Article pubs.acs.org/IECR
Cite This: Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
Gas Flow Behavior through Inorganic Nanopores in Shale Considering Confinement Effect and Moisture Content Zheng Sun,*,†,‡ Juntai Shi,†,‡ Keliu Wu,†,‡ and Xiangfang Li†,‡ †
MOE Key Laboratory of Petroleum Engineering, China University of Petroleum (Beijing), Beijing 102249, P. R. China State Key Laboratory of Petroleum Resources and Engineering in China University of Petroleum at Beijing, Beijing 102249, China
‡
ABSTRACT: Previous attempts to characterize the gas transport through inorganic nanopores were not fully successful. The presence of an adsorption water film within nanopores is generally overlooked. Moreover, the compound influences of moisture content and confinement effect on critical properties of the gas phase have not been considered before. With the intent of overcoming these deficiencies, a fully coupled analytical model has been developed, in which complex bulk-gas transport mechanisms, moisture content, confinement effect, and various cross-section shapes of nanopores are incorporated. Results show that the confinement effect will significantly enhance the apparent gas permeability when the pore radius is smaller than 5 nm, and the real-gas effect can achieve an average increase of 4.38% when the pore radius falls in the range 1−2 nm. The stress dependence will greatly decrease the apparent gas permeability and the corresponding degree for slitlike inorganic nanopores will slightly increase with the increasing aspect ratio.
1. INTRODUCTION Currently, due to the improved horizontal drilling and hydraulic fracturing technologies, commercial gas production of shale formation emerges as a significant source of energy in the United States, Canada, and China.1−6 And the global remaining technically recoverable resource of shale gas is estimated as 213 tcm (1 tcm = 1 × 1012 m3), which plays a significant role on the balance of the rapidly rising energy demand all over the world.7,8 Nanopores are abundant in the shale matrix, which has a large surface area, and their pore sizes approach the molecular mean free path. Therefore, gas flow behavior within nanopores cannot be simulated as a continuum process. The accurate knowledge of gas flow behavior through nanopores expects to cement the theoretical basis for the numerical simulation and production prediction, which will greatly advance the development of shale gas reservoirs. To our best knowledge, the majority of current evaluations on gas transport capacity in nanopores are based on the dry condition, which is conflicted with the actual atmosphere of the shale gas reservoirs.9−18 Statistical data show that the initial saturation of Barnett Basin, Marcellus Basin, Haynesville Basin, Fayetteville Basin is 0.25−0.35, 0.12−0.35, 0.15−0.35, and 0.25−0.5, respectively.19−21 The presence of water will adversely affect the gas transport capacity and is generally overlooked by previous researchers. Thus, it is significantly necessary to shed light on the effect of water on gas flow behavior through nanopores. It is a well-established fact that the shale formation is heterogeneous rock characterized by both organic and inorganic matters.22−24 And it is generally believed that the organic pores in kerogen formed during hydrocarbon generation process is hydrophobic and without water in terms of existing literature. On the contrary to the feature of © XXXX American Chemical Society
organic pores, the inorganic pores are considered as hydrophilic; thus the water molecules can adsorb on the pore surface. Thus, it can be concluded that the presence of water will mainly affect the gas flow behavior through inorganic nanopores. Note that gas is stored in shale formation as free gas and adsorbed gas. Under dry conditions, both the organic and inorganic nanopores serve as the adsorption media due to the large internal surface area. Experimental evidence report that the adsorption capacity of inorganic pores will decrease by 80−95% under moist conditions,25−27 which can be explained that water molecules occupy the adsorption sites on the surface of inorganic nanopores. Therefore, adsorption gas within inorganic pores can be neglected. For the realistic shale gas reservoir, adsorption gas and free gas coexist in the organic pore, and only free gas transports through inorganic nanopores. During the depressurization development process of shale gas reservoirs, it can be concluded that the organic pores serve as the source of the hydrocarbon and the inorganic pores can be regarded as the bridge between organic pores and artificial hydraulic fractures. Therefore, the gas transport capacity of inorganic nanopores is crucial for the gas production process in shale and is significantly necessary to be investigated. In terms of the aforementioned gas−water distribution characteristics in organic and inorganic nanopores of shale gas reservoirs, it can be demonstrated that there exists a large difference for gas flow behavior in the two kinds of pores. Gas flow behavior through organic nanopores should take the bulkReceived: Revised: Accepted: Published: A
January 18, 2018 February 20, 2018 February 21, 2018 February 21, 2018 DOI: 10.1021/acs.iecr.8b00271 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
Article
Industrial & Engineering Chemistry Research
