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An Analytical Model of Apparent Gas Permeability for Tight Porous Media

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Abstract

Simulation of fluid flow in tight rocks, such as shale gas reservoirs, has been a challenging task because of the coexistence of various flow regimes, including the continuum flow, slippage, transition flow, and Knudsen diffusion within the porous structure. Currently, both numerical and analytical methods have been applied to address this issue. In this paper, we have extended the application of most widely used analytical solution for single uniform capillary proposed by Beskok and Karniadakis (Microscale Thermophys Eng 3(1):43–77,1999) to the porous media. The porous structure is represented by a bundle of tortuous capillary tubes with different diameters. Fractal theory is applied to mathematically express the capillary diameter distribution and their tortuosity. For shale gas and coal seam gas formations where adsorption gas is present, the effect of surface diffusion is also included in the analytical solution. Thus, the presented analytical model has allowed us to study the effect of pore size distribution, fractal dimensions for pore size and tortuosity, porosity, surface diffusivity and Langmuir parameters on flow processes. Experimental data from 100 tight gas sand samples and one shale sample, with equivalent liquid permeability ranging from nanodarcy to millidarcy, are used to effectively evaluate the application of the analytical model in the study of flow behavior of gas in tight rocks. The results of this study also show that in tight formation, there has been an increase in apparent gas permeability through the life of production by a factor of 2.2.

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Abbreviations

A :

Total cross section area (\(\hbox {m}^{2}\))

\(A_\mathrm{p}\) :

Total porous area in the cross section (\(\hbox {m}^{2}\))

b :

Langmuir constant (\(\hbox {Pa}^{-1}\))

\(b_\mathrm{K}\) :

Gas slippage factor (Pa)

C :

Free gas concentration (\(\hbox {mol/m}^{3}\))

\(C_\mathrm{L}\) :

Langmuir capacity (\(\hbox {mol/m}^{3}\))

\(C_\mathrm{s}\) :

Adsorbed gas concentration (\(\hbox {mol/m}^{3}\))

\(\widehat{G}_\mathrm{K}\) :

Knudsen diffusivity (\(\hbox {m}^{2}/\hbox {s}\))

\(\widehat{G}_\mathrm{s}\) :

Surface diffusivity (\(\hbox {m}^{2}/\hbox {s}\))

\(\widehat{G}_\mathrm{s,0}\) :

Surface diffusivity at zero loading (\(\hbox {m}^{2}/\hbox {s}\))

\(\dot{D}_\mathrm{p}\) :

Fractal dimension for pore size

\(\dot{D}_\mathrm{t}\) :

Fractal dimension for tortuosity

\(d_\mathrm{rw}\) :

Jump distance in random walk simulation (m)

\(J_\mathrm{K}\) :

Knudsen diffusion flux (\(\hbox {mol}/\hbox {m}^{2}/\hbox {s}\))

\(J_\mathrm{s}\) :

Surface diffusion flux (\(\hbox {mol}/\hbox {m}^{2}/\hbox {s}\))

\(K_{n}\) :

Knudsen number

\(k_{\infty }\) :

Equivalent liquid permeability (\(\hbox {m}^{2}\) or mD)

\(k_\mathrm{app}\) :

Apparent gas permeability (\(\hbox {m}^{2}\) or mD)

\(k_\mathrm{app,p}\) :

Apparent gas permeability of porous flow (\(\hbox {m}^{2}\) or mD)

\(k_\mathrm{app,s}\) :

Apparent gas permeability of surface diffusion (\(\hbox {m}^{2}\) or mD)

\(k_\mathrm{ratio}\) :

Permeability ratio of apparent gas permeability to equivalent liquid permeability

\(k_\mathrm{B}\) :

Boltzmann constant (\(1.3805 \times 10^{-23}\,\hbox {J/K}\))

\(L_\mathrm{t}\) :

Tortuous length of the pore (m)

\(L_{0}\) :

Representative/straight length of the pore (m)

M :

Molecular weight of gas (kg/mol)

P :

Pressure (MPa)

Q :

Mass flow rate through the cross section (kg/s)

q :

Mass flow rate through the single capillary (kg/s)

R :

Molar gas constant (8.314 J/mol/K)

\(r_{35\,\%}\) :

