diffusion terms of temperature field; i.e., the term zT/zx i which was not included in equation (19) only considered the effect of temperature on adsorption/desorption terms within their study on gas transport in fissures and coal matrix in the selfheating of coal seams. e mathematical model of heat-fluid-solid coupling effect considering effect of temperature on coalbed methane seepage, gas-water coexistence, and many other factors in the paper, compared with the existing mathematical models [27,28], mainly has the main characteristics as follows: at the left end of the equation, refers to the gas migration term caused by temperature gradient, with a large effect on transport of coalbed methane under large temperature gradient, which cannot be ignored, and this term is not considered in existing models.
(2) Terms II and III at the right end of the equation, respectively, consider the actions of solid deformation and gas-water relative saturation. (3) Terms IV and V at the right end of the equation refer to the effect of pore pressure variation of free and absorbed coalbed methane on the seepage process, respectively. (4) Term VI at the right end of the equation refers to the effect of temperature variation on free coalbed methane transport, while terms VII and VIII refer to the effect of temperature on adsorbed-phase coalbed methane, further affecting the mechanisms of seepage process: (5) Terms X and XI in the solid deformation equation of the model refer to the action of pore pressure on solid deformation and the action of adsorbed-phase coalbed methane on solid deformation, respectively.
From the analysis of the mathematical model of the solid-fluid-heat coupling effect on coalbed methane transport proposed in the paper, the distinctive characteristics and developments mentioned above are not included in existing mathematical models, which is another important progress in the theoretical model of coalbed methane transport. e main idea of program design is to perform time loops on various physics fields (see Figure 1). In each time loop, each physics field is analyzed separately, and then the relevant parameters are coupled and iterated to solve. e specific design process is as follows: ① According to the existing initial and boundary conditions, we first calculate the temperature field distribution of the coal and rock mass at time t 0 , so as to obtain the relative saturation of two-phase gas-liquid fluid. At the same time, according to the temperature value at this time, the physical parameters of fluid and coal matrix are reassigned, and the above parameters are substituted into the seepage equation. ② e pressure and velocity of the fluid are calculated based on the seepage equation, and then the calculated values are substituted into the seepage equation of coalbed methane. ③ Calculate the pore pressure of each node on coal seam, and obtain the gas content of each unit. Correspondingly, the expansion stress of coal and rock mass caused by adsorption can be obtained. ④ Calculate the stress field distribution of coal mass under the coupled process of coalbed methane adsorption, geostatic stress, pore pressure, thermal stress, and expansion stress. ④ When t 1 � t 0 +Δt, repeat the above steps ①-③. In this way, the distribution and variational trend of various physical parameters with multiphysics coupling process during in situ heat injection mining process of coalbed methane can be obtained. e flowchart of the calculation program according to the above method is shown in Figure 1.  33%. e thickness of the 2# coal seam used here as an example is 6°m. In the crossheading of the working face, the horizontal drilling was taken with an interval of 100 m, to form drilling holes for heat injection and production.

Model Simplification.
Hexahedron isoparametric elements were used to divide the area into 2,448 nodes and 1,650 units.
According to the mathematical model of the "solid-fluidheat" coupling effect on coalbed methane extraction using heat injection technology proposed in formula (35) and the parameters of the test area, it was assumed that the physical and mechanical characteristics of the coal mass in each section were consistent along the drainage aperture, and the section perpendicular to the drill holes was used as the calculation model, which is 100°m in length, 0.3 m in width, and 6°m in height. e simplified geometrical-physical model can be seen in Figures 2 and 3. Initial and boundary conditions can be seen in Table 1. Basic physical and mechanical parameters of the test model can be seen in Table 2.

Numerical Test.
is paper establishes a numerical model based on the 2# coal seam of Tunlan Coal Mine, with the heat injection hole set on the left of the model and the drainage hole set on the right, and conducts numerical simulation calculation with the consideration of "solid-fluid-heat" coupling.

