Energy Release and Heat Generation During the 1999 Ms7.6 Chi-Chi,
Taiwan, Earthquake
Jeen-Hwa Wang
Institute of Earth Sciences, Academia Sinica
P.O. Box 1-55, Nangang, Taipei, 115 Taiwan
Tel: 886-2-27839910 (ext. 326)
E-mail: [email protected]
(submitted: March 2, 2005; First Revision: September 5, 2005;
Second Revision: March 22, 2006; Third Revision: June 21, 2006;
Accepted: July 21, 2006)
Abstract.
On September 20, 1999, the Ms7.6 Chi-Chi earthquake ruptured the
Chelungpu fault in central Taiwan. Integrating observed and inversed results of source
parameters, the fracture energy, Eg, and frictional energy, Ef, on the fault and its
northern and southern segments are estimated. Together with given values of strain
energy, ∆E, and seismic radiation energy, Es, the seismic efficiency, i.e., η=Es/∆E, and
the radiation efficiency, i.e., ηR=Es/(Es+Eg), are evaluated. The average fracture
energy per unit area, G, is also calculated from Eg. The frictional heat caused by
dynamic frictional stress is calculated from Ef. Results show a marked difference in
source properties between the two segments. The average frictional and ambient stress
levels on the two segments are estimated. The total energy budget of and heat
generated by the earthquake are elucidated based on a 2-D faulting model with
frictional heat. Both observed and calculated results suggest the possible existence of
fluids, which produced suprahydrostatic gradients, on the fault during faulting.
Lubrication and thermal fluid pressurization might play a significant role on rupture.
List of used symbols (Included also are the unit and the section or sub-section, where
the symbol is used the first time.)
Energy
∆E: Strain energy (J, which is the abbreviation of Joule) (Section 1)
Es: Seismic radiation energy (J) (Section 1)
Eg: Fracture energy (J) (Section 1)
Es: Frictional energy (J) (Section 1)
1
η=Es/∆E: Seismic efficiency (Section 1)
ηR=Es/(Es+Eg): Radiation efficiency (Section 1)
Fracture Energy Density
G=Eg/A: Fracture energy density (J/m2) (Sub-section 2.4)
ζ: Geometry factor (Sub-section 3.4)
K: Stress intensity factor K (m1/2) (Sub-section 3.4)
Y: Young modulus: (MPa) (Sub-section 3.4)
Geometry and Kinematic Parameters of a Fault
θ: Dip angle of a fault (in degree) (Sub-section 3.6)
L: The faulting-striking direction (Sub-section 2.2)
W: The faulting-dipping direction (Sub-section 2.2)
W: Fault width (m) (Sub-section 3.2)
A: Fault area (m2) (Sub-section 2.4)
D: Displacement (or slip) (m) (Sub-section 2.1)
Dmax: Maximum Displacement (m) (Sub-section 3.3)
PGV: Peak ground velocity (m/s) (Sub-section 3.6)
vR: Rupture velocity (m/s) (Sub-section 2.4)
Symbols of segments: “N” for the north and “S” for the south (Section 2)
Mo: Seismic moment (Newton-m, Nm) (Sub-section 2.1)
v: Sliding velocity v (m/s) (Sub-section 3.6)
Friction
µf: Static frictional coefficient (Sub-section 3.6)
µfe: Effective static frictional coefficient (Sub-section 3.6)
µd: Dynamic frictional coefficient (Sub-section 3.6)
ξ=µd/µf: Ratio of dynamic to static frictional coefficients (Sub-section 3.6)
µv: Frictional coefficient at the sliding velocity v (Sub-section 3.6)
µo: Frictional coefficient at the reference velocity vo (Sub-section 3.6)
a: Coefficient for the direct effect (Sub-section 3.6)
b: Coefficient for the evolution effect (Sub-section 3.6)
Dc: Length scale parameter of the friction law (m) (Section 1)
vo: Reference velocity (m/s) (Sub-section 3.6)
ϕ: State variable (Sub-section 3.6)
µss: Steady-state frictional coefficient when v=vo (Sub-section 3.6)
2
Dc: Characteristic slip distance (m) (Section 1)
Heat
∆T: Temperature rise (oC) (Sub-section 2.5)
C: Heat capacity (J/kg-oC) (Sub-section 2.5)
Q: Heat strength (oC-m) (Sub-section 2.5)
h: A thickness within which heat is distributed (m) (Sub-section 2.5)
Lubrication
H: Average width of the gap between two fault walls (m) (Sub-section 3.2)
L: Horizontal length scale (m) (Sub-section 3.2)
Lc: Critical lubrication length (m) (Sub-section 3.2)
Marone and Kilgore‘s model for Dc
κ: Constant (≈10-2) (Sub-section 3.3)
tsb: Shear band thickness (cm) (Sub-section 3.3)
Physical properties of crustal materials
ρ: Density (kg/m3) (Sub-section 2.5)
λ: Lame constant (MPa) (Sub-section 3.2)
µ: Rigidity (MPa) (Sub-section 2.1)
α: P-wave velocity (m/s) (Sub-section 3.7)
β: S-wave velocity (m/s) (Sub-section 2.4)
Pressure
σL: The lithostatic pressure (MPa) (Sub-section 3.5)
H: Depth (m) (Sub-section 3.5)
g: Gravity acceleration (=9.8 m/s2) (Sub-section 3.5)
pw: Pore pressure (MPa) (Sub-section 3.6)
γ: Pore-fluid factor (Sub-section 3.6)
Stress
σo: Static frictional stress (strength) (MPa) (Section 1)
σd: Dynamic frictional stress (MPa) (Section 1)
σf: Final stress (MPa) (Section 1)
σn: Normal stress (MPa) (Sub-section 3.6)
σs: Shear stress (MPa) (Sub-section 3.6)
σ1: Maximum (horizontal) principal stress (MPa) (Sub-section 3.6)
3
σ3: Minimum (vertical) principal stress (MPa) (Sub-section 3.6)
σT: Additional tectonic stress (MPa) (Sub-section 3.6)
Stress Drop
∆σs= σo-σf: Static stress drop (MPa) (Section 1)
∆σd= σo-σd: Dynamic stress drop (MPa) (Section 1)
1. Introduction
After an earthquake ruptures, the frictional stress, σ(t), which is a function of
time and slip on a fault plane decreases from an initial σo to a dynamic σd, and finally
becomes σf (see Figure 1). In general σd is equal to or smaller than σf [Kanamori and
Heaton, 2000]. According to the slip- and rate-weakening frictional law, the frictional
stress changes from σo to σd in a characteristic slip displacement, Dc [cf. Marone,
1998; Wang, 2002]. From numerical simulations based on a crack model in the
presence of rate- and state-dependent friction, Bizzarri and Cocco [2003] stated that
fault friction decreases with slip in a slip-weakening distance, Dc, and they also
stresses Dc>Dc. Dc is 10-5–10-3 m from laboratory results [Okubo and Dieterich,
1984; Ohnaka and Yamashita, 1989; Marone, 1998], and Dc of from 0.1 to few meters
for real earthquakes [Guatteri and Spudich, 2000; Mikumo et al., 2003; Zhang, 2003;
Tinti et al., 2005]. The friction law to describe the frictional stress is complicated [cf.
