- Email: [email protected]
Postseismic stress and pore pressure readjustment and aftershock distributions
Tectonophysics,
144 (1987) 37-54
Elsevier Science Publishers
37
B.V., Amsterdam
Postseismic
VICTOR
in The Netherlands
stress and pore pressure readjustment and aftershock distributions C. LI *, SANDRA
’ Department ’ Department
- Printed
of Ciwl Engineering,
of Earth, Atmosphere (Received
H. SEALE
January
’ and TIANQING
M. I. T., Cambridge,
and PlanetaT
2
MA 02139 (U.S.A.)
Science, M.I. T., Camhrrdge,
14. 1986; revised version
CA0
accepted
October
MA 02139 (U.S.A.)
10. 1986)
Abstract Li. V.C., Seale, S.H. and Cao, T.. 1987. Postseismic In: R.L. Wesson
(Editor),
The time and spatial
Mechanics
readjustment
stress and pore pressure
of Earthquake
Faulting.
of stress and pore pressure
of a model which treats the earth’s crust as a linear poro-elastic fundamental
solution
compressible.
The main
maintained
Rice
and
is modeled
from regions
Cleary
who
treated
as a sudden
uniform
introduced
dislocation
distribution
undergoing
compression
It is assumed pressure
that aftershocks
in normal
increase)
of spatial
exceeds the shear strength
distribution
of aftershocks
aftershock
zone
aftershock
-zones for off-fault
provides
from
the compression
a means
into the dilation
aftershocks
to study
appear
the association
the resulting
undergoing
The coseismic
at the two ends
high
as separately
slip dislocation
Hence
Coulomb
the stress and pore
coefficient,
the suggestion
a small rotation
regions. increase
6, and 7 are
and p is the pore
confirms
predicts
of the main
stress
is
fields. Diffusion
of the rock medium.
elastic solution solution
by means
is based on a
as ec = r + ~(0” + p), where
and shift into the dilating
between
constituents
dilation.
p is a friction
The transient
region
to expand
fluid
sets up stress and pore pressure
slip surface,
Lisowski.
are investigated
and
on the flow parameters
of the slip surface.
by Stein and
rupture
and
drop
stress (defined
stress and shear stress across a potential
distributions.
by fluids. The analysis
the solid
to regions
occur when the Coulomb
infiltrated
stress
and aftershock
144: 37-54.
due to a strike-slip material
fields evolve over time, with time scales which depend
the increase
which
by
shock
in time. This suddenly
of fluid takes place pressure
obtained
readjustment
Tectonophysics,
rupture.
of the
whereas
the
A new plot is developed with off-fault
aftershock
locations. The aftershock are compared
distributions
of three earthquakes
to the zones predicted
and in some cases form distinct provide
a primary
in the contexts
aftershock
zones and suggests
inaccuracy
in location
confirm
or disprove
off-fault
mechanism
of these sample
aftershock
clusters
in the areas
and the implications
earthquakes.
an explanation
of aftershocks the theory
(Pasinler,
1983: Borrego
by the model. The earthquake
The model
that aftershock
distributions
Mountain,
of high Coulomb
predicts change
stress.
in predicted
are controlled
observed
of off-fault
the dilatant
The ability
for material
the generally
or absence
1968: and Haicheng.
are skewed towards
of this mechanism
for the presence
and the small magnitude
aftershocks
Coulomb
of the model
properties spatial
aftershock
1975) regions
expansion
clusters.
to
are discussed of the
However,
stress makes it difficult
the to
by fluid flow.
Introduction
these patterns
It has been observed that for some earthquakes with dominant strike-slip focal mechanism, the spatial distribution of aftershocks has distinct patterns when the aftershocks can be located accurately. Some commonly observed characteristics of
aftershock zones beyond the ends of the main rupture. In addition, clusters of aftershocks are sometimes located off the fault plane and skewed towards different ends of the rupture on either side of the fault (the dilatant quadrant). Examples of off-fault aftershock patterns include the
0040.1951/87/$03,50
B 1987 Elsevier Science Publishers
B.V.
include
the linear
extension
of the
38
Homestead
Valley
earthquake
discussed
by Stein
range
from days
and Lisowski (1983) and Das and Scholz (1981)and
least
for some
the Borrego
nism
may
Mountain
and Managua
earthquakes
discussed
by Das and Scholz (1981). Although
temporal
development
been
of such patterns
well identified,
aftershocks
a large
the
has not
proportion
of the
do occur with some time delay.
Attempts
at explaining
the spatial
distributions
of aftershocks
have
made
several
re-
Das
and
searchers.
Gzovsky
been
et al. (1974)
Scholz (1981) analyzed a plane that
strain
Mode
at distances
by and
the stress field induced II (shear)
about
crack
one rupture
and length
by
found per-
pendicular to the fault the shear stress increased by about 10% of the stress drop on the crack. Das and Scholz associate
such stress change
with off-
fault aftershock clusters. Based on work by Chinnery (1963), Stein and Lisowski (1983) analyzed the Coulomb stress induced by the Homestead Valley earthquake, which they modelled as vertical dislocation patches extending from the ground surface to 5 km depth. The Coulomb stresses were calculated on the ground (free) surface and included the effect of friction proportional to induced normal stresses. This causes the Coulomb stress to shift into the dilatant quadrant. The references cited above are based on elastic analyses, with the implication that the stresses are induced coseismically (on the time scale of elastic wave travel time from the rupture source to the aftershock site). In reality, however, a great proportion of aftershocks have delay times of days, weeks or months depending on the magnitude of the main shock. In this paper. we propose a simple model which incorporates many of the essential ideas of Das and Scholz (1981) and of Stein and Lisowski (1983) and also includes a time component. The time source is due to the diffusion process of water induced by a pressure disequilibrium set up by the main rupture. The diffusion process allows water to flow from the high compression region to the dilatant region, causing a time-dependent stress change. The result is a gradual enhancement of the Coulomb shear stress and also an enlargement of the area of high Coulomb shear stress in the dilating quadrant. The time scale of the flow process depends on the rupture length and diffusivity of the rock mass and may
aftershocks patterns
to months. earthquake
be responsible
and full development
as mentioned
The suggestion
Booker
and
out that the diffusion
may
such
of
of their spatial
was originally (1972).
process
They
produces
root time decay in pressure
to the aftershock nisms
delay
that water flow may be respon-
posed
Here we hasten
at
earlier. generation
by Nur
that
this mecha-
for the time
sible for aftershock
square
We suggest ruptures
pro-
pointed
an inverse
change
similar
event decay law of Omori (1894). to add that other physical
as stress
also operate
corrosion
simultaneously,
mecha-
or viscoelasticity although
they
may have quite different time scales. Hull (1983) gave a thorough review of various plausible aftershock
mechanisms.
