- Email: [email protected]
Estimation and modeling of pressure-dependent gas diffusion coefficient for coal: A fractal theory-based approach
Fuel 253 (2019) 588–606
Contents lists available at ScienceDirect
Fuel journal homepage: www.elsevier.com/locate/fuel
Full Length Article
Estimation and modeling of pressure-dependent gas diffusion coefficient for coal: A fractal theory-based approach
T
Yun Yang, Shimin Liu
⁎
Department of Energy and Mineral Engineering, G3 Center and EMS Energy Institute, The Pennsylvania State University, University Park, PA 16802, USA
ARTICLE INFO
ABSTRACT
Keywords: Diffusion coefficient Knudsen and bulk diffusion Fractal theory Unipore model
Diffusion coefficient is one of the key parameters determining the coalbed methane (CBM) reservoir economic viability for exploitation. Diffusion coefficient of coal matrix controls the long-term late production performance for CBM wells as it determines the gas transport effectiveness from matrix to fracture/cleat system. Pore structure directly relates to the gas adsorption and diffusion behaviors, where micropore provides the most abundant adsorption sites and meso- and macro-pore serve as gas diffusive pathway for gas transport. Gas diffusion in coal matrix is usually affected by both Knudsen diffusion and bulk diffusion. A theoretical porestructure-based model was proposed to estimate the pressure-dependent diffusion coefficient for fractal porous coals. The proposed model dynamically integrates Knudsen and bulk diffusion influxes to define the overall gas transport process. Uniquely, the tortuosity factor derived from the fractal pore model allowed to quantitatively take the pore morphological complexity to define the diffusion for different coals. Both experimental and modeled results suggested that Knudsen diffusion dominated the gas influx at low pressure range (< 2.5 MPa) and bulk diffusion dominated at high pressure range (> 6 MPa). For intermediate pressure ranges (2.5–6 MPa), combined diffusion should be considered as a weighted sum of Knudsen and bulk diffusion, and the weighing factors directly depended on the Knudsen number. The proposed model was validated against experimental data, where the developed automated computer program based on the Unipore model can automatically and time-effectively estimate the diffusion coefficients with regressing to the pressuretime experimental data. The theoretical determined diffusion coefficients well predicted the pressure-dependent variations of the experimental measured diffusion coefficient for different coals. This theoretical model is the first-of-its-kind to link the realistic complex pore structure into diffusion coefficient based on the fractal theory. The experimental results and proposed model can be coupled into the commercially available simulator to predict the long-term CBM well production profiles.
1. Introduction Diffusion is the process that matter (gases, liquids, and solids) tends to migrate and eliminate the spatial difference in composition in such a way to approach a uniform equilibrium state with maximum entropy [1–3]. The study of diffusion in nanoporous solids came to prominence as such materials have sufficient surface area required for high capacity and activity with extensive application in the petroleum and chemical process industries [4]. For transport through the pores with size comparable to diffusing gas molecules, diffusion effects may even dominate the overall transport rate [5]. A comprehensive understanding of the complex diffusional behavior lies the foundation for the technological development of porous materials in adsorption and catalytic processes [6]. As a natural polymer-like porous material, coal behaves like man-
⁎
made nanoporous materials for its exceptional sorption capacity contributed by nano- to micro-scale pores [7–9]. Dual porosity model proposed by Warren and Root [10] well represents the broad size distribution of coal pores, where macro- (> 50 nm) and mesopores (2–50 nm) most likely provide transport pathway, and micropores (< 2 nm) provide the greatest internal surface area and gas storage capacity [8,11–13]. The International Union of Pure and Applied Chemistry (IUPAC) [14] classification of pores is closely related to the different types of forces controlling the overall adsorption behavior in the different sized pore spaces. Surface force dominates the adsorption mechanism in micropores, and even at the center of the pore, the adsorbed molecules cannot break from the force field of the pore surfaces. For larger pores, capillary force becomes important [4]. Different diffusion mechanisms occur in different sized pores governing the overall
Corresponding author. E-mail address: [email protected] (S. Liu).
https://doi.org/10.1016/j.fuel.2019.05.009 Received 22 March 2019; Received in revised form 27 April 2019; Accepted 2 May 2019 Available online 16 May 2019 0016-2361/ © 2019 Elsevier Ltd. All rights reserved.
Fuel 253 (2019) 588–606
Y. Yang and S. Liu
gas mass influx through coal matrix [8,15–17]. Gas transport within coal can occur via diffusion through either pore volumes and/or along pore surface. Surface diffusion describes the movement of adsorbed molecules along the pore surface and is an activated process stimulated by diffusional activation energy [18]. At temperatures significantly higher than the normal boiling point of sorbate, diffusion mainly happens in pore volume, where the diffusional activation energy is negligible compared with the heat of adsorption forming localized layers of adsorbed molecules [4,19–21]. As a result, it is reasonable to neglect surface diffusion flux for CBM production until extremely low pressure is reached [22]. In this study, we limited our discussion to diffusion occurring in the center region of the pore. In other words, the pore volume diffusions, both Knudsen and bulk diffusions, will be considered and discussed in this work. Two forms of diffusion modes are generally considered in diffusion in pore volume, which are bulk and Knudsen diffusions [23–25]. The relative importance of the two diffusion modes depends on Knudsen number (Kn), which is the ratio of the mean free path (λ) to pore diameter (d ) for porous rocks [26,27]. Two extreme scenarios are given in the discussion of the prevalence of the two diffusion mechanisms [4,20]. For nanopores with d , the frequency of molecule-wall collisions far exceeds the intermolecular collisions resulting in the dominance of Knudsen diffusion. In the reverse case (i.e. d ), the contribution from molecule-wall collisions fades relative to the intermolecular collisions and the diffusivity approaches the molecular diffusivity. As a rule of thumb, molecular diffusion prevails when the pore diameter is greater than ten times the mean free path; Knudsen diffusion may be assumed when the mean free path is greater than ten times the pore diameter [28,29]. In the intermediate regime, both the Knudsen and molecular diffusivities contribute to the effective diffusivity. Most real cases of diffusion in CBM are intermediate between these two limiting cases [30]. The mean free path of gas molecules is a function of pressure [31], and as a result, a transition of flow regime from Knudsen diffusion to molecular diffusion will occur as pressure evolves. Diffusion coefficient (D ) governs the rate of diffusion, and in CBM, it can be determined from desorption time [32,33]. A significant amount work [17,22,34–48] has reported the diffusion coefficient (D ) of methane in coal at different pressures as summarized in Table 1. The measured diffusion coefficient of methane ranges from 10 11 to 10 15 m2 /s (Table 1). In this study, we are particularly interested in the influence of pressure as it determines the mean free path and the dominant diffusion regime.
Due to the complex pore morphology of coal, gas transport in coal is closely related to the coal pore structure [49]. To our best knowledge, limited efforts have been devoted to study the quantitative inter-relationship between pore structure and gas transport in coal. Yao et al. [50] observed a strong negative correlation between the permeability and heterogeneity, that is quantitatively obtained from the fractal dimension for high-rank coals whereas a slightly negative relationship was found for low-rank coals. However, the work does not provide detailed quantitative analyses to define the fundamental mechanism for the experimental observations. A study conducted by Li et al. [51] found that coals with higher fractal dimensions have smaller gas permeability because of complex pore shape for tectonically deformed coals. During a tectonic event such as deformation, open pores or semiopen pores may develop into ink-bottle-shaped pores or narrow slit pores. These pore morphological modifications result in a loss of pore inter-connectivity and a more heterogenous pore structure (i.e. high fractal dimension). In these previous studies [50,51], permeability was used to characterize gas flow deliverability of coalbed. In terms of coal matrix gas transmissivity, diffusivity rather than permeability should be examined to delineate gas transport in micro-scale pore structure [5]. As shown in Table 1, numerous experimental works have investigated on gas diffusion characteristics of coal. Specifically, moisture content [52], coal types [53,54], coal rank [55], sample size [56,57] were the key parameters that influence the gas diffusion behavior through micropore structures. But all these studies required the measurement of sorption rate to estimate diffusion coefficients, where they ignored the pore-structure determined diffusion mechanism. As a result, quantitative linkages between coal properties and gas diffusion coefficient are still lack in the literature, which is an important aspect in predicting the gas transport behavior in CBM reservoirs. Pore structural characteristics such as pore volume and its distribution directly depends on coal composition [58]. Therefore, this current study used pore structural parameters to define coal diffusion behavior and uncovered the relationship between the micropore structure and gas diffusion characteristics. In this current study, four coal samples were collected from four different coal mines in China. Low-pressure N2 gas adsorption and desorption data were analyzed through fractal analysis to characterize the pore structure of these four coal samples as many previous literatures [59,60]. High-pressure methane sorption data were used to obtain the gas kinetic parameters by using the unipore model. The pore
Table 1 Sorption kinetic experiments of methane performed in the various literature [17,22,34–48]. HVB and LVB are high and low volatile bituminous coals. Sub is subbituminous coals. Diffusion coefficient is derived from unipore model. List of Works
Year
Location
Rank
Nandi and Walker [43]
1975
U.S. coals
Anthracite to HVB
Smith and Williams [47] Mavor et al. [42]
Marecka and Mianowski [41] Clarkson and Bustin [37] Busch et al. [35]
Cui et al. [38] (further reworked by [22]) Kumar [40] Pone et al. [45]
Charrière et al. [36]
Pillalamarry et al. [22]
Salmachi and Haghighi [46] Bhowmik and Dutta [34] Zhao et al. [48]
Wang and Liu [17] Naveen et al. [44]
1984 1990 1998
1999
2004
2004
2007 2009
2010 2011
2012
2013
2014 2016 2017
Fruitland, San Juan Basin
Sub
Fruitland, San Juan Basin
Sub to LVB
Unknown
Lower Cretaceous Gates Formation, Canada
Silesian Basin of Poland Unknown
Illinois Basin
Western Kentucky Coalfield Lorraine Basin, France
Semi-anthracite Bituminous
HVB
Australian coal seam
Raniganj Coalfield, Jharia Coalfield, Gondwana Basin of India Shanxi, China San Juan Basin and Pittsburgh
Jharia Coalfield, Gondwana Basin of India
589
Pressure, MPa
Range of D , m2/s
0.315
1.14
10
13
10
14
1.9
5.7 0.1
10
13
10
14
10
13
10
10
10
15
10
11
10
13
0.25
2.52 13.6
1.25, 0.62, 0.2, 0.032
0–0.1
0.21
0.9
1.1
3
3.38
10
11
HVB
0.25
0.54
7.82
10
13
10
14
Bituminous
4.76
10 10
13
10
15
10
13
10
13
10
14
10
12
4.61
10
12
10
13
4.56
0.125
0.30
Bituminous
0.25
3.1 0.1
Bituminous
0.143
HVB
Illinois Basin
Avg Particle size, mm
0.48
5.3
11
0.294
0 7 0.014
Sub to HVB
0.1245
0.36
Bituminous
0.225
1.05
10
11
10
12
0.5
0
9
10
13
10
14
0.23
0
7
10
13
HVB
Bituminous HVB
4.678
Fuel 253 (2019) 588–606
Y. Yang and S. Liu
structure-gas diffusion model is proposed to relate pore fractal dimensions to gas diffusion coefficient, where the multiscale gas tsnsport CBM is modeled by superposing different flow regimes. Wu and Chen [61] summarized relevant studies that combines different transport mechanisms based on weighing coefficients [61–71]. However, no validation was provided in their work as it cannot satisfy the ideal conditions made in the theoretical models [69]. This work developed a unified model that utilized the easily accessible pore structural data to estimate gas diffusion coefficient at in-situ condition. The proposed model is an intuitive and mechanism-based model to define the gas diffusion behavior in coal and it can serve as a bridge from pore-scale structure of mass transport for the CBM gas production prediction.
coefficient and permeability of coal [40,72]. At the late stage of a CBM production well (i.e. mature wells), coal permeability might not be the bottle-neck for the overall gas production as commonly believed, and instead, diffusion process dominates overall well production performance since the matrix to cleat influx is limited [17]. Fick’s second law of diffusion for spherically symmetric flow [2] is widely applied to describe gas diffusion process across pore space, where a diffusion coefficient (D ) governs the rate of diffusion. The adopted system of measurement in this work is all in international system of units (SI). Mathematically, the diffusion can be described as:
2. Theoretical background and gas transport modeling
where, r is the radius of the pore, C is the adsorbate concentration, and t is the diffusion time. ‘Unipore’ and ‘bidisperse pore’ models are two widely adapted solutions to the above partial differential equation (PDE) to quantify the diffusive flow [30,43]. As the name suggests, the unipore model assumes a unimodal pore size distribution while the bidisperse model considers a bimodal pore size distribution. The bidisperse model can provide a better modeling result to the entire sorption rate curve than the unipore model for most of the coals [38,47]. Different from unipore model, the bidisperse model separates the macropore diffusivity from the micropore diffusivity, and a ratio of micropore/macropore relative contribution to overall gas mass transfer has been quantified in the model. The bidisperse model is a more robust model than the unipore model because it reflects the heterogeneous nature of the coal pore structure. Nevertheless, the bidisperse model requires to regress
D C C r2 = r2 r r t
2.1. Multiscale and multistage gas transport process for CBM Methane migration within a CBM reservoir starts with gas desorption from the internal pore surfaces of coal followed by the diffusion through the matrix defined by diffusion coefficient under various gas concentration gradients [72,73]. Followed with diffusion process, the gas reaches the naturally occurring fractures (cleats) and the flow regime evolves to pressure gradient driven Darcian flow described by the coal permeability as illustrated in Fig. 1. The rate of viscous Darcian flow through the cleat network depends on the distribution of cleat presented in coalbed. The rate of gas diffusion depends on the pore properties of the coal matrix. Production of gas from a CBM reservoir is intuitively affected by both diffusion
Fig. 1. Illustration of multi-scale and multimechanism gas flows in a CBM reservoir. CBM production data source: DringInfo.inc.
590
(1)
Fuel 253 (2019) 588–606
Y. Yang and S. Liu
multiple modeling parameters (i.e. micropore diffusivity, macropore diffusivity and volume ratio of micropore to macropore) to the experimental data and it may potentially encounter non-uniqueness solution sets [37]. Besides, the bidisperse model is not applicable at high pressure stage for its assumption on the linear isotherm, where the methane adsorption is known to be non-linear [37]. It also assumes an independent process of rapid macropore diffusion and slow micropore diffusion, which cannot be always true [74]. The unipore model is simple and has been successfully used to coal kinetic analysis of CH 4 sorption in several previous studies as summarized in Table 1. In this study, the unipore model was chosen to analyze the sorption data with two reasons: (1) unipore model gives reasonable accuracy over the whole range of coal desorption [75] and (2) unipore model is the model adapted by commercial production simulators [22]. In unipore model [76], constant gas surface concentration is assumed at the external surface, and the corresponding boundary condition can be expressed as:
where,
( ) Mt M
De = D / re2
6 2
n=1
1 exp ( De n2 2t ) n2
Mt M
exp
and
( ) Mt M
2
model
model
(5)
are experimentally measured and analy-
exp
obtained experimentally as described in Section 3.3. Then the
user specifies a search window of the apparent diffusion coefficient as upper (Dhigh ) and lower (Dlow ) limits for the targeted value. Dhigh and Dlow should be a reasonable range of typical values of diffusion coefficient. Based on the reported data as shown in Table 1, the measured diffusion coefficients (D) of various coal samples range from 10 15 to 10 10 m2/s . As suggested by Mavor et al. [42], 60 mesh is the recommended particle size for isotherm measurement. Therefore, the recommend settings for effective diffusivity Dhigh and Dlow are estimated to be between 1e-3 and 1e-8 1/s . The last required input is the number of terms in the infinite summation term (nmax ) of the unipore model (Eq. (3)) to fit the experimental data. A good entry of nmax is 50 to truncate the infinite summation term and the rest terms with large n are negligible. Following the Golden Section Search Algorithm, the diffusion coefficient is determined at the best fit that minimizes the difference between experimental and analytical sorption rate data modeled by unipore model. The flowchart (Fig. 3) shows the algorithm of the automated computer program.
where, C0 is the concentration at the external surface of the pore. In the sorption experiment, this is known to be valid since the coal particles will have a constant pressure at the surface of the particle throughout the experimental procedure. With assumption on uniform pore size distribution, the unipore model is given by
Mt =1 M
( )
exp
Mt M
tically determined sorption fraction. In this computer program, the primary input is the experimental sorption rate data stored in” diffusion.txt” composed of two columns of experimental data. The first column of entry is the sorption time, and the second column is the corresponding sorption fraction
(2)
C (r , t > 0) = C0
Mt M
S=
(3) (4)
where, re is the effective diffusive path, Mt is the sorption fraction, and M De is apparent diffusivity. In order to automatically and time-effectively analyze the sorptiondiffusion data, we develop a matlab-based computer program (Fig. 2) in this study based on a least-squares criterion to regress the experimental gas sorption kinetic data and determine the corresponding diffusion coefficient. An automated computer code was programmed to estimate the apparent diffusivity and the program is listed in the appendix. The apparent diffusivity (D /re2 ) was adjusted using the Golden Section Search algorithm [77] until the global minimum of the objective function was reached. The least-squares function (S ) was chosen to be the objective function and described as:
Fig. 2. User interface of unipore model based effective diffusion coefficient estimation in MATLAB GUI.
Fig. 3. Flowchart of the automated computer program for effective diffusion coefficient estimation in MATLAB GUI. 591
Fuel 253 (2019) 588–606
Y. Yang and S. Liu
2.2. Coal pore fractal analysis The concept of fractal geometry described in Mandelbrot [78] is a powerful tool to characterize the pore structure of coal as traditional Euclidean geometric methods fail to define the micro-structure of coal pores [56]. It provides a quantification approach to define the heterogeneity of pore surface and volume of coal and fractal dimension (Df ) herein is derived and the value is found to be between 2 and 3 for different sized pores [79]. A more heterogenous structure is typically accompanied by a rougher surface and Df approaches 3. Although recent progress makes imaging analysis become a new and convenient method to determine the microstructure of coal [80], they can only provide localized interpret image of a sample and induce great artifacts and uncertainty [81,82]. On the other hand, fractal analysis provides a simple and effective approach to study the tortuous gas diffusive path and it well serves the purpose in the development of a practical and quantitative diffusion model. The fractal dimensions have physical meaning and they can serve as the effective and equivalent pore structure based on the direction experimental data. In this work, Df of the four coal samples are determined by the fractal Frenkel–Halsey–Hill (FHH) model [83–85] using N2 sorption data, which is given by:
ln
V P = Aln ln 0 V0 P
+B
Fig. 4. Fractal pore model. L is the axial length of the pore segment corresponding to the actual travel distance of diffusive path and r is the end-to-end distance (Modified from Coppens [90]).
by
c=
(7)
and
D f = 3A + 3
DK , pm =
(8)
( ) V V0
( ( ) ) corresponding to high- and low-pressure adsorption. The
vs. ln ln
P0 P
slope of the linear parts is used to estimate the fractal dimension of the coal samples. 2.3. Diffusion coefficient in coal
2.3.1. Knudsen diffusion coefficient Knudsen diffusion [4,5] is the dominant diffusion regime when the mean free path is about or even greater than the equivalent effective pore diameter at which the pore wall-molecular collisions outnumber molecular-molecular collisions. For the gas transport in coal, Knudsen diffusion dominates the overall mass transport in small pores or under low pressure. A critical point about Knudsen diffusion is that when a molecule hits and exchanges energy with the pore wall, the velocity of molecule leaving the surface is independent of the velocity of molecule hitting the surface, and the reflecting direction is arbitrary. As a result, Knudsen diffusivity (Dk) is only a function of pore size and mean molecular velocity and can be expressed as [26]:
DK =
1 dc 3
(10)
DK
(11)
where, is the porosity, and is the tortuosity factor. Eq. (11) relates the diffusivity in a porous medium to the diffusivity in a straight cylindrical pore with a diameter equal to the mean pore diameter under comparable physical condition by a simple tortuosity parameter ( ). considers the combined effects of increased diffusive path length, the effect of connectivity and variation of pore diameter. However, the definition of the tortuosity factor is not universally accepted [89]. Instead of using simple bodies from Euclidean geometry, Coppens [90] successfully applied fractal geometry to describe the convoluted pore structure of amorphous porous catalysts and quantified the effect of the fractal surface morphology on diffusion. In this current study, we would use the fractal pore model proposed by Wheatcraft and Tyler [89] to determine the tortuosity of the diffusive path of the pore within coal matrix. A schematic of the fractal pore model is shown in Fig. 4. The key concept behind this model is that the tortuosity is induced by the surface roughness. This model provides a practical and explicit approach to quantify tortuosity by relating it to the surface fractal dimension as developed below. This model depicted in Fig. 4 considers a line having a true length, F, and fractal dimension, Df , which is an intensive property and independent of the size of the measuring yardstick molecules ( ). The expression of F is given by [91],
Two equations (Eqs. (7) and (8)) have been used to estimate Df from the slope ( A ), whereas, Eq. (8) would consistently generate an unreasonably high value of fractal dimension [60]. In this study, we would use Eq. (7) to obtain the fractal dimension of the tested coal samples.