Marone, 1998; Wang, 2002]. However, it can be approximated by a piece-wise linear
function as displayed in Figure 1. The static stress drop ∆σs=σo-σf and the dynamic
stress drop ∆σd=σo-σd are usually used to specify the change of stresses on a fault.
The strain energy, ∆E, per unit area, which results from tectonic loading, can be
approximated by the area of a trapezoid underneath the linearly decreasing function of
stress versus slip (Figure 1). ∆E is transferred into, at least, three parts (see Figure 1),
i.e., the seismic radiation energy (Es), fracture energy (Eg), and frictional energy (Ef),
that is ∆E=Es+Eg+Ef. Es is the energy radiated through seismic waves. Eg is the energy
used to extend the fault plane. Ef, which results from the dynamic friction stress, can
generate heat. Because of incomplete data there are high uncertainties in measuring
the energies, especially for Ef.
On September 20 1999, the Ms7.6 Chi-Chi earthquake ruptured the Chelungpu
fault, which is a ～100-km-long and east-dipping thrust fault, with a dip angle of 30o,
4
in central Taiwan [cf. Ma et al., 1999; Shin and Teng, 2001]. The epicenter, fault trace,
and the fault plane are displayed in Figure 2. Hwang et al. [2001a] measured Es from
near-fault seismograms. Wang [2004] evaluated ∆E from the slip distribution inferred
by Dominguez et al. [2003] and revised the value of Es measured by Hwang et al.
[2001a] by removing finite frequency bandwidth limitation. Wang [2004] also
obtained the seismic efficiency η=Es/∆E=0.14 for the event. From far-field surface
waves, Hwang et al. [2001b] obtained Es＝1.9×1017 J. Venkataraman and Kanamori
[2004] evaluated the radiation efficiency, i.e., ηR=Es/(Es+Eg), from teleseismic data.
Their value is 0.8, thus giving Eg=0.25Es. From local seismograms, Zhang et al. [2003]
and Ma and Mikumo [2004] inferred the spatial distribution of Dc. The averages of Dc
are: (1) ～1 m in the south and ～6 m in the north by the former; and (2) ～1 m in
the south and ～10 m in the north by the latter. The maximum value of Dc in the
north is about 10 m by the former and 12 m by the latter. However, they might
over-estimate Dc as mentioned below. Wang [2003; 2004; 2006] summarized the
observed and inferred results of different source parameters, and he stressed the
differences in source parameters between the northern and southern segments, which
are separated at a locality near Station TCU065 as shown in Figure 2. In addition,
from seismic reflection experiments, Wang et al [2004] also pointed out a difference
in sub-surface fault geometry between the two segments.
In 2000, two shallow boreholes near the Chelungpu fault (see Figure 2) were
drilled [cf. Tanaka et al., 2002]. The distances from the drilling site to the fault trace
are 500 m and 250 m, respectively, for the northern and southern boreholes. From
core samples, two fractures zones can be recognized. Huang et al. [2002] stated that
the two boreholes encountered the fault plane of the event, and assumed that the
possible fracture zone of the Chi-Chi earthquake is at 225–330 m and 177–180 m,
respectively, in the northern and southern boreholes. The main results reported by
several authors [Huang et al., 2002; Otsuki et al., 2001; Tanaka et al., 2002; and
Tanikawa et al., 2004] are: (1) For the northern borehole: (a) a random fabric breccia
distribution of several tens of centimeters thick; (b) the presence of fault gouge of
0.5–2 cm thick; (c) intrusion of soft clay into layers of fault breccia; (d) Dc=～1 m;
(e) permeability=10-16–10-19 m2; and (f) a water content up to 45 vol%; and (2) For
the southern one: (a) the existence of foliated fault breccia with ultracataclasites and
5
pseudotachylites; (b) a lack of any injection structure; and (c) permeability =10-14–
10-17 m2. In addition, the temperature rise in the two boreholes was measured about
1.4 years after the earthquake. The peak values of temperature rise on the fault plane
are 0.5 oC and 0.1 oC, respectively, in the southern and northern boreholes. The
temperature rise decreases with increasing distance from the fault plane as described
by a 1-D cooling equation, from which Ito [2004, personnel communication] and
Mori [2004] estimated the frictional coefficients. Results are: (1) 0.7–1.0, with an
average 0.85, at the 182-m depth in the south and 0.1–0.2, with an average 0.15, at
the 320-m depth in the north [Ito, 2004]; and (2) an average 0.45 for the two segments
[Mori, 2004]. Mori [2004] also obtained Ef=3.6×1016 J for the whole fault from model
computations. His value might be the lower bound of Ef, because he used a small
frictional coefficient for the whole fault plane.
In this study, the observed and inferred values of source parameters of the
earthquake are first reviewed. According to the strain and seismic radiation energies
evaluated by Wang [2004], the fracture and frictional energies are estimated. Based on
the four kinds of energies, an attempt is made to investigate the energy budget and
heat generation during the earthquake by integrating observed and inferred results of
other parameters. The seismic efficiency, radiation efficiency, and average ambient
(static) and dynamic frictional stress levels will be estimated from given data. Zhang
et al. [2003] and Ma and Mikumo [2004] might over-estimate Dc on the northern
segment. Since Dc controls dynamic friction and, thus, rupture, reliable evaluations of
Dc are necessary. I propose a way based on a physical model to examine the possible
range of Dc. The results from this study and others show differences in energies and
heat between the two segments. In order to explore the reasons to produce such
differences, I propose a 2-D faulting model with frictional heat in the presence of pore
pressures to study physical behavior on a fault plane, including interplay between
frictional strength and pore pressure. Results will be applied to elucidate the
differences in observed and inferred source parameters between the northern and
southern segments of the fault. In addition, lubrication and thermal fluid
pressurization will be taken into account to interpret the observations. Also included
will be a description of indirect evidences of the existence of fluids.
2. Measurements of Source Parameters
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The observed and inferred results of physical parameters of the fault and its two
segments are described below. The capital letters ‘N’ and ‘S’ are used, respectively, to
represent the northern and southern segments.
2.1 Average Displacements and Seismic Moment
According to the slip distribution inverted by Dominguez et al. [2003] from GPS
data and SPOT images, the average displacement, D, on the fault plane is evaluated
through the following expression: D=∑Di/n, where Di=the displacement at the i-th
rectangular dislocation and n=the number of rectangular dislocations in use. Results
are: DN=7.15 m and DS=4.88 m (listed in column 2 of Table 1), and, thus, D=5.89 m
for the fault. Seismic moment, Mo, is measured by Mo=∑Moi, where Moi=µDiAi is the
value at the i-th rectangular dislocation. Results are: MoN=8.14 × 1019 Nm and
MoS=7.12×1019 Nm, and, thus, Mo=MoN+MoS=1.53×1020 Nm for the entire fault.
2.2 Strain Energy
The strain energy ∆E can be expressed by ∆E=[(σo+σf)/2]DA [cf. Knopoff
[1958]. Owing to this, Wang [2004] measured ∆E of the earthquake from ∆E=
∆EL+∆EW, where ∆EL=[(σoL+σdL)/2]DLA and ∆EW=[(σoW+σdW)/2]DWA, from the slip
distribution inferred by Dominguez et al. [2003]. He used σd instead of σf by
assuming σd≈σf as made by others [e.g. Kanamori and Brodsky, 2004]. In the
formulae, “L” and “W” denote the fault-striking and the fault-dipping directions,
respectively. Details of the methodology used to measure ∆E are given in Wang
[2004]. The estimated strain energies for the two segments are shown in column 4 of
Table 1. Since the rotation components and the higher-order derivatives of
deformations were not included in the calculations, only a minimum estimate of strain
energy was made. The rotation components are usually considered to be very small
and ignored by seismologists. Except for the areas with abnormally large changes in
displacements, the variation of slip on the fault is smooth, and thus, the higher-order
derivatives of deformations would be small. In the practical inversion, the
displacement on a grid is set to be a constant. This makes the higher-order derivatives
of deformations be zero. Therefore, the difference between the estimated and real
strain energies caused by excluding the two components should be small.
