Geoscience Reference
In-Depth Information
where
p
a
and
p
w
denote the average air and water pres-
sures (in Pa), respectively. The capillary head (suction),
Ψ
In the fully saturated condition, the Darcy velocity
(also called the filtration velocity) is defined as the time
derivative of the filtration displacement. In these condi-
tions, the generalized Darcy
, is defined as
'
s law is given by (Jardani
ρ
w
g
=
p
w
−
p
c
p
a
ρ
w
g
et al., 2010)
Ψ
=
−
3 2
k
0
η
f
∇
p
+
ρ
f
u
+
ρ
f
α
ϕ
w
=
−
−
w
F
f
3 5
In unsaturated flow conditions, the gradient of the cap-
illary head is given by
where
k
0
denotes the quasistatic permeability of the
porous material (in m
2
),
F
=
1
ρ
w
g
∇
∇Ψ
=
p
w
3 3
denotes the electrical
formation factor (dimensionless), which is the ratio of
the bulk tortuosity
α ϕ
of the pore space to the connected
porosity, and
F
f
denotes the body force applied to the
pore water phase (in Nm
−
3
, e.g., the gravitational body
force or the electrical force acting on the excess of electri-
cal charges of the pore water). For the following equa-
tions, to keep the notation as light as possible, we will
not distinguish the variables expressed in the time
domain from those in the frequency domain. However,
it will be easy enough to recognize the domain of
the formulations when the equations are written in the
frequency domain versus the time domain. The switch
from one domain to the other is done by a simple Fourier
transform or its associated inverse Fourier transform.
In unsaturated conditions, the filtration displacement
and the mass density of the fluid (subscript
f
) phase
are given by
α
This assumption is used to avoid dealing with the flow
of the air phase, simplifying the problem solution. In
unsaturated conditions, the capillary pressure is positive,
the capillary head is negative, and the pressure of the
water phase is smaller than the atmospheric pressure.
The total head
h
includes the gravity force and is defined
by
+
z
where
z
denotes the elevation head (gravity
effect). Our model will be restricted to the capillary
regime, which has a saturation level that is above the
irreducible water saturation level. There are also many
mechanisms of electrical polarization in porous media
including the diffusion polarization, the polarization of
the Stern layer coating the surface of the mineral grains,
and the polarization of the metallic particles (e.g., pyrite
and magnetite) acting as semiconductors. At low fre-
quencies (<1MHz) and in the absence of metallic parti-
cles, the so-called
Ψ
-polarization prevails (Revil, 2013a,
b), which is due to the polarization of the Stern layer.
Finally, attenuation of the seismic waves associated with
squirt-flow dissipation mechanisms will be neglected
despite the fact that this mechanism is known to control
the attenuation of seismic waves in the frequency band
usually used in the field for seismic investigations (see
Rubino & Holliger, 2012). Neglecting squirt flow makes
the problem simpler and more tractable.
In saturated conditions, the (averaged) filtration dis-
placement is defined as (Morency & Tromp, 2008)
α
w
w
=
s
w
ϕ
u
w
−
u
3 6
ρ
f
=1
−
s
w
ρ
g
+
s
w
ρ
w
3 7
respectively. In these equations,
s
w
denotes the degree of
water saturation (
s
w
= 1 at full saturation). The mass den-
sity of the gas phase can be neglected, and therefore,
ρ
f
≈
s
w
ρ
w
. From Equation (3.4),
the porosity can be
replaced by
s
w
ϕ
), dealing with
the solid phase, remain unchanged). The Darcy velocity
associated with the water phase is given by
(of course, terms in (1
−
ϕ
w
=
ϕ
u
w
−
u
3 4
k
r
k
0
η
w
∇
ρ
w
s
w
α
w
ϕ
w
w
=
−
p
w
+
ρ
w
s
w
u
+
F
w
−
F
w
3 8
w
where
denotes the connected porosity and
u
w
and
u
correspond to the averaged displacement of the water
and solid phases, respectively. All the disturbances con-
sidered in the following text will be harmonic,
exp
ϕ
where
k
r
denotes the relative permeability (dimension-
less),
η
w
denotes the dynamic viscosity of the pore water
(in Pa s), and
p
w
is the pressure of the water phase (it will
be replaced later by the suction head defined previously
−
i
ω
t
, where
ω
=2
π
f
denotes the angular frequency
(in rad s
−
1
) and
f
=
ω
2
π
is the frequency (in Hertz).
