Abstract
In this paper, results from a combined network/averaging study are presented. The emphasis is placed on understanding the flow phenomena, rather than predicting results for real porous media. Idealized porous media, consisting of networks of tubes, are used to interpret two of the terms in the averaged momentum equation. In particular, it is demonstrated that the ‘pressure term’ accounts for microscopic cross flow, and that the magnitude of this term is proportional to the variation of the cross-sectional areas of the tubes in the macroscopic flow direction. For one-dimensional macroscopic flow in these idealized porous media, the agreement of ‘network theory’ and ‘averaging theory’ permeabilities depends on ‘areosity’ (a term related to the area open to flow in a direction) remaining constant in the macroscopic flow direction; it may vary in other directions.
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Abbreviations
- a i :
-
gravitational acceleration
- A :
-
area
- A k :
-
cross-sectional area of the REV normal to directionk
- A β :
-
A σβ +A βe
- A βe :
-
interfacial area between theΒ phase inside and outside of the REV
- A β1 :
-
area open to flow in directionk=1
- A σβ :
-
interfacial area between the σ andΒ phases contained within the REV
- b I (r, η):
-
distribution function for the microscopic velocity inside tubeI
- f I :
-
fraction of volumetric flow rate in tubeI (=q I /Q 1)
- g I :
-
π δ 4 I /(128μS I )
- k s :
-
permeability of standard network model
- k 1 :
-
permeability in directionk=1
- L k :
-
length of the REV in directionk
- n k :
-
unit outwardly normal vector
- I n k :
-
orientation of tubeI
- N :
-
number of tubes
- p :
-
thermodynamic pressure
- p h :
-
pressure at the upstream end of the REV
- P Ia ,P Ib :
-
pressures at the two ends of tubeI
- p l :
-
pressure at the downstream end of the REV
- q I :
-
volumetric flow rate in tubeI
- q IJ :
-
volumetric flow rate inside tubeI going in or coming out of junctionJ
- Q k :
-
total volumetric flow rate in the REV in directionk
- r :
-
radial distance to a point in a tube
- r k :
-
coordinate on the microscopic scale
- R I (η):
-
local radial distance to the wall of the tubeI
- S I :
-
length of tubeI
- t :
-
time
- υ k :
-
bulk or macroscopic velocity in directionk
- V :
-
volume
- V b :
-
total volume of the REV
- w k :
-
microscopic velocity of the fluid
- (w σβ ) k :
-
velocity of surfaceA σβ
- I w k :
-
microscopic velocity of the fluid inside the tubeI
- X k :
-
coordinate on the macroscopic scale
- δ a ,δ b ,δ c :
-
tube diameters used in networks models
- δ I :
-
diameter of tubeI
- η :
-
angle that the radius through a point inside a tube makes with the reference
- η A :
-
azimuthal angle of a tube
- η p :
-
planar angle of a tube
- µ :
-
dynamic viscosity of the fluid
- μ δ :
-
mean tube diameter
- µ S :
-
mean tube length
- ξ :
-
areosity
- ρ :
-
density of the fluid
- σ δ :
-
standard deviation of the tube diameters
- σ S :
-
standard deviation of the tube lengths
- τ :
-
tortuosity
- τ :
-
porosity
- ψ :
-
a general variable
- 〈 〉:
-
phase average
- 〈 〉 α :
-
intrinsic phase average
- ∼:
-
deviation from intrinsic phase average
- α :
-
a general phase
- Β :
-
the fluid phase
- σ :
-
the solid phase
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Ruth, D., Suman, R. The râle of microscopic cross flow in idealized porous media. Transp Porous Med 7, 103–125 (1992). https://doi.org/10.1007/BF00647392
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DOI: https://doi.org/10.1007/BF00647392