2.3 Seismic Radiation Energy
Hwang et al. [2001] measured Es from near-fault seismograms. Wang [2004]
revised their values by eliminating finite frequency bandwidth limitation. The revised
7
results are shown in column 5 of Table 1. Hence, the values of η are: ηN=0.21 and
ηS=0.03 (listed in column 7 of Table 1), and, thus, η=0.14 for the whole fault. The
values of ηN and ηS lead to that about 80% of ∆EN and 97% of ∆ES were transferred
into the non-seismic radiation energies, mainly including Eg and Ef.
2.4 Fracture Energy
Eg can be evaluated from the following expression:
Eg=[(1-vR/β)/(1+vR/β)]1/2∆σdDA/2,
(1)
where vR and β are, respectively, the rupture and S-wave velocities [Kanamori and
Heaton, 2000]. This equation is valid only for a crack-like rupture model [Tintn et al.,
2005a], and Eg computed from Eq. (1) is an average global value, because ∆σd and D
are both average values on the fault plane. Obviously, Eg depends on vR/β, and is
much smaller than ∆E, because of vR/β=0.75–0.85 [Kanamori and Heaton, 2000].
Define G=Eg/A to be the fracture energy density (per unit area). From the original
definition, G must be a local parameter. But, now only the value of G on a rectangular
dislocation is measured and then the global averages are calculated. To calculate Eg
and G, the values of vR/β, ∆σd, and A are, respectively, taken from Ma et al. [2001],
Huang et al. [2001], and Wang [2004]. The values of related parameters are:
(vR/β)S=0.75, ∆σdS=6.52 MPa, DS=4.88 m, and AS=4.551×108 m2 for the northern
segment; and (vR/β)N=0.80, ∆σdN=29.7 MPa, DN=7.15 m, and AN=3.615×108 m2 for
the northern segment. The values of Eg, ηR, and G for the two segments are listed,
respectively, in columns 6, 8, and 9 of Table 1.
2.5 Frictional Energy and Heat
From Ef=∆E-(Es+Eg), the values of Ef for the two segments are shown in column
10 of Table 1., On a fault area of A, heat produced by σd in a displacement D during
faulting is Ef=σdDA, and Ef yields a temperature rise, ∆T. Assuming that heat is
distributed within a layer of thickness h around the ruptured plane, ∆T is
∆T=Ef/CρAh,
(2)
where C and ρ are, respectively, the specific heat and density [Kanamori and Heaton,
8
2000]. For crustal rocks, C=103 J/kg-oC and ρ=2.6×103 kg/m3. The heat strength is
Q=Ef/CρA=∆T·h.
(3)
The values of QS and QN are shown in column 11 of Table 1, and (∆T)S=(102/h)oC
and (∆T)N=(154/h)oC. From teleseismic data, Venkataraman and Kanamori [2004]
obtained Es=0.88×1016 J and ηR=0.8, thus leading to Eg=0.22×1016 J, for the overall
fault plane. This gives Es+Eg=1.10×1016 J, which is one fifth of Es+Eg=5.47×1016 J of
this study. Using the value of ∆E of this study and their value of Es+Eg, we have
Ef=3.09×1017 J, which is 1.12 times larger than 2.65×1017 J of this study. The related
values of Q are 144.3 oC-m and 123.9 oC-m, respectively, with a difference of 20.4
o
C-m. The difference is clearly small.
3. Discussion
3.1 Seismic Moment
The value of Mo reported by USGS for the earthquake is 2.4×1020 Nm, which is
about 1.5 times larger than Mo=1.53×1020 Nm of this study. The values of Mo
estimated by Hwang et al. [2001] from local seismic data are (0.35–1.5)×1020 Nm at
two northern stations and (5.3–5.8)×1018 Nm at two southern ones. Their values are
comparable with the present one for the northern segment, yet not for the southern
one. Although the fault area is larger on the southern segment than on the northern
one, MoS is close to MoN, because ∆σs is smaller on the former than on the latter
[Huang et al., 2001; Hwang et al., 2001a]. This indicates that Mo is not a significant
factor in producing the differences in source parameters between the two segments.
3.2 Seismic Efficiency
Wang [2004] evaluated ∆E by using the approximations of first spatial
derivatives of displacements. This leads to a fact that the average stress, i.e., (σo+σd)/2,
is proportional to strain. How does the approximation affect the evaluation of η? In
spite of a smaller strike-slip component, the Chelungpu fault is a very long dip-slip
fault. Hence, only the dip-slip component is considered. For an infinite length dip-slip
fault with a width W, ∆E=µ(DW/W)DWA. Since ∆σs is proportional to strain,
Es+Eg=∆σsDWA/2 is also proportional to strain. Venkataraman and Kanamori [2004]
9
defined the radiation efficiency ηR=Es/(Es+Eg). This gives Es=ηR(Es+Eg), and, thus,
Es=ηR∆σsDWA/2. For an infinite length dip-slip fault, ∆σs=µ[4(λ+µ)/π(λ+2µ)](DW/W)
[cf. Scholz, 1990], where and λ is the Lame constant and usually equal to µ. Hence,
the relation of η versus ηR is approximated by η=4ηR/3π=0.42ηR, because of λ=µ.
From ηRS=0.61 and ηRN=0.81, the modeled values of η are ηS=0.26 and ηN=0.34,
which are larger than the respective measured values (see Table 1). The difference is
larger on the southern segment than on the northern one.
Four reasons could cause the above-mentioned difference. First, η=0.42ηR is
obtained from an infinite length dip-slip fault. But, the real fault is a finite one.
Secondly, the ignorance of the strike-slip component can affect the results. Thirdly,
for a static model vR is regarded as infinity, while for a symmetrical circular crack
model [Sato and Hirasawa, 1973], Ide [2002] stated that η increases with vR/β. Hence,
the static model will produce larger η. Fourthly, the directivity effect [Aagaard et al.,
2004]] is also a significant factor. This effect shows that directivity decreases when
the angle between the propagation direction of ruptures and the slip direction. Higher
directivity leads to larger-amplitude velocity pulses. From the inverted source rupture
processes [e.g. Ma et al., 2001], Huang and Wang [2002] stated that the earthquake
ruptured mainly along the fault-dipping (W-) direction on the southern fault plane and
along both the fault-striking (L-) and W-directions on the northern one. On the two
segments the W-direction is perpendicular to the propagation direction of ruptures.
This reduced Es, thus decreasing η.
As mentioned above, ηN>ηS. This inequality can be explained by two reasons.
First, although the W-direction slip is perpendicular to the propagation direction of
ruptures on the two segments: the L-direction slip is in parallel with the propagation
direction of ruptures on the northern segment. Thus, the decrease in Es is larger on the
southern segment than on the northern one. The directivity effect leads to ηN>ηS.
