Transport in Porous Media

, Volume 129, Issue 3, pp 885–899 | Cite as

Stress-Dependent Porosity and Permeability of Porous Rocks Represented by a Mechanistic Elastic Cylindrical Pore-Shell Model

  • Faruk CivanEmail author


The stress dependency of the porosity and permeability of porous rocks is described theoretically by representing the preferential flow paths in heterogeneous porous rocks by a bundle of tortuous cylindrical elastic tubes. A Lamé-type equation is applied to relate the radial displacement of the internal wall of the cylindrical elastic tubes and the porosity to the variation of the pore fluid pressure. The variation of the permeability of porous rocks by effective stress is determined by incorporating the radial displacement of the internal wall of the cylindrical elastic tubes into the Kozeny–Carman relationship. The fully analytical solutions of the mechanistic elastic pore-shell model developed by combining the Lamé and Kozeny–Carman equations are shown to lead to very accurate correlations of the stress dependency of both the porosity and the permeability of porous rocks.


Porosity Permeability Stress dependency Lamé equation Kozeny–Carman equation 

List of Symbols

A1 and Ab

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

a, b, c, d and e

Empirical parameters

a′, b′, c′ and D

Empirical parameters

A, B, C, D and F

Empirical parameters


Young’s modulus (Pa)


Intrinsic permeability of porous rock (m2)


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

L1 and 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 and p2

Pressures applied over the inside surface radius r1 and the outside surface 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 and Vb

Pore volume and bulk volume of porous rock (m3)

Greek Symbols

\( \alpha ,\beta \)


\( \sigma \)

Effective stress (Pa)

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

Total confining stress (Pa)

\( \delta_{\text{r}} \)

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. 2019

Authors and Affiliations

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

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