We do have velocity variations (with reference to magnitude and direction) within an individual pore; as well as along the various (tortuous) pores (along the direction of fluid flow).
But still, all these microscopic variations in magnitude and velocity (somehow or magically) get cancelled or nullified; and what we have is the (mostly) well defined - only one - resultant direction of fluid flow - in a saturated groundwater system at the macroscopic scale.
Is there an effort to prove mathematically that this final - single - resultant fluid flow direction at the macroscopic-scale results from the conglomeration of velocity variations at the microscopic-scale - in a typical saturated groundwater system?
Mathematical Models of Flow in Porous Media - Springer:
https://www.springer.com/cda/content/document/cda.../9783642235580-c2.pdf?...
Dear Suresh,
I would say that this is quite well understood. Essentially, the problem you describe is nothing but the scale transition (from the pore scale to the averaged, Darcy scale). There are various techniques to make this transition rationally. Volume averaging is quite popular allowing you to derive formally the upscaled equations satisfied by the Darcy scale velocity. Homogenisation provides the same results, but by this technique can also provide rigorous mathematical proofs.
Either way you choose, the result should be the same. For example, if you assume a Stokes or Navier Stokes model for the fluid flow at the scale of pores (through the pores), the corresponding averaged fluid velocity and pressure will satisfy the Darcy law at the larger scale of interest (e.g. laboratory scale, or even field scale). The pore scale variations are averaged out.
Good luck,
Sorin
Dear Dr. Sorin Pop,
Thanks for your reply.
We know that the volume averaging of microscopic equations of flow/transport processes may lead to extremely complicated results.
And, that’s the reason why I was asking about any earlier attempt on its actual mathematical derivation.
Of course, it will not be having any practical value.
However, as you have pointed out, we can always make use of the control volume analysis to resolve some of the difficulties associated with the volume averaging of the porous media.
Thanks.
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