Variation of the Coal Temperature Field with Heat
Injection. Figure 4(a) shows that the state of heat transfer in coal seams was clear after heat injection for 3 days; that is, under the condition of hydraulic fracturing and convective heat transfer, coal mass and high-temperature water reached thermal equilibrium very quickly, and temperature rose rapidly in the water injection areas. e temperature of the coal seams in the area of the heat injection holes reached 250°C, with the fissure zone as the boundary, and the temperature of the upper and lower coal seams gradually rose. e temperature rises in    Figure 5 is the temperature field distribution profile of the model on the y � 0 plane. After 30 days, the temperature reached almost 280°C within 2 m of the fissure zone up and down. Later, it gradually expanded from the fissure zone to the bottom and top beds, and the temperature accelerated desorption of coalbed methane in the coal mass.  All sides of the model are airtight boundaries (except the extraction hole): zp g /zn � 0 Extraction hole: p g (x 1 , y 1 , z 1 , t) � 1 atm As seen from Figure 6, along with heat injection, the adsorbed gas near the fracture belt of coal seams underwent quick desorption, and the gas content in the area decreased rapidly. Bordered by the fissure, the gas content of the upper and lower parts of the coal seams weakened. On the 3rd day of heat injection, as shown in Figure 6(a), the absorbed gas content of the fissure zone clearly decreased, and the absorbed gas content of the area surrounding the heat injection holes decreased from 22.7 m 3 /t to 2 m 3 /t; the absorbed gas content still reached 17.8 m 3 /t at a distance of 3 m from the fissure zone. Most of the gas in other areas had not been desorbed at this point in time. Figures 6(b) and 6(c) are diagrams of absorbed gas content of the coal seams after heat injection for 10 days and 20 days, respectively, and the desorbed area gradually expanded along with heat injection of absorbed gas. In Figure 6(d), after heat injection and extraction for one month, the absorbed gas content of the coal mass in the simulated area clearly decreased, with the highest content only 2.5 m 3 /t. Figure 7 is the distribution of adsorbed gas content near the vertical heat injection holes, and the absorbed gas content along with heat injection varied similarly as those in Figure 6. Figure 8 shows the variation of the total gas content of the coal seams. e gas content in one ton of coal in the primitive coalbeds in the model reached 23.8 m 3 ; after heat injection for 3 days, the total gas content of the fissure zone decreased rapidly, with a minimum value of 1.2 m 3 /t coal . After 10 days, the total gas content of the coal seams continuously decreased, with the   total content at top beds down to 17.3 m 3 /t coal and the minimum near the extraction holes. After 30 days, the maximum gas content of the coal seams reached 6.1 m 3 /t coal , with a minimum value of 1.2 m 3 /t coal . According to the area weighted average, the total gas content reached 3.5 m 3 /t coal , and the mining rate reached 85% within one month in the heat injection and extraction area of coalbed methane, which is an effect that has not been achieved by single extraction technology to date. Figure 8 shows the contour map of total gas content, gas content decreased gradually along with heat injection from the top and bottom beds to the fissure zone, forming a pressure decrease funnel around the extraction hole, and the minimum value of the gas content in coal seams only reached 1.2 m 3 /t coal .

Variation Trends of Coalbed Gas Pore Pressure with
Heat Injection Process. As seen in Figure 10(a), after heat injection for 3 days, the pore pressure within 0∼60°m of the fissure zone in the coal seams increased rapidly in the x direction, the maximum gas pore pressure was 61 atm, and the pore pressure gradually decreased to 39 atm from the fissure zone towards the top and bottom beds. In the vicinity of heat injection holes, the pore pressure of the coalbed gas was affected by extraction, which was lowered by 13.25 atm compared to the original pore pressure of the coalbed gas. From Figures 10(b)-10(d), the pore pressure of the coalbed gas had a greater uplift compared to the original pore pressure along with heat injection, and the pressure gradient between the extraction holes increased, conducive to extraction.
As seen from Figure 11, the pressure of the coalbed methane from the heat injection hole towards the drainage hole changed from high to low, allowing for the coalbed methane to flow towards the drainage hole. Figure 11(a) clearly shows that the pressure of coalbed methane was distributed along the fissure zone, in both directions vertically, which was the result of coalbed methane desorption caused by the temperature field variation. Due to drainage holes, a pressure decrease area formed near the center coordinate area of semicircular pressure decrease area (center coordinate x, y � 100°m, 3°m) that formed by extraction in Figure 11(a). Such a pressure gradient variation is reflected in Figures 11(b)-11(d); namely, the pressure gradient caused by extraction of coalbed methane always existed. e left half of Figures 11(b)-11(d) contains high-pressure for up to 30 days, and the coalbed methane pressure increased to 40 atm to the utmost within 0∼50 m in x direction. is was because the desorbed coalbed methane was temporarily stored in the micropores and was not drained in a       Mathematical Problems in Engineering timely manner under high temperature, resulting in the rise in local pressure. Effects of gradual extraction moved from the drainage holes towards the heat injection well.