Perhaps the most complete analysis of pore fluid flow as a mechanism for aftershock generation has been performed by Booker (1974). who solved for the time-dependent stress field induced by a plane fracture in an infinite poro-elastic body. He investigated the case of a uniform slip distribution modelled by two suddenly applied edge dislocations equal in magnitude but opposite in sign; and the material was assumed to have incompressible constituents. The primary concern of Booker’s work was to explain aftershocks on the rupture plane induced by the reloading of the fault. Calculations of time-dependent stresses and pore pressure for a slip distribution consisting of two edge dislocations (constant slip) using the fundamental solutions of Rice and Cleary (1976) were performed by Hull (1983). In this case the material was assumed to have compressible constituents. The primary differences between the results of this work and those of Booker (1974) are the values of the coseismic stresses and the length of time required to reach steady state. The model employed here is based on representing the main shock as a suddenly introduced two-dimensional continuous distribution of shear edge dislocations in an infinite poro-elastic medium. As in Das and Scholz (1981), the slip distribution on the main rupture is chosen as that due to a crack, with maximum slip in the middle and tapering off towards the ends of the rupture. Our analysis is based on a fundamental solution
39
due to Rice and fluid
and
pressible.
solid
Cleary
(1976)
constituents
who treated
as separately
While we have not evaluated
of improvements material
adopted
in the present
they may nevertheless
be expected
reality
to uniform
(in comparison
pressible
are,
however,
For example,
strain
1
comand
the
model,
to be closer to slip and incom-
constituents).
There model.
r-
the degree
with the slip geometry
behavior
the
deformation
to reality, reasonable
many
limitations
in the
the two-dimensional
plane
can only be an approximation
although when
this the
approximation
fault
assumed smooth and constant bution on the main rupture with observations
width
may
is large.
be The
the slip distribution In the following,
for locations
assumed
of fault slip; slip distrlbutlon Coulomb
off the fault,
may be sensible.
we shall briefly
describe
ulus
and
linear bution
the
y) and pore pressure
of the 1983 Pasinler Mt. earthquake and be discussed in the with the latter two
a,,(~, y. t)=
earthquake, the 1968 Borrego the Haicheng earthquake will context of the model results, in some detail. The presenta-
tion of data for these three earthquakes is meant to illustrate some of the ideas suggested by the theoretical model. Lack of accuracy in earthquake data (such as fault length) and in material parameters precludes a strong verification of the theoretical predictions. We have also not oughly the sensitivity of theoretical parameters
studied thorresults to the
used in the model.
Poisson
the time and
of the induced
are:
material
6(y)
given by
and pore pressure
V, is the undrained
superposition,
theoretical model and discuss some numerical results based on this model. The aftershock patterns
various
stress q
p.
(in time) slip distriis also inconsistent
in surface breaks and aftershocks
on the fault. However,
Fig. 1. Geometry eqn. (1) induces
stresses
ratio.
spatial
By
distri-
u,, with (i, j = x,
p due to the main
rupture
atqx’)
y. r)rdx’
-j-;rG,,(x-x'.
and:
p(x,
t) = -/;,G&-x'.
4‘.
where rupture
we have
already
x, r)%&$dx’
assumed
that
occurs at time zero. The Green’s
the main function
_Y, r) and G,(x - x’, J’, t) represents
G,,(x -x’, the stress and pressure change at a point (x, J’) and at time t due to a unit dislocation suddenly introduced
at (x’, 0) and at time zero in a poro-
Model description and results
elastic medium. Such fundamental solutions have been previously obtained by Rice and Cleary (1976) and they have been included in Appendix
The model is based on representing the main shock as a suddenly introduced two-dimensional continuous distribution of shear edge dislocations
A. The factor (as( x’)/ax’) dx’ may be interpreted as the accumulated slip within an element dx’
in an infinite poro-elastic medium (Fig. 1). The slip magnitude 6(x) is determined from elastic crack theory in which a uniform stress drop Au is
ual stress components and pressure may be computed using eqns. (1) and (2), and the numerical implementation is described in Appendix A. Following Stein and Lisowski (1983) we assume that the occurrence of aftershocks is associated with the increase of Coulomb stress, defined as:
imposed S(x)
=
along the length of the rupture: 2(L -;)A0
dm
(1)
where Au is the main rupture stress drop, I is the half-length of rupture. G is the elastic shear mod-
along the length of the rupture.
u,=u,,
+/J(u,.+p)
Hence the individ-
(3)
L
Fig. 2. a. Contours about
origin.
of Coulomb
b. Contours
stress at t = 0: shown
of Coulomb
atress
only for J > 0. Stress contours
for 1’ i 0 can be obtained
by 180°
rotation
I --* m
where ~1 is a coefficient of friction (chosen as 0.75 for the following discussion) and the pore pressure p has been included to reflect its influence on
I).Contours are not shown for I: < O.l/ for lack of numerical accuracy. Due to symmetry, we have chosen to show only the space y > 0. Increase of
reducing frictional resistance to sliding. Of course u, ,,, (I,. and p are all time dependent functions given by eqn. (2) although it might be expected that the time change of pore pressure p would have the strongest effect on a,. The association of increase a, as defined in eqn. (3) with aftershock
Coulomb stress is indicated by the solid contours and decrease of Coulomb stress is indicated by the dashed contours. Several characteristics in this
generation implicitly assumes that the aftershock focal mechanisms are similar to that of the main rupture.
This may explain the observation of aftershocks being distributed along a zone longer than the main rupture. The off-fault peak identified (as
In Fig. 2a, we show a contour plot of coseismic Coulomb stress change (normalized by the stress drop) at t = 0, immediately after the main rupture, due to a right-lateral rupture lying along the x-axis between - 1 and 1 (all length dimensions are normalized with respect to the half-rupture length
symmetrical) by Das and Scholz (1981) has been rotated into the dilatant zone and is located at about J = 1.21 from the main rupture, at an angle of 25” to the y-axis (point D). The Coulomb stress increase there attains a maximum of 15% of the stress drop. Interestingly, another peak exists
figure are worth mentioning. Along the fault, a very large Coulomb stress increase exists beyond the rupture zone at both ends as may be expected.