Pore throat radius at 35 % cumulative pove volume during MICP (m)

T :

Temperature (K)

D :

Pore diameter (m)

\(D_\mathrm{max}\) :

Maximum pore diameter (m)

\(D_\mathrm{min}\) :

Minimum pore diameter (m)

\(\alpha \) :

Rarefaction coefficient

\({\varGamma }\) :

Jump frequency of the gas molecules (\(\hbox {s}^{-1}\))

\(\delta \) :

Collision diameter of the gas molecules (m)

\(\theta \) :

Surface coverage on the pore wall (fraction)

\(\rho _\mathrm{avg}\) :

Average gas density in the system (\(\hbox {kg/m}^{3}\))

\(\mu \) :

Dynamics viscosity of the fluid (\(\hbox {Pa}\,\hbox {s}\))

\(\tau \) :

Tortuosity of pore

\(\phi \) :

Porosity, fraction

\(\lambda \) :

Diameter

References

  • Aguilera, R.: Incorporating capillary pressure, pore throat aperture radii, height above free-water table, and Winland r35 values on Pickett plots. AAPG Bull. 86(4), 605–624 (2002)

    Google Scholar 

  • Akkutlu, I.Y., Fathi, E.: Multiscale gas transport in shales with local Kerogen heterogeneities. SPE J. 17(04), 1002–1011 (2012)

    Article  Google Scholar 

  • Ambrose, R.J., Hartman, R.C., Diaz-Campos, M., et al.: Shale gas-in-place calculations part I: new pore-scale considerations. SPE J. 17(01), 219–229 (2012)

    Article  Google Scholar 

  • Bai, B., Elgmati, M., Zhang, H., et al.: Rock characterization of Fayetteville shale gas plays. Fuel 105, 645–652 (2013)

    Article  Google Scholar 

  • Beskok, A., Karniadakis, G.E.: Report: a model for flows in channels, pipes, and ducts at micro and nano scales. Microscale Thermophys. Eng. 3(1), 43–77 (1999)

    Article  Google Scholar 

  • Chalmers, G.R., Bustin, R.M., Power, I.M.: Characterization of gas shale pore systems by porosimetry, pycnometry, surface area, and field emission scanning electron microscopy/transmission electron microscopy image analyses: examples from the Barnett, Woodford, Haynesville, Marcellus, and Doig units. AAPG Bull. 96(6), 1099–1119 (2012)

    Article  Google Scholar 

  • Chen, Y., Yang, R.: Concentration dependence of surface diffusion and zeolitic diffusion. AIChE J. 37(10), 1579–1582 (1991)

    Article  Google Scholar 

  • Choi, J.-G., Do, D., Do, H.: Surface diffusion of adsorbed molecules in porous media: Monolayer, multilayer, and capillary condensation regimes. Ind. Eng. Chem. Res. 40(19), 4005–4031 (2001)

    Article  Google Scholar 

  • Civan, F.: Effective correlation of apparent gas permeability in tight porous media. Transp. Porous Media 82(2), 375–384 (2010)

    Article  Google Scholar 

  • Civan, F., Rai, C.S., Sondergeld, C.H.: Determining shale permeability to gas by simultaneous analysis of various pressure tests. SPE J. 17(03), 717–726 (2012)

    Article  Google Scholar 

  • Curtis, M., Ambrose, R., Energy, D., et al.: Structural characterization of gas shales on the micro- and nano-scales. In: Proceedings of Canadian Unconventional Resources and International Petroleum Conference, Calgary, Alberta, Canada (2010)

  • Curtis, M.E., Cardott, B.J., Sondergeld, C.H., et al.: Development of organic porosity in the Woodford Shale with increasing thermal maturity. Int. J. Coal Geol. 103, 26–31 (2012)

    Article  Google Scholar 

  • Do, D.D.: Adsorption Analysis: Equilibria and Kinetics, vol. 2. Imperial College Press, London (1998). (Reprint)

    Google Scholar 

  • Dongari, N., Sharma, A., Durst, F.: Pressure-driven diffusive gas flows in micro-channels: from the Knudsen to the continuum regimes. Microfluid. Nanofluidics 6(5), 679–692 (2008)

    Article  Google Scholar 

  • Einstein, A.: Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Ann. Phys. 322(8), 549–560 (1905)

