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Mass Transport of Adsorbed-Phase in Stochastic Porous Medium with Fluctuating Porosity Field and Nonlinear Gas Adsorption Kinetics


Using an upscaling approach based on small perturbation theory, the authors have previously investigated the influence of local heterogeneities in matrix porosity on Darcy flow and Fickian-type pore diffusion in the presence of linear non-equilibrium gas adsorption Fathi and Akkutlu, J. Transp. Porous Med. 80, 281–3044 (2009). They identified non-trivial macro-transport and -kinetics effects of the heterogeneity which significantly retard gas release from the matrix and influence the ultimate gas recovery adversely. The work was a unique fundamental approach for our understanding of gas production and sequestration in unconventional reservoirs; however, it was simplified and did not consider (i) the presence of nonlinear sorption kinetics and (ii) a transport mechanism for the adsorbed phase. In this article, we incorporate the nonlinearity and surface diffusion effects of the adsorbed-phase into their formulation and apply the same upscaling approach to further study the heterogeneity effects. Gas sorption involves the so-called Langmuir kinetics, which is reduced to the well-known Langmuir isotherm in the equilibrium limit. It is found that the nonlinearity participates into both macro-transport and -kinetics, promoting primarily the surface diffusion effects. Whereas surface diffusion, although commonly ignored during modeling subsurface phenomena, brings an intricate nature to the gas release dynamics. Through macro-transport effect of the heterogeneity, it increases ultimate gas recovery and, through the macro-kinetics effect of the heterogeneity, it significantly decreases the time needed to reach the ultimate recovery. As the consequence of these effects, it is shown that the gas–matrix system practically does not reach the equilibrium adsorption limit during any stage of the matrix gas release.

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B 0 :

Absolute matrix permeability (cm2)

C :

Free gas amount (mol/cc pore)

C μ :

Adsorbed gas amount (mol/cc solid)

C μ s :

Maximum adsorbed gas amount (mol/cc solid)

D :

Molecular diffusion coefficient (cm2/s)

\({\mathcal{D}}\) :

Apparent diffusion coefficient (cm2/s)

D s :

Surface diffusion coefficient (cm2/s)

E :

Potential energy (J)

g :

Average molar density of free gas (mol/cc)

K :

Partition (distribution) coefficient

k f :

Gas adsorption rate coefficient (1/s)

k r :

Gas desorption rate coefficient (1/s)

R g :

Universal gas constant (J K−1 mol−1)

t :

Time coordinate(s)

T :


x :

Space coordinate (cm)

z :

Gas compressibility factor

α :

Effective drift velocity (m/s)

\({\phi}\) :


\({\phi}\) :

Solid-to-bulk volume ratio

\({\sigma_{f}^2}\) :

Variance of porosity fluctuations

μ :

Gas viscosity (kg/cm s)

λ :

Porosity correlation length (cm)

ν :

vibration frequency factor (1/s)


  1. Ambrose R.J., Hartman R.C., Campos M.D., Akkutlu,I. Y., Sondergeld, C. H.: New pore-scale considerations for shale gas in place calculations. SPE 131772 (2010)

  2. Alvarado V., Scriven L.E., Davis H.T.: Stochastic-perturbation analysis of a one-dimensional dispersion reaction equation: effects of spatially-varying reaction rates. Transp. Porous Med. 32, 139–161 (1998)

  3. Pope C.G.: Flow of adsorbable gases and vapours in a microporous medium. Proc. R. Soc. Lond A271, 1–18 (1963)

  4. Aylmore L.A.G., Barrer R.M.: Surface and volume flow of single gases and of binary gas mixtures in a microporous carbon membrane. Proc. R. Soc. A290, 477 (1966)

  5. Brusseau M.L., Jessup R.E., Rao P.S.C.: Nonequilibrium sorption of organic chemicals: elucidation of rate limiting processes. Environ. Sci. Technol. 25, 134–142 (1991)

  6. Carman P.C., Raal F.A.: Diffusion and flow of gases and vapours through micropores. III. Surface diffusion coefficients and activation energies. Proc. R. Soc. Lond. A209, 38 (1951)

