# Flow in porous media I: A theoretical derivation of Darcy's law

## 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 volume-averaged pressure and velocity. The equation of motion contains integrals involving spatial deviations of the pressure and velocity, the Brinkman correction, and other lower-order 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 volume-averaged 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 β-σ interface contained within the macroscopic system, m2

A βe :

area of entrances and exits for the β-phase contained within the macroscopic system, m2

Aβσ:

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

A *β σ:

interfacial area of the β-σ interface contained within a unit cell, m2

e :

area of entrances and exits for the β-phase contained within a unit cell, m2

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/s2

K :

Darcy's law permeability tensor, m2

L :

characteristic length scale for volume averaged quantities, m

β :

characteristic length scale for the β-phase (see Figure 2), m

σ :

characteristic length scale for the σ-phase (see Figure 2), m

n βσ :

unit normal vector pointing from the β-phase toward the σ-phase (n βσ=−n σβ)

n βe :

unit normal vector for the entrances and exits of the β-phase contained within a unit cell

p β :

pressure in the β-phase, N/m2

p ββ :

intrinsic phase average pressure for the β-phase, N/m2

$$\tilde p_\beta$$ :

p β−〈p ββ, spatial deviation of the pressure in the β-phase, N/m2

r 0 :

radius of the averaging volume and radius of a capillary tube, m

v β :

velocity vector for the β-phase, m/s

v β〉:

phase average velocity vector for the β-phase, m/s

v ββ :

intrinsic phase average velocity vector for the β-phase, m/s

$$\tilde v_\beta$$ :

v β−〈v ββ, spatial deviation of the velocity vector for the β-phase, m/s

V :

averaging volume, m3

Vβ :

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

β :

Vβ/V, volume fraction of the β-phase

ρβ :

mass density of the β-phase, kg/m3

μβ :

viscosity of the β-phase, Nt/m2

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