41
(b)
Fig. 2 (continued).
close
to the dilatant
(point stress
A). Between these two peaks, the Coulomb is continuously enhanced with time. (The
locations
of peaks
end
that
of the
occur
near
main
rupture
the plane
of
rupture depend on the particular slip distribution and thus may vary between earthquakes.) It might be expected that aftershock development will be particularly active in this sector. The long term enhanced Coulomb stress is shown in Fig. 2b. After sufficient time the off-fault peak merges with the high stress zone at the end of the fault. To understand
this behavior,
we show
in Fig. 3 the coseismic spatial distribution of pore pressure. The negative pore pressure in the dilatant zone will induce flow from the compression zone and sets up a diffusion process. Of course the induced pore pressure vanishes in the long term equilibrium state. The pore pressure change at
four locations
(labelled
plotted as a function been normalized by
A, B, C, D in Fig. 2) is of time (Fig. 4). Time has a characteristic relaxation
time, defined by t, = 12/4c where c is the diffusivity of the rock mass. The pore pressure rises with time from an initial negative value to zero in this fault quadrant. The locations close to the fault end (e.g. location A) have shorter delay time before pressure sponding
picks up. Figure 5 shows the correchange in Coulomb stress with time is
greater the closer the location is to the dilatant fault end (A, B, C, D in decreasing order). This suggests an apparent expansion of aftershock zones (Utsu, 1969) particularly if an off-fault cluster of small quakes is absent or not recognized as aftershocks of the main rupture. The Coulomb stress along the line A-D is shown in Fig. 6 at various fractions of the characteristic time. Again
-----.\
*,
-.04 .\ ‘\\ \ -.06
A/---
--“I
-\
-
-1
+1
Fig. 3. Pore pressure
0.00
,-0.4Y5 (1 $0.10
I
I
II c
t&j-0. 15 $0.20
-
F-0.2;
-
g-c.“”
.-
“-0.35
-
-0.40
at time r = 0.
B
-
-CT.45 ,
i) Li? - ‘1
-G
/ _;
A -‘ 1 loglo
Fig. 4. Pore pressure
vs. time at locations
I -3
1’
/
-2
I - !
(4ct/&2) A. B. C. and D shown
in Fig. 2a, b
D
1
43
,
I
I
I
lo’;;10 (4Ct/&!l Fig. 5. Coulomb
stress vs. time
Q,. 0. 0. 0.
0. 0. 0. 0. 0.
KORIIALIZED DISTAFICE FROM FAULT EbD ALOllG LINE A - E Fig. 6. Coulomb
stress vs. distance
along
A-E.
(0.0) to very long time scale (co). The dotted pre-stress
states. See text for more details.
The four curves
and dashed
show fluid pressure-deformation
lines indicate
effects on spatial
distribution
coupling
effects
of aftershocks
from coseismic for two different
44
the off-fault
peak at D is clearly
(and
is still maintained
seen
that
most
between
shown
at f = O.lt,).
of the Coulomb
A and D occurs within
time. For example, km, and diffusivity
for a rupture range
at t = 0
It may
stress
be
increase
one characteristic length
In the following patterns
quakes in the context
sections,
of several
we discuss
strike-slip
of these theoretical
(see, e.g. Li, 1984/1985), the characteristic time is between i to 50 yrs. If the larger diffusivity ap-
consider-
ations.
zones may
Pasinler earthquake (October 1983) The Pasinler center
earthquake
40.3” N and
event which occurred
(M, = 7.1 (NEIS);
42.2” E (USGS)) in northeast
Turkey.
It has a
well-constrained
diffusivity
of strike N42OE dip 80” which is consistent
hundred
extend
these time lengths
by a
fold.
Figure
6 offers another,
interpretation
perhaps
of aftershock
more general,
patterns.
Suppose
the
fault plane
solution
epi-
is a recent
be fully developed within half a month (O.lt,) to half a year after the main rupture. The smaller would
the
earth-
of 21= 50
of 10’ to lo5 cm2 s-’
plies, it may be seen that the aftershock
different. aftershock
with attitude with
the trend of observed surface faulting and aftershock patterns (Toksoz, 1984). The aftershock distribution shown
of 10 days after
in Fig. 7. The data
the main
rupture
were collected
is
by a
pre-rupture stress state and the material properties of the rock mass in the vicinity of the main
network
rupture are such that a 20% of Au increase in Coulomb stress is required for aftershock occur-
instruments deployed is much larger than
rence (indicated by the horizontal dashed line), then no off-fault aftershock clusters will be observed, and an expansion of the aftershock zone will be recorded. Aftershocks will be confined to about 0.11 immediately after the rupture and gradually expanded to about 0.61 (measured along the line A-E). If, however, the required minimum in Coulomb stress is 10% of Au, then immediately after the main rupture, near the fault end, the
where faulting was observed. Because the lengths of the surface faulting and the observed surface displacements (10 to 80 cm) are too small to represent the complete faulting of an earthquake of magnitude M, = 7.1, and because the most prominent fault break falls to one side of the
aftershocks
are confined
to about
0.21 and simul-
taneously an off-fault cluster occurs between 0.71 and 1.71. With time (say t = O.lt,) the near fault
of eight portable
analog
and three digital
locally. The aftershock zone the 10 to 12 km long zone
maximum intensity contour, Toksoz (1984) suggested that the primary slip may have occurred at depth. We have placed the fault line visually with respect to the aftershock gated isoseismals. Fault
locations and the elonmotion is left-lateral as
aftershock zone expands outwards. At one characteristic time, the expanded zone merges with the
indicated by the arrows. Admittedly the fault length shown in Fig. 7 is at best dubious and for this reason, we have not carried out detailed anal-
off-fault
ysis
clusters.
From the above discussion, it should be clear that the stress state prior to the main rupture, the material properties (particularly fracture properties). and the main shock stress drop dictate the presence or absence of off-fault clusters, although in both cases, aftershock zone expansion (with time) may be observed. Furthermore, the area1 size and location of the off-fault clusters depend on the rupture length, while the time scale of expansion and merging with the off-fault cluster depend on the diffusivity of the rock mass. Because these dependent parameters can be different from earthquake to earthquake, the exact temporal and spatial pattern of aftershocks may also look quite
for
this
case.