    Article  Google Scholar 

  • Ertekin, T., King, G.A., Schwerer, F.C.: Dynamic gas slippage: a unique dual-mechanism approach to the flow of gas in tight formations. SPE Form. Eval. 1(1), 43–52 (1986)

    Article  Google Scholar 

  • Etminan, S.R., Javadpour, F., Maini, B.B., et al.: Measurement of gas storage processes in shale and of the molecular diffusion coefficient in kerogen. Int. J. Coal Geol. 123, 10–19 (2014)

    Article  Google Scholar 

  • Fathi, E., Akkutlu, I.Y.: Matrix heterogeneity effects on gas transport and adsorption in coalbed and shale gas reservoirs. Transp. Porous Media 80(2), 281–304 (2009)

    Article  Google Scholar 

  • Feder, J.: Fractals. Plenum, New York (1988). (Reprint)

  • Freeman, C., Moridis, G., Blasingame, T.: A numerical study of microscale flow behavior in tight gas and shale gas reservoir systems. Trans. Porous Media 90(1), 253–268 (2011)

    Article  Google Scholar 

  • Gad-el-Hak, M.: The MEMS Handbook. CRC, Boca Raton (2002)

    Google Scholar 

  • He, Y.-L., Tao, W.-Q.: Multiscale simulations of heat transfer and fluid flow problems. J. Heat Transf. 134(3), 031018 (2012)

    Article  Google Scholar 

  • Heid, J., McMahon, J., Nielsen, R., et al.: Study of the permeability of rocks to homogeneous fluids. In: Proceedings of Drilling and Production Practice, New York, New York (1950)

  • Higashi, K., Ito, H., Oishi, J.: Surface diffusion phenomena in gaseous diffusion, (I) surface diffusion of pure gas. J. Atomic Energy Soc. Jpn. 5, 846–853 (1963)

    Article  Google Scholar 

  • Hirschfelder, J.O., Curtiss, C.F., Bird, R.B., et al.: Molecular Theory of Gases and Liquids, vol. 26. Wiley, New York (1954). (Reprint)

    Google Scholar 

  • Javadpour, F.: Nanopores and apparent permeability of gas flow in mudrocks (shales and siltstone). J. Can. Pet. Technol. 48(8), 16–21 (2009)

    Article  Google Scholar 

  • Javadpour, F., Fisher, D., Unsworth, M.: Nanoscale gas flow in shale gas sediments. J. Can. Pet. Technol. 46(10), 55–61 (2007)

    Article  Google Scholar 

  • Johnson, P.K.: A Method for Calculating Viscosity and Thermal Conductivity of a Helium-Xenon Gas Mixture. Paper No. NASA CR-2006–214394 (2006)

  • Jones, F.O., Owens, W.: A laboratory study of low-permeability gas sands. J. Pet. Technol. 32(09), 1631–1640 (1980)

    Article  Google Scholar 

  • Kadono, R., Kiefl, R., Brewer, J., et al.: Quantum diffusion of muonium in GaAs. Hyperfine Interact. 64(1–4), 635–640 (1991)

    Article  Google Scholar 

  • Kang, S.M., Fathi, E., Ambrose, R.J., et al.: Carbon dioxide storage capacity of organic-rich shales. SPE J. 16(04), 842–855 (2011)

    Article  Google Scholar 

  • Kapoor, A., Yang, R.: Surface diffusion on energetically heterogeneous surfaces—an effective medium approximation approach. Chem. Eng. Sci. 45(11), 3261–3270 (1990)

    Article  Google Scholar 

  • King, D.A.: Surface diffusion of adsorbed species: a review. J. Vac. Sci. Technol. 17(1), 241–247 (1980)

    Article  Google Scholar 

  • Klinkenberg, L.J.: The permeability of porous media to liquids and gases. In: Proceedings of Drilling and Production Practice, New York, New York (1941)

  • Knudsen, M.: The law of the molecular flow and viscosity of gases moving through tubes. Ann. Phys. 28, 75–130 (1909)

    Article  Google Scholar 

  • Koponen, A., Kataja, M., Timonen, J.: Tortuous flow in porous media. Phys. Rev. E 54(1), 406 (1996)

