Bull. Earthq. Res. Inst.
Univ. Tokyo
Vol. 12 ,**-
pp. /3ῌ0/
Fracture surface energy of natural earthquakes from
the viewpoint of seismic observations
Satoshi Ide*
Department of Earth and Planetary Science, University of Tokyo
Abstract
Fracture surface energy, Gc, is one of the fundamental parameters that governs earthquake
rupture propagation and seismic energy radiation. Unlike other parameters of friction, it is
relatively easily estimated using seismic observations. In this paper we consider two aspects of Gc,
+) the implications of the minimum Gc for the scaling of seismic energy, and ,) spatial distribution
of Gc calculated from kinematic source models. Analyses of Gc based on seismic observations are
important when comparing in-situ and laboratory observations of fault materials, which will be
obtained in the Nankai-drilling project.
Key words : fracture surface energy, Gc, earthquake scaling, energy budget
+.
Introduction
the slip component whose direction is perpendicular
The earthquake source process includes rupture
to the main slip vector should be included in the
propagation, which consumes energy to create new
fracture energy.
Inelastic attenuation of seismic
fracture surfaces. Although an earthquake source is
waves near the fault zone is also indistinguishable
also regarded as slippage of preexisting weak planes,
from other forms of energy consumption.
the strength of such planes may be recovered by a
tively, the width of the zone where such a physical
healing process during an interseismic period in the
processes take place increases as event size increases.
geological scale, and a considerable amount of en-
This is also consistent with the fact that the width of
ergy is required for them to slip again. This energy
microcracking zone scales with fault length from
is called fracture surface energy and is written as Gc.
field observations (Vermilye and Scholz, +332).
If we consider Gc as a material property such as free
Therefore, Gc scales not only with fault area but also
surface energy or fracture energy of tensile fracture,
with another dimension such as fault zone width.
this must be a constant and total fracture surface
Thus, Gc is considered to be a -D property, although
energy of an earthquake scales with the fault area as
we call it fracture surface energy. It is not obvious
,D quantity. Potential energy release and seismic
whether or not this is negligible compared to seismic
energy by earthquake scale with seismic moment
energy for natural earthquakes.
Intui-
and are -D quantities. This leads to the conclusion
Gc scales with event size in laboratory experi-
that for a large earthquake, the fracture surface en-
ments (Ohnaka, ,**-) and field observations (Scholz,
ergy of ,D is negligible compared to -D potential
,**,). There have been attempts to measure the Gc of
energy release and seismic energy (e. g., Kostrov,
the overall rupture process of large earthquakes (e. g.,
+31.).
Husseini, +31/ ; Papageorgiou and Aki, +32-), and
However, Gc cannot be considered to be a mate-
these values, +*.2 J/m,, are usually much higher
rial property because we cannot distinguish the en-
than those obtained in a laboratory experiment. Gc
ergy consumption required to create a main fault
can be estimated locally based on rupture models of
plane among the various kinds of energy consump-
large events (Beroza and Spudich, +322 ; Guatteri et
tion, such as microcracking, plastic deformation, and
al., ,**+), and its maximum is about +*0 J/m, for M1
abrasion within the fault zone. Frictional work by
class earthquakes.
* e-mail : [email protected] (1ῌ-ῌ+, Hongo, Bunkyo, Tokyo, ++-ῌ**--, Japan)
59 S. Ide
In this paper we discuss two aspects of the sur-
,.
