Geoscience Reference
In-Depth Information
computation with the new kernel matrix. For each dipole
position, a series of statistically generated dipole moments
are used to compute the forward model simulation of the
data which is then used in a cost function that calculates
the error between the data and the model. The GA com-
putes numerous model realizations and converges to a
minimum of the cost function for each dipole position.
After all dipoles in the matrix have had their cost functions
minimized, the resulting error function is searched for
the minimum of all model realizations, the dipole solution
for each event. A GA was selected for its computational
parallelization potential, thereby generating a systematic
approach that can approach real-time speeds with parallel
computing resources.
P
=
1
2
ρ
∇
J
S
M
xP
,
M
dV
+
1
2
E
M
xP
,
M
dV
ψ
∇
ln
ρ
M
π
Ω
π
Ω
5 38
where
x
denotes the distance from the source at position
M
to the electrode located at position
P
where the
electrical potential signal is recorded. In Equation (5.38),
the two contributions associated with the primary field
(first term of the right-hand side of Eq. 5.38) and the
secondary potential (the second term of the right-hand
side of Eq. 5.38) are separated. The primary source term
is due to the hydromechanical disturbances, while the
second term is due to the heterogeneities in the resistivity
distribution of the resistive block.
The solution can be written in a more compact form
using the following convolution integral:
5.3.5.1 Electrical and hydromechanical coupling
The coupling between the hydromechanical equations
and the electromagnetic equations is described in
Mahardika et al. (2012) including dynamic terms. The
governing equation for the occurrence of self-potential
signals is obtained by combining a constitutive equation
with a continuity equation. The constitutive equation
corresponds to a generalized Ohm
ψ
P
=
K
P
,
M
J
S
M dV
5 39
Ω
where
K
(
P
,
M
) is called the kernel or the leading field and
dV
is a small volume around the source point
M
and
(
P
)
denotes the vector of self-potential measurement at a set
of stations
P
. We use this equation in the following com-
putations in our attempt to localize the causative source
of the electrical bursts shown in Figure 5.17. Generally,
the elements of the kernel are Green
ψ
'
s law for the total
current density
J
(in A m
−
2
):
J
=
σ
E
+
J
S
5 35
s functions that
mathematically relate voltage measurements at an array
of observation stations,
P
, located at the variously defined
measurement locations to each of the individual sources
of current density defined by a set of source points,
M
,
located in the conducting volume. In this system, the ker-
nel computation accounts for the electrical resistivity dis-
tribution and the boundary conditions that are present
within the test block. In the following computations, a
uniform resistivity distribution is used within the forward
model domain volume, excluding the holes that were
drilled into the block. Indeed, these holes represent infi-
nite impedance zones within the volumetric resistivity
distribution of the modeled block and are explicitly
accounted for in the computation of the kernel. In fact,
accounting for the presence of the holes within the model
volume is crucial to properly compute the kernel and
localize the source current density. The kernel matrix
used in Equation (5.39) takes the form of Green
'
where
denotes the low-frequency electrical conductivity
of the porous material (in S m
−
1
),
E
=
σ
the electrical
field in the quasistatic limit of the Maxwell equations (in
Vm
−
1
), and
−∇
φ
the electrical potential (in V). The source
current density is given by
J
S
=
Q
V
w
where
w
denotes
the Darcy velocity and
Q
V
the excess of charge (of the
diffuse layer) per unit pore volume of the porous or
fractured material (in C m
−
3
) that can be dragged by
the flow of the pore water. As seen in Chapter 1, the Pois-
son equation for the electrical disturbance
φ
ψ
(expressed
in V) is written as
∇
σ
∇
ψ
=
5 36
where denotes the volumetric current density (in
Am
−
3
). This volumetric current density is given by
J
S
=
Q
0
V
Q
0
≡∇
∇
w
+
∇
V
w
5 37
s func-
tions of the resistivity in the block and has units of resis-
tivity, conforming to Ohm
'
The electrical potential distribution at an observation
point
P
is given by
'
s law. Figure 5.21 shows the
