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spheroid model (Berryman, 1980) with an aspect
ratio
ν
S
=
0.25
(b)
of 0.1. A general proof for the equivalence
of these three geometries in the predictions of
V
P
and
V
S
has been given by Takei (2002), and
α
=
α
10
2
10
1
water melt
0.1 is called the ''equivalent aspect ratio'' of
the other two geometries.
The
V
P
data can be discussed by considering the
ratio of the variations in P and S-wave velocities.
Let
R
SP
be the ratio of the fractional changes in
V
S
and
V
P
caused by the liquid phase:
10
0
gas (0 km)
10
−
1
thin cracks and dikes
filled with melt
(a)
k
S
/
k
f
=
1
2
texturally equilibrated
partially molten region
30 km
30 km
V
S
d
ln
V
S
d
ln
V
P
=
V
S
/
2
water
50 km
R
SP
=
(3.22)
V
P
V
P
/
10 km
0 km
3
5
10
25
melt
0 km
V
P
0
are nearly
proportional to the liquid volume fraction
V
S
0
Because both
V
S
/
and
V
P
/
1
5 km
φ
,
R
SP
50
100
is almost independent of
φ
. Figure 3.9a shows
R
SP
0 km
400
10
5
versus pore geometry for
ν
S
=
0.25 and for various
gas
texturally equilibrated
rock
+
aqueous fluid system
values of
k
S
/
k
f
, where pore geometry is gener-
ally represented by the equivalent aspect ratio
.
Figure 3.9a demonstrates a strong dependence of
R
SP
on
α
0
0.001
0.01
0.1
1
α
and
k
S
/
k
f
. For the partially molten peri-
Aspect ratio,
α
V
P
0
dotite,
V
P
/
=
7
.
90%
−
9.03% at
φ
=
0.049
Fig. 3.9
(a)
R
SP
, showing the ratio of the fractional
changes in
V
S
and
V
P
caused by the liquid phase, versus
the pore aspect ratio
and P
1 GPa (Murase & Fukuyama, 1980), and
hence
R
SP
=
=
1.3-1.5. The measured
R
SP
agrees
well with theoretically predicted value of
R
SP
for a
texturally equilibrated partially molten peridotite
characterized by
k
f
represents
the ratio of solid and fluid bulk moduli. A rock
α
. The parameter
k
S
/
+
melt
system at 50-0 km depth corresponds to
k
S
/
k
f
∼
5-10,
0.25
(Figure 3.9a). For the texturally equilibrated bor-
neol + melt system, characterized by
α
=
0.1,
k
S
/
k
f
∼
5, and
ν
S
=
arock
aqueous fluid system at 30-0 km depth
corresponds to
k
S
/
+
k
f
∼
10-40, and a rock
+
gas system
α
=
0.1,
50-10
5
.
R
SP
is almost independent of the liquid volume fraction
at 30-0 km depth corresponds to
k
S
/
k
f
∼
k
S
/
0.37,
R
SP
is predicted to
be approximately 4, which agrees well with the
R
SP
measured for this system (Takei, 2000, 2002).
Therefore, the
V
S
and
V
P
data for partially
molten peridotite at 1GPa, and for the partially
molten rock analogue at
k
f
∼
1
.
16, and
ν
S
=
φ
.
−
ln
V
S
/φ
α
(b)
. After Takei (2002). Reproduced
with permission of the American Geophysical Union.
versus
(
η
melt
∼
1-10Pa s),
f
sq
∼
1-10 MHz for
α
=
0.1
22
◦
−
17
◦
, are con-
θ
=
α
=
and 100 kHz-1MHz for
0.05 (Schmeling,
sistent with
0.1, which is equivalent to the
contiguity model with A
α
=
1985). Therefore, if
0.1, the assumption of
a relaxed state of melt squirt flow is not valid.
Although a possible deviation from the relaxed
state is difficult to estimate, the deviation is not
considered to be significant, because the effect
of decreasing
α<
3, and hence equiv-
alent to the ''standard model'' of the equilibrium
geometry for
=
2
.
20
◦
(Figure 3.1d). As discussed
in Section 3.6.1, the theoretical results shown
in Figure 3.8 assume a relaxed state of melt
squirt flow. This assumption is valid for borneol
+
θ
=
η
melt
by increasing
T
(T
=
1523 K
for
φ
=
0.01, and T
=
1573 K for
φ
=
0.16) is not
melt, for which the melt viscosity is low
10
−
3
Pa s), and the characteristic
frequency of the squirt flow high (
f
sq
>
(
η
melt
∼
7
×
evident in Figure 3.8.
Faul
et al
.
10MHz;
Takei, 2000). For partially molten peridotite
(1994) measured the aspect ratio
α
directly from microstructural observations,
