Gas transport study in the confined microfractures of coal reservoirs

Journal of Natural Gas Science and Engineering 68 (2019) 102920

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Gas transport study in the confined microfractures of coal reservoirs a,∗

a

Fanhui Zeng , Fan Peng , Jianchun Guo Bing Zhangb a b

a,∗∗

a

b

b

, Dayong Wang , Shouren Zhang , Ping Zhang ,

T

State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu, 610500, China China United Coalbed Methane Company, Beijing, 100015, China

ARTICLE INFO

ABSTRACT

Keywords: Coalbed gas Confined microfracture Stress sensitivity Gas effective viscosity Gas transport Influencing factors

Microfractures are commonly observed in coal reservoirs. During the coalbed gas production process, stress sensitivity and coalbed gas viscosity changes are significant factors that affect gas transport. By using research methods based on desorption theory and elastic-plastic mechanics, a coal confined microfracture gas transport model that considers dynamic microfracture width variations and gas effective viscosity is established in this paper. This model comprehensively fuses the Knudsen diffusion model, the slip flow model, the surface diffusion model, and the cubic grid model. The reliability of this model is verified via molecular simulations, and the influence factors of gas transport capacity in confined microfractures of coal reservoirs are then discussed in detail. The results demonstrate the following findings. (1) The analyzed flows are well simulated in coal confined microfractures by the model established in this paper, which considers stress sensitivity and coalbed methane viscosity change. (2) Under low formation pressure (less than 5 MPa), the effective gas viscosity rapidly decreases with the decrease in formation pressure, and the negative contribution of gas viscosity change to the microfracture permeability of coal is large. The smaller the initial fracture width of coal, the more obvious this negative effect. (3) When the initial fracture width is fairly small (near 1 nm), Knudsen flow and surface diffusion greatly contribute to the microfracture permeability of coal. However, the larger the initial microfracture width, the smaller the contribution of the two flow regimes to the microfracture permeability, and the permeability is mainly provided by slip flow. (4) Under given conditions, the microfracture permeability of coal is positively related to rock mechanical parameters (Young's modulus and Poisson's ratio) and negatively related to fracture compressibility. Under low formation pressures (less than 15 MPa), the microfracture permeability of coal is positively related to gas desorption. When the formation pressure exceeds 15 MPa, the influence of gas desorption performance on permeability is nearly constant.

1. Introduction Coalbed methane (CBM), as a new, clean and high-quality unconventional natural gas resource (Fu et al., 2007), is an abundant resource with enormous development potential. In coal reservoirs, nanoscopic, microscopic, mesoscopic and macroscopic channel flows show multiscale characteristics, which cause the coalbed methane transport mechanism to be exceptionally complex. Based on the Knudsen number (Kn), the coalbed methane flow regimes are generally divided into continuous flow, slip flow, transition flow and Knudsen flow (Shahri et al., 2012; Singh et al., 2014). There are many microfractures in coal reservoirs, and their widths, which are mostly on the microscale and nanoscale, vary greatly (Cai et al., 2013). The presence of a large number of microfractures has a

∗

significant effect on coalbed methane transport (Laubach et al., 1998). When a gas transforms from a macroscopically continuous flow into a discontinuous flow, the N-S equation based on the continuity hypothesis and the no-slip condition will fail (Roy et al., 2003). To establish a multiscale unified transport equation coupled with various flow regimes for coalbed methane, two main treatment methods are used. The first method modifies the slip boundary conditions to consider different transport mechanisms; however, the method contains more empirical coefficients. The second method is to use a direct linear summation to superimpose different transport mechanisms. Previous researchers have proposed gas continuous, slippage, Knudsen and surface diffusion transport equations (Singh et al., 2014; Karniadakis et al., 2005; Freeman et al., 2011). Based on the Knudsen diffusion and slip flow models, Wu et al. (2015) established a gas transport model for

Corresponding author. Corresponding author. E-mail addresses: [email protected] (F. Zeng), [email protected] (J. Guo).

∗∗

https://doi.org/10.1016/j.jngse.2019.102920 Received 30 January 2019; Received in revised form 15 June 2019; Accepted 18 June 2019 Available online 22 June 2019 1875-5100/ © 2019 Elsevier B.V. All rights reserved.

