- Email: [email protected]
Development and validation of THM coupling model of methane-containing coal
International Journal of Mining Science and Technology 22 (2012) 879–883
Contents lists available at SciVerse ScienceDirect
International Journal of Mining Science and Technology journal homepage: www.elsevier.com/locate/ijmst
Development and validation of THM coupling model of methane-containing coal Tao Yunqi ⇑, Xu Jiang, Liu Dong, Liang Yongqing State Key Laboratory of Coal Mine Disaster Dynamics and Control, Chongqing University, Chongqing 400044, China State and Local Joint Engineering Laboratory of Methane Drainage in Complex Coal Gas Seam, Chongqing University, Chongqing 400044, China
a r t i c l e
i n f o
Article history: Received 3 April 2012 Received in revised form 4 May 2012 Accepted 15 June 2012 Available online 24 January 2013 Keywords: Coal containing methane Temperature field Seepage field Stress field Fluid–solid-heat coupling
a b s t r a c t Based on nine necessary basic assumptions for THM coupling model, this research comprehensively applied the theories of elastic mechanics, seepage mechanics and heat transfer, and established a real three-field and two-way coupled mathematical model to reveal the connections among seepage field, deformation field and temperature field within the system of methane-containing coal. In comparison between numerical and analytical solutions, the coupling modeling for THM of methane-containing coal was proved to be correct by model application in the physical simulation experiment of coal and gas outburst. The model established in this paper was the improvement of traditional seepage theory of methane-containing coal and fluid–solid coupled model theory, which can be widely used in prevention of coal and gas outburst as well as exploitation of coal bed methane. Ó 2012 Published by Elsevier B.V. on behalf of China University of Mining & Technology.
1. Introduction With the increase in the depth of coal mining, thermal effect has become one of the critical influencing factors of gas flow in the coal seam. Therefore, it is required to consider the thermal effect induced by an increase of mining depth when involving researches on coal seam flow, namely THM coupling. Plenty of research results of the THM coupling obtained by scholars at home and abroad focused on fields such as development and utilization of the geothermal resources, oil heat extraction and nuclear waste repositories, etc., which did not efficiently implement a real three-field and two-way coupling [1–6]. For example, Liu et al. studied THM coupling of gas flow in the coal seam under the non-isothermal condition, and established a three-field coupled model [7–10]. However, it is a pity that the deformation equation did not reflect the temperature field, and the temperature equation did not reflect the deformation field. In addition, the model did not implement the two-way coupling. In this paper, based on the auctorial researches in recent years, a comparatively perfect mathematical model was developed for THM coupling of methane-containing coal in order to achieve two-way fully coupling model of the temperature, seepage and stress fields [11].
methane-containing coal should consist of three parts: temperature field, seepage field and the stress field. To make it more convenient, the following nine assumptions were proposed on the basis of former results: (1) Gas-containing coal was treated as a homogeneous and isotropic body, and the heat transfer coefficients of coal mass and gas did not change with the temperature. (2) Gas-containing coal was saturated with single-phase methane. (3) The inertial force and volume force of movement induced by free gas seepage and coal mass deformation were neglected. (4) The change of effective stress of containing-gas coal was in conformity with the modified Terzaghi’s law. (5) Volumetric deformation of the saturated porous media was equal to the pore deformation [12]. (6) The law of gas flow in the coal seam was in accord with Darcy law.
q¼
k
l
rp
ð1Þ
According to the result obtained by Tao et al. [13]:
3 k0 e ðbDT K Y DpÞð1 u0 Þ eP 1þ 1þe u0 u0 u0
2. Basic assumptions
k¼
The coal seam deformation and gas flow in coal seam is affected by fluid–solid-heat coupling, so the THM coupling model of
where k is the permeability; k0 the initial permeability; q the velocity vectors of gas flow; Dp the gas pressure gradient; l the gas dynamic viscosity; KY the volume compressibility coefficient; b the volume thermal expansion coefficient of coal mass; u0 the initial coal porosity; e the volume strain of coal mass; DT the
⇑ Corresponding author. Tel.: + 86 15003829708. E-mail address: [email protected] (Y. Tao).
2095-2686/$ - see front matter Ó 2012 Published by Elsevier B.V. on behalf of China University of Mining & Technology. http://dx.doi.org/10.1016/j.ijmst.2012.12.009
ð2Þ
880
Y. Tao et al. / International Journal of Mining Science and Technology 22 (2012) 879–883
absolute temperature variation; and eP the expansion strain induced by gas adsorption and desorption. (7) The amount of adsorption gas in the coal mass matched the modified Langmuir adsorption model, and free gas accorded conformed with the real gas state equation.
