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The Annals of Frontier and Exploratory Science **

Hierarchical Scaled Physics and Technologies (HSPT), Rheinbach, Germany, Denver, CO, USA

**Fundamentals of Hierarchical Scaled (VAT) Description of Fluid Mechanics in Porous Media**

If the porous medium which
overall properties are sought, recognized as being in dependence on lower (smaller) scale physical phenomena at the pore level
then its physical and mathematical descriptions need to be considered and constructed in a way incorporating the interdependence
of higher (larger) scale descriptions and mathematical modeling for flow of substances into the lower and vice versa. Governing
equations for momentum transport in porous media mathematically correct are described using the hierarchical non-local Volume
Averaging Theory (VAT). Most often arising the situation when the two levels of hierarchy and two - or three phase in media
allows to connect phenomena in the neighboring scales. The hierarchical VAT governing mathematical equations for scaled (porous)
medium resulting from the VAT based analysis in many fields are the parabolic partial differential equations (PPDE) and they have
additional integral and integro-differential terms.
This theory, among methods in use today, has special attributes that enable explanations of hierarchical phenomena to be obtained
because it is able to combine pore scale, porous medium scale, interfacial and morphology features simultaneously with
mathematical rigor. More in - Nonlinear Effects in Multiple Regime Transport of
Momentum in Longitudinal Capillary Porous Medium Morphology (665K)

**Why is it Different from Homogeneous and other Fluid Mechanics in Porous Media Descriptions ?**

The main differences are due to different governing equations for the UPPER scale physics description. Those are mainly
because the Heterogeneous Whitaker-Slattery-Anderson-Marle (WSAM) kind of theorems explain why those equations are different.
Fundamentals of Hierarchical Scaled Description in Physics and Technology.

**What is the VAT Fluid Mechanics in Porous Media?**

The VAT description of momentum transport in porous medium includes the mathematical models, governing equations for both (at
least) scales of the medium - lower spatial scale homogeneous governing equations and the upper scale heterogeneous governing
equations - Fundamentals of Hierarchical Scaled Description in Physics and Technology

Laminar Flow in Porous Media Turbulent Transport

Turbulent Transport

Capillary morphology of porous medium can be treated for some morphology classes with the actually exact solution of
momentum transport problem

Nonlinear Effects in Multiple Regime Transport of Momentum in
Longitudinal Capillary Porous Medium Morphology (665K)

Exact Closure Procedures of Hierarchical VAT Capillary
Thermo-Convective Problem for Turbulent and Laminar Regimes (589K)

**Are there any other Methods and Theories available ?**

There were many theories and methods devoted to porous (heterogeneous) media Fluid Mechanics.
Numerous books and textbooks are based on bulk one scale consideration approximate methods.
That is the conventional (and wrong, obsolete for porous, multiporous, heterogeneous media) kind of analysis and modeling. See more explanations in -

Fundamentals of Hierarchical Scaled Description in Physics and Technology.

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