Transport in Porous Media

, Volume 127, Issue 3, pp 573–585 | Cite as

Stress Dependency of Permeability Represented by an Elastic Cylindrical Pore-Shell Model: Comment on Zhu et al. (Transp Porous Med (2018) 122:235–252)

  • Faruk CivanEmail author


Stress dependency of permeability of porous rocks is described by means of a theoretical elastic cylindrical pore-shell model. This model is developed based on a bundle of elastic capillary tubes representation of the preferential flow paths formed in heterogeneous porous rocks. The radial displacement caused in tubes by the pore fluid pressure applied over the surface of the elastic cylindrical flow tubes is expressed by a Lamé-type equation. The radial displacement is incorporated into the Kozeny–Carman relationship to determine the variation of the permeability of porous rocks by variation of the pore fluid pressure. The solution of this equation yields a semi-analytical equation which provides accurate correlations of the stress dependency of the permeability data of porous rocks. The errors associated with the previous formulation of this problem by Zhu et al. (Transp Porous Med 122:235–252, 2018) are explained in view of the present formulation.


Porous rocks Permeability Stress dependency Kozeny–Carman equation Lamé equation 

List of symbols

a, b, c

Empirical parameters

aʹ, bʹ, cʹ, Dʹ

Empirical parameters

A, B, D, C

Empirical parameters

A1, Ab

Pore and bulk cross-sectional areas of porous rock (m2)


Young’s modulus (Pa)


Intrinsic permeability of porous rock (m2)


Intrinsic permeability at a reference effective stress \( \sigma_{\text{o}} \) (m2)

L1, Lb

Length of actual tortuous flow path and bulk length of porous rock (m)


Number of flow paths formed in porous rock


Pore fluid pressure (Pa)

p1, p2

Pressures applied over the inside surface of radius r1 and the outside surface of radius r2 of a hollow elastic cylindrical tube (Pa)


Flowing fluid volumetric flow rate (m3/s)


Average internal radius of the bundle of elastic capillary tubes (m)


Average external radius of influence of the variation of the pressure inside the flow tube beyond which no deformation occurs (m)


Coefficients of regression, dimensionless


Biot–Willis poroelastic coefficient, dimensionless

V1, Vb

Pore volume and bulk volume of porous rock (m3)

Greek symbols

\( \sigma \)

Effective stress (Pa)

\( \sigma_{\text{c}} \)

Total confining stress (Pa)

\( \delta_{r} \)

Radial displacement at any radius r (m)

\( \mu \)

Fluid viscosity (Pa s)

\( \upsilon \)

Poisson’s ration of the reservoir rock formation, dimensionless

\( \tau \)

Tortuosity, dimensionless


Porosity of porous formation, fraction


Reference porosity, fraction



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Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Mewbourne School of Petroleum and Geological EngineeringThe University of OklahomaNormanUSA

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