Geoscience Reference
In-Depth Information
p
f
components of the elastic and viscous contributions of
the Maxwell fluid are given by
p
f
=
ωτ
m
,
2 5
1
−
i
η
f
1
−
i
ωτ
m
e
e
e
T
ij
=
λ
f
e
kk
δ
ij
+2
G
f
e
ij
,
2 1
η
f
=
2 6
v
v
=2
η
f
e
ij
−
p
f
δ
ij
,
T
ij
2 2
η
f
denote effective fluid pressure
(in Pa) and dynamic viscosity (in Pa s), respectively. At
low frequencies
The quantities
p
f
and
where the superscripts
“
e
”
and
“
v
”
stand for
“
elastic
”
and
η
f
, the
third term on the right-hand side of Equation (2.4) is
negligible, and the stress tensor is given by (e.g., De Groot
and Mazur, 1984)
ω
ω
m
, we have
p
f
=
p
f
and
η
f
=
“
contributions to the stress tensor and to thedefor-
mation tensor, the dot above the symbol denotes the first
time derivative,
viscous
”
λ
f
and
G
f
(both in Pa) are the Lamé and
shearmoduli of the fluid,
η
f
is thedynamic (shear) viscosity,
and
p
is the local fluid pressure. We use the (Einstein) con-
vention that repeated indices represent summation.
The bulk modulus of the fluid
K
f
(in Pa) is defined by
the relationship
v
T
f
=
K
f
∇
u
f
I
+2
η
f
d
f
,
2 7
v
where the
d
f
corresponds to the viscous contribution
to the fluid strain deviator,
u
f
I
where
I
denotes the
identity tensor and
u
f
(in m) the displacement of the fluid
phase. The bulk modulus of the fluid is related to the
Lamé constants by
K
f
=
−
p
f
I
=
K
f
∇
=
1
1
3
∇
v
f
+23
G
f
. For a Maxwell fluid,
the components of the stress tensor of the fluid,
T
f
, are
given by
T
ij
=
T
ij
λ
T
2
∇
v
f
+
∇
−
d
f
v
f
v
f
I
2 8
e
v
e
v
=
T
ij
and
e
ij
=
e
ij
+
e
ij
. Using
The vector
d
f
(dimensionless) denotes the fluid strain
deviator,
v
f
(m s
−
1
) corresponds to the velocity of the
fluid, and the superscript
T
means transpose. The defor-
mation tensor is related to the deviator by
e
v
Equations (2.1) and (2.2) with
e
ij
=
e
ij
+
e
ij
yields
T
ij
+
G
f
η
f
e
T
ij
+
p
f
δ
ij
=2
G
f
e
ij
+
λ
f
e
kk
δ
ij
2 3
1
3
∇
Assuming harmonic variations of the stress (i.e., the
stress oscillates in time as
e
−
i
ω
t
,
i
denotes the pure imag-
inary number as in Chapter 1), we can use the previous
equation in the frequency domain using a Fourier trans-
form (we keep the same notations of the time and fre-
quency domain variables). The characteristic frequency
is defined as
e
f
=
u
f
I
+
d
f
2 9
At low frequencies, the fluid behaves therefore as a
Newtonian fluid. At high frequencies
ω
>>
ω
m
, the stress
tensor is given from Equation (2.4) by
ω
m
=
G
f
η
f
and the associated relaxation time
T
f
=
K
f
∇
u
f
I
+2
G
f
d
f
2 10
is
η
f
G
f
. Typically, for the Mexican crude oil investi-
gated by Dante et al. (2007), the critical frequency
depends on the temperature and is in the range
10
τ
m
=
Therefore, at high frequencies, the pore fluid behaves
as an elastic material.
A singular perfect Maxwell fluid has a relaxation time
distribution described by a Dirac (delta) function. How-
ever, the real fluids we consider here are complex mix-
tures of various different fluids, and as such, these
mixtures do not behave like a single, perfect Maxwell
fluid. Instead, they behave more like a generalized
Maxwell fluid with a variety of relaxation times. From
this, we can create a more realistic and generalizedmodel
using a distribution of relaxation times. For instance, a
Cole
40 C. For heavy
oils in tar sands, for example, the critical frequency of
resonance can occur at much lower frequencies. Usually,
heavy oils are heated to decrease their dynamic viscosity,
and as a result, the resonance frequency can shift into the
seismic frequency band (see Behura
et al
., 2007), depend-
ing on the resulting change in dynamic viscosity.
In the frequency domain, Equation (2.3) is written as
-
100 Hz in the temperature range 20
-
v
p
f
δ
η
f
T
ij
=
−
ij
+2
−
i
ω
e
ij
2 4
Cole distribution of relaxation times can be used
as a generalized model (e.g., Revil et al., 2006; Behura
-
ωτ
m
1
−
i
ωτ
m
λ
i
e
e
−
f
e
kk
δ
ij
+2
G
f
e
ij
,
