Darcy’s Permeability

Darcy’s Law 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 the 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.

File:Darcy's Law.png From Wikipedia

The permeability $latex \kappa $ in this equation is truly an intrinsic property 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 function of the ultrastructure of the media (porosity, pore size, etc.).

It is important for us to distinguish Darcy’s permeability from our commonly used  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 relationship it is clear that $latex \epsilon $ is not an intrinsic material property because changing the value of L (the membrane thickness) or $latex \mu $ will change its value. Because by “hydraulic” permeability we exclusively mean the passage of water $latex \mu $ will always have the same value and this dependence is of no consequence. The dependence of $latex \epsilon $ on length however, is key. We get higher flow rates for the same pressure and area precisely because of membrane thinness and so we want the permeability values that we use for comparisons to other materials to capture this important material difference.


Professor McGrath holds a BS degree in Mechanical Engineering from Arizona State and a MS degree in Mechanical Engineering from MIT. He earned a PhD in Biological Engineering from Harvard/MIT's Division of Health Sciences and Technology. He then trained as a Distinguished Post-doctoral Fellow in the Department of Biomedical Engineering at the Johns Hopkins University. Professor McGrath has been on the Biomedical Engineering faculty at the University of Rochester since 2001 where he also serves as the director of the graduate program BME and associate director of the URNano microfab and metrology core. McGrath's research was focused on the phenomena of cell migration until 2007 when he founded the interdisciplinary Nanomembrane Research Group to development and apply silicon membrane technologies. Professor McGrath is also a co-founder and past president of SiMPore Inc., a company founded to commercially manufacture silicon membranes and related technologies.

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