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

Abstract

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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Abbreviations

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 :

Temperature(K)

x :

Space coordinate (cm)

z :

Gas compressibility factor

α :

Effective drift velocity (m/s)

\({\phi}\) :

Porosity

\({\phi}\) :

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)

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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). https://doi.org/10.1007/s11242-011-9830-x

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Keywords

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