Secondly, the existence of lubrication on the northern segment should play a
significant role. Brodsky and Kanamori [2001] proposed a lubrication model, from
which a fault zone is modeled as a thin viscous fluid sandwiched between two fault
walls with undulations having a horizontal length scale L. On average, L increases
with D. Let H be the average width of the gap between two fault walls. According to
the Navier-Stokes equation, the viscous stresses are balanced by a change of pressure
10
in the thin viscous fluid. They defined a critical lubrication length, Lc, which is the
length at which elastic deformation is comparable to the initial gap height. The
asperities on the fault walls were initially in full contact. For the areas with D>Lc, an
increased lubrication pressure widened the fault gap and, thus, remarkably reduced
the frictional strengths, thus decreasing the resistance to relative motions of two fault
planes. This results in a large stress drop, thus leading to a bigger kinetic energy.
Theoretically, Wang [1995, 1996] also obtained the same conclusion. Brodsky and
Kanamori [2001] assumed that lubrication occurred on the northern segment, yet not
on the southern one. From measured Lc, Ma et al. [2003] claimed that the lubrication
pressures in areas with D>Lc increased during the earthquake. This reduced the
asperity contact and high-frequency accelerations, especially on the northern fault
plane. Huang and Wang [2002] reported larger fallout of near-fault displacement
spectral amplitudes at high frequencies in the north than in the south. Their results are
consistent with the conclusion made by Ma et al. [2003]. This produces ηN>ηS
3.3 Radiation Efficiency and Characteristic Slip Displacement
For a Mode-III crack, ηR is a function of vR/β:
ηR=1-[(1-vR/β)/(1+vR/β)]1/2
(4)
[cf. Venkataraman and Kanamori, 2004]. Based on a symmetrical circular crack
model [Sato and Hirasawa, 1973], Ide [2002] stated that ηR increases with vR/β and
larger than 0.6 when vR/β>0.7. Using a different model, Dong and Papageorgiou
[2002] also stated that ηR increases with vR/β and larger than 0.6 when vR/β>0.4.
Figure 3 shows the variation of ηR with vR/β from Eq. (4) when ηR=0.5 to 1.0. Clearly,
ηR increases with vR/β. Ma et al. [2001] inferred (vR/β)S=0.75 and (vR/β)N=0.80. The
calculated values of ηR from Eq. (4) are ηRCS=0.62 at (vR/β)S=0.75 and ηRCN=0.67 at
(vR/β)N=0.80, and denoted by crosses in Figure 3. From the core sample, Ma [2005]
obtained ηRN=0.25–0.9, whose upper bound is larger than ηRN=0.81 of this study.
For the whole fault plane, Venkataraman and Kanamori [2004] obtained ηR=0.8
for vR/β=0.66 [Ji et al., 2003] from teleseismic data. This ηR value, which is displayed
by a symbol ‘V’ in Figure 3, is very similar to ηR=0.79 of this study. The calculated
value of ηRC at vR/β=0.66 from Eq. (4) is 0.65, which is smaller than 0.8 and is shown
11
by a symbol ‘J’ in Figure 3. For the southern segment, the measured and calculated
values are, respectively, ηRS=0.61 (displayed by a symbol ‘O’) and ηRCS=0.62. The
two values are similar. For the northern segment, the two values are, respectively,
ηRN=0.81 (demonstrated also by a symbol ‘O’) and ηRCN=0.66. The former is 0.14
larger than the latter. This inconsistency might be due to either an over-estimate of Es
or an under-estimate of ηR using Eq. (4). Since Wang [2004] eliminated possible
effects on the estimate of Es due to several factors, including the finite frequency
bandwidth limitation, site effect, path effect, and radiation pattern, so the estimated
values should be reliable. Hence, I assume that inconsistency is due to an
under-estimate of ηR using Eq. (4), which was obtained based on dry rocks. Thus, the
mechanisms due to fluids, for example lubrication [Brodsky and Kanamori, 2001] or
thermal fluid pressurization [Sibson, 1973; Mase and Smith, 1984/1985], must be
taken into account. Since the ratio vR/β in Eq. (4) cannot represent such mechanisms,
an advanced model is needed.
Since Dc is usually in the range of 0.1 m to few meters from earthquake data [Ide
and Takeo, 1997; Mikumo et al., 2003], the maximum values of DcN, i.e., 10 and 12 m,
and the average ones, i.e., 6 and 10 m, inferred, respectively, by Zhang et al. [2003]
and Ma and Mikumo [2004] seem to be unusually large. In principle, at any locality
on the fault plane, Dmax must be larger than Dc, and, thus, on the whole fault plane
average Dmax should be also larger than average Dc. The maximum value of Dc
inferred by Ma and Mikumo [2004] is larger than DN=7.15 m calculated in this study,
even though it is less than DNmax=15 m at H=8000 m inferred by Ma et al. [2001].
Scholz [1988] stressed that the fault would be stable and could not generate large
earthquakes if Dc is too long. Hence, DcN=6 and 10 m might be over-estimated.
Fukuyama et al. [2001] stated that there is high uncertainty in estimates of Dc from
the inferred stress-slip function. In addition, estimates of Dc could be affected by the
source time function [Piatanesi et al., 2004; Tinti et al., 2005b] and filtration [Spudich
and Guatteri, 2004]. Hence, it is necessary to explore the range of DcN.
I propose a way to explore the range of Dc. This way is described below. From
Figure 1, Eg can be approximately evaluated by
Eg≈∆σdDcA/2.
(5)
12
Of course, Eg estimated from Eq. (5) is slightly smaller than that done from Eq. (1).
The difference between the values of Eg evaluated from the two equations depends on
several factors, including the stress drops, frictional strength, frictional law etc, the
nonlinear decrease of stress with increasing slip etc. A comparison between Eq. (1)
and Eq. (5) proposes an approximate relation of Dc versus D. The approximation is:
Dc≈[(1-vR/β)/(1+vR/β)]1/2D
(6)
From Eq. (5), DcS=1 m gives EgS=0.15×1016 J and, thus, ηRS=0.69, which is only
slightly larger ηRCS=0.62 and represented by ‘S1’ in Figure 3 for the southern segment.
Eq. (6) gives a modeled value of DcS=1.4 m when DS=4.88 m. The two values of DcS
are close to each other. Hence, DcS=1 m is acceptable. The related values of EgS, GS,
and ηRS are shown in Table 1.
For the northern segment, there are several particular values of DcN. From the
analyses of core samples obtained in the northern borehole, Tanikawa et al. [2004]
assumed DcN=1 m by using numerical simulations based on the thermal pressurization
model. Of course, their value is for the shallow depth, where velocity-hardening
friction, usually with smaller DcN, controls faulting [Scholz, 1990]. From laboratory
measurements, Marone and Kilgore [1993] suggested that within fault gouge, Dc
scales with the thickness of active shear: Dc=κtsb, where κ is a constant (≈10-2) and tsb
is the shear band thickness. From the synoptic model of a shear zone, tsb increases
with depth in most upper crust and decreases with increasing depth in the lowest
upper crust [Scholz, 1990]. Anyway, tsb is larger at depths than at the topmost 2 km. If
the laboratory results can be applied to real shear zones, Dc increases with depth.
Hence, average DcN at depths would be longer than 1 m. Hence, DcN=1 m at shallow
depths cannot work at depths.
