Civil Engineering Reference
In-Depth Information
5.4
Models for the effective density and the bulk modulus
of the saturating fluid
5.4.1 Pride
et al.
model for the effective density
A model that can be used for identical cylindrical pores with different cross-sectional
shapes has been described in Chapter 4. This model predicts the right asymptotic
behaviour at high and low frequencies, and gives good predictions in the intermediate
frequency range, at least for slits and circular cross-sectional shaped pores. More general
models have been suggested by Johnson
et al
. (1987), and Pride
et al
. (1993). The
effective density suggested by Johnson
et al
. (1987) has the simplest expression for
the high-frequency limit and the low-frequency limit of the imaginary part previously
indicated in this chapter, which satisfies the physical constraint due to causality
concerning the singularities which must be located on the positive imaginary frequency
axis. The model has been modified by Pride
et al
. (1993) to adjust the low-frequency
limit of the real part of the effective density with a parameter denoted as
b
in what
follows. The ratio
ρ/ρ
0
defined as the dynamic tortuosity is given by
$
1
/
2
'
(
+
b
1
+
2
α
∞
q
0
bφ
2
jω
ν
νφ
jωq
0
α(ω)
=
1
−
b
+
α
∞
(5.32)
%
B
2
ν
,B
2
being the Prandtl number. The limit of the real part of
the effective density when
ω
tends to zero is
ρ
0
[
α
∞
+
where
ν
=
η/ρ
0
=
q
0
/(bφ
2
)
]. The right
low-frequency limit
α
0
for the real part of
ρ
is obtained by Lafarge (2006) for
b
,
given by
2
α
2
∞
2
q
0
α
2
∞
φ
2
(α
0
−
b
=
(5.33)
α
∞
)
3
/
4. Simulations on simple geome-
tries performed by Perrot (2006), and experiments with air-saturated porous media show
that Equation (5.32) can provide very precise predictions of the effective density. Nev-
ertheless, there is a limit to the applicability of Equations (5.32) and (5.33). Simulations
performed by Cortis
et al
. (2003) show that the general formulations of Pride
et al
.
(1993) with
given by Equation (5.24) become inadequate for porous structures with
sharp edges.
The limit for the circular pores is obtained for
b
=
5.4.2 Simplified Lafarge model for the bulk modulus
In Johnson
et al
. (1987) the dynamic tortuosity
α
(
ω
) is the elementary function used
to express the dynamic viscous permeability, and in the present work, the effective
density. Similar functions exist for the description of the thermal exchanges and the
incompressibility. The function denoted as
α
(
ω
), related to the bulk modulus
K
by
P
0
/
1
−
−
1
γα
(ω)
γ
K
=
(5.34)
