# The Forchheimer equation: A theoretical development

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## Abstract

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

*A*_{βσ}area of the

*β*-σ interface contained with the macroscopic region, m^{2}*A*_{βe}area of the entrances and exits of the

*β*-phase at the boundary of the macroscopic region, m^{2}*A*_{βσ}area of the

*β*-σ interface contained within the averaging volume, m^{2}*A*_{p}surface area of a particle, m

^{2}**b**the vector field that maps μ

_{β}〈v_{β}〉^{β}onto \(\tilde p\)_{β}when inertial effects are negligible, m^{−1}**B**tensor that maps \(\tilde v\)

_{β}onto 〈v_{β}〉^{β}when inertial effects are negligible*d*_{p}6

*V*_{p}/A_{p}, effective particle diameter, m- g
gravitational acceleration, m/s

^{2}**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 μ

_{β}〈v_{β}〉^{β}onto \(\tilde p\)_{β}, m^{−1}**M**tensor that maps \(\tilde v\)

_{β}onto μ_{β}〈v_{β}〈^{β}- n
_{ασ} unit normal vector directed from the

*β*-phase toward the*σ*-phase*p*_{β}total pressure in the

*β*-phase, Pa- 〈
*p*_{β}〉^{β} 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

*r*0radius of the averaging volume, m

*t*time, s

*t*^{*}characteristic process time, s

- v
_{β} velocity in the

*β*-phase, m/s- 〈v
_{β}〉^{β} intrinsic average velocity in the

*β*-phase, m/s- 〈v
_{β}〉 superficial average velocity in the

*β*-phase, m/s- \(\tilde v\)
_{β} v

_{β}− 〈v_{β}〉^{β}, spatial deviation velocity, m/s- ν
local averaging volume, m

^{3}- V
_{β} volume of the

*β*-phase contained within the averaging volume, m^{3}*V*_{p}volume of a particle, m

^{3}**x**position vector locating the centroid of the averaging volume, m

- y
_{β} position vector locating points in the

*β*-phase relative to the centroid of the averaging volume, m

## Greek Symbols

- ε
_{β} V

_{β}/ν, volume fraction of the*β*-phase- ρ
_{β} density of the

*β*-phase, kg/m^{3}- μ
_{β} viscosity of the

*β*-phase, Pa s

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