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Fig. 1. Ideal matrix and fracture cubic grid model and simplified microfracture model (Assuming all pores in the coal reservoir are produced by microfractures).

microfractures by combining several different transport mechanisms. However, this model does not consider the effects of microfracture width changes, viscosity changes of real gas, gas desorption and surface diffusion on gas transport. Because of the large compressibility of coal, the stress-sensitivity effect has a great influence on the permeability change of microfractures (Seidle et al., 1992; Zeng and Guo, 2016; Zeng et al., 2018; Zeng et al., 2019a, 2019b, 2019c). In addition, when the formation pressure decreases to the critical desorption pressure, gas adsorbed on organic matter surface begins to desorb (Yao and Zhou, 1988; Fathi and Akkutlu, 2012). Some researchers have pointed out that gas adsorption induces coal matrix expansion, and gas desorption causes matrix shrinkage, resulting in microfracture width changes (Zhu et al., 2007; Palmer, 2009; Pan and Connell, 2007; Xu et al., 2013; Li and Zhang, 2018; Zhang et al., 2018). Beskok (1996) highlighted that the contribution of the macroscopic gas viscosity to gas transport comes from the collisions between gas molecules. However, an increase in the concentration of a gas in a microchannel reduces the collisions between gas molecules, and the collisions between gas molecules and the channel surface play a major role. Therefore, gas viscosity in a microchannel is different from that in a macrochannel and is specifically expressed as the effective viscosity. When studying the transport rules of coalbed methane in microfractures, the effects of the stress sensitivity, viscosity change of a real gas and gas desorption on gas transport should be fully considered. The traditional stress-sensitive model is mostly based on test fittings, and exponential or power exponential models are generally used to describe the change in permeability with the pore pressure (Palmer and Mansoori, 1998; Shi and Durucan, 2005; Cui and Bustin, 2005). These models have the disadvantages of containing large empirical components and have poor universality, and thus, they require considerable experimental data support. To overcome these disadvantages, Robertson and Christiansen (2006) classified a coal microfracture reservoir into cubic grids and studied the influence of stress sensitivity on the change in microfracture width according to the elasticity theory, and they concluded that fracture compressibility, matrix compressibility and gas desorption together lead to microfracture width changes. However, they did not study the gas transport characteristics in coal microfractures. In a study on gas effective viscosity, Veijola and Turowski (2001) presented an empirical formula for calculating the effective viscosity of gas, which is closely related to the Knudsen number. Based on the work in previous studies, this paper examines the gas transport rules in the confined microfractures of coal reservoirs. First, by considering the dynamic changes in the coal microfracture width, a coalbed methane effective viscosity model is established for confined microfractures. Second, a coalbed methane transport model in confined

microfractures in consideration of the stress sensitivity and gas effective viscosity is established, and then, molecular simulation data are used to verify the model's reliability. Finally, this paper analyzes the factors affecting gas transport capacity in the confined microfractures of coal reservoirs. The stress sensitivity, real gas viscosity changes and other transport mechanisms (slip flow, Knudsen diffusion and surface diffusion) greatly affect the gas transport in tiny coal fractures under certain conditions. If these factors are ignored, the calculated coal permeability will greatly deviate from the actual value. The model in this paper comprehensively considers these factors such that the gas transport laws of coal reservoirs can be described under real formation conditions. Permeability calculation based on this model will yield accurate results. Moreover, this model has important significance for the calculation of CBM productivity and rational development. 2. Model establishment 2.1. Physical model Coal reservoirs generally develop tiny natural fractures. As coalbed gas is extracted from a reservoir and the formation pressure drops, the effective stress acting on the microfracture surface increases. A coal formation is typically treated as a fractured reservoir, meaning that the sole contributor to the overall permeability of the reservoir is the fracture system and the contribution of diffusion through the matrix to total flow can be neglected. A coal reservoir can be divided into grids to obtain a number of cubic matrix fracture grids (Robertson and Christiansen, 2006). Among these grids, the matrix units are cubes of length a , and between the matrix units, tiny fractures of width b are observed; together, these components eventually form an ideal cubic grid model (Fig. 1a). Considering that the pores in the matrix are extremely small and that all the pores in a reservoir are produced by microfractures, this paper focuses on the study of the fractures between the matrix units. Through further simplification, we can obtain a microfracture model for a length and height of a and a width of b (Fig. 1b). It is assumed in this paper that the coalbed methane migrating in microfractures is a single-phase compressible fluid and that the reservoir temperature remains unchanged during coalbed methane production, which is the same as not considering the vertical flow of gas. 2.2. Mathematical model The following section establishes the relevant mathematical models: microfracture gas transport models for different transport mechanisms in coal reservoirs, the coalbed methane effective viscosity model in coal confined microfractures, and the coalbed methane transport model in 2

