New Coupled Apparent Permeability Models for Gas Transport in

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New coupled apparent permeability models for gas transport in inorganic nanopores of shale reservoirs considering multiple effects Shan Wang, Juntai Shi, Ke Wang, Zheng Sun, and Zhengfu Zhao Energy Fuels, Just Accepted Manuscript • DOI: 10.1021/acs.energyfuels.7b02948 • Publication Date (Web): 21 Nov 2017 Downloaded from http://pubs.acs.org on December 4, 2017

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New coupled apparent permeability models for gas transport in inorganic nanopores of shale reservoirs considering multiple effects Shan Wanga,b**, Juntai Shia,b*, Ke Wangc, Zheng Suna,b, Zhengfu Zhaoc a

MOE Key Laboratory of Petroleum Engineering in China University of Petroleum at Beijing,

Beijing 102249, China b

State Key Laboratory of Petroleum Resources and Engineering in China University of Petroleum at

Beijing, Beijing 102249, China c

State Key Laboratory of Petroleum Resources and Prospecting in China University of Petroleum at

Beijing, Beijing 102249, China * Corresponding author: Email: Juntai Shi ([email protected]) & Shan Wang ([email protected]) Tel: Juntai Shi ((010) 89732193); Shan Wang (+86-18811368396)

Abstract The study of seepage mechanism in shale gas reservoir has been paid more and more attention. The shale gas reservoirs are rich in nano-sized pores. The pores in shale matrix can be divided into organic nanopores and inorganic nanopores. At present, there are many literatures focusing on establishing model to analyze gas transport mechanism in shale organic pores. Some researchers also considered the difference of gas transport in organic and inorganic matrix pores, and mathematical model of inorganic nanopores has been established. However, for inorganic nanopores, most of the models ignore the effect of irreducible water distribution on gas transport, which lead to overestimating of gas transport capability. In this paper, firstly, based on the weighting coefficient proposed by Wu, the apparent permeability models are established for inorganic nanopores with two different cross-section shapes, which are known as cylindrical capillary and slit nanopores. The influence of irreducible water distribution, real gas effect and stress dependence is also taken into account in the models. Then, the proposed model is verified, and the results show that the model and the experimental data can be well fitted. Finally, the effect of each factor on apparent gas permeability is analyzed and discussed. The results indicate that the apparent permeability of nanopores with different cross-section shapes decreases with the increase of relative humidity. When the relative humidity increases to a critical value, the apparent permeability decreases sharply and the 1

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pores will be blocked with capillary water. The gas transport capability in cylindrical capillaries and slit nanopores at the same cross-sectional area is different, and the pore pressure, pore size, effective stress and the aspect ratio of the slit nanopores are important factors affecting the transport process. Under high temperature and low pressure conditions, methane transport capacity is significantly higher than ethane and carbon dioxide. The results of this paper can provide a reference for researchers in the study of gas seepage mechanism in shale gas reservoirs. Keywords: Shale gas; Inorganic nanopores; Irreducible water; Apparent permeability; Real gas transport 1. Introduction Shale gas is an important unconventional natural gas resource and its exploration and development have received extensive attention in recent years.1-3 Shale gas reservoirs have the characteristics of self-generation and self-storage, and the gas is mainly stocked as free gas and adsorbed gas, which are different from conventional gas reservoirs.4,5 Shale porosity and permeability are extremely low.6,7 Although the use of hydraulic fracturing technology can effectively improve the shale gas production, the imperfect understanding of the seepage mechanism of shale gas reservoir is also constrained the economic and reasonable development of shale gas resource to some extent. Shale gas reservoirs usually have a certain water saturation in actual condition, while the water production in high-quality shale gas wells is extremely low.8,9 It is generally believed that such shale gas reservoirs are in the state of “sub-irreducible water saturation” and the water in these gas shale usually exists in the form of irreducible water.9,10 In shale gas reservoirs, the matrix pores can be divided into organic and inorganic matrix pores, while gas storage patterns are quite different in organic pores and inorganic pores.11-14 The organic pores are considered to be hydrophobic in traditional view,15 and the water in shale organic matrix pores is expelled during hydrocarbon generation process, which forms a continuous gas phase in the pores. While some researchers point out that water also exists in kerogen material by their experiment studies.16 Thus, there is still controversy about water distribution in organic pores. However, the inorganic pores are usually considered to have strong hydrophilic ability, and the pore surface absorbs water molecules to form bound water during hydrocarbon generation process.17,18 The evaporation of bound water film on the pore surface is one of the primary reasons for the formation of “sub-irreducible water saturation”. At present, most studies on gas transport through shale nanopores are mainly aimed at researching 2


