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On the derivation of the Forchheimer equation by means of the averaging theorem

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Abstract

The averaging theorem is applied to the microscopic momentum equation to obtain the macroscopic flow equation. By examining some very simple tube models of flow in porous media, it is demonstrated that the averaged microscopic inertial terms cannot lead to a meaningful representation of non-Darcian (Forchheimer) effects. These effects are shown to be due to microscopic inertial effects distorting the velocity and pressure fields, hence leading to changes in the area integrals that result from the averaging process. It is recommended that the non-Darcian flow regime be described by a Forchheimer number, not a Reynolds number, and that the Forchheimer coefficientΒ be more closely examined as it may contain information on tortuosity.

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Abbreviations

a i :

gravitational acceleration (m/s2)

A fs :

interfacial area between the fluid and solid phases (m2)

Fo:

Forchheimer number

k :

permeability (m2)

k 0 :

permeability at zero velocity (m2)

p :

thermodynamic pressure (Pa)

r i :

coordinate on the microscopic scale (m)

Re:

Reynolds number

t :

time (s)

u i ,u :

bulk velocity (m/s)

V :

volume (m3)

V f :

fluid volume (m3)

w i ,w :

microscopic velocity (m/s)

x i ,x :

coordinate on the macroscopic scale (m)

Β :

the Forchheimer coefficient (1/m)

λ ij :

extra (viscous) stress tensor (Pa)

Λij :

stress tensor (Pa)

Μ :

Viscosity (Pa. s)

ρ :

density (kg/m3)

Φ :

porosity

ψ :

a general variable

< >:

phase average

< >f :

intrinsic phase average

\(\widetilde\cdot\) :

the fluctuating part of a variable

References

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Authors and Affiliations

  1. Department of Mechanical Engineering, University of Manitoba, Winnepeg, Manitoba, Canada

    Douglas Ruth & Huiping Ma

Authors
  1. Douglas Ruth
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  2. Huiping Ma
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Ruth, D., Ma, H. On the derivation of the Forchheimer equation by means of the averaging theorem. Transp Porous Med 7, 255–264 (1992). https://doi.org/10.1007/BF01063962

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  • DOI: https://doi.org/10.1007/BF01063962

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