Q¼
abpc p qn þu 1 þ bp pn
qg ¼
ð3Þ
qn p
ð4Þ
pn Z
where Q is the gas content per unit volume of coal mass; qg the gas density at the pressure of p; qn the gas density in the standard state; pn the gas pressure in the standard state; Z the compressibility factor, and its value was approximate to 1; c the correction parameter of the coal quality; a the limiting adsorbed amount per unit mass of coal under the specific pressure; and b the adsorption equilibrium constant. (8) Deformation of coal mass was tiny, and coal mass stayed at the line elastic deformation stage, in accord with the generalized Hooke law [14]:
r0ij ¼ kdij e þ 2Geij
ð5Þ
where k is the Lame constant; G the shear modulus; and eij the strain tensor. (9) The symbol laws of stress–strain was the same as elasticity, and compressive stress and compressive strain were negative, but the tensile stress and extensional strain were positive; decrease of right angles in shear deformation progress was positive, conversely negative. Displacement in the positive axis direction is positive, conversely negative. 3. THM coupling model of containing-methane coal 3.1. Stress equation (1) Balance equation: Gas-containing coal is the dual porosity media composed by the coal skeleton containing molecular scale pores as well as the fissures between coal particles. Due to the effect of load, coal mass will produce stress, and the deformation or relative displacement of coal skeletons will occur. The gas not only moves along with the coal mass skeleton, but also flows relative to coal mass skeleton. According to the principle of elasticity, differential equation of equilibrium of gas-containing coal is as follows:
rij;j þ F i ¼ 0 ði; j ¼ 1; 2; 3Þ
ð6Þ
where Fi is the volume force. According to the basic assumption (4), the effective stress change of gas-containing coal skeleton coincided with the modified Terzaghi’s law:
r=ij ¼ rij apdij
ð7Þ
where
a¼
rð1 uÞ p
þu
ð8Þ
According to the conclusions resulted from Tao et al. [15], following equation was obtained:
r ¼ Ee ¼ u¼1
2aqRTð1 2tÞ EbDT ð1 2tÞDp lnð1 þ bpÞ þ 3V m 3
ð1 u0 Þ 2aqRTK Y lnð1 þ bpÞ 1 þ bDT K Y Dp þ 1þe 3V m ð1 u0 Þ =
ð9Þ
ð10Þ
where rij is the total stress tensor; rij the effective stress tensor; p the gas pressure; a the pore compressibility coefficient; r the total
swelling stress in all direction; e the total linear expansion strain; E the elasticity modulus; t the Poisson ratio; u the coal porosity; u0 the initial coal porosity; e the volume strain of coal mass; b the volume thermal expansion coefficient of coal mass; DT the absolute temperature variation; Dp the variable quantity of gas pressure; p the gas pressure; KY the volume compressibility coefficient; eP the expansion strain induced by gas adsorption and desorption; q the apparent density of the coal mass; Vm the gas molar volume; R the universal gas constant; a the limiting adsorbed amount per unit mass of coal under the specific pressure, with a = a(T) as a function of temperature [16]; b the adsorption equilibrium constant, with b = b(T) as a function of temperature T [13]; and dij the Kronecker symbol.
2
1 0 0
3
7 6 7 6 dij ¼ 6 0 1 0 7 5 4 0 0 1
ð11Þ
By substitution of Eq. (7) into Eq. (6), differential equation of equilibrium expressed by the effective stress was obtained as follows:
r=ij;j þ ðapdij Þ;j þ F i ¼ 0
ð12Þ
(2) Geometric equation:
1 2
eij ¼ ðuij þ uji Þ ði; j ¼ 1; 2; 3Þ
ð13Þ
(3) THM constitutive equation: Based on the assumption of linear thermoelasticity, the constitutive equation was established. That is to say, the total strain of gas-containing coal was composed of thermal strain, compressive strain of coal caused by gas pressure, the strain caused by swell that was induced gas adsorption, and the strain induced by exterior stress. According to the relationship of thermoelasticity and stress–strain, constitutive equation (Eq. (14)) for fluid–solid-heat coupling was deduced by a series of derivation and transformation, in which effective stress was represented by strain.
r0ij ¼ kedij þ 2Geij hT DTdij hPY Dpdij hPX aT lnð1 þ bpÞdij
ð14Þ
where hT, hPY, hPX are the coefficient of thermal stress, the stress coefficient caused by gas pressure and the stress coefficient of gas adsorption, respectively. And:
8 Et 2Gt k ¼ ð1þtÞð12 > tÞ ¼ 12t > > > > > > E > G ¼ 2ð1þ > tÞ > > > < ð3kþ2GÞb hT ¼ 3 > > > > > ð3k2GÞK Y > hPY ¼ > > 3 > > > > : qRK Y Þ hPX ¼ ð3kþ2GÞð2 9V m
ð15Þ
(4) Stress equation: Substituting Eqs. (13) and (14) into Eq. (12) could obtain THM coupling stress field equations of gas-filled coal:
G uj;ji hT ðDTÞ;i hPY ðDpÞ;i hPX aT½lnð1 þ bpÞ;i þ ap;i 1 2t þ Fi ¼ 0 ð16Þ
Gui;jj þ
Eq. (16) contains coupling terms ap,i that reflects the influence of seepage field and temperature field synchronously, hT(DT),i and hPXaT[ln (1 + bp)],i that reflect the influence of temperature field.
Y. Tao et al. / International Journal of Mining Science and Technology 22 (2012) 879–883
3.2. Coupling seepage equation of coal containing gas (1) Continuity equation: According to basic assumption (6) and the law of mass conservation, continuity equation of gas flow in the coal seam considering source sink term was obtained as follows:
@Q þ r ðqg qÞ ¼ I @t
4. Definite conditions
" # @e 2ð1 uÞ 2abcpn 2abcpn @p þ 2 uþ r pþ þ @t ks @t ð1 þ bpÞ2 1 þ bp k rp2
2ap
l
ð18Þ
where permeability k and porosity u are the function of strain, temperature and gas pressure; adsorption constants a, b the function of temperature T. Therefore, the seepage equation consisted of the coupling term simultaneously reflecting stress field and temperature field of coal containing gas. 3.3. Coupling temperature equation of gas-containing coal Both field observation and experimental studies show that thermal effect is present in the progress of gas adsorption–desorption and seepage in the coal seam, as well as deformation progress of the induced by external force. So coupling temperature equation of gas-containing coal should also be coupled with stress field and seepage field. By comprehensive analysis of the energy conversion and balance within gas-containing coal mass, the total work done by stress on per unit volume of gas-containing coal in terms of the first law of thermodynamics and the Gauss formula.
dW ¼ rij deij
ð19Þ
The specific entropy was introduced by the second law of thermodynamic, namely s:
ds ¼
dQ d T
ð20Þ
Another thermodynamic state function was also introduced, namely Helmholz free energy F:
F ¼ U Ts
deformation work partly. In addition, it is the coupling term simultaneously reflecting seepage field, temperature field and stress field. Therefore, solution of the temperature equation could be achieved unless simultaneous equation with seepage equation and the stress equation.
ð17Þ
(2) Seepage equation: On the basis of the assumptions, Eqs. (6), (8), and (9) were substituted into Eq. (17). The coupling seepage equation of gascontaining coal (Eq. (18)) was obtained, which included source sink term I.
¼I
881
4.1. Definite conditions of stress field for THM model (1) Boundary conditions: The boundary condition of stress field generally includes three types. The first type was displacement boundary condition, and the boundary displacement of coal was already known to us. The second type was stress boundary condition, and the surface force of coal boundary was known. The component surface forces along x, y, z three directions were Fx, Fy, Fz, respectively. The angles of boundary normal directions to the three directions in orthogonal coordinates were represented by n, w and -; and direction cosines were represented by l, m and n (l = cos n, m = cos w, n = cos -). Therefore, the stress boundary condition was as follows:
8 r l þ syx m þ szx n ¼ F x > < x ry l þ szy m þ sxy n ¼ F y > : rz l þ sxz m þ syz n ¼ F z
ð23Þ
The third type was the mixed boundary condition, of which the displacement of some boundary and the boundary stress of the coal mass skeleton were known. (2) Initial condition: The initial condition of stress field in methane-containing coal was the initial value of displacement or particle velocity when t is zero. 4.2. Definite conditions of seepage field for THM model The boundary condition of seepage field included three types, too. The first one was the constant pressure on the boundary; the second type was the constant flow on the boundary; the third one was the mixed boundary condition, of which flow rate on partial boundary was given and pressure on the other part of boundary was also given; the forth one was that flow rates at the interface were equal if there was inter boundary, that was:
k1
@p1 @p2 ¼ k 2 @n1 L1 @n2 L2
ð24Þ
ð21Þ
According to the second law of thermodynamics, coupling temperature equation of gas-containing coal was determined by simultaneous equation of Eqs. (19)–(21) and THM constitutive equation of gas-containing coal.
@T @e þ T 0 hT @t @t @a ap @b @e þ hPX T 0 T 0 lnð1 þ bpÞ þ T 0 þ alnð1 þ bpÞ @T 1 þ bp @T @t
gr2 T þ qQ ¼ qC V
ð22Þ where qQ is the additional item caused by gas desorption. That is to say, temperature shows an decrease, which is induced by gas desorption. And this term is also the coupling term simultaneously reflecting seepage field and temperature field. The term qC V @T þ T 0 hT @e is the additional term caused by the deformation @t @t work due to thermoelastic and gas. That is to say, the heat given to coal body led to not only increase in temperature, but also
4.3. Definite conditions of temperature field for THM model (1) Boundary conditions: Generally, there are three types of boundary condition for temperature field. The first one was that temporal and spatial variation of temperature at coal mass the boundary was given. The second one was that the heat flux (qn)C at the outward normal of the boundary had the certain functional relation with space and time. The third one was that the temperature Tf of fluid medium and the heat transfer coefficient h of boundary surface were certain, if the convective heat transfer existed between fluid and object surface. (2) Initial condition: If the temperature field was unstable, the initial condition referred to the temperature T when time t was zero and the initial temperature value could be a certain value, or a function of space.
882
Y. Tao et al. / International Journal of Mining Science and Technology 22 (2012) 879–883
5. THM coupling model validation
1.2 1.0
@ k1 ðp þ BÞ @p ¼0 @x @x l
experimental solution
0.8 0.6 0.4
numerical solution
0.2 0.0
0
20
40
60
80
100
120
140
t/s
(a) Variation of gas pressure 296
experimental solution
295
numerical solution
T /K
To verify the THM coupling model of methane-containing coal established in this paper, it is necessary to comparatively analyze analytic and the numerical solution. Because of the nonlinear complexity of THM model, it is difficult to calculate the analytical solutions. Consequently, THM coupling model was simplified in order to get the analytical solutions, and then verified the THM coupling model by comparative analysis between the analytic and numerical solutions. The correctness and validity of the THM model and the resolution method based on COMSOL Multiphysics were proved by the example of one-dimensional gas seepage considering the Klingberg effect. Assuming that porous ratio u was constant under the condition of constant temperature. Considering the Klingberg effect, Eq. (18) could be simplified as one-dimension stable flow equation [17]:
p /MPa
5.1. Analytic solutions validation
294
ð25Þ 293
Under the one-dimensional case, when the inlet flow rate Qm of sample and the outlet pressure pL were constant, Eq. (26) could be deduced from Eq. (25):
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Q m lðL xÞ pðxÞ ¼ B þ B2 þ p2L þ 2BpL þ k1
400
800 t/s
1200
1600
2000
(b) Variation of temperature Fig. 2. Correlation curve of gas pressure and temperature variable curve.
