Geoscience Reference
In-Depth Information
Fig. 10.4
Schematic plot of a disk-shaped crack expanding at velocity V
c
. The extrinsic current
flowing through the surface S is shown with
red arrow
Consider first an individual disk-shaped microcrack increasing in size at constant
velocity V
c
. A schematic plot of the microcrack with radius R and half-width
R is shown in Fig.
10.4
.Letdq be a small charge generated at the fresh crack
sides forming for a short time dt due to the microcrack opening. As a result the
extrinsic current i
e
flows through the surface S asshowninFig.
10.4
with red arrow.
Taking into account that the fresh microcrack surface equals 2RV
c
dt, the charge
increment for the time dt is equal to dq
D
2RV
c
†
c
dt, where †
c
denotes the
surface charge density. For simplicity, we have ignored the rock conductivity that
may lead to the charge relaxation due to conduction currents. Whence we find the
extrinsic current produced by a single crack
i
e
D
2RV
c
†
c
:
(10.1)
Now we estimate the macroscopic current density J
e
caused by the ensemble of
expanding microcracks
J
e
i
e
nR
D
2RRV
c
†
c
n;
(10.2)
where n is the number density of mobile cracks. Let n be the total number density
of all the cracks, that is, mobile and stationary ones. Here we ignore the distribution
in the microcrack sizes. The mobile crack number density can be thus estimated as
n
@
t
n where
R=V
c
. The rate of crack number density can be estimated
as follows: @
t
n
n
=t
, where n
R
3
denotes the extreme crack concentration
at which the multiple crack intersection happens that results in the complete rock
disruption. Here the parameter t
stands for the time scale of disruption process.
Now we leave out of account the field attenuation due to the skin effect in order
to estimate the upper bound of the magnetic effect caused by microfracturing in the
earthquake focal zone. According to
Biot-Savart-Laplace
law and Eqs. (
7.2
) and
(
7.3
), the magnetic field due to the current J
e
is on the order of
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