However.
two
loops
of
aftershocks can be seen to skew towards the dilatant quadrants (compare with Fig. 2). There is no distinctive clustering at an off-fault location, and this may be indicative of the case of required (for aftershock occurrence) Coulomb stress alteration exceeding the induced off-fault peak Coulomb reference
stress alteration, as described to the dashed line in Fig. 6.
earlier
in
Borrego Mountain earthquake (April 1968) The main shock of the 1968 Borrego Mountain earthquake is shown in Fig. 8, together with the spatial distribution of 533 aftershocks for 91 days
45 40.6
40.5
40.4
~
40.3
z ;: 4
40.2
/
Narman
/
..
\
: .,z:\.
40.1
40
39.9 LONGITUDE
Fig. 7. Aftershock
distributions
Note the concentration
of the Pasinler
of aftershocks
earthquake
in the dilatant
(after
ToksGz. 1984). Symbol
star indicates
location
of the main shock.
quadrants
15’
30’
116”OO’
45’ I-
\
j’-
‘..
Q
25 KM
0 -1.
s,
I
“..,,
X”
i’LONGITUDE Fig. 8. Aftershock the dilatant
distributions
quadrants.
of the Borrego
Mountain
earthquake
(after
Hamilton,
1972). Note the off-fault
clusters
shifted
into
Bon-ego
Aftershocks
METERS Fig. 9. Early aftershocks the basis for choosing
of the Borrego
Mountain
the half fault length,
earthquake
in rotated
I = 20 km. Arrow
at origin
after the main rupture (from Hamilton, 1972). The length of rupture (dashed line in Fig. 8) is esti-
chosen
rupture
length
tion
is
of main shock.
zone ruptures..
Without
a better
we have used the 5 day aftershocks
with an arrow. The
of 40 km is longer
5 days after the main shock. This distribution
location
observed surface break of 31 km (Fig. 8). The early-day aftershocks defining the rupture zone is a generally accepted notion, especially for subduc-
mated to be 40 km from the extent of the aftershock distribution within 5 days of the main event (Fig. 9). The location of the main shock at the origin of Fig. 9 is indicated
coordinates, indicates
rupture
than the
plane
for the Borrego
to define
Mountain
quake. Figure 8 shows the tendency
Borrcgo
alternative, the
earth-
of the off-fault
Aftershocks
iI 0
c
00
0
-i.s
Fig. 10. Borrego at 0.81 i length
Mountain
1y 11.51associated
I = 20 km. Diffusivity
aftershocks
-‘l
plotted
with positive c = 0.1 m* s-’
-6.5
in a distance-Coulomb
Coulomb used.
stress change
_l
4
00
015
1
stress space. Note the clear clustering of approximately
10% of stress drop
2
1.5
of off-fault
of main
shock.
aftershocks Half
fault
47
aftershocks the right
to fall into lateral
fault
also quite distinctive, approximately Coulomb
the dilatant motion.
quadrant
for
The clustering
and the clusters
in the location
Coulomb
is
aftershocks,
are located
of peak
induced
stress. This may be seen by comparing
Fig. 8 to Fig. 2a. To further
investigate
the correlation
between Random
u 000
as a function
as shown
obtained
using the following
in Fig.
10. These
i the earthquake
catalogue
triplet
y,, t,). After normalizing
from stress
plots
procedure:
aftershock (x,,
distance
of Coulomb
of
are
For each provides
Aftershocks
08 O _R
v-
”
B”
0
00
O”B
0
0
‘0
ts, O cs0
Too -0
.nn
0
0
O
0
!
0
00
00 0 0 01 0
80 0
0
o”oo
.
!
-2x104
-1 B 0
0
0
0 003
3” g
0 o0o rnc
0
$OO”
0
0
10’
0
0 %J 0 0
0 !
+..
-10’
0
0
-
“_ 80
au 0
.m.
oooo 0
0
n
!
.
q
4
@ .
2x104
Meters
(a)
Random
distributions
of aftershocks
with same number
Also I = 20 km and c = 1.0 m2 s-r. in the positively
case shown in Fig. IOa, b.
stressed
region
Aftershocks
of events (533) and cover same spatial
This random
catalogue
is seen for this random
produces
case, in contrast
area as for the Borrego
the distance-Coulomb to the distinct
a
with the half
-Oo
0
in (b). No clustering
the occurrence
we replot the aftershock
increase,
8
-
earthquake.
and
trace
0
Fig. 11. (a) Random
increase
the fault
0
Mountain
stress
stress plot shown
clustering
for the actual
48
rupture (x,/l,
length y,/l,
and
t/t,)
relaxation
time
t,, the triplet
may then be used as arguments
in eqn. (3) to calculate
the normalized
Coulomb
stress (u,/Aa),. Each circle in Fig. 10 represents a pair ( 1y 1,/I, (~,/Au)~). In other words, each aftershock in the catalogue is plotted at its absolute normalized
distance
from the rupture
stress. The tight clustering 0.1 - 0.15Au suggests stress
increase
and Anderson, In
order
the Coulomb stress computed for the event. The time distribution of the aftershocks is lost in this
pattern
representation;
aftershocks
however,
off-fault
clusters
are en-
Figure Mountain
that the required
for aftershock
at uc = Coulomb
occurrence
in this
region is 1.5 to 2.3 bar, based on an estimated stress drop of 15 bar for the main shock (Kanamori
versus
hanced.
of aftershocks
of
1975). to bring
out
the
subtleties
in Fig. 10, we used an artificial
spatially
uniform
random
of the catalogue
distribution
with the same number
of
of events in an
area of 50 km x 50 km which covers all of the real 10 shows
such pairs
aftershock
from
catalogue.
the Borrego
The
distinctive
features to look for in these plots are the association of off-fault aftershocks at about half a fault length distance on either side of the main rupture ( 1y I/l = 1) with
positive
increase
Aftershock
in Coulomb
aftershocks,
shown
in
Fig.
lla.
The
result
of
applying the same procedure of reducing these data to the u,/Au -y/I plane is shown in Fig. 1 I b. The right-ward opening half-funnel shape is an artifact of the increasing spatial occupation the positively stressed zone with distance y/l,
Aftershock
Migration
of as
Migration
x
0
Ldk??-i
0
0
Distance
Aftershock
Aftershock
Migration
i 0
Day 11-29
0
Fig. 12. Four “time-windows” the main rupture
of the Borrego
km
Day 30-91 0
0
of logarithmically Mountain
Rupture,
Migration
s 8 0
0
From Initial
equal time periods
earthquake.
showing
number
of earthquakes
as a function
of distance
from
49
seen in the stress contour the real catalogue fault
aftershocks
tive Coulomb
associating
much slip
These
to the details and
to
(Fig.
theory does
are
are presumably
breaking
of “barriers”
Hence
the aftershocks
investigate
the
with
the
as described
by
lying close to
time-dependence
of after-
shock occurrence, we checked to see if the fluid flow rate may be correlated directly with the time occurrence of aftershocks. One method of doing this is by varying the relaxation time t, (= r2/4c) through controlling the diffusivity c. If there is direct correlation, then the clustering effect shown in Fig. 10 will be optimized. Unfortunately the characteristic be sensitive
pattern enough
in Fig. 10 does not appear to to show major distinction
through two orders of magnitude change in c. This is apparently due to the small change in Coulomb stress magnitude condition,
the day after the main shock and persisted
for some time afterwards. enough number
If the catalogue
is long
(i.e., beyond 91 days) we expect that the of aftershocks everywhere will eventually
subside at later time periods. These observations are consistent with the time-dependent enhancement of Coulomb
stress by fluid diffusion.