ð26Þ
For the one-dimension model, the geometric model was a columniform raw coal, which was 10 m in length and 1 m2 in cross-sectional area. The outlet pressure and the inlet gas flow rate were both constant. The numerical and analytical solutions deduced by the COMSOL Multiphysics are shown in Fig. 1. The calculated results indicate that the numerical solution agrees well with analytical solution. Therefore, the computational method induced in this paper is correct and the model is reliable. 5.2. Validation in physical experiment To assure the correctness of coupling THM model, we calculated the change curve of gas pressure and temperature during aeration and absorption progress before outburst. By numerical analysis of THM model, and then this study compared it with measured curve in the physical experiment of coal and gas outburst. The geometric size and boundary conditions for numerical calculation were in accord with that for physical simulation experiment, which was 570 mm in length and 365 mm in height. The boundary conditions for temperature field in geometric model were all restrained by certain temperature, 293 K; the initial temperature in coal sample was determined to 293 K, too; the initial gas pressure in coal sample was determined to 0.1 MPa. As for boundary conditions for stress field, the upper boundary was applied with normal stress, r1 = 4.0 MPa; on the right boundary, there was only horizontal stress, namely r2 2.4 MPa; and other boundaries were all under analytical solution numerical solution
p (p-1.0e5) /Pa
0
L /m
Fig. 1. Comparison of numerical and analytic solutions of one-dimensional steady gas flow.
displacement constrains. The result of numerical analysis is shown in Fig. 2. It is found that it required 120 s for gas pressure increasing from 0.1 to 1.0 MPa, in comparison to 108 s in the physical experiment, and the pressure did not change obviously within the former 10 s, in comparison to 8 s in the physical experiment. When time of filling gas lasted for 1490 s, the temperature increased by 2.24 °C (from 293.00 to 295.24 K), compare with 2.64 °C (from 293.00 to 295.64 K) in the physical experiment. The reason for difference of temperature change is the frictional heat between coal particles when applying loading, which was not considered in numerical calculation. Although less better than similar degree between numerical and analytical solutions, the simulated result is basically in accord with the result from physical experiment. Therefore, it can be considered that the regular conclusions of numerical simulation by COMSOL Multiphysics based on THM coupling model are credible. 6. Conclusions (1) Based on theories of elastic mechanics, seepage mechanics and heat transfer, a real three-field and two-way coupled mathematical model is established to reveal the connections among seepage, deformation and temperature difference within methane-containing coal. (2) It is considered that the total strain of gas-containing coal is composed of thermal strain, compressive strain of coal caused by gas pressure, the strain due to gas adsorption and the strain induced by exterior stress. Based on this, the modified THM constitutive equation of gas-filled coal is established. (3) The coal mass deformation and gas flow in coal seam is affected by fluid–solid-heat coupling, because of the effect of gas adsorption–desorption on the stress field and the temperature field. (4) The coupling modeling for THM of methane-containing coal is proved to correct, by comparison of numerical and analytical solutions of the simplified model, and by model application under the background of physical simulation experiment of coal and gas outburst.
Y. Tao et al. / International Journal of Mining Science and Technology 22 (2012) 879–883
Acknowledgments Thanks for financially supported in part by the State Key Basic Research Program of China (No. 2011CB201203), the General Project of the National Natural Science Foundation of China (No. 50974141), the Key Project of the National Natural Science Foundation of China (No. 50534080), and the Key Special Subjects National Science and Technology of China (No. 2011ZX05034-004). References [1] Bear J, Corapcioglu MY. A mathematical model for comsolidation in at hermoelastic aquifer due to hot water injection or pumping. Water Resource Res 1981;17:723–36. [2] Lewis RW, Sukirman Y. Finite element modeling of three phase flow in deforming saturated oil reservoirs. Int J Num Anal Methods Geomech 1993;17:577–98. [3] Lewis RW. Finite element modeling of two phase heat and fluid flow in deforming porous media. Trans Porous Media 1989;4:319–34. [4] He Yulong, Yang Lizhong, Yang Jiyi. Governing equations for coupled thermo– hydro-mechanical behaviors in unsaturated rock mass. J Southwest Jiao Tong Univ 2006;41(4):419–23. [5] Gutierrez M, Makurat A. Coupled HTM modelling of cold water injection in fractured hydrocarbon reservoirs. Int J Rock Mech Min Sci Geomech Abstr 1997;34(3/4):429.
