Geoscience Reference
In-Depth Information
Table 2.1
Nomenclature of the nonmechanical material
properties.
Table 2.2
Nomenclature of the mechanical material properties.
Symbol
Meaning
Unit
Symbol Meaning
Unit
C
Biot modulus
Pa
F
Formation factor
Dimensionless
C
s
Biot modulus at saturation
Pa
f
Fraction of counterions in the Stern layer
Dimensionless
M
Biot modulus
Pa
m
Cementation exponent
Dimensionless
M
s
Biot modulus at saturation
Pa
n
Saturation exponent
Dimensionless
α
Biot coefficient at saturation
Dimensionless
Brooks and Corey exponent
Dimensionless
λ
α
w
Biot coefficient at partial saturation
Dimensionless
r
Coupling coefficient saturation exponent
Dimensionless
G=G
fr
Shear modulus of the solid frame
Pa
q
Characteristic time saturation exponent
Dimensionless
K=K
fr
Bulk modulus of the solid frame
Pa
m
2
k
S
Permeability at saturation
Lamé coefficient
Pa
λ
m
2
k
Low-frequency permeability
λ
u
Undrained Lamé coefficient
Pa
0
k
r
Relative permeability
Dimensionless
K
a
Bulk modulus of the air
Pa
m
2
k
Complex permeability
K
o
Bulk modulus of the oil
Pa
m s
−
1
K
h
Hydraulic conductivity
K
w
Bulk modulus of the water
Pa
m s
−
1
K
Hydraulic conductivity at saturation
K
S
Bulk modulus of the solid phase
Pa
s
V m
−
1
C
S
Coupling coefficient at saturation
K
f
Bulk modulus of the fluid phase
Pa
V m
−
1
C
Low-frequency coupling coefficient
K
u
Undrained bulk modulus
Pa
0
C
r
Relative coupling coefficient
Dimensionless
Vm
−
1
C
p
Complex streaming potential coupling
coefficient
C m
−
3
C
os
Electroosmotic coupling coefficient
the generalized Maxwell fluid (see discussion in
Section 2.1.1). The body force applied to the pore fluid
corresponds to the body force due to the gravity field plus
the electrostatic force associated with the existence of the
local net electrical charge density
ϕ
Porosity
Dimensionless
S m
−
1
σ
Electrical conductivity
S m
−
1
σ
In-phase electrical conductivity
S m
−
1
σ
Quadrature conductivity
(Cm
−
3
)
ρ
S m
−
1
σ
eff
Complex effective electrical conductivity
S m
−
1
σ
eff
(Real) effective electrical conductivity
E
e
−
i
ω
t
,
F
f
=
ρ
f
g
+
ρ
2 30
S m
−
1
σ
S
Surface conductivity of the solid phase
σ
∗
S m
−
1
Complex electrical conductivity
ρ
f
(in kg m
−
3
) is the mass density of the pore water,
g
(in m s
−
2
) the acceleration of the gravity, and
E
is the
electric field (in Vm
−
1
). The average force
F
f
denotes a
source of momentum in the momentum conservation
equation (see Section 2.1.4). Next, we average Equation
(2.28) over the fluid phase which yields
F m
−
1
where
ε
eff
(Real) effective dielectric constant
F m
−
1
ε
Dielectric constant
A m
−
2
L
∗
ω
Streaming current coupling coefficient
Cm
−
3
0
V
Moveable charge density at low
frequency
Q
∞
V
Cm
−
3
Moveable charge density at high
frequency
Q
C m
−
3
Q
V
Total charge density from the CEC
2
v
f
+
F
f
C m
−
3
−
i
ωρ
f
v
=
−∇
p
f
+
i
ωρ
f
v
s
+
η
f
∗
∇
2 31
Charge density of the diffuse layer
Q
V
H m
−
1
μ
Magnetic permeability
In the low frequency viscous laminar flow regime, the
no-slip boundary condition at the surface of the pores is
written as
v
=0.
As discussed in Chapter 1, the local charge density,
the phase average velocity of the solid phase. In the fre-
quency domain, using Equation (2.24), we have
ρ
(in Cm
−
3
), is associated with the electrical diffuse layer
part of the electrical double layer (see Figure 1.2) and
is therefore dependent on the position inside the pore
space. This local charge density is related to the concen-
trations of the ionic species that are affected by the
coulombic field created by the effective surface charge
density of the mineral surface. As discussed in Chapter 1,
η
f
∗
∇
2
v
f
+
F
f
,
−
i
ωρ
f
v
=
−∇
p
f
+
i
ωρ
f
v
s
+
2 28
η
f
η
f
∗
ω
=
ωτ
m
c
,
2 29
1
−
i
where
c
is the Cole
Cole distribution parameter corre-
sponding to the distribution of the relaxation times for
-