mechanisms. However, the model was designed for gas flow in natural fractures and therefore can only be utilized for slitlike nanopores. Incorporating the water distribution characteristic in the shale matrix, Sun developed apparent permeability model for real-gas transport through inorganic nanopores, which considered the bulk-gas transport mechanisms, stress dependence, real-gas effect, and thickness of the water film.28 However, Sun’s model can be only applied for slitlike pores. The thickness of the water film adsorbed on the slit nanopores was considered as uniform; however, there exists a difference in the walls for width side and height side. Moreover, the confinement effect was neglected and the influence of moisture content on the gas transport capacity was also overlooked in Sun’s research. In this paper, similar to previous published literature, the gas transport capacity through inorganic nanopores is described by the concept of apparent gas permeability. Moreover, with the intent of considering various cross-section shapes, two apparent permeability models are developed for the cylindrical nanopores and slitlike nanopores, respectively. Notably, the thickness of the water film adsorbed on pore walls of different cross-section shapes is quantified by Li’s model, which is based on the thermodynamic equilibrium between liquid phase and vapor phase.40 The confinement effect in the nanopores is captured on the basis of Wu’s fitting equations, which are based on data from dozens of molecular dynamics simulation and experimental results in the previous literature.41 The influence of moisture content in the gas phase on methane critical properties is quantified through the well-established calculation formulas from the technical book written by Sun,42 and Beskok’s model is utilized to characterize the bulk-gas transport mechanisms, which has a simple formula form and can cover all the known bulk-gas transport mechanisms.43 More features, such as the real-gas effect and stress dependence are coupled in both models. Subsequently, on the basis of the proposed models, the influences of several key factors on transport capacity are identified, including cross-section shapes, moisture content, confinement effect, and so on. The proposed model can achieve more precise knowledge of the gas flow behavior through inorganic nanopores and can serve as the theoretical basis for the next generation numerical simulation for shale gas reservoirs.
gas transport mechanisms, surface diffusion, and gas adsorption/desorption into account.28−33 And sufficient research has been performed to investigate the gas flow features in organic pores and achieved desirable results and mature insights. However, gas transport behavior through inorganic pores has not received due attention. Previous attempts to establish models for nanoconfined gas flow in inorganic pores were not fully successful. One issue can be attributed to the fact that some important physical phenomena at the nanoscale inorganic pore were not incorporated, including the presence of a water film, the confinement effect, and various cross-section shapes. In addition to these factors, gas flow behavior through inorganic nanopores is also influenced by bulk-gas transport mechanisms, stress dependence, and the real-gas effect. Moreover, all these factors are closely associated and affect each other, which greatly aggravates the complexity of this issue. The moisture content, confinement effect, and various cross-section shapes will simultaneously influence the critical properties of methane, which can also affect the gas flow behavior through inorganic nanopores. Both experimental measurements and MD simulation are employed by scholars to investigate the nanoconfined gas flow characteristics.34−36 However, two methods suffer fatal deficiencies, because they are expensive, data-intensive, and time-consuming. In contrast, on the basis of some reasonable assumptions, an analytical model not only can provide instantaneous calculation results but also is convenient to identify the effect of each physical key parameter. At present, a model that can consider the aforementioned essential complexity for gas transport through realistic inorganic nanopores is still lacking. Some scholars have proposed correlated models for describing the gas transport capacity through inorganic nanopores in shale. Javadpour, for the first time, presented images of nanopores in mudrocks (shales and slitstone) by utilizing the atomic force microscopy (AFM).37 Subsequently, Javadpour proposed an apparent permeability for gas flow in the nanopores based on the linear superposition of Knudsen diffusion and slip flow.29 Note that the concept of apparent permeability takes the form of the Darcy equation so that it can be easily implemented for the numerical simulation. However, the model established by Javadpour failed to capture multiple key effects mentioned above. Civan presented a rigorously apparent permeability model for gas transport through nanopores, which accounted for the effects of characteristic parameters, the rarefaction coefficient, and the Klinkenberg gas slippage factor.31 However, although all the known bulk-gas transport mechanisms can be described by Civan’s model, the presence of water, the confinement effect, and different crosssection shapes are not incorporated. Employing the apparent permeability model proposed by Javadpour, Darabi applied several modifications to upscale the model from a straight cylindrical nanotube to ultratight natural porous media.30 Thus, Darabi’s model suffered the same deficiencies of Javadpour’s model. Rahmanian proposed a novel unified model for gas transport through tight gas reservoirs, which considered the nanopore geometry and could cover entire bulk-gas flow mechanisms.38 However, the moist condition and confinement effect were still not considered. Wu proposed a model for gas transport in microfractures of shale or tight gas reservoirs, which is based on the weight superposition of slip flow and Knudsen diffusion.39 Wu’s model can reasonably describe the process of the mass transform of different bulk-gas transport