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[6] Ziming Wang. Reservoir fluid–solid-heat coupling model research and preliminary application. Chengdu: Southwest Petroleum University; 2002. [7] Liu Jianjun, Liang Bing, Zhang Mengtao. Research on low the gas flow under conditions of unequal temperature. J Xi’an Univ Sci Technol 1999;19(4): 302–8. [8] Liang Bing, Liu Jianjun, Wang Jinshan. The study of coupling effect of coal and gas under unequal temperature. J Liaoning Tech Univ 1999;18(5):483–6. [9] Liang Bing, Liu Jianjun, Fan Houbin, Zhang Mengtao. The mathematical model and its numerical solution of gas flow under unequal temperatures. Chin J Rock Mech Eng 2000;19(1):1–5. [10] Liu Jianjun. Mathematic models for thermo–hydro-mechanical coupled flow of coal-bed methane. J Wuhan Polytech Univ 2002;20(2):91–4. [11] Tao Yunqi, Xu Jiang, Li Shuchun, Peng Shoujian. Advances in study on seepage flow property of coal bed methane. Coal Geol Explor 2009;37(2):1–5. [12] Bear J. Porous medium fluid dynamics. Beijing: China Architecture & Building Press; 1998. [13] Tao Yunqi, Xu Jiang, Cheng Mingjun, Li Shuchun, Peng Shoujian. The recital analysis and experimental study on permeability of gas-bearing coal. Chin J Rock Mech Eng 2009;8(9):3363–70. [14] Sun Peide. SUN model and its application. Hang Zhou: Zhejiang University Press; 2002. [15] Tao Yunqi, Xu Jiang, Peng Shoujian, Yuan Mei. Experimental study of influencing factor of porosity and effective stress of gas-filled coal. Rock Soil Mech 2010;31(11):3417–22. [16] Tao Yunqi. Study on the gassy coal THM coupling model and coal and gas outburst simulation. Chongqing: Chongqing University; 2009. [17] Boit MA. General theory of three-dimensional consolidation. J Appl Phys 1941;12:155–64.
Contents lists available at SciVerse ScienceDirect
International Journal of Mining Science and Technology journal homepage: www.elsevier.com/locate/ijmst
Development and validation of THM coupling model of methane-containing coal Tao Yunqi ⇑, Xu Jiang, Liu Dong, Liang Yongqing State Key Laboratory of Coal Mine Disaster Dynamics and Control, Chongqing University, Chongqing 400044, China State and Local Joint Engineering Laboratory of Methane Drainage in Complex Coal Gas Seam, Chongqing University, Chongqing 400044, China
a r t i c l e
i n f o
Article history: Received 3 April 2012 Received in revised form 4 May 2012 Accepted 15 June 2012 Available online 24 January 2013 Keywords: Coal containing methane Temperature field Seepage field Stress field Fluid–solid-heat coupling
a b s t r a c t Based on nine necessary basic assumptions for THM coupling model, this research comprehensively applied the theories of elastic mechanics, seepage mechanics and heat transfer, and established a real three-field and two-way coupled mathematical model to reveal the connections among seepage field, deformation field and temperature field within the system of methane-containing coal. In comparison between numerical and analytical solutions, the coupling modeling for THM of methane-containing coal was proved to be correct by model application in the physical simulation experiment of coal and gas outburst. The model established in this paper was the improvement of traditional seepage theory of methane-containing coal and fluid–solid coupled model theory, which can be widely used in prevention of coal and gas outburst as well as exploitation of coal bed methane. Ó 2012 Published by Elsevier B.V. on behalf of China University of Mining & Technology.
1. Introduction With the increase in the depth of coal mining, thermal effect has become one of the critical influencing factors of gas flow in the coal seam. Therefore, it is required to consider the thermal effect induced by an increase of mining depth when involving researches on coal seam flow, namely THM coupling. Plenty of research results of the THM coupling obtained by scholars at home and abroad focused on fields such as development and utilization of the geothermal resources, oil heat extraction and nuclear waste repositories, etc., which did not efficiently implement a real three-field and two-way coupling [1–6]. For example, Liu et al. studied THM coupling of gas flow in the coal seam under the non-isothermal condition, and established a three-field coupled model [7–10]. However, it is a pity that the deformation equation did not reflect the temperature field, and the temperature equation did not reflect the deformation field. In addition, the model did not implement the two-way coupling. In this paper, based on the auctorial researches in recent years, a comparatively perfect mathematical model was developed for THM coupling of methane-containing coal in order to achieve two-way fully coupling model of the temperature, seepage and stress fields [11].
methane-containing coal should consist of three parts: temperature field, seepage field and the stress field. To make it more convenient, the following nine assumptions were proposed on the basis of former results: (1) Gas-containing coal was treated as a homogeneous and isotropic body, and the heat transfer coefficients of coal mass and gas did not change with the temperature. (2) Gas-containing coal was saturated with single-phase methane. (3) The inertial force and volume force of movement induced by free gas seepage and coal mass deformation were neglected. (4) The change of effective stress of containing-gas coal was in conformity with the modified Terzaghi’s law. (5) Volumetric deformation of the saturated porous media was equal to the pore deformation [12]. (6) The law of gas flow in the coal seam was in accord with Darcy law.
q¼
k
l
rp
ð1Þ
According to the result obtained by Tao et al. [13]:
3 k0 e ðbDT K Y DpÞð1 u0 Þ eP 1þ 1þe u0 u0 u0
2. Basic assumptions
k¼
The coal seam deformation and gas flow in coal seam is affected by fluid–solid-heat coupling, so the THM coupling model of
where k is the permeability; k0 the initial permeability; q the velocity vectors of gas flow; Dp the gas pressure gradient; l the gas dynamic viscosity; KY the volume compressibility coefficient; b the volume thermal expansion coefficient of coal mass; u0 the initial coal porosity; e the volume strain of coal mass; DT the
⇑ Corresponding author. Tel.: + 86 15003829708. E-mail address: [email protected] (Y. Tao).
2095-2686/$ - see front matter Ó 2012 Published by Elsevier B.V. on behalf of China University of Mining & Technology. http://dx.doi.org/10.1016/j.ijmst.2012.12.009
ð2Þ
880
Y. Tao et al. / International Journal of Mining Science and Technology 22 (2012) 879–883
absolute temperature variation; and eP the expansion strain induced by gas adsorption and desorption. (7) The amount of adsorption gas in the coal mass matched the modified Langmuir adsorption model, and free gas accorded conformed with the real gas state equation.