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single-phase flow in organic matrix pores, while there is few study focusing on gas transport in inorganic pores especially for clay mineral which absorb water molecules to form water film on the pore surface.19-21 This is partly due to the fact that it is difficult to carry out molecular simulation or nano-scale experimental investigation on gas-water two-phase flow. Physics of fluid flow in shale reservoirs which depends on the flow and porous medium conditions is quite different from conventional gas reservoirs.22,23 The Knudsen number which represents the relative degree of gas molecules collision with the gas molecules and pore walls is usually used to divide the gas transfer mechanism into continuum flow, slip flow, transition flow and free-molecular flow.24-26 Furthermore, much research on apparent permeability has been carried out. Klingenberg27 first proposed the presence of gas slippage in porous media. He found that the flow rate of real gas at different pressures was higher than that predicted by Darcy's law and introduced apparent permeability which was corrected by a slipper factor. Beskok and Karniadakis28 proposed the expression for gas apparent permeability considering slip coefficient, rarefaction coefficient and Knudsen number, which was suitable for gas flow through a narrow capillary tube for all flow regimes. Javadpour29 considered both Knudsen diffusion and slippage effect of shale gas flow in nanopores and added them linearly to obtain gas apparent permeability. Civan30 proposed the expression for rarefaction coefficient and derived a model for apparent gas permeability considering the effect of the characteristic parameters of porous media. Recently, it is generally considered that the transport mechanism of shale gas in organic and inorganic nanopores is different.31-33 Inorganic matrix in shale often has high clay-material content, and the shale with higher clay content can adsorb more water molecules.34,35 For inorganic nanopores, the pore surface absorbs water molecules to form bound water film, and the gas flow in the pore center. The adsorption of methane on the water film surface belongs to gas-liquid interface adsorption. However, the presence of water will significantly decrease methane adsorption capacity. Compared with organic matter, the adsorbed gas in inorganic nanopores can be neglected. Thus, the surface diffusion and desorption of adsorbed gas is not considered in inorganic pores, and the transport capacity is dominated by bulk gas (free gas) transport. What is more, the real gas effect and stress dependence coexist during depressurization process,24 which also have significant effects on gas transport behavior. Soeder36 performed core analysis and found that gas permeability of the Marcellus was highly stress-dependent and doubling the net stress could reduce the permeability by nearly 70%. The 3


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experiment carried out by Dong et al.37 also showed that the permeability of silty-shale at Pc=3 MPa was one to two orders of magnitude larger than that at Pc=10 MPa. The effect of confining pressure on shale permeability was apparent. However, some of the shale reservoirs are deeply-buried, and the influence of real gas effect on gas transport mechanism under high pressure cannot be ignored. Wu et al.38 concluded that the real gas effect is controlled by pressure, temperature, nanopore scale and gas type, and its effect on gas transport can be as high as 23%. Pressure increasing or temperature decreasing will lead to the interaction between the gas molecules stronger, then the real gas effect on gas transport is more obvious. Wang et al.39 utilized lattice Boltzmann method to establish a pore-scale model and concluded that ignoring the real gas effect may result in overestimating shale gas production at the early stage. Furthermore, the water distribution and cross-section shape also affect the effective pore radius of inorganic nanopores, which in turn affect the gas transport capacity. Shi et al.11 developed the apparent permeability model for inorganic nanopores considering the presence of water, but the model is only applicable to the gas transport through circular cross-section pores in the Darcy flow regime or the slip flow regime without considering the influence of effective stress and real gas effect. Li et al.17 established a model to quantify the thickness of water film, and then derived gas transport model by weighted superposition of slip flow and Knudsen diffusion. The model took into account the effect of water distribution and was suitable for all flow regimes. However, Li’s model also neglected stress dependence and cross-section shape. The apparent gas permeability model proposed by Wu43 considered the effect of varying cross-section type on the transport of ideal gas and real gas, but the model is only suitable for organic matrix pores and does not analyze the effect of effective stress on pore size. Song et al.12 clarified that gas transport mechanism in inorganic pores should involve viscous flow and Knudsen diffusion. They derived apparent permeability model coupled with stress dependence, real gas effect and phase behavior. However, the model is based on gas flow in circular cross-section pores and water distribution is not taken into account. In conclusion, gas transport in inorganic nano-sized pores is influenced by many factors such as real gas effect, stress dependence, water distribution and cross-section shape. In order to establish an apparent permeability model for all flow regimes, the coupling of different flow mechanism and multiple factors need comprehensive consideration. However, it is still lacking this kind of model to accurately characterize gas transport in shale inorganic pores. Generally, the cross-sectional shapes of shale matrix pores are complex and diverse, and they can be observed as ellipsoidal, triangular, circular, rectangular and other irregular shapes under the 4


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scanning electron microscope (SEM).41-43 Afsharpoor et al.41 proposed the shape factor to characterize the different pore shapes in shale. They considered the basic cross-sectional shapes of shale matrix pores are rectangular and circular. The smaller the shape factor is, the closer the pore geometry is to rectangular shape. And the larger the shape factor is, the closer the pore geometry is to circular shape. The other irregular pore shapes can be obtained by changing the shape factor. Wu et al.43 proposed that the proportion of capillary-like nanopores and slit-like nanopores in shale matrix pores can be determined by SEM, while the others are the proportion of irregular-shaped pores. And the total mass flux model based on proportion combination of gas mass flux in capillary-like nanopores and slit-like nanopores can meet the engineering accuracy requirements in shale gas reservoirs. What is more, it is difficult and almost impossible to study the gas flow behavior in all nanopores with various cross-section shapes. Therefore, in this paper, we divide the pore cross-section shapes into circular and rectangular to simplify the model and compare the differences of gas transport behavior in different cross-section pores. Nanopores with a cicular cross-section and a rectangular cross-section are also referred to as cylindrical capillary and slit nanopores, respectively. In this paper, we first develop fully coupled apparent permeability models for inorganic matrix pores with different cross-section shape based on the weighting coefficient proposed by Wu40, the models consider the impact of real gas effect, stress dependence, irreducible water distribution on gas transport. Subsequently, the proposed model is verified by experimental data published. Further, the effect of each factor on apparent gas permeability is analyzed and discussed. 2. Mathematical model construction A lot of research44-47 indicate that molecular mean free path of gas and physical properties of nanopores are important factors influencing gas flow in nanopores. The Knudsen number is defined as the ratio of the molecular mean free path to the nanopore characteristic length, which is usually used to divide gas flow regime. According to the research results by Shi11 (see Figure 1), the gas transport mechanism in shale pores mainly includes continuum flow, slip flow and transition diffusion. The mean free path of ideal gas is given as30 λ=