2. MODEL DESCRIPTION AND ESTABLISHMENT To develop a fully coupled analytical model for gas transport through inorganic nanopores with various cross-section shapes, several reasonable assumptions are performed. First, the basic cross-section shapes of nanopores are classified as circular and rectangular. Second, both proposed models are based on the effective hydraulic radius assumption, which is widely used by previous apparent permeability models. In this paper, the effective hydraulic radius will decrease due to the presence of an adsorption water film and the effect of stress dependence, and the confinement effect and moisture content will evolve the critical properties of the gas phase, which will influence the realgas effect and further affect gas transport capacity. Finally, the shale matrix is assumed as a connected system and therefore the relative humidity (RH) in the nanopores can be considered as uniform, which is defined as the ratio of the vapor partial pressure to saturate vapor pressure. 2.1. Physical Models. As depicted in Figure 1, the presence of a water film is incorporated in both physical models for gas transport through cylindrical and slitlike nanopores. Note that B
DOI: 10.1021/acs.iecr.8b00271 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
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formation pressure, MPa; and η is the porosity stressdependence factor, MPa−1. Similarly, under the effect of stress dependence, the relationship between the matrix permeability and formation pressure has the following expression.18,44
K (P) = K ini e−ψ (Pini − P)
(2)
where Ψ is the permeability stress-dependence factor, MPa−1, and Kini is the intrinsic permeability when the effective stress is zero, m2. The formulas of intrinsic permeability for nanopores with different cross-section shapes are different; eq 3 is for the cylindrical nanopores and eq 4 is for the slitlike nanopores.45 Figure 1. Diagram of the inorganic nanopores with various crosssection shapes.
K ini,tube =
gas transport capacity of slitlike nanopores depends heavily on the aspect ratio (AR), which is defined as the ratio of the width and the height. As mentioned above, the thickness of the water film adsorbed on the width side and the height side are different, and the previous research implemented by Sun assumed the thicknesses of the water film on both sides were same in the slitlike nanopores, which would be corrected in this work. Subsequently, after considering the stress dependence and the presence of a water film, the aspect ratio will change and further influence the apparent permeability for slitlike nanopores. Moreover, for gas flow behavior in cylindrical and slitlike nanopores, the decrease of the pore scale will result in a more obvious confinement effect, which will decrease the critical properties of gas phase. More importantly, the presence of water vapor in the bulk-gas phase is overlooked in all current apparent permeability models, the effect of which on gas transport capacity has not been investigated before. According to the statistical data, the reservoir depth of shale is generally 3000−6000 m with up to 60 MPa of initial reservoir pressure and approximately 450 K of the reservoir temperature.18 The high temperature corresponds to the high saturate water vapor pressure. During the depressurization development process of shale gas reservoirs, the moisture content will increase with the decrease of the pressure and therefore its effect will become gradually obvious with production. Thus, it is rather attractive and necessary to shed light on the mechanism. All these effects are incorporated to establish the gas transport model for realistic shale matrix in section 2, and the validation efforts are implemented in section 3. After that, the proposed apparent permeability models are utilized to identify the influence of each key property in section 4. Finally, conclusions are in section 5. 2.2. Effect of Stress Dependence. The permeability and porosity of the shale matrix will decrease with the decrease of formation pressure, which can be described as the stressdependence effect. As illustrated in Figure 1, the stress dependence will decrease the effective pore size of nanopores, which has a negative effect on the gas transport capacity. The correlation formula to represent the relationship between porosity and formation pressure is44,45 ϕ(P) = ϕr + (ϕini − ϕr)e−η(Pini − P)
K ini,slit =
ϕiniR ini 2 8τ
(3)
ϕinih ini 2 12τ
(4)
where Rini is the initial radius for the cylindrical nanopores, m; hini is the initial height for the slitlike nanopores, m; and τ is the touristy of the nanopores, dimensionless. Furthermore, according to eqs 3 and 4, the relationship between intrinsic petro-properties and pore size can be obtained. R ini =
h ini =
8τ
K ini ϕini
12τ
K ini ϕini
(5)
(6)
Substituting the permeability and porosity under the effect of stress dependence into eqs 5 and 6, the relationship between the effective pore size and formation pressure considering stress dependence can be derived. Note that an assumption is made here that the width of the slitlike nanopores is insensitive to the effect of stress dependence and remains unchanged under different effective stress condition. K (P ) ϕ(P)