Haicheng
earthquake
(February 1975)
of the seismic
associated
the horizontal axis (y = 0) in Figs. 10 and 11 should be disregarded for the present discussion. To
peared
on or very close to the
aftershocks
more sensitive distributions.
Aki (1984).
with posi-
catalogue
that our present
the aftershocks
rupture.
postseismic
distinctly
for the artificial
It should be noted main
itself with the off-
stress change (Fig. 10) in contrast
the distribution lib). not predict
plot in Fig. 2. However,
distinguishes
The Haicheng lateral
earthquake
strike-slip
best known
motion)
recent
(M, = 7.3 with left-
in China
great
ruptures
is one of the because
of the
successful short term prediction by Chinese scientists. The spatial pattern of 1239 (M > 3.0) aftershock events for 1421 days after the main rupture is shown in Fig. 13. The fault trace is obtained by a least square fit of aftershocks visually judged to be close to or on the fault. While surface
breakage
due to the Haicheng
earthquake
has been reported to run as much as 70 km long (Gu et al., 1976) the early aftershocks within 5 hrs after the main rupture occupy a length of only 60 km (Fig. 14). We have adopted the shorter estimate as the actual rupture length for this analysis. The fault length estimated from the aftershock zone
5 days after
from the undrained
to the drained
same
(60 km) although
and may also be related
to the inaccu-
occurred especially close to the main shock. This fault trace and rupture length is plotted as the
racies in aftershock locations. An alternative method to show the time-depen-
the main
rupture
many
more
remains events
the have
dotted line on Fig. 13. Note that the epicentre (white star in Fig. 13 and arrow in Fig. 14) of the main shock is not located at the middle of the
dent component of aftershock patterns is by means We divided the 91 day of “ time-windows”. aftershock catalogue from Hamilton (1972) into four logarithmically equal periods (day 1-3, day 4-10, day 11-29, and day 30-91). The number of aftershocks as a function of distance from the main rupture is plotted in Fig. 12. It is seen that there is a general decay in the number of
stress increase scribed earlier
aftershocks from the main rupture outwards. Two other features should be noted. A great number of
aftershocks, and the result is shown in Fig. 15. A cluster of aftershocks is seen at approximately one
aftershocks
half-rupture-length
occur with some time delay,
i.e., not
coseismically. In the third and fourth period (day 11-29 and day 30-91) the number of events both near the main rupture and particularly off-fault (24-32 km from main-rupture) actually increases. Indeed Hamilton (1972) and Allen and Nordquist (1972) have shown that the off-fault clusters ap-
assumed rupture (and there is no reason to do so). It is shown in Fig. 13 that the off-fault aftershocks are again skewed towards the dilatant quadrant. The procedure for correlating Coulomb and aftershock occurrence dewas used to study the Haicheng
away
accompanied
by
a
Coulomb stress increase of about 10 to 15% of the stress drop. This amounts to CI== 5 to 8 bar necessary for triggering the off-fault aftershocks, for an estimated stress drop of 53 bar (Cipar, 1979). To show the time dependence of aftershock occurrence, the 333 days after the main rupture
50
Hoicheng
1 I.
.
.
I
.
.
’
Aftershocks
1
-.
I
”
.
’
, 1 --
I
.
.
.
.
I
1121.5
SOKM
0
0
.
.
.
.
t
.
.
I.
.
123
122.5
122
!,
I..
L
123.5
Longitude
Fig. 13. Aftershock
distributions
of the Haicheng
earthquake.
Note
the off-fault
clusters
of aftershocks
shifted
into the dilatant
quadrants Hoicheng
Aftershocks
I
I
I
0
1
0
k
0
0
0
0
0
Q1
I
k---f--
I
METERS Fig. 14. Early aftershocks the basis for choosing
of the Haicheng
the half fault length
earthquake
in rotated
coordinates,
I = 30 km for this earthquake.
5 hrs after main shock located
at origin (arrow).
This is
51
Halcheng cy
Aftershocks
,‘.“!‘.‘.I‘..’
.~‘~l’~.‘l~‘.~~~~‘~
3 --* -
0 It
Fig. 15. Haicheng associated
aftershocks
with positive
are divided
a.5
0
-<
plotted
Coulomb
in a distance-Coulomb
stress change
into four logarithmically
stress space. Note the clear clustering
of approximately
equal periods
1.5
1
of aftershocks
10 to 15% of main shock stress drop.
present.
These general
at 0.81<
1y 1 < 1.41
I = 30 km, c = 10 m2 s-‘.
findings
are consistent
with
(day 1-4, day 5-18, day 19-78 and day 79-333) and the number of aftershocks are shown in Fig. 16 as a function of distance from the main rup-
observations of aftershock distributions. Three earthquakes having distinctive aftershock patterns are shown. They are the Pasinler, the
ture. As for the Borrego Mountain earthquake, the off-fault aftershocks appear to show up distinctly in the third period (day 19-78) after some time
Borrego Mountain and the Haicheng earthquakes. The latter two are analyzed in a bit more detail. Tight clustering of off-fault aftershocks where
delay.
