The Darcy law is as follows: $$u=-\frac{k}{\mu}\nabla p.$$

Assume we have a gas, then $\mu$ is about $10^{-5}$. For packed spheres a few $\rm mm$ in diameter, $k$ is of order $10^{-8}\ {\rm m^2}$. Say the pressure difference across a $0.1\ \rm m$ bed is $10\ \rm Pa$. Then we get $u= 0.1\ \rm m/s$, which seems really high for such a small pressure difference. I mean, tens of pascals is a variation which is comparable to random pressure fluctuations. Is this reasonable?

  • 4
    $\begingroup$ It's difficult to assess how reasonable your figures are unless you specify units for all of them. This will also help to check that you've done the right calculation. $\endgroup$
    – N. Virgo
    Commented Jun 20, 2012 at 20:18
  • $\begingroup$ Everything in SI. $\endgroup$
    – tiam
    Commented Jun 27, 2012 at 13:37
  • $\begingroup$ What makes you say that the speed is unreasonably high? $\endgroup$ Commented Jun 27, 2012 at 18:10
  • $\begingroup$ Because I gave a hard time imagining 10 Pa pressure fluctuations cause a 10 cm/s speed in a packed bed. I am modelling combustion in porous media and it is very sensitive to velocity: $u=0.05$ m/s and $u=0.25$ m/s can yeild quite different results. And accoring to Darcy's law it seems that velocities would fluctate randomly in those value ranges because it is a matter of a few Pa fluctuations in pressure. $\endgroup$
    – tiam
    Commented Jun 28, 2012 at 13:05

1 Answer 1


For the input parameters provided, your velocity estimate is reasonable but probably not accurate. The reason is that Darcy's law assumes Stokes (small Reynolds number) flow.

For the parameters provided, together with a density of $1\ \rm kg/m^3$, and substituting a flow length scale of 0.001 m, the Reynolds number reaches a value around 10. This means that the flow in the pore space is non-laminar and inertial effects can not be ignored. A straightforward engineering approach is to apply so-called Forchheimer corrections.

  • $\begingroup$ In that case, when is Darcy's law ever applicible for gas flow anyway? I mean, if even with a perssure difference of 10 Pa it is already unreasonable.. $\endgroup$
    – tiam
    Commented Jun 28, 2012 at 13:00
  • $\begingroup$ Additionaly, would using Forchheimer instead prinicpaly change the pressure diffreence magnitudes required to generate a given speed? $\endgroup$
    – tiam
    Commented Jun 28, 2012 at 13:07
  • 1
    $\begingroup$ Darcy's law is applicable for gas flow in tighter (lower permeability) porous materials. In other words, if your packed bed consists of spheres much smaller in diameter than a millimeter. For the case you are studying, applying the Forchheimer correction will not change the order of magnitude of the velocity derived (hence my remark "reasonable but not accurate"). $\endgroup$
    – Johannes
    Commented Jun 28, 2012 at 20:28

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