In order to explore the problem, ηRN is calculated for several values of DcN. First,
DcN=1 m leads to EgN=0.54×1016 J. This gives ηRN=0.99, which is 0.32 larger than
ηRCN=0.67 and represented by ‘N1’ in Figure 3. Secondly, considering DcN=6 and 10
m, which were inferred, respectively, by Zhang et al. [2003] and Ma and Mikumo
[2004], EgN=3.22×1016 J and EgN=5.37×1017 J, which lead to ηRN=0.55 and 0.43,
respectively. The two values of ηRN are both smaller than ηRCN=0.67 and shown by
‘N6’ and ‘N10’ in Figure 3. Although the two values of ηRN are in the range 0.2–0.9
13
evaluated from core samples at the 1111-m depth, they are both smaller than ηRS=0.69
and those (0.75–0.95) of normal earthquakes [Kanamor and Brodsky, 2004]. Hence,
DcN=6 and 10 m could be over-estimated. Thirdly, considering the average
displacement on the northern fault plane, i.e., DN=7.15 m, DcN=1.8 m. This value
could be the lower bound of DcN. Fourthly, considering EsN=3.98×1016 J with
∆σdN=2.97×107 N/m2 and AN=3.929×108 m2, DcN=3.7 m. This value is related to
ηRN=0.67 at (vR/β)N=0.80. Its data point is denoted by a symbol ‘N3’, which is much
closed to the theoretical curve, in Figure 3. Obviously, ηRN=0.81 and 0.67,
respectively, related to DcN=1.8 m and 3.7 m are both larger than ηRS=0.61. This
retains the positive correlation between ηR and vR/β. Hence, it is reasonable to
consider DcN to be in between 1.8 m and 3.7 m. From near-fault seismograms Mori
[2005] inferred DcN=2.3 m. This confirms the present way of examining DcN.
3.4 Fracture Energy Density
Several authors [Scholz, 1990; Ide, 2003; Rice et al., 2005; Tinti et al., 2005a]
reported G=106–107 J/m2 for earthquakes. The values of GN and GS of this event (see
Table 1) are in this range. From local seismograms, Zhang et al. [2003] and Ma and
Mikumo [2004] reported that G increases from south to north, and GS=105–108 J/m2
in the south and GN up to 3 × 108 J/m2 in the north. Their values are about
one-order-of-magnitude larger than those of this study. From the core sample on the
1111-m slip zone of a 2000-m deep hole, Ma et al. [2005] reported that the thickness
of the slip zone is about 0.02 m and the grain size is in the range (50–1000)×10-9 m.
From the grain size, they obtained average G=4.8×106 J/m2, which is about one fifth
of that in this study and one-order-of-magnitude smaller than those estimated by
Zhang et al. [2003] and Ma and Mikumo [2004]. Obviously, the value of GN of this
study seems better than theirs.
From laboratory experiments, Wong [1982] stated that temperature, pressure,
rock type etc. can all change Eg (as well as G) up to an order of magnitude. Since the
temperature and lithostatic pressure on the fault both increase with depth, G should
increase with depth. The spatial distribution of G inferred by Ma and Mikumo [2004]
obviously confirms this point. Hence, average G of the northern fault plane should be
larger than that at the 1111-m depth.
Table 1 and the results by both Zhang et al. [2003] and Ma and Mikumo [2004]
show GN>GS. The reason to cause this difference is explained below. From the
14
dislocation theory, G=ζK2/Y [cf. Kostrov and Das, 1988], where ζ is a geometry
factor, K is the stress intensity factor, and the Y is Young modulus. Since the core
samples are different between the two shallow boreholes, K and Y could be different
between the two segments. This causes GN≠GS.
3.5 Ambient Stress Levels
For the Chi-Chi earthquake, we have a good chance to evaluate average σd
through σd=Ef/AD and average σo from σo=σd+∆σd on the Chelungpu fault, when σf
is consider to be σd. Huang et al. [2001] reported ∆σdS=6.5 MPa and ∆σdN=30 MPa.
Hence, we have σdS=55 MPa and σdN=57 MPa, thus leading to σoS=62 MPa and
σoN=87 MPa, respectively. Obviously, σd is almost the same on the two segments, and
σo is higher on the northern segment than on the southern one. Hence, it is more
difficult to trigger an earthquake on the former than on the latter. This is the same as
the concluding point made by Wang [2003]. This leads to a longer return period of
events on the former than on the latter as pointed out by Wang [2005]. In order to
exceed a higher frictional strength on the northern segment, either melting of
materials or a larger pore pressure due to fluids is necessary. The lithostatic stress at a
depth H is σL=ρgH, where g (=9.8 m/s2) is the gravity acceleration. Thus, average σL
calculated from the average depth H=8000 m of the fault plane is 204 MPa. It is
obvious that the estimated value of σo is 52% and 70% less than average σL,
respectively, on the southern and northern segments. This implies that the Chelungpu
fault is weaker, and the southern segment is weaker than the northern one.
For the two segments, the strain energy can also be re-computed from the
evaluated values of σo and σd through ∆E=[(σo+σd)/2]DA. Results are ∆ES=1.30×1017
J and ∆EN=1.86×1017 J, which are both similar to those mentioned above. The values
of ∆σd used for the calculations were independently evaluated by Huang et al. [2001].
Hence, similarity between the two sets of strains energies strongly suggests a high
degree of robustness of the results.
3.6 Frictional Strengths and Pore Pressures on the Chelungpu Fault
In order to study the relationship among frictional strength, pore pressure, and
heat, we construct a 2-D faulting model with frictional heat. As depicted in Figure 3,
the lithostatic pressure σL at H is ρgH. The (maximum) horizontal principal stress σ1
is ρgH plus an additional tectonic stress, and the (minimum) vertical principal stress is
σ3=σL. The normal and shear stresses, i.e., σn and σs, on the fault plane with a dip
15
angle of θ are both a function of σ1 and σ3. The relation of σn versus σs is in the form:
|σs|=µf(σn-pw), where µf is the frictional coefficient and pw is the pore pressure. Let
pw=γρgH, where γ is the pore-fluid factor [cf. Sibson, 1992]. At shallow depths, where
the fluid gradient is hydrostatic and γ is the ratio of fluid to rock density, typically
∼0.4. At depths, where the fluid pressure may become suprahydrostatic, γ>0.4, with
an extreme of γ→1. When σn=ρgH, |σs|=µf(1-γ)σn. Thus, the term µf(1-γ) behaves like
the effective frictional coefficient and denoted by µfe. The values of µf inferred by H.
Ito (personal communication, 2004) and Mori [2004] were made based on this
condition and, thus, should be µfe
Based on Anderson theory of faulting [cf. Turcotte and Schubert, 1982], for
thrust faults σs is:
σs=-µfρgH(1-γ)sin(2θ)/[(1+µf2)1/2-µf].
(7)
(Since σn is not used below, the related formula is not given here.) As mentioned
above, dynamic friction yields Ef=σdAD. In order to link Ef with Eq. (7), a relation
between σd and σo is needed. However, such a relation is lack. The experimental
results by Byerlee [1967] shows that σd is a fraction of σo. This makes us able to
assume σd=ξσo, where ξ<1. Inserting the absolute value of Eq. (7) into Eq. (3) with
Ef=ξσoAD leads to
Q=ξµf(1-γ)gHDsin(2θ)/C[(1+µf2)1/2-µf].