Coulomb stress increases are seen in a new plot which merges theoretical calculations with
Concluding discussions
aftershock
data. The Borrego
shown to require A model
is presented
of postseismic
stress and
pore pressure readjustment in time and space. The model is based on a theory by Rice and Cleary (1976) of coupled deformation-diffusion process in a linear poro-elastic medium subjected to sudden loading. In the present study, the sudden loading is associated with a seismic rupture modelled as a uniform stress drop on the rupture surface, resulting in a continuous distribution of dislocations. These imposed dislocations set up a stress and pore pressure pute the time-dependent
field from which we comCoulomb stress. The con-
an increase
Mountain in Coulomb
region
is
stress of
2 bars to trigger the off-fault aftershocks, while the Haicheng earthquake apparently requires 5-8 bars. While the time-dependence of fluid flow provides a plausible explanation for the delay of aftershocks, the results presented have not been able to correlate them irrefutably. Even so, the “time window” plots (Fig. 12 and Fig. 16) show that many aftershocks occur with some time delay and that clusters may form subsequent to and off the main fault. As an additional example, Hutton et al. (1980) noted that aftershocks off the main fault
emerged
as clusters
6 hrs after
the main
is made by increase with
shock of the 1979 Homestead Valley earthquake. A purely elastic rheology cannot explain such
As general results, we found that the aftershock zone expands with time in the dilating quadrant. Depending on initial stress field (before rupture), rupture magnitude and length, and material parameters, off-fault clusters may or may not be
time-transient phenomenon. The time and spatial distribution of aftershocks generated by postseismic stress and pore pressure readjustments due to fluid flow may be consistent with such observed aftershock patterns. Slight modifications of the calculated stress
nection to aftershock distributions associating high Coulomb stress aftershock
occurrence.
Aftershock
Fig. 16. Four “time-windows” the main rupture
Migration
Aftershock
of logarithmically
for the Haicheng
equal time periods
showing
number
Migration
of aftershocks
as a function
of distance
from
‘earthquake.
fields are possible by assuming an impermeable fault, as was recently suggested by Rudnicki (1985). Also, variations in the coefficient of friction would change the computed Furthermore, direct comparisons
stress contours. with field obser-
vations are hampered by complexities in the slip distributions of the main rupture, heterogeneity in the pre-existing stress field, and existence of neighboring branch faults, as in the case of the Borrego Mountain earthquake (Fig. 8).
States Geological Survey and the Exxon Education Foundation is gratefully acknowledged. APPENDIX
A
Numerical implementation
of eqn. (2)
Based on the slip distribution mulated slip within an element length
of the rupture
&3(x’) -___
dx,
is:
2(1-
=
3X'
(1) the accudx’ along the
v")Au G
x’
dx’
JS
Acknowledgements
The authors would like to thank D. Veneziano and J. Rudnicki for helpful discussions. Support from the National Science Foundation, the United
(‘41) In normalized (I
‘J_
A0
-
form, eqn. (2) may be written
20-v”) G
/
1 G,,(F-T', -1
j,
i)
as:
53
Y, i>
P(%
A0 P -= AU
2(1-v,)
1
G
/ -1
X
(KS
where X = x/l,
(AZ)
G,(X-X’,
2(1 i vu) 3 F Gp(X - Xk,
=
j, 7)
i)Xk
j,
k=l
(A5)
dx’
j = y/l,
i = ct/12,and from Rice
and Cleary (1976):
where
N = 1,. . , N - 1,
X, = cos(an/N),
cos x(2k - 1)/2N, number
k = l,...,
of collocation
and
chosen
Equation
N
is the
to the degree
of accuracy
required.
to compute
the Figs. 2 to 6. The numerical
used for the material
277r( 1 - V”)(l - V)
N,
points
Xk =
(A5) has been used
constants
values
introduced
in eqn.
(A3) are v = 0.2, vu = 0.33 and n = 0.3. These values also give B = 0.61.
i
sin 0 2 exp( --r*/4ct) i
x [l - exp(
’
l-v 4ct - - _ v, - v r2
References Aki, K.. 1984. Asperities,
-r2/4ct)])
quakes.
l-v x ( cos e --F[l-exp(-r2/4cr)l) L ”
)
J. Geophys.
Allen, C.R. and Nordquist, and late aftershocks in “The
sin 8 y
[l - exp( -r*2/4ct)]
i
{sin 8n-‘[l
- ei ”
Geol.
Borrego
Surv.
I (A3)
and 0 = tan-‘(y/x)
and n =
V”)(l - v)]. 3(V” - v)/[2B(l+ To obtain the stress components in a rectangular coordinate system, the following transformation rules may be used: G 1.X
G,,
=
i G,, :I
cos%
sin28
- 2 sin e cos e
sin*B
sin28
2 sin e cos e
sin e cos e
-sin e cos e
Chinnery, Cipar,
caused
~,,(X”,
Y, t>
by
Office,
The
faulting
of
Res.. 79: 2037-2044
stress
changes
that
accompany
Bull. Seismol. Sot. Am., 53: 921-932. processes
of the
observations C.H.,
shear
Haicheng,
of p and
1981. Off-fault
stress?
F. and Gupta. integral
M.B..
Kudreshova,
Bull.
China
s waves.
aftershock
Seismol.
Sot.
Bull.
clusters Am..
H-D.
Chen.
February Hamilton,
G.D., 1972. On the numerical equations,
71:
Osokina, results).
D.N..
Y-T,
Gao.
and
Study of Seismic Regime,
X-L
and
Liaoming
R.B., 1972. Aftershocks
Mountain
A.A.
and earthquake
(in Russian).
4. 1975. Acta Geophys.
earthquake
Lomakin. ruptures
Regional
pp. 113-124
of Haicheng,
solution
Q. Appl. Math., 29: 5255534.
B.B., 1974. Stresses,
Stinitz, Kishniev. mechanism
term in the
Printing
1669-1675.
Gu.
singular
Gov.
strain following
J. Geophys.
Source from
Das, S. and Scholz,
(A4)
- x”)
earthquake
of April 9. 1968”.
Seismol. Sot. Am., 69 (6): 190331916.
Gzovsky,
The Gauss-Chebyshev integration scheme (Erdogan and Gupta, 1972) is most suitable for solving eqn. (A2) as it takes advantage of the presence of the l/$(1 integrand. Hence:
J., 1979.
of singular
: GrB I
U.S.
1963.
foci (modelling
Go,,
Pap.
faulting.
earthquake
GV x
Earthquake
787,
main shock
Mountain
Mountain
medium. M.A.,
Erdogan,
c0s2e - sin20 1
J.M., 1972. Foreshock.
Booker, J.R., 1974. Time dependent a porous
of earth-
D.C.. pp. 16-23.
strike-slip
where r = \lm
and characteristics
of the Borrego
Prof.
Washington,
- ex p( --*2/4ct)l
barriers
Res., 89: 5867-5872.
Yi. Z.,
Province. Sinica,
1976.
Focal
earthquake
of
19 (4): 270-285.
of the Borrego
Mountain
from April 12 to June 12, 1968, in the Borrego earthquake
of April
Pap. 787, U.S. Gov. Printing
9, 1968. Geol.
