Numerical assessment of the effect of equilibration time on coal permeability evolution characteristics

Fuel 140 (2015) 81–89

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Numerical assessment of the effect of equilibration time on coal permeability evolution characteristics Qingquan Liu a, Yuanping Cheng a,b,⇑, Wang Haifeng a, Zhou Hongxing a, Wang Liang a, Li Wei a, Liu Hongyong a a b

National Engineering Research Center for Coal Gas Control, China University of Mining & Technology, Xuzhou 221116, China State Key Laboratory of Coal Resources and Safe Mining, China University of Mining & Technology, Xuzhou 221008, China

h i g h l i g h t s  A novel permeability model derived based on the dual poroelastic theory is proposed.  Permeability evolution in a coal sample variation with time is investigated.  Effect of equilibration time on coal permeability evolution is evaluated.

a r t i c l e

i n f o

Article history: Received 8 August 2014 Received in revised form 16 September 2014 Accepted 21 September 2014 Available online 7 October 2014 Keywords: Coal permeability experiment Equilibration time Dual poroelasticity Numerical modeling

a b s t r a c t Although equilibration time has great significance in experiments of sorption-induced strain and coal permeability, most permeability models in which only one gas pressure has been taken into account cannot be used to calculate the permeability evolution during the adsorption phase, even though they have been successful in reservoir simulations. Here, a new mathematical model for coupled gas migration and coal deformation is developed to investigate the dynamics of CH4 adsorption in a coal sample when conducting a coal permeability experiment, and a novel permeability model based on the dual poroelastic theory is formulized to investigate the relationships between equilibration time and coal permeability evolution characteristics during the adsorption phase. A finite element model is applied to investigate the evolution characteristics of two gas pressures and permeability in a coal sample under laboratory conditions. Results illustrate that there is a large difference between the two gas pressures during the adsorption phase; before adsorption equilibrium is achieved, permeability, which is greater than that obtained when adsorption equilibrium is achieved, first increases and then decreases with increasing equilibration time. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Coalbed methane (CBM) is an abundant, low-cost fuel that has significant long-term potential for discovery and development [1]. Worldwide CBM reserves have been estimated at 84–262 trillion m3, and the majority are located in Russia (17–113 trillion m3), Canada (6–76 trillion m3), China (30–35 trillion m3), Australia (8– 14 trillion m3), and THE USA (11 trillion m3) [2,3]. Productivity evaluation and prediction are important steps in the development of CBM reservoirs. There are many factors affecting CBM produc-

⇑ Corresponding author at: National Engineering Research Center for Coal Gas Control, China University of Mining & Technology, Xuzhou 221116, China. Tel.: +86 516 83885948; fax: +86 516 83995097. E-mail address: [email protected] (Y. Cheng). http://dx.doi.org/10.1016/j.fuel.2014.09.099 0016-2361/Ó 2014 Elsevier Ltd. All rights reserved.

tion, among which the permeability of coal is recognized as the most important parameter [4]. The permeability of coal is more complicated than that of conventional gas reservoirs. Coal permeability is highly sensitive to effective stress and sorption-based volume changes [5,6], and the evolution of coal permeability is controlled by the competing influences of effective stress and sorption-based volume changes [7]. Thus, accurate sorption-induced strain data are needed for building a coal permeability model. Obtaining accurate experimental sorption-induced strain data is closely related to equilibration time, and many researchers have studied this issue. Battistutta et al. [8] conducted a series of swelling and sorption experiments using four different gases and found that the equilibration time depends on gas, temperature of the system and sizes of coal sample used in the experiments, and the time to achieve equilibrium for the four gases is increasing in the following order: He, N2, CH4 and CO2.

