Geoscience Reference
In-Depth Information
by Eq. 3.2). The ratio
α
w
/
ϕ
corresponds to the bulk tortu-
ρ
w
=
ρ
w
F
3 14
osity of the water phase,
α
w
, divided by the connected
porosity. This ratio can be replaced by Archie
'
s second
where
ρ
w
denotes an apparent pore water mass density.
The relationship between the permeability and the water
saturation can be expressed with the Brooks and Corey
(1964) relationship:
law
Fs
w
where
n
is called the second Archie
s exponent
(
n
> 1, dimensionless) (see, for instance, Archie, 1942,
and Revil et al., 2007). From now on, the constitutive
equations are described in the frequency domain. There-
fore,
w
w
is replaced by
'
w
w
and so on. The Darcy
equation, Equation (3.8), can be rewritten as
−
i
ω
2+3
λ
k
r
=
s
3 15
w
where
λ
is termed the Brook and Corey exponent. Using
Equation (3.11) and Equations (3.13)
-
(3.15), the follow-
ing relationship between the relaxation time and the
saturation is obtained:
k
∗
ω
η
w
∇
2
−
i
ω
w
w
=
−
p
w
−
ω
ρ
w
s
w
u
−
F
w
3 9
where
k
∗
ω
is a complex-valued permeability given by
ρ
w
b
τ
k
=
s
w
2+3
λ
+1
−
n
k
r
k
0
3 16
λ
k
∗
ω
=
3 10
1
−
i
ωτ
k
In order to write a hydrodynamic equation coupled
with the electrical field, the body force
and where the relaxation time is given by
F
w
entering
Equation (3.9) should be expressed by Coulomb
'
s law:
τ
k
=
k
r
k
0
ρ
w
F
η
w
s
1
−
n
w
3 11
F
w
=
Q
V
ω
E
3 17
The relaxation time
τ
k
represents the transition
between the viscous laminar flow regime and the inertial
laminar flow regime. The critical frequency associated
with this relaxation time is given by
where
Q
V
ω
denotes the frequency-dependent (effec-
tive) excess charge that can be dragged by the flow of
pore water through the pore space of the material
(dynamic excess charge density of the pore space) and
E
denotes the electrical field, in Vm
−
1
. The charge density
Q
V
ω
is frequency dependent because there are more
charges dragged in the inertial laminar flow regime than
in the viscous laminar flow regime, in agreement with
the model of Pride (1994). In the following, the para-
meters
Q
0
η
w
1
k
r
k
0
ρ
w
F
s
n
−
1
f
k
=
πτ
k
=
3 12
w
2
2
π
0. Note
that the measurement of this relaxation frequency can
be used to estimate the permeability of the material.
In this analysis, low frequencies (
Note that because
n
≥
1 (Archie, 1942),
n
−
1
≥
V
and
Q
V
are the volumetric charge density
dragged in the low
ωτ
k
1 and high
ωτ
k
1 fre-
quency regimes, respectively. Because the transition
between low- and high-frequency regimes is governed
by the relaxation time,
τ
k
, the following functional can
be used to compute the effective charge density as a func-
tion of the frequency:
k
1,
f f
k
) cor-
respond to the viscous laminar flow regime where the
flow in a cylindrical pore obeys Poiseuille law. High fre-
quencies (
ωτ
k
1,
f f
k
) correspond to the inertial lam-
inar flow regime for which the pore water flow represents
a potential-flowproblem. Note that some authors (like Biot
in his earlier works) prefer to include the inertial effect in
an apparent (or effective) dynamic viscosity
ωτ
η
∗
ω
rather
than defining an apparent (or effective) permeability
k
∗
ω
1
Q
V
ω
1
Q
V
1
Q
0
1
Q
V
1
=
+
V
−
3 18
1
−
i
ωτ
k
. This choice (arbitrary) was abandoned later on.
In poroelasticity, it is customary to define the following
two variables (e.g., Morency & Tromp, 2008):
The form of this function is derived and explained fur-
ther in Revil and Mahardika (2013). We need to find
expressions for the low- and high-frequency charge den-
sities,
Q
0
b
=
η
w
k
0
3 13
V
and
Q
V
, respectively. We note that:
