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

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## 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 a*no 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 a*closure 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 of*abrupt changes* in the structure of a porous medium is not considered.

## Key words

Volume averaging Brinkman correction closure## Nomenclature

## Roman Letters

*A*_{βσ}interfacial area of the β-σ interface contained within the macroscopic system, m

^{2}*A*_{βe}area of entrances and exits for the β-phase contained within the macroscopic system, m

^{2}- Aβσ
interfacial area of the β-σ interface contained within the averaging volume, m

^{2}- A
_{β}^{*}σ interfacial area of the β-σ interface contained within a unit cell, m

^{2}- Aβ
*e* area of entrances and exits for the β-phase 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 β-phase (see Figure 2), m

- ℓ
_{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/m

^{2}- 〈
*p*_{β}〉^{β} intrinsic phase average pressure for the β-phase, N/m

^{2}- \(\tilde p_\beta \)
*p*_{β}−〈*p*_{β}〉^{β}, spatial deviation of the pressure in the β-phase, N/m^{2}*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**_{β}〉^{β} intrinsic phase average velocity vector for the β-phase, m/s

- \(\tilde v_\beta \)
**v**_{β}−〈**v**_{β}〉^{β}, spatial deviation of the velocity vector for the β-phase, m/s*V*averaging volume, m

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

^{3}

## Greek Letters

- ∈
_{β} V

_{β}/*V*, volume fraction of the β-phase- ρ
_{β} mass density of the β-phase, kg/m

^{3}- μ
_{β} viscosity of the β-phase, 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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