Minimum event and Gc
face fracture energy of natural earthquakes. First,
As illustrated in Fig. +, the potential energy re-
based on recent studies of seismic energy scaling (Ide
leased by an earthquake is divided into two parts,
and Beroza, ,**+ ; Ide et al., ,**-), we consider what
seismically radiated energy (seismic energy) Es, and
we expect in the presence of the minimum value of
unradiated energy consumption (fracture energy) Ef,
Gc. Second, we estimate the spatial distribution of Gc
which is the Gc integrated across fault area ῏,
for an earthquake, the +33/ Kobe earthquake of Mw
Ef῎῍ GcdSῌ
῎ῌ
0.2, based on a study of the energy balance using a
kinematic model (Ide, ,**,). We also briefly discuss
ῌ
Many studies have estimated seismic energy, Es,
the relation with possible outcomes from the Nankai
for earthquakes of various sizes, showing the size
drilling project.
dependences of energy and moment ratio, Es/Mo, or
apparent stress, sa῎mEs/Mo, where m is the rigidity of
the source region (e. g., Abercrombie, +33/ ; Kanamori
and Brodsky, ,**+).
However, it is still an open
question whether or not apparent stress depends on
earthquake size, and all previous estimates have presented little evidence of a breakdown of constant
apparent stress scaling (Ide and Beroza, ,**+). Fig. ,
is a summary of the apparent stress estimation. In
the study of small events by Gibowicz et al. (+33+)
and Jost (+332), the energies seem to be substantially
underestimated due to the band limitation, and we
show the values after adding the energy that is
probably missing as described in Ide and Beroza
Fig. +. Stress and stress relation during faulting
process and energy balance.
Shaded areas represent the amount of seismic energy (Es) and
fracture energy (EF). EF is given by the integral of
Gc over the fault plane as defined by equation (+).
(,**+). For events between Mw *./ and -, we include
recent studies using empirical Green’s function
method for Long Valley, California, (Ide et al., ,**-)
Fig. ,. Summary of studies of seismic energy. For small events of Gibowicz et al. (+33+) and Jost et al. (+332),
gray symbols are the values in the original papers and black symbols are those after adding potentially missing
energy described by Ide and Beroza (,**+).
῍ 60 ῍
Fracture surface energy of natural earthquakes from the viewpoint of seismic observations
and western Nagano, Japan, (Matsuzawa et al., ,**,).
ῐ
For the studies of intermediate and large earth-
Gc
ῒs
ῐ
,m
*.//Mo+ῌ-ῒs,ῌ-ῌ
῏
quakes by Mayeda and Walter (+330) and Kanamori
where Gc is the average Gc over the crack. This
et al. (+33-), we limit the minimum event size to Mw .
relationship of energy/moment ratio and seismic
to avoid an event selection bias as suggested by Ide
moment defines the minimum seismic moment for
and Beroza (,**+).
given Gc, and ῒs.
All estimates of energy/moment ratio in Fig. ,
Although Gc may scale with event size, we now
are located between +*ῐ0 and +*ῐ-. Although it is not
consider only the minimum of Gc, Gcmin for simplicity.
apparent in Fig. ,, there should be a breakdown of
Fig. - shows the relations between Es/Mo and Mo for
such constant energy/moment ratio scaling because
some combinations of Gcmin and ῒs in equation (/).
fault motion consumes fracture surface energy, Gc, to
For example, Gcminῑ+** J/m, and ῒsῑ*.+ MPa pre-
create slipping surfaces. The energy can be quite
dict that there are no earthquakes smaller than Mw *.
small and may scale with event size as described
From observations of microearthquakes, this is a
later, but cannot be zero. For example, Gc is about *./
reasonable lower limit, but there are certainly many
J/m in a frictional slip experiment between two
smaller events, for example, in mines.
granite rocks of -* cm (Ohnaka and Yamashita, +323).
implies that Gcmin is much smaller or ῒs is much
Because earthquakes occur on an interface which has
higher for such events. From observation of recur-
healed for a long time and is stronger than in such
rence intervals of repeated earthquakes, Nadeau and
,
This fact
frictional experiments, it is di$cult to imagine that
Johnson (+332) calculated the stress drop for M,
the Gc of a natural fault is smaller than this value.
earthquakes to be ,1* MPa.
For a mode I fracture, Gc is estimated to be +ῐ+** J/
mated a high stress drop for small events based on
m, (Atkinson, +32.) and for a triaxial mode III frac-
his fractal asperity model. Such a large stress drop
ture, Gc is on the order of +* J/m (Wong, +32,). The
can explain the existence of small events, without
minimum is comparable to or less than these values,
assuming quite small fracture surface energy.
