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Knudsen’s Permeability Correction for Tight Porous Media

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Abstract

Various flow regimes including Knudsen, transition, slip and viscous flows (Darcy’s law), as applied to flow of natural gas through porous conventional rocks, tight formations and shale systems, are investigated. Data from the Mesaverde formation in the United States are used to demonstrate that the permeability correction factors range generally between 1 and 10. However, there are instances where the corrections can be between 10 and 100 for gas flow with high Knudsen number in the transition flow regime, and especially in the Knudsen’s flow regime. The results are of practical interest as gas permeability in porous media can be more complex than that of liquid. The gas permeability is influenced by slippage of gas, which is a pressure-dependent parameter, commonly referred to as Klinkenberg’s effect. This phenomenon plays a substantial role in gas flow through porous media, especially in unconventional reservoirs with low permeability, such as tight sands, coal seams, and shale formations. A higher-order permeability correlation for gas flow called Knudsen’s permeability is studied. As opposed to Klinkenberg’s correlation, which is a first-order equation, Knudsen’s correlation is a second-order approximation. Even higher-order equations can be derived based on the concept used in developing this model. A plot of permeability correction factor versus Knudsen number gives a typecurve. This typecurve can be used to generalize the permeability correction in tight porous media. We conclude that Knudsen’s permeability correlation is more accurate than Klinkenberg’s model especially for extremely tight porous media with transition and free molecular flow regimes. The results from this study indicate that Klinkenberg’s model and various extensions developed throughout the past years underestimate the permeability correction especially for the case of fluid flow with the high Knudsen number.

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Abbreviations

DSMC:

Direct-simulation of Monte Carlo

LBS:

Linearized Boltzmann solution

MEMS:

Micro electro-mechanical systems

NEMS:

Nano electro-mechanical systems

b k :

Klinkenberg’s slippage factor (psi)

b :

Slip coefficient in microflow model

D :

Diffusion coefficient (ft2/day)

f c :

Correction factor (dimensionless)

H :

Microchannel height (ft)

k :

Porous medium permeability to fluid (md)

k a :

Apparent gas permeability (md)

K B :

Boltzmann constant (1.3805 × 10−23 J/K)

k ∞ :

Absolute (Klinkenberg’s corrected) permeability (md)

Kn :

Knudsen number (dimensionless)

L :

Pipe length (ft)

M :

Molecular weight (kg/kmol)

P :

Pressure (psi)

r :

Pore-throat radius (μm)

R g :

Universal gas constant (8.314 J/kg-mole/K)

T :

Temperature (K)

q :

Flow rate (ft3/day)

u :

Velocity in x-direction (ft/s)

ΔP :

Pressure drop (psi)

a:

Apparent

ads:

Adsorbed

B:

Boltzmann

c:

Correction

g:

Gas

k:

Klinkenberg

α :

Dimensionless rarefaction coefficient

σ :

Collision diameter (m)

ρ :

Fluid density (lb/ft 3)

μ :

Dynamic viscosity of fluid (cp)

\({\phi}\) :

Porosity (fraction)

τ :

Tortuosity (dimensionless)

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Authors and Affiliations

  1. Department of Chemical and Petroleum Engineering, Schulich School of Engineering, University of Calgary, 2500 University Dr., NW, Calgary, T2N 1N4, AB, Canada

    Ali S. Ziarani & Roberto Aguilera

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  1. Ali S. Ziarani
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  2. Roberto Aguilera
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Correspondence to Ali S. Ziarani.

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Ziarani, A.S., Aguilera, R. Knudsen’s Permeability Correction for Tight Porous Media. Transp Porous Med 91, 239–260 (2012). https://doi.org/10.1007/s11242-011-9842-6

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