(8)
The value of Q with ξ=1 is the upper bound of heat strength. Inserting Eq. (8) into Eq.
(2) leads to
∆T=Q/h=ξµf(1-γ)gHDsin(2θ)/hC[(1+µf2)1/2-µf].
(9)
The value of ∆T with ξ=1 is the upper bound of temperature rise. Eq. (9) also shows
that as γ→1, ∆T→0, because the frictional stress is balanced by the pore pressure.
Although ξ<1, it would be close to 1. From the experimental result by Byerlee
16
[1967], the frictional coefficient decreases from 0.72 to 0.6, thus showing ξ=0.83.
Owing to the rate- and state-dependent friction law proposed by Ruina [1983], the
frictional coefficient µv at a sliding velocity v is: µv=µo+aln(v/vo)+bln(ϕvo/Dc), where
ϕ and vo are, respectively, the state variable and reference velocity. Contributions to
the total friction coefficient is scaled by a for the direct effect and b for the evolution
effect. Two one-state-variable friction laws, i.e., dϕ/dt=-(ϕv/Dc)ln(ϕv/Dc) and
dϕ/dt=1-ϕv/Dc, are commonly used to describe the state-dependent evolution effect
[e.g., Marone, 1998]. The steady-state friction coefficient µss is µo+(a-b)ln(v/vo).
Hence, µss is µss1=µo+(a-b)ln(v1/vo) at v1 and µss2=µo+(a-b)ln(v2/vo) at v2, thus giving
µss1-µss2=(a-b)ln(v1/v2). The average of a-b for weakening friction in the seismogenic
ss
ss
zone is about -0.005 [cf. Scholz, 1998]. This results in µ 1-µ 2=0.11 when
v1=1.58×10-10 m/sec, which is the long-term average sliding velocity across the fault
[Wang, 2005], and v2=1 m/sec, which is almost the upper bound of slip velocity
[Brune, 1970] and the average PGV on the northern segment [Wang et al., 2002]. The
value µss1-µss2=0.11 is very similar to the laboratory one [Bylerlee, 1967]. At low v, a
fault is almost at rest, and, thus, µd (=µv) is close to µf. This leads to ξ≈µss2/µss1<
ss
ss
ss
ss
(µ 1-0.1)/µ 1 =1-0.11/µ 1. When µ 1=0.6, which is a common value of µf [Byerlees,
1978], ξ≈0.82; and the maximum value of ξ is 0.89 when µss1=1. Obviously, Eqs. (8)
and (9) can work well when ξ=1, because ξ is close to 1.
Since σs is proportional to H, average Q and ∆T calculated along the W-direction
are almost equal to those at H=8000 m, which is not only the focal depth of the
earthquake, but also the average fault width. We assume that µfeS=0.85 and µfeN=0.15
inferred by Ito [2004] and Mori [2004] can be extrapolated to H=8000 m. From Eqs.
(8) and (9), the estimated values are: QN=153 oC-m and ∆TN=(153/h) oC for DN=7.15
m and QS=347 oC-m and ∆TS=(347/h) oC for DS=4.88. The values are close to those
estimated from seismic data on the northern segment, but about 3 times larger than
those on the southern one. A reason to produce inconsistency on the latter is
17
inappropriate extrapolation of µfeS=0.85 to H=8000 m. Hence, a question arises: What
should be the value of µf on the fault during faulting? This is explored below.
At shallow depths, the averages of µfe inferred from measured ∆T are µfeS=0.85
and µfeN=0.15, which represent the stronger and weaker faults, respectively. From a
synoptic model of seismogenic zones [Scholz, 1990], at shallow depths, which could
be down to 2 km, material is velocity-strengthening.Large value of µfeS=0.85, which
could be higher than µf, implies that at southern shallow depths of the Chelungpu fault
material was still in the strengthening state 1.4 years after the earthquake. Whereas, at
northern shallow depths µfeN=0.15, which is much smaller than 0.85, might indicate
the presence of high pore pressures. Hence, it is necessary to explore the possible
effect on µfe due to pore pressures. Inserting sin(2θ)=1/(1+µf2)1/2 into Eq. (8) leads to
µf(1-γ)=(QC/ξgHD)(1+µf2)1/2[(1+µf2)1/2-µf].
(10)
In Eq. (10), the left-hand side is a linearly increasing function of µf, whose slope is
1-γ, with a maximum value of 1 when γ=0. The linear functions for γ=0–0.8, with a
unit of 0.1, are displayed with solid lines in Figure 5. The right-hand side of Eq. (10),
which is denoted by F(µf) hereafter, is a monotonously decreasing function of µf, with
the maximum value of QC/ξgHD. The two functions are both defined in the domain
of 0≦µf≦1. For a certain γ, the two functions intersect at a point, which represents
the solution of Eq. (10). Since the values of γ on the two segments are not clear, we
cannot solve Eq. (10) exactly. Inserting QS=102 oC-m plus DS=4.88 m and QN=154
o
C-m plus DN=7.15 m, respectively, into Eq. (10), with C=103 J/kg-oC and H=8000 m,
lead to two solid curves for the northern and southern segments in Figure 5. In
addition to ξ=1, included also are the curves for ξ=0.8 and 0.9. This figure shows: (1)
a decrease in µf with γ; (2) µfeN>µfeS, with a small difference, at any γ, thus implying
that the northern segment was slightly stronger than the southern one under the same
pore pressure during faulting; and (3) an increase in the solution of Eq. (10), i.e., µf,
with decreasing ξ on both segments. Point (3) indicates that when ξ=1, Eq. (10) gives
the lower bound of µfe. When ξ=1, the points, at which the two curves intersecting the
line with γ=0 denote the lower bounds of µf, i.e., µflS=0.22 and µflN=0.23.
At southern shallow depths, µfeS=0.85 is much larger than F(µf) and, thus, not
18
shown in Figure 5. The condition of making µfS(1-γ)=0.85 hold is γS<0.2. This leads
to hydrostatic fluid gradients at southern shallow depths. When γS is small, µfeS=0.85
is close to µfS. At northern shallow depths, µfeN=0.15 is smaller than F(µf) and, thus,
also not displayed in Figure 5. Any γN in between 0 and 0.85 makes the equality
µfN(1-γ)=0.15 held. But, it is not sure only from this equality whether the fluid
gradients are suprahydrostatic or not. If the value of µfN at γN=0 is set to be 0.85, γN is
0.82 when µfeN=0.15. In an area on the topmost 5 km and near the northern shallow
hole, Ma et al. [2003] inferred low Lc associated with high lubrication during faulting.
This gives large γN, thus implicating that during faulting the fluid gradients at northern
shallow depths, at least in such an area, could be suprahydrostatic.
At depths, we consider two particular cases. First, µf=0.85, which is shown by a
vertical dashed-dotted line in Figure 5, is taken for the whole fault plane. This line
intersects F(µf) at the points with γS=0.87 and γN=0.92, related to µfeS=0.74 and
µfeN=0.78, respectively, when ξ=1. Secondly, µf=0.58, which is displayed by a vertical
dashed line in Figure 5, associated with the dip angle of 30o determined from the
fault-plane solution [cf. Ma et al., 1999] is also taken into account. This line intersects
F(µf) at the points with γS=0.80 and γN=0.88, related to µfeS=0.46 and µfeN=0.51,
respectively, when ξ=1. For the two particular cases, γN and γS both increase with
decreasing ξ. The difference between γN and γS for ξ=0.8–1.0 is less than 0.05. This
indicates that the proposed 2-D faulting model can work well even ξ=1. Meanwhile,
for the two particular cases large suprahydrostatic fluid gradients would appear at
depths during faulting. But, µfeS and µfeN are smaller from µf=0.85 than from µf=0.58.
