Civil Engineering Reference
In-Depth Information
scale
p
(
1
)
and the other physical quantities in Equations (5.89) and (5.93) do not depend
on X
3
(Auriault 1986). Therefore Equations (5.89) and (5.94) will provide, under the
long-wavelength condition, a description of sound propagation in cylindrical pores similar
to the one given in Chapter 4.
Moreover the justification of the semi-phenomological macroscopic description of the
dynamic flow of a viscous fluid in a porous medium with a viscous permeability tensor
q
ij
which reduces to a dimensionless viscous permeability
q
∗
for isotropic media has
been obtained from Equations (5.89) - (5.92) by Levy (1979) and Auriault (1983)
q
ij
(ω)
η
∗
∂p
(
0
)
∂x
j
0
i
φ
υ
=−
(5.95)
A return to dimensional quantities is obtained by multiplying the right-hand side of
this equation by
l
2
p
c
/(Lη
c
v
c
)
=
1. This leads to the linear relation
q
ij
(ω)
η
∂p
∂X
j
φ
υ
i
=−
(5.96)
q
ij
/l
2
q
ij
=
where
f
·
d
1
|
f
|
· =
(In the main papers concerning homogenization,
φ
is removed because the velocity
is averaged over the total representative elementary volume). It has been shown by
Sanchez-Palencia (1980) that the tensor
q
ij
is symmetrical. Similarly, Auriault (1980) has
shown that Equations (5.92) -(5.93) lead to the following linear relation
q
(ω)
κ
φ
τ
=
jωp
(5.97)
The use of an equivalent fluid at the macroscopic scale is justified, under the
long-wavelength condition, by Equations (5.96) - (5.97). The homogenization method
for periodic structures justifies the use of the semi-phenomenological models where an
equivalent fluid is described from an effective density and a bulk modulus.
5.10
Double porosity media
5.10.1 Definitions
In this section, the basic definitions and some results of the work by Olny and Boutin
(2003) about sound propagation in double porosity media are presented. Detailed calcu-
lations can be found in Olny and Boutin (2003), Olny (1999) and Boutin
et al
. (1998).
In these media two networks of pores of very different characteristic sizes are intercon-
nected, as for instance in an ordinary porous medium with a microporous frame. The
macroscopic size characteristic of wave propagation is defined by
O
λ
2
π
L
=
(5.98)
