Darcy’s Law for a relates the volumetric flow rate Q through a porous media in response to a pressure gradient $latex \Delta P $. …

$latex Q = \frac{\kappa A}{\mu}\frac{\Delta P}{L} $

Where $latex \kappa $ is an intrinsic permeability of the media, $latex \mu$ is the fluid viscosity, L is the length of the media that fluid must travel through to exit, and A is the cross section normal to the fluid flow direction.

*From Wikipedia*

The permeability $latex \kappa $ in this equation is an intrinsic permeability of the media. Although the flow rate will vary in proportion to the media cross section, and inversely with fluid viscosity, and length; these quantities are excluded from $latex \kappa $ so that $latex \kappa $ is strictly a measure of the ultrastructure of the media (porosity, pore size, etc.).

For us it is important to distinguish this definition of permeability from Hydraulic Permeability:

# $latex \epsilon = \frac{Q}{\Delta P \, A_m}$

The relationship between the two permeabilities is

$latex \epsilon = \frac{\kappa }{L \, \mu} $

From this it is clear that our standard definition is not an intrinsic material property because changing the value of L (the membrane thickness) or $latex \mu $ will change the value. However because by “hydraulic” permeability we mean the passage of water $latex \mu $ will always have the same value.

The reason we use $latex \epsilon $ instead of $latex \kappa $ is precisely because only $latex \epsilon $ quantifies the advantages of using thin membranes. We get higher flow rates for the same pressure and area precisely because of membrane thinness, so we can report values that are permeability values that are independent of membrane length.

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