Abstract
Stokes flow through a rigid porous medium is analyzed in terms of the method of volume averaging. The traditional averaging procedure leads to an equation of motion and a continuity equation expressed in terms of the volumeaveraged pressure and velocity. The equation of motion contains integrals involving spatial deviations of the pressure and velocity, the Brinkman correction, and other lowerorder terms. The analysis clearly indicates why the Brinkman correction should not be used to accommodate ano slip condition at an interface between a porous medium and a bounding solid surface.
The presence of spatial deviations of the pressure and velocity in the volumeaveraged equations of motion gives rise to aclosure problem, and representations for the spatial deviations are derived that lead to Darcy's law. The theoretical development is not restricted to either homogeneous or spatially periodic porous media; however, the problem ofabrupt changes in the structure of a porous medium is not considered.
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Abbreviations
 A _{βσ} :

interfacial area of the βσ interface contained within the macroscopic system, m^{2}
 A _{βe } :

area of entrances and exits for the βphase contained within the macroscopic system, m^{2}
 Aβσ:

interfacial area of the βσ interface contained within the averaging volume, m^{2}
 A ^{*}_{β} σ:

interfacial area of the βσ interface contained within a unit cell, m^{2}
 Aβe :

area of entrances and exits for the βphase contained within a unit cell, m^{2}
 B :

second order tensor used to represent the velocity deviation (see Equation (3.30))
 b :

vector used to represent the pressure deviation (see Equation (3.31)), m^{−1}
 d :

distance between two points at which the pressure is measured, m
 g :

gravity vector, m/s^{2}
 K :

Darcy's law permeability tensor, m^{2}
 L :

characteristic length scale for volume averaged quantities, m
 ℓ_{β} :

characteristic length scale for the βphase (see Figure 2), m
 ℓ_{σ} :

characteristic length scale for the σphase (see Figure 2), m
 n _{βσ} :

unit normal vector pointing from the βphase toward the σphase (n _{βσ}=−n _{σβ})
 n _{βe } :

unit normal vector for the entrances and exits of the βphase contained within a unit cell
 p _{β} :

pressure in the βphase, N/m^{2}
 〈p _{β}〉^{β} :

intrinsic phase average pressure for the βphase, N/m^{2}
 \(\tilde p_\beta \) :

p _{β}−〈p _{β}〉^{β}, spatial deviation of the pressure in the βphase, N/m^{2}
 r _{0} :

radius of the averaging volume and radius of a capillary tube, m
 v _{β} :

velocity vector for the βphase, m/s
 〈v _{β}〉:

phase average velocity vector for the βphase, m/s
 〈v _{β}〉^{β} :

intrinsic phase average velocity vector for the βphase, m/s
 \(\tilde v_\beta \) :

v _{β}−〈v _{β}〉^{β}, spatial deviation of the velocity vector for the βphase, m/s
 V :

averaging volume, m^{3}
 V_{β} :

volume of the βphase contained within the averaging volume, m^{3}
 ∈_{β} :

V_{β}/V, volume fraction of the βphase
 ρ_{β} :

mass density of the βphase, kg/m^{3}
 μ_{β} :

viscosity of the βphase, Nt/m^{2}
 ψ:

arbitrary function used in the representation of the velocity deviation (see Equations (3.11) and (B1)), m/s
 ξ:

arbitrary function used in the representation of the pressure deviation (see Equations (3.12) and (B2)), s^{−1}
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Whitaker, S. Flow in porous media I: A theoretical derivation of Darcy's law. Transp Porous Med 1, 3–25 (1986). https://doi.org/10.1007/BF01036523
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DOI: https://doi.org/10.1007/BF01036523
Key words
 Volume averaging
 Brinkman
 correction
 closure