µg

π RT

P

2M w

(1)

where λ is the molecular mean free path, m; µg is the gas viscosity, mPa·s; P is the pore pressure, 5


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MPa; R is the universal gas constant, J/(mol k); T is the reservoir temperature, K; Mw is the gas molar weight, kg/mol. and the expression for mean free path of real gas is30 λ=

µg

πZRT

P

2M w

(2)

where Z is gas compressibility factor, dimensionless. Figure 2 and Figure 3 depict the molecular mean free path of gas varying with pressure and temperature, respectively. The results show that the mean free path of different gas molecules is quite different. The molecular mean free under low pressure condition changes significantly with pressure variations, while the effect of temperature on the mean free path of different gas is little. The Knudsen number in circular nanopores can be calculated as follows43 Knc =

λ

(3)

D

where D is the pore diameter, m. In slit nanopores, it can be expressed as43 Kns =

λ

(4)

w

where w is the height of slit nanopores, m. and the aspect ratio for nanopores with a rectangular cross section is given below

η=

H w

(5)

where H is the width of slit nanopores, m. Figure 4 is a schematic diagram of the gas transport in nanopores with different cross-section shape. As shown in Figure 4, water molecules adsorb on the internal surface wall of pores as irreducible water. The gas adsorbed at the gas-water interface is ignored and only transport of bulk gas (free gas) exists in nanopores. Hence, our model is proposed based on the mechanism of bulk gas transport. It is also noticeable that the effective pore size of inorganic nanopores especially for the clay mineral will decrease due to the adsorption of water film and water distribution characteristic on the pore surface. Meanwhile, the pore scale is easily changed with the change of formation pressure during the production process, which affects gas transport capacity. Considering all these factors, the basic assumptions for apparent permeability models of shale gas flow in inorganic nanopores are as follows: (1) the cross-section shape of pore is circular or rectangular; (2) the pores have strong hydrophilicity with water film adsorbed on the surface, and the water film is incompressible; (3) the 6


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temperature and humidity in the nanopores are constant; (4) the dissolved gas in the water film is neglected; (5) the rock is compressible and the effect of gravity on the gas flow is ignored; (6) the height and width of the slit nanopores are changed during depressurization process, and the aspect ratio remains constant. 2.1 Real gas effect Under different temperature and pressure conditions, the mean free path of shale gas molecules is different and the slip effect will be enhanced or weakened, which affects the gas transport capacity. Thus, the real gas effect cannot be ignored. The gas compressibility factor is a function of reduced temperature and reduced pressure and can be calculated by the following expressions48 Pr = P / Pc

(6)

Tr = T / Tc

(7)

Z = 1+

  Pr  2.16  1  + 1 − 1  10.24Tr  Tr  Tr  

(8)

where Pr is the reduced pressure, dimensionless; Pc is the critical pressure, MPa; Tr is the reduced temperature, dimensionless; Tc is the critical temperature, K. The gas viscosity can be written as follows40

(

µ g = 10 −4 ζ exp Xρ g Y

ζ =

)

(9)

(9.379 + 0.01607M w )T 1.5

(10)

209.2 + 19.26M w

X = 3.448 +

986 .4 + 0.01009 M w T

(11)

Y = −0.2224X + 2.447 ρg =

(12)

PM w ZRT

(13)

2.2 Water distribution In inorganic nanopores, when the pore surface is adsorbed by water molecules, the effective pore size will be reduced. The water film is assumed to be homogeneous. According to water film thickness quantification model proposed by Li, which is derived from the thermodynamic equilibrium theory between the liquid water and vapor in the pores and the mechanical equilibrium theory, the calculation of water film thickness in capillaries is as follows17,49 7


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Π c (hc ) = −

P RT ln v∗ Vm P

Π c (hc ) = Π (hc ) + Π c (hc ) =

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(14)

