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Transport in Porous Media

, Volume 12, Issue 2, pp 185–206 | Cite as

Formation factor and tortuosity of homogeneous porous media

  • Ravi Suman
  • Douglas Ruth
Article

Abstract

In this paper, volume averaging in porous media is applied to the microscopic electric charge conservation equation (differential form of Ohm's law) and an expression is derived for the formation factor of a homogeneous porous medium saturated with an electrically conductive fluid. This expression consists of two terms; the first term involves the integral of the current density over the fluid volume and the second term involves the integral of the electric potential over the solid-fluid interface. The physical meaning of the two terms is discussed with the help of three idealized porous media. The results for these media indicate a definite relation between the second term and tortuosity. These results also demonstrate the simplistic nature of the classical definition of the tortuosity as a ratio of geometric lengths. An exact relation between the formation factor and tortuosity is presented. It is shown that the assumed equivalence of the electrical and hydraulic tortuosities is not valid. The general application of the expression for the formation factor is discussed briefly.

Key words

Ohm's law formation factor volume averaging REV network theory idealized porous media tortuosity 

Notation

A

area

Ak

cross-sectional area of the REV normal to directionk

Aβ

A βσ +A βe

Aβe

interfacial area between theΒ phase inside and outside of the REV

Aβ

effective cross-sectional area open to flow in thek=1 direction

Aσβ

interfacial area between theσ andΒ phases contained within the REV

CI

current in theIth tube

C1

macroscopic current in thek=1 direction

Ek

electric field intensity

fI

fraction of macroscopic current in theIth tube (=c I /C1)

F

formation factor

Fs

formation factor of a standard RUC

gI

πδ I 2 /(4R w S I )

Jk

current density in directionk

k1

permeability in thek=1 direction

Le

effective average length in the definition of tortuosity

Lk

length of the REV in directionk

m

hydraulic mean radius

nk

unit outwardly normal vector

rk

coordinate on the microscopic scale

R0

resistivity of REV saturated with an electrically conductive fluid

Rw

resistivity of the fluid phase

SI

length of theIth tube

Tc

current term in Equation (19)

Tu

potential term in Equation (19)

V

volume

Vb

total volume of the REV

Vv

electric potential

Vhv

electric potentials at the upstream face of the REV

VIav,VIbv

electric potential at the two ends of a tube

Vlv

electric potential at the downstream face of the REV

Vβ

fluid volume inside the REV

xk

coordinate on the macroscopic scale

Greek Letters

δ

tube diameter in RUC 1

δa,δb,δc

tube diameters in RUC 2

δI

diameter of theIth tube

ξ

areosity

Τa,Τa

two types of tortuosity

Τh

hydraulic tortuosity

Φ

porosity

ψ

a general variable

Symbols

<>α

intrinsic phase average

Subscripts

α

a general phase

Β

the fluid phase

σ

the solid phase

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Copyright information

© Kluwer Academic Publishers 1993

Authors and Affiliations

  • Ravi Suman
    • 1
  • Douglas Ruth
    • 1
  1. 1.Department of Mechanical and Industrial EngineeringUniversity of ManitobaWinnipegCanada

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