R stress =
8τ
hstress =
12τ
K (P ) ϕ(P)
(7)
(8)
where Rstress is the pore radius of cylindrical nanopores considering stress dependence, m, and hstress is the height of slitlike nanopores considering stress dependence, m. 2.3. Thickness of Adsorption Water Film. Due to the moisture condition within realistic shale matrix, the water molecules can be stored in the reservoir with two forms: the water film adsorbed on the pore surface on one hand and the water vapor in the bulk-gas phase on the other hand. Thus, it is significantly necessary to quantify the effect of water storage mechanisms on the gas transport capacity. The effect of the water film will be incorporated in this section and the influence of water vapor will be considered in section 2.5. In terms of Li’s model based on the thermodynamic equilibrium theory between the liquid phase and vapor phase within nanopores, the thickness of the water film in the slitlike and cylindrical nanopores can be determined by the following formula.40
(1)
where ϕ(P) is the shale matrix porosity considering stress dependence, dimensionless; ϕr is the porosity under high effective stress, dimensionless; ϕini is the initial matrix porosity when the effective stress is zero, dimensionless; Pini is the initial C
DOI: 10.1021/acs.iecr.8b00271 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
Article
Industrial & Engineering Chemistry Research Π(hw )Vm = −RT ln
Pv Po
For the water film inside cylindrical nanopores, besides the disjoining pressure caused by the pore surface, the additional cylindrical capillary pressure caused by wetting film should also be taken into account. The effective disjoining pressure inside capillaries is given below.40
(9)
where Vm is the water molar volume, m /mol; Π(hw) is the disjoining pressure between the solid phase and water film, MPa; Po is the saturated water vapor pressure, which is dependent on the reservoir temperature, MPa; and Pv is the partial pressure of water vapor in the bulk-gas phase, MPa. For the slitlike nanopores, the total disjoining pressure encompasses the London-van der Waals force, electrical force, and structural force, which has the following expression. 3
Π(hw ) = Π1(hw ) + Π 2(h w) + Π3(hw ) Π1(hw ) =
AH hw
3
+
εεo(ξ1 − ξ2)2 8πhw
2
Π(hw ) =
Π flat(hw ) = Π m(hw ) + Πe(hw ) + Πs(hw )
(10)
Π m (h w ) = −
+ κ e −h w / Γ (11)
Notably, there is a difference for the thickness of the water film adsorbed on the width-side and height-side walls for the slitlike nanopores. For the width-side wall, eqs 12 and 13 should be applied, and eqs 14 and 15 are for the height-side wall. Π 2( h w ) = Π3(hw ) =
Π 2( h w ) =
Π3(hw ) =
AH (hstress − hw )3
+
(12)
AH * (hstress − 2hw )3 AH (wini − hw )3
+
(13)
εεo(ξ1 − ξ2)2 8π (wini − hw )2
(14)
where Π1(hw) is the disjoining pressure between the water film and surface at the same side by considering both the shortrange structural force and long-range molecular force, MPa; Π2(hw) is the disjoining pressure between the water film and surface at the opposite side, MPa; Π3(hw) is the disjoining pressure between two water films, MPa; AH is the Hamaker constant for solid−gas−liquid interactions, J; AH* is the Hamaker constant for liquid−gas−liquid interactions, J; εo is electric constant in vacuum, F/m; ε is the relative dielectric permittivity of liquid, dimensionless; ξ1 is the electric potentials of the solid−liquid interfaces, mV; ξ2 is the electric potentials of the liquid−air interfaces, mV; κ is the coefficient for the strength of structural force, N/m2; and Γ is the characteristic length of water molecules, nm. Therefore, the effective height and width of the slitlike nanopores considering both stress dependence and thickness of the adsorption water film can be described below: (16)
heff = hstress − 2hw,height
(17)
(
) )
(20)
+2
12πhw 3 1 + 5.32
hw 2 l
(21)
Πe(hw ) = 2εrεo(kBT /χe)2 ζ 2
(22)
Πs(hw ) = k e−hw / ω
(23)
(24)
where hw,capillary is the thickness of the water film inside cylindrical nanopores at a given relative humidity, m. 2.4. Confinement Effect. It is well-established that the gas flow behavior under the confinement effect is significantly different from the bulk-gas flow due to the non-negligible van der Waals forces, and pore size and geometry influence both the constraint limiting gas molecules number and the van der Waals forces exerted by nanopores walls. Subsequently, the critical properties of the gas phase will change due to the confinement effect. Moreover, research has reported that the critical pressure and temperature will significantly decrease when the pore size approaches the molecular diameter. In this paper, the varying extent of critical properties of the confined gas is quantified by Wu’s formulas, which are fitting equations in terms of sufficient MD simulation results. The equations for the variation of gas critical properties within cylindrical nanopores are given below:41
(15)
weff = wini − 2hw,width
hw l
R eff = R stress − hw,capillary
AH * (wini − 2hw )3
(
Agws 15.96
(19)
where γ is the gas−water surface tension, mN/m; Agws is the Hamaker constant in the gas−water−solid system, J; l is the London wavelength, m; kB is the Boltzmann constant, J/K; T is the formation temperature, K; e is the electron charge, C/ number; χ is the ion valence, dimensionless; ζ is a parameter related to the surface charge density and film thickness, dimensionless; and ω is the characteristic decay length, m. Considering both the stress dependence and the thickness of the water film, the effective radius for cylindrical nanopores can be described as follows.