γ

(15)

r − hc

− c AH εε 0 ∆ζ 2 ω − + κ e hc3 8πhc2 h

(16)

where Πc(hc) is the disjoining pressure between water film and solid surface for capillaries, MPa; Vm is the molar volume of water, cm3/mol; Pv is the partial pressure of water vapor, MPa; P* is the saturated vapor pressure of liquid film, MPa; Pv/P* is the relative humidity of gas phase, fraction; Π(hc) is the total disjoining pressure, MPa; γ is the gas-water interfacial tension, mN/m; hc is the thickness of water film in cylindrical capillary, nm; AH is the Hamaker constant for solid–gas–liquid interactions, J; ε is the relative dielectric permittivity of liquid, dimensionless; ε0 is the electric constant in vacuum, F/m; ∆ζ is the potentials difference between solid-liquid interface and liquid-air interface, mV; κ is the coefficient for the strength of structural force, N/m2; ω is the characteristic length of water molecules, nm. The expression of ∆ζ is ∆ζ = ζ 1 − ζ 2

(17)

where ζ1 is the electric potentials of the solid–liquid interfaces, mV; ζ2 is the electric potentials of the liquid–air interfaces, mV. Meanwhile, the film thickness in slit nanopores can be calculated by17,49 P RT ⋅ ln v∗ Vm P

(18)

Π s (hs ) = Π 1 (hs ) + Π 2 (hs ) + Π 3 (hs )

(19)

Π s (hs ) = −

Π1 (hs ) =

− s AH εε 0 ∆ζ 2 + + κe ω 3 2 hs 8πhs

(20)

Π 2 (hs ) =

εε 0 ∆ζ 2 AH + (w − hs )3 8π (w − hs )2

(21)

Π 3 (hs ) =

AH* (w − 2hs )3

(22)

h

where Πs(hs) is the disjoining pressure for slit nanopores, MPa; Π1(hs) is the disjoining pressure 8


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between water film and solid surface at the same side, MPa; Π2(hs) is the disjoining pressure between water film and solid surface at the opposite side, MPa; Π3(hs) is the disjoining pressure between two water films, MPa; hs is the thickness of water film in slit nanopores, nm; AH* is the Hamaker constant for liquid–gas–liquid interactions, J. Subsequently, substituting the thickness of water film calculated by eqs 15 and 19 into eqs 23 and 24, respectively. The water saturation in capillaries and slit nanopores can be obtained. Sw_c =

Sw_ s =

[

]

π r 2 − (r − rc )2 Lt  h  = 1 − 1 − c  r  πr 2 Lt 

[2whs + 2hs (H − 2hs )]Lt wHLt

=

2

(23)

2hs (w + H − 2hs ) wH

(24)

2.3 Stress dependence The porosity and permeability of shale are sensitive to the change of effective stress. The effect of stress dependence on rock porosity and permeability can be expressed by the power law37

φ = φ0 (Pe / P0 )− n

(25)

k = k0 (Pe / P0 )

(26)

−m

where ɸ0 is the porosity under atmospheric pressure, fraction; Pe is the effective stress, MPa; P0 is the atmospheric pressure, MPa; n is the porosity material constant, dimensionless; k0 is the permeability under atmospheric pressure, mD; m is the permeability material constant, dimensionless. in which Pe = Pc − χ P

(27)

where Pc is the confining pressure, MPa; χ is the effective stress constant, dimensionless. If the effect of effective stress is not taken into account, the radius of circular pore and the width of slit pore can be expressed by the intrinsic permeability and porosity.

r=

w=

8k∞ _ cτ

(28)

φ 12k∞ _ sτ

(29)

φ

According to eqs (25)-(29), the pore size of circular pore and the width of slit pore under the influence of stress dependence are given by eqs 30 and 31.

9


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rstress

k 0τ  Pe  =2 2 φ 0  P0

w stress

12 k 0τ  Pe  = φ 0  P0

  

  

n−m

n−m

P = r0  e  P0

P = w0  e  P0

  

  

n−m 2

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(30)

n−m 2

(31)

Thus, the effective pore size for inorganic pores considering the effect of water distribution and stress dependence can be written as

reff = rstress − hc

(32)

weff = wstress − 2hs

(33)

2.4 Bulk gas transport (1) Viscous flow in nanopores When Kn1, the collision between gas molecules and internal wall surface dominates the transport process. The Knudsen diffusion mass flux for real gas transports in nanopores with different cross-section shape can be described by24 J Kn _ c = −

φ 2r τ 3

J Kn _ s = −

φ w B (η ) τ η

8 ZRT PM w C g ∇P πM w ZRT

(43)

ZRT PM w C g ∇P 2π M w ZRT

(44)

Therefore, the apparent gas permeability in Knudsen diffusion regime can be derived as follow k Kn _ c =

φ 2r τ 3

k Kn _ s =

φ w B (η ) τ η

Cg =

8 ZRT µ C πM w g g

(45)

ZRT µgCg 2π M w

(46)

1 1 dZ − P Z dP

(47)

where B(η) is the section shape factor for Knudsen diffusion and is calculated by

 1 1  (1 + η 3 )− (η 2 + 1)1.5  B(η ) = η 2 ln + 1 + 2  + η ln η + 1 + η 2 + η  3   η

(

)

(48)

2.5 Coupled model As mentioned above, gas transport in the pores of shale matrix requires consideration of the multiple mechanisms coupling. At present, there are two main methods to establish the apparent gas permeability model for shale nanopores. One is the linear superposition of the permeability obtained by different mechanisms. The other is calculating the contribution of multiple mechanisms to gas 11