Variation Law of Volume Stress.
As seen from Figure 12, along with heat injection, coalbed gas was gradually desorbed, and the local pore pressure of the desorbed coal matrix block increased. is in turn decreased the volume stress of coal mass under constant situation of total stress, and the temperature of the coal seams in the fissure zone more rapidly increased. Also, the absorbed gas underwent more desorption, so the volume stress from the fissure zone outwards vertically gradually decreased from 200 kg/cm 2 to 170 kg/cm 2 . As seen from Figure 13, the adsorbed gas content of coal mass gradually was desorbed, the pore pressure rose, and the volume stress decreased.
To sum up, the results of the simulation show extreme nonlinearity, which is closely related to the extreme nonlinearity of the mathematical model, and there is square term p 2 in the seepage equation of CBM. ere is an exponent e − αT affected by temperature in the content equation of CBM. e exponential item e cC in the volume expansion law is caused by CBM adsorption. e permeability of coalbed methane and water density are nonlinear, which makes the model appear extremely nonlinear. At the same time, the coupling effects of heat transfer, water seepage, CBM absorption, and coal-rock deformation are considered, so that the mathematical model of CBM migration coupled with deformation, seepage, and heat transfer is more comprehensive and complete. e nonlinearity of the calculation results is the reflection of the above reasons.

Conclusions
rough in-depth analysis on the mechanisms of coalbed methane extraction using heat injection in coal seams and industrial schemes, this paper analyzed the complex coupling process of temperature field, gas-water seepage field, and coal mass deformation field for desorption and seepage of coalbed methane using heat injection. e following can be concluded: (1) A solid-fluid-heat coupling mathematical model of coalbed methane extraction using heat injection was established, with distinctive characteristics that the right end of the coalbed methane equation contained the pore variation action term zn/zt, the action term zS g /zt of gas-liquid relative saturation, the action term zp g /zt of free gas pore pressure variation, temperature variation action term zT/zt, and the action term of coalbed methane migration cause by temperature gradient under high temperature. ese are a key development in the theory of coalbed methane transport. (2) Under water pressure, the temperature of coal seams increased rapidly along the fissure zone. After 30 days, the temperature reached 260°C within 2°m of the fissure zone outwards vertically. e minimum temperature of coal seams on both sides of the fissure zone reached 140°C. (3) e pore pressure of the coal seams gradually increased with the increase of temperature. After heat injection for 10 days, the pore pressure of the coal seams in the heat injection area increased to 5.5 MPa, and after 30 days, the pressure remained constant at 3.5 Mpa. Such a high fracturing gradient promoted the rapid flow and drainage of gas. e coalbed methane mainly took seepage in the direction parallel to the fissures, forming a pressure decrease round funnel near the drainage holes. is funnel area gradually expanded along with heat injection and extraction. (4) Under high temperature, the coalbed gas was desorbed quickly, the absorbed gas content formed an oval funnel from the heat injection hole towards the extraction hole, centered by the fractured fissure and outwards vertically towards the coalbed upper and lower boundaries. Along with heat injection and extraction, the absorbed gas content rapidly decreased. On the 30th day, the absorbed gas content of the entire heat injection area decreased to 1.5 m 3 /t, only 7% of the original content.