Office, Washington,
Surv. Prof. D.C., pp.
31-54. Hull,
S., 1983. The mechanics
of aftershock.
R83-6, Dep. of Civil Engineering, Hutton,
L.K., Johnson,
C.E., Pechmann,
J.W., Cole. D.H. and Gernan, tions for the Homestead
110-114.
No. Mass.
J.C.. Ebel, J.E., Gwen,
P.T., 1980. Epicentral
Valley earthquake
15. 1979. Calif. Geol.. May:
Res. Rep.
M.I.T.. Cambridge,
sequence,
locaMarch
54
Kanamori,
H. and Anderson,
some empirical
relations
D.L., 1975. Theoretical in seismology.
basis of
Bull. Seismol.
Sot.
of rock masses. Nur,
R. and
Estimation J.R.,
pore fluid flow? Science, Omori,
F., 1894. Investigation
Comm..
2: 103-139
Rice, J.R. and Cleary,
diffusitity
122: 545-559.
1972. Aftershocks
caused
by the
of aftershocks.
Rep. Earth
Inv.
M., 1976. Some basic
for fluid-saturated
pressible
constituents.
elastic porous
Rev. Geophys.
stress diffusion media with com-
Space
Phys.,
14 (2):
1985.
Coupled
accompanying
sequence,
postseismic
deformations.
M.N.,
1984.
Earth,
Atmosphere
bridge,
Mass.
Utsu.
control
J. Geophys.
Seismicity Rep., Earth and
5th
New York.
California:
T., 1969. Aftershocks
Fat. Sci.. Hokkaido
diffusion fault.
M., 1983. The 1979 Homestead
earthquake
studies in Turkey.
deformation-fluid
slip on an impermeable
Ewing Symposium,
Stein, R. and Lisowski,
Toksoz,
175: 885-887.
(in Japanese).
solutions 227-241.
of in-situ hydraulic
Pure Appl. Geophys.,
Booker,
J.W.,
effects Maurice
Am., 65: 1073-1095. Li, V.C., 1984/1985.
Rudnicki,
and
and
Valley and
Res.. 88: 6477-6490. earthquake
Resources
Planetary
of aftershocks
Sciences,
earthquake
prediction
Laboratory, M.I.T., statistics
Univ., Jpn.. 3 (3): 129-195.
Dep. Cam(I). J.
144 (1987) 37-54
Elsevier Science Publishers
37
B.V., Amsterdam
Postseismic
VICTOR
in The Netherlands
stress and pore pressure readjustment and aftershock distributions C. LI *, SANDRA
’ Department ’ Department
- Printed
of Ciwl Engineering,
of Earth, Atmosphere (Received
H. SEALE
January
’ and TIANQING
M. I. T., Cambridge,
and PlanetaT
2
MA 02139 (U.S.A.)
Science, M.I. T., Camhrrdge,
14. 1986; revised version
CA0
accepted
October
MA 02139 (U.S.A.)
10. 1986)
Abstract Li. V.C., Seale, S.H. and Cao, T.. 1987. Postseismic In: R.L. Wesson
(Editor),
The time and spatial
Mechanics
readjustment
stress and pore pressure
of Earthquake
Faulting.
of stress and pore pressure
of a model which treats the earth’s crust as a linear poro-elastic fundamental
solution
compressible.
The main
maintained
Rice
and
is modeled
from regions
Cleary
who
treated
as a sudden
uniform
introduced
dislocation
distribution
undergoing
compression
It is assumed pressure
that aftershocks
in normal
increase)
of spatial
exceeds the shear strength
distribution
of aftershocks
aftershock
zone
aftershock
-zones for off-fault
provides
from
the compression
a means
into the dilation
aftershocks
to study
appear
the association
the resulting
undergoing
The coseismic
at the two ends
high
as separately
slip dislocation
Hence
Coulomb
the stress and pore
coefficient,
the suggestion
a small rotation
regions. increase
6, and 7 are
and p is the pore
confirms
predicts
of the main
stress
is
fields. Diffusion
of the rock medium.
elastic solution solution
by means
is based on a
as ec = r + ~(0” + p), where
and shift into the dilating
between
constituents
dilation.
p is a friction
The transient
region
to expand
fluid
sets up stress and pore pressure
slip surface,
Lisowski.
are investigated
and
on the flow parameters
of the slip surface.
by Stein and
rupture
and
drop
stress (defined
stress and shear stress across a potential
distributions.
by fluids. The analysis
the solid
to regions
occur when the Coulomb
infiltrated
stress
and aftershock
144: 37-54.
due to a strike-slip material
fields evolve over time, with time scales which depend
the increase
which
by
shock
in time. This suddenly
of fluid takes place pressure
obtained
readjustment
Tectonophysics,
rupture.
of the
whereas
the
A new plot is developed with off-fault
aftershock
locations. The aftershock are compared
distributions
of three earthquakes
to the zones predicted
and in some cases form distinct provide
a primary
in the contexts
aftershock
zones and suggests
inaccuracy
in location
confirm
or disprove
off-fault
mechanism
of these sample
aftershock
clusters
in the areas
and the implications
earthquakes.
an explanation
of aftershocks the theory
(Pasinler,
1983: Borrego
by the model. The earthquake
The model
that aftershock
distributions
Mountain,
of high Coulomb
predicts change
stress.
in predicted
are controlled
observed
of off-fault
the dilatant
The ability
for material
the generally
or absence
1968: and Haicheng.
are skewed towards
of this mechanism
for the presence
and the small magnitude
aftershocks
Coulomb
of the model
properties spatial
aftershock
1975) regions
expansion
clusters.
to
are discussed of the
However,
stress makes it difficult
the to
by fluid flow.
Introduction
these patterns
It has been observed that for some earthquakes with dominant strike-slip focal mechanism, the spatial distribution of aftershocks has distinct patterns when the aftershocks can be located accurately. Some commonly observed characteristics of
aftershock zones beyond the ends of the main rupture. In addition, clusters of aftershocks are sometimes located off the fault plane and skewed towards different ends of the rupture on either side of the fault (the dilatant quadrant). Examples of off-fault aftershock patterns include the
0040.1951/87/$03,50
B 1987 Elsevier Science Publishers
B.V.
include
the linear
extension
of the
38
Homestead
Valley
earthquake
discussed
by Stein
range
from days
and Lisowski (1983) and Das and Scholz (1981)and
least
for some
the Borrego
nism
may
Mountain
and Managua
earthquakes
discussed
by Das and Scholz (1981). Although
temporal
development
been
of such patterns
well identified,
aftershocks
a large
the
has not
proportion
of the
do occur with some time delay.