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transport as weighting factors, and then the model is derived considering these weighting factors. Among them, the weighting factors of slip flow and the Knudsen diffusion proposed by Wu43 are based on the ratio of gas intermolecular collision and the collision frequency between molecules and wall to the total collision frequency, which have a clear physical meaning. The expression is given as43

δ slip _ c =

1 1 + Knc

δ slip _ s =

1 Kn s (1 + 1 η ) 1+ 2

δ Kn _ c =

1 1 + 1 Knc

δ Kn _ s =

1

(49) (50)

(51)

(52)

2 1+ Kn s (1 + 1 η )

where δslip_c is the weighting coefficient for slip flow in cylindrical capillaries, dimensionless; δslip_s is the weighting coefficient for slip flow in slit nanopores, dimensionless; δkn_c is the weighting coefficient for Knudsen diffusion in cylindrical capillaries, dimensionless; δkn_s is the weighting coefficient for Knudsen diffusion in slit nanopores, dimensionless. Therefore, apparent gas permeability for capillary and slit nanopores can be obtained by eqs 53 and 54, respectively. k app _ c = δ slip _ c k slip _ c + δ Kn _ c k Kn _ c

(53)

k app _ s = δ slip _ s k slip _ s + δ Kn _ s k Kn _ s

(54)

Figure 5 shows that the weighting coefficient for capillaries and slit nanopores with different aspect ratio change with the Knudsen number. It can be seen that when the Knudsen number is less than 0.1, the slip flow dominates the flow behavior. With the increase of Knudsen number, the contribution of slip flow decreases, while Knudsen diffusion gradually dominates. Therefore, the weighting coefficients calculated by eqs 49-52 are suitable for all flow regimes.

3. Model Validation In order to validate the reliability of the proposed model, the experimental results published by Wu et al.51 are used. The experiment selected high-purity nitrogen as gas phase and was performed in water-wet nanochannels at room temperature. For the gas-displacing-water experiment, they 12


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observed that gas occupied the middle of the nanochannels and water layer attached to the edge of the channel wall, which is similar to the assumption of the model proposed in this paper. The parameters used for validation are shown in Table 1. Figure 6 depicts experimental results and apparent permeability of slit nanopores calculated by proposed model at different pressure, which indicate that the proposed model matches the experimental data very well.

4. Results and discussion Bulk gas transport in shale inorganic pores is the result of coupling multiple mechanisms. In this section, we mainly carry out the sensitive analysis of apparent permeability based on different humidity, pore pressure, cross-section shape, stress dependence, pore size, and gas type. The parameters used in the model calculation are shown in Table 2. Here, we assume that the pores of shale matrix used in analysis are inorganic nanopores. Also, the difference of pore distribution in the shale matrix is ignored, that is, all pore sizes are consistent. In addition, inorganic nanopores are considered having strong hydrophilicity, and capillary condensation phenomenon will not occur in the pores.

4.1 Influence of pore pressure The apparent permeability of cylindrical capillaries and slit nanopores with the same cross-sectional area varying with pressure are calculated, as shown in Figure 7. Meanwhile, the change of weighting coefficient is also shown in Figure 8. As shown in Figure 7a and Figure 8a, when the pore size is small, the intermolecular force is strong at high pressure, and the slip flow dominates. With the decrease of pressure, the apparent permeability is almost constant. When the pressure is lower than 20 MPa, the gas transport is mainly dominated by Knudsen diffusion. The lower the pressure, the greater the weighting factor of Knudsen diffusion. Hence, the apparent permeability increases rapidly with the decrease of pressure. As shown in Figure 7b and Figure 8b, the cross-sectional area of pores is larger, and the slip flow is the main transport mechanism. The weighting coefficient of slip flow is more than 90%. When the pressure is higher than 10 MPa, the transport capacity is affected by stress dependence and the mean free path of gas molecules and decreases gradually during depressurization process. We can also see that when the pressure is lower than 10 MPa, both the weighting factor of Knudsen diffusion and apparent gas permeability increase with pressure decreasing. In general, the effect of pore pressure on the gas transport capacity in large pores is greater than that in small pores. The change of apparent permeability for small pores is almost invisible at high pressure, while the change under low pressure is significant. 13


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Thus, we can also see that the influence of stress dependence and pore size cannot be ignored, and we will continue to study the effect of these two factors on gas transport capacity in the following sections.