  7. Do D.D.: Adsorption analysis: equilibria and kinetics. Imperial college press, London (1998)

  8. Do D.D., Wang K.: A new model for the description of adsorption kinetics in heterogeneous activated carbon. Carbon 36(10), 1539–1554 (1998)

  9. Farooq S., Ruthven D.M.: Numerical simulation of a kinetically controlled pressure swing adsorption bulk separation process based on a diffusion model. Chem. Eng. Sci 46, 2213 (1991)

  10. Fathi E., Akkutlu I.Y.: Matrix heterogeneity effects on gas transport and adsorption in coalbedand shale gas reservoirs. J. Transp. Porous Med. 80, 281–3044 (2009)

  11. Forster D.: Hydrodynamics fluctuations broken symmetry and correlation functions. Benjamin-Cummings, Reading, MA (1977)

  12. Gelhar L.W.: Stochastic subsurface hydrology. Prentice Hall, Englewood Cliffs (1993)

  13. Gilliland E.R., Baddour R.F., Russell J.L.: Rates of flow through microporous solids. AICHE J. 4, 90–96 (1958)

  14. Hu B.X., Deng F., Cushman J.H.: Non-local reactive transport with physical and chemical heterogeneity: linear non-equilibrium sorption with random K d . Water Resour. Res. 31(9), 2239–2252 (1995)

  15. Kang, S.M., Fathi, E. Akkutlu, I.Y., Sigal, R.F.: CO2 storage capacity of organic-rich shales. SPE 134583 (2010)

  16. Koss, V.A., Wickens D., Cucka, P., La Cava, A.I.: Proc Carbon 86 (4 internationale Kohlensttofftagung), Deutsche Keramische Gesellschaft, Baden-Baden Germany, p. 388 (1986)

  17. L’Heureux I.: Stochastic reaction-diffusion phenomena in porous media with nonlinear kinetics: effects of quenched porosity fluctuations. Phys. Rev. Lett. 93(18), 180602 (2004)

  18. McCabe L.W., Smith J.C., Harriott P.: Unit operations of chemical engineering. Chemical engineering series. McGraw-Hill Inc., New York (1993)

  19. Schlebaum W., Scharaa G., Vanriemsdijk W.H.: Influence of nonlinear sorption kinetics on the slow-desorbing organic contaminant fraction in soil. Environ. Sci. Technol. 33, 1413–1417 (1999)

  20. Sevenster P.G.: Diffusion of gas through coal. Fuel 38, 403–415 (1959)

  21. Siemons N., Wolf Karl-Heinz A.A., Bruining J.: Interpretation of carbon dioxide diffusion behavior in coals. Int. J. Coal Geol. 72, 315–324 (2007)

  22. Srinivasan R., Auvil S.R., Schork J.M.: Mass transfer in carbon molecular sieves—an interpretation of langmuir kinetics. Chem. Eng. J. 57, 137–144 (1995)

  23. Thimons E.P., Kissell F.N.: Diffusion of methane through coal. Fuel 52, 274–280 (1973)

  24. Tiselius A.: Die diffusion von Wasser in einem Zeolith-Kristall. Ein Beitrag zur Frage der Beweglichkeit adsorbierter Molekule. Z. Phys. Chem. A169, 425 (1934)

  25. Tiselius A.: Sorption and diffusion von Ammoniak in Analcim. Z. Phys. Chem. A174, 401 (1935)

  26. 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)

  27. Yi J., Akkutlu I.Y., Karacan C.Ö, Clarkson C.R.: Gas sorption and transport in coals: a poroelastic medium approach. Int. J. Coal Geol. 77(1–2), 137–144 (2009)

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Correspondence to I. Yucel Akkutlu.

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Fathi, E., Akkutlu, I.Y. Mass Transport of Adsorbed-Phase in Stochastic Porous Medium with Fluctuating Porosity Field and Nonlinear Gas Adsorption Kinetics. Transp Porous Med 91, 5–33 (2012).

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  • Adsorption
  • Diffusion
  • Heterogeneity
  • Upscaling
  • Macro-transport
  • Macro-kinetics
  • Langmuir isotherm
  • Shale gas