εεo(ξ1 − ξ2)2 8π (hstress − hw )2
R stress γ Π flat(hw ) + R stress − hw R stress − hw
where hw,width is the thickness of the water film adsorbed on width-side wall, m, and hw,height is the thickness of the water film adsorbed on height-side wall, m. And the correspondence effective aspect ratio of slitlike nanopores can be calculated, which is the key value affecting the gas transport capacity. w AR eff = eff heff (18)
−1/0.88 ⎛D d ⎞ Tcc/Tcb = 1 − 1.2⎜ − a ⎟ ⎝σ σ⎠
(25)
−1/1.6 ⎛D d ⎞ Pcc/Pcb = 1 − 1.5⎜ − a ⎟ ⎝σ σ⎠
(26)
D = 2R eff
(27)
where D is the effective pore diameter of the cylindrical nanopores, m; da is the layer thickness of the adsorbed gas in nanopores, which can be assumed as zero due to the presence of a water film; σ is the Leonard-Jones parameter for nanopores, which is assigned as 0.28 nm in this paper; Tcc is the critical temperature of gas considering the confinement D
DOI: 10.1021/acs.iecr.8b00271 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
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Industrial & Engineering Chemistry Research effect, K; Pcc is the critical pressure of gas considering the confinement effect, MPa; Tcb is the critical temperature of bulk gas, K; and Pcb is the critical pressure of bulk-gas, MPa. In terms of the aforementioned text, the geometry of nanopores will also influence the variation regularity of critical properties of nanoconfined gas. For the slitlike nanopores, the corresponding formulas have the following expressions.
molecular is comparable with the pore size of nanopores and its volume will significantly influence the gas flow behavior. Hence, the real-gas effect cannot be neglected for the nanoconfined gas transport, which can be described by accounting for the gas compressibility factor, viscosity. The mentioned above parameters are functions of pseudopressure and pseudotemperature, which can be described by following formulas.18,42
−1/0.96 ⎛h d ⎞ Tcc/Tcb = 1 − 0.4⎜ eff − a ⎟ ⎝ σ σ⎠
(28)
Pr =
P Pc
(35)
−1/2 ⎛h d ⎞ Pcc/Pcb = 1 − 0.65⎜ eff − a ⎟ ⎝ σ σ⎠
(29)
Pr =
P Pc
(36)
According to the equations in this section, the critical properties for the gas transport through cylindrical or slit nanopores can be directly calculated. Therefore, the confinement effect within the nanopores can be captured and the concrete effect on the apparent gas permeability will be discussed in section 4. 2.5. Moisture Content in the Shale Matrix. As mentioned above, water storage mechanisms within nanopores include an adsorption water film and water vapor in the gas phase. The water vapor in the gas phase can be quantified utilizing the concept of moisture content, which is dependent on the relative humidity (RH), formation pressure, and formation temperature (eq 30). The effect of water vapor on the methane critical properties will become obvious when the atmosphere falls in the high-pressure range (>10 MPa). Moreover, the formation pressure of shale gas reservoirs is generally in the range 5−30 MPa. Thus, it is necessary to consider the effect of moisture content on methane critical properties, which have the following formulas.42 Po·RH P
(30)
DevTc = 222.22·ymoi
(31)
DevPc = 8.7542·ymoi
(32)
ymoi =
Tc =
Pc =
Tcc − 647.2·ymoi 1 − ymoi
Pcc − 22.058·ymoi 1 − ymoi
Z = 0.702Pr 2e−2.5Tr − 5.524Pr e−2.5Tr + 0.044Tr 2 − 0.164Tr (37)
+ 1.15
μ = (1 × 10−4)ξ exp(XρY )
(38)
ρ=
pM ZRT
(39)
ξ=
(22.65 + 0.0388M )T1.5 (209.2 + 19.26M + 1.8T )
(40)
X = 3.448 + 548 + 0.01M
(41)
Y = 2.447 − 0.224X
(42)
where Z is the gas compressibility factor, dimensionless, and μ is the gas viscosity, mPa·s. In terms of these equations, the gas compressibility factor and gas viscosity can be obtained at a given reservoir pressure and temperature. It should be noted that the critical properties of the gas phase are corrected by considering the confinement effect and moisture content. These physical properties will be further utilized to calculate the mean free path and subsequently the Knudsen number, which can determine the bulk-gas transport mechanisms. 2.7. Gas Transport Capacity through Inorganic Nanopores with Various Cross-Section Shapes. On the basis of the Knudsen number, the bulk-gas transport mechanisms can be classified as continuum flow (Kn < 0.001), slip flow (0.001 < Kn < 0.1), transition flow (0.1 < Kn < 10), and Knudsen diffusion (Kn > 10). The definition of the Knudsen number is the ratio of mean free path to the characteristic length of cylindrical or slit nanopores. The mean free path of real gas can be expressed as13