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where Jvs0 is the microfracture gas flow for slip flow (kg·s−1), m is the gas slippage constant (dimensionless), and is the rarefaction effect factor (dimensionless).

confined microfractures. 2.2.1. Gas transport model for different transport mechanisms in the confined microfractures of coal reservoirs The microfractures of coal reservoirs have different shapes. For convenience, the microfracture cross section can be treated as equivalent to a rectangular cross section. The geometrical parameter is defined as follows:

=

a b

2.2.1.3. Knudsen diffusion. The collisions between gas molecules and the walls dominate when Kn ≥ 10, and the gas transport mechanism is Knudsen diffusion. For circular tubes of radius r , gas transport can be described by the Knudsen equation (Singh et al., 2014):

(1)

r3

Jk0 =

M 2 RT

0.5

dp da

(8)

where represents the microfracture aspect ratio (dimensionless), a represents the microfracture height (m), and b represents the microfracture width (m). The Knudsen number, Kn (dimensionless), in microfracture gas transport can be expressed as follows (Thompson and Owens, 1975):

where Jk0 is the microfracture gas flow for Knudsen diffusion (kg·s−1), and r is the radius of the circular tubes (m). For a rectangular cross-section microfracture, the flow of Knudsen diffusion can be expressed as follows (Thompson and Owens, 1975):

K n=

Jk0 =

(2)

b

where represents the mean molecule free path (m). The mean molecule free path can be expressed as follows (Loeb, 1934):

=

kB T 2 2p

=

Jsurface0 =

192 5

i = 1,3,5

tanh(i /2) i5

(5)

(6)

2.2.1.2. Slip flow. When 10−3 < Kn < 0.1, the collisions between gas molecules and the collisions between gas molecules & the microfracture walls have a great influence on gas transport. The gas molecules show a slippage effect during transport. In conjunction with the modified slippage boundary conditions, the gas slip flow can be expressed as follows (Karniadakis et al., 2005):

A( )

dp 6K n ab3 pM (1 + K n) 1 + 12µ RT 1 mK n da

1+

1 2

+ ln( +

1+

2)

(

2

+ 1)3/2 1+ + 3 3

3

MDs

Cs max pL dp (p + pL )2 da

(11)

2.2.2. Coalbed methane effective viscosity model in confined microfractures Previous studies have shown that the viscosity of coalbed methane in confined microfractures changes significantly under different conditions (Beskok, 1996), and the viscosity change has a large impact on coalbed methane transport. In this section, by calculating the microfracture width change, the Knudsen number considering the change in the microfracture width is obtained, and then, the coalbed methane effective viscosity model in coal confined microfractures is established. From the viewpoint of elastoplastic mechanics, Robertson and Christiansen (2006) studied the effect of stress sensitivity on the variation of microfracture width. The width variation of the microfracture caused by the decrease in the formation pressure is influenced by microfracture compressibility, matrix compressibility and gas desorption. The effects of these factors on the changes in the microfracture width are discussed in the following sections. The influence of the compressibility of a microfracture can be expressed as follows:

In these equations, A ( ) is a microfracture section-shape factor that affects the continuous flow (dimensionless), and tanh(x ) represents the ex e x hyperbolic tangent function, with tanh(x ) = e x + e x .

