Transport in Porous Media

, Volume 116, Issue 2, pp 613–644 | Cite as

Effective Permeability of a Porous Medium with Spherical and Spheroidal Vug and Fracture Inclusions

  • Mojdeh Rasoulzadeh
  • Fikri J. KuchukEmail author


Vugs and fractures are common features of carbonate formations. The presence of vugs and fractures in porous media can significantly affect pressure and flow behavior of a fluid. A vug is a cavity (usually a void space, occasionally filled with sediments), and its pore volume is much larger than the intergranular pore volume. Fractures occur in almost all geological formations to some extent. The fluid flow in vugs and fractures at the microscopic level does not obey Darcy’s law; rather, it is governed by Stokes flow (sometimes is also called Stokes’ law). In this paper, analytical solutions are derived for the fluid flow in porous media with spherical- and spheroidal-shaped vug and/or fracture inclusions. The coupling of Stokes flow and Darcy’s law is implemented through a no-jump condition on normal velocities, a jump condition on pressures, and generalized Beavers–Joseph–Saffman condition on the interface of the matrix and vug or fracture. The spheroidal geometry is used because of its flexibility to represent many different geometrical shapes. A spheroid reduces to a sphere when the focal length of the spheroid approaches zero. A prolate spheroid degenerates to a long rod to represent the connected vug geometry (a tunnel geometry) when the focal length of the spheroid approaches infinity. An oblate spheroid degenerates to a flat spheroidal disk to represent the fracture geometry. Once the pressure field in a single vug or fracture and in the matrix domains is obtained, the equivalent permeability of the vug with the matrix or the fracture with matrix can be determined. Using the effective medium theory, the effective permeability of the vug–matrix or fracture–matrix ensemble domain can be determined. The effect of the volume fraction and geometrical properties of vugs, such as the aspect ratio and spatial distribution, in the matrix is also investigated. It is shown that the higher volume fraction of the vugs or fractures enhances the effective permeability of the system. For a fixed-volume fraction, highly elongated vugs or fractures significantly increase the effective permeability compared with shorter vugs or fractures. A set of disconnected vugs or fractures yields lower effective permeability compared with a single vug or fracture of the same volume fraction.



Major semi-axis of spheroid


Minor semi-axis of spheroid


Confocal distance

\(\mathbf {e}_{1,2,3}\)

Normal unit vectors of the curvilinear coordinate system

\(\epsilon \)

Rate of strain tensor


Gegenbauer function of first kind

\(\varGamma _{p,v}\)

Interface of porous and vuggy domain


Scale factors of curvilinear coordinate system


Gegenbauer function of second kind

\(\eta \)

Spheroidal coordinate



\(\lambda \)

Beavers–Joseph–Saffman empirical coefficient

\(\mu \)

Fluid viscosity

\(\mathbf {n}\)

Unit normal vector to the interface


Legendre polynomial of second kind and order n




Legendre polynomial of first kind and order n

\(\phi \)

Spheroidal coordinate

\(\psi \)



Distance from the origin

\(\rho \)

Fluid density and spherical coordinate


Spheroidal coordinate

\(\varvec{\tau }\)

Unit tangential vector to the interface


Spheroidal coordinate

\( U_\infty \)

Constant velocity field at the infinity


Velocity element

\(\mathbf {u}\)

Velocity vector


Velocity element


Cartesian coordinate

\(\xi \)

Spheroidal coordinate

\(\chi \)

Curvilinear coordinate


Dummy variable


Cartesian coordinate


Cartesian coordinate

\(\varOmega _\mathrm{m}\)

Porous matrix

\(\varOmega _\mathrm{v}\)

Vug domain



Parameter in matrix


Parameter in vug


Parameter in porous inclusion


Parameter for spherical shape of vug or inclusion



The authors are grateful to Schlumberger for permission to publish this article.


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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Schlumberger-Doll ResearchCambridgeUSA

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