Q¼
abpc p qn þu 1 þ bp pn
qg ¼
ð3Þ
qn p
ð4Þ
pn Z
where Q is the gas content per unit volume of coal mass; qg the gas density at the pressure of p; qn the gas density in the standard state; pn the gas pressure in the standard state; Z the compressibility factor, and its value was approximate to 1; c the correction parameter of the coal quality; a the limiting adsorbed amount per unit mass of coal under the specific pressure; and b the adsorption equilibrium constant. (8) Deformation of coal mass was tiny, and coal mass stayed at the line elastic deformation stage, in accord with the generalized Hooke law [14]:
r0ij ¼ kdij e þ 2Geij
ð5Þ
where k is the Lame constant; G the shear modulus; and eij the strain tensor. (9) The symbol laws of stress–strain was the same as elasticity, and compressive stress and compressive strain were negative, but the tensile stress and extensional strain were positive; decrease of right angles in shear deformation progress was positive, conversely negative. Displacement in the positive axis direction is positive, conversely negative. 3. THM coupling model of containing-methane coal 3.1. Stress equation (1) Balance equation: Gas-containing coal is the dual porosity media composed by the coal skeleton containing molecular scale pores as well as the fissures between coal particles. Due to the effect of load, coal mass will produce stress, and the deformation or relative displacement of coal skeletons will occur. The gas not only moves along with the coal mass skeleton, but also flows relative to coal mass skeleton. According to the principle of elasticity, differential equation of equilibrium of gas-containing coal is as follows:
rij;j þ F i ¼ 0 ði; j ¼ 1; 2; 3Þ
ð6Þ
where Fi is the volume force. According to the basic assumption (4), the effective stress change of gas-containing coal skeleton coincided with the modified Terzaghi’s law:
r=ij ¼ rij apdij
ð7Þ
where
a¼
rð1 uÞ p
þu
ð8Þ
According to the conclusions resulted from Tao et al. [15], following equation was obtained:
r ¼ Ee ¼ u¼1
2aqRTð1 2tÞ EbDT ð1 2tÞDp lnð1 þ bpÞ þ 3V m 3
ð1 u0 Þ 2aqRTK Y lnð1 þ bpÞ 1 þ bDT K Y Dp þ 1þe 3V m ð1 u0 Þ =
ð9Þ
ð10Þ
where rij is the total stress tensor; rij the effective stress tensor; p the gas pressure; a the pore compressibility coefficient; r the total
swelling stress in all direction; e the total linear expansion strain; E the elasticity modulus; t the Poisson ratio; u the coal porosity; u0 the initial coal porosity; e the volume strain of coal mass; b the volume thermal expansion coefficient of coal mass; DT the absolute temperature variation; Dp the variable quantity of gas pressure; p the gas pressure; KY the volume compressibility coefficient; eP the expansion strain induced by gas adsorption and desorption; q the apparent density of the coal mass; Vm the gas molar volume; R the universal gas constant; a the limiting adsorbed amount per unit mass of coal under the specific pressure, with a = a(T) as a function of temperature [16]; b the adsorption equilibrium constant, with b = b(T) as a function of temperature T [13]; and dij the Kronecker symbol.
2
1 0 0
3
7 6 7 6 dij ¼ 6 0 1 0 7 5 4 0 0 1
ð11Þ
By substitution of Eq. (7) into Eq. (6), differential equation of equilibrium expressed by the effective stress was obtained as follows:
r=ij;j þ ðapdij Þ;j þ F i ¼ 0
ð12Þ
(2) Geometric equation:
1 2
eij ¼ ðuij þ uji Þ ði; j ¼ 1; 2; 3Þ
ð13Þ
(3) THM constitutive equation: Based on the assumption of linear thermoelasticity, the constitutive equation was established. That is to say, the total strain of gas-containing coal was composed of thermal strain, compressive strain of coal caused by gas pressure, the strain caused by swell that was induced gas adsorption, and the strain induced by exterior stress. According to the relationship of thermoelasticity and stress–strain, constitutive equation (Eq. (14)) for fluid–solid-heat coupling was deduced by a series of derivation and transformation, in which effective stress was represented by strain.
r0ij ¼ kedij þ 2Geij hT DTdij hPY Dpdij hPX aT lnð1 þ bpÞdij
ð14Þ
where hT, hPY, hPX are the coefficient of thermal stress, the stress coefficient caused by gas pressure and the stress coefficient of gas adsorption, respectively. And:
8 Et 2Gt k ¼ ð1þtÞð12 > tÞ ¼ 12t > > > > > > E > G ¼ 2ð1þ > tÞ > > > < ð3kþ2GÞb hT ¼ 3 > > > > > ð3k2GÞK Y > hPY ¼ > > 3 > > > > : qRK Y Þ hPX ¼ ð3kþ2GÞð2 9V m
ð15Þ
(4) Stress equation: Substituting Eqs. (13) and (14) into Eq. (12) could obtain THM coupling stress field equations of gas-filled coal:
G uj;ji hT ðDTÞ;i hPY ðDpÞ;i hPX aT½lnð1 þ bpÞ;i þ ap;i 1 2t þ Fi ¼ 0 ð16Þ
Gui;jj þ
Eq. (16) contains coupling terms ap,i that reflects the influence of seepage field and temperature field synchronously, hT(DT),i and hPXaT[ln (1 + bp)],i that reflect the influence of temperature field.