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Harpalani and Schraufnagel [9] noted that desorption was an extremely slow process and it took a long time for the reading on the strain indicator to stabilize. Seidle and Huitt [10] measured the sorption-induced strain of coal samples and found that it took nearly three months for the coal matrix strain to stabilize during the adsorption phase at each pressure step. Zutshi and Harpalani [11] also found that equilibration times of over 75 days were needed during the gas adsorption phase. As shown in Fig. 1a and b, van Bergen et al. [12] measured the sorption-induced linear strain of coal samples which were exposed to CO2 and CH4; the gas pressures were constant during the measurements, but the linear strain did not stabilize after 35 h for CH4 and 25 h for CO2. Robertson [13] measured and modeled sorption-induced strain and permeability changes in coal and used extended Langmuir theory to model strain to avoid the long time needed for experiments. As shown in Fig. 1c and d, St, the equilibrated strain (extrapolated to infinite time), was obtained by fitting experimental data measured within the first 24 h of equilibration. As shown in Fig. 1e and f, Pan et al. [5] and Majewska et al. [14] also measured the sorption-induced strain of coal samples exposed to CO2 or CH4 and measured the relationship between the sorption-induced strain and equilibration time. From the above-mentioned experimental studies, it can be seen that sorption-induced strain requires a long time to stabilize. Therefore, equilibration time must also have significance in coal permeability experiments. Siriwardane et al. [15] found that the permeability of coal samples decreased significantly with equilibration time when CO2 was used as the fluid medium, and the equilibration time can range from 1.5 days to a week or typically approximately two days under laboratory conditions. However, the importance of equilibration time has not drawn considerable attention in coal permeability experiments, mainly because there is no suitable mathematical model to obtain equilibration time, which is usually estimated based on experience. Qu et al. [16] introduced a concept of matrix swelling transition from local to global under stress conditions, and pointed out that the local equilibrium condition has not been achieved under common laboratory conditions. Coal samples are typically dual-porosity systems that consist of coal matrix surrounded by intersecting fractures. In such a system, two pressures are present at every point: one in the fractures, pf, and the other in the coal matrix, pm. In a coal permeability experiment, sorption-induced strain will stabilize during the adsorption phase only when the two pressures are both equal to the target pressure. The difference between pf and pm results in the long time for the stabilization of the sorption-induced strain. However, only one gas pressure has been used in most coal permeability models. These models cannot be used to calculate the permeability evolution during the adsorption phase, even though they have been successful in reservoir simulations. The primary objectives of this study are to formulize a fullycoupled gas migration and solid deformation model and to use the model to investigate the relationships between equilibration time and coal permeability evolution characteristics during the adsorption phase of coal permeability experiments. The principal goal of the study is to formulize a coal permeability model based on the dual-poroelastic theory which can be used to calculate the permeability evolution during the adsorption phase.

at one point: one in the fractures, pf, MPa; and the other in coal matrix, pm, MPa [18]. The gas pressure pm is defined as the ‘‘virtual’’ pressure that would be in equilibrium with the current concentration of adsorbate in the matrix blocks [19]. The permeability of a coal specimen is a function of its fracture system, and the coal specimen can be treated as a dual-porosity, single-permeability scheme [20,21]. The uniaxial deformation of depleting reservoir was first proposed by Geertsma [22] who hypothesized that, with continued production, an oil reservoir having a high lateral dimension compared to vertical dimension deforms mainly in vertical direction. Many permeability models are developed based on the assumptions that the coal seam is under uniaxial strain condition which is similar to the in-situ condition. Mitra et al. [23] is the first reported experimental study where flow measurement were made while coal was held under uniaxial strain condition. The coal sample setup under uniaxial strain condition for a typical coal permeability experiment is illustrated in Fig. 2. Methane is charged from both the top and the bottom of the coal sample during the adsorption phase. Two phases of methane gas migration in a coal sample are involved in the study: the first phase is the Darcy flow of free methane gas through the fractures in the coal sample, and the second phase involves the Fickian diffusion of free gas from the fractures into the adsorbed phase within the coal matrix blocks [24]. During the adsorption phase, two distinct phenomena are associated with two gradually increasing gas pressure [4,25]. The first phenomenon is a decrease in the effective horizontal stress under uniaxial strain conditions; the second is methane adsorption into the coal matrix, resulting in coal matrix swelling and thus a rise in the horizontal stress. 2.2. Governing equations 2.2.1. Coal deformation The presence of methane in coal modifies the mechanical response of coal. As two distinct gas pressures are present in a coal specimen which is a dual-porosity media, the effective stress law for dual-porosity media rather than that for single-porosity media is more suitable for obtaining the effective stress [26]:

