Geoscience Reference
In-Depth Information
∇
v
f
=0,
2 18
T
s
=
K
s
∇
u
s
I
+2
G
s
d
s
,
2 26
respectively, where
ρ
f
denotes the local mass density of
the fluid,
F
f
is the body force, and
d
s
=
1
1
3
∇
u
s
2
∇
u
s
+
∇
−
u
s
I
,
2 27
where
d
s
is the solid strain deviator (dimensionless),
K
s
(in Pa) is the bulk modulus of the solid phase assumed
to be isotropic, and
G
s
is the shear modulus (in Pa).
i
ωτ
m
e
π
=
T
f
+
p
f
I
+
ωτ
m
K
f
∇
u
f
I
+2
G
f
d
f
2 19
1
−
i
represents the viscous (deviatoric) stress tensor of the
fluid. For a purely viscous Newtonian fluid, we have
∇
π
2.1.2 Properties of the porous material
Next, we derive the coupled constitutive equations
between the Darcy velocity and the total current density
for a linear poroelastic material saturated by a polar (wet)
oil (Figure 2.2). The non-mechanical properties used in
this section are provided in Table 2.1 while the mechan-
ical properties are summarized in Table 2.2. We look for
an average force balance equation on the fluid in relative
motion with respect to the solid phase. We introduce the
relative flow velocity
v
=
v
f
−
2
v
f
. Assuming a linearized Maxwell model and
using Equation (2.4), the viscous stress tensor
=
η
f
∇
π
obeys
G
f
η
f
π
π
+
=2
G
f
e
f
,
2 20
τ
m
π
+
π
=
η
f
∇
v
f
2 21
In the frequency domain, Equation (2.21) yields
v
s
(nm s
−
1
) where
v
s
=
u
s
is
1
−
i
ωτ
m
π
=
η
f
∇
v
f
,
2 22
L
η
f
1
−
i
ωτ
m
∇
2
v
f
∇
π
=
2 23
Inserting Equation (2.23) inside the Navier
-
Stokes
equation, Equation (2.17), yields
z
2
R
η
f
∇
2
v
f
+
F
f
−
i
ωρ
f
v
f
=
−∇
p
f
+
2 24
-
Stokes equation of a viscous Newtonian fluid. This equa-
tion can be used by replacing the classical viscosity
Equation (2.24) has the same form than the Navier
z=L
z
=0
η
f
by
an effective (time- or frequency-dependent) viscosity
(a)
η
f
.
The result of the analysis in this section is that a Cole
REV
-
Cole distribution of relaxation times can be used to
describe a generalized Maxwell fluid (see Figure 2.1b).
We consider that the solid phase is formed by a mono-
mineralic isotropic solid material assumed to be perfectly
elastic. The local elastic equation of motion for the solid
phase is
Solid
Fluid
2
u
s
∂
ρ
s
∂
∇
T
s
+
F
s
=
,
2 25
t
2
(b)
Figure 2.2
Sketch of the porous material.
a)
The representative
elementary volume (REV) of the porous material is an averaging
disk of radius
R
and length
L
.
b)
The REV corresponds to a porous
body with an elastic skeleton filled with a viscoelastic fluid
characterized by an extended Maxwell rheology.
where
u
s
(in m) denotes the displacement of the solid
phase,
ρ
s
(in kg m
−
3
) is the bulk density of the solid phase,
and
F
s
is the body force applied to the grains (Pa m
−
1
or
Nm
−
3
). The microscopic solid stress tensor is given by
