Civil Engineering Reference
In-Depth Information
A calculation of
q
dp
is performed in Olny and Boutin (2003) for a microporous
medium drilled with parallel circular cylindrical pores. The calculation is restricted to the
direction of the pores. The semi-phenomenological models cannot be used. For
ω<ω
νp
the velocity in the pores is much larger than in the micropores and the contribution of
the micropores to the flow can be neglected. From Equations (5.2) and (5.43) the density
of the fluid which replaces the porous medium is given, on the macroscopic scale by
η
jωq
dp
(ω)
ρ
dp
(ω)
=
(5.107)
The bulk modulus defined by Equation (5.44) is given by
−
1
1
K
p
(ω)
+
1
K
m
(ω)
K
dp
(ω)
=
(
1
−
φ
p
)
(5.108)
φ
p
)φ
m
in the micropores. The bulk modulus of air in the pores is
φ
p
K
p
(ω)
and
φ
m
K
m
(ω)
in
the micropores and
The volume of air in the pores is
φ
p
per unit volume of material and is
(
1
−
1
K
dp
(ω)
=
φ
p
φ
p
K
p
(ω)
+
(
1
−
φ
p
)φ
m
φ
m
K
m
(ω)
.
5.10.5 High permeability contrast
For the selected medium, considering that
ω
O(ε
3
)
.Asinthecase
of low-contrast media at
ω<ω
νm
the flow in the micropores is mainly viscous, and the
bulk modulus is isothermal. The viscous permeability is given by
=
O(ω
d
)
yields
ε
0
=
q
dp
(ω)
=
(
1
−
φ
p
)q
m
(ω)
+
q
p
(ω)
(5.109)
At first-order development the pressure
p
p
in the pores only varies on the macroscopic
scale. The main difference with the low-contrast media concerns the first-order pressure
p
m
in the micropores which can vary as a function of the mesoscopic scale variable
y
,
due to the poor transmission of the acoustic field in the micropores. The pressure satisfies
a diffusion equation in the micropore domain
sp
(see Equation 91 in Olny and Boutin
2003). As indicated in Boutin
et al
. (1998) the description of the pressure field in the
microporous domain
sp
around the pores in the double porosity medium is similar to
the description of the temperature field in the fluid around the pores in a simple porosity
medium. The temperature is equal to 0 at the surface of the pores in contact with air, and
the pressure is equal to the pore pressure
p
p
at the surface of the pores in contact with
the microporous medium. The thermal skin depth is
δ
=
(
2
ν
/ω)
1
/
2
and the skin depth
for the pressure is given by (see Equation 91 in Olny and Boutin 2003)
2
P
0
q
0
m
φ
m
ηω
1
/
2
δ
d
=
(5.110)
In Olny and Boutin (2003) the factor 2 on the right-hand side is removed due to
the different definition of the skin depth. In order to generalize the use of the simplified