Jvs0 =

+

where Jsurface0 is the microfracture gas flow for surface diffusion (kg·s−1), pL is the Langmuir pressure (MPa), Ds represents the diffusion coefficient of the surface (m2/s), and Csmax is the maximum concentration of the surface adsorption gas (mol/m3).

where

A( ) = 1

1

2.2.1.4. Surface diffusion. Surface diffusion is the phenomenon in which coalbed methane migrates along the adsorption surface wall of a microfracture, and it can be described using the Langmuir isothermal adsorption equation (Freeman et al., 2011):

(4)

ab3 pM dp 12µ RT da

2 ln

(10)

where Jv0 is the microfracture gas flow for continuous flow (kg·s−1), is the microfracture porosity (dimensionless), M is the gas molar mass (kg·mol−1), is the microfracture tortuosity (dimensionless), µ is the gas viscosity (Pa·s), and R is the universal gas constant (J·mol−1 K−1). Considering the influence of the microfracture geometry on gas flow, equation (4) can be further expressed as follows (Karniadakis et al., 2005):

A( )

(9)

In these equations, B ( ) is the microfracture shape factor that can affect Knudsen diffusion (dimensionless).

2.2.1.1. Continuous flow. When Kn < 10−3, the flow of gas in a microfracture is continuous, and the collisions among gas molecules are the dominant transport mechanism. The Hagen-Poiseuille equation can be used to describe continuous flow (Singh et al., 2014):

Jv0 =

dp da

B( )

where kB is the Boltzmann constant (J/K), T is the formation temis the molecular collision diameter (m), and p is the perature (K), formation pressure (Pa). Different transport models for different flow regimes of coalbed methane flowing through the confined microfractures can be expressed as follows:

Jv0 =

0.5

M 2 RT

where

(3)

ab3 pM dp 12µ RT da

B ( ) b3

bf = b0 cf (pp

pp0 )

(12)

where bf is the width variation of the microfracture caused by the microfracture compressibility (m), b0 is the initial microfracture width (m), cf is the fracture compressibility factor (MPa−1), pp is the current pore pressure (MPa), and pp0 is the initial pore pressure (MPa). The influence of matrix compressibility can be expressed as follows:

(7) 3

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a 0 (1

bm =

2 )(pp

pp0 )

where fm is the collision frequency among gas molecules (s−1), n is the number of gas molecules per unit volume (m−3), and v represents the average gas thermal kinematic velocity (m·s−1). In microfractures, the collision frequency between gas molecules and the microfracture wall, fw (s−1), is as follows (Thompson and Owens, 1975):

(13)

E

where bm is the change in the microfracture width caused by matrix compressibility (m), a0 is the original matrix length (m), E is Young's modulus (MPa), and is Poisson's ratio (dimensionless). The influence of gas desorption can be expressed as follows:

a 0 S L pL (p (pL + pp0 )(pL + pp ) p

bs =

pp0 )

(14)

pp0 ) +

a0 SL pL pL + pp0

(pp

pL + pp

a0 (1 2 ) (pp E

wvs0 =

b + bt

=

kB T 2p (b

2

pp0 )

pp0 ) (15)

w k0 =

b

µ 1 + 2K n+ 0.2K n0.788 exp

(

Kn 10

)

µ 1 + 2K n b +

0.2K n0.788 b

exp

(

K nb 10

nabda

=

Jvs =

(18)

(21)

= 1+

2 K n(1 + 1/ )

1

(22)

a b + bt

(23)

1

K nb (1 + 1/ b) 2

(24) 1

2 K nb (1 + 1/ b)

(25)

a (b + bt )3 pM dp 6K nb (1 + K nb) 1 + 12µeff RT 1 mK nb da

A ( b)

(26)

where Jvs represents the microfracture gas flow of the slippage flow, considering the coalbed methane viscosity change and the microfracture width dynamic change (kg·s−1).

Jk =

B ( b)

(b + bt )3

M 2 RT

0.5

dp da

(27)

where Jk represents the microfracture gas flow of Knudsen diffusion, considering the microfracture width dynamic change (kg·s−1).