Typically, two linear parts were observed in the log-log plot of ln
8RT / M
where, R is the universal gas constant, T is the ttemperature, and M is the gas molar mass. The Knudsen diffusivity (Dk ) for porous media have been proposed and applied to numerous pervious works [4,88], where the porous media is assumed to consist of open pores (i.e. porosity) of the mean pore diameter and have a degree of interconnection resulting in a tortuous diffusive path longer than an end to end distance (i.e. tortuosity). The Knudsen diffusion coefficient in porous and rough media is derived as:
(6)
where, V / V0 is the relative adsorption at the equilibrium pressure P , V0 is a monolayer adsorption volume, P0 is gas saturation pressure, B is the yintercept in the log-log plot, and A is the power-law exponent used to determine the fractal dimension of the coal surface (D ) [86]. Two distinct formulas were proposed to relate A to Df by [87]:
Df = A + 3
Assuming a Maxwell-Boltzmann distribution of velocity, c is given
F=N
Df
= constant
(12)
where, N is the number of yardsticks required to pave completely the line and varies with . The number of yardsticks (N ) multiplied by the size of a yardstick ( ) is an approximate or measured length (L ( )) of the line and can be expressed as:
(9)
where, d is the pore diameter, and c is the average molecular velocity determined from gas kinetic theory.
592
Fuel 253 (2019) 588–606
Y. Yang and S. Liu
(13)
L( ) = N
coal as only one species of gas is involved. Considering Meyer’s theory [94], the bulk or self-diffusion coefficient (DB ) was derived neglecting the difference in size and weight of the diffusing molecules as [23,95]:
Combining Eqs. (12) and (13), the measured length (L ( )) is related to the fractal dimension as: (14)
1 Df
L( ) = F
DB =
The characteristic length (Ls ) is defined as the length of the line segment holding a constant Df . If = Ls , then N = 1 and the expression of F can be written as: Df
Then Ls is determined as: Df 1 D f
(16)
L ( ) = Ls
At Df = 1, Ls is the end-to-end distance ( r ). For practical application, the axial length of the pore segment ( L ) was approximated by L ( ) [23]. The tortuosity factor ( ), the ratio of the measured length to the endto-end distance, is then determined to be D
=
L Ls f 1 = r Ls
Df
=
1 1 1 = + Dp DK DB
(17)
where, Df is the fractal dimension of a line with a value between 1 and 2. The fractal dimension derived from the Nitrogen sorption data is the surface fractal dimension with a value ranging from 2 to 3 [83]. Taking this into account, the expression of can be updated to:
=
1 1 = wK + (1 Dp DK , pm
2 Df
(18)
Ls
Ls
2 Df
DK =
2 2 dR0.5T 0.5 Ls 3 0.5M 0.5
2 Df
(19)
where, DK is the Knudsen diffusion coefficient in a smooth cylindrical pore [92]. Eq. (19) has the same formula to Eq. (7) in Coppens [90] except porosity was introduced to consider that mass transport only happens in pore space, not through the solid matrix. Ls is the outer cutoff of the fractal scaling regime, i.e. the size of the largest fjords [90]. In this current study, as the structural parameters were obtained from low pressure nitrogen sorption data, Ls was treated as the largest cutoff of the pore size (i.e. maximum pore diameter) in the pore size distribution (PSD). The other parameter, is the molecular diameter of adsorbed molecules. At reservoir condition, methane diffusion in free phase and pore volume dominates the overall mass transport process [4,19–21] and as a result, was estimated to be the mean free path of transport gas molecules as the distance between successive collisions and the effective diffusive diameter of the gas molecules. The mean free path ( ) for real gas given in Chapman et al. [93] is determined as:
=
5µ 8P
RT 2M
wK )
1 DB
(23)
The relative contribution of individual diffusion regime is dependent on the Knudsen number (Kn), which is the ratio of pore diameter to mean free path. It is critical to identify the lower and upper limits of Kn where pure Knudsen and bulk diffusion can be reasonably assumed. Commonly, when Kn is smaller than 0.1, the diffusion regime can be considered as pure bulk diffusion [14,29]. Then wK is written in a piecewise function, f (Kn) and takes the form as:
Eq. (18) provides an intuitive estimation of the tortuosity factor through the correlation with surface fractal dimension. Combing Eqs. (9), (11) and (18), the Knudsen diffusion coefficient of porous media (DK , pm ) is then found as:
DK , pm =
(22)
Eq. (22) stated that the resistance to transport the diffusing species the is a sum of resistance generated by wall collisions and by intermolecular collisions [97–99]. One main implicit assumption behind this reciprocal addictive relationship is that Knudsen diffusion and bulk diffusion acts independently on the overall diffusion process. In reality, the probabilities between gas molecules colliding with each other and colliding with pore wall should be considered [70,100]. Then a dimensionless weighing factor (wK ) was introduced to consider the relative importance of the two diffusion mechanisms as referred to Wu et al. [70], Wang et al. [101].
1 Df
Ls
(21)
2.3.3. Relative contributions of bulk and Knudsen diffusions and related weighing factors When gas transport includes both aforementioned diffusion modes, the relative contribution on the overall gas influx should be quantified. For free gas phase, the combined transport diffusivity (Dp ) including the transfer of momentum between diffusing molecules and between molecules and the pore wall is given as [96]:
(15)
F = Ls
1 c 3
1 (Kn > Kn ), pure Knudsen diffusion wK = f (Kn) = (0, 1) (0.1 < Kn < Kn ), transition flow 0(Kn < 0.1), pure bulk diffusion
(24)
where, Kn is the critical Knudsen number of pure Knudsen diffusion. To estimate the contribution of each mechanism, one should examine the manner in which Dp 1 varies with pressure. From general kinetic theory [94], the bulk diffusion coefficient is inversely proportional to pressure whereas the Knudsen diffusion coefficient is independent of pressure. A diagnostic plot of Dp 1 obtained at a single temperature vs. various pressures (Fig. 5(a)) is useful to identify the diffusion mechanism as suggested by Evans III et al. [100]. A horizontal line corresponds to pure Knudsen flow, a straight line with a positive slope passing the origin represents pure bulk flow, and a straight line with an appreciable intercept depicts a combine mechanism as illustrated in Fig. 5(a). These interpretations are based on Eq. (22) rather than Eq. (23). In fact, the diagnostic plot simplifies the real case as it does not consider the dependence of DK , pm and wK at various pressures. The weighing factor is subject to Kn and pressure, and a straight line will not persist for a combined diffusion. Besides, the combined diffusion should be a weighted sum of pure bulk and Knudsen diffusion. The line of combined diffusion will lie between rather than above the pure bulk and Knudsen diffusion. On the other hand, Knudsen diffusion in porous media also depends on the tortuosity factor, which varies with pressure. As a result, a horizontal line will not present for pure Knudsen diffusion. It should be noted that DK , pm is not that sensitive to the change in pressure as DB , and a relative flat line may still occur at low pressure corresponding to pure Knudsen flow. But it needs to be further justified through our experimental data as the flat region is important to
(20)
where, µ is the viscosity of the transport molecules in kg m/s , P is the pressure. The factor, 5/8, considers the Maxwell-Boltzmann distribution of molecular velocity and correct the problem that exponent of temperature has a fixed value of 1/2 [31]. 2.3.2. Bulk diffusion coefficient Bulk diffusion is the dominant diffusion regime when the mean free path is far less than the pore diameter, which is usually found in large pores or for high pressure gas transport. Gas-gas collisions outnumber gas-pore wall collision. The present work focuses on gas self-diffusion in 593
Fuel 253 (2019) 588–606
Y. Yang and S. Liu
Fig. 5. Diagnostic plot of reciprocal diffusion coefficient (Dp 1) vs. P to determine the dominant diffusion regime. Plot (b) is updated from plot (a) by considering the weighing factor of individual diffusion mechanisms and Knudsen diffusion coefficient for porous media.
specify the critical Knudsen number (K n ) for pure Knudsen diffusion. Considering the effect of weighing factor and tortuosity factor on the overall diffusion process, the diagnostic plot is updated from Fig. 5(a) to (b).
earlier work Yang et al. [102]. The measurement of sorption isotherm has also been reported in Yang et al. [102]. The particle method was adopted to quantify the diffusive flow for pulverized coal samples. Numerous studies [17,22] have used this technique to characterize the gas diffusion behavior of coal. This method requires pulverizing the coal to powders and ensures gas transport is purely driven by diffusion. However, grinding the coal increases the surface area for gas adsorption. The change is considered to be minimal as the increase for 40–100 mesh coal size ranges from 0.1 to 0.3% [22,110]. Other studies [38,75,111] suggested that grain size has no effect on diffusion when the particle radius is greater 0.1 mm. The coal particle in this study has a size of 60–80 mesh (0.177 mm–0.250 mm). Therefore, pulverization of the coal samples is not believed to affect the accuracy of diffusion measurements and still meets the purpose of this experiment to reduce the diffusion time and ensure diffusion-driven in nature. In the adsorption experiment, the pressure in the cell section was continuously monitoring through the data acquisition system (DAS). After each dose of methane, the pressure in the reference cell was higher than in the sample cell. When they were connected, a step increase in pressure occurred following by a gradual decrease in pressure until equilibrium was reached. The decrease in pressure was generated by the adsorption of methane occurring at the pore surface of coal matrix, and was measured very precisely. Constant pressure boundary condition was controlled by isolating the cell section from the gas supply system. This ensures a direct application of the diffusion models and the simplest solution of diffusion coefficient (D ) is given when the constant concentration is maintained at the external surface [52]. The real-time pressure data were used to calculate the sorption fraction versus time data, which is a required input of the unipore model.
3. Experimental method and procedures 3.1. Coal sample procurement and preparation Four coal samples named as Xiuwu-21, Luling-9, Luling-10, Sijiazhuang-15 were collected from three different coal mines in China. The geological information of these four coal samples can be found in elsewhere [102]. The coal samples were grounded and sieved to the desired sample size for the proximate analysis (10 g of the sample; 70–200 mesh), N2 adsorption–desorption test (1 g; 60–80 mesh) and methane sorption/diffusion testing (40 g; 40–60 mesh). Moisture and ash analysis of the four coal samples was conducted following the standard ISO 17246:2010 (Coal, Proximate analysis) [103] with a 5EMAG6600 proximate analyzer. 3.2. N2 adsorption/desorption test Nitrogen adsorption/desorption test is a common technique to obtain the pore size distribution of porous materials [104,105], where N2 serves as the adsorbate and porous media is the adsorbent. The ASAP 2020 apparatus at Material Research Institute, Penn State University was used to conduct the N2 sorption test following the ISO 159013:2007 (Pore size distribution and porosity of solid materials by mercury porosimetry and gas adsorption, Part 3: Analysis of micropores by gas adsorption) [106]. Each coal sample weighing 1 g was initially loaded into a sample tube, and automatically degassed by heating at ∼110 °C in a vacuum to pore structure analysis [107,108]. The degas procedure removes the adsorbed moisture and volatile matter of the coal samples. The coal samples were kept at a constant temperature of liquid N2 (77 K) and the adsorption isotherms were then obtained for the relative pressure (P /P0 ) ranging from 0.009 to 0.994. The instruments software also can determine surface areas, pore volumes and pore size distributions (PSD) based on the Brunauer-Emmett-Teller (BET) model and density functional theory (DFT) model [109].
At the ith pressure stage, the sorption fraction
( ) was gradually Mt M
increasing with time corresponding to a gradual decrease in pressure. The sorption rate data was calculated from the pressure-time data (Psi (t ) ), injection pressure (Pri ), equilibrium pressure in the previous pressure stage (Psei 1), and saturated or maximum amount of adsorbed i gas molecules in the current pressure stage (nsat ).
Psei 1 (Vs Vsam ) P i Vr Mt 1 = i + r M (Z ) Psei 1 (Z ) Pri nsat RT
Psi (t )(Vr + Vs (Z ) Psi
Vsam ) (25)
3.3. Diffusion coefficient estimation
where Mt is the adsorbed amount of the diffusing gas in time t and M is the adsorbed amount in infinite time. T is the temperature of the adsorption system. R is the universal gas constant. Vs , Vr , Vsam are the volume of sample cell, reference cell, and coal sample, respectively. Z
The sorption capacity and diffusion coefficient were measured simultaneously using the same experimental apparatus shown in the 594
Fuel 253 (2019) 588–606
Y. Yang and S. Liu
with appropriate subscript is symbolled as the gas compressibility i factor at current pressure. nsat is a maximum adsorbed amount at the ith pressure stage and directly obtainable from the adsorption isotherm as the step change in cumulative adsorption amount of the two neighboring equilibrium points. The experimentally measured value of Mt M was then fitted by the analytical solution of unipore model [42] to determine the diffusion coefficient of the coal samples at the best match.
on the low-pressure N2 adsorption data for Luling-10. Two distinct linear segments were observed at the P /P0 interval of 0–0.5 and 0.5–1, which indicates different gas adsorption mechanisms at different pressure stage. The linear fittings of two segments exhibited convincible correlation with R2 greater than 0.95 for all four tested coal samples, which has been previously reported in our previous work [102]. Typically, Van der Waals force formed between gas molecules and coal surfaces is the dominant adsorption mechanism at low pressure within micropores. Capillary condensation becomes important as pressure increases and it commonly occurs in mesopores and macropores. An average fractal dimension (Df ) was estimated by linear fitting the sorption data from entire pressure interval, which describes the overall heterogeneity of the pore structure. This value provides a quantitative description of the tortuous diffusive path in the complex pore structure through the fractal pore model (Fig. 4). From Eq. (18), the tortuosity factor (τ) derived from the fractal pore model depends on the fractal dimension and a normalized parameter (i.e. / dmax ). Apparently, mean free path (λ) varies with pressure. In this study, the diffusion coefficients were measured at six different pressures, which are 0.55, 1.38, 2.48, 4.14, 6.07, and 8.07 MPa. Along with the pore structural parameters, the pressures were used to calculate the mean free path and corresponding tortuosity factors. The results were listed in Table 3. The average fractal dimension of the four coal samples ranges from 2.229 to 2.496. From fractal results, Luling-10 provides the most complex pore structure with the Df of 2.496. Combing with the pore structural information from PSD, we could see that Sijiazhuang-15 provides the most tortuous diffusive path with a highest value of τ for all pressures. As a result, the diffusion time in Sijaizhuang-15 is expected to be longest and this was confirmed by our experimental results.
4. Results and discussion 4.1. Quantitative analyses of pore structure Low-pressure N2 gas adsorption was known to be a time- and costeffective technique to characterize micropores in the porous media [112]. Typically, it was commonly used to evaluate the complexity and morphological characteristics of coal pores [113]. The results of the N2 adsorption/desorption isotherms at 77 K along with hysteresis pattern were presented in our previous work [102] and it was used to investigate pore structure characteristics of the coal. Following the classification of IUPAC to classify the shape of hysteresis loop N2 adsorption–desorption isotherm [114,115], the tested coal samples can be classified as Type H3 suggesting the prevalence of slit-shaped pores. In this section, pore size distribution (PSD) and fractal analysis were carefully analyzed to have a comprehensive understanding of how pore structural parameters affect the diffusion coefficient of coal. 4.1.1. Pore size distribution (PSD) In this study, we used the classical pore size model developed by Barret, Joyner, and Halenda (BJH) in 1951 [116] to obtain the pore size distribution of the coal samples. This model is adjusted for multi-layer adsorption and based on the Kelvin equation. The ready accessibility in commercial software makes the BJH model be extensively applied to determine the PSD of microporous material [117]. The desorption branch of the hysteresis loop considers the evaporation of condensed liquid [109] and thus, the shape of desorption branch was directly dependent on the PSD of adsorbent [118]. The bimodal nature of PSDs is apparent from the two peaks observed in most samples. The pore volume was primarily contributed by adsorption pores for all coal samples (i.e. pore diameter < 100 nm). According to the IUPAC classification, the pore volumes of different sized pores (micro-, meso- and macro-pores) were listed in Table 2. Meanwhile, it also reports the average pore diameter (d ), and lower and upper cutoff of pore diameter (dmin , dmax , respectively) for the studied four coal samples. Fig. 6 presents the PSDs of the four coal samples obtained from the BJH desorption branch.
4.2. Quantitative analyses of sorption kinetics Following the procedure depicted in the particle method [22], highpressure methane adsorption rate data were collected at six different pressure steps from initial pressure at 0.55 MPa up to the final pressure at 8.07 MPa. With eight transducers connecting to the data acquisition system, twenty-four sorption rate measurements were performed in this study. For each pressure, the apparent diffusion coefficient is assumed to be constant. As a result, the estimated diffusion coefficient is an average of the intrinsic diffusivity at a specific pressure interval. The stepwise adsorption pressure-time data were modeled by the unipore model described in Section 2 (Eq. (3)) and the pressure-dependence apparent diffusivity (D/ re2 ) was estimated by pressure and time regression using our proposed automate Matlab program. Fig. 8 shows two of the twenty-four rate measurements with modeled results based on the unipore model. These measurements were for Xiuwu-21 and Luling-10 at 0.55 MPa. It can be seen that the unipore model can accurately predict the trend of the sorption rate data with less than 1% percent error. Due to the assumption on uniform pore size distribution, the unipore model was found to be more applicable at high pressure steps [37,42,47]. The lowest pressure stage in this study was 0.55 MPa and
4.1.2. Fractal analysis In this study, the fractal dimension was obtained using the FHH model, and the detailed procedure can be found elsewhere [60].As shown in Fig. 7, log-log scale of ln
( ) vs. ln (ln ( ) ) was plotted based V V0
P0 P
Table 2 Average pore diameter, lower and upper cutoff of pore diameter, and pore volume of the coal samples estimated from the BJH model. Coal sample
Xiuwu-21 Luling-9 Luling-10 Sijiangzhuang-15
d (nm)
7.61 12.49 15.05 4.6
Pore Volume (cm3/100 g) Vtotal
Vmicro
Vmeso
Vmacro
1.178 0.395 0.393 2.772
0.0247 0.00330 0.00372 0.0537
0.703 0.172 0.149 0.456
0.451 0.220 0.240 2.262
dmin (nm)
dmax (nm)
1.741 1.880 1.870 1.565
83.759 115.440 112.430 132.447
d : average pore diameter; Vtotal: total pore volume; Vmicro: micropore volume; Vmeso: mesopore volume; Vmacro: macropore volume.