Although the two particular cases can both produce large displacements on the fault,
the amount is larger from a bigger suprahydrostatic gradient than from a smaller one.
In addition, they can both generate larger displacements on the northern segment than
on the southern one. This is consistent with the observations [cf. Wang, 2003]. For a
wide variety of rocks and surface types, Byerlee [1978] proposed µf=0.6 at σn=200
MPa and 0.85 at lower σn. In addition, µf=0.85 might be due to velocity-strengthening
as mentioned above. Hence, µf=0.58 is the preference for the whole fault plane.
Sibson [1977] attributed the formation of the in-cohesive rocks and the random
fabric cataclasite series to the elastico-frictional processes and that of the foliated
mylonite series to quasi-plastic processes at depths equivalent to ambient
19
temperatures in excess of ～300oC. He also differentiated those types of fault rocks
generated by rapid seismic slip from those produced during slow deformations.
Sibson [1975] argued that pseudotachylites are the result of frictional melting during
seismic slip and are usually formed at a temperature of around 1100 oC. Hence, the
existence of pseudotachylites is an indication of melting of the fault zone. From
average QS and QN, average ∆TS and ∆TN are both 1100 oC when hS=0.09 m and
hN=0.14 m. Hence, on average pseudotachylites resulted from heating would be
thicker in the north than in the south. At shallow depths, the initial values of Q and ∆T
in the two boreholes immediately after the earthquake happened are: (1) QS=1.3 oC-m
and ∆ΤS=(1.3/h) oC, because of HS=180 m, µfeS=0.85, and DS=1 m; and (2) QN=1.5
o
C-m and ∆ΤN=(1.5/h) oC, because of HN=300 m, µfeN=0.15, and DN=4 m. Hence, the
thickness of pseudotachylites is less than 0.0012 m in the southern borehole and about
0.0014 m in the northern one. The two thicknesses are close to each other. Sibson
[1975] proposed a formula to correlate h with D: h=(4.36/D)1/2, where h and D are
measured in meters. The values of h at shallow depths are 0.008 m for D=3 m in the
south and 0.007 m for D=9 m in the north. The two thicknesses are very similar, but
about 5 times larger than the previous two. However, pseudotachylites were found
only in the southern borehole. The absence of pseudotachylites in the northern
borehole might be due to the absence of frictional melting because of weakening of
the northern fault zone caused by high pore pressures.
3.7 Fluid Effects
Mase and Smith [1984/1985] addressed the following viewpoint: When the
permeability exceeds 10-15 m2, frictional melting may occur on the fault surface
before thermal pressurization becomes significant. During an earthquake the failure
surface is heated to a temperature required for the thermal expansion of pore fluids to
balance the rate of fluid loss due to flow and the fluid-volume changes because of
pore dilatation. Once the condition is reached, the pore fluids pressurize and the shear
strength decreases rapidly to a value sufficient to maintain the thermal pressurization
of pore fluids at the near-lithostatic stresses. If the permeability is less than 10-17 m2,
fluid pressurization is most likely to occur with a temperature rise of less than 200 oC,
and friction will drop significantly. Based on the physical properties of the fault-zone
materials, the permeability should be different between the northern and southern
segments. Tanikawa et al. [2004] reported that the permeability is in the range 10-16–
20
10-19 m2 in the northern borehole and 10-14–10-17 m2 in the southern one. The values
of permeability are almost less than 10-15 m2. The permeability, which usually
exponentially decreases with depth [Morrow and Byerlee, 1992], would not exceed
10-15 m2 in the most part of the Chelungpu fault. Thus, it is likely that in addition to
lubrication, thermal fluid pressurization also played a significant role on faulting,
especially at depths. Thermal fluid pressurization would be more important on the
northern fault plane than on the southern one. Of course, interplay between lubrication
or thermal fluid pressurization should exist.
Although there is a lack of direct evidence to show the existence of fluids on the
Chelungpu fault, it is able to examine the possibility using an indirect way. A fault
zone is usually not wide, but it links with some sub-faults. Hence, fluids could
exchange between the fault zone and the wall rocks through the sub-faults. This could
result in a change of α (the P-wave velocity) and β in the area surrounding the fault
before and after the earthquake. In addition, the difference between γS and γN indicates
different contents of fluids on the two segments, thus leading to a difference in the
change of α and β before and after the event between the two segments. The
difference can be detected using the 3D tomography methodology. Larger α/β might
indicate the presence of fluids, which mainly decreases β. From Chen et al. [2001],
the change of the pre- and post-event values of α/β was the smallest in the topmost 2
km, increased with depth up to 10 km, and then decreased again. Such a change is
larger on the northern segment than on the southern one. In the depth range of 5–10
km, (α/β)N>(α/β)S before the earthquake and the pre-event values of α/β were larger
than after the post-event ones on the two segments. The inequality is comparable with
γN>γS as mentioned above. After the earthquake, at depths below 2 km α/β
remarkably reduces. The observed results would indicate that the fluids came from
either the lower crust or the upper mantle as expected by Sibson [1992], the fluid
content was higher on the northern segment than on the southern one, and fluids
existed mainly at depths during faulting. After the earthquake, a decrease in fluid
content at depths should be due to post-failure fluid discharge [cf. Sibson, 1992]. A
small change of pre- and post-event values of α/β at shallow depths implies that the
measured water content of 45 vol% and a small one can be regarded as the original
values, respectively, on the northern and southern segments during faulting.
3.8 Brief Summary
21
Results show that there are marked differences in σo, ∆σs, ∆σd, D, Dc, ∆E, Es, Eg,
Ef, G, η, ηR, Q, and ∆T between the northern and southern segments. The amount of
each parameter is larger on the former than on the latter. Stress, friction and pore
pressure are three major factors in producing the differences.
When tectonic loading exceeded the static frictional strength on the Chelungpu
fault, the Chi-Chi earthquake was triggered. Of course, the pore pressure could also
play a significant role in reducing µfe. Average ambient static stress levels, σo, are
about 62 and 87 MPa, respectively, on the southern and northern fault planes. A
higher stress level caused a longer return period on the former than on the latter.
When the Chelungpu fault broke, ruptures started first from the hypocenter on the
southern segment, and then propagated outward. The northward ruptures, which were
much stronger than the southward ones, do not seem able to initiate failure of the
northern segment, because the static frictional stress is about 25 MPa higher on the
northern segment than on the southern one. But, the former actually broke. This might
be due to, at least, three reasons occurred on the northern segment: an excess of the
sum of long-term tectonic loading plus additional stresses caused by the southern
ruptures over the static frictional strengths, reduction of frictional strengths due to
frictional melting, a decreases in frictional strengths caused by the superhydrostatic
pressures due to injection of fluids. However, it is not yet clear how fluids did inject
into the fault, even though indirect evidence suggests the existence of fluids in the
source area.
