Abstract
This paper provides critical insights into the rigorous formulation of moderately rarefied gas transport through narrow channels in naturally and induced fractured porous media, such as gas shale rocks, approximated as round (cylindrical) and flat (slit) types. This formulation considers critical improvements over the previous attempts by proper implementation of the effective equivalent mean hydraulic radius of tight flow channels, the wall-slip effect accommodation of Maxwell (Philos Trans R Soc Lond A 170:231–256, 1879), the variable cross-section hard sphere model of gas molecules and the modified bulk mean free-path of Bird (Phys Fluids 26(11):3222–3223, 1983. https://doi.org/10.1063/1.864095), the apparent viscosity and mean free path for the confined-state gas behavior modification, the flow through narrow capillary tubes represented by a Hagen–Poiseuille-type equation, the Knudsen diffusivity, and an improved relationship between the apparent permeability and the intrinsic permeability. The description of gas transport through extremely tight channels is accomplished by superposition of the Poiseuille bulk flow (convection) and the Knudsen transport (diffusion) mechanisms. This approach is applied to investigate the accuracy of several previous studies on the modeling of gas transport through extremely tight narrow channels of round and flat types under moderately rarefied conditions. Although the simulation results reported by the previous studies of Roy et al. (J Appl Phys 93(8):4870–4879, 2003), Javadpour (J Can Pet Technol 48(8):16–21, 2009), and Veltzke and Thöming (J Gas Mech 698:406–422, 2012) appear to follow the trends observed in the experimental studies of Roy et al. (2003) flowing argon gas through a round channel (tube) and Ewart et al. (J Gas Mech 584:337–356, 2007. https://doi.org/10.1017/S0022112007006374) flowing helium gas through a single straight flat narrow channel, it is concluded that these results are not actually accurate for several reasons. The accuracy of the basic model presented by Javadpour (2009) suffers from some formulation issues and the low-order accuracy of the numerical approximations. The complicated model presented by Veltzke and Thöming (2012) is impractical and difficult because of the two-dimensional solutions of the Navier–Stokes equations with no-slip boundary condition and produces inaccurate solutions because of the improper definition of the effective radius of the straight flat narrow channel. The improved pressure equation of the compressible rarefied gas flow in tight channels developed in the present paper is highly nonlinear. The possibility of numerical calculation errors associated with the solution of the differential pressure equation was eliminated completely by an application of an integral transformation by facilitating a pseudo-transfer or flow potential function, and the solution of this equation was obtained accurately and fully analytically. It is shown that the proper formulation and accurate analytical solution developed in this paper can indeed lead to significantly accurate matches of the same experimental data than those reported by the previous studies. Thus, the deviation of the previous simulation results from the experimental data cannot be attributed simply to possible experimental errors associated with the laboratory tests but also to the limitations in formulation and inaccuracies in numerical solution. The exercises presented in this paper reveal that the previous modeling efforts certainly involve various types of errors and the experimental data cannot be matched by the gas transport models simply by adjusting the values of the unknown model parameters unless the models and their parameters are theoretically meaningful.
