XVI International Conference on Computational Methods in Water Resources (CMWRXVI) Ingeniørhuset  

Scale Dependent Fluid Momentum and Solute Mass Macroscopic Balance Equations: Theory and Observations  Paper   

Author:  Shaul Sorek <sorek@bgu.ac.il> (BenGurion University of the Negev, ISRAEL) 
 Daniel Ronen <danronen@bgu.ac.il> (BenGurion University of the Negev, ISRAEL) 
Presenter:  Shaul Sorek <sorek@bgu.ac.il> (BenGurion University of the Negev, ISRAEL) 
Date:
 20060618
Track:
Special Sessions
Session:
Multiscale methods for flow in porous media 
DOI:  10.4122/1.1000000651 
DOI:  10.4122/1.1000000652 

A mathematical development is presented concerning extensions to the macroscopic
momentum balance equation for compressible Newtonian fluids flowing through
saturated porous matrices, and the macroscopic mass balance equation of solutes
transported with the fluids.
It is shown that each of these balance equations is composed of a dominant
macroscopic equation associated with a larger spatial scale, coupled with a
secondary macroscopic balance equation valid at a smaller spatial scale. The
dominant fluid momentum balance equation can govern the propagation of shock waves,
conform to Forchheimer's law or to Darcy's law when friction at the solidfluid
interface is dominant. Concurrently, the secondary momentum balance equation is
governed by inertia flow that conforms to a wave equation propagating the intensive
momentum and the dispersive momentum flux, both deviating from their corresponding
dominant average terms. The dominant macroscopic solute mass balance equation
accounts for advection and hydrodynamic dispersion. The secondary macroscopic solute
mass balance equation describes pure advection of the product of deviations from the
average solute mass fraction and the average fluid density.
Field observations under natural gradient flow conditions show excessive high
concentration of colloids (average of 50 mg/L) under land irrigated by sewage
effluents. The high concentration of colloids in a macroscopic flow field where
specific discharge varies between 4 to 16 m/yr, is suggested to be due to the
enhancement of colloidal mobility resulting from the secondary fluid momentum
equation governed by inertia and the secondary solute mass equation of pure
advection, both proven to be valid at a scale smaller then the one considered for
spatial averaging. 