.
,
and the existence of the minimum suggests a breakdown of scaling.
Seno (,**-) also esti-
For real earthquakes, it is di$cult to determine
the static stress drop. It should be noted that stress
To see the e#ects of the minimum fracture sur-
drops determined by corner frequencies such as the
face energy, consider a simple circular crack with a
Brune stress drop (Brune, +31*, +31+) are just alterna-
constant stress drop ῒs and a radius of r*.
tive measures of seismic energy, and the static stress
The
average slip of this crack is
uῑ
+0 ῒs
r*ῌ
1p m
drop should be measured from other information
such as aftershock distribution or stopping phases
ῌ
(Imanishi and Takeo, ,**,). Once the information on
where m is the rigidity. It is also represented
using the seismic moment Moῑmpr*,u as,
mum event size within the event detection limit, we
can estimate the value Gcmin for the region. For this
,/0῎+ῌ- +
ῒs,ῌ-Mo+ῌuῑῌ
῍ .3 ῏ pm
ῐ
*.//
ῒs,ῌ-Mo+ῌ-῍
m
stress drop is available and we can define the mini-
purpose, high-sensitivity and high-sampling seismometers in deep boreholes with low noise play an
῍
important role. In particular, earthquake observa-
The summation of Ef and Es is given from Fig. +
as,
tions at boreholes of great depths in the Nankai
subduction zone will provide us with an opportunity
Ef῏Esῑ
ῒs
Mo
,m
to estimate Gcmin along the subducting plate interface.
῎
which corresponds to the potential energy re-
-.
lease minus frictional work for all fault area. If we
take energy/moment ratio,
Spatial distribution of Gc on fault planes
Energy balance on a fault plane during an earth-
quake is expressed using local slip and stress relations (Kostrov, +31.). Using these relations and the
finite di#erence method, Ide (,**,) determined the
spatial distribution of seismic energy for kinematic
῎ 61 ῎
S. Ide
Fig. -. The relation between the seismic energy/moment ratio and seismic moment predicted from the stress
drop and the minimum fracture surface energy Gcmin, superimposed on Fig. ,. Dashed, solid, and gray lines
represent the stress drop of +*, +, and *.+ MPa, respectively. Assumed Gcmin is shown for each line.
source models of the +33/ Kobe, Japan, earthquake.
Gcaῑ
It is also possible to calculate fracture surface energy,
t,ῒx
῍
,m
as well as seismic energy. However, the estimation of
If the stress drop is +* MPa, which is the maxi-
fracture energy is not as straightforward as that of
mum in this calculation, Gca is about /ῐ+*/ J/m,.
seismic energy, because we need to define dynamic
Because there is a broad area of Gc that is larger than
stress level.
+*0 J/m,, it cannot be explained merely by an artifact,
Fig. . shows distributions of seismic energy den-
and such a large fracture surface energy is essential
sity and fracture surface energy, and slip-stress rela-
to explain the rupture propagation behavior of the
tions at di#erent points on the fault plane for the
Kobe earthquake.
+33/ Kobe earthquake. It is not clear what the dy-
Fig. . shows areas of large Gc near the hypocen-
namic stress level is in each relation. In the present
ter (A) and shallow region around C. Theoretically,
calculation, we assume dynamic stress level to be the
the hypocenter is a point where the surface fracture
minimum stress value in each slip-stress relation,
energy consumed after the hypocentral time is zero.
which corresponds to the light gray area in the slip-
However, if we use a grid system, Gc is averaged with
stress relation in Fig. ..
the neighboring grids and we cannot obtain zero
Such a calculation includes the contribution
fracture energy, and there are artifacts as described,
from artificial slip-weakening distance due to in-
too. Therefore, the high Gc around the hypocenter is
su$cient resolution (Ide and Takeo, +331).