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Abbreviations
- a0, a1, a2, and a3 :
-
Empirical constants, dimensionless
- A b :
-
Bulk surface area normal to transport direction (m2)
- AR:
-
Aspect ratio of the cross-sectional area of a flat channel, dimensionless
- b :
-
Empirical slip coefficient, dimensionless
- c T :
-
Coefficient of isothermal compressibility (1/Pa)
- C(AR):
-
Correction factor, dimensionless
- \( D_{\text{K}}^{*} \) :
-
Knudsen diffusivity (m2/s)
- D K :
-
Knudsen diffusivity modified for fractional thermal accommodation to wall temperature (m2/s)
- \( {\text{Ei}}( - \gamma ) \) :
-
Exponential integral function, dimensionless
- \( f_{s}^{{D_{\text{K}} }} \) :
-
Fractional thermal accommodation to wall temperature, dimensionless
- \( f_{s}^{F} \) :
-
Fraction of gas molecules diffusely reflected after colliding with the pore wall, dimensionless
- \( f({\text{Kn}}) \) :
-
Permeability correction factor as a function of the Knudsen number, dimensionless
- F :
-
Wall-slip correction factor, dimensionless
- H :
-
Thickness of flat channel (m)
- \( j_{\text{A}} \) :
-
Convective mass flux of gas (kg/m2/s)
- \( j_{\text{D}} \) :
-
Dispersive mass flux of gas (kg/m2/s)
- j T :
-
Total mass flux of gas (kg/m2/s)
- K :
-
Apparent permeability (m2)
- \( K^{*} \) :
-
Intrinsic permeability (m2)
- K A :
-
Gas transport permeability (m2)
- Kn:
-
Knudsen number, dimensionless
- L :
-
Length of transport channel (m)
- L p :
-
Perimeter (m)
- M :
-
Molecular mass (kg/kmol)
- n :
-
Number density of molecules (kmol/m3)
- n h :
-
Number of capillary tubes, number
- p :
-
Absolute gas pressure (Pa)
- p1 and p2 :
-
Gas pressures at the inlet and outlet of the transport channel (Pa)
- q h :
-
Convective volumetric transport rate (m3/s)
- R :
-
Round channel radius (m)
- R e :
-
Equivalent hydraulic radius of a channel (m)
- R g :
-
Universal gas constant, 8314 J/kmol/K
- S :
-
Cross-sectional area (m2)
- x :
-
Straight Cartesian distance along transport direction (m)
- x h :
-
Distance covered along actual tortuous transport path (m)
- T :
-
Absolute temperature (K)
- u :
-
Volumetric gas flux (m3/m2/s)
- W :
-
Width of flat channel (m)
- \( \rho \) :
-
Bulk gas density (kg/m3)
- \( \mu \) :
-
Gas viscosity in a confined medium (Pa s)
- \( \mu^{*} \) :
-
Gas viscosity in an unconfined bulk medium (Pa s)
- \( \phi \) :
-
Porosity, fraction
- \( \tau_{\text{h}} \) :
-
Tortuosity of capillary transport paths, dimensionless
- \( \sigma \) :
-
Average cross section of collision area of a real gas (m2)
- \( \omega \) and \( \zeta \) :
-
Empirical exponents, dimensionless
- \( \lambda^{*} \) :
-
Unconfined-state bulk mean free path of gas (m)
- \( \lambda \) :
-
Confined-state mean free path of molecules (m)
- \( \varPsi \) :
-
Pseudo-transfer or flow potential function (kg/m/s)
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Appendices
Appendix 1: Unconfined Gas Viscosity and Mean Free Path
The unconfined-state gas viscosity \( \mu^{*} \) is correlated by a power-law function of temperature T (Bird 1983):
The average cross section of collision area \( \sigma \) of a real gas is correlated by an inverse power-law function of temperature T (Bird 1983):
The empirical exponents of \( \omega \) and \( \zeta \) are related by (Bird 1983):
The unconfined-state Chapman–Enskog bulk viscosity \( \mu^{*} \) of gas is expressed by (Bird 1983):
where T is the absolute temperature, M is the molecular mass, and Rg is the universal gas constant. The number density of the molecules is given by, where \( \rho \) is the gas density:
The mean free path is the average of the distances covered by the molecules of gas until collisions occur with each other or with the confining wall of transport path along the straight lines of different lengths, different for various molecules of the gas. Considering a variable cross-section hard sphere model (VHS), the unconfined-state bulk mean free path \( \lambda^{*} \) of gas is given by (Bird 1983):
The modified bulk mean free path is determined by eliminating \( \sigma \) between Eqs. (40) and (42) (Bird 1983):
Appendix 2: Modeling Gas Transport Through a Bundle of Narrow Capillary Channels
The gas transport through narrow hydraulic channels of porous materials is formulated by modifications of the previous various approaches (Beskok and Karniadakis 1999, Javadpour 2009, and Civan 2010, 2011, 2017, 2019a). Bundle of round tubes and flat slit channels are considered for approximating the interconnected pores in naturally and induced fractured porous materials as tight parallel hydraulic channels for gas transport. The moderately rarefied gas transport through the narrow channels is formulated by applying the variable cross-section hard sphere model of gas molecules (Bird 1983), confined-state gas behavior modification, equivalent radius of flow channels, and wall-slip effect accommodation of Maxwell (1879).
