Geoscience Reference
In-Depth Information
It should be noted that the effective magnetic moment is antiparallel to Earth's
magnetic field irrespective of crack plane orientation. Notice the same property
follows from Eq. (
7.56
).
Considering a random crack population, one should average the magnetic
moment over the random orientations of the vector normal to the crack plane. As the
normal may take any directions in space with equal probability then we obtain the
following average values:
ǝ
cos
2
0
Ǜ
D
1=3 and
ǝ
sin
2
0
Ǜ
D
2=3. Whence it follows
that the average magnetic moment reads
g
1
C
T
a
r
2
dr:
Z
r
l
2
w
2
g
2
3
h
M
iD
B
0
A
w
4
3
(7.73)
0
As is seen from this equation, the function g
does enter this equation. This means
that the contribution of transverse velocity component V
to the average magnetic
moment is equal to zero.
Substituting Eq. (
7.71
)forg
1
, g
2
and A into Eq. (
7.73
) and performing integra-
tion by parts, gives
1
Z
r
l
P
u
l
z
r
dT
a
dr
C
3T
a
rdr:
4
w
2
3
B
0
S
3
h
M
i
D
(7.74)
0
The average magnetic moment thus depends only on the radial motion due to
longitudinal waves radiated by the cracks.
Now we assume that the discontinuity of the vertical displacement at the crack
surface, Œ
u
z
.t/, is the increasing or slightly oscillating function that tends to a
constant value
u
0
as t
!1
. The rise time of this function and typical time of
its oscillations are supposed to be much smaller than the arrival time t
D
r=C
l
of the longitudinal wave. This implies that the function
P
u
l
z
D
Œ
P
u
z
.t
r=C
l
/
under the integral sign in Eq. (
7.74
) is close to zero everywhere except for a short
interval in the vicinity of point r
D
r
l
. Consequently, one may take the factor
.rdT
a
=dr
C
3T
a
/r at the point r
D
r
l
and then move it through the integral sign.
As a result, we come to the following estimate for the average value of the effective
magnetic moment
1
r
l
dT
a
.r
l
/
dr
C
3T
a
r
l
:
4
w
2
3
B
0
SC
l
u
0
3
h
M
i
(7.75)
Since the average magnetic moment is proportional to the volume S
u
0
arising
due to the crack opening, the net electromagnetic effect produced by the tension
crack population can vary in direct proportion to the total volume of all the cracks
and pores generated due to the rock fracture and dilatancy effect (Surkov
1999
).
The GMPs caused by tension cracks can be roughly estimated by substituting
h
M
i
into Eq. (
7.5
). Additionally one should take into account the crack distribution over
sizes. We study this problem in Chap.
10
in more detail.
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