Jsurface =

2.2.3. Coalbed methane transport model in confined microfractures To consider the influence of different gas transport mechanisms on the contribution of total microfracture gas transport in coal based on the slippage flow and Knudsen diffusion, the collision frequency among gas molecules and the collision frequency between the gas molecules and the microfracture walls can be regarded as the contribution coefficient of the slippage flow and Knudsen diffusion, respectively (Wu et al., 2015). In microfractures, the collision frequency of the gas molecules is as follows (Thompson and Owens, 1975):

v

1

Therefore, under different pressures, considering the viscosity change in coalbed methane and the microfracture width variations in coal, the gas transport equations of various transport mechanisms can be written as follows:

where µeff is the gas effective viscosity considering the change in the microfracture width (Pa·s). It can be seen from equation (18) that the effective viscosity of a gas in a microchannel is related to the Knudsen number. As the Knudsen number increases, the effective viscosity of a gas decreases. When the Knudsen number approaches zero, the effective viscosity of the gas in the microchannel tends toward the macroscopic ideal gas viscosity of the gas. The maximum relative error of equation (18) at any value of Knb, is less than approximately 0.6%.

fm =

fm + fw

wk = 1 +

(17)

)

fw

wvs = 1 +

where µeff0 is the gas effective viscosity (Pa·s), and µ is the gas ideal viscosity (Pa·s). The gas effective viscosity considering the change in microfracture width is presented as follows:

µeff =

K n(1 + 1/ ) 2

Based on the weight coefficient of slippage flow and Knudsen diffusion considering the microfracture width change, wvs (dimensionless) and w k (dimensionless) can be expressed as

where Knb is the Knudsen number, considering the change in microfracture width (dimensionless). It is a basic requirement for a gas flow simulation to accurately reflect the gas viscosity coefficient in a microfracture. Veijola and Turowski (2001) presented the formula for calculating the gas effective viscosity:

µeff0 =

fm + fw

= 1+

Combining equations (1) and (15), the microfracture aspect ratio with the change in microfracture width, b , can be obtained:

(16)

+ bt )

fm

The Knudsen diffusion weight coefficient, w k0 (dimensionless), can be expressed as the ratio of the collision frequency between gas molecules and the microfracture wall to the total collision frequency:

During a pressure drop in a coal reservoir, the change in fracture width caused by the pressure drop will lead to a Knudsen number change:

K nb =

(20)

The weight coefficient of slippage flow, wvs0 (dimensionless), can be expressed as the ratio of the collision frequency among gas molecules to the total collision frequency:

where bs is the change in microfracture width caused by gas desorption (m), SL represents the Langmuir strain (m), and pL represents the Langmuir pressure (MPa). From equations (12)–(14), the total change in the microfracture width caused by the pressure change bt (m) can be expressed as follows:

bt = bf + bm + bs = b0 cf (pp

1 nv a + b)da 2

fw =

MDs

Cs max pL dp (p + pL )2 da

(28)

where Jsurface represents the microfracture gas flow of the surface diffusion, considering the microfracture width dynamic change (kg·s−1). By combining equations (24)–(28), the total transport equation of the microfracture gas in coal considering the coalbed methane viscosity change and microfracture width dynamic change can be expressed as follows:

Jt = wvs Jvs + w k Jk + Jsurface

(29)

where Jt is the total gas transport flow in a confined microfracture considering the coalbed methane viscosity change and microfracture width dynamic change (kg·s−1). For the sake of discussion, the apparent permeability is used to

(19) 4

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indicate the transport capacity of each transport mechanism:

K vs =

Jvs wvs Ma (b + bt ) dp /da

= wvs Vstd A ( b )

Kk =

Table 1 Basic parameters.

µeff Vstd

wk

(b + b t ) 2 p 6K nb (1 + K nb) 1 + 12 RT 1 mK nb

µeff Vstd Jk = w k µeff Vstd B ( b) Ma (b + bt ) dp /da

(30)

(b + bt )2 1 a 2 M RT

0.5

(31)

K surface = MDs

µeff Cs max pL

(32)

(p + p L ) 2

(33)

