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3.5 Comparison with
experimental data
and Smits (1968) model for unsaturated siliciclastic
materials. The saturation dependence of Model B was
recently proposed for the seismoelectric coupling in
unsaturated conditions byWarden et al. (2013; see equa-
tion 13). Later, we check the consistency of these two
models with respect to the available electrokinetic data.
The effect of saturation on the excess of electrical
charges per unit volume is developed by extending the
empirical relationship between these two parameters in
unsaturated conditions (see Linde et al., 2007; Revil
et al., 2007):
3.5.1 The effect of saturation
We consider an unsaturated material with two immisci-
ble fluid phases: (1) water that is considered to be the
wetting phase for the solid and (2) a nonwetting and
insulating fluid phase like air or oil. The saturation can
be related to the capillary pressure by the Brook and
Corey (1964) relationship:
λ
,
p
c
≥
p
e
p
e
p
c
s
r
+1
−
s
r
Q
V
s
w
=
Q
V
s
w
s
w
=
3 191
3 195
1,
p
c
<
p
e
or alternatively
p
c
=
p
e
s
e
−
1
λ
(or
s
e
=
p
c
p
e
−
λ
,
p
c
This equation has been recently challenged by Jougnot
et al. (2012) but seems to be consistent with experimen-
tal data as checked by Linde et al. (2007), Revil et al.
(2007), and Mboh et al. (2012). We will show later that
this model is also consistent with the data from Guichet
et al. (2003), Revil et al. (2011), and Vinogradov and
Jackson (2011).
p
e
; see
equation 12 of Brooks & Corey, 1964) where
s
e
=
s
w
≥
s
r
denotes the effective or reduced
water saturation,
s
r
is the irreducible water saturation,
and
p
c
represents the capillary pressure. The capillary
entry pressure,
p
e
, is the critical pressure needed to dis-
place the water phase by the gas phase when the porous
material is fully water saturated. The capillary entry pres-
sure is related to the media permeability in the following
full saturation condition.
The saturation dependence of the dynamic mass den-
sity is given by
−
s
r
1
−
3.5.2 Additional scaling relationships
In this section, we develop a unified set of scaling rela-
tionships between the hydraulic and electrical properties
in order to reduce the number of input parameters. Three
scaling laws are developed later, one for the relative
water permeability, one for the capillary entry pressure,
and one for the streaming potential coupling coefficient.
For each new scaling law, we will show that it is in agree-
ment with existing experimental data or empirical scaling
laws based on fitting experimental data.
Johnson (1986) developed the following equation for
the electrical conductivity for a water-saturated rock:
ρ
f
s
w
=
1
F
s
w
1
−
s
w
ρ
g
+
s
w
ρ
w
3 192
The saturation dependence of the permeability-related
constant,
k
ω
, and the coefficients,
S
ω
, can be easily calcu-
lated. The dependence of the Biot coefficient on satura-
tion can be found in Revil and Mahardika (2013).
Now, we need to determine the effect of the saturation
on the electrical conductivity. We investigate here two
models:
ρ
=
1
F
σ
w
+
2
σ
Λ
Σ
S
3 196
=
1
β
S
Q
V
s
w
F
s
w
σ
σ
w
+
Model A
3 193
where
Λ
is a characteristic length scale of the pore space.
A comparison of Equations (3.192) and (3.194) with
Equation (3.196) implies the following scaling laws for
the dependence of the formation factor and length scale,
Λ
, with the relative water saturation:
=
1
Q
V
s
w
F
s
w
σ
σ
w
+
β
S
Model B
3 194
where
n
is called the saturation exponent (Archie, 1942).
Model A is discussed in detail in Revil (2013a) and was
used for the seismoelectric problem by Revil and
Mahardika (2013). It is also consistent with the Waxman
Fs
−
w
F
3 197
Λ
Λ
s
w
Model A
3 198
