We focus on thefirst author’s previous work addressing macroscopic balance equations developed
for different spatial and temporal scales. We elaborate on previous findings so
as to orient the reader to fundamental concepts with which the mathematical
formulations are developed. The macroscopic balance Partial Differential Equations
(PDE’s) are obtained from their microscopic counterparts by volume averaging
over a Representative Elementary Volume (REV), considering a non-Brownian
motion. The macroscopic quantity of phase/component intensive quantities
product, is the premise of two concurrent decomposed macroscopic balance PDE’s
of the corresponding extensive quantity. These are concurrently valid at the
primary REV order of length and at a significantly smaller secondary length.
The hydrodynamic characteristic at the smaller spatial scale was found to
always be described by pure hyperbolic PDE’s, the solution of which presents
displacement of sharp fronts. Reported field observations of condensed
colloidal parcels motion, validate the suggestion of hyperbolic PDE’s
describing fluid momentum and components mass balance at the smaller spatial
scale. Controlled experiments supplemented by numerical predication can yield
the hydrodynamic interrelation between the two adjacent spatial scales.
Further, we focus
on the first author past developments concerning dominant macroscopic balance
PDE’s of a phase mass and momentum and a component mass following an onset of
abrupt pressure change. These account for the primary REV order of length and
for evolving temporal scales. Numerical simulations were found to be consistent
in excellent agreement with experimental observations. During the second time
increment and in view of the aforementioned developments, we presently
elaborate on new findings addressing theoretically the efficiency of expansion
wave for extracting solute from a saturated matrix. Simulations comparing
between pumping using an approximate analytical form based on Darcy’s equation
and numerical prediction addressing the emitting of an expansion wave, suggest
that the latter extracts by far more solute mass for a spectrum of different
porous media.
Application of spatial averaging rules,
referring to a REV, leads to the formulation of the macroscopic balance
equations addressing phase interactions such as fluids carrying components and
a deformable porous matrix.
Further elaborations by Sorek et al. Sorek
and Ronen and Sorek et al.
prove that the phases and components
macroscopic balance PDE’s can be decomposed into a primary part that refers to
the REV length scale and, concurrently, a secondary part valid at a length
scale smaller than that of the corresponding REV length. The secondary
macroscopic balance equation always conforms to a hyperbolic PDE. Geometrical
patterns of different spatial scales that prevail in various porous media are
exemplified. Such patterns support the notion of the need to
implement macroscopic balance equations addressing different spatial scales.
Observations verify that the hydrodynamic characterization of colloidal
transport comply with the developed fluid and component macroscopic balance
equations for the smaller spatial scale.