Kt = K vs + K k+ K surface

where Vstd is the gas molar volume under standard conditions (m3·mo1−1), K vs is the microfracture gas apparent permeability of slippage flow considering the dynamic variation in microfracture width (m2), Kk is the microfracture gas Knudsen diffusion apparent permeability considering the coalbed methane viscosity change and the dynamic variation in microfracture width (m2), K surface is the microfracture gas apparent permeability of surface diffusion considering the coalbed methane viscosity change and the dynamic variation in microfracture width (m2), and Kt is the microfracture gas transport total apparent permeability considering the coalbed methane viscosity change and the dynamic variation in microfracture width (m2). 3. Model validation To validate the model established in this paper, molecular simulation and experimental results are used in this paper. Sone and Hasegawa (1987) used molecular dynamic theory to obtain the molecular simulation data used in this paper. In detail, using the linear Boltzmann method with a boundary of diffusion refection, the gas transport flow over the entire range of Knudsen numbers is obtained on the basis of macroscopic variables, including gas velocity, density and temperature. The experimental study carried out by Tison (1993) utilized a micro-capillary with finite-length tubes (length/radius = 200). The validation results are presented in Figs. 3 and 4. When we use the molecular simulation to validate the model established in this paper, based on equation (29), we use the degeneration model to exclude the surface diffusion flow for better comparison with the molecular simulation because the molecular simulation does not include surface diffusion. In contrast, when we conduct the analysis of the influencing factors in Section 4, we consider all flow mechanisms, including surface flow. The degeneration model is described by the following equations:

Symbol

Unit

Value

Gas type Boltzmann constant Molecular collision diameter Initial microfracture width

CH4 kB b0

– J/K m m

Initial microfracture height

ɑ0

m

Aspect ratio of microfracture Initial fracture compressibility factor Compressibility change rate Poisson ratio Young's modulus Langmuir strain Initial formation pressure Final formation pressure Universal gas constant Temperature Gas molar mass Gas ideal viscosity Gas molar volume under standard conditions Gas density Maximum concentration of surface adsorption gas Langmuir pressure Surface diffusion coefficient Microfracture porosity Microfracture tortuosity Gas slippage constant Rarefaction effect factor

ζ c0

– Pa−1

– 1.3805 × 10−23 0.42 × 10−9 1 × 10−9 (5 × 10−9, 1 × 10−8) 1 × 10−7 (5 × 10−7, 1 × 10−6) 100 7.19 × 10−4

δ ν E SL Pp0 Pfinal R T M μ Vstd

Pa−1 – Pa – Pa Pa J/(mol·K) K kg/mol Pa·s m3·mo1−1

3.04 × 10−3 0.3 3 × 109 0.00251 2 × 107 0.55 × 106 8.314 423 1.6 × 10−2 1.75 × 10−5 22.4 × 10−3

ρ Csmax

kg/m3 mol/m3

0.655 25040

PL Ds

Pa m2/s – – – –

7.7 × 106 2.89 × 10−10 0.03 1.2 −1 1.5

τ m α

swelling (or shrinkage) of the matrix blocks. In this paper, methane is mainly discussed. Furthermore, the model developed herein is general and can be applied to the transportation of other gases such as carbon dioxide and nitrogen. Because their abilities to adsorb (or desorb) differ, the results will also show some differences. Our main concern is methane, which is of great value; therefore, CH4 is selected as the main gas type. Figs. 2 and 3 show the results of model validation. The results indicate the following. (1) The results of this model fit well with the molecular simulation and experiment results, which shows that the model in this paper can reasonably describe the coalbed gas transport of different flow regimes in microfractures. (2) When the value of Kn is approximately 1, low dimensionless gas transport is observed, which is consistent with previous studies that observed a minimum value of gas transport (Kogan, 1961; Williams, 1971; Cercignani, 1977). (3) Compared with the situation considering gas viscosity change, there is a

(34)

Jt0 = wvs0 Jvs0 + w k Jk0 Jt0 6K n = wvs0 (1 + K n) 1 + Jv0 1 mK n

Parameters

+ w k0

12 B ( ) aA ( )

Jt0 a A( ) 6K n = wvs0 (1 + K n) 1 + Jk0 12 B ( ) 1 mK n

(35)

+ w k0

(36) −1

where Jt0 is the total flow of the microfracture gas (kg·s ), Jt0 / Jv0 represents the dimensionless gas transport flow according to continuous flow (dimensionless), and Jt0 / Jk0 represents the dimensionless gas transport flow according to Knudsen diffusion (dimensionless). Table 1 lists all input parameters necessary for this paper. All possible values of parameters are from the open literature. The fracture compressibility factor (cf) is a function of net stress, and it can be derived from the initial fracture compressibility factor (c0) and the fracture compressibility change rate (δ) (McKee et al., 1998). Coalbeds have the ability to adsorb (or desorb) large amounts of gas, such as methane, carbon dioxide, and nitrogen, which causes

Fig. 2. Results of model validation (based on continuous flow). 5

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Fig. 3. Results of model validation (based on Knudsen diffusion).

