Geoscience Reference
In-Depth Information
=
η
f
k
0
and
k
0
(in m
2
) is the permeability for
quasistatic flow. In the following,
where
b
saturation exponent),
σ
w
denotes the electrical conduc-
tivity of the pore water (in S m
−
1
), and
σ
s
stands for
the surface conductivity (in S m
−
1
) associated with con-
duction in the electrical double layer coating the surface
of the grains. The dimensionless formation factor,
F
,is
related to the connected porosity,
ρ
=1
−
ϕ ρ
S
+
ϕρ
f
,
7 20
(in kg m
−
3
) denotes the bulk density of material,
ϕ
repre-
sent the connected porosity,
ρ
S
and
ρ
f
are the mass
densities of the solid and pore fluid phases, respectively.
Here, we provide the equations describing the electro-
kinetic theory in unsaturated conditions using the theory
developed by Revil and Mahardika (2013). In the follow-
ing, the electrical field is given by
E
=
−∇
φ
in the quasi-
static limit of the Maxwell equations, and
σ
(in S m
−
1
)
represents the bulk electrical conductivity of the porous
medium. The electric potential
ψ
(in V) is obtained by
solving the following elliptic equations (see details in
Jardani et al., 2010):
ϕ
(dimensionless), by
ϕ
−
m
Archie
1,
dimensionless) is usually called the cementation expo-
nent and more precisely the porosity exponent or first
Archie
'
s law,
F
=
(Archie, 1942), where
m
(
≥
s exponent. In the following example, we use
m
=
n
= 1.8, and the surface conductivity is reported in
Table 7.1.
The effect of the saturation on the other properties is
given by (see Chapter 3)
'
ρ
f
=1
−
s
w
ρ
g
+
s
w
ρ
w
,
7 25
K
f
=
1
−
1
s
w
K
g
+
s
w
∇
σ
∇
ψ
=
,
7 21
ℑ
K
w
,
7 26
where the volumetric source current density is given by
ℑ
s
w
η
g
η
w
η
g
J
S
. As discussed in Chapter 2, the dynamic stream-
ing current density,
J
S
(in Am
−
2
), is related to the proper-
ties of the seismic waves by
=
∇
η
f
=
,
7 27
ρ
f
=
1
F
s
w
ρ
f
,
7 28
J
S
=
Q
0
Q
0
V
s
w
k
ω
∇
2
V
s
w
w
=
−
i
ω
p
−
ω
ρ
f
u
7 22
k
0
s
w
=
k
r
s
w
k
S
=
s
n
+
w
k
S
7 29
In Equation (7.22),
Q
0
V
is the effective charge per unit
pore volume that is dragged by the flow of the pore water
relatively to the solid phase, and
w
symbolizes the Darcy
velocity (see Chapters 1 and 2). The charge density at sat-
uration,
Q
0
The model parameters used for the simulation are
given in Table 7.1.
V
, is related to the low-frequency permeability
at saturation,
k
0
, by (Figure 1.11)
Table 7.1
Material properties used in the numerical modeling.
Parameter
Symbol
Value
Units
log
10
Q
V
=
−
9 2349
−
0 8219log
10
k
0
7 23
kg m
−
3
Matrix density grains
ρ
s
2650
This relationship avoids the introduction of additional
parameters other than those used to describe the seismic
wave propagation part of the overall formulation.
The last effect to account for is the variability of fluid
saturation. The bulk electrical conductivity,
σ
, is deter-
mined by (Revil, 2013a,b)
kg m
−
3
Water density
ρ
w
1000
Matrix bulk modulus
K
s
36.5
GPa
Frame bulk modulus
K
fr
0.65
GPa
Shear modulus
G
0.09
GPa
Water bulk modulus
K
w
0.20
GPa
1×10
−
3
Water viscosity
η
w
Pa s
1×10
−
14
Permeability
k
0
Pa s
6.8 × 10
−
2
Sm
−
1
Water conductivity
σ
w
=
1
1×10
−
5
Sm
−
1
F
σ
w
s
w
+
σ
s
s
n
−
1
w
Surface conductivity
σ
s
σ
7 24
kg m
−
3
Air density
ρ
g
1.205
Air bulk modulus
K
g
0.140
MPa
where
s
w
is the water saturation,
n
(
≥
1, dimensionless)
symbolizes
2×10
−
5
Air viscosity
η
Pa s
g
the second Archie
'
s exponent
(or
the