4.2 Influence of cross-section shape In this paper, the cross-section shape of shale inorganic nanopres is divided into circular and rectangular to simplify the model, and we will discuss the difference of gas transport capability in these two types of nanopores. The results in Figure 9 indicate the gas transport capability in capillaries and slits with the same cross-sectional area. For the smaller nanopores, the apparent permeability under high pressure condition is slightly higher than slits. However, when the pressure drops to critical pressure, the apparent permeability of slit nanopores is significantly lower than capillaries. When the pore radius is greater than 50 nm, the apparent permeability for pores with circular cross section at any pressure is always higher than pores with rectangular cross section. In addition, for slit nanopores with the same cross-sectional area, if they have different aspect ratio, the gas transport capability is also different. This is due to the fact that when they have the same cross-sectional area, the larger the aspect ratio, the greater the height of slit nanopores. It can be found from eq 4 that the smaller the pore height, the greater the Knudsen number. This means the contribution of Knudsen diffusion is greater, while the contribution of slip flow in total gas transport is less. Hence, under low pressure condition, if the pore size for slit nanopores is small, the greater the aspect ratio, the larger the apparent permeability. In contrast, for the large pores with slip flow as the main transport mechanism, the apparent permeability decreases with the increase of aspect ratio, especially under high pressure condition. The results in Figure 10 indicate the gas transport capability in capillaries and slits with the same pore size. It is clear that the apparent permeability in slit nanopors with different aspect ratio is always higher than that in cylindrical capillaries. This is because the cross-sectional area of slits is higher than capillaries when they have the same pore size, and the higher the aspect ratio, the greater the cross-sectional area. Hence, the nanopores with rectangular cross section have higher transport capability, especially for those with large aspect ratio.

4.3 Influence of humidity As mentioned above, inorganic pores especially for clay mineral are hydrophilic, and the thickness of water film is related to relative humidity of gas phase. Therefore, the gas transport 14


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behavior in shale matrix pores with varying pressure and relative humidity of 0, 0.2, 0.4 and 0.6 is calculated. The pore radius of capillaries utilized in calculation is 20 nm, and the aspect ratio of slit nanopores which have the same cross-sectional area with capillaries is 2. As shown in Figure 11 and Figure 12, the apparent permeability of capillaries under high pressure is more sensitive to pressure change, however, the effect of pressure variation on gas transport in slit nanopores is more significant when then pressure is lower than 10 MPa. In addition, the relative humidity of gaseous phase affects the thickness of the water film formed in the pores, thus affecting the effective pore size, which leads to a decrease in apparent permeability. The higher the relative humidity, the greater the thickness of the water film. Thus, the apparent permeability is indirectly affected by relative humidity. What is more, under different relative humidity and pressure conditions, the change in gas transport capacity is different. We can see from Figs. 11b and 12b that comparing with the condition that clay bound water distribution is not taken into account, when the relative humidity in nanopores reaches 0.6, its effect on apparent permeability of circular and rectangular cross-section pores is up to 10% and 14%, respectively. It can be seen from Figure 13 and Figure 14 that the apparent permeability of nanopores with different cross-section shape decreases with the increase of relative humidity, and the relative humidity has a greater effect on large pores. When the relative humidity reaches a certain value, the curve drops sharply. This is because the thickness of the adsorbed water film will gradually increase with the increase of water vapor pressure.52 When the water film increases to a certain critical thickness, and the pores began to occur capillary condensation phenomenon. If the relative humidity is further increased, the pores will be blocked or entirely filled with capillary water, which may prevent gas flow. Thus, the point at which the apparent permeability drops sharply can be used to characterize the stability of water film. The critical relative humidity of capillaries and slit nanopores changing with pore size is shown in Figure 15. The results show that the capillary condensation is easier to occur in capillary-like nanopores than in slits, and the stability of water film is related to pore size. With the increase of the pore size, the relative humidity required for condensation is higher and it will remain constant when the pore is large enough.

4.4 Influence of stress dependence As depicted in Figure 16 and Figure 17, after considering the stress dependence, the apparent gas permeability is significantly reduced. For cylindrical capillaries, when the effective stress is less than 30 MPa, the effective pore radius drops rapidly as a result of effective stress increasing, which 15


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reduces gas transport capability at the same time. And the influence of effective stress on apparent permeability of nanopores with different cross-sectional area has no evident difference. When the effective stress is higher than 30 MPa, the larger the pore radius, the more significant the effect of the effective stress. When the effective stress surpasses 50 MPa, the Knudsen diffusion gradually dominates the flow behavior, and the apparent permeability increases with an increasing effective stress. It is also clear that the gas transport capability is more sensitive to the change of effective stress. For slit-like nanopores with the same cross-sectional area, the aspect ratio is assumed as 1. The change of apparent permeability with pressure variations is similar to capillary-like nanopores. Under the same influence of effective stress, the smaller the pore size, the less the apparent permeability decreases.

4.5 Influence of pore size Figure 18 and Figure 19 shows the change of apparent permeability in cylindrical capillaries and slit nanopores with pore size under different pressure, respectively. It is clear that apparent gas permeability and pore size are positively correlated in both capillary-like nanopore and slit-like nanopore. This is because there is no adsorption layer in inorganic nanopores, and the transport mechanism is mainly dominated by free gas. Therefore, the larger the cross-sectional area of pores, the greater the flux of gas, which means the corresponding apparent permeability is higher. When the pore size is constant, the apparent permeability which is affected by stress dependence, real gas effect, the properties of nanopores and bulk gas is different with formation pressure variation.