Fig. 5. Comparison between the apparent permeability with the ideal gas viscosity and the apparent permeability with the effective gas viscosity.

with that for ideal gas viscosity, and the following results are obtained. As shown in Fig. 5, when the formation pressure is high, the permeability of microfractures under the two conditions (considering viscosity change and not considering viscosity change) is not much different because of the small change in gas viscosity. Under low formation pressure (less than 5 MPa), the permeability of microfractures is obviously different in the two cases because of the sharp decrease in the gas effective viscosity. This trend becomes more obvious with a smaller initial microfracture width. At low formation pressures, gas desorption leads to an increase in fracture width; thus, the permeability without considering the viscosity change shows an increasing trend. The gas viscosity has a negative effect on permeability with the consideration of a gas viscosity change, especially under low formation pressure. Overall, permeability shows a downward trend with the comprehensive influence of the two factors. Obviously, the comprehensive result is more in line with our general understanding of the permeability change of coal reservoirs with the formation pressure change.

Fig. 4. Comparison of gas viscosity changes with different initial microfracture widths.

certain deviation in dimensionless gas transport without considering the change in gas viscosity; thus, the influence of gas viscosity on gas transport cannot be neglected. In addition, equations (30) and (31) show that the continuous flow is less affected by gas viscosity and that the Knudsen flow is more affected by gas viscosity. This trend is also reflected in Figs. 2 and 3.

4.2. Initial microfracture width Taking 1 × 10−9 m and 1 × 10−8 m as the initial microfracture widths, basic data are substituted into equations (30)–(33) to obtain the values of Kvs, Kk, Ksurface and Kt. The results are shown in Figs. 6 and 7, respectively. As shown in Figs. 6 and 7, with a decrease in the formation pressure, the effective stress acting on the fracture surface increases and the microfracture width decreases gradually; thus, the microfracture permeability presents a downward trend. When the initial microfracture width is fairly small (1 nm), Knudsen flow and surface diffusion have an

4. Results and discussion During the production of coalbed gas with formation pressure drop, many basic parameters, such as gas viscosity, initial microfracture width, rock mechanical parameters (Poisson ratio and Young's modulus of the rock), desorption of gas, and fracture compressibility, affect the microfracture gas transport capacity. In the following section, these basic parameters are analyzed. 4.1. Gas viscosity Using 1 × 10−9 m, 5 × 10−9 m, and 1 × 10−8 m as the initial microfracture widths, we can use basic data as inputs for equation (18) to obtain the value of μ/μeff. As shown in Fig. 4, gas viscosity decreases with decreasing formation pressure and decreases rapidly at low formation pressure (less than 5 MPa). The smaller the initial microfracture width, the more obvious the viscosity changes with the pressure change. Furthermore, by substituting the basic parameters into equations (30)–(33), the microfracture permeability with effective viscosity can be obtained. The permeability with effective gas viscosity is compared

Fig. 6. Each transport mechanism contribution to the apparent permeability when the initial microfracture width is 1 nm. 6

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Fig. 7. Each transport mechanism contribution to the apparent permeability when the initial microfracture width is 10 nm.

Fig. 9. Comparison of apparent permeability with different Poisson ratios.

obvious contribution to permeability. As the initial fracture width increases gradually, it can be seen from equation (16) that the Knudsen number Knb gradually decreases; thus, the influence of the Knudsen flow and surface diffusion on permeability gradually decreases. 4.3. Rock mechanical parameters In this section, we study the effect of rock mechanical parameters (Young's modulus and Poisson ratio) on apparent permeability. The results are shown in Figs. 8 and 9, respectively. According to Figs. 8 and 9, the rock mechanical parameters have a relatively large influence on microfracture permeability, and the two parameters have the most significant influence on the permeability near the medium formation pressure (5 MPa). When the other conditions are the same, the larger the Poisson ratio and Young's modulus, the greater is the permeability of coal. Overall, the permeability of coal is positively related to rock mechanical parameters.

