Geoscience Reference
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and obtained
0.05, which was then used
by Hammond and Humphreys (2000) to predict
values of
V
P
,
V
S
,and
R
SP
. From Figure 3.9b,
α
∼
see, therefore, that similar results for
have been
obtained in multiple studies with multiple mate-
rials, and
λ
therefore provides a good benchmark
for checking the validity of theoretical models.
The parameter
λ
ln
V
S
/φ
∼−
4.5 for
α
=
0
.
05, whereas
ln
V
S
/φ
∼
−
0.1. As shown in Figure 3.8, the rapid
reduction is consistent with the data for an ice
+
2.2 for
α
∼
η
0
, in Equation (3.23), used to be
simply interpreted as the viscosity of a melt-free
sample. However, Faul and Jackson (2007) pointed
out that
0
◦
(Spetzler & Anderson,
1968; Stocker & Gordon, 1975), but not with the
data for samples with moderate dihedral angles.
Faul
et al
. (1994) measured
NaCl brine with
θ
∼
η
0
, determined by fitting the data of
partially molten samples to Equation (3.23), rep-
resents the viscosity of a ''nominally melt-free''
sample derived from San Carlos olivine, and it
is 1-2 orders smaller in value than the viscosity
of a ''genuinely melt-free'' sol-gel-derived olivine
sample, which is written as
, and used the oblate
spheroid model to connect microstructure and
the mechanical properties. However, a direct
comparison between such an oversimplified
geometry and the actual 3D melt geometry is
difficult, and can be a source of significant errors.
Contiguity is considered to be more suitable
for quantifying microstructures and predicting
mechanical properties (e.g., Yoshino
et al
., 2005).
α
0). Similarly,
for the partially molten rock analogue, the value
of
η
(
φ
=
η
0
determined from a sample with a small
amount of melt (
0.0025) was smaller by a
factor of 40 compared with the value of
φ
=
η
φ
=
0),
measured for a high-purity borneol sample at the
same temperature and with the same grain size
(McCarthy & Takei, 2011). These results suggest
the presence of a singularity such that a small
amount of melt significantly decreases viscosity.
Figure 3.10 shows the viscosity plotted against
the fraction of melt, as predicted by multiple
models. In this plot, viscosity is normalized to
the shear viscosity of the melt-free system, which
corresponds to
(
3.5.2 Viscous rheology
Experimental data for the shear and bulk vis-
cosities of partially molten rocks with nearly
equilibrium textures can be closely fitted by the
empirical formula
ξ
=
η
0
e
−
λφ
,
η
,
(3.23)
0) in the above discussion.
Although these models assume similar melt
geometries, the results depend significantly
on the particular model used. The thick line
shows the result of the contiguity model, in
which the diffusion of matter through the grain
boundaries is treated explicitly (Equation 3.16),
and
η
(
φ
=
where
η
0
are used as the fitting parameters.
The parameter
λ
and
plays an important role as it
records the amplitude of the weakening caused
by porosity. As shown below,
λ
η
0
is also impor-
tant in assessing the effects of a small amount
of melt. I summarize the experimental results for
λ
η
0
. From the shear deformation of olivine-
basalt aggregates in the diffusion and dislocation
creep regimes, Mei
et al
. (2002) reported
and
=
η
cc
(Coble-creep viscosity). Thin
lines show the results of the mixture models
in which a solid-liquid two-phase system is
modeled as a mixture of two Newtonian fluids
(Sumita
et al
., 1996; Takei, 1998a; Simpson
et al
.,
2010a,b). The ''mixture model 1'' (thin solid line)
shows the model developed by Simpson
et al
.
(2010a,b) based on the homogenization method.
''Mixture model 2'' (thin dotted line) is obtained
from the elastic contiguity model by substituting
ν
S
=
η
(
φ
=
0)
to be
25 and 37, respectively, and Hirth and Kohlstedt
(2003) reported
λ
to be 30 and 45, respectively.
For the bulk viscosity
λ
ξ
in the diffusion creep
regime, Renner
et al
.
(2003) obtained
λ
=
33
.
7
by using olivine
+
Li-silicate samples, where the
30
◦
, and the low viscosity
of the melt makes it suitable for measuring
dihedral angle is
θ
=
.
Similarly, for the shear viscosity in the diffu-
sion creep regime,
ξ
30 was obtained for the
partially molten rock analogue (borneol
λ
∼
5 (incompressible) into Equation (3.18)
(Takei, 1998a,b). Similar results are obtained
from the two mixture models. In summary,
0
.
+
melt)
35
◦
(McCarthy&Takei, 2011). One can
where
θ
=
η
is
