Geoscience Reference
In-Depth Information
To take into account the distribution of cracks over their sizes, we replace N by
the function N .l/ which denotes the number of cracks arising per unit time with a
length greater than l occurring in a specified area. Instead of Eq. (
10.9
), we come to
the following:
Z
l
max
d N .l/
dl
h
ı
B
t
iD
b
.
r
;l/dl;
(10.10)
0
where
Z
b
.
r
;l/
D
h
ı
B
1
.
r
;t;
n
;l/
i
dt;
(10.11)
0
and l
max
is the maximal crack size. The angular brackets in Eq. (
10.11
) denote the
statistical averaging over the angles which determine the random orientation of the
vectors
n
. Below we show that the small cracks make a little contribution to the inte-
gral in Eq. (
10.10
) due to strong damping of the acoustic waves radiated by the small
cracks. In this picture a minimal crack size in Eq. (
10.10
) is unimportant. Because of
this statement the considered theory differs from that by Molchanov and Hayakawa
(
1994
,
1995
) where the main emphasis was on microcracks.
As before, the ground is supposed to be a uniform conductor immersed in the
constant geomagnetic field
B
0
. Consider first the electromagnetic perturbations, ı
B
1
and
E
1
, caused by acoustic emission of a single crack. The magnetic perturbations
.ıB
1
B
0
/ satisfy the quasi-stationary Maxwell equation (
7.12
). For convenience,
we introduce the vectorial and scalar potentials by Eqs. (
5.73
) and (
5.74
); that is,
ı
B
1
Dr
A
1
and
E
1
Dr
dž
1
@
t
A
1
. These potentials satisfy the standard gauge
for a conductive medium (e.g., see Molchanov et al.
2002
)
r
A
1
C
0
dž
1
D
const:
(10.12)
Substituting the above representations of the electromagnetic field through the
potentials into Eq. (
7.12
), taking into account Eq. (
10.12
) and rearranging, we obtain
2
A
1
C
V
B
0
;
@
t
A
1
D
m
r
(10.13)
where
m
D
.
0
/
1
is the coefficient of magnetic diffusion. The mass medium
velocity,
V
.
r
;t/, can be expressed through the vector of medium displacement,
u
.
r
;t/,via
V
D
@
t
u
.
In order to obtain the time-integrated magnetic perturbation
b
.
r
;l/in Eq. (
10.11
)
one should integrate Eq. (
10.13
) with respect to time from 0 to infinity under the
condition that
A
1
.0/
D
A
1
.
1
/
D
0 and then take the mean value over the crack
orientation. Thus, we get
2
a
Ch
u
s
i
B
0
D
0;
m
r
(10.14)
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