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The Forchheimer equation: A theoretical development

Abstract

In this paper we illustrate how the method of volume averaging can be used to derive Darcy's law with the Forchheimer correction for homogeneous porous media. Beginning with the Navier-Stokes equations, we find the volume averaged momentum equation to be given by

$$\langle v_\beta \rangle = - \frac{K}{{\mu _\beta }} \cdot (\nabla \langle p_\beta \rangle ^\beta - \rho _\beta g) - F\cdot \langle v_\beta \rangle .$$

The Darcy's law permeability tensor, K, and the Forchheimer correction tensor, F, are determined by closure problems that must be solved using a spatially periodic model of a porous medium. When the Reynolds number is small compared to one, the closure problem can be used to prove that F is a linear function of the velocity, and order of magnitude analysis suggests that this linear dependence may persist for a wide range of Reynolds numbers.

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Abbreviations

A βσ :

area of the β-σ interface contained with the macroscopic region, m2

A βe :

area of the entrances and exits of the β-phase at the boundary of the macroscopic region, m2

A βσ :

area of the β-σ interface contained within the averaging volume, m2

A p :

surface area of a particle, m2

b :

the vector field that maps μβ〈vββ onto \(\tilde p\) β when inertial effects are negligible, m−1

B :

tensor that maps \(\tilde v\) β onto 〈vββ when inertial effects are negligible

d p :

6V p/Ap, effective particle diameter, m

g:

gravitational acceleration, m/s2

I :

unit tensor

ℓβ:

characteristic length for the β-phase, m

ℓί:

ί = 1,2,3, lattice vectors, m

L :

characteristic length for macroscopic quantities, m

L ρ :

inertial length, m

m:

the vector field that maps μβ〈vββ onto \(\tilde p\) β, m−1

M :

tensor that maps \(\tilde v\) β onto μβ〈vββ

nασ :

unit normal vector directed from the β-phase toward the σ-phase

p β :

total pressure in the β-phase, Pa

p ββ :

intrinsic average pressure in the β-phase, Pa

p β〉:

superficial average pressure in the β-phase, Pa

\(\tilde p\) β :

p β − 〈p β〉 spatial deviation pressure, Pa

r:

position vector, m

r0:

radius of the averaging volume, m

t :

time, s

t * :

characteristic process time, s

vβ :

velocity in the β-phase, m/s

〈vββ :

intrinsic average velocity in the β-phase, m/s

〈vβ〉:

superficial average velocity in the β-phase, m/s

\(\tilde v\) β :

vβ − 〈vββ, spatial deviation velocity, m/s

ν:

local averaging volume, m3

Vβ :

volume of the β-phase contained within the averaging volume, m3

V p :

volume of a particle, m3

x :

position vector locating the centroid of the averaging volume, m

yβ :

position vector locating points in the β-phase relative to the centroid of the averaging volume, m

εβ :

Vβ/ν, volume fraction of the β-phase

ρβ :

density of the β-phase, kg/m3

μβ :

viscosity of the β-phase, Pa s

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Whitaker, S. The Forchheimer equation: A theoretical development. Transp Porous Med 25, 27–61 (1996). https://doi.org/10.1007/BF00141261

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Key words

  • Forchheimer equation
  • Darcy's law
  • volume averaging
  • closure