4.6 Influence of gas type The main component of shale gas is alkane, where methane accounts for the majority. In addition, it also contains a small amount of carbon dioxide, nitrogen, sulfur dioxide and so on. The physical properties of different gas molecules are different, which lead to different gas transport capability. Therefore, methane, ethane and carbon dioxide are chosen to analyze the effect of gas type on apparent gas permeability. As shown in Figure 20 and Figure 21, when the pressure is larger than 10 MPa, there is little difference on the apparent permeability of the three gases in capillaris or slit nanopores. When the pressure decreases from 60 MPa to 10 MPa, the apparent permeability of carbon dioxide has the most decreasing. When the pressure is lower than 10 MPa, the apparent permeability increases with the decrease of pressure, and the transport capability of methane is more sensitive to pressure change. This is because the reduced temperature of ethane is the greatest among these gases at the same 16


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temperature condition, while the methane has the minimal reduced temperature. The temperature utilized in calculation is 400 K. Under high temperature and low pressure conditions, the reduced temperature is the main factor affecting the real gas effect.53 Hence, the influence of real gas effect on methane is more significant when the pressure is lower than 10 MPa. Furthermore, the molecular mean free path of methane is apparently higher than ethane and carbon dioxide under low pressure (see Figure 22), which means the Knudsen diffusion of methane molecules is greater. Thus, the influence of pore pressure on methane is greater at a low pressure condition.

5. Conclusions In this paper, apparent permeability models for cylindrical capillaries and slit nanopores are proposed. The inorganic pores are considered to be hydrophilic, and water molecules adsorb on the internal surface wall to form irreducible and homogeneous water film. The real gas effect and stress dependence are also considered in the models. According to this study, the following major conclusions can be drawn: (1) The apparent permeability of capillaries under high pressure is more sensitive to pressure change than slit nanopores, however, the change of pore pressure has a greater effect on the apparent permeability of slit nanopores when the pressure is lower than 10 MPa. What is more, the gas transport capability decreases with the increase of relative humidity, and the relative humidity has a greater effect on large pores. The results also indicate that the capillary condensation phenomenon is easier to occur in cylindrical capillaries. (2) For the smaller pores, the apparent permeability of capillaries in a high pressure condition is larger than that in slit nanopores at the same cross-sectional area. When the pressure is less than a critical pressure, the gas transport capability of slit nanopores is larger. However, when the pore radius is larger than 50 nm, the gas transport capability in capillaries is always greater than that in slits with the decrease of pressure, and the greater the aspect ratio, the lower the apparent permeability. (3) The stress dependence has a significant effect on gas transport in both cylindrical capillaries and slit nanopores. With the increase of effective stress, apparent permeability decreases first and increases later. When the effective stress is less than the critical value, the influence of effective stress on apparent permeability of nanopores with different cross-sectional area has no evident difference. When it exceeds critical value, the greater the pore size, the more the gas transport capacity decreases. 17


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(4) Under a high pressure condition, there is little difference on the apparent permeability of methane, ethane and carbon dioxide in nanopores with different cross-section shape. Under a low pressure condition, the influence of real gas effect on methane is the largest among the three gases, and the gas transport capability of methane is obviously larger than that of ethane and carbon dioxide.

Acknowledgements This study was supported by the National Natural Science Foundation Projects of China (51504269 and 51490654), and the National Science and Technology Major Projects of China (2016ZX05042) and (2017ZX05009-003). The authors acknowledge the MOE Key Laboratory of Petroleum Engineering at China University of Petroleum (Beijing) for the permission to publish this paper.

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1

P/T (MPa/K)

Kn=0.001

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Transition flow

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Figure 1. Shale gas transport regimes in matrix pores.11

5 methane

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3 2.5 2 1.5 1 0.5 0 0

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Figure 2. The effects of pressure on mean free path of shale gas.

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0.6 methane

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0.45 0.4 0.35 0.3 250

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Figure 3. The effects of temperature on mean free path of shale gas.

(a)

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Figure 4. The schematic of gas transport in shale inorganic pores. (a) cylindrical capillary nanopore, (b) slit nanopore.

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1.2 capillary

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700

Experimental data (Sw=15%) Proposed model (Sw=15%) Experimental data (Sw=19%) Proposed model (Sw=19%) Experimental data (Sw=30%) Proposed model (Sw=30%) Experimental data (Sw=42%) Proposed model (Sw=42%)

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100 80 60

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Figure 7. The effects of pore pressure on apparent permeability with (a) r0=5 nm and (b) r0=50 nm (η=2). 25


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1

0.9

0.9

0.8

0.8

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0.7 capillary(slip) capillary(kn) slit(slip) slit(kn)

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Figure 8. The contribution of slip flow and Knudsen diffusion on total apparent permeability with (a) r0=5 nm and (b) r0=50 nm (η=2).

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500 400

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Figure 9. Apparent permeability of cylindrical capillaries and slit nanopres with the same cross-sectional area. (a) r0=5 nm, (b) r0=10 nm, (c) r0=20 nm, (d) r0=50 nm.

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10000 capillary

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slit(η=5））

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Figure 10. Apparent permeability of cylindrical capillaries and slit nanopres with the same pore size.

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-2 (kapp-ki)× × 100/ki (%)

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Figure 11. The effects of humidity on apparent permeability of cylindrical capillary. (a) apparent permeability at different pore pressure, (b) the deviation degree of apparent permeability in cylindrical capillary after considering the different water distribution. 800

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500 400 300 RH=0 RH=0.2 RH=0.4 RH=0.6

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60

70

Energy & Fuels (a)

(b)

Figure 12. The effects of humidity on apparent permeability of slit nanopore. (a) apparent permeability at different pore pressure, (b) the deviation degree of apparent permeability after considering the different water distribution (η=2).

