Macroscopic Balance Equations for Spatial or Temporal Scales of Porous Media Hydrodynamic Modeling

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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.