Abstract
In coalbeds and shales, gas transport and storage are important for accurate prediction of production rates and for the consideration of subsurface greenhouse gas sequestration. They involve coupled fluid phenomena in porous medium including viscous flow, diffusive transport, and adsorption. Standard approach to describe gas–matrix interactions is deterministic and neglects the effects of local spatial heterogeneities in porosity and material content of the matrix. In this study, adopting weak-noise and mean-field approximations and using a statistical approach in spectral domain, matrix heterogeneity effects are investigated in the presence of non-equilibrium adsorption with random partition coefficient. It is found that the local heterogeneities can generate non-trivial transport and kinetic effects which retard gas release from the matrix and influence the ultimate gas recovery adversely. Macro-transport shows 1/[1 + N Pe /(1 + N Pe )] dependence on the Péclet number, and persists at the diffusive ultra-low permeability limit. Macro-kinetics is directly related to Thiele modulus by the following expression: N Th /(1 + 2N Pe ). It leads to trapping of gas in the adsorbed phase during its release from the matrix, and to an adsorption threshold during the gas uptake by the matrix. Both effects are proportional to the initially available adsorbed gas amount and becomes more pronounced with the increasing variance of the porosity field. Consequently, a new upscaled deterministic gas mass balance is proposed for practical purposes. Numerical results are presented showing free and adsorbed gas distributions and fractional gas sorption curves for unipore coal matrix exhibiting Gaussian porosity distribution. This study is a unique approach for our further understanding of the coalbeds and gas shales, and it is important for the development of sound numerical gas production and sequestration models.
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Abbreviations
- B 0 :
-
Absolute coal permeability (cm2)
- C :
-
Free gas concentration (mol/cc pore)
- C μ :
-
Adsorbed gas concentration (mol/cc solid)
- C μ s :
-
Maximum adsorbed gas concentration (mol/cc solid)
- D :
-
Molecular diffusion coefficient (cm2/s)
- \({\mathcal{D}}\) :
-
Apparent diffusion coefficient (cm2/s)
- E :
-
Adsorbate adsorbent interaction energy (J/mol)
- g :
-
Average free gas concentration (mol/cc)
- K :
-
Partition (distribution) coefficient (fraction)
- k f :
-
Gas adsorption rate coefficient (1/s)
- k r :
-
Gas desorption rate coefficient (1/s)
- k r∞ :
-
Gas desorption rate constant at zero energy level (1/s)
- R :
-
Universal gas constant (J K−1 mol−1)
- r :
-
Pore half width (cm)
- t :
-
Time coordinate (s)
- T :
-
Temperature (K)
- x :
-
Space coordinate (cm)
- α :
-
Effective drift velocity (m/s)
- \({\phi}\) :
-
Porosity (fraction)
- \({\phi}\) :
-
Solid-to-bulk volume ratio (fraction)
- \({\sigma_{\rm f}^2}\) :
-
Variance of porosity fluctuations
- μ :
-
Gas viscosity (kg/cm s)
- λ :
-
Porosity correlation length (cm)
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Fathi, E., Akkutlu, I.Y. Matrix Heterogeneity Effects on Gas Transport and Adsorption in Coalbed and Shale Gas Reservoirs. Transp Porous Med 80, 281–304 (2009). https://doi.org/10.1007/s11242-009-9359-4
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DOI: https://doi.org/10.1007/s11242-009-9359-4