2.1 Bundle of Tight Round Tubes Gas Transport
The number nh of capillary tubes in porous media is estimated by:
where \( \phi \) and Ab denote the porosity and bulk surface area of transport in porous media and Re is the effective equivalent hydraulic radius. The value calculated by Eq. (44) is rounded up to the closest integer because the number of tubes is an integer.
The volumetric convective transport rate qh of gas through a bundle of nh tortuous round capillary tubes is determined by a Hagen–Poiseuille-type equation (Hagen 1839, 1869; Poiseuille 1835, 1840):
where qh is the volumetric transport, F indicates the wall-slip correction factor given by Eq. (9), and \( \mu \) is the confined-state gas viscosity given by Eq. (13).
The tortuosity \( \tau_{\text{h}} \) of capillary transport paths is expressed by:
where x is the straight Cartesian distance along the transport direction and xh is the length of the tortuous transport channel.
The convective gas mass flux \( j_{\text{A}} \) of gas is given by:
where u is the volumetric gas flux.
The dispersive gas mass flux \( j_{\text{D}} \) of gas is given by, which simplifies for a constant porosity \( \phi \) media as:
where DK is the Knudsen diffusivity coefficient and cT is the isothermal compressibility given by Eq. (2).
The total gas mass flux jT is expressed by a Darcy-type equation (Darcy 1856) using Eqs. (48) and (49):
The apparent permeability K of porous media is obtained from Eq. (50):
Consider a Darcy-type equation (Darcy 1856) for gas transport as:
where KA is the permeability:
where F denotes the wall-slip correction factor.
Thus, the intrinsic permeability \( K^{*} \) is determined by combining Eqs. (47), (52), and (53) (Carman 1956):
where F and DK are given by Eqs. (9) and (11), respectively.
Alternatively, consider the unified Hagen–Poiseuille-model given as (Beskok and Karniadakis 1999; Civan 2010):
where b is an empirical slip coefficient which assumes a value of b = − 1 for fully developed slip flow. Equations (50) and (56) lead to a correction factor (dimensionless) \( f({\text{Kn}}) \) as a function of the Knudsen number:
2.2 Bundle of Tight Flat Slits Gas Transport
The number of flat slits of thickness H and width W in porous media is determined by:
where \( \phi \) and Ab denote the porosity and bulk surface area of transport in porous media, respectively. The value calculated by Eq. (58) is rounded to the closest integer as the number of tubes is an integer number.
The convective gas mass flux \( j_{\text{A}} \) through the tortuous flat slits is given by a Hagen–Poiseuille-type equation (Beskok and Karniadakis 1999):
where C(AR) is a correction factor:
AR denotes the aspect ratio of the cross-sectional area Ab = WH of a flat channel:
By means of Eq. (62):
where u is the volumetric flux, \( \mu \) is the viscosity of gas, x is the straight Cartesian distance, and \( \rho \) is the density.
The dispersive gas mass flux \( j_{\text{D}} \) is given by, which simplifies as by assuming constant porosity \( \phi \):
The total mass flux of gas jT is expressed by a Darcy-type equation (Darcy 1856) using Eqs. (63) and (64):
where K denotes the apparent permeability of porous media obtained from Eq. (65):
Next, consider a Darcy-type equation (Darcy 1856):
where the permeability KA is given by:
where F denotes the wall-slip correction factor.
The intrinsic permeability K* is obtained from Eqs. (62) and (67) as:
Comparing Eqs. (69) and (66) provides:
For extremely narrow flat slits, Eq. (9) gives F = 1 as the curvature approaches Rh\( \to \infty \) and Eq. (60) gives:
Therefore, Eq. (70) simplifies as:
Alternatively, consider the unified Hagen–Poiseuille-model given as (Beskok and Karniadakis 1999):
where \( \mu \) and b are defined above in “Bundle of Tight Round Tubes Gas Transport” section. Then, Eqs. (65) and (73) lead to a correction factor (dimensionless) \( f({\text{Kn}}) \) as a function of the Knudsen number:
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Civan, F. Can Moderately Rarefied Gas Transport Through Round and Flat Tight Channels of Fractured Porous Media be Described Accurately?. Transp Porous Med 132, 157–181 (2020). https://doi.org/10.1007/s11242-020-01385-1
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DOI: https://doi.org/10.1007/s11242-020-01385-1