Civil Engineering Reference
In-Depth Information
Mesoscopic
scale
Microscopic
scale
Γ
sp
Ω
sm
Ω
fp
Ω
fm
Ω
sp
l
p
l
m
Figure 5.10
A double porosity medium on the mesoscopic scale and on the microscopic
scale.
Two characteristic sizes
l
p
,l
m
are defined for the pore and the micropore structure,
related to two representative elementary volumes, REVp for the pores and REVm for the
micropores (see Figure 5.10).
The subscript
p
is used for the pores, the subscript
m
for the micropores and the
subscript
dp
for the double porosity medium. Two parameters,
ε
defining the (macro-
scopic/mesoscopic) separation of scale, and
ε
0
defining the (mesoscopic/microscopic)
separation are given by
ε
=
l
p
/L
(5.99)
ε
0
=
l
m
/l
p
(5.100)
The homogenization method for periodic structures can be used under the condition
that these parameters must be much smaller than 1. Two fictive single porosity materials
are introduced, the mesoporous medium defined from the REVp with an impervious
skeleton, and the microporous medium defined from the REVm without pores. Let
φ
p
be the porosity of the mesoporous medium and
φ
m
be the porosity of the microporous
medium. The porosity
φ
of the actual porous medium is
φ
=
φ
p
+
(
1
−
φ
p
)φ
m
.
5.10.2 Orders of magnitude for realistic double porosity media
Macroscopic size for a simple porosity medium
In a first step the Johnson
et al
. model is used to evaluate an order of magnitude of
the wavelength and of the macroscopic characteristic size of a simple porosity medium.
Two regimes characterize the viscous interaction. At low frequencies the dynamic viscous
permeability is close to the static viscous permeability. The static viscous permeability
q
0
is linked to the flow resistivity by
q
0
=
η/σ
(Equation 5.3), and the order of magnitude
O(η/φl
2
)
where
l
is the characteristic size of the pores (see
of the flow resistivity is
σ
=
=
8
η/(R
2
φ))
. The order of magnitude of the static viscous
for instance Equation 4.79,
σ
permeability is
O(φl
2
)
q
0
=
(5.101)
At large
ω
, the viscous permeability has the following limit
ηφ
jωρ
0
α
∞
q(
∞
)
=
(5.102)