Y. Tao et al. / International Journal of Mining Science and Technology 22 (2012) 879–883
3.2. Coupling seepage equation of coal containing gas (1) Continuity equation: According to basic assumption (6) and the law of mass conservation, continuity equation of gas flow in the coal seam considering source sink term was obtained as follows:
@Q þ r ðqg qÞ ¼ I @t
4. Definite conditions
" # @e 2ð1 uÞ 2abcpn 2abcpn @p þ 2 uþ r pþ þ @t ks @t ð1 þ bpÞ2 1 þ bp k rp2
2ap
l
ð18Þ
where permeability k and porosity u are the function of strain, temperature and gas pressure; adsorption constants a, b the function of temperature T. Therefore, the seepage equation consisted of the coupling term simultaneously reflecting stress field and temperature field of coal containing gas. 3.3. Coupling temperature equation of gas-containing coal Both field observation and experimental studies show that thermal effect is present in the progress of gas adsorption–desorption and seepage in the coal seam, as well as deformation progress of the induced by external force. So coupling temperature equation of gas-containing coal should also be coupled with stress field and seepage field. By comprehensive analysis of the energy conversion and balance within gas-containing coal mass, the total work done by stress on per unit volume of gas-containing coal in terms of the first law of thermodynamics and the Gauss formula.
dW ¼ rij deij
ð19Þ
The specific entropy was introduced by the second law of thermodynamic, namely s:
ds ¼
dQ d T
ð20Þ
Another thermodynamic state function was also introduced, namely Helmholz free energy F:
F ¼ U Ts
deformation work partly. In addition, it is the coupling term simultaneously reflecting seepage field, temperature field and stress field. Therefore, solution of the temperature equation could be achieved unless simultaneous equation with seepage equation and the stress equation.
ð17Þ
(2) Seepage equation: On the basis of the assumptions, Eqs. (6), (8), and (9) were substituted into Eq. (17). The coupling seepage equation of gascontaining coal (Eq. (18)) was obtained, which included source sink term I.
¼I
881
4.1. Definite conditions of stress field for THM model (1) Boundary conditions: The boundary condition of stress field generally includes three types. The first type was displacement boundary condition, and the boundary displacement of coal was already known to us. The second type was stress boundary condition, and the surface force of coal boundary was known. The component surface forces along x, y, z three directions were Fx, Fy, Fz, respectively. The angles of boundary normal directions to the three directions in orthogonal coordinates were represented by n, w and -; and direction cosines were represented by l, m and n (l = cos n, m = cos w, n = cos -). Therefore, the stress boundary condition was as follows:
8 r l þ syx m þ szx n ¼ F x > < x ry l þ szy m þ sxy n ¼ F y > : rz l þ sxz m þ syz n ¼ F z
ð23Þ
The third type was the mixed boundary condition, of which the displacement of some boundary and the boundary stress of the coal mass skeleton were known. (2) Initial condition: The initial condition of stress field in methane-containing coal was the initial value of displacement or particle velocity when t is zero. 4.2. Definite conditions of seepage field for THM model The boundary condition of seepage field included three types, too. The first one was the constant pressure on the boundary; the second type was the constant flow on the boundary; the third one was the mixed boundary condition, of which flow rate on partial boundary was given and pressure on the other part of boundary was also given; the forth one was that flow rates at the interface were equal if there was inter boundary, that was:
k1
@p1 @p2 ¼ k 2 @n1 L1 @n2 L2
ð24Þ
ð21Þ
According to the second law of thermodynamics, coupling temperature equation of gas-containing coal was determined by simultaneous equation of Eqs. (19)–(21) and THM constitutive equation of gas-containing coal.
@T @e þ T 0 hT @t @t @a ap @b @e þ hPX T 0 T 0 lnð1 þ bpÞ þ T 0 þ alnð1 þ bpÞ @T 1 þ bp @T @t
gr2 T þ qQ ¼ qC V
ð22Þ where qQ is the additional item caused by gas desorption. That is to say, temperature shows an decrease, which is induced by gas desorption. And this term is also the coupling term simultaneously reflecting seepage field and temperature field. The term qC V @T þ T 0 hT @e is the additional term caused by the deformation @t @t work due to thermoelastic and gas. That is to say, the heat given to coal body led to not only increase in temperature, but also
4.3. Definite conditions of temperature field for THM model (1) Boundary conditions: Generally, there are three types of boundary condition for temperature field. The first one was that temporal and spatial variation of temperature at coal mass the boundary was given. The second one was that the heat flux (qn)C at the outward normal of the boundary had the certain functional relation with space and time. The third one was that the temperature Tf of fluid medium and the heat transfer coefficient h of boundary surface were certain, if the convective heat transfer existed between fluid and object surface. (2) Initial condition: If the temperature field was unstable, the initial condition referred to the temperature T when time t was zero and the initial temperature value could be a certain value, or a function of space.