reij ¼ rij  ðbf pf þ bm pm Þdij

ð1Þ

where reij is the effective stress. rij is the total stress (positive in compression). dij is the Kronecker delta tensor. bf and bm are effective stress coefficients for the fractures and the matrix, respectively, and can be expressed as [26]

bf ¼ 1 

bm ¼

K Km

K K  Km Ks

ð2Þ

ð3Þ

2. Physical model and governing equations

where K is the bulk modulus of the coal, MPa, where K = E/3(1  2t). Km is the bulk modulus of the coal grains, MPa, where Km = Em/ 3(1  2t). Ks is the bulk modulus of the coal skeleton, MPa. Ks usually cannot be directly measured, however, it can be calculated using the equation [27] K s ¼ K m f1  3/m ð1  tÞ=½2ð1  2tÞg. E is the Young’s modulus of the coal, MPa. Em is the Young’s modulus of the coal grains, MPa. t is the Possion’s ratio of the coal. The strain–displacement relationship is defined as:

2.1. Physical model

eij ¼ ðui;j þ uj;i Þ

The common conceptual model applied to coal is that it is a dual-porosity reservoir that consist of coal matrix surrounded by intersecting fractures [17]. This leads to two distinct gas pressures

where eij denotes the component of the total strain tensor. ui denotes the displacement component in the i-direction. The equilibrium equation is defined as

1 2

ð4Þ

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Fig. 1. Sorption-induced strain plotted against equilibration time [5,12–14].

rij;j þ F i ¼ 0

ð5Þ

where Fi denotes the body force component in the i-direction. Based on the dual-poroelastic theory, the constitutive relation for the coal seam can be expressed as [28]:

rij ¼ 2Geij þ

2Gt ev dij  bf pf dij  bm pm dij 1  2t

ð6Þ

where G is the shear modulus of coal, MPa, where G = E/2(1 + t). ev ¼ e11 þ e22 þ e33 is the volumetric strain of the coal matrix.

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Fig. 2. Migration process of methane gas in a coal sample under laboratory conditions.

Combining Eqs. (4)–(6) yields the Navier-type equation for coal seam deformation:

Gui;jj þ

G uj;ji  bf pf ;i  bm pm;i þ F i ¼ 0 1  2t

and

d/f ¼ ð7Þ

1 ðdr  bf dpf  bm dpm Þ  M |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Effect of effectiv e stress



   K d eL pm dpm 1 M dp pm þ PL |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflmffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl}

Effect of sorption induced coal deformation

ð11Þ 2.2.2. Dynamic permeability model for fractures The permeability of a coal sample is a function of its fracture system, and the relationship between fracture porosity and permeability can be defined by the following cubic equation:

k ¼ k0

/f /f 0

!3 ð8Þ

where k is the fracture permeability, m2. k0 is the initial fracture permeability, m2. /f is the fracture porosity, %. /f0 is the initial fracture porosity, %. The change in strain on the coal sample is small when under uniaxial strain conditions; therefore, the relationship between fracture porosity, coal volume strain, grain volume strain and pore volume strain can be determined by the poroelastic theory as follows [29]:

!

dep ¼

1  /f dec deg  /f /f

ð9Þ

where ep is the pore volume strain. ec is the coal volume strain. eg is the grain volume strain. A change in pore volume strain ep leads to a change in fracture porosity which is controlled by the competing influence of effective stress changes and sorption-induced volume changes. The effective stress changes are defined by Eq. (1), and the sorption-induced volume strain can be calculated using a Langmuir-type equation [30]:

es ¼

eL pm pm þ PL

ð10Þ

where es is the sorption-induced volume strain. PL is the Langmuir pressure constant at which the measured volumetric strain is equal to 0.5 eL , Pa. eL is the Langmuir volumetric strain. Eq. (11) can be easily converted to a total differential:

d/f ðpf ; pm Þ ¼

     bf b K d eL pm dpm 1 dpf þ m þ M dpm pm þ PL M M

ð12Þ

Based on the theorems of multivariable differential calculus, we have

/f 1 ¼1þ ½b ðp  pf 0 Þ þ bm ðpm  pm0 Þ M/f 0 f f /f 0    eL K pm pm0 þ 1  /f 0 M PL þ pm PL þ pm0

