According to Darcy's Law, the volumetric flow rate Q of a fluid occuring due to a pressure difference $\Delta p$ over a distance L and through a cross-sectional area A of a porous medium (volume V) is given by:

$$Q = \frac{k\cdot A}{\mu}\cdot \frac{\Delta p}{L}\tag{1}$$

When calculating the pressure increase caused by that inflow of Q over a time $\Delta t$, it is typically considered for porous media that only a fraction of its total volume can actually be filled by the fluid, where the fraction of "free space" is characterized by the porosity $n = V_{free}/V_{total}$:

$$\Delta p = K\cdot Q\cdot \frac{\Delta t}{n\cdot V_{real}}\tag{2}$$

My question now is: Why is the porosity considered in the last equation, but not in the cross-sectional area of the first equation? By the very same logic I could say, that only a fraction of the cross-sectional area is "open" for fluid to flow into, couldn't I? Especially since (1) is often used for the determination of material permeabilities, but always the total cross-sectional area is used for the calculation.

This is just something that seems odd to me and maybe one of you guys has some insight on that and tell me where I'm thinking wrong.


1 Answer 1


The porosity is needed if you are interested in the flow velocity within the porous medium. If you look at $Q/A$ this is also a velocity but with reference to the void or empty channel (this is also referred to as superficial velocity), as $A$ represents void and matrix. To get the velocity within the medium (in straight flow direction) one needs to correct $A$ by the porosity or void fraction $n$: $$v_{medium}= \frac {Q} {An}$$


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