595
Fuel 253 (2019) 588–606
Y. Yang and S. Liu
Fig. 6. The pores size distribution of the selected coal samples calculated from the desorption branch of nitrogen isotherm by the BJH model.
the unipore model gave convincible accuracy to model the sorption rate data (Fig. 9). Thus, for higher pressure stage, the unipore model should still retain its legitimacy in this application. In this work, other measurements exhibited the same or even higher accuracy when applying
the unipore model although they had different length of adsorption equilibrium time. Fig. 9 shows the results of the estimated diffusion coefficients at different pressures for the four tested coal samples, where the effective
Fig. 7. Fractal analyses of N2 desorption isotherms for the studied four coal samples (Ref: [102]). 596
Fuel 253 (2019) 588–606
Y. Yang and S. Liu
Table 3 The fractal dimension, mean free path and tortuosity factor based on the fractal pore model and estimated at the specified pressure stage (i.e. 0.55, 1.38, 2.48, 4.14, 6.07, and 8.07 MPa). Coal sample
A
Df = A + 3
R2
P (MPa) Mean free path, λ (nm)
0.55 6.595
1.38 2.660
2.48 1.503
4.14 0.924
6.07 0.656
8.07 0.516
Xiuwu-21 Luling-9 Luling-10 Sijiangzhuang-15
−0.772 −0.505 −0.504 −0.537
2.229 2.495 2.496 2.463
0.967 0.989 0.975 0.932
Tortuosity factor, τ
1.787 4.128 4.078 5.606
2.199 6.472 6.395 9.444
2.506 8.587 8.486 13.111
2.800 10.924 10.798 17.336
3.029 12.948 12.800 21.114
3.199 14.576 14.409 24.223
diffusive path was estimated to be the radius of the particle [42]. The values of diffusion coefficient exhibited an overall negative trend when the gas pressure was above 2.48 MPa. The decreasing trend is consistent with the theoretical bulk diffusion coefficient in open space (Eq. (21)), which is dependent on the mean free path of the gas molecule and gas pressure. The diffusion coefficient became relatively small at pressures higher than 6 MPa when the coal matrix had high methane concentration and a low concentration gradient. The initial slight increasing trend were observed in the diffusion curves when the pressure was below 2.48 MPa. The same experimental trend was reported in Wang and Liu [17], and they explained that as the exerted gas pressure on the coal samples may open the previously closed pores and more gas pathways were created to enhance the diffusion flow. Besides, the relative contribution of Knudsen and bulk diffusions to the gas transport process changes at various gas pressures. Knudsen diffusion loses its importance in the overall diffusion process as gas pressure increases and molecular-molecular collisions are more frequent. At the same time, bulk diffusion becomes important at higher pressure and typically, it has the relatively fast diffusion rate than the Knudsen diffusion, which explains the increase in diffusion coefficient with the increase in pressure when pressure was less than 2.48 MPa. Overall, a decreasing trend was found between pressure and diffusion coefficient in this study. Some previously reported data [22,35,38] also confirmed the same decreasing trend, where the reduction in diffusivity is due to the dominance of bulk diffusion and potential greater swelling effect caused by gas adsorption at higher pressure. However, other studies [52,119,120] showed a reverse trend due to the associated increase in surface diffusion, which increases as surface coverage increases. The underlying fundamental mechanism will be further discussed in the subsequent section. The values of diffusivity range from 1.05 × 10 13 to
Fig. 9. Variation of the experimentally measured methane diffusion coefficients with pressure.
9.77 × 10 12 m2/s . At all pressure steps, Xiuwu-21 had the highest diffusivity and two Luling coals have low diffusivity because both Luling coals have high Df as reported in Table 3. 4.3. Effect of pore structure on the diffusion coefficient in coal As the diffusion process controls the gas influx from matrix towards the cleat/fracture system, it dominates the long-term well performance
Fig. 8. Sample experimental results of CH4 sorption rate at 0.55 MPa for Xiuwu-21 and Luling-10.
597
Fuel 253 (2019) 588–606
Y. Yang and S. Liu
Table 4 Theoretically calculated bulk diffusion coefficient (DB ) and Knudsen diffusion coefficent of porous media (DK , pm ). Pressure [MPa]
0.55
Theoretical Diffusion Coefficient
DB
DK,pm
DB
DK,pm
DB
DK,pm
DB
DK,pm
DB
DK,pm
DB
DK,pm
Xiuwu-21 Luling-9 Luling-10 Sijiazhuang-15
104.77 41.87 38.47 262.48
67.60 19.22 21.54 51.02
42.27 16.89 15.52 105.89
54.94 12.26 13.73 30.29
23.88 9.54 8.77 59.82
48.22 9.24 10.35 21.81
14.69 5.87 5.39 36.79
43.15 7.26 8.13 16.50
10.42 4.16 3.83 26.11
39.90 6.13 6.86 13.55
8.20 3.28 3.01 20.56
37.77 5.44 6.10 11.81
[× 1010 m2/s ]
1.38
2.48
4.14
6.07
8.07
of CBM after the fracture storage is depleted [17]. The estimation of diffusion coefficient based on pore structure is critical to determine the production potential of a given coal formation. Apparently, diffusion process is slower for coal pore in a smaller size or having a more complex structure. As mentioned above, the diffusive gas influx is controlled by combined Knudsen and bulk diffusions. The theoretical values of the diffusivity under these two diffusion modes was calculated based Eq. (19) and Eq. (21) and the results are listed in Table 4. It should be noted that the expression of DB given in Eq. (21) is derived for open space and independent of the solid structure. For porous media, a multiplication of porosity is added to the expression of DB that considers volume not occupied by the solid matrix [121–123]. The overall diffusion coefficient (Dp ) was then defined as a weighted sum of Knudsen diffusion and bulk diffusion given in Eq. (23). To estimate the weighing factor (wK ) of each mechanism, it is critical to determine the critical Knudsen number (Kn ) and for Kn > Kn , a pure Knudsen diffusion can be assumed. Examination of the manner in which Dp varies with pressure using the diagnostic plot (Fig. 5(b)) is intuitively helpful to identify the pressure interval for pure Knudsen flow. One challenging aspect of applying the diagnostic plot is the uncertainty about the sensitivity of DK , pm to the change in pressure. If DK , pm is not very sensitive to pressure, a small variation in pressure will not have an apparent change of Dp at low pressure stages and under pure Knudsen diffusion. Then a relative flat line can be found in a plot of Dp 1 vs. P at low pressure. It corresponds to a pressure interval of pure Knudsen flow, and the contribution from bulk diffusion is ignored as the intermolecular collision strongly correlated with pressure. Fig. 10 shows the change in DB and DK , pm with pressure for Sijiazhuang-15 sample. Fig. 11 demonstrates the application of using diagnostic plot to identify diffusion mechanism. In Fig. 11, bulk diffusion was subject to much greater variation than Knudsen diffusion over the pressure range of interest. Consequently, a relatively flat line was found at low pressure interval (P < P ) in the diagnostic plot (Fig. 11) for a pure Knudsen diffusion. Effective diffusion coefficient (Dp 1) is then equivalent to DK , pm and weighing factor (wK ) equals to one. The critical Knudsen number (Kn ) is determined at the inflection point, where P = P . As pressure increases, pore wall effect diminishes as mean free path of gas molecules shortens and bulk diffusion becomes important. Then at about 2.5 MPa, Dp 1 was subject to a greater variation in terms of pressure variation since DB is directly proportional to mean free path and inversely proportional to the pres-
Fig. 10. Variation of bulk diffusion coefficient (DB ) and Knudsen diffusion coefficient (DK , pm ) at different pressure stages for Sijiazhuang-15.
Fig. 11. Diagnostic plot of reciprocal diffusion coefficient (Dp 1) vs. P to specify pressure interval of pure Knudsen flow (P < P ) and critical Knudsen number (Kn =Kn(P ) )).
Fig. 12. Diffusion regimes at different pressure interval and range of Knudsen number.
598
Fuel 253 (2019) 588–606
Y. Yang and S. Liu
Fig. 13. A plot of wK as a piecewise function of Kn. The horizontal tails at the low and high interval of Kn correspond to pure bulk and Knudsen diffusion, respectively.
Fig. 15. Comparison between experimental and theoretical calculated diffusion coefficients of the studied four coal samples at same ambient pressure.
coefficient in coal matrix. For catalysis, Wheeler [124] proposed an empirical combination of Knudsen and bulk diffusion coefficient to determine the effective diffusion coefficient of combined diffusion as
Dp = DB (1
e
1/Kn)
(26)
In Eq. (26), Dp approaches to DB as Kn approaches to zero and mean free path is far less than the pore diameter. Dp approaches to DK as Kn approaches infinity since e 1/Kn 1 1/Kn . Correspondingly, the weighing factor of Knudsen diffusion (wK ) grows towards higher Kn . However, some built-in limitations are also observed for this theoretical formula. First, it fails to consider the change in the effective diffusive path at different pressures as DK , pm rather than DK should be involved to describe the diffusion rate under Knudsen regime. Besides, it underestimates wK as Eq. (26) implicitly states that pure Knudsen diffusion only occurs for flow with infinite value of Kn . In fact, Knudsen flow dominates the overall diffusion once Kn is reached as illustrated in Fig. 11. Instead, wK is assumed to have a linear dependence on Kn in the transition pressure range and for a combined diffusion. This assumption would be further justified by comparing with the experimental data. Fig. 13 is a plot of wK vs. Kn applied to quantify the relative contribution of each diffusion mechanism. Once the wK is given, the overall diffusion coefficient can be theoretically determined by Eq. (23). Experimentally measured diffusion coefficients for methane are presented in Fig. 10. The results were then compared with theoretical values predicted by the relationships proposed by Wheeler [124] and this study as given in Eq. (26) and Eq. (23), respectively. Fig. 14 indicates that the theory of wK developed in this study provided better fit to the experimental measured diffusion coefficient than the one proposed by Wheeler [124]. The improvement in the prediction of diffusivity was more obvious towards low pressure and Knudsen diffusion became predominant. This is because our method allows for the expected changes in the effective diffusion path. Nevertheless, great discrepancy was still found at low pressure stages compared with the experimental measured diffusion coefficients. The source of error originated from the accuracy in the estimation of pore
Fig. 14. Comparison between experimental and theoretical calculated diffusion coefficient for methane diffusion in Xiuwu-21. Wheeler [124] is described by Eq. (26), and this work is given by Eq. (23).
sure. The dividing pressure between pure Knudsen diffusion and combined diffusion for tested coal samples were all determined to be 2.5 MPa, i.e. P = 2.5MPa . For even higher pressure, the effect of pore wall-molecular collisions can be neglected and Dp 1 was estimated by DB 1. As a result, a linear trend was noted at pressure greater than 6 MPa when bulk diffusion dominates the overall diffusion and wK equals to zero. Using Fig. 11, we would be able to identify the dominant diffusion mechanism at different pressure stages and evaluate the relative contribution of each mechanism or wK as dictated by Eq. (24). Undoubtfully, wK equals to one for pure Knudsen diffusion and zero for pure bulk diffusion as shown in Fig. 12. In the transition regime, no theoretical development has been made on the prediction of diffusion
599
Fuel 253 (2019) 588–606
Y. Yang and S. Liu
structural parameters, which is critical in Knudsen diffusion when pore morphology is important. Besides, the scale of measured diffusion coefficient is three order of magnitudes smaller than the predicted one. This is caused by the presence of surface diffusion. Movement of gas molecules along the pore wall surface contributes significantly to the gas transport of adsorbed species in micropores, where gas molecules cannot escape from the potential field of pore surface [125,126]. The relative contribution of surface diffusion and diffusion in pore volume is related to the volume ratio of gas in adsorbed phase and free phase [4]. The primary purpose of this work is to predict diffusion behavior of coal based on pore structure. Surface diffusion as an activated diffusion is mainly a function of adsorbate properties, rather than adsorbent properties. To eliminate the variation in surface diffusion, we conducted the analysis under the same ambient pressure. In Fig. 15, the experimental measured diffusion coefficients are plotted against the theoretical values determined by Eq. (23) for the four coal samples at each pressure stages. The experimental diffusion coefficients were measured at six pressure stages ranging from 0.55 MPa to 8.07 MPa. Therefore, six isobaric lines are presented in Fig. 15 and each line is composed of 4 points corresponding to the four studied coal samples. The theoretical diffusion coefficient derived from Eq. (23) is a function of pore structural parameters. Overall, it provides good fits to the experimental diffusion coefficients. Due to the presence of surface diffusion, the scale of the theoretical values does not agree with it of the experimental values. But the linear relationships in Fig. 15 inherently illustrates that pore structure has negligible effect on the transport of gas molecules along the pore surface. Otherwise, the contribution from surface diffusion should vary for different coal samples, and the four points will not stay in the same line. There is a compelling mechanism that determines the steepness of the linear relationships. Generally, surface diffusion becomes predominant as surface coverage increases and multilayer of adsorption builds up at higher pressure stages. The slope is reduced towards high pressures due to an increase in the contribution from surface diffusion. On the contrary, as the pore surface is smoothed, and the effective diffusive path is shortened with a reduction in the induced tortuosity. This leads to a faster diffusion process with greater mass transport occurring in pore volume, and the lines are expected to be steeper as pressure increases. Under these mechanisms, the lines are steeper at lower pressure stages (P < 4 MPa ) in Fig. 15. For higher pressures, reverse trend can be found as the lines tend to be horizontal as pressure increases.
samples. Experimentally, adsorption and transport of methane in coal particles were analyzed with the particle method. By applying the unipore model, the diffusion coefficients were then estimated from the analytical modeling of experimental sorption pressure-time data. An automated computer program was proposed that can automatically and time-effectively estimate the diffusion coefficients with regressing to the pressure-time experimental results. A theoretical pore-structure-based model was proposed to estimate the pressure-dependent diffusion coefficient for fractal coals. The proposed model took the pore structure parameters, including, porosity, pore size distribution and fractal dimension, as inputs to predict the pressure dependent gas diffusion behavior. In the proposed model, we integrated Knudsen and bulk diffusion influxes to dynamically define the overall gas transport process. The proposed diffusion coefficient model was validated against the experimental measured diffusion coefficient results. Both experimental and modeled results suggested that Knudsen diffusion dominated the gas influx at low pressure range (< 2.5 MPa) and bulk diffusion dominated at high pressure range (> 6 MPa). For intermediate pressure ranges (2.5–6 MPa), combined diffusion should be considered as a weighted sum of Knudsen and bulk diffusion, and the weighing factor directly depended on the Knudsen number. Uniquely, we took the tortuosity as an input parameter for the theoretical diffusion model and this allows modelers to quantitatively take the pore morphological complexity to define the diffusion for different coals. Based on the fractal pore model, the distribution of pores at various size and their associated heterogeneity i.e., PSD and Df along with the mean free path of gas molecules (λ) are the essential parameters to determine the tortuosity factor. The determined tortuosity factors of the four coal samples ranged from 1.787 to 24.223 for the tested pressure interval between 0.55 MPa and 8.07 MPa. The results suggested that the increase in pressure and pore structural heterogeneity resulted in a longer effective diffusion path and a higher value of tortuosity factor affecting the Knudsen diffusion influx in porous media. The pore structural parameters lose their significance in controlling the overall mass transport process as bulk diffusion dominates. The overall diffusion coefficient was then evaluated as a weighted sum of Knudsen and bulk diffusion coefficient. At individual pressure stages from 0.55 MPa and 8.07 MPa, it provided good fits to the experimentally measured overall diffusion coefficient, which varied from 1.05 × 10 13 to 9.77 × 10 12m2 /s . This theoretical model is the first-of-its-kind to link the realistic complex pore structure into diffusion coefficient based on the fractal theory. Based on the proposed model, we propose a diagnostic tool that illustrates and examines the pressure-dependent variations of diffusion coefficient for different coals. The experimental results and proposed model can be coupled into the commercially available simulator to predict the long-term CBM well production profiles.
5. Summary and conclusions This works presents a series of experimental measurements and theoretical modeling on the diffusion coefficient for four different coal
Appendix. User interface in MATLAB GUI for the estimation of effective diffusivity The code below is named as “UniporeModel. Fig”. In the command window of MATLAB, type ‘open UniporeModel.fig’. A user interface should pop up as shown in Fig. A1.
600
Fuel 253 (2019) 588–606
Y. Yang and S. Liu
Fig. A1. User Interface (Unipore Model).
The required input is the experimental sorption rate data (i.e., Mt vs. t). The data should be stored in a txt file in the same directory as the M ‘UniporeModel.fig’ and named as ‘diffusion.txt’. The next entry is the search interval of Gold Section Search method for the apparent diffusivity (De ), which are marked as Dhigh and Dlow in the unit of s 1.The last required input is the number of terms in the infinite summation of unipore model denoted as nmax . In the infinite summation, the value of each individual term decreases as the index of the term increases. Thus, an entry of 50 for nmax is good enough to truncate the infinite summation. Once all the required inputs are entered in the program, hit the calculate button. Then the value of apparent diffusivity (De) will pop up along with the percentage error. The error of the fitting by unipore model is determined as the average sum of squared difference, which is the ratio of the result from least-square function (Eq. (5)) over the number of sorption rate datapoints. With the determined apparent diffusivity, the sorption rate data is fitted by the unipore model (Eq. (3)). A figure of the experimental sorption data with the regressed curve is shown at the bottom of the window. Fig. A2 is an example of applying the ‘UniporeModel.fig’ to determine the apparent diffusion coefficient.
Fig. A2. Typical example of applying ‘UniporeModel.fig’ to determine.
In the attached code, y denotes as the sorption fraction, x denotes as the apparent diffusion coefficient. Subscript ‘exp’ is the abbreviation of experimental and ‘model’ means sorption rate data estimated by the unipore model. yexp and ymodel are the experimental measured and analytical determined sorption fraction. De _true is the determined diffusion coefficient that provides the best fit to the experimental data.
