Well,

The attached pdf examined the multiple-phase fluid flow through a porous medium.

Sincerely

This paper describes a case in which calculation and modeling errors resulted in disastrous (very expensive) consequences, which could have been avoided http://dudleybenton.altervista.org/publications/Contaminant%20Transport%20in%20Granular%20Media.pdf You will find several interesting animations including groundwater flow through fractured media at this location http://dudleybenton.altervista.org/projects/Remediation/Remediation.html and http://dudleybenton.altervista.org/projects/Ptrax/ParticleTracking.html

This question is very important. Darcy's law applied on real life core material with multiple phases represents a macroscopic average behavior of very heterogeneous behavior at the pore scale. It is implicitly assumed that the same flow pattern will take place as long as the same pressure gradients or fluxes are applied and if one can demonstrate repeatability in the lab it seems reliable to extrapolate that to larger scale. So, although the ideal conditions of Darcy's system are not held, it is observed that effective parameters such as relative permeability can capture the role of generalizing the assumptions. Unfortunately, this is mainly a correction parameter without real physical meaning and it is still challenging to predict trends or absolute values from theory. From a modelling perspective it does however capture first order effects in a manageable way.

Kind regards,

Pål

I quite agree with Pal. When evaluating contamination of the environment or design of management and clean-up systems, we hope our calculations will accurately represent what is occurring in the world. When it doesn't, disaster may result and the cost may greatly increase, as in the paper I referred to previously. If the process (diffusion) behaves as expected (in a continuum) then we can safely apply properties determined from laboratory experiments. If the processes in situ don't correspond to laboratory experiments, our properties and scaling parameters may be inadequate. This can be a very expensive lesson to learn the hard way. One such example gave rise to the MADE (MAcro Dispersion Experiment) funded by EPRI and USDoD, for which I provided most of the mathematical support and published several papers, which are available on RG and at the location noted previously. Determining appropriate bulk properties for the specific combination of substances and media is a very expensive, lengthy, and complex process. I have provided several related papers and some software.

Thanks Dr Zeng.

The point is that it is possible for us to upscale any process from a pore-scale multi-phase fluid dynamics to a REV-based Darcy-scale fluid dynamics only when we have geometric-similarity, kinematic-similarity and dynamic similarity.

We cannot afford to have differing physical/chemical/biological processes at pore-scale and at Darcy-scale, and then, going ahead with homogenisation to up-scale it - may not be correct.