Fig. 10. Comparison of apparent permeability with different gas desorption performances.

4.4. Desorption of gas The results of the study on the influence of gas desorption on apparent permeability are shown in Fig. 10. As shown in Fig. 10, the effect of gas desorption on Kt is significant at low formation pressures (less than 10 MPa). When the formation pressure is less than 15 MPa, the microfracture permeability is positively related to gas desorption. When the formation pressure is greater than 15 MPa, the influence of different gas desorption performance on permeability is nearly the same.

Fig. 11. Comparison of apparent permeability with different fracture compressibility values.

4.5. Fracture compressibility The results regarding the influence of fracture compressibility on apparent permeability are shown in Fig. 11. Fig. 11 shows that fracture compressibility has a large influence on microfracture permeability. Under low formation pressure (less than 5 MPa), the influence of different fracture compressibility values on permeability gradually decreases. When other conditions are the same, a greater fracture compressibility (the greater the value of c0, the smaller the value of δ) results in easier compression of the fractures and a smaller value of permeability. In general, permeability is negatively related to fracture compressibility.

Fig. 8. Comparison of apparent permeability with different Young's modulus values. 7

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5. Conclusions

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(1) The analyzed flows can be well simulated in coal confined microfractures by the model established in this paper, which considers stress sensitivity and coalbed methane viscosity change. (2) Under low formation pressure (less than 5 MPa), with a decrease in formation pressure, the effective gas viscosity rapidly decreases, and its negative contribution to the microfracture permeability of coal increases. The smaller the initial fracture width of coal, the more obvious is this negative effect. (3) When the initial fracture width is fairly small (near 1 nm), Knudsen flow and surface diffusion contribute greatly to the microfracture permeability of coal. However, the larger the initial microfracture width, the smaller is the contribution of the two flow regimes to the microfracture permeability, and the permeability is mainly determined by slip flow. (4) Under the given conditions, the microfracture permeability of coal is positively related to rock mechanical parameters (Young's modulus and Poisson ratio) and negatively related to fracture compressibility. Under low formation pressures (less than 15 MPa), the microfracture permeability of coal is positively related to gas desorption. When the formation pressure exceeds 15 MPa, the influence of gas desorption performance on permeability is nearly constant. Acknowledgment This research was supported by the National Key Research and Development Plan of China (Grant No. 2018YFB0605602), the Natural Science Foundation of China (Grant Nos. 51504203, 51525404, and 51374178), and the National Key Research and Development Program of China (Grant No. 2017ZX05037-004). Appendix A. Supplementary data Supplementary data to this article can be found online at https:// doi.org/10.1016/j.jngse.2019.102920. References Beskok, A., 1996. Simulation and Models for Gas Flows in Microgeometries. Ph.D Dissertation. Princeton University. Cai, Y., Liu, D., Pan, Z., Yao, Y., Li, J., Qiu, Y., 2013. Pore structure and its impact on CH4 adsorption capacity and flow; capability of bituminous and subbituminous coals from northeast China. Fuel 103, 258–268. Cercignani, C., 1977. Theory and application of the Boltzmann equation. Phys. Today 30 (1), 66–68. Cui, X., Bustin, R.M., 2005. Volumetric strain associated with methane desorption and its impact on coalbed gas production from deep coal seams. AAPG Bull. 89 (9), 1181–1202. Fathi, E., Akkutlu, I.Y., 2012. Mass transport of adsorbed-phase in stochastic porous medium with fluctuating porosity field and nonlinear gas adsorption kinetics. Transp. Porous Media 91 (1), 12–20. Freeman, C.M., Moridis, G.J., Blasingame, T.A., 2011. A numerical study of microscale flow behavior in tight gas and shale gas reservoir systems. Transp. Porous Media 90 (1), 253–268.

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