601
Fuel 253 (2019) 588–606
Y. Yang and S. Liu
602
Fuel 253 (2019) 588–606
Y. Yang and S. Liu
603
Fuel 253 (2019) 588–606
Y. Yang and S. Liu
604
Fuel 253 (2019) 588–606
Y. Yang and S. Liu
References
2004;83(3):293–303. [39] Kelemen SR, Kwiatek LM. Physical properties of selected block Argonne Premium bituminous coal related to CO2, CH4, and N2 adsorption. Int J Coal Geol 2009;77(1):2–9. [40] Kumar A. Methane diffusion characteristics of Illinois coals. Southern Illinois University at Carbondale; 2007. [41] Marecka A, Mianowski A. Kinetics of CO2 and CH4 sorption on high rank coal at ambient temperatures. Fuel 1998;77(14):1691–6. [42] Mavor M, Owen L, Pratt T. Measurement and evaluation of coal sorption isotherm data. SPE annual technical conference and exhibition. 1990. [43] Nandi SP, Walker PL. Activated diffusion of methane from coals at elevated pressures. Fuel 1975;54(2):81–6. [44] Naveen P, Asif M, Ojha K, Panigrahi DC, Vuthaluru HB. Sorption kinetics of CH4 and CO2 diffusion in coal: theoretical and experimental study. Energy Fuels 2017;31(7):6825–37. [45] Pone JDN, Halleck PM, Mathews JP. Sorption capacity and sorption kinetic measurements of CO2 and CH4 in confined and unconfined bituminous coal. Energy Fuels 2009;23(9):4688–95. [46] Salmachi A, Haghighi M. Temperature effect on methane sorption and diffusion in coal: application for thermal recovery from coal seam gas reservoirs. APPEA J 2012;52(1):291–300. [47] Smith DM, Williams FL. Diffusional effects in the recovery of methane from coalbeds. Soc Petrol Eng J 1984;24(05):529–35. [48] Zhao W, Cheng Y, Yuan M, An F. Effect of adsorption contact time on coking coal particle desorption characteristics. Energy Fuels 2014;28(4):2287–96. [49] Cui X, Bustin A, Bustin RM. Measurements of gas permeability and diffusivity of tight reservoir rocks: different approaches and their applications. Geofluids 2009;9(3):208–23. [50] Yao Y, Liu D, Tang D, Tang S, Huang W, Liu Z, et al. Fractal characterization of seepage-pores of coals from China: an investigation on permeability of coals. Comput Geosci 2009;35(6):1159–66. [51] Li A, Ding W, He J, Dai P, Yin S, Xie F. Investigation of pore structure and fractal characteristics of organic-rich shale reservoirs: a case study of Lower Cambrian Qiongzhusi formation in Malong block of eastern Yunnan Province South China. Mar Petrol Geol 2016;70:46–57. [52] Pan Z, Connell LD, Camilleri M, Connelly L. Effects of matrix moisture on gas diffusion and flow in coal. Fuel 2010;89(11):3207–17. [53] Karacan CÖ. Heterogeneous sorption and swelling in a confined and stressed coal during CO2 injection. Energy Fuels 2003;17(6):1595–608. [54] Crosdale PJ, Beamish BB, Valix M. Coalbed methane sorption related to coal composition. Int J Coal Geol 1998;35(1):147–58. [55] Keshavarz A, Sakurovs R, Grigore M, Sayyafzadeh M. Effect of maceral composition and coal rank on gas diffusion in Australian coals. Int J Coal Geol 2017;173:65–75. [56] Han F, Busch A, Krooss BM, Liu Z, Yang J. CH4 and CO2 sorption isotherms and kinetics for different size fractions of two coals. Fuel 2013;108:137–42. [57] Busch A, Gensterblum Y, Krooss BM, Littke R. Methane and carbon dioxide adsorption–diffusion experiments on coal: upscaling and modeling. Int J Coal Geol 2004;60(2–4):151–68. [58] Clarkson CR, Bustin RM. The effect of pore structure and gas pressure upon the transport properties of coal: a laboratory and modeling study. 1. Isotherms and pore volume distributions. Fuel 1999;78(11):1333–44. [59] Li Z, Hao Z, Pang Y, Gao Y. Fractal dimensions of coal and their influence on methane adsorption. J China Coal Soc 2015;40(4):863–9. [60] Yao Y, Liu D, Tang D, Tang S, Huang W. Fractal characterization of adsorptionpores of coals from North China: an investigation on CH4 adsorption capacity of coals. Int J Coal Geol 2008;73(1):27–42. [61] Wu K, Chen Z. Real gas transport through complex nanopores of shale gas reservoirs. SPE Europec featured at 78th EAGE conference and exhibition. 2016. p. 22. [62] Azom PN, Javadpour F. Dual-continuum modeling of shale and tight gas reservoirs. SPE Annual Technical Conference and Exhibition. 2012. [63] Darabi H, Ettehad A, Javadpour F, Sepehrnoori K. Gas flow in ultra-tight shale strata. J Fluid Mech 2012;710:641–58. [64] Ertekin T, King GA, Schwerer FC. Dynamic gas slippage: a unique dual-mechanism approach to the flow of gas in tight formations. SPE-12258-PA 1986;1(01):43–52. [65] Javadpour F. Nanopores and Apparent Permeability of Gas Flow in Mudrocks (Shales and Siltstone). 2009. [66] Liu Q, Shen P, Yang P. Pore scale network modelling of gas slippage in tight porous media. Contemp Math 2002;295:367–76. [67] Rahmanian M, Aguilera R, Kantzas A. A new unified diffusion–viscous-flow model based on pore-level studies of tight gas formations. SPE J 2012;18(01):38–49. [68] Singh H, Javadpour F. Nonempirical apparent permeability of shale. Unconventional resources technology conference (URTEC). 2013. [69] Wu K, Li X, Guo C, Chen Z. Adsorbed gas surface diffusion and bulk gas transport in nanopores of shale reservoirs with real gas effect-adsorption-mechanical coupling. SPE reservoir simulation symposium. 2015. [70] Wu K, Li X, Wang C, Chen Z, Yu W. Apparent permeability for gas flow in shale reservoirs coupling effects of gas diffusion and desorption. Unconventional resources technology conference, Denver, Colorado, 25-27 August 2014. 2014. p. 2328–45. [71] Wu K, Li X, Wang C, Chen Z, Yu W. A model for gas transport in microfractures of shale and tight gas reservoirs. AIChE J 2015;61(6):2079–88. [72] King GR. Numerical simulation of the simultaneous flow of methane and water through dual porosity coal seams during the degasification process. University Park (USA): Pennsylvania State Univ; 1985.
[1] Thomas Graham Memorial S, Sherwood JN, University of S. Diffusion processes; proceedings of the Thomas Graham Memorial Symposium, University of Strathclyde. London; New York: Gordon and Breach, 1971. [2] Fick A. Ueber diffusion. Ann Phys 1855;170(1):59–86. [3] Philibert J. One and a half century of diffusion: Fick, Einstein, before and beyond. Diffus Fundam 2005;2(1):1–10. [4] Kärger J, Ruthven DM, Theodorou DN. Diffusion in nanoporous materials. John Wiley & Sons; 2012. [5] Kärger J, Chmelik C, Heinke L, Valiullin R. A new view of diffusion in nanoporous materials. Chem Ing Tech 2010;82(6):779–804. [6] Kainourgiakis M, Steriotis TA, Kikkinides E, Romanos G, Stubos A. Adsorption and diffusion in nanoporous materials from stochastic and process-based reconstruction techniques. Colloids Surf, A 2002;206(1–3):321–34. [7] Gray I. Reservoir engineering in coal seams: part 1-The physical process of gas storage and movement in coal seams. SPE Reservoir Eng 1987;2(01):28–34. [8] Harpalani S, Chen G. Influence of gas production induced volumetric strain on permeability of coal. Geotech Geol Eng 1997;15(4):303–25. [9] Levine JR. Model study of the influence of matrix shrinkage on absolute permeability of coal bed reservoirs. Geol Soc London Spec Publ 1996;109(1):197–212. [10] Warren J, Root PJ. The behavior of naturally fractured reservoirs. Soc Petrol Eng J 1963;3(03):245–55. [11] Laubach S, Marrett R, Olson J, Scott A. Characteristics and origins of coal cleat: a review. Int J Coal Geol 1998;35(1):175–207. [12] George JS, Barakat M. The change in effective stress associated with shrinkage from gas desorption in coal. Int J Coal Geol 2001;45(2):105–13. [13] Ceglarska-Stefańska G, Zarębska K. The competitive sorption of CO2 and CH4 with regard to the release of methane from coal. Fuel Process Technol 2002;77:423–9. [14] Schüth F, Sing KSW, Weitkamp J. Handbook of porous solids. Weinheim, Germany: Wiley-VCH; 2002. [15] Clarkson CR, Pan Z, Palmer ID, Harpalani S. Predicting sorption-induced strain and permeability increase with depletion for coalbed-methane reservoirs. SPE J 2010. [16] Liu S, Harpalani S. Permeability prediction of coalbed methane reservoirs during primary depletion. Int J Coal Geol 2013;113:1–10. [17] Wang Y, Liu S. Estimation of pressure-dependent diffusive permeability of coal using methane diffusion coefficient: laboratory measurements and modeling. Energy Fuels 2016;30(11):8968–76. [18] Collins R. New theory for gas adsorption and transport in coal. Proceedings of the 1991 coalbed methane symposium, Tuscaloosa, USA. 1991. p. 425–31. [19] Dvoyashkin M, Valiullin R, Kärger J. Temperature effects on phase equilibrium and diffusion in mesopores. Phys Rev E 2007;75(4):041202. [20] Evans III R, Watson G, Mason E. Gaseous diffusion in porous media at uniform pressure. J Chem Phys 1961;35(6):2076–83. [21] Valiullin R, Kortunov P, Kärger J, Timoshenko V. Concentration-dependent selfdiffusion of liquids in nanopores: a nuclear magnetic resonance study. J Chem Phys 2004;120(24):11804–14. [22] Pillalamarry M, Harpalani S, Liu S. Gas diffusion behavior of coal and its impact on production from coalbed methane reservoirs. Int J Coal Geol 2011;86(4):342–8. [23] Welty JR, Wicks CE, Wilson RE. Fundamentals of momentum, heat, and mass transfer. New York: Wiley; 1984. [24] Mason EA, Malinauskas A. Gas transport in porous media: the dusty-gas model. Elsevier Science Ltd; 1983. [25] Zheng Q, Yu B, Wang S, Luo L. A diffusivity model for gas diffusion through fractal porous media. Chem Eng Sci 2012;68(1):650–5. [26] Knudsen M. Die Gesetze der Molekularströmung und der inneren Reibungsströmung der Gase durch Röhren. Ann Phys 1909;333(1):75–130. [27] Steckelmacher W. Knudsen flow 75 years on: the current state of the art for flow of rarefied gases in tubes and systems. Rep Prog Phys 1986;49(10):1083. [28] Yang RT. Gas separation by adsorption processes. Butterworth-Heinemann; 2013. [29] Nie B, He X, Wang B. Mechanism and modes of gas diffusion in coal seams. Chin Saf Sci J 2000;10(6):24–8. [30] Shi JQ, Durucan S. A bidisperse pore diffusion model for methane displacement desorption in coal by CO2 injection. Fuel 2003;82(10):1219–29. [31] Bird G. Definition of mean free path for real gases. Phys Fluids 1983;26(11):3222–3. [32] Lama R, Bodziony J. Management of outburst in underground coal mines. Int J Coal Geol 1998;35(1):83–115. [33] Wei X, Wang G, Massarotto P, Golding S. A case study on the numerical simulation of enhanced coalbed methane recovery. SPE Asia Pacific oil & gas conference and exhibition. 2006. [34] Bhowmik S, Dutta P. Adsorption rate characteristics of methane and CO2 in coal samples from Raniganj and Jharia coalfields of India. Int J Coal Geol 2013;113:50–9. [35] Busch A, Gensterblum Y, Krooss BM, Littke R. Methane and carbon dioxide adsorption–diffusion experiments on coal: upscaling and modeling. Int J Coal Geol 2004;60(2):151–68. [36] Charrière D, Pokryszka Z, Behra P. Effect of pressure and temperature on diffusion of CO2 and CH4 into coal from the Lorraine basin (France). Int J Coal Geol 2010;81(4):373–80. [37] Clarkson CR, Bustin RM. The effect of pore structure and gas pressure upon the transport properties of coal: a laboratory and modeling study. 2. Adsorption rate modeling. Fuel 1999;78(11):1345–62. [38] Cui X, Bustin RM, Dipple G. Selective transport of CO2, CH4, and N2 in coals: insights from modeling of experimental gas adsorption data. Fuel
605
Fuel 253 (2019) 588–606
Y. Yang and S. Liu
[103] Thommes M, Morlay C, Ahmad R, Joly J. Assessing surface chemistry and pore structure of active carbons by a combination of physisorption (H 2 O, Ar, N 2, CO 2), XPS and TPD-MS. Adsorption 2011;17(3):653. [104] Gaylord NG, Eirich FR. The structure and properties of porous materials. J Polym Sci 1960;45(145):278–9. [105] Mastalerz M, He L, Melnichenko YB, Rupp JA. Porosity of coal and shale: insights from gas adsorption and SANS/USANS techniques. Energy Fuels 2012;26(8):5109–20. [106] Standard B, ISO B. Pore size distribution and porosity of solid materials by mercury porosimetry and gas adsorption. BS ISO 2005:15901-1. [107] Bustin RM, Clarkson CR. Geological controls on coalbed methane reservoir capacity and gas content. Int J Coal Geol 1998;38(1–2):3–26. [108] Busch A, Gensterblum Y, Krooss BM, Siemons N. Investigation of high-pressure selective adsorption/desorption behaviour of CO2 and CH4 on coals: an experimental study. Int J Coal Geol 2006;66(1–2):53–68. [109] Gregg SJ, Sing KSW, Salzberg H. Adsorption surface area and porosity. J Electrochem Soc 1967;114(11):279C-C. [110] Jones AH, Bell G, Schraufnagel R. A review of the physical and mechanical properties of coal with implications for coal-bed methane well completion and production. Geological and coal-bed methane resources of the northern San Juan Basin, Colorado and New Mexico. 1988. p. 169–81. [111] Siemons N, Busch A, Bruining H, Krooss B, Gensterblum Y. Assessing the kinetics and capacity of gas adsorption in coals by a combined adsorption/ diffusion method. SPE annual technical conference and exhibition. 2003. p. 8. [112] Clarkson CR, Solano N, Bustin RM, Bustin AMM, Chalmers GRL, He L, et al. Pore structure characterization of North American shale gas reservoirs using USANS/ SANS, gas adsorption, and mercury intrusion. Fuel 2013;103:606–16. [113] Fu H, Tang D, Xu T, Xu H, Tao S, Li S, et al. Characteristics of pore structure and fractal dimension of low-rank coal: a case study of Lower Jurassic Xishanyao coal in the southern Junggar Basin, NW China. Fuel 2017;193:254–64. [114] Everett DH, Stone FS, Colston Research S. The Structure and properties of porous materials. London: Butterworths, 1960. [115] Sing KS. Reporting physisorption data for gas/solid systems with special reference to the determination of surface area and porosity (Recommendations 1984). Pure Appl Chem 1985;57(4):603–19. [116] Barrett EP, Joyner LG, Halenda PP. The determination of pore volume and area distributions in porous substances. I. Computations from nitrogen isotherms. J Am Chem Soc 1951;73(1):373–80. [117] Groen JC, Pérez-Ramı́rez J. Critical appraisal of mesopore characterization by adsorption analysis. Appl Catal A 2004;268(1):121–5. [118] Oulton TD. The pore size-surface area distribution of a cracking catalyst. J Phys Colloid Chem 1948;52(8):1296–314. [119] Cai Y, Pan Z, Liu D, Zheng G, Tang S, Connell LD, et al. Effects of pressure and temperature on gas diffusion and flow for primary and enhanced coalbed methane recovery. Energy Explor Exploit 2014;32(4):601–19. [120] Zhang XG, Ranjith PG, Perera MSA, Ranathunga AS, Haque A. Gas transportation and enhanced coalbed methane recovery processes in deep coal seams: a review. Energy Fuels 2016;30(11):8832–49. [121] Maxwell JC. A treatise on electricity and magnetism. Clarendon Press; 1881. [122] Rayleigh LLVI. On the influence of obstacles arranged in rectangular order upon the properties of a medium. London Edinburgh Dublin Philos Mag J Sci 1892;34(211):481–502. [123] Weissberg HL. Effective diffusion coefficient in porous media. J Appl Phys 1963;34(9):2636–9. [124] Wheeler A. Reaction rates and selectivity in catalyst pores. In: Emmett PH, editor. Catalysis, fundamental principles (Part 2). New York: Reinhold Publishing Corporation; 1955. p. 105–65. [125] Do DD. Adsorption analysis: equilibria and kinetics. Imperial College Press; 1998. [126] Dutta A. Multicomponent gas diffusion and adsorption in coals for enhanced methane recovery. Stanford University; 2009.
[73] King GR, Ertekin T, Schwerer FC. Numerical simulation of the transient behavior of coal-seam degasification wells. SPE-12258-PA 1986;1(02):165–83. [74] Wang G, Ren T, Qi Q, Lin J, Liu Q, Zhang J. Determining the diffusion coefficient of gas diffusion in coal: development of numerical solution. Fuel 2017;196:47–58. [75] Nandi SP, Walker PL. Activated diffusion of methane in coal. Fuel 1970;49(3):309–23. [76] Crank J. The mathematics of diffusion. Oxford, GB: Clarendon Press; 1975. [77] Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical recipes in Fortran 77, the art of scientific computing. Cambridge University Press; 1992. [78] Mandelbrot BB. The fractal geometry of nature. New York: WH Freeman; 1983. [79] Pfeifer P, Avnir D. Chemistry in noninteger dimensions between two and three. I. Fractal theory of heterogeneous surfaces. J Chem Phys 1983;79(7):3558–65. [80] Wang HP, Yang YS, Wang YD, Yang JL, Jia J, Nie YH. Data-constrained modelling of an anthracite coal physical structure with multi-spectrum synchrotron X-ray CT. Fuel 2013;106:219–25. [81] Sing KSW. Characterization of porous materials: past, present and future. Colloids Surf, A 2004;241(1):3–7. [82] Wang H, Yang Y, Yang J, Nie Y, Jia J, Wang Y. Evaluation of multiple-scale 3D characterization for coal physical structure with DCM method and synchrotron Xray CT. Sci World J 2015;2015:414262-. [83] Avnir D, Jaroniec M. An isotherm equation for adsorption on fractal surfaces of heterogeneous porous materials. Langmuir 1989;5(6):1431–3. [84] Cai Y, Liu D, Yao Y, Li J, Qiu Y. Geological controls on prediction of coalbed methane of No. 3 coal seam in Southern Qinshui Basin, North China. Int J Coal Geol 2011;88(2–3):101–12. [85] Brunauer S, Emmett P, Teller E. Absorption of gases in multimolecular layers. J Am Chem Soc 1938;60:309–19. Find this article online. [86] Qi H, Ma J, Wong P-Z. Adsorption isotherms of fractal surfaces. Colloids Surf, A 2002;206(1):401–7. [87] Liu X, Nie B. Fractal characteristics of coal samples utilizing image analysis and gas adsorption. Fuel 2016;182:314–22. [88] Javadpour F, Fisher D, Unsworth M. Nanoscale gas flow in shale gas sediments. J Can Petrol Technol 2007. [89] Wheatcraft SW, Tyler SW. An explanation of scale-dependent dispersivity in heterogeneous aquifers using concepts of fractal geometry. Water Resour Res 1988;24(4):566–78. [90] Coppens M-O. The effect of fractal surface roughness on diffusion and reaction in porous catalysts–from fundamentals to practical applications. Catal Today 1999;53(2):225–43. [91] Avnir D, Farin D, Pfeifer P. Molecular fractal surfaces. Nature 1984;308:261. [92] Coppens M-O, Froment GF. Diffusion and reaction in a fractal catalyst pore—I Geometrical aspects. Chem Eng Sci 1995;50(6):1013–26. [93] Chapman S, Cowling TG, Burnett D. The mathematical theory of non-uniform gases: an account of the kinetic theory of viscosity, thermal conduction and diffusion in gases. Cambridge University Press; 1990. [94] Meyer OE. Die kinetische Theorie der Gase. Maruschke & Berendt; 1899. [95] Jeans JH. The dynamical theory of gases. University Press; 1921. [96] Sd S, DFA L. Diffusion of ideal gases in capillaries and porous solids. AIChE J 1962;8(1):113–7. [97] Pollard WG, Present RD. On gaseous self-diffusion in long capillary tubes. Phys Rev 1948;73(7):762–74. [98] Bosanquet C. British TA Report BR-507. 1944. [99] Mistler TE, Correll GR, Mingle JO. Gaseous diffusion in the transition pressure range. AIChE J 1970;16(1):32–7. [100] Evans III R, Truitt J, Watson G. Superposition of forced and diffusive flow in a large-pore graphite. Tenn.: Oak Ridge National Lab.; 1961. [101] Wang Y, Liu S, Zhao Y. Modeling of permeability for ultra-tight coal and shale matrix: a multi-mechanistic flow approach. Fuel 2018;232:60–70. [102] Yang Y, Liu S, Zhao W, Wang L. Intrinsic relationship between Langmuir sorption volume and pressure for coal: experimental and thermodynamic modeling study. Fuel 2019;241:105–17.
