Civil Engineering Reference
In-Depth Information
as indicated by Auriault (2005), a random medium and a periodic medium built with a
characteristic cell of the random medium present similar properties at the macroscopic
scale. Under the long-wavelength condition, different steps which justify the use of an
equivalent fluid are presented. Some properties of the double porosity media obtained
with the homogenization method by Olny and Boutin (2003), Boutin
et al
. (1998) and
Auriault and Boutin (1994) are summarized.
5.2
Viscous and thermal dynamic and static permeability
5.2.1 Definitions
Viscous permeability
More rigorously the viscous permeability should be defined as the visco-inertial perme-
ability. The viscous dynamic permeability
q
has been defined by Johnson
et al
. (1987).
The viscous dynamic permeability is a complex parameter that relates the pressure gra-
dient and the fluid velocity
υ
in an isotropic porous medium by
>
−
q(ω)
∇
p
= η
φ<
υ
(5.1)
where
η
is the viscosity and
φ
is the porosity. The symbol
<>
denotes an average over
the fluid part
f
of a representative elementary volume
. This leads to (see Equation
4.38)
ηφ
jωρ(ω)
,
q(ω)
=
(5.2)
where
ρ
is the effective density. From the definition of the flow resisivity
σ
, the limit
q
0
of
q
when
ω
tends to zero is
η
σ
q
0
=
(5.3)
The static viscous permeability
q
0
is a intrinsic parameter depending only on the
microgeometry of the porous frame.
Thermal permeability
The dynamic thermal permeability
q
has been defined by Lafarge (1993). The thermal
permeability is a complex parameter that relates the pressure time derivative to the mean
temperature by
q
(ω)jωp
=
φκ <τ>
(5.4)
where
κ
is the thermal conductivity. The use of the same denomination for the thermal
and the viscous parameters is justified by the analogy which appears at the microscopic
scale between Equations (4.1), (4.3) and Equations (4.2), (4.4) which are valid under the
long-wavelength condition. In Equation (4.1) the source term is
p
and in Equation
(4.2) the source term is
∂p/∂t
. In Equation (4.2) the thermal inertia
ρ
0
c
p
replaces the
density
ρ
0
in Equation (4.1). Moreover,
v
and
τ
are equal to 0 at the air - frame contact
−
∇