882
Y. Tao et al. / International Journal of Mining Science and Technology 22 (2012) 879–883
5. THM coupling model validation
1.2 1.0
@ k1 ðp þ BÞ @p ¼0 @x @x l
experimental solution
0.8 0.6 0.4
numerical solution
0.2 0.0
0
20
40
60
80
100
120
140
t/s
(a) Variation of gas pressure 296
experimental solution
295
numerical solution
T /K
To verify the THM coupling model of methane-containing coal established in this paper, it is necessary to comparatively analyze analytic and the numerical solution. Because of the nonlinear complexity of THM model, it is difficult to calculate the analytical solutions. Consequently, THM coupling model was simplified in order to get the analytical solutions, and then verified the THM coupling model by comparative analysis between the analytic and numerical solutions. The correctness and validity of the THM model and the resolution method based on COMSOL Multiphysics were proved by the example of one-dimensional gas seepage considering the Klingberg effect. Assuming that porous ratio u was constant under the condition of constant temperature. Considering the Klingberg effect, Eq. (18) could be simplified as one-dimension stable flow equation [17]:
p /MPa
5.1. Analytic solutions validation
294
ð25Þ 293
Under the one-dimensional case, when the inlet flow rate Qm of sample and the outlet pressure pL were constant, Eq. (26) could be deduced from Eq. (25):
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Q m lðL xÞ pðxÞ ¼ B þ B2 þ p2L þ 2BpL þ k1
400
800 t/s
1200
1600
2000
(b) Variation of temperature Fig. 2. Correlation curve of gas pressure and temperature variable curve.
ð26Þ
For the one-dimension model, the geometric model was a columniform raw coal, which was 10 m in length and 1 m2 in cross-sectional area. The outlet pressure and the inlet gas flow rate were both constant. The numerical and analytical solutions deduced by the COMSOL Multiphysics are shown in Fig. 1. The calculated results indicate that the numerical solution agrees well with analytical solution. Therefore, the computational method induced in this paper is correct and the model is reliable. 5.2. Validation in physical experiment To assure the correctness of coupling THM model, we calculated the change curve of gas pressure and temperature during aeration and absorption progress before outburst. By numerical analysis of THM model, and then this study compared it with measured curve in the physical experiment of coal and gas outburst. The geometric size and boundary conditions for numerical calculation were in accord with that for physical simulation experiment, which was 570 mm in length and 365 mm in height. The boundary conditions for temperature field in geometric model were all restrained by certain temperature, 293 K; the initial temperature in coal sample was determined to 293 K, too; the initial gas pressure in coal sample was determined to 0.1 MPa. As for boundary conditions for stress field, the upper boundary was applied with normal stress, r1 = 4.0 MPa; on the right boundary, there was only horizontal stress, namely r2 2.4 MPa; and other boundaries were all under analytical solution numerical solution
p (p-1.0e5) /Pa
0
L /m
Fig. 1. Comparison of numerical and analytic solutions of one-dimensional steady gas flow.
displacement constrains. The result of numerical analysis is shown in Fig. 2. It is found that it required 120 s for gas pressure increasing from 0.1 to 1.0 MPa, in comparison to 108 s in the physical experiment, and the pressure did not change obviously within the former 10 s, in comparison to 8 s in the physical experiment. When time of filling gas lasted for 1490 s, the temperature increased by 2.24 °C (from 293.00 to 295.24 K), compare with 2.64 °C (from 293.00 to 295.64 K) in the physical experiment. The reason for difference of temperature change is the frictional heat between coal particles when applying loading, which was not considered in numerical calculation. Although less better than similar degree between numerical and analytical solutions, the simulated result is basically in accord with the result from physical experiment. Therefore, it can be considered that the regular conclusions of numerical simulation by COMSOL Multiphysics based on THM coupling model are credible. 6. Conclusions (1) Based on theories of elastic mechanics, seepage mechanics and heat transfer, a real three-field and two-way coupled mathematical model is established to reveal the connections among seepage, deformation and temperature difference within methane-containing coal. (2) It is considered that the total strain of gas-containing coal is composed of thermal strain, compressive strain of coal caused by gas pressure, the strain due to gas adsorption and the strain induced by exterior stress. Based on this, the modified THM constitutive equation of gas-filled coal is established. (3) The coal mass deformation and gas flow in coal seam is affected by fluid–solid-heat coupling, because of the effect of gas adsorption–desorption on the stress field and the temperature field. (4) The coupling modeling for THM of methane-containing coal is proved to correct, by comparison of numerical and analytical solutions of the simplified model, and by model application under the background of physical simulation experiment of coal and gas outburst.
Y. Tao et al. / International Journal of Mining Science and Technology 22 (2012) 879–883
Acknowledgments Thanks for financially supported in part by the State Key Basic Research Program of China (No. 2011CB201203), the General Project of the National Natural Science Foundation of China (No. 50974141), the Key Project of the National Natural Science Foundation of China (No. 50534080), and the Key Special Subjects National Science and Technology of China (No. 2011ZX05034-004). References [1] Bear J, Corapcioglu MY. A mathematical model for comsolidation in at hermoelastic aquifer due to hot water injection or pumping. Water Resource Res 1981;17:723–36. [2] Lewis RW, Sukirman Y. Finite element modeling of three phase flow in deforming saturated oil reservoirs. Int J Num Anal Methods Geomech 1993;17:577–98. [3] Lewis RW. Finite element modeling of two phase heat and fluid flow in deforming porous media. Trans Porous Media 1989;4:319–34. [4] He Yulong, Yang Lizhong, Yang Jiyi. Governing equations for coupled thermo– hydro-mechanical behaviors in unsaturated rock mass. J Southwest Jiao Tong Univ 2006;41(4):419–23. [5] Gutierrez M, Makurat A. Coupled HTM modelling of cold water injection in fractured hydrocarbon reservoirs. Int J Rock Mech Min Sci Geomech Abstr 1997;34(3/4):429.
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