Civil Engineering Reference
In-Depth Information
log(
λ
/
2π
)
log(
λ
/
2π
)
6
6
log(
l
0
/
2p
)
log(
l
0
/
2p
)
4
4
2
2
0
0
log(
l
p
)
−
2
−
2
−
4
−
4
−
6
10
−
3
−
6
ω
vP
ω
vP
ω
d
ω
vm
10
4
(rad s
−
1
)
10
−
3
10
3
(rad s
−
1
)
1
ω
1
ω
vm
ω
(a)
(b)
Figure 5.11
Asymptotic modulus of the wavelength in the pores (thick line) and in the
micropores (broken line) compared to the wavelength in the free air: (a) low contrast,
(b) high contrast (Olny and Boutin 2003). Reprinted with permission from Olny, X., &
Boutin, C. Acoustic wave propagation in double porosity media.
J. Acoust. Soc. Amer.
114
, 73 - 89. Copyright 2003, Acoustical Society of America.
and the microscopic scale. Three dimensionless space variables describe at each scale the
pressure and the velocity fields. Let
X
be the ordinary space variable. The macroscopic
space variable
x
is defined by
x
=
X
/L
, the mesoscopic space variable
y
and the micro-
scopic space variable
z
are defined by
y
=
X
/l
p
and
z
=
X
/l
m
. The first separation of
scales is defined by
ε
=
l
p
/L
, the second separation of scales is defined by
ε
0
=
l
m
/l
p
.
1and
ε
0
1. The acoustic pres-
sure and velocity are described with (
x, y, z
,
ω
) in the micropores, and with (
x, y
,
ω
)in
the pores except in a thin layer at the interface (
sp
)
with the micropores. In Section 5.7,
the porous material as a whole is replaced by an equivalent fluid of bulk modulus
K
and density
ρ
which occupies the full volume of the layer. This formalism is used in
the next subsections because its use is easier when two different porosities are present
than the initial model, where an equivalent free fluid replaces the air in the pores. Some
results are summarized in what follows.
The parameters
ε
and
ε
0
must satisfy the relations
ε
5.10.4 Low permeability contrast
For the selected example, estimating
L
from Equation (5.98) for
ω
O(ω
νm
)
yields
ε
=
O(ε
0
)
. The pressure in the micropores only varies at the macroscopic scale. In
a REVp at first order, the pressure denoted as
p
dp
(x)
, is uniform. At the macroscopic
level, the macroscopic flow law is similar to the macroscopic flow law given by Equation
(5.1) in a simple porosity medium
=
q
dp
(ω)
η
∗
υ
dp
=−
x
p
dp
φ
∇
1
|
fm
U
fp
|
(5.106)
.
=
.d
fm
U
fp