In the
the combined result of the artifact and the averaging
finite di#erence calculation of inplane shear crack,
e#ect. Rather, a broad area with a large Gc to the
stress, t, just after initiation of slip, u, is approxi-
northeast of the hypocenter is important. This area
mated by,
is necessary to decelerate northeastward rupture
tῑ῏m
u
ῌ
ῒx
where ῒx represents grid interval.
propagation at the initial stage of this earthquake.
ῌ
The area of high Gc around C corresponds to the
Artificial
Akashi strait, where the rupture process is separated
fracture surface energy, Gca, is the product of arti-
into northeast and southwest. The surface traces of
ficial slip-weakening distance and stress drop di-
active faults are also discontinuous.
vided by , :
graphical irregularity may be responsible for the
῎ 62 ῎
Such a geo-
Fracture surface energy of natural earthquakes from the viewpoint of seismic observations
Fig. .. Distribution of seismic energy density and fracture surface energy. Relations between slip and stress for
three selected points are shown at the bottom. In each slip-stress relation, the dark gray area corresponds to
seismic energy and the light gray area corresponds to fracture surface energy estimated in this paper.
high fracture surface energy.
the Nankai drilling project, we plan to drill through a
The distribution of Gc in Fig. . is quantitatively
megathrust fault to / km or deeper depth. In-situ
consistent with the estimation using rate and state
observations of density, porosity, and microcrack
friction law (Guatteri et al., ,**+). The large Gc is also
density of the fault zone will be useful to distinguish
suggested by a dynamic simulation of the +33, Lan-
which process is dominant in subduction earth-
ders earthquake of Mw 1.- (Olsen et al., +331 ; Peyat et
quakes and may enable the estimation of Gc based on
al., ,**,). They used the slip weakening friction law
these observable quantities.
with a slip-weakening distance of 2* cm (Olsen et al.,
Although recent improvements in observation
+331). Since the stress drop is on the order of +* MPa,
systems have been remarkable, it is still di$cult to
Gc is on the order of +*1 J/m,. The values of seismic
obtain su$cient data to determine details of slip
energy density in these studies are almost of the
history for future Tonankai or Nankai earthquakes.
same order as the fracture energy, so we cannot
In such cases, the parameter estimated is fracture
neglect Gc in the discussion of earthquake dynamics.
surface energy rather than other dynamic parameters such as breakdown stress and characteristic dis-
Discussion and Conclusion
placement of slip-weakening friction law (Guatteri
Fracture energy represents energy consumption
and Spudich, ,*** ; Ide ,**,). Using the method dis-
from real physical processes, such as production of
cussed in this paper, it is possible to measure the
microcracks, plastic deformation and abrasion. In
spatial distribution of large future events. Moreover,
..
ῌ 63 ῌ
S. Ide
if the size of the minimum event is well-defined, we
can estimate the overall Gc of this area. An independent estimate of Gc from in-situ observation is useful
for calibration, although we should be careful about
the di#erence of scale.
Information about frictional properties, a and b
of rate-and state-friction law, or breakdown stress
drop and critical displacement of slip-weakening friction law, is critical to determine rupture propagation
behavior. Experiments using rock samples from the
actual fault zone are informative, but these are properties of almost one point on a large fault surface,
and overall slip behavior may be irrelevant from
these properties. On the other hand, what we can
estimate from seismic waves are spatially averaged
properties.
The seismic observations and in-situ
measurement in the Nankai trough will provide a
good opportunity to obtain the values of both small
and large scales, and consider scaling between them.
/.
Acknowledgements
I thank Dr. Shingo Yoshida for suggesting errors
in the manuscript. Figures are prepared using Generic Mapping Tool.
This work is supported by
Grant-in-Aid for Scientific Research of the Ministry
of Education, Sports, Science and Technology and
Special Project for Earthquake Disaster Mitigation in
urban areas.
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῍Received February -, ,**-῎
῍Accepted April ,/, ,**-῎
ῌ 65 ῌ