606
Contents lists available at ScienceDirect
Fuel journal homepage: www.elsevier.com/locate/fuel
Full Length Article
Estimation and modeling of pressure-dependent gas diffusion coefficient for coal: A fractal theory-based approach
T
Yun Yang, Shimin Liu
⁎
Department of Energy and Mineral Engineering, G3 Center and EMS Energy Institute, The Pennsylvania State University, University Park, PA 16802, USA
ARTICLE INFO
ABSTRACT
Keywords: Diffusion coefficient Knudsen and bulk diffusion Fractal theory Unipore model
Diffusion coefficient is one of the key parameters determining the coalbed methane (CBM) reservoir economic viability for exploitation. Diffusion coefficient of coal matrix controls the long-term late production performance for CBM wells as it determines the gas transport effectiveness from matrix to fracture/cleat system. Pore structure directly relates to the gas adsorption and diffusion behaviors, where micropore provides the most abundant adsorption sites and meso- and macro-pore serve as gas diffusive pathway for gas transport. Gas diffusion in coal matrix is usually affected by both Knudsen diffusion and bulk diffusion. A theoretical porestructure-based model was proposed to estimate the pressure-dependent diffusion coefficient for fractal porous coals. The proposed model dynamically integrates Knudsen and bulk diffusion influxes to define the overall gas transport process. Uniquely, the tortuosity factor derived from the fractal pore model allowed to quantitatively take the pore morphological complexity to define the diffusion for different coals. Both experimental and modeled results suggested that Knudsen diffusion dominated the gas influx at low pressure range (< 2.5 MPa) and bulk diffusion dominated at high pressure range (> 6 MPa). For intermediate pressure ranges (2.5–6 MPa), combined diffusion should be considered as a weighted sum of Knudsen and bulk diffusion, and the weighing factors directly depended on the Knudsen number. The proposed model was validated against experimental data, where the developed automated computer program based on the Unipore model can automatically and time-effectively estimate the diffusion coefficients with regressing to the pressuretime experimental data. The theoretical determined diffusion coefficients well predicted the pressure-dependent variations of the experimental measured diffusion coefficient for different coals. This theoretical model is the first-of-its-kind to link the realistic complex pore structure into diffusion coefficient based on the fractal theory. The experimental results and proposed model can be coupled into the commercially available simulator to predict the long-term CBM well production profiles.
1. Introduction Diffusion is the process that matter (gases, liquids, and solids) tends to migrate and eliminate the spatial difference in composition in such a way to approach a uniform equilibrium state with maximum entropy [1–3]. The study of diffusion in nanoporous solids came to prominence as such materials have sufficient surface area required for high capacity and activity with extensive application in the petroleum and chemical process industries [4]. For transport through the pores with size comparable to diffusing gas molecules, diffusion effects may even dominate the overall transport rate [5]. A comprehensive understanding of the complex diffusional behavior lies the foundation for the technological development of porous materials in adsorption and catalytic processes [6]. As a natural polymer-like porous material, coal behaves like man-
⁎
made nanoporous materials for its exceptional sorption capacity contributed by nano- to micro-scale pores [7–9]. Dual porosity model proposed by Warren and Root [10] well represents the broad size distribution of coal pores, where macro- (> 50 nm) and mesopores (2–50 nm) most likely provide transport pathway, and micropores (< 2 nm) provide the greatest internal surface area and gas storage capacity [8,11–13]. The International Union of Pure and Applied Chemistry (IUPAC) [14] classification of pores is closely related to the different types of forces controlling the overall adsorption behavior in the different sized pore spaces. Surface force dominates the adsorption mechanism in micropores, and even at the center of the pore, the adsorbed molecules cannot break from the force field of the pore surfaces. For larger pores, capillary force becomes important [4]. Different diffusion mechanisms occur in different sized pores governing the overall
Corresponding author. E-mail address: [email protected] (S. Liu).
https://doi.org/10.1016/j.fuel.2019.05.009 Received 22 March 2019; Received in revised form 27 April 2019; Accepted 2 May 2019 Available online 16 May 2019 0016-2361/ © 2019 Elsevier Ltd. All rights reserved.
Fuel 253 (2019) 588–606
Y. Yang and S. Liu
gas mass influx through coal matrix [8,15–17]. Gas transport within coal can occur via diffusion through either pore volumes and/or along pore surface. Surface diffusion describes the movement of adsorbed molecules along the pore surface and is an activated process stimulated by diffusional activation energy [18]. At temperatures significantly higher than the normal boiling point of sorbate, diffusion mainly happens in pore volume, where the diffusional activation energy is negligible compared with the heat of adsorption forming localized layers of adsorbed molecules [4,19–21]. As a result, it is reasonable to neglect surface diffusion flux for CBM production until extremely low pressure is reached [22]. In this study, we limited our discussion to diffusion occurring in the center region of the pore. In other words, the pore volume diffusions, both Knudsen and bulk diffusions, will be considered and discussed in this work. Two forms of diffusion modes are generally considered in diffusion in pore volume, which are bulk and Knudsen diffusions [23–25]. The relative importance of the two diffusion modes depends on Knudsen number (Kn), which is the ratio of the mean free path (λ) to pore diameter (d ) for porous rocks [26,27]. Two extreme scenarios are given in the discussion of the prevalence of the two diffusion mechanisms [4,20]. For nanopores with d , the frequency of molecule-wall collisions far exceeds the intermolecular collisions resulting in the dominance of Knudsen diffusion. In the reverse case (i.e. d ), the contribution from molecule-wall collisions fades relative to the intermolecular collisions and the diffusivity approaches the molecular diffusivity. As a rule of thumb, molecular diffusion prevails when the pore diameter is greater than ten times the mean free path; Knudsen diffusion may be assumed when the mean free path is greater than ten times the pore diameter [28,29]. In the intermediate regime, both the Knudsen and molecular diffusivities contribute to the effective diffusivity. Most real cases of diffusion in CBM are intermediate between these two limiting cases [30]. The mean free path of gas molecules is a function of pressure [31], and as a result, a transition of flow regime from Knudsen diffusion to molecular diffusion will occur as pressure evolves. Diffusion coefficient (D ) governs the rate of diffusion, and in CBM, it can be determined from desorption time [32,33]. A significant amount work [17,22,34–48] has reported the diffusion coefficient (D ) of methane in coal at different pressures as summarized in Table 1. The measured diffusion coefficient of methane ranges from 10 11 to 10 15 m2 /s (Table 1). In this study, we are particularly interested in the influence of pressure as it determines the mean free path and the dominant diffusion regime.
Due to the complex pore morphology of coal, gas transport in coal is closely related to the coal pore structure [49]. To our best knowledge, limited efforts have been devoted to study the quantitative inter-relationship between pore structure and gas transport in coal. Yao et al. [50] observed a strong negative correlation between the permeability and heterogeneity, that is quantitatively obtained from the fractal dimension for high-rank coals whereas a slightly negative relationship was found for low-rank coals. However, the work does not provide detailed quantitative analyses to define the fundamental mechanism for the experimental observations. A study conducted by Li et al. [51] found that coals with higher fractal dimensions have smaller gas permeability because of complex pore shape for tectonically deformed coals. During a tectonic event such as deformation, open pores or semiopen pores may develop into ink-bottle-shaped pores or narrow slit pores. These pore morphological modifications result in a loss of pore inter-connectivity and a more heterogenous pore structure (i.e. high fractal dimension). In these previous studies [50,51], permeability was used to characterize gas flow deliverability of coalbed. In terms of coal matrix gas transmissivity, diffusivity rather than permeability should be examined to delineate gas transport in micro-scale pore structure [5]. As shown in Table 1, numerous experimental works have investigated on gas diffusion characteristics of coal. Specifically, moisture content [52], coal types [53,54], coal rank [55], sample size [56,57] were the key parameters that influence the gas diffusion behavior through micropore structures. But all these studies required the measurement of sorption rate to estimate diffusion coefficients, where they ignored the pore-structure determined diffusion mechanism. As a result, quantitative linkages between coal properties and gas diffusion coefficient are still lack in the literature, which is an important aspect in predicting the gas transport behavior in CBM reservoirs. Pore structural characteristics such as pore volume and its distribution directly depends on coal composition [58]. Therefore, this current study used pore structural parameters to define coal diffusion behavior and uncovered the relationship between the micropore structure and gas diffusion characteristics. In this current study, four coal samples were collected from four different coal mines in China. Low-pressure N2 gas adsorption and desorption data were analyzed through fractal analysis to characterize the pore structure of these four coal samples as many previous literatures [59,60]. High-pressure methane sorption data were used to obtain the gas kinetic parameters by using the unipore model. The pore
Table 1 Sorption kinetic experiments of methane performed in the various literature [17,22,34–48]. HVB and LVB are high and low volatile bituminous coals. Sub is subbituminous coals. Diffusion coefficient is derived from unipore model. List of Works
Year
Location
Rank
Nandi and Walker [43]
1975
U.S. coals
Anthracite to HVB
Smith and Williams [47] Mavor et al. [42]
Marecka and Mianowski [41] Clarkson and Bustin [37] Busch et al. [35]
Cui et al. [38] (further reworked by [22]) Kumar [40] Pone et al. [45]
Charrière et al. [36]
Pillalamarry et al. [22]
Salmachi and Haghighi [46] Bhowmik and Dutta [34] Zhao et al. [48]
Wang and Liu [17] Naveen et al. [44]
1984 1990 1998
1999
2004
2004
2007 2009
2010 2011
2012
2013
2014 2016 2017
Fruitland, San Juan Basin
Sub
Fruitland, San Juan Basin
Sub to LVB
Unknown
Lower Cretaceous Gates Formation, Canada
Silesian Basin of Poland Unknown
Illinois Basin
Western Kentucky Coalfield Lorraine Basin, France
Semi-anthracite Bituminous
HVB
Australian coal seam
Raniganj Coalfield, Jharia Coalfield, Gondwana Basin of India Shanxi, China San Juan Basin and Pittsburgh
Jharia Coalfield, Gondwana Basin of India
589
Pressure, MPa
Range of D , m2/s
0.315
1.14
10
13
10
14
1.9
5.7 0.1
10
13
10
14
10
13
10
10
10
15
10
11
10
13
0.25
2.52 13.6
1.25, 0.62, 0.2, 0.032
0–0.1
0.21
0.9
1.1
3
3.38
10
11
HVB
0.25
0.54
7.82
10
13
10
14
Bituminous
4.76
10 10
13
10
15
10
13
10
13
10
14
10
12
4.61
10
12
10
13
4.56
0.125
0.30
Bituminous
0.25
3.1 0.1
Bituminous
0.143
HVB
Illinois Basin
Avg Particle size, mm
0.48
5.3
11
0.294
0 7 0.014
Sub to HVB
0.1245
0.36
Bituminous
0.225
1.05
10
11
10
12
0.5
0
9
10
13
10
14
0.23
0
7
10
13
HVB
Bituminous HVB
4.678
Fuel 253 (2019) 588–606
Y. Yang and S. Liu
structure-gas diffusion model is proposed to relate pore fractal dimensions to gas diffusion coefficient, where the multiscale gas tsnsport CBM is modeled by superposing different flow regimes. Wu and Chen [61] summarized relevant studies that combines different transport mechanisms based on weighing coefficients [61–71]. However, no validation was provided in their work as it cannot satisfy the ideal conditions made in the theoretical models [69]. This work developed a unified model that utilized the easily accessible pore structural data to estimate gas diffusion coefficient at in-situ condition. The proposed model is an intuitive and mechanism-based model to define the gas diffusion behavior in coal and it can serve as a bridge from pore-scale structure of mass transport for the CBM gas production prediction.
coefficient and permeability of coal [40,72]. At the late stage of a CBM production well (i.e. mature wells), coal permeability might not be the bottle-neck for the overall gas production as commonly believed, and instead, diffusion process dominates overall well production performance since the matrix to cleat influx is limited [17]. Fick’s second law of diffusion for spherically symmetric flow [2] is widely applied to describe gas diffusion process across pore space, where a diffusion coefficient (D ) governs the rate of diffusion. The adopted system of measurement in this work is all in international system of units (SI). Mathematically, the diffusion can be described as:
2. Theoretical background and gas transport modeling
where, r is the radius of the pore, C is the adsorbate concentration, and t is the diffusion time. ‘Unipore’ and ‘bidisperse pore’ models are two widely adapted solutions to the above partial differential equation (PDE) to quantify the diffusive flow [30,43]. As the name suggests, the unipore model assumes a unimodal pore size distribution while the bidisperse model considers a bimodal pore size distribution. The bidisperse model can provide a better modeling result to the entire sorption rate curve than the unipore model for most of the coals [38,47]. Different from unipore model, the bidisperse model separates the macropore diffusivity from the micropore diffusivity, and a ratio of micropore/macropore relative contribution to overall gas mass transfer has been quantified in the model. The bidisperse model is a more robust model than the unipore model because it reflects the heterogeneous nature of the coal pore structure. Nevertheless, the bidisperse model requires to regress
D C C r2 = r2 r r t
2.1. Multiscale and multistage gas transport process for CBM Methane migration within a CBM reservoir starts with gas desorption from the internal pore surfaces of coal followed by the diffusion through the matrix defined by diffusion coefficient under various gas concentration gradients [72,73]. Followed with diffusion process, the gas reaches the naturally occurring fractures (cleats) and the flow regime evolves to pressure gradient driven Darcian flow described by the coal permeability as illustrated in Fig. 1. The rate of viscous Darcian flow through the cleat network depends on the distribution of cleat presented in coalbed. The rate of gas diffusion depends on the pore properties of the coal matrix. Production of gas from a CBM reservoir is intuitively affected by both diffusion
Fig. 1. Illustration of multi-scale and multimechanism gas flows in a CBM reservoir. CBM production data source: DringInfo.inc.
590
(1)
Fuel 253 (2019) 588–606
Y. Yang and S. Liu
multiple modeling parameters (i.e. micropore diffusivity, macropore diffusivity and volume ratio of micropore to macropore) to the experimental data and it may potentially encounter non-uniqueness solution sets [37]. Besides, the bidisperse model is not applicable at high pressure stage for its assumption on the linear isotherm, where the methane adsorption is known to be non-linear [37]. It also assumes an independent process of rapid macropore diffusion and slow micropore diffusion, which cannot be always true [74]. The unipore model is simple and has been successfully used to coal kinetic analysis of CH 4 sorption in several previous studies as summarized in Table 1. In this study, the unipore model was chosen to analyze the sorption data with two reasons: (1) unipore model gives reasonable accuracy over the whole range of coal desorption [75] and (2) unipore model is the model adapted by commercial production simulators [22]. In unipore model [76], constant gas surface concentration is assumed at the external surface, and the corresponding boundary condition can be expressed as:
where,
( ) Mt M
De = D / re2
6 2
n=1
1 exp ( De n2 2t ) n2
Mt M
exp
and
( ) Mt M
2
model
model
(5)
are experimentally measured and analy-
exp
obtained experimentally as described in Section 3.3. Then the
user specifies a search window of the apparent diffusion coefficient as upper (Dhigh ) and lower (Dlow ) limits for the targeted value. Dhigh and Dlow should be a reasonable range of typical values of diffusion coefficient. Based on the reported data as shown in Table 1, the measured diffusion coefficients (D) of various coal samples range from 10 15 to 10 10 m2/s . As suggested by Mavor et al. [42], 60 mesh is the recommended particle size for isotherm measurement. Therefore, the recommend settings for effective diffusivity Dhigh and Dlow are estimated to be between 1e-3 and 1e-8 1/s . The last required input is the number of terms in the infinite summation term (nmax ) of the unipore model (Eq. (3)) to fit the experimental data. A good entry of nmax is 50 to truncate the infinite summation term and the rest terms with large n are negligible. Following the Golden Section Search Algorithm, the diffusion coefficient is determined at the best fit that minimizes the difference between experimental and analytical sorption rate data modeled by unipore model. The flowchart (Fig. 3) shows the algorithm of the automated computer program.
where, C0 is the concentration at the external surface of the pore. In the sorption experiment, this is known to be valid since the coal particles will have a constant pressure at the surface of the particle throughout the experimental procedure. With assumption on uniform pore size distribution, the unipore model is given by
Mt =1 M
( )
exp
Mt M
tically determined sorption fraction. In this computer program, the primary input is the experimental sorption rate data stored in” diffusion.txt” composed of two columns of experimental data. The first column of entry is the sorption time, and the second column is the corresponding sorption fraction
(2)
C (r , t > 0) = C0
Mt M
S=
(3) (4)
where, re is the effective diffusive path, Mt is the sorption fraction, and M De is apparent diffusivity. In order to automatically and time-effectively analyze the sorptiondiffusion data, we develop a matlab-based computer program (Fig. 2) in this study based on a least-squares criterion to regress the experimental gas sorption kinetic data and determine the corresponding diffusion coefficient. An automated computer code was programmed to estimate the apparent diffusivity and the program is listed in the appendix. The apparent diffusivity (D /re2 ) was adjusted using the Golden Section Search algorithm [77] until the global minimum of the objective function was reached. The least-squares function (S ) was chosen to be the objective function and described as:
Fig. 2. User interface of unipore model based effective diffusion coefficient estimation in MATLAB GUI.
Fig. 3. Flowchart of the automated computer program for effective diffusion coefficient estimation in MATLAB GUI. 591
Fuel 253 (2019) 588–606
Y. Yang and S. Liu
2.2. Coal pore fractal analysis The concept of fractal geometry described in Mandelbrot [78] is a powerful tool to characterize the pore structure of coal as traditional Euclidean geometric methods fail to define the micro-structure of coal pores [56]. It provides a quantification approach to define the heterogeneity of pore surface and volume of coal and fractal dimension (Df ) herein is derived and the value is found to be between 2 and 3 for different sized pores [79]. A more heterogenous structure is typically accompanied by a rougher surface and Df approaches 3. Although recent progress makes imaging analysis become a new and convenient method to determine the microstructure of coal [80], they can only provide localized interpret image of a sample and induce great artifacts and uncertainty [81,82]. On the other hand, fractal analysis provides a simple and effective approach to study the tortuous gas diffusive path and it well serves the purpose in the development of a practical and quantitative diffusion model. The fractal dimensions have physical meaning and they can serve as the effective and equivalent pore structure based on the direction experimental data. In this work, Df of the four coal samples are determined by the fractal Frenkel–Halsey–Hill (FHH) model [83–85] using N2 sorption data, which is given by:
ln
V P = Aln ln 0 V0 P
+B
Fig. 4. Fractal pore model. L is the axial length of the pore segment corresponding to the actual travel distance of diffusive path and r is the end-to-end distance (Modified from Coppens [90]).
by
c=
(7)
and
D f = 3A + 3
DK , pm =
(8)
( ) V V0
( ( ) ) corresponding to high- and low-pressure adsorption. The
vs. ln ln
P0 P
slope of the linear parts is used to estimate the fractal dimension of the coal samples. 2.3. Diffusion coefficient in coal
2.3.1. Knudsen diffusion coefficient Knudsen diffusion [4,5] is the dominant diffusion regime when the mean free path is about or even greater than the equivalent effective pore diameter at which the pore wall-molecular collisions outnumber molecular-molecular collisions. For the gas transport in coal, Knudsen diffusion dominates the overall mass transport in small pores or under low pressure. A critical point about Knudsen diffusion is that when a molecule hits and exchanges energy with the pore wall, the velocity of molecule leaving the surface is independent of the velocity of molecule hitting the surface, and the reflecting direction is arbitrary. As a result, Knudsen diffusivity (Dk) is only a function of pore size and mean molecular velocity and can be expressed as [26]:
DK =
1 dc 3
(10)
DK
(11)
where, is the porosity, and is the tortuosity factor. Eq. (11) relates the diffusivity in a porous medium to the diffusivity in a straight cylindrical pore with a diameter equal to the mean pore diameter under comparable physical condition by a simple tortuosity parameter ( ). considers the combined effects of increased diffusive path length, the effect of connectivity and variation of pore diameter. However, the definition of the tortuosity factor is not universally accepted [89]. Instead of using simple bodies from Euclidean geometry, Coppens [90] successfully applied fractal geometry to describe the convoluted pore structure of amorphous porous catalysts and quantified the effect of the fractal surface morphology on diffusion. In this current study, we would use the fractal pore model proposed by Wheatcraft and Tyler [89] to determine the tortuosity of the diffusive path of the pore within coal matrix. A schematic of the fractal pore model is shown in Fig. 4. The key concept behind this model is that the tortuosity is induced by the surface roughness. This model provides a practical and explicit approach to quantify tortuosity by relating it to the surface fractal dimension as developed below. This model depicted in Fig. 4 considers a line having a true length, F, and fractal dimension, Df , which is an intensive property and independent of the size of the measuring yardstick molecules ( ). The expression of F is given by [91],
Two equations (Eqs. (7) and (8)) have been used to estimate Df from the slope ( A ), whereas, Eq. (8) would consistently generate an unreasonably high value of fractal dimension [60]. In this study, we would use Eq. (7) to obtain the fractal dimension of the tested coal samples.