    Article  Google Scholar 

  • Kowalczyk, P., Gauden, P.A., Terzyk, A.P., et al.: Microscopic model of carbonaceous nanoporous molecular sieves-anomalous transport in molecularly confined spaces. Phys. Chem. Chem. Phys. 12(37), 11351–11361 (2010)

    Article  Google Scholar 

  • Krohn, C.E.: Fractal measurements of sandstones, shales, and carbonates. J. Geophys. Res. Solid Earth (1978–2012) 93(B4), 3297–3305 (1988)

    Article  Google Scholar 

  • Kuila, U., Prasad, M., Derkowski, A., et al.: Compositional controls on mudrock pore-size distribution: an example from Niobrara formation. In: Proceedings of SPE Annual Technical Conference and Exhibition, San Antonio, Texas, USA (2012)

  • Kuila, U., Prasad, M., Kazemi, H.: Application of Knudsen flow in modeling gas-flow in shale reservoirs. In: Proceedings of 9th Biennial International Conference and Exposition on Petroleum Geophysics. Hyderabad (2013)

  • Langmuir, I.: The adsorption of gases on plane surfaces of glass, mica and platinum. J. Am. Chem. Soc. 40(9), 1361–1403 (1918)

    Article  Google Scholar 

  • Lee, A.L., Gonzalez, M.H., Eakin, B.E.: The viscosity of natural gases. J. Pet. Technol. 18(08), 997–1000 (1966)

    Article  Google Scholar 

  • Letham, E.A.: Matrix Permeability Measurements of Gas Shales: Gas Slippage and Adsorption As Sources of Systematic. Bachelor Bachelor Thesis, University of British Columbia (2011)

  • Li, K., Horne, R.N.: Experimental study and fractal analysis of heterogeneity in naturally fractured rocks. Transp. Porous Media 78(2), 217–231 (2009)

    Article  Google Scholar 

  • Loyalka, S., Hamoodi, S.: Poiseuille flow of a rarefied gas in a cylindrical tube: solution of linearized Boltzmann equation. Phys. Fluids A Fluid Dyn. (1989–1993) 2(11), 2061–2065 (1990)

    Article  Google Scholar 

  • Mandelbrot, B.B.: The Fractal Geometry of Nature, vol. 173. Macmillan, London (1983). (Reprint)

    Google Scholar 

  • Mason, E.A., Malinauskas, A.: Gas Transport in Porous Media: the Dusty-Gas Model, vol. 17. Elsevier, Amsterdam (1983). (Reprint)

  • Medved, I., Cerny, R.: Surface diffusion in porous media: a critical review. Microporous Mesoporous Mater. 142(2), 405–422 (2011)

    Article  Google Scholar 

  • Mitchell, B.S.: An Introduction to Materials Engineering and Science for Chemical and Materials Engineers. Wiley, New Jersey (2004). (Reprint)

  • Miyabe, K., Guiochon, G.: Kinetic study of the mass transfer of bovine serum albumin in anion-exchange chromatography. J. Chromatogr. A 866(2), 147–171 (2000)

    Article  Google Scholar 

  • Miyabe, K., Guiochon, G.: Measurement of the parameters of the mass transfer kinetics in high performance liquid chromatography. J. Sep. Sci. 26(3–4), 155–173 (2003)

    Article  Google Scholar 

  • Roy, S., Raju, R., Chuang, H.F., et al.: Modeling gas flow through microchannels and nanopores. J. Appl. Phys. 93(8), 4870–4879 (2003)

    Article  Google Scholar 

  • Sakhaee-Pour, A., Bryant, S.: Gas permeability of shale. SPE Reserv. Eval. Eng. 15(04), 401–409 (2012)

    Article  Google Scholar 

  • Sevenster, P.: Diffusion of gases through coal. Fuel 38(4), 403–418 (1959)

    Google Scholar 

  • Sing, K., Everett, D., Haul, R., et al.: Physical and biophysical chemistry division commission on colloid and surface chemistry including catalysis. Pure Appl. Chem. 57(4), 603–619 (1985)

    Article  Google Scholar 

  • Smith, D.M., Williams, F.L.: Diffusional effects in the recovery of methane from coalbeds. Soc. Pet. Eng. J. (United States) 24(5), 529–535 (1984)