Attempts
at explaining
the spatial
distributions
of aftershocks
have
made
several
re-
Das
and
searchers.
Gzovsky
been
et al. (1974)
Scholz (1981) analyzed a plane that
strain
Mode
at distances
by and
the stress field induced II (shear)
about
crack
one rupture
and length
by
found per-
pendicular to the fault the shear stress increased by about 10% of the stress drop on the crack. Das and Scholz associate
such stress change
with off-
fault aftershock clusters. Based on work by Chinnery (1963), Stein and Lisowski (1983) analyzed the Coulomb stress induced by the Homestead Valley earthquake, which they modelled as vertical dislocation patches extending from the ground surface to 5 km depth. The Coulomb stresses were calculated on the ground (free) surface and included the effect of friction proportional to induced normal stresses. This causes the Coulomb stress to shift into the dilatant quadrant. The references cited above are based on elastic analyses, with the implication that the stresses are induced coseismically (on the time scale of elastic wave travel time from the rupture source to the aftershock site). In reality, however, a great proportion of aftershocks have delay times of days, weeks or months depending on the magnitude of the main shock. In this paper. we propose a simple model which incorporates many of the essential ideas of Das and Scholz (1981) and of Stein and Lisowski (1983) and also includes a time component. The time source is due to the diffusion process of water induced by a pressure disequilibrium set up by the main rupture. The diffusion process allows water to flow from the high compression region to the dilatant region, causing a time-dependent stress change. The result is a gradual enhancement of the Coulomb shear stress and also an enlargement of the area of high Coulomb shear stress in the dilating quadrant. The time scale of the flow process depends on the rupture length and diffusivity of the rock mass and may
aftershocks patterns
to months. earthquake
be responsible
and full development
as mentioned
The suggestion
Booker
and
out that the diffusion
may
such
of
of their spatial
was originally (1972).
process
They
produces
root time decay in pressure
to the aftershock nisms
delay
that water flow may be respon-
posed
Here we hasten
at
earlier. generation
by Nur
that
this mecha-
for the time
sible for aftershock
square
We suggest ruptures
pro-
pointed
an inverse
change
similar
event decay law of Omori (1894). to add that other physical
as stress
also operate
corrosion
simultaneously,
mecha-
or viscoelasticity although
they
may have quite different time scales. Hull (1983) gave a thorough review of various plausible aftershock
mechanisms.
Perhaps the most complete analysis of pore fluid flow as a mechanism for aftershock generation has been performed by Booker (1974). who solved for the time-dependent stress field induced by a plane fracture in an infinite poro-elastic body. He investigated the case of a uniform slip distribution modelled by two suddenly applied edge dislocations equal in magnitude but opposite in sign; and the material was assumed to have incompressible constituents. The primary concern of Booker’s work was to explain aftershocks on the rupture plane induced by the reloading of the fault. Calculations of time-dependent stresses and pore pressure for a slip distribution consisting of two edge dislocations (constant slip) using the fundamental solutions of Rice and Cleary (1976) were performed by Hull (1983). In this case the material was assumed to have compressible constituents. The primary differences between the results of this work and those of Booker (1974) are the values of the coseismic stresses and the length of time required to reach steady state. The model employed here is based on representing the main shock as a suddenly introduced two-dimensional continuous distribution of shear edge dislocations in an infinite poro-elastic medium. As in Das and Scholz (1981), the slip distribution on the main rupture is chosen as that due to a crack, with maximum slip in the middle and tapering off towards the ends of the rupture. Our analysis is based on a fundamental solution
39
due to Rice and fluid
and
pressible.
solid
Cleary
(1976)
constituents
who treated
as separately
While we have not evaluated
of improvements material
adopted
in the present
they may nevertheless
be expected
reality
to uniform
(in comparison
pressible
are,
however,
For example,
strain
1
comand
the
model,
to be closer to slip and incom-
constituents).
There model.
r-
the degree
with the slip geometry
behavior
the
deformation
to reality, reasonable
many
limitations
in the
the two-dimensional
plane
can only be an approximation
although when
this the
approximation
fault
assumed smooth and constant bution on the main rupture with observations
width
may
is large.
be The
the slip distribution In the following,
for locations
assumed
of fault slip; slip distrlbutlon Coulomb
off the fault,
may be sensible.
we shall briefly
describe
ulus
and
linear bution
the
y) and pore pressure
of the 1983 Pasinler Mt. earthquake and be discussed in the with the latter two
a,,(~, y. t)=
earthquake, the 1968 Borrego the Haicheng earthquake will context of the model results, in some detail. The presenta-
tion of data for these three earthquakes is meant to illustrate some of the ideas suggested by the theoretical model. Lack of accuracy in earthquake data (such as fault length) and in material parameters precludes a strong verification of the theoretical predictions. We have also not oughly the sensitivity of theoretical parameters
studied thorresults to the
used in the model.
Poisson
the time and
of the induced
are:
material
6(y)
given by
and pore pressure
V, is the undrained
superposition,
theoretical model and discuss some numerical results based on this model. The aftershock patterns
various
stress q
p.
(in time) slip distriis also inconsistent
in surface breaks and aftershocks
on the fault. However,
Fig. 1. Geometry eqn. (1) induces
stresses
ratio.
spatial
By
distri-
u,, with (i, j = x,
p due to the main
rupture
atqx’)
y. r)rdx’
-j-;rG,,(x-x'.
and:
p(x,
t) = -/;,G&-x'.
4‘.
where rupture
we have
already
x, r)%&$dx’
assumed
that
occurs at time zero. The Green’s
the main function
_Y, r) and G,(x - x’, J’, t) represents
G,,(x -x’, the stress and pressure change at a point (x, J’) and at time t due to a unit dislocation suddenly introduced
at (x’, 0) and at time zero in a poro-
Model description and results
elastic medium. Such fundamental solutions have been previously obtained by Rice and Cleary (1976) and they have been included in Appendix
The model is based on representing the main shock as a suddenly introduced two-dimensional continuous distribution of shear edge dislocations
A. The factor (as( x’)/ax’) dx’ may be interpreted as the accumulated slip within an element dx’
in an infinite poro-elastic medium (Fig. 1). The slip magnitude 6(x) is determined from elastic crack theory in which a uniform stress drop Au is
ual stress components and