Microscopic Darcy flux with the Macroscopic Darcy's law:
How exactly do we bridge the gap between these two different scales?
The Darcy flux (or the apparent velocity) pertains to the microscopic-scale scenario, while the Darcy's law pertains to the macroscopic scale.
The apparent velocity at the microscopic-scale can be converted into an equivalent actual velocity of fluid pertaining to the macroscopic-scale by dividing the apparent velocity with the effective porosity.
I don't know if this bridges the gap, but I've done a lot of work with particle tracking through domains that can't be adequately characterized by macroscopic processes. I devoted much effort to modeling diffusion and dispersion in amorphous as well as fractured media and validating the results. I have posted several papers on RG and more can be found on my web site, along with fully-functional software. Experience has shown that we must not ignore large or small scale effects in the case of contaminant transport through the ground or surface waters. The same could be said for air.
As per the Darcy's law flow velocity is proportional to pressure gradient. Here flow velocity very low. So the flow is microscopic behavior only.
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