    Article  Google Scholar 

  • Sutera, S.P., Skalak, R.: The history of Poiseuille’s law. Ann. Rev. Fluid Mech. 25(1), 1–20 (1993)

    Article  Google Scholar 

  • Tang, G., Tao, W., He, Y.: Gas slippage effect on microscale porous flow using the lattice Boltzmann method. Phys. Rev. E 72(5), 056301 (2005)

    Article  Google Scholar 

  • Tsien, H.-S.: Superaerodynamics, mechanics of rarefied gases. J. Aeronaut. Sci. (Institute of the Aeronautical Sciences) 13(12), 653–664 (1946)

  • Valiullin, R., Kortunov, P., Kärger, J., et al.: Surface self-diffusion of organic molecules adsorbed in porous silicon. J. Phys. Chem. B 109(12), 5746–5752 (2005)

    Article  Google Scholar 

  • Van den Berg, A., Bromley, S., Flikkema, E., et al.: Molecular-dynamics analysis of the diffusion of molecular hydrogen in all-silica sodalite. J. Chem. Phys. 120(21), 10285–10289 (2004)

    Article  Google Scholar 

  • Wang, F.P., Reed, R.M.: Pore networks and fluid flow in gas shales. In: Proceedings of SPE Annual Technical Conference and Exhibition, New Orleans, Louisiana (2009)

  • Xu, P., Yu, B.: Developing a new form of permeability and Kozeny-Carman constant for homogeneous porous media by means of fractal geometry. Adv. Water Resour. 31(1), 74–81 (2008)

    Article  Google Scholar 

  • Yang, F., Ning, Z., Liu, H.: Fractal characteristics of shales from a shale gas reservoir in the Sichuan Basin, China. Fuel 115, 378–384 (2014)

    Article  Google Scholar 

  • Yang, R., Sikavitsas, V.: Kinetic theory for predicting multicomponent diffusivities from pure-component diffusivities for surface diffusion and diffusion in molecular sieves. Chem. Eng. Sci. 50(20), 3319–3322 (1995)

    Article  Google Scholar 

  • Yang, R.T., Fenn, J.B., Haller, G.L.: Modification to the Higashi model for surface diffusion. AIChE J. 19(5), 1052–1053 (1973)

    Article  Google Scholar 

  • Yi, J., Akkutlu, I.Y., Deutsch, C.V.: Gas transport in bidisperse coal particles: investigation for an effective diffusion coefficient in coalbeds. J. Can. Pet. Technol. 47(10), 20–26 (2008)

    Article  Google Scholar 

  • Yu, B.: Fractal character for tortuous streamtubes in porous media. Chin. Phys. Lett. 22(1), 158 (2005)

    Article  Google Scholar 

  • Yu, B.: Analysis of flow in fractal porous media. Appl. Mech. Rev. 61(5), 050801 (2008)

    Article  Google Scholar 

  • Yu, B., Cheng, P.: A fractal permeability model for bi-dispersed porous media. Int. J. Heat Mass Transf. 45(14), 2983–2993 (2002)

    Article  Google Scholar 

  • Yu, B., Li, J.: Some fractal characters of porous media. Fractals 9(03), 365–372 (2001)

    Article  Google Scholar 

  • Zheng, Q., Yu, B., Wang, S., et al.: A diffusivity model for gas diffusion through fractal porous media. Chem. Eng. Sci. 68(1), 650–655 (2012)

    Article  Google Scholar 

  • Ziarani, A.S., Aguilera, R.: Knudsen’s permeability correction for tight porous media. Transp. Porous Media 91(1), 239–260 (2012)

    Article  Google Scholar 

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Acknowledgments

The authors would like to acknowledge Prof. Boming Yu from Huazhong University of Science and Technology, China, for his help in deep understanding of fractal theory.

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  1. School of Petroleum Engineering, University of New South Wales, Sydney, NSW, Australia

    Yudong Yuan, Nima Gholizadeh Doonechaly & Sheik Rahman

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  1. Yudong Yuan
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Yuan, Y., Gholizadeh Doonechaly, N. & Rahman, S. An Analytical Model of Apparent Gas Permeability for Tight Porous Media. Transp Porous Med 111, 193–214 (2016). https://doi.org/10.1007/s11242-015-0589-3

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