ð13Þ

The resulting constitutive equation for permeability, derived from Eqs. (8) and (13), takes the form:

8 > > > < 1 k ¼ k0 1 þ ½b ðp  pf 0 Þ þ bm ðpm  pm0 Þ > M/f 0 f f > > |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} : Effect of effectiv e stress



þ



93 > > > =

eL K pm pm0 1  /f 0 M PL þ pm PL þ pm0 > > |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}> ; Effect of sorption induced coal deformation

ð14Þ

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2.2.3. Gas flow The transfer of free methane gas through the fractures in a coal sample is governed by a mass conservation equation:

@ ð/ q Þ ¼ rðqg VÞ þ Q s ð1  /f Þ @t f g

ð15Þ

3

where qg is the gas density, kg/m . V is the gas velocity in fractures, m/s. Qs is the gas exchange rate per volume of matrix blocks, kg/ (m3 s). By applying the ideal gas law, the relation between gas density and pressure can be described as:

M qg ¼ c pf RT

ð16Þ

where Mc is the molar mass of methane, kg/mol. R is the universal gas constant, J/(mol K). T is the temperature, K. Volumetric flow in the fractures is governed by Darcy’s law

V¼

k

l

rpf

ð17Þ

where l is the methane viscosity, Pa s. 2.2.4. Gas diffusion By applying a mass balance equation of methane in coal matrix, we have:

@m ¼ Q s @t

ð18Þ

where t is time, s. m is the quantity of adsorbed gas and free gas per volume of coal matrix blocks, kg/m3, which can be calculated using the Langmuir equation and the ideal gas law:

m¼

V L pm M c M q þ /m c pm pm þ P L V M c RT

ð19Þ

where VL denotes the maximum adsorption capacity of the coal, m3/kg. PL denotes the Langmuir pressure constant, Pa. VM is the molar volume of methane under standard conditions, m3/mol. qc is the gas density at standard conditions, kg/m3. /m is the coal matrix porosity, %. Diffusion between the coal matrix and fractures is driven by the concentration gradient, and the gas exchange rate can be expressed as [31,32]

  Q s ¼ D rc c m  c f

ð20Þ 2

where D is the gas diffusion coefficient, m /s. rc is the coal matrix block shape factor, m2. cm is the concentration of gas in the matrix blocks, kg/m3. cf is the concentration of gas in fractures, kg/m3. According to the ideal gas law:

cm ¼

Mc p RT m

ð21Þ

and

Mc cf ¼ p RT f

ð22Þ

In general, obtaining the diffusion parameters D and rc is difficult. Thus, current models incorporate the diffusion parameters using the parameter ‘‘sorption time’’, which is commonly used to approximate the diffusivity of the coal matrix blocks [32–35]:

s¼

1

rc D

ð23Þ

where s is the sorption time of coal matrix and it is numerically equivalent to the time during which 63.2% of the coal gas desorbed, s. Substituting Eqs. (21)–(23) into Eq. (20), the governing equation for the gas exchange rate can be expressed as:

Qs ¼

 1 Mc  p p s RT m f

2.2.5. Coupled equations Substituting Eqs. (14), (16), (17), and (24) into the balanced Eq. (15), the governing equation for the transfer of free methane gas through the fractures in a coal sample can be expressed as:

ð24Þ

/f ðpf ; pm Þ

  @pf @/f ðpf ; pm Þ kðpf ; pm Þ þ pf ¼r pf rpf @t @t l

  1 þ 1  /f ðpf ; pm Þ pm  pf

s

ð25Þ The partial derivative of /f with respect to time, which is needed in Eq. (25), can be obtained from Eq. (13):

  @/f ðpf ; pm Þ 1 @pf @p ¼ bf þ bm m M @t @t @t   eL PL K @pm þ  1 @t ðPL þ pm Þ2 M

ð26Þ

Substituting Eqs. (18) and (24) into the balanced Eq. (18), the governing equation for the diffusion between the coal matrix and fractures in a coal sample can be expressed as:

V M ðpm  pf Þðpm þ PL Þ2 @pm ¼ @t sV L RTPL qc þ s/m V M ðpm þ PL Þ2

ð27Þ

Eqs. (25) and (27) describe the two-phase state of methane gas migration in the coal sample and the mass transfer between them. Eq. (7) describes the influence of pressures pm and pf on coal deformation and the influences of coal deformation on gas migration has been achieved by the permeability model which is deduced based on the dual poroelastic theory. Therefore, Eqs. (7), (14), (25)-(27) define a mathematical model of coupled gas flow and coal deformation with gas diffusion effects. 2.2.6. Boundary and initial conditions For completeness, the standard boundary and initial conditions are defined bellow. The displacement and stress conditions on the boundaries for the Navier equation, Eq. (7), are given as:

ui ¼ ui ðtÞ on @ X

ð28Þ

rij ! n ¼ f i ðtÞ on @ X

ð29Þ

where ui(t) and fi(t) are the known displacement and stress on the ! boundary, respectively. n is the outward normal unit vector on the boundary. For the gas flow process, the Dirichlet and Neumann boundary conditions are defined as:

pm ¼ pf ¼ ps on @ X

ð30Þ

rpm  ! n ¼ 0 on @ X

ð31Þ

rpf  ! n ¼ 0 on @ X

ð32Þ

where ps is the known constant pressure on the boundaries, Pa. Eqs. (31) and (32) are also known as a no-flow boundary condition. The initial conditions for displacement, stress and gas flow in the domain are defined as:

ui ð0Þ ¼ u0 in X

ð33Þ

rij ð0Þ ¼ ro in X

ð34Þ

pm ð0Þ ¼ pf ð0Þ ¼ p0 in X

ð35Þ

where u0, r0 and p0 are the initial values of displacement, stress and gas pressure in the domain, respectively.

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3. Numerical simulation and discussions Eqs. (7), (14), (25)-(27) define the mathematical model that can be used to implement numerical simulations of the effect of equilibration time on coal permeability evolution characteristics. All of these governing equations are implemented and solved using COMSOL Multiphysics finite element code, which provides a complete and integrated modeling environment for creating, analyzing, and visualizing multiphysics models [36]. These numerical simulations use three modules of COMSOL Multiphysics: the diffusion of methane is implemented through the ‘‘PDE module’’; the Darcy flow of methane is implemented through ‘‘Darcy’s Law module’’ and the stress and strain of coal is calculated using the ‘‘Solid Mechanics module’’. The diffusion of methane into fractures is defined as a ‘‘Mass Source’’ and its value is calculated using Eq. (24) in the ‘‘Darcy’s Law module’’. The effect of the free methane on the coal sample is implemented by the body load caused by the gas pressure gradient. 3.1. Model description and input parameters To investigate the effect of equilibration time on coal permeability evolution characteristics, a simulation model was constructed. The geometry and boundary conditions of the simulation model

are shown in Fig. 3. The 2D analyzed zone measures 50 mm across by 100 mm tall as the geometric model is a simplification of a cylindrical coal sample which is 100 mm in height and 50 mm in diameter. The 2D model simplifies the stress condition, and therefore, the simulation model is focused on the analysis of the effect of equilibration time on coal permeability evolution characteristics. Suitable boundary conditions were applied to the simulation model based on the analysis described in Section 2.2.6. To achieve the uniaxial strain condition for the solid deformation model, the left and right sides and base are rollered, and constant stress is applied to the top. For the gas flow model, a constant gas pressure is applied to both the top and the base, and no flow conditions are applied to both the right and left sides. For the initial condition, gas pressures pm and pf are both equal to 0.1 MPa. The target pressure is 2.0 MPa. The input parameters used in these simulations are listed in Table 1; most were chosen from an appropriate range obtained from recently published studies [7,13,33]. For example, the permeabilities of U.S. coal range between 0.1 mD and 250 mD, whereas Chinese coal ranges from 0.002 mD to 450 mD; the sorption time of Chinese coal ranges from several hours to many days (up to 600 days) [37,38]. 3.2. Variation of gas pressures with equilibration time It is clear that the coal permeability changes constantly with changes in effective stress and sorption-induced strain during the adsorption phase in a coal permeability experiment. The changes in the effective stress and sorption-induced strain are closely related to the two gas pressures pf and pm. The variation of the two gas pressures with equilibration time will be discussed first. pf and pm obtained at different equilibration times of 1 h, 6 h, 12 h, 24 h and 48 h are illustrated in Fig. 4 using the same color range. The minimum and maximum values of the color range are 0.1 MPa and 2.0 MPa, respectively. The maximum values of the two gas pressures pf and pm are located along the top and bottom boundaries; the minimum values of the two gas pressures pf and pm are located in the central domain. Both pf and pm increase with the increasing equilibration time, and they simultaneously distribute more equally in the whole model. However, there are still many differences between them: pf is far greater than pm at equilibration times of 1 h and 6 h; pm obtained at an equilibration time of 12 h is less than pf obtained at the equilibration time of 1 h. The difference between pf and pm decreases with the increasing equilibration time. It is hard to distinguish the difference between pf and pm at an equilibration time of 48 h. The two gas pressures located along detection line AB are illustrated in Fig. 5 to enable a comparison of their values. Coordinates

Table 1 Property parameters used in the numerical simulation.

Fig. 3. Geometry and boundary conditions for the coupled gas migration and coal deformation in a coal sample (subscript ‘‘⁄’’ denotes ‘‘m’’ and ‘‘f’’ simultaneously).

Parameter

Value

Young’s modulus of coal, E Young’s modulus of coal grains, Em Bulk modulus of coal skeleton, Ks Possion’s ratio of coal, t Initial porosity of coal matrix, /m0 Initial porosity of fractures, /f0 Initial gas permeability, k0 Sorption time, s Langmuir pressure constant, PL Langmuir volume constant, VL Langmuir volumetric strain constant, eL Density of coal, qc Stress, Fy Molar mass of methane, Mc Temperature, T

2713 MPa 8139 MPa 10335 MPa 0.339 0.06 0.012 5  104 mD 8h 1 MPa 0.02 m3/kg 0.012 1250 kg/m3 6 MPa 0.016 kg/mol 293 K

Q. Liu et al. / Fuel 140 (2015) 81–89

87

3.3. Permeability variation with equilibration time

Fig. 4. Gas pressure distribution in a coal sample at different equilibration times.

Fig. 5. Gas pressure distribution along the detection line AB [A (2.5 mm, 0 mm) and B (2.5 mm, 10 mm)].

of the two endpoints are A (2.5 mm, 0 mm) and B (2.5 mm, 10 mm). The minimum pressures in fractures are 1.28 MPa, 1.36 MPa, 1.60 MPa, 1.93 MPa and 2.00 MPa at the equilibration times of 1 h, 6 h, 12 h, 24 h and 48 h, respectively; at the same times, the minimum pressures in coal matrix are 0.16 MPa, 0.54 MPa, 1.08 MPa, 1.83 MPa and 2.00 MPa, respectively. It is clear that at the initial stage of the adsorption phase, pf is at a high level, while pm is at a fairly low level. It can be concluded that the pressure pf can quickly reach the target, and the equilibration time is determined by the pressure pm, which is mainly influenced by the diffusion characteristic of the coal sample. The variable v is defined to measure the ratio between pf and pm:

vðtÞ ¼

pf ðtÞ pm ðtÞ

As two gas pressures pf and pm have been included in the permeability model (Eq. (14)) which is established based on dual poroelastic theory, the model can be used to investigate permeability evolution during the adsorption phase. Relative permeability along detection line AB at different equilibration times is illustrated in Fig. 6. It is well known that the evolution of permeability is controlled by the competing influences of effective stress and sorption-induced volume change. Relative permeabilities are greater than 1.0 at equilibration times of 1/12 h and 1/2 h and render a concave form. At the same time, it can be concluded that the effect of effective stress is greater than that of sorption-induced volume change and that the effect of effective stress at the top and bottom of the coal sample is greater than that at the center of the coal sample as the pressure pf is greater at the top and bottom of the coal sample. Relative permeability is approximately equal to 1.0 at an equilibration time of 1 h and renders a slight convex form, indicating that the effect of sorption-induced volume change is greater than that of effective stress at the top and bottom of the coal sample and at the center of the coal sample, and vice versa. On the whole, the two effects are with small dominant in different parts of the coal sample, and permeability is approximately equal to initial permeability. Relative permeabilities are less than 1.0 at equilibration time of 6 h, 12 h and 24 h and render a convex form, indicating that the effect of sorption-induced volume change is greater than that of effective stress and that the effect of sorption-induced volume change at the top and bottom of the coal sample is greater than that at the center of the coal sample, as the pressure pm is greater at the top and bottom of the coal sample. The permeabilities at these time points are less than the initial permeability and are less at the top and bottom of the coal sample than at the center of the coal sample. The distribution of permeability is uniform and equal inside the coal sample at an equilibration time of 48 h, as adsorption equilibrium has been achieved. To consider the overall relationships between equilibration time and coal permeability evolution characteristics, permeability evolution characteristics of three different points inside the coal sample during the adsorption phase in a coal permeability experiment have also been studied. As shown in Fig. 7, the coordinates of the three detection points are A (2.5 mm, 9.0 mm), B (2.5 mm, 8.0 mm) and C (2.5 mm, 5.0 mm). The patterns of permeability variation with time at the three detection points are consistent: per-