Typically, two linear parts were observed in the log-log plot of ln
8RT / M
where, R is the universal gas constant, T is the ttemperature, and M is the gas molar mass. The Knudsen diffusivity (Dk ) for porous media have been proposed and applied to numerous pervious works [4,88], where the porous media is assumed to consist of open pores (i.e. porosity) of the mean pore diameter and have a degree of interconnection resulting in a tortuous diffusive path longer than an end to end distance (i.e. tortuosity). The Knudsen diffusion coefficient in porous and rough media is derived as:
(6)
where, V / V0 is the relative adsorption at the equilibrium pressure P , V0 is a monolayer adsorption volume, P0 is gas saturation pressure, B is the yintercept in the log-log plot, and A is the power-law exponent used to determine the fractal dimension of the coal surface (D ) [86]. Two distinct formulas were proposed to relate A to Df by [87]:
Df = A + 3
Assuming a Maxwell-Boltzmann distribution of velocity, c is given
F=N
Df
= constant
(12)
where, N is the number of yardsticks required to pave completely the line and varies with . The number of yardsticks (N ) multiplied by the size of a yardstick ( ) is an approximate or measured length (L ( )) of the line and can be expressed as:
(9)
where, d is the pore diameter, and c is the average molecular velocity determined from gas kinetic theory.
592
Fuel 253 (2019) 588–606
Y. Yang and S. Liu
(13)
L( ) = N
coal as only one species of gas is involved. Considering Meyer’s theory [94], the bulk or self-diffusion coefficient (DB ) was derived neglecting the difference in size and weight of the diffusing molecules as [23,95]:
Combining Eqs. (12) and (13), the measured length (L ( )) is related to the fractal dimension as: (14)
1 Df
L( ) = F
DB =
The characteristic length (Ls ) is defined as the length of the line segment holding a constant Df . If = Ls , then N = 1 and the expression of F can be written as: Df
Then Ls is determined as: Df 1 D f
(16)
L ( ) = Ls
At Df = 1, Ls is the end-to-end distance ( r ). For practical application, the axial length of the pore segment ( L ) was approximated by L ( ) [23]. The tortuosity factor ( ), the ratio of the measured length to the endto-end distance, is then determined to be D
=
L Ls f 1 = r Ls
Df
=
1 1 1 = + Dp DK DB
(17)
where, Df is the fractal dimension of a line with a value between 1 and 2. The fractal dimension derived from the Nitrogen sorption data is the surface fractal dimension with a value ranging from 2 to 3 [83]. Taking this into account, the expression of can be updated to:
=
1 1 = wK + (1 Dp DK , pm
2 Df
(18)
Ls
Ls
2 Df
DK =
2 2 dR0.5T 0.5 Ls 3 0.5M 0.5
2 Df
(19)
where, DK is the Knudsen diffusion coefficient in a smooth cylindrical pore [92]. Eq. (19) has the same formula to Eq. (7) in Coppens [90] except porosity was introduced to consider that mass transport only happens in pore space, not through the solid matrix. Ls is the outer cutoff of the fractal scaling regime, i.e. the size of the largest fjords [90]. In this current study, as the structural parameters were obtained from low pressure nitrogen sorption data, Ls was treated as the largest cutoff of the pore size (i.e. maximum pore diameter) in the pore size distribution (PSD). The other parameter, is the molecular diameter of adsorbed molecules. At reservoir condition, methane diffusion in free phase and pore volume dominates the overall mass transport process [4,19–21] and as a result, was estimated to be the mean free path of transport gas molecules as the distance between successive collisions and the effective diffusive diameter of the gas molecules. The mean free path ( ) for real gas given in Chapman et al. [93] is determined as:
=
5µ 8P
RT 2M
wK )
1 DB
(23)
The relative contribution of individual diffusion regime is dependent on the Knudsen number (Kn), which is the ratio of pore diameter to mean free path. It is critical to identify the lower and upper limits of Kn where pure Knudsen and bulk diffusion can be reasonably assumed. Commonly, when Kn is smaller than 0.1, the diffusion regime can be considered as pure bulk diffusion [14,29]. Then wK is written in a piecewise function, f (Kn) and takes the form as:
Eq. (18) provides an intuitive estimation of the tortuosity factor through the correlation with surface fractal dimension. Combing Eqs. (9), (11) and (18), the Knudsen diffusion coefficient of porous media (DK , pm ) is then found as:
DK , pm =
(22)
Eq. (22) stated that the resistance to transport the diffusing species the is a sum of resistance generated by wall collisions and by intermolecular collisions [97–99]. One main implicit assumption behind this reciprocal addictive relationship is that Knudsen diffusion and bulk diffusion acts independently on the overall diffusion process. In reality, the probabilities between gas molecules colliding with each other and colliding with pore wall should be considered [70,100]. Then a dimensionless weighing factor (wK ) was introduced to consider the relative importance of the two diffusion mechanisms as referred to Wu et al. [70], Wang et al. [101].
1 Df
Ls
(21)
2.3.3. Relative contributions of bulk and Knudsen diffusions and related weighing factors When gas transport includes both aforementioned diffusion modes, the relative contribution on the overall gas influx should be quantified. For free gas phase, the combined transport diffusivity (Dp ) including the transfer of momentum between diffusing molecules and between molecules and the pore wall is given as [96]:
(15)
F = Ls
1 c 3
1 (Kn > Kn ), pure Knudsen diffusion wK = f (Kn) = (0, 1) (0.1 < Kn < Kn ), transition flow 0(Kn < 0.1), pure bulk diffusion
(24)
where, Kn is the critical Knudsen number of pure Knudsen diffusion. To estimate the contribution of each mechanism, one should examine the manner in which Dp 1 varies with pressure. From general kinetic theory [94], the bulk diffusion coefficient is inversely proportional to pressure whereas the Knudsen diffusion coefficient is independent of pressure. A diagnostic plot of Dp 1 obtained at a single temperature vs. various pressures (Fig. 5(a)) is useful to identify the diffusion mechanism as suggested by Evans III et al. [100]. A horizontal line corresponds to pure Knudsen flow, a straight line with a positive slope passing the origin represents pure bulk flow, and a straight line with an appreciable intercept depicts a combine mechanism as illustrated in Fig. 5(a). These interpretations are based on Eq. (22) rather than Eq. (23). In fact, the diagnostic plot simplifies the real case as it does not consider the dependence of DK , pm and wK at various pressures. The weighing factor is subject to Kn and pressure, and a straight line will not persist for a combined diffusion. Besides, the combined diffusion should be a weighted sum of pure bulk and Knudsen diffusion. The line of combined diffusion will lie between rather than above the pure bulk and Knudsen diffusion. On the other hand, Knudsen diffusion in porous media also depends on the tortuosity factor, which varies with pressure. As a result, a horizontal line will not present for pure Knudsen diffusion. It should be noted that DK , pm is not that sensitive to the change in pressure as DB , and a relative flat line may still occur at low pressure corresponding to pure Knudsen flow. But it needs to be further justified through our experimental data as the flat region is important to
(20)
where, µ is the viscosity of the transport molecules in kg m/s , P is the pressure. The factor, 5/8, considers the Maxwell-Boltzmann distribution of molecular velocity and correct the problem that exponent of temperature has a fixed value of 1/2 [31]. 2.3.2. Bulk diffusion coefficient Bulk diffusion is the dominant diffusion regime when the mean free path is far less than the pore diameter, which is usually found in large pores or for high pressure gas transport. Gas-gas collisions outnumber gas-pore wall collision. The present work focuses on gas self-diffusion in 593
Fuel 253 (2019) 588–606
Y. Yang and S. Liu
Fig. 5. Diagnostic plot of reciprocal diffusion coefficient (Dp 1) vs. P to determine the dominant diffusion regime. Plot (b) is updated from plot (a) by considering the weighing factor of individual diffusion mechanisms and Knudsen diffusion coefficient for porous media.
specify the critical Knudsen number (K n ) for pure Knudsen diffusion. Considering the effect of weighing factor and tortuosity factor on the overall diffusion process, the diagnostic plot is updated from Fig. 5(a) to (b).
earlier work Yang et al. [102]. The measurement of sorption isotherm has also been reported in Yang et al. [102]. The particle method was adopted to quantify the diffusive flow for pulverized coal samples. Numerous studies [17,22] have used this technique to characterize the gas diffusion behavior of coal. This method requires pulverizing the coal to powders and ensures gas transport is purely driven by diffusion. However, grinding the coal increases the surface area for gas adsorption. The change is considered to be minimal as the increase for 40–100 mesh coal size ranges from 0.1 to 0.3% [22,110]. Other studies [38,75,111] suggested that grain size has no effect on diffusion when the particle radius is greater 0.1 mm. The coal particle in this study has a size of 60–80 mesh (0.177 mm–0.250 mm). Therefore, pulverization of the coal samples is not believed to affect the accuracy of diffusion measurements and still meets the purpose of this experiment to reduce the diffusion time and ensure diffusion-driven in nature. In the adsorption experiment, the pressure in the cell section was continuously monitoring through the data acquisition system (DAS). After each dose of methane, the pressure in the reference cell was higher than in the sample cell. When they were connected, a step increase in pressure occurred following by a gradual decrease in pressure until equilibrium was reached. The decrease in pressure was generated by the adsorption of methane occurring at the pore surface of coal matrix, and was measured very precisely. Constant pressure boundary condition was controlled by isolating the cell section from the gas supply system. This ensures a direct application of the diffusion models and the simplest solution of diffusion coefficient (D ) is given when the constant concentration is maintained at the external surface [52]. The real-time pressure data were used to calculate the sorption fraction versus time data, which is a required input of the unipore model.
3. Experimental method and procedures 3.1. Coal sample procurement and preparation Four coal samples named as Xiuwu-21, Luling-9, Luling-10, Sijiazhuang-15 were collected from three different coal mines in China. The geological information of these four coal samples can be found in elsewhere [102]. The coal samples were grounded and sieved to the desired sample size for the proximate analysis (10 g of the sample; 70–200 mesh), N2 adsorption–desorption test (1 g; 60–80 mesh) and methane sorption/diffusion testing (40 g; 40–60 mesh). Moisture and ash analysis of the four coal samples was conducted following the standard ISO 17246:2010 (Coal, Proximate analysis) [103] with a 5EMAG6600 proximate analyzer. 3.2. N2 adsorption/desorption test Nitrogen adsorption/desorption test is a common technique to obtain the pore size distribution of porous materials [104,105], where N2 serves as the adsorbate and porous media is the adsorbent. The ASAP 2020 apparatus at Material Research Institute, Penn State University was used to conduct the N2 sorption test following the ISO 159013:2007 (Pore size distribution and porosity of solid materials by mercury porosimetry and gas adsorption, Part 3: Analysis of micropores by gas adsorption) [106]. Each coal sample weighing 1 g was initially loaded into a sample tube, and automatically degassed by heating at ∼110 °C in a vacuum to pore structure analysis [107,108]. The degas procedure removes the adsorbed moisture and volatile matter of the coal samples. The coal samples were kept at a constant temperature of liquid N2 (77 K) and the adsorption isotherms were then obtained for the relative pressure (P /P0 ) ranging from 0.009 to 0.994. The instruments software also can determine surface areas, pore volumes and pore size distributions (PSD) based on the Brunauer-Emmett-Teller (BET) model and density functional theory (DFT) model [109].
At the ith pressure stage, the sorption fraction
( ) was gradually Mt M
increasing with time corresponding to a gradual decrease in pressure. The sorption rate data was calculated from the pressure-time data (Psi (t ) ), injection pressure (Pri ), equilibrium pressure in the previous pressure stage (Psei 1), and saturated or maximum amount of adsorbed i gas molecules in the current pressure stage (nsat ).
Psei 1 (Vs Vsam ) P i Vr Mt 1 = i + r M (Z ) Psei 1 (Z ) Pri nsat RT
Psi (t )(Vr + Vs (Z ) Psi
Vsam ) (25)
3.3. Diffusion coefficient estimation
where Mt is the adsorbed amount of the diffusing gas in time t and M is the adsorbed amount in infinite time. T is the temperature of the adsorption system. R is the universal gas constant. Vs , Vr , Vsam are the volume of sample cell, reference cell, and coal sample, respectively. Z
The sorption capacity and diffusion coefficient were measured simultaneously using the same experimental apparatus shown in the 594
Fuel 253 (2019) 588–606
Y. Yang and S. Liu
with appropriate subscript is symbolled as the gas compressibility i factor at current pressure. nsat is a maximum adsorbed amount at the ith pressure stage and directly obtainable from the adsorption isotherm as the step change in cumulative adsorption amount of the two neighboring equilibrium points. The experimentally measured value of Mt M was then fitted by the analytical solution of unipore model [42] to determine the diffusion coefficient of the coal samples at the best match.
on the low-pressure N2 adsorption data for Luling-10. Two distinct linear segments were observed at the P /P0 interval of 0–0.5 and 0.5–1, which indicates different gas adsorption mechanisms at different pressure stage. The linear fittings of two segments exhibited convincible correlation with R2 greater than 0.95 for all four tested coal samples, which has been previously reported in our previous work [102]. Typically, Van der Waals force formed between gas molecules and coal surfaces is the dominant adsorption mechanism at low pressure within micropores. Capillary condensation becomes important as pressure increases and it commonly occurs in mesopores and macropores. An average fractal dimension (Df ) was estimated by linear fitting the sorption data from entire pressure interval, which describes the overall heterogeneity of the pore structure. This value provides a quantitative description of the tortuous diffusive path in the complex pore structure through the fractal pore model (Fig. 4). From Eq. (18), the tortuosity factor (τ) derived from the fractal pore model depends on the fractal dimension and a normalized parameter (i.e. / dmax ). Apparently, mean free path (λ) varies with pressure. In this study, the diffusion coefficients were measured at six different pressures, which are 0.55, 1.38, 2.48, 4.14, 6.07, and 8.07 MPa. Along with the pore structural parameters, the pressures were used to calculate the mean free path and corresponding tortuosity factors. The results were listed in Table 3. The average fractal dimension of the four coal samples ranges from 2.229 to 2.496. From fractal results, Luling-10 provides the most complex pore structure with the Df of 2.496. Combing with the pore structural information from PSD, we could see that Sijiazhuang-15 provides the most tortuous diffusive path with a highest value of τ for all pressures. As a result, the diffusion time in Sijaizhuang-15 is expected to be longest and this was confirmed by our experimental results.
4. Results and discussion 4.1. Quantitative analyses of pore structure Low-pressure N2 gas adsorption was known to be a time- and costeffective technique to characterize micropores in the porous media [112]. Typically, it was commonly used to evaluate the complexity and morphological characteristics of coal pores [113]. The results of the N2 adsorption/desorption isotherms at 77 K along with hysteresis pattern were presented in our previous work [102] and it was used to investigate pore structure characteristics of the coal. Following the classification of IUPAC to classify the shape of hysteresis loop N2 adsorption–desorption isotherm [114,115], the tested coal samples can be classified as Type H3 suggesting the prevalence of slit-shaped pores. In this section, pore size distribution (PSD) and fractal analysis were carefully analyzed to have a comprehensive understanding of how pore structural parameters affect the diffusion coefficient of coal. 4.1.1. Pore size distribution (PSD) In this study, we used the classical pore size model developed by Barret, Joyner, and Halenda (BJH) in 1951 [116] to obtain the pore size distribution of the coal samples. This model is adjusted for multi-layer adsorption and based on the Kelvin equation. The ready accessibility in commercial software makes the BJH model be extensively applied to determine the PSD of microporous material [117]. The desorption branch of the hysteresis loop considers the evaporation of condensed liquid [109] and thus, the shape of desorption branch was directly dependent on the PSD of adsorbent [118]. The bimodal nature of PSDs is apparent from the two peaks observed in most samples. The pore volume was primarily contributed by adsorption pores for all coal samples (i.e. pore diameter < 100 nm). According to the IUPAC classification, the pore volumes of different sized pores (micro-, meso- and macro-pores) were listed in Table 2. Meanwhile, it also reports the average pore diameter (d ), and lower and upper cutoff of pore diameter (dmin , dmax , respectively) for the studied four coal samples. Fig. 6 presents the PSDs of the four coal samples obtained from the BJH desorption branch.
4.2. Quantitative analyses of sorption kinetics Following the procedure depicted in the particle method [22], highpressure methane adsorption rate data were collected at six different pressure steps from initial pressure at 0.55 MPa up to the final pressure at 8.07 MPa. With eight transducers connecting to the data acquisition system, twenty-four sorption rate measurements were performed in this study. For each pressure, the apparent diffusion coefficient is assumed to be constant. As a result, the estimated diffusion coefficient is an average of the intrinsic diffusivity at a specific pressure interval. The stepwise adsorption pressure-time data were modeled by the unipore model described in Section 2 (Eq. (3)) and the pressure-dependence apparent diffusivity (D/ re2 ) was estimated by pressure and time regression using our proposed automate Matlab program. Fig. 8 shows two of the twenty-four rate measurements with modeled results based on the unipore model. These measurements were for Xiuwu-21 and Luling-10 at 0.55 MPa. It can be seen that the unipore model can accurately predict the trend of the sorption rate data with less than 1% percent error. Due to the assumption on uniform pore size distribution, the unipore model was found to be more applicable at high pressure steps [37,42,47]. The lowest pressure stage in this study was 0.55 MPa and
4.1.2. Fractal analysis In this study, the fractal dimension was obtained using the FHH model, and the detailed procedure can be found elsewhere [60].As shown in Fig. 7, log-log scale of ln
( ) vs. ln (ln ( ) ) was plotted based V V0
P0 P
Table 2 Average pore diameter, lower and upper cutoff of pore diameter, and pore volume of the coal samples estimated from the BJH model. Coal sample
Xiuwu-21 Luling-9 Luling-10 Sijiangzhuang-15
d (nm)
7.61 12.49 15.05 4.6
Pore Volume (cm3/100 g) Vtotal
Vmicro
Vmeso
Vmacro
1.178 0.395 0.393 2.772
0.0247 0.00330 0.00372 0.0537
0.703 0.172 0.149 0.456
0.451 0.220 0.240 2.262
dmin (nm)
dmax (nm)
1.741 1.880 1.870 1.565
83.759 115.440 112.430 132.447
d : average pore diameter; Vtotal: total pore volume; Vmicro: micropore volume; Vmeso: mesopore volume; Vmacro: macropore volume.