120 100

r=2 nm

r=3 nm

r=4 nm

r=5 nm

kapp (nD)

80 60 40 20 0 0

0.2

0.4

0.6

0.8

1

RH Figure 13. The apparent permeability of cylindrical capillary varies with relative humidity for different pore radius.

40 35

w=2 nm

w=3 nm

w=4 nm

w=5 nm

30 kapp (nD)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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25 20 15 10 5 0 0

0.2

0.4

0.6

0.8

1

RH Figure 14. The apparent permeability of slit nanopore varies with relative humidity for different pore height (η=2).

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1 0.8

RH

0.6 0.4 capillary

0.2

slit

0 2

16

30

44 58 72 Pore size (nm)

86

100

Figure 15. The critical relative humidity for capillary condensation in nanopores with different cross section.

-50

r=5 nm r=10 nm r=20 nm r=30 nm

-55 (kapp-ki)× ×100/ki (%)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Energy & Fuels

-60 -65 -70 -75 0

20

40 Pe (MPa)

60

Figure 16. The deviation degree of apparent permeability after considering stress dependence varies with the effective stress in capillary.

29


Energy & Fuels

(kapp-ki)× ×100/ki (%)

-50

S=78.54 nm2 S=314.16 nm2 S=1256.64 nm2 S=2827.43 nm2

-55 -60 -65 -70 -75 0

20

40 Pe (MPa)

60

Figure 17. The deviation degree of apparent permeability after considering stress dependence varies with the effective stress in slit nanopore (η=1).

10000

P=5 MPa P=20 MPa

8000

P=30 MPa P=50 MPa

kapp (nD)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 30 of 34

6000 4000 2000 0 0

20

40

60

80

100

r (nm) Figure 18. Apparent permeability of cylindrical capillary varies with pore radius under different pore pressure.

30


Page 31 of 34

10000

P=5 MPa P=20 MPa

8000

P=30 MPa

kapp (nD)

P=50 MPa

6000 4000 2000 0 0

20

40

60

80

100

w (nm) Figure 19. Apparent permeability of slit nanopore varies with pore height under different pore pressure (η=2).

600 Methane

500

Ethane Carbon dioxide

400 kapp (nD)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Energy & Fuels

300 200 100 0 0

10

20

30 40 P (MPa)

50

60

70

Figure 20. The effects of different gas type on apparent permeability of cylindrical capillary.

31


Energy & Fuels

800 Methane

700

Ethane

kapp (nD)

600

Carbon dioxide

500 400 300 200 100 0 0

10

20

30 40 P (MPa)

50

60

70

Figure 21. The effects of different gas type on apparent permeability of slit nanopore (η=2).

5 Methane

4.5

Ethane

4

Carbon dioxide

3.5

λ (nm)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 32 of 34

3 2.5 2 1.5 1 0.5 0 0

10

20

30 40 P (MPa)

50

60

Figue 22. The molecular mean free path versus pore pressure.

32


70

Page 33 of 34

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Energy & Fuels Table 1. The parameters used in the proposed model for validation

Parameters

Value

Gas type

N2

Porosity, ɸ

0.02

Tortuosity, τ

1

Fitting constant, ɑ

4

Fitting constant, b

0.4

Gas slip constant, β

-1

Temperature, T (K)

298

Critical temperature, Tc (K)

126.1

Critical pressure, Pc (MPa)

3.4

Molecular weight, Mw (kg/mol)

0.028

Unicersal gas constant, R (J/(mol·K))

8.314

Water saturation, Sw

0.15~0.42

Model length, L (µm)

200

33


Energy & Fuels

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 34 of 34

Table 2. Parameters used in calculation and discussions

Parameters

Value

Gas type

CH4

Porosity, ɸ0

0.06

Pore radius of capillary, r (nm)

2~50

Tortuosity, τ

2

Temperature, T (K)

400

Molecular weight, Mw (kg/mol)

0.016

Unicersal gas constant, R (J/(mol·K))

8.314

Aspect ratio of slit nanopore, η

1~5

Confining pressure, Pc (MPa)

60

Atmospheric pressure, P0 (MPa)

0.1

Permeability material constant, m

0.3

Porosity material constant, n

0.04

Effective stress constant, χ

0.85

Hamaker constant for solid–gas–liquid interactions, AH (J)

1×10-20

Hamaker constant for liquid–gas–liquid interactions, AH* (J)

1.5×10-21

Gas-water interfacial tension, γ (N/m)

0.072

Electric constant in vacuum, ε0 (F/m)

8.85×10-12

Relative dielectric permittivity of liquid, ε

81.5

Potentials difference between solid-liquid interface and liquid-air 50 interface, ∆ζ (mV) Coefficient for the strength of structural force, κ (N/m2)

1×107

Characteristic length of water molecules, ω (nm)

2

Molar volume of water, Vm (m3/mol)

1.8×10-5

Fitting constant, ɑ

4

Fitting constant, b

0.4

Gas slip constant, β

-1

Critical temperature, Tc (K)

190.4

Critical pressure, Pc (MPa)

4.59

34


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