Geoscience Reference
In-Depth Information
1
M
p
With the poromechanical equations defined, we can
now define the electrical equations. Computing the por-
oelastic problem is a necessary step prior to determining
the volumetric electrical source current density =
δ
pd
Ω−
k
ω
∇
p
∇
δ
pd
Ω
+
θ
ω
∇
pud
Ω
Ω
Ω
Ω
+
k
ω
∇
p
n
δ
P
+
θ
ω
u
n
d
Γ
=0
4 6
J
S
(expressed in Am
−
3
). The volumetric electrical source
current density will be used as a source term when
solving the following Poisson equation for the electrical
potential
∇
Γ
s
is the
solid strain tensor, and
n
represents the unit outward
normal vector to the surface
where
u
n
denotes the normal component of
u
,
ε
ψ
(see Eq. (2.215)):
. In the next step, we dis-
cretize these integrals and each of their elements can be
represented in the following matrix form:
Γ
∇
σ
∇
ψ
= ,
4 15
Q
0
2
J
S
=
−
i
ω
V
k
w
∇
p
−
ω
ρ
f
u
,
4 16
u
e
=
N
s
u
n
e
,
p
e
=
N
f
p
n
e
,
4 7
where
denotes the electrical conductivity of the
material. We reformulate the electric part of the problem
using variational discretized equations. This yields (e.g.,
Soueid et al., 2013)
σ
where {
u
n
}
e
and {
p
n
}
e
denote the solid displacement
and fluid pressure at the finite-element grid nodes,
respectively, and [
N
s
] and [
N
f
] correspond to the shape
functions of the solid phase displacement and the fluid
pressure within each element, noted by the superscript e.
In the next step, we rebuild each integral under the
following discretized forms:
m
j =
1
ψ
i
φ
i
4 17
e
i
e
i
e
=
Ω
σ
∇
ψ
∇
φ
j
=
φ
j
+
Γ
σ
∇
ψ
n
φ
j
,
with
ψ
Ω
=
Jn
m
u
T
M
u
,
s
ω
ρ
u
δ
ud
Ω
=
δ
4 8
1
ψ
i
Ω
σ
∇
φ
i
∇
φ
j
=
φ
j
+
J
n
φ
j
4 18
Ω
Γ
j
=
Ω
u
T
K
u
,
s
u d
T
u
ε
Ω
=
δ
4 9
e
represents the discrete formulation of the
electrical potential
where
ψ
Ω
φ
i
is an interpolation function.
In terms of matrices, we write
ψ
and
u
T
C
p
,
θ
ω
∇
p
δ
ud
Ω
=
δ
4 10
Ω
ψ
=
Q
e
,
K
e
4 19
p
T
H
p
,
k
w
∇
p
∇
δ
pd
Ω
=
δ
4 11
Ω
e
ij
=
K
Ω
σ
∇
φ
i
∇
φ
j
,
4 20
1
M
p
p
T
Q
p
,
δ
pd
Ω
=
δ
4 12
Q
e
=
Ω
φ
j
+
J
n
φ
j
4 21
Ω
Γ
p
T
C
u
,
θ
ω
∇
Ω
=
δ
pud
4 13
Equation (4.19) can be solved using a Gaussian elimi-
nation approach with partial pivoting to determine the
distribution of the electrical field (Kaw and Kalu, 2008).
Ω
where
M
denotes a mass matrix and
K
a stiffness matrix.
The matrices
H
and
Q
are kinetic and compression
energymatrices for the fluidphase, and
C
is thevolumetric
hydromechanical coupling matrix. This leads to the
following linear system of equations:
4.1.2 Perfectly matched layer
boundary conditions
Equations (4.1) and (4.2) describe the propagation of
the seismic waves in a poroelastic framework within an
infinite (unbounded) medium. However, when one
performs numerical simulations, the domain investi-
gated is always bounded. A finite domain will always
produce reflections at the domain boundaries unless the
F
s
F
f
−
ω
2
M
+
K
−
C
u
p
=
,
4 14
C
T
−
H
+
Q
where
F
s
s loading vector, while
F
f
is the skeleton
'
is the
kinematic coupling vector for the interstitial fluid.