Higher dynamic friction produced stronger heat strength, thus resulting in higher
temperature rise, on the northern segment than on the southern one. However, the
temperature rise measured at two shallow holes 1.4 years after the earthquake was
lower in the north than in the south. This might be due to either a larger thermal
diffusivity or higher loss of heat through fluids in the former than in the latter. Ma et
al. [2003] inferred the existence of a small area with abnormally high pore pressures
in the topmost 5 km on the northern fault plane. The northern hole was located at such
an area. Thus, loss of heat through fluids is very possible. The field measures show
lower permeability in the northern shallow hole than in the southern one. Since
permeability usually decreases with depth, it is expected that higher thermal
pressurization existed, thus yielding stronger loss of heat, on the northern segment
than on the southern one.
22
The values of µfe inferred from temperature rise measured at two shallow holes
cannot be applied at depths. The value of µf inferred from the focal-plane solution is
about 0.58. This is an acceptable value for the overall fault plane. The rate- and
state-dependent friction law can interpret the stress-slip functions inferred from
seismic data. Dc is longer on the northern segment than on the southern one. Average
Dc is in between 1.8 and 3.7 m on the former and about 1 m on the latter.
Higher static stress, larger dynamic stress drop, stronger lubrication and thermal
pressurization, thus leading to larger displacements, slip velocities, strain energy,
seismic radiation energy, frictional energy, and heat strength on the northern segment
than on the southern one. In addition, higher lubrication and/or thermal pressurization
could result in more kinetic energies and lower high-frequency content in signals, thus
leading to lower radiation efficiency and a larger exponent of source scaling law on
the former than on the latter.
4. Conclusions
[1]. There are marked differences in ∆E, Es, Eg, Ef, G, η, ηR, Q, and ∆T between the
northern and southern segments of the Chelungpu fault. Different static tress
levels, dynamic stress drops, frictional strengths, rupture modes, degrees of
melting, lubrication, and thermal pressurization in the two segments would lead
to the differences.
[2]. As for the characteristic slip displacement, DcS=1 m inferred either by Tanikawa
et al. [2005] or from this study for the southern segment is acceptable. On the
other hand, DcN could be over-estimated by Zhang et al. [2003] and Ma and
Mikumo [2004], and the modeled value is in between 1.8 and 3.7 m.
[3]. Average ambient stress levels, σo, are about 62 and 87 MPa, respectively, on the
southern and northern segments. The two values are, respectively, 52% and 70%
less than the lithostaic pressure at H=8000 m, which is an average depth of the
fault plane. This implies that the Chelungpu fault would be weak.
[4]. The values of frictional coefficient inferred from temperature rise measured at
two shallow holes cannot be applied to the overall fault plane, and the acceptable
value is 0.58, which is evaluated from the fault-plane solution, with a dip angle
of 30o.
23
[5]. Ef occupied the majority of ∆E, and led to heat on the fault. For the same
thickness within which heat is distributed, temperature rise is higher on the
northern segment than on the southern one.
[6]. According to a 2-D faulting model with frictional heat, during faulting the fluid
gradients on the fault plane would be hydrostatic at southern shallow depths and
suprahydrostatic elsewhere.
[7]. Permeability would not exceed 10-15 m2 in the most part of the Chelungpu fault.
Meanwhile, permeability usually decreases with depth. Thus, it is likely that in
addition to lubrication, thermal fluid pressurization also played a significant role
on faulting, especially at depths. Thermal fluid pressurization would be more
important on the northern fault plane than on the southern one. At southern
shallow depths, frictional melting would produce very thin pseudotachylites.
[8] Indirect evidence suggests the presence of fluids in the source area during faulting.
Acknowledgments. The authors would like to express his thanks to two
associated editors and four reviewers Profs. R.H. Sibson, N. Boness, R.I. Madariaga,
and E. Tinti for constructive suggestions and comments. The study was financially
supported by Academia Sinica (Taipei) and the National Sciences Council under
Grant Nos. NSC94-2116-M-001-016 and NSC95-2116-M-001-008.
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Table 1. ∆E and Es were evaluated by Wang [2004]. D, Dc, Eg, η, ηR, G, Ef, and Q are
estimated in this study. Eg is evaluated in this study through two ways: (1)
from Eq. (2) with DS=4.88 m and DN=7.15 m; and (2) from Eq. (6) with
DcS=1 m and DcN=1, 3.7, 6, and 10 m. (Subscripts: “S” and “N” for the
southern and northern segments of the Chelungpu fault, respectively)
D (m)
Dc (m)
∆E
(1017 J)
Es
(1016 J)
Eg
(1016 J)
η
ηR
S
4.88
7.15
1.26
1.26
1.94
1.94
1.94
1.94
1.94
0.33
0.33
3.98
3.98
3.98
3.98
3.98
0.21
0.15
0.95
0.54
1.99
3.22
5.37
0.03
N
1.4
1.0
1.8
1.0
3.7
6.0
10.0
0.61
0.69
0.81
0.99
0.67
0.55
0.43
30
0.14
G
Ef
(107 J/m2) (1017 J)
0.45
0.33
2.59
1.44
5.34
8.64
14.4
Q
(oC-m)
1.21
102
1.45
154
Figure 1. The stress-slip function: lines AC and CD represent slip-weakening friction,
Dc=the characteristic slip displacement, Dmax=the maximum slip, σo=initial stress
(or static frictional stress), σd=dynamic frictional stress, and σf=final stress. The
strain energy, ∆E, per unit area is the area of a trapezoid below line AD,
Es=seismic radiation energy, Eg=fracture energy, and Ef=frictional energy.
31
Figure 2. A figure to show the epicenter (in a solid star), the surface trace of the
Chelungpu fault (in a solid line), the fault plane (bounded by four dashed lines), the
nine near-fault seismic station sites (in open triangles), and the borehole sites (in solid
circles). The northern and southern segments of the fault are separated at a locality
near Station TCU065.
32
Figure 3. The curve shows the variation of the radiation efficiency, ηR, with vR/β
calculated from Eq. (4) in the text when vR/β=0.5–1.0. Two crosses denote ηR at
vR/β=0.75 and 0.80 [Ma et al., 2001]. ‘J’ displays ηR=0.65 at vR/β=0.66 [Ji et al.,
2003]. ‘V’ displays ηR=0.8 [Venkataraman and Kanamori, 2004]. ‘O’ shows both
ηRS=0.61 at vR/β=0.75 and ηRN=0.81 at vR/β=0.80 of this study. ‘S1’ denotes ηRS=0.69
at vR/β=0.75 for DcS=1 m. The symbols ‘N’ with numbers represent ηRN at vR/β=0.80
when DcN=1, 3.7, 6, and 10 m, which are individually displayed with an integer near
the symbols.
33
Figure 4. Figure shows the thrust fault with a dip angle of θ. The depth is denoted by
H. The principal stresses along the horizontal vertical axes are σ1 and σ3, respectively.
The σn and σs represent the normal stress and shear stress, respectively.
34
Figure 5. The variations of µf(1-γ) versus µf for γ=0–0.8, with a unit of 0.1, are
displayed with thin solid lines and the functions of F(µf) are shown by two solid
curves: ‘N’ for the northern segment and ‘S’ for the southern one. The numbers
(.0–.8) denote the values of γ. The values of µf related to γ=0 are µflN and µflS for the
northern and southern segments, respectively. The dashed and dashed- dotted lines
denote µf=0.58 and 0.85, respectively.
35