595
Fuel 253 (2019) 588–606
Y. Yang and S. Liu
Fig. 6. The pores size distribution of the selected coal samples calculated from the desorption branch of nitrogen isotherm by the BJH model.
the unipore model gave convincible accuracy to model the sorption rate data (Fig. 9). Thus, for higher pressure stage, the unipore model should still retain its legitimacy in this application. In this work, other measurements exhibited the same or even higher accuracy when applying
the unipore model although they had different length of adsorption equilibrium time. Fig. 9 shows the results of the estimated diffusion coefficients at different pressures for the four tested coal samples, where the effective
Fig. 7. Fractal analyses of N2 desorption isotherms for the studied four coal samples (Ref: [102]). 596
Fuel 253 (2019) 588–606
Y. Yang and S. Liu
Table 3 The fractal dimension, mean free path and tortuosity factor based on the fractal pore model and estimated at the specified pressure stage (i.e. 0.55, 1.38, 2.48, 4.14, 6.07, and 8.07 MPa). Coal sample
A
Df = A + 3
R2
P (MPa) Mean free path, λ (nm)
0.55 6.595
1.38 2.660
2.48 1.503
4.14 0.924
6.07 0.656
8.07 0.516
Xiuwu-21 Luling-9 Luling-10 Sijiangzhuang-15
−0.772 −0.505 −0.504 −0.537
2.229 2.495 2.496 2.463
0.967 0.989 0.975 0.932
Tortuosity factor, τ
1.787 4.128 4.078 5.606
2.199 6.472 6.395 9.444
2.506 8.587 8.486 13.111
2.800 10.924 10.798 17.336
3.029 12.948 12.800 21.114
3.199 14.576 14.409 24.223
diffusive path was estimated to be the radius of the particle [42]. The values of diffusion coefficient exhibited an overall negative trend when the gas pressure was above 2.48 MPa. The decreasing trend is consistent with the theoretical bulk diffusion coefficient in open space (Eq. (21)), which is dependent on the mean free path of the gas molecule and gas pressure. The diffusion coefficient became relatively small at pressures higher than 6 MPa when the coal matrix had high methane concentration and a low concentration gradient. The initial slight increasing trend were observed in the diffusion curves when the pressure was below 2.48 MPa. The same experimental trend was reported in Wang and Liu [17], and they explained that as the exerted gas pressure on the coal samples may open the previously closed pores and more gas pathways were created to enhance the diffusion flow. Besides, the relative contribution of Knudsen and bulk diffusions to the gas transport process changes at various gas pressures. Knudsen diffusion loses its importance in the overall diffusion process as gas pressure increases and molecular-molecular collisions are more frequent. At the same time, bulk diffusion becomes important at higher pressure and typically, it has the relatively fast diffusion rate than the Knudsen diffusion, which explains the increase in diffusion coefficient with the increase in pressure when pressure was less than 2.48 MPa. Overall, a decreasing trend was found between pressure and diffusion coefficient in this study. Some previously reported data [22,35,38] also confirmed the same decreasing trend, where the reduction in diffusivity is due to the dominance of bulk diffusion and potential greater swelling effect caused by gas adsorption at higher pressure. However, other studies [52,119,120] showed a reverse trend due to the associated increase in surface diffusion, which increases as surface coverage increases. The underlying fundamental mechanism will be further discussed in the subsequent section. The values of diffusivity range from 1.05 × 10 13 to
Fig. 9. Variation of the experimentally measured methane diffusion coefficients with pressure.
9.77 × 10 12 m2/s . At all pressure steps, Xiuwu-21 had the highest diffusivity and two Luling coals have low diffusivity because both Luling coals have high Df as reported in Table 3. 4.3. Effect of pore structure on the diffusion coefficient in coal As the diffusion process controls the gas influx from matrix towards the cleat/fracture system, it dominates the long-term well performance
Fig. 8. Sample experimental results of CH4 sorption rate at 0.55 MPa for Xiuwu-21 and Luling-10.
597
Fuel 253 (2019) 588–606
Y. Yang and S. Liu
Table 4 Theoretically calculated bulk diffusion coefficient (DB ) and Knudsen diffusion coefficent of porous media (DK , pm ). Pressure [MPa]
0.55
Theoretical Diffusion Coefficient
DB
DK,pm
DB
DK,pm
DB
DK,pm
DB
DK,pm
DB
DK,pm
DB
DK,pm
Xiuwu-21 Luling-9 Luling-10 Sijiazhuang-15
104.77 41.87 38.47 262.48
67.60 19.22 21.54 51.02
42.27 16.89 15.52 105.89
54.94 12.26 13.73 30.29
23.88 9.54 8.77 59.82
48.22 9.24 10.35 21.81
14.69 5.87 5.39 36.79
43.15 7.26 8.13 16.50
10.42 4.16 3.83 26.11
39.90 6.13 6.86 13.55
8.20 3.28 3.01 20.56
37.77 5.44 6.10 11.81
[× 1010 m2/s ]
1.38
2.48
4.14
6.07
8.07
of CBM after the fracture storage is depleted [17]. The estimation of diffusion coefficient based on pore structure is critical to determine the production potential of a given coal formation. Apparently, diffusion process is slower for coal pore in a smaller size or having a more complex structure. As mentioned above, the diffusive gas influx is controlled by combined Knudsen and bulk diffusions. The theoretical values of the diffusivity under these two diffusion modes was calculated based Eq. (19) and Eq. (21) and the results are listed in Table 4. It should be noted that the expression of DB given in Eq. (21) is derived for open space and independent of the solid structure. For porous media, a multiplication of porosity is added to the expression of DB that considers volume not occupied by the solid matrix [121–123]. The overall diffusion coefficient (Dp ) was then defined as a weighted sum of Knudsen diffusion and bulk diffusion given in Eq. (23). To estimate the weighing factor (wK ) of each mechanism, it is critical to determine the critical Knudsen number (Kn ) and for Kn > Kn , a pure Knudsen diffusion can be assumed. Examination of the manner in which Dp varies with pressure using the diagnostic plot (Fig. 5(b)) is intuitively helpful to identify the pressure interval for pure Knudsen flow. One challenging aspect of applying the diagnostic plot is the uncertainty about the sensitivity of DK , pm to the change in pressure. If DK , pm is not very sensitive to pressure, a small variation in pressure will not have an apparent change of Dp at low pressure stages and under pure Knudsen diffusion. Then a relative flat line can be found in a plot of Dp 1 vs. P at low pressure. It corresponds to a pressure interval of pure Knudsen flow, and the contribution from bulk diffusion is ignored as the intermolecular collision strongly correlated with pressure. Fig. 10 shows the change in DB and DK , pm with pressure for Sijiazhuang-15 sample. Fig. 11 demonstrates the application of using diagnostic plot to identify diffusion mechanism. In Fig. 11, bulk diffusion was subject to much greater variation than Knudsen diffusion over the pressure range of interest. Consequently, a relatively flat line was found at low pressure interval (P < P ) in the diagnostic plot (Fig. 11) for a pure Knudsen diffusion. Effective diffusion coefficient (Dp 1) is then equivalent to DK , pm and weighing factor (wK ) equals to one. The critical Knudsen number (Kn ) is determined at the inflection point, where P = P . As pressure increases, pore wall effect diminishes as mean free path of gas molecules shortens and bulk diffusion becomes important. Then at about 2.5 MPa, Dp 1 was subject to a greater variation in terms of pressure variation since DB is directly proportional to mean free path and inversely proportional to the pres-
Fig. 10. Variation of bulk diffusion coefficient (DB ) and Knudsen diffusion coefficient (DK , pm ) at different pressure stages for Sijiazhuang-15.
Fig. 11. Diagnostic plot of reciprocal diffusion coefficient (Dp 1) vs. P to specify pressure interval of pure Knudsen flow (P < P ) and critical Knudsen number (Kn =Kn(P ) )).
Fig. 12. Diffusion regimes at different pressure interval and range of Knudsen number.
598
Fuel 253 (2019) 588–606
Y. Yang and S. Liu
Fig. 13. A plot of wK as a piecewise function of Kn. The horizontal tails at the low and high interval of Kn correspond to pure bulk and Knudsen diffusion, respectively.
Fig. 15. Comparison between experimental and theoretical calculated diffusion coefficients of the studied four coal samples at same ambient pressure.
coefficient in coal matrix. For catalysis, Wheeler [124] proposed an empirical combination of Knudsen and bulk diffusion coefficient to determine the effective diffusion coefficient of combined diffusion as
Dp = DB (1
e
1/Kn)
(26)
In Eq. (26), Dp approaches to DB as Kn approaches to zero and mean free path is far less than the pore diameter. Dp approaches to DK as Kn approaches infinity since e 1/Kn 1 1/Kn . Correspondingly, the weighing factor of Knudsen diffusion (wK ) grows towards higher Kn . However, some built-in limitations are also observed for this theoretical formula. First, it fails to consider the change in the effective diffusive path at different pressures as DK , pm rather than DK should be involved to describe the diffusion rate under Knudsen regime. Besides, it underestimates wK as Eq. (26) implicitly states that pure Knudsen diffusion only occurs for flow with infinite value of Kn . In fact, Knudsen flow dominates the overall diffusion once Kn is reached as illustrated in Fig. 11. Instead, wK is assumed to have a linear dependence on Kn in the transition pressure range and for a combined diffusion. This assumption would be further justified by comparing with the experimental data. Fig. 13 is a plot of wK vs. Kn applied to quantify the relative contribution of each diffusion mechanism. Once the wK is given, the overall diffusion coefficient can be theoretically determined by Eq. (23). Experimentally measured diffusion coefficients for methane are presented in Fig. 10. The results were then compared with theoretical values predicted by the relationships proposed by Wheeler [124] and this study as given in Eq. (26) and Eq. (23), respectively. Fig. 14 indicates that the theory of wK developed in this study provided better fit to the experimental measured diffusion coefficient than the one proposed by Wheeler [124]. The improvement in the prediction of diffusivity was more obvious towards low pressure and Knudsen diffusion became predominant. This is because our method allows for the expected changes in the effective diffusion path. Nevertheless, great discrepancy was still found at low pressure stages compared with the experimental measured diffusion coefficients. The source of error originated from the accuracy in the estimation of pore
Fig. 14. Comparison between experimental and theoretical calculated diffusion coefficient for methane diffusion in Xiuwu-21. Wheeler [124] is described by Eq. (26), and this work is given by Eq. (23).
sure. The dividing pressure between pure Knudsen diffusion and combined diffusion for tested coal samples were all determined to be 2.5 MPa, i.e. P = 2.5MPa . For even higher pressure, the effect of pore wall-molecular collisions can be neglected and Dp 1 was estimated by DB 1. As a result, a linear trend was noted at pressure greater than 6 MPa when bulk diffusion dominates the overall diffusion and wK equals to zero. Using Fig. 11, we would be able to identify the dominant diffusion mechanism at different pressure stages and evaluate the relative contribution of each mechanism or wK as dictated by Eq. (24). Undoubtfully, wK equals to one for pure Knudsen diffusion and zero for pure bulk diffusion as shown in Fig. 12. In the transition regime, no theoretical development has been made on the prediction of diffusion
599
Fuel 253 (2019) 588–606
Y. Yang and S. Liu
structural parameters, which is critical in Knudsen diffusion when pore morphology is important. Besides, the scale of measured diffusion coefficient is three order of magnitudes smaller than the predicted one. This is caused by the presence of surface diffusion. Movement of gas molecules along the pore wall surface contributes significantly to the gas transport of adsorbed species in micropores, where gas molecules cannot escape from the potential field of pore surface [125,126]. The relative contribution of surface diffusion and diffusion in pore volume is related to the volume ratio of gas in adsorbed phase and free phase [4]. The primary purpose of this work is to predict diffusion behavior of coal based on pore structure. Surface diffusion as an activated diffusion is mainly a function of adsorbate properties, rather than adsorbent properties. To eliminate the variation in surface diffusion, we conducted the analysis under the same ambient pressure. In Fig. 15, the experimental measured diffusion coefficients are plotted against the theoretical values determined by Eq. (23) for the four coal samples at each pressure stages. The experimental diffusion coefficients were measured at six pressure stages ranging from 0.55 MPa to 8.07 MPa. Therefore, six isobaric lines are presented in Fig. 15 and each line is composed of 4 points corresponding to the four studied coal samples. The theoretical diffusion coefficient derived from Eq. (23) is a function of pore structural parameters. Overall, it provides good fits to the experimental diffusion coefficients. Due to the presence of surface diffusion, the scale of the theoretical values does not agree with it of the experimental values. But the linear relationships in Fig. 15 inherently illustrates that pore structure has negligible effect on the transport of gas molecules along the pore surface. Otherwise, the contribution from surface diffusion should vary for different coal samples, and the four points will not stay in the same line. There is a compelling mechanism that determines the steepness of the linear relationships. Generally, surface diffusion becomes predominant as surface coverage increases and multilayer of adsorption builds up at higher pressure stages. The slope is reduced towards high pressures due to an increase in the contribution from surface diffusion. On the contrary, as the pore surface is smoothed, and the effective diffusive path is shortened with a reduction in the induced tortuosity. This leads to a faster diffusion process with greater mass transport occurring in pore volume, and the lines are expected to be steeper as pressure increases. Under these mechanisms, the lines are steeper at lower pressure stages (P < 4 MPa ) in Fig. 15. For higher pressures, reverse trend can be found as the lines tend to be horizontal as pressure increases.
samples. Experimentally, adsorption and transport of methane in coal particles were analyzed with the particle method. By applying the unipore model, the diffusion coefficients were then estimated from the analytical modeling of experimental sorption pressure-time data. An automated computer program was proposed that can automatically and time-effectively estimate the diffusion coefficients with regressing to the pressure-time experimental results. A theoretical pore-structure-based model was proposed to estimate the pressure-dependent diffusion coefficient for fractal coals. The proposed model took the pore structure parameters, including, porosity, pore size distribution and fractal dimension, as inputs to predict the pressure dependent gas diffusion behavior. In the proposed model, we integrated Knudsen and bulk diffusion influxes to dynamically define the overall gas transport process. The proposed diffusion coefficient model was validated against the experimental measured diffusion coefficient results. Both experimental and modeled results suggested that Knudsen diffusion dominated the gas influx at low pressure range (< 2.5 MPa) and bulk diffusion dominated at high pressure range (> 6 MPa). For intermediate pressure ranges (2.5–6 MPa), combined diffusion should be considered as a weighted sum of Knudsen and bulk diffusion, and the weighing factor directly depended on the Knudsen number. Uniquely, we took the tortuosity as an input parameter for the theoretical diffusion model and this allows modelers to quantitatively take the pore morphological complexity to define the diffusion for different coals. Based on the fractal pore model, the distribution of pores at various size and their associated heterogeneity i.e., PSD and Df along with the mean free path of gas molecules (λ) are the essential parameters to determine the tortuosity factor. The determined tortuosity factors of the four coal samples ranged from 1.787 to 24.223 for the tested pressure interval between 0.55 MPa and 8.07 MPa. The results suggested that the increase in pressure and pore structural heterogeneity resulted in a longer effective diffusion path and a higher value of tortuosity factor affecting the Knudsen diffusion influx in porous media. The pore structural parameters lose their significance in controlling the overall mass transport process as bulk diffusion dominates. The overall diffusion coefficient was then evaluated as a weighted sum of Knudsen and bulk diffusion coefficient. At individual pressure stages from 0.55 MPa and 8.07 MPa, it provided good fits to the experimentally measured overall diffusion coefficient, which varied from 1.05 × 10 13 to 9.77 × 10 12m2 /s . This theoretical model is the first-of-its-kind to link the realistic complex pore structure into diffusion coefficient based on the fractal theory. Based on the proposed model, we propose a diagnostic tool that illustrates and examines the pressure-dependent variations of diffusion coefficient for different coals. The experimental results and proposed model can be coupled into the commercially available simulator to predict the long-term CBM well production profiles.
5. Summary and conclusions This works presents a series of experimental measurements and theoretical modeling on the diffusion coefficient for four different coal
Appendix. User interface in MATLAB GUI for the estimation of effective diffusivity The code below is named as “UniporeModel. Fig”. In the command window of MATLAB, type ‘open UniporeModel.fig’. A user interface should pop up as shown in Fig. A1.
600
Fuel 253 (2019) 588–606
Y. Yang and S. Liu
Fig. A1. User Interface (Unipore Model).
The required input is the experimental sorption rate data (i.e., Mt vs. t). The data should be stored in a txt file in the same directory as the M ‘UniporeModel.fig’ and named as ‘diffusion.txt’. The next entry is the search interval of Gold Section Search method for the apparent diffusivity (De ), which are marked as Dhigh and Dlow in the unit of s 1.The last required input is the number of terms in the infinite summation of unipore model denoted as nmax . In the infinite summation, the value of each individual term decreases as the index of the term increases. Thus, an entry of 50 for nmax is good enough to truncate the infinite summation. Once all the required inputs are entered in the program, hit the calculate button. Then the value of apparent diffusivity (De) will pop up along with the percentage error. The error of the fitting by unipore model is determined as the average sum of squared difference, which is the ratio of the result from least-square function (Eq. (5)) over the number of sorption rate datapoints. With the determined apparent diffusivity, the sorption rate data is fitted by the unipore model (Eq. (3)). A figure of the experimental sorption data with the regressed curve is shown at the bottom of the window. Fig. A2 is an example of applying the ‘UniporeModel.fig’ to determine the apparent diffusion coefficient.
Fig. A2. Typical example of applying ‘UniporeModel.fig’ to determine.
In the attached code, y denotes as the sorption fraction, x denotes as the apparent diffusion coefficient. Subscript ‘exp’ is the abbreviation of experimental and ‘model’ means sorption rate data estimated by the unipore model. yexp and ymodel are the experimental measured and analytical determined sorption fraction. De _true is the determined diffusion coefficient that provides the best fit to the experimental data.
