Transport in Porous Media

, Volume 25, Issue 1, pp 27–61

# The Forchheimer equation: A theoretical development

• Stephen Whitaker
Article

## 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.

### Key words

Forchheimer equation Darcy's law volume averaging closure

### Nomenclature

Aβσ

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

Aβe

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

Aβσ

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

Ap

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

dp

6Vp/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

Lρ

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

pβ

total pressure in the β-phase, Pa

pββ

intrinsic average pressure in the β-phase, Pa

pβ

superficial average pressure in the β-phase, Pa

$$\tilde p$$β

pβ − 〈pβ〉 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

Vp

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

### Greek Symbols

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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