ð36Þ

The minimum values of the two pressures have been used in the calculation, and v equals to 800%, 252%, 148% and 105% at the equilibration times of 1 h, 6 h, 12 h and 24 h, respectively. Large differences exist during most of the adsorption time. However, only one gas pressure has been taken into account in most coal permeability models which are formulized based on the assumption that little difference between pf and pm exists. Therefore, these models cannot be used to calculate the permeability evolution during the adsorption phase even though they have been successful in reservoir simulations.

Fig. 6. Relative permeability along the detection line AB [A (2.5 mm, 0 mm) and B (2.5 mm, 10 mm)].

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Q. Liu et al. / Fuel 140 (2015) 81–89

(2) Before adsorption equilibrium is achieved during the adsorption phase, the coal permeability is greater than that obtained when the adsorption equilibrium is achieved, as both gas pressures pf and pm are less than the target value. (3) The evolution of permeability is controlled by the competing influence of effective stress and sorption-induced volume change, thus permeability first increases and then decreases with the increasing equilibration time before adsorption equilibrium is achieved during the adsorption phase. It will take the most time for the permeability of the center part of a coal sample to stabilize, and the typical time can be treated as the suitable equilibration time.

Acknowledgements

Fig. 7. Gas pressure distribution of three detection points inside the coal sample [A (2.5 mm, 9.0 mm), B (2.5 mm, 8.0 mm) and C (2.5 mm, 5.0 mm)].

meability first increases and then decreases with an increase in equilibration time during the adsorption phase before adsorption equilibrium is achieved. Both the effects of effective stress and sorption-induced volume change increase with an increase in the two pressures before reaching adsorption equilibrium. The effect of effective stress is greater than that of sorption-induced volume change when the permeability is greater than the initial value; however, the effect of effective stress is less than that of sorption-induced volume change when the permeability is less than the initial value. There are also some differences in the permeability evolution characteristics of the three detection points. The order of maximum permeabilities at the three detection points is k(A) > k(B) > k(C). However, the order of the permeabilities at the three detection points is k(C) > k(B) > k(A) during most of the equilibration time. The permeability of point C is the last to stabilize. The differences in the permeability evolution characteristics of the three detection points are induced by the two phases of methane gas migration in the coal sample. 4. Conclusions In this study, a new mathematical model for coupled gas migration and coal deformation is developed to investigate the dynamics of CH4 adsorption in a coal sample when conducting a coal permeability experiment. The coal sample has been treated as a dualporosity and single permeability continuum, and at every point, two pressures are present: one in the fractures, pf, and the other in the coal matrix, pm. Due to the unique features of the coal sample, the effective stress for a dual-porosity medium and two phases of methane gas migration in a coal sample have been included in the study. A novel permeability model based on the dual poroelastic theory is formulized to investigate the relationships between equilibration time and coal permeability evolution characteristics during the adsorption phase. Major findings are summarized as follows: (1) There is a large difference between the two gas pressures pf and pm during the adsorption phase in coal permeability experiments. Therefore, permeability models in which only one gas pressure has been taken into account cannot be used to calculate the permeability evolution during the adsorption phase even though they have been successful in reservoir simulations.

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