601
Fuel 253 (2019) 588–606
Y. Yang and S. Liu
602
Fuel 253 (2019) 588–606
Y. Yang and S. Liu
603
Fuel 253 (2019) 588–606
Y. Yang and S. Liu
604
Fuel 253 (2019) 588–606
Y. Yang and S. Liu
References
2004;83(3):293–303. [39] Kelemen SR, Kwiatek LM. Physical properties of selected block Argonne Premium bituminous coal related to CO2, CH4, and N2 adsorption. Int J Coal Geol 2009;77(1):2–9. [40] Kumar A. Methane diffusion characteristics of Illinois coals. Southern Illinois University at Carbondale; 2007. [41] Marecka A, Mianowski A. Kinetics of CO2 and CH4 sorption on high rank coal at ambient temperatures. Fuel 1998;77(14):1691–6. [42] Mavor M, Owen L, Pratt T. Measurement and evaluation of coal sorption isotherm data. SPE annual technical conference and exhibition. 1990. [43] Nandi SP, Walker PL. Activated diffusion of methane from coals at elevated pressures. Fuel 1975;54(2):81–6. [44] Naveen P, Asif M, Ojha K, Panigrahi DC, Vuthaluru HB. Sorption kinetics of CH4 and CO2 diffusion in coal: theoretical and experimental study. Energy Fuels 2017;31(7):6825–37. [45] Pone JDN, Halleck PM, Mathews JP. Sorption capacity and sorption kinetic measurements of CO2 and CH4 in confined and unconfined bituminous coal. Energy Fuels 2009;23(9):4688–95. [46] Salmachi A, Haghighi M. Temperature effect on methane sorption and diffusion in coal: application for thermal recovery from coal seam gas reservoirs. APPEA J 2012;52(1):291–300. [47] Smith DM, Williams FL. Diffusional effects in the recovery of methane from coalbeds. Soc Petrol Eng J 1984;24(05):529–35. [48] Zhao W, Cheng Y, Yuan M, An F. Effect of adsorption contact time on coking coal particle desorption characteristics. Energy Fuels 2014;28(4):2287–96. [49] Cui X, Bustin A, Bustin RM. Measurements of gas permeability and diffusivity of tight reservoir rocks: different approaches and their applications. Geofluids 2009;9(3):208–23. [50] Yao Y, Liu D, Tang D, Tang S, Huang W, Liu Z, et al. Fractal characterization of seepage-pores of coals from China: an investigation on permeability of coals. Comput Geosci 2009;35(6):1159–66. [51] Li A, Ding W, He J, Dai P, Yin S, Xie F. Investigation of pore structure and fractal characteristics of organic-rich shale reservoirs: a case study of Lower Cambrian Qiongzhusi formation in Malong block of eastern Yunnan Province South China. Mar Petrol Geol 2016;70:46–57. [52] Pan Z, Connell LD, Camilleri M, Connelly L. Effects of matrix moisture on gas diffusion and flow in coal. Fuel 2010;89(11):3207–17. [53] Karacan CÖ. Heterogeneous sorption and swelling in a confined and stressed coal during CO2 injection. Energy Fuels 2003;17(6):1595–608. [54] Crosdale PJ, Beamish BB, Valix M. Coalbed methane sorption related to coal composition. Int J Coal Geol 1998;35(1):147–58. [55] Keshavarz A, Sakurovs R, Grigore M, Sayyafzadeh M. Effect of maceral composition and coal rank on gas diffusion in Australian coals. Int J Coal Geol 2017;173:65–75. [56] Han F, Busch A, Krooss BM, Liu Z, Yang J. CH4 and CO2 sorption isotherms and kinetics for different size fractions of two coals. Fuel 2013;108:137–42. [57] Busch A, Gensterblum Y, Krooss BM, Littke R. Methane and carbon dioxide adsorption–diffusion experiments on coal: upscaling and modeling. Int J Coal Geol 2004;60(2–4):151–68. [58] Clarkson CR, Bustin RM. The effect of pore structure and gas pressure upon the transport properties of coal: a laboratory and modeling study. 1. Isotherms and pore volume distributions. Fuel 1999;78(11):1333–44. [59] Li Z, Hao Z, Pang Y, Gao Y. Fractal dimensions of coal and their influence on methane adsorption. J China Coal Soc 2015;40(4):863–9. [60] Yao Y, Liu D, Tang D, Tang S, Huang W. Fractal characterization of adsorptionpores of coals from North China: an investigation on CH4 adsorption capacity of coals. Int J Coal Geol 2008;73(1):27–42. [61] Wu K, Chen Z. Real gas transport through complex nanopores of shale gas reservoirs. SPE Europec featured at 78th EAGE conference and exhibition. 2016. p. 22. [62] Azom PN, Javadpour F. Dual-continuum modeling of shale and tight gas reservoirs. SPE Annual Technical Conference and Exhibition. 2012. [63] Darabi H, Ettehad A, Javadpour F, Sepehrnoori K. Gas flow in ultra-tight shale strata. J Fluid Mech 2012;710:641–58. [64] Ertekin T, King GA, Schwerer FC. Dynamic gas slippage: a unique dual-mechanism approach to the flow of gas in tight formations. SPE-12258-PA 1986;1(01):43–52. [65] Javadpour F. Nanopores and Apparent Permeability of Gas Flow in Mudrocks (Shales and Siltstone). 2009. [66] Liu Q, Shen P, Yang P. Pore scale network modelling of gas slippage in tight porous media. Contemp Math 2002;295:367–76. [67] Rahmanian M, Aguilera R, Kantzas A. A new unified diffusion–viscous-flow model based on pore-level studies of tight gas formations. SPE J 2012;18(01):38–49. [68] Singh H, Javadpour F. Nonempirical apparent permeability of shale. Unconventional resources technology conference (URTEC). 2013. [69] Wu K, Li X, Guo C, Chen Z. Adsorbed gas surface diffusion and bulk gas transport in nanopores of shale reservoirs with real gas effect-adsorption-mechanical coupling. SPE reservoir simulation symposium. 2015. [70] Wu K, Li X, Wang C, Chen Z, Yu W. Apparent permeability for gas flow in shale reservoirs coupling effects of gas diffusion and desorption. Unconventional resources technology conference, Denver, Colorado, 25-27 August 2014. 2014. p. 2328–45. [71] Wu K, Li X, Wang C, Chen Z, Yu W. A model for gas transport in microfractures of shale and tight gas reservoirs. AIChE J 2015;61(6):2079–88. [72] King GR. Numerical simulation of the simultaneous flow of methane and water through dual porosity coal seams during the degasification process. University Park (USA): Pennsylvania State Univ; 1985.
[1] Thomas Graham Memorial S, Sherwood JN, University of S. Diffusion processes; proceedings of the Thomas Graham Memorial Symposium, University of Strathclyde. London; New York: Gordon and Breach, 1971. [2] Fick A. Ueber diffusion. Ann Phys 1855;170(1):59–86. [3] Philibert J. One and a half century of diffusion: Fick, Einstein, before and beyond. Diffus Fundam 2005;2(1):1–10. [4] Kärger J, Ruthven DM, Theodorou DN. Diffusion in nanoporous materials. John Wiley & Sons; 2012. [5] Kärger J, Chmelik C, Heinke L, Valiullin R. A new view of diffusion in nanoporous materials. Chem Ing Tech 2010;82(6):779–804. [6] Kainourgiakis M, Steriotis TA, Kikkinides E, Romanos G, Stubos A. Adsorption and diffusion in nanoporous materials from stochastic and process-based reconstruction techniques. Colloids Surf, A 2002;206(1–3):321–34. [7] Gray I. Reservoir engineering in coal seams: part 1-The physical process of gas storage and movement in coal seams. SPE Reservoir Eng 1987;2(01):28–34. [8] Harpalani S, Chen G. Influence of gas production induced volumetric strain on permeability of coal. Geotech Geol Eng 1997;15(4):303–25. [9] Levine JR. Model study of the influence of matrix shrinkage on absolute permeability of coal bed reservoirs. Geol Soc London Spec Publ 1996;109(1):197–212. [10] Warren J, Root PJ. The behavior of naturally fractured reservoirs. Soc Petrol Eng J 1963;3(03):245–55. [11] Laubach S, Marrett R, Olson J, Scott A. Characteristics and origins of coal cleat: a review. Int J Coal Geol 1998;35(1):175–207. [12] George JS, Barakat M. The change in effective stress associated with shrinkage from gas desorption in coal. Int J Coal Geol 2001;45(2):105–13. [13] Ceglarska-Stefańska G, Zarębska K. The competitive sorption of CO2 and CH4 with regard to the release of methane from coal. Fuel Process Technol 2002;77:423–9. [14] Schüth F, Sing KSW, Weitkamp J. Handbook of porous solids. Weinheim, Germany: Wiley-VCH; 2002. [15] Clarkson CR, Pan Z, Palmer ID, Harpalani S. Predicting sorption-induced strain and permeability increase with depletion for coalbed-methane reservoirs. SPE J 2010. [16] Liu S, Harpalani S. Permeability prediction of coalbed methane reservoirs during primary depletion. Int J Coal Geol 2013;113:1–10. [17] Wang Y, Liu S. Estimation of pressure-dependent diffusive permeability of coal using methane diffusion coefficient: laboratory measurements and modeling. Energy Fuels 2016;30(11):8968–76. [18] Collins R. New theory for gas adsorption and transport in coal. Proceedings of the 1991 coalbed methane symposium, Tuscaloosa, USA. 1991. p. 425–31. [19] Dvoyashkin M, Valiullin R, Kärger J. Temperature effects on phase equilibrium and diffusion in mesopores. Phys Rev E 2007;75(4):041202. [20] Evans III R, Watson G, Mason E. Gaseous diffusion in porous media at uniform pressure. J Chem Phys 1961;35(6):2076–83. [21] Valiullin R, Kortunov P, Kärger J, Timoshenko V. Concentration-dependent selfdiffusion of liquids in nanopores: a nuclear magnetic resonance study. J Chem Phys 2004;120(24):11804–14. [22] Pillalamarry M, Harpalani S, Liu S. Gas diffusion behavior of coal and its impact on production from coalbed methane reservoirs. Int J Coal Geol 2011;86(4):342–8. [23] Welty JR, Wicks CE, Wilson RE. Fundamentals of momentum, heat, and mass transfer. New York: Wiley; 1984. [24] Mason EA, Malinauskas A. Gas transport in porous media: the dusty-gas model. Elsevier Science Ltd; 1983. [25] Zheng Q, Yu B, Wang S, Luo L. A diffusivity model for gas diffusion through fractal porous media. Chem Eng Sci 2012;68(1):650–5. [26] Knudsen M. Die Gesetze der Molekularströmung und der inneren Reibungsströmung der Gase durch Röhren. Ann Phys 1909;333(1):75–130. [27] Steckelmacher W. Knudsen flow 75 years on: the current state of the art for flow of rarefied gases in tubes and systems. Rep Prog Phys 1986;49(10):1083. [28] Yang RT. Gas separation by adsorption processes. Butterworth-Heinemann; 2013. [29] Nie B, He X, Wang B. Mechanism and modes of gas diffusion in coal seams. Chin Saf Sci J 2000;10(6):24–8. [30] Shi JQ, Durucan S. A bidisperse pore diffusion model for methane displacement desorption in coal by CO2 injection. Fuel 2003;82(10):1219–29. [31] Bird G. Definition of mean free path for real gases. Phys Fluids 1983;26(11):3222–3. [32] Lama R, Bodziony J. Management of outburst in underground coal mines. Int J Coal Geol 1998;35(1):83–115. [33] Wei X, Wang G, Massarotto P, Golding S. A case study on the numerical simulation of enhanced coalbed methane recovery. SPE Asia Pacific oil & gas conference and exhibition. 2006. [34] Bhowmik S, Dutta P. Adsorption rate characteristics of methane and CO2 in coal samples from Raniganj and Jharia coalfields of India. Int J Coal Geol 2013;113:50–9. [35] Busch A, Gensterblum Y, Krooss BM, Littke R. Methane and carbon dioxide adsorption–diffusion experiments on coal: upscaling and modeling. Int J Coal Geol 2004;60(2):151–68. [36] Charrière D, Pokryszka Z, Behra P. Effect of pressure and temperature on diffusion of CO2 and CH4 into coal from the Lorraine basin (France). Int J Coal Geol 2010;81(4):373–80. [37] Clarkson CR, Bustin RM. The effect of pore structure and gas pressure upon the transport properties of coal: a laboratory and modeling study. 2. Adsorption rate modeling. Fuel 1999;78(11):1345–62. [38] Cui X, Bustin RM, Dipple G. Selective transport of CO2, CH4, and N2 in coals: insights from modeling of experimental gas adsorption data. Fuel
605
Fuel 253 (2019) 588–606
Y. Yang and S. Liu
[103] Thommes M, Morlay C, Ahmad R, Joly J. Assessing surface chemistry and pore structure of active carbons by a combination of physisorption (H 2 O, Ar, N 2, CO 2), XPS and TPD-MS. Adsorption 2011;17(3):653. [104] Gaylord NG, Eirich FR. The structure and properties of porous materials. J Polym Sci 1960;45(145):278–9. [105] Mastalerz M, He L, Melnichenko YB, Rupp JA. Porosity of coal and shale: insights from gas adsorption and SANS/USANS techniques. Energy Fuels 2012;26(8):5109–20. [106] Standard B, ISO B. Pore size distribution and porosity of solid materials by mercury porosimetry and gas adsorption. BS ISO 2005:15901-1. [107] Bustin RM, Clarkson CR. Geological controls on coalbed methane reservoir capacity and gas content. Int J Coal Geol 1998;38(1–2):3–26. [108] Busch A, Gensterblum Y, Krooss BM, Siemons N. Investigation of high-pressure selective adsorption/desorption behaviour of CO2 and CH4 on coals: an experimental study. Int J Coal Geol 2006;66(1–2):53–68. [109] Gregg SJ, Sing KSW, Salzberg H. Adsorption surface area and porosity. J Electrochem Soc 1967;114(11):279C-C. [110] Jones AH, Bell G, Schraufnagel R. A review of the physical and mechanical properties of coal with implications for coal-bed methane well completion and production. Geological and coal-bed methane resources of the northern San Juan Basin, Colorado and New Mexico. 1988. p. 169–81. [111] Siemons N, Busch A, Bruining H, Krooss B, Gensterblum Y. Assessing the kinetics and capacity of gas adsorption in coals by a combined adsorption/ diffusion method. SPE annual technical conference and exhibition. 2003. p. 8. [112] Clarkson CR, Solano N, Bustin RM, Bustin AMM, Chalmers GRL, He L, et al. Pore structure characterization of North American shale gas reservoirs using USANS/ SANS, gas adsorption, and mercury intrusion. Fuel 2013;103:606–16. [113] Fu H, Tang D, Xu T, Xu H, Tao S, Li S, et al. Characteristics of pore structure and fractal dimension of low-rank coal: a case study of Lower Jurassic Xishanyao coal in the southern Junggar Basin, NW China. Fuel 2017;193:254–64. [114] Everett DH, Stone FS, Colston Research S. The Structure and properties of porous materials. London: Butterworths, 1960. [115] Sing KS. Reporting physisorption data for gas/solid systems with special reference to the determination of surface area and porosity (Recommendations 1984). Pure Appl Chem 1985;57(4):603–19. [116] Barrett EP, Joyner LG, Halenda PP. The determination of pore volume and area distributions in porous substances. I. Computations from nitrogen isotherms. J Am Chem Soc 1951;73(1):373–80. [117] Groen JC, Pérez-Ramı́rez J. Critical appraisal of mesopore characterization by adsorption analysis. Appl Catal A 2004;268(1):121–5. [118] Oulton TD. The pore size-surface area distribution of a cracking catalyst. J Phys Colloid Chem 1948;52(8):1296–314. [119] Cai Y, Pan Z, Liu D, Zheng G, Tang S, Connell LD, et al. Effects of pressure and temperature on gas diffusion and flow for primary and enhanced coalbed methane recovery. Energy Explor Exploit 2014;32(4):601–19. [120] Zhang XG, Ranjith PG, Perera MSA, Ranathunga AS, Haque A. Gas transportation and enhanced coalbed methane recovery processes in deep coal seams: a review. Energy Fuels 2016;30(11):8832–49. [121] Maxwell JC. A treatise on electricity and magnetism. Clarendon Press; 1881. [122] Rayleigh LLVI. On the influence of obstacles arranged in rectangular order upon the properties of a medium. London Edinburgh Dublin Philos Mag J Sci 1892;34(211):481–502. [123] Weissberg HL. Effective diffusion coefficient in porous media. J Appl Phys 1963;34(9):2636–9. [124] Wheeler A. Reaction rates and selectivity in catalyst pores. In: Emmett PH, editor. Catalysis, fundamental principles (Part 2). New York: Reinhold Publishing Corporation; 1955. p. 105–65. [125] Do DD. Adsorption analysis: equilibria and kinetics. Imperial College Press; 1998. [126] Dutta A. Multicomponent gas diffusion and adsorption in coals for enhanced methane recovery. Stanford University; 2009.
[73] King GR, Ertekin T, Schwerer FC. Numerical simulation of the transient behavior of coal-seam degasification wells. SPE-12258-PA 1986;1(02):165–83. [74] Wang G, Ren T, Qi Q, Lin J, Liu Q, Zhang J. Determining the diffusion coefficient of gas diffusion in coal: development of numerical solution. Fuel 2017;196:47–58. [75] Nandi SP, Walker PL. Activated diffusion of methane in coal. Fuel 1970;49(3):309–23. [76] Crank J. The mathematics of diffusion. Oxford, GB: Clarendon Press; 1975. [77] Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical recipes in Fortran 77, the art of scientific computing. Cambridge University Press; 1992. [78] Mandelbrot BB. The fractal geometry of nature. New York: WH Freeman; 1983. [79] Pfeifer P, Avnir D. Chemistry in noninteger dimensions between two and three. I. Fractal theory of heterogeneous surfaces. J Chem Phys 1983;79(7):3558–65. [80] Wang HP, Yang YS, Wang YD, Yang JL, Jia J, Nie YH. Data-constrained modelling of an anthracite coal physical structure with multi-spectrum synchrotron X-ray CT. Fuel 2013;106:219–25. [81] Sing KSW. Characterization of porous materials: past, present and future. Colloids Surf, A 2004;241(1):3–7. [82] Wang H, Yang Y, Yang J, Nie Y, Jia J, Wang Y. Evaluation of multiple-scale 3D characterization for coal physical structure with DCM method and synchrotron Xray CT. Sci World J 2015;2015:414262-. [83] Avnir D, Jaroniec M. An isotherm equation for adsorption on fractal surfaces of heterogeneous porous materials. Langmuir 1989;5(6):1431–3. [84] Cai Y, Liu D, Yao Y, Li J, Qiu Y. Geological controls on prediction of coalbed methane of No. 3 coal seam in Southern Qinshui Basin, North China. Int J Coal Geol 2011;88(2–3):101–12. [85] Brunauer S, Emmett P, Teller E. Absorption of gases in multimolecular layers. J Am Chem Soc 1938;60:309–19. Find this article online. [86] Qi H, Ma J, Wong P-Z. Adsorption isotherms of fractal surfaces. Colloids Surf, A 2002;206(1):401–7. [87] Liu X, Nie B. Fractal characteristics of coal samples utilizing image analysis and gas adsorption. Fuel 2016;182:314–22. [88] Javadpour F, Fisher D, Unsworth M. Nanoscale gas flow in shale gas sediments. J Can Petrol Technol 2007. [89] Wheatcraft SW, Tyler SW. An explanation of scale-dependent dispersivity in heterogeneous aquifers using concepts of fractal geometry. Water Resour Res 1988;24(4):566–78. [90] Coppens M-O. The effect of fractal surface roughness on diffusion and reaction in porous catalysts–from fundamentals to practical applications. Catal Today 1999;53(2):225–43. [91] Avnir D, Farin D, Pfeifer P. Molecular fractal surfaces. Nature 1984;308:261. [92] Coppens M-O, Froment GF. Diffusion and reaction in a fractal catalyst pore—I Geometrical aspects. Chem Eng Sci 1995;50(6):1013–26. [93] Chapman S, Cowling TG, Burnett D. The mathematical theory of non-uniform gases: an account of the kinetic theory of viscosity, thermal conduction and diffusion in gases. Cambridge University Press; 1990. [94] Meyer OE. Die kinetische Theorie der Gase. Maruschke & Berendt; 1899. [95] Jeans JH. The dynamical theory of gases. University Press; 1921. [96] Sd S, DFA L. Diffusion of ideal gases in capillaries and porous solids. AIChE J 1962;8(1):113–7. [97] Pollard WG, Present RD. On gaseous self-diffusion in long capillary tubes. Phys Rev 1948;73(7):762–74. [98] Bosanquet C. British TA Report BR-507. 1944. [99] Mistler TE, Correll GR, Mingle JO. Gaseous diffusion in the transition pressure range. AIChE J 1970;16(1):32–7. [100] Evans III R, Truitt J, Watson G. Superposition of forced and diffusive flow in a large-pore graphite. Tenn.: Oak Ridge National Lab.; 1961. [101] Wang Y, Liu S, Zhao Y. Modeling of permeability for ultra-tight coal and shale matrix: a multi-mechanistic flow approach. Fuel 2018;232:60–70. [102] Yang Y, Liu S, Zhao W, Wang L. Intrinsic relationship between Langmuir sorption volume and pressure for coal: experimental and thermodynamic modeling study. Fuel 2019;241:105–17.
606