+ DevTc (33)
+ DecPc (34)
where ymoi is the moisture content in the gas phase, dimensionless; Tc is the critical temperature of gas considering both the confinement effect and moisture content, K; Pc is the critical pressure of gas considering both the confinement effect and moisture content, MPa; and RH is the relative humidity in the shale gas reservoirs, dimensionless. Analyzing introduced equations in sections 2.3 and 2.5, it can be found that the relative humidity can be utilized not only to calculate the thickness of the water film but also to quantify the effect of the moisture content in the nanopores. Moreover, the relative humidity of a reservoir is a common petro-property that can be easily determined. Therefore, the relative humidity is selected to perform the sensitive analysis for the water storage mechanisms on the gas transport capacity in section 4. 2.6. Real-Gas Effect. Due to the existence of abundant nanopores in shale gas reservoirs, the diameter of a gas
λ=
μ P
πZRT 2M
(43)
where R is the universal gas constant, J/(mol·K), and M is the gas molar weight, kg/mol. For the nanotubes, the formula for calculating the Knudsen number can be expressed as43
Kn =
λ Reff
(44)
Subsequently, on the basis of Beskok’s model, the apparent gas permeability for cylindrical nanopores can be developed: E
DOI: 10.1021/acs.iecr.8b00271 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
Article
Industrial & Engineering Chemistry Research Kcylin = α = α0
2 ⎛ ϕ(P) R eff 4Kn ⎞ ⎟ (1 + αKn)⎜1 + ⎝ τ 8 1 + Kn ⎠
2 tan−1(α1Kn β ) π
Boltzmann solutions and Tison’s experiments. The influence of the existence of water on the gas transport capacity has been seriously considered in this work. Hence, another experiment regarding the gas−water two-phase flow behavior in nanoscale channels performed by Wu is selected.48 The displacement of two-phase flow in slitlike channels with a dimension of 100 nm depth and 5000 nm width are performed in Wu’s experiments and the flow mechanism and capacity in the nanoscale slitlike channels are reported. In this view, Wu’s experimental data are fairly appropriate for the validation process for the proposed model for slitlike nanopores. The basic parameters can be found in Table 1. To validate the proposed model for
(45)
(46)
where αo is the rarified effect coefficient when Kn → ∞, dimensionless, and α1 and β are the fitting constants, dimensionless. For the slitlike nanopores, the Knudsen number has the following expression.
Kn =
λ heff
(47)
Table 1. Summary of Parameters in the Validation Process
Similarly, the apparent permeability for gas transport through slitlike nanopores can be derived according to Beskok’s basic model. K slit
parameters (unit) gas type initial porosity tortuosity methane molar mass (kg/mol) critical pressure for bulk-gas (MPa) critical temperature for bulk-gas (K) universal gas constant (J/(mol·)) temperature (K) saturated water vapor pressure (MPa) radius for cylindrical nanopores (nm) aspect ratio for slitlike nanopores rarefaction effect coefficient fitting constant fitting constant permeability stress-dependence factor (Pa−1) porosity stress-dependence factor (Pa−1) initial pressure Hamaker constant (solid−gas−liquid) (J) Hamaker constant (liquid−gas−liquid) (J) electric constant in vacuum (F/m) electric potentials (solid−liquid) (mV) electric potentials (liquid−gas) (mV) coefficient for structural force (N/m2) characteristic length of water molecules (nm)
⎛ h 2 ϕ(P) 6Kn ⎞ ⎟ = C(AR eff ) eff (1 + αKn)⎜1 + ⎝ τ 12 1 + Kn ⎠ (48)
C(AR eff ) = 1 −
192 AR eff ·π 5
∞
∑ i = 1,3,5,...
tanh(iπ ·AR eff /2) i5 (49)
where C(AReff) is the correction factor for the apparent permeability of slitlike nanopores with diverse aspect ratios. The analytical permeability model for cylindrical and slitlike inorganic nanopores are developed; the concrete formulas are eq 45 and eq 48, respectively. It can be observed that the proposed models are similar to the normal bulk-gas transport model for nanopores, which possess simple and concise forms. Moreover, all known key effects are incorporated in these models. Specifically, the stress dependence and thickness of the water film are incorporated to calculate the effective pore size and the effective aspect ratio of nanopores, and the confinement effect and moisture content are considered to obtain the critical properties of the gas phase and calculate the mean free path and the Knudsen number. Notably, different formulas of these key mechanisms for nanopores with various cross-section shapes are comprehensively considered, including an adsorption water film, stress dependence, confinement effect, and bulk-gas transport mechanisms.
symbol
value
ϕ τ M Pcb Tcb R T Po Rini AR α0 α1 β Ψ η Pini AH AH* εo ξ1 ξ2 κ Γ
CH4 0.05 2.3 0.016 4.6 190.6 8.314 413 0.361 2−100 50 1.19 4.0 0.4 2.5 × 10−8 5 × 10−8 50 1 × 10−20 1.5 × 10−21 8.85 × 10−12 100 50 1 × 10−7 1.5
cylindrical nanopores, the relationship between the ratios of the apparent gas permeability to the initial permeability versus Knudsen number is illustrated. During the validation process of the proposed model for slitlike nanopores, the relationship between apparent permeability and water saturation is plotted, which is consistent with the form of the experimental results collected from Wu’s literature. As illustrated in Figure 2, it can be demonstrated that the reliability of the proposed model for cylindrical nanopores is successfully verified with excellent agreement with the Boltzmann solutions and experimental data, and the reliability of the proposed model for slitlike nanopores can be also clarified through the comparison with the experimental results, which are illustrated in Figure 3. In these regards, it can be concluded that the proposed two models can match the existed experimental data, the reliability of which are well clarified. The experiments are considered as the closest to the reality. However, experiments for gas transport through nanopores are generally expensive, time-consuming, and therefore hard to identify effects of key parameters. In contrast, analytical models established in this research can serve as the powerful tool to
3. MODEL VALIDATION With the objective of verifying the reliability of the proposed models, the existing linearized Boltzmann solutions and two sets of experimental data are collected. In detail, Loyalka studied the rarefied gas flow in a cylindrical tube by utilizing the linearized Boltzmann equation.46 Through a series of experiments implemented by Tison, the conductance of capillary leaks has been quantified for a variety of gases over a wide range of pressure.47 It can be concluded that the Boltzmann solutions and experimental data from Tison are focused on the gas transport capacity through nanotubes. Thus, these data are appropriate to clarify the reliability of the proposed permeability model for cylindrical nanopores in this work. Although both proposed models are established in a similar way, the model for slitlike nanopores still requires verification with additional convincing data due to the utilization of different concrete formulas. Moreover, it should be noted that the presence of water has not been considered in the above F
DOI: 10.1021/acs.iecr.8b00271 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
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Industrial & Engineering Chemistry Research
depicted in Figure 4, it can be observed that the relationships between apparent gas permeability and pressure for various
Figure 2. Validation of the proposed model for gas transport through cylindrical nanopores.
Figure 4. Gas transport capacity through nanopores with various cross-section shapes.
cross-section shapes show the similar variation tendency. The gas transport capacity will first decrease and then increase with the decreasing pressure, which can be explained by stress dependence and Knudsen diffusion. At a relatively highpressure condition, it can be reasonably demonstrated that the bulk-gas transport mechanism belongs to the continuum flow or slip flow, the gas transport capacity of which is heavily dependent on the pore size. Moreover, the stress dependence will decrease the effective pore size with the decrease of the pressure; therefore, the apparent gas permeability will first decrease with the decreasing pressure. With a further decrease of pressure, the bulk-gas transport mechanism transitions to the transition flow or Knudsen diffusion, which is heavily dependence on the value of Knudsen number (eqs 45 and 48). At a low-pressure condition, the Knudsen number will significantly increase with the decrease of pressure; thus the apparent gas permeability will then increase with the decreasing pressure. More importantly, as depicted in Figure 4, it should be noted that the gas transport capacity of slitlike pores exceeds that of the cylindrical nanopores at a certain pressure, and it can also be concluded that the slitlike nanopores with larger aspect ratios will achieve the stronger gas transport capacity. 4.2. Real-Gas Effect. Within the nanoscale pores, the volume of molecules itself cannot be overlooked and therefore the real-gas effect must be considered. In this section, the constant pressure condition is set and the value is 20 MPa, and the inorganic nanopores are assumed as cylindrical nanopores. As illustrated in Figure 5, the relationship between the apparent gas permeability and pore scale is presented. It can be observed that the real-gas effect will enhance the gas transport capacity when the pore radius is in the range 1−2 nm. After further calculation, the average enhancement degree is 4.38%. It can be also observed from Figure 5 that the real-gas effect has little influence for the larger inorganic nanopores (>2 nm). Notably, through the water sorption experiments, Zolfaghari found that the average inorganic nanopores in the shale matrix approximate 5 nm,51 which is larger than 1−2 nm. Thus, the conclusion seems to be generated that the real-gas effect has little effect on the gas transport capacity and can be neglected. However, experimental results represent the pore scale without considering stress dependence and the thickness of the water
Figure 3. Validation of the proposed model for gas transport through slitlike nanopores.
yields generally instantaneous predictions and observations. The related sensitivity analysis will be performed in section 4.
4. RESULTS AND DISCUSSION From the aforementioned context, multiple key mechanisms coexist during the depressurization development process of shale gas reservoirs, which all have influences on gas transport behavior in nanopores. In this section, on the basis of the proposed models for gas transport capacity through inorganic nanopores, the effects of various cross-section shapes, the realgas effect, the confinement effect, the relative humidity, and stress dependence are quantified. The basic physical properties can be found in Table 1. 4.1. Various Cross-Section Shapes. Different crosssection shapes will significantly change the mass transfer process within nanopores. Afsharpoor confirmed that the assumption of simplified cylindrical nanopores would result in overestimation for the apparent liquid permeability.49,50 Thus, it is crucial to identify the effect of cross-section shapes of inorganic nanopores on gas transport capacity. In this section, both the initial pore diameter of cylindrical nanopores and the initial height of the slitlike nanopores are set as 20 nm. As G
DOI: 10.1021/acs.iecr.8b00271 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
Article
Industrial & Engineering Chemistry Research
than 5 nm and can be neglected when the pore radius is greater than 5 nm. The phenomenon can be attributed to the fact that the confinement effect has significant influence on the critical properties of the gas phase when the pore radius is smaller than 5 nm and has little influence when the pore radius is larger than 5 nm. 4.4. Relative Humidity. As mentioned above, the water molecules can be stored in the nanopores in the form of an adsorption water film and in the form of moisture content in the gas phase, both of which can influence the gas transport capacity through inorganic nanopores. The water film will decrease the pore size of the nanopores, which can directly decrease the apparent gas permeability. The thickness of the water film can be quantified in terms of section 2.3, and the moisture content in the gas phase will influence the critical properties of gas (section 2.5), which can also affect the apparent gas permeability. In this section, the initial pore radius is set as 5 nm and the cross-section shape is assumed as circle. As illustrated from Figure 7, it can be observed that the
Figure 5. Real-gas effect on the gas transport capacity through inorganic nanopores.
film. Considering these effects in the realistic shale gas reservoirs, the effective pore size will decrease and become close to the above range (1−2 nm), especially for the lowpressure condition with a strong effect of stress dependence. Thus, for the development process of realistic shale gas reservoirs, the enhancement phenomenon caused by the realgas effect cannot be neglected, and it should be noted that the real-gas effect is based on the critical properties (section 2.6), which are influenced by both the confinement effect and moisture content. A concrete effect of both factors will be seriously discussed in the following sections. 4.3. Confinement Effect. The critical properties will significantly decrease when the pore scale is close to the molecule diameter, which is the important feature of the confinement effect. The influence on the gas transport capacity is fairly attractive and significantly required to be investigated. As illustrated in Figure 6, the confinement effect will greatly contribute to the increase of the gas transport capacity when the pore radius is 1 nm, and it can be observed that the enhanced degree will increase with the decreasing pressure, and the average enhancement degree is calculated as 28.21%. Meanwhile, it can be found that the confinement effect has little influence on the apparent permeability when the pore radius is 5 nm and can be neglected. Thus, the confinement effect can only affect the flow behavior when the pore radius is smaller
Figure 7. Effect of the water film on the gas transport capacity through inorganic nanopores.
apparent permeability will decrease with the increasing relative humidity at a certain pressure, which is consistent with the above analyzed statement regarding the water film. Moreover, it can be demonstrated from Figure 8 that the effect of moisture has little effect on the apparent gas permeability for nanopores and can be neglected. Analyzing the reason behind this
Figure 6. Confinement effect on the gas transport capacity through inorganic nanopores. H
DOI: 10.1021/acs.iecr.8b00271 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
Article
Industrial & Engineering Chemistry Research
Figure 8. Effect of moisture content on the gas transport capacity through inorganic nanopores.
Figure 10. Average influencing degree of stress dependence for nanopores with various cross-section shapes.
phenomenon, it can be attributed to the fact that the pressure of the shale gas reservoirs is relatively high (>30 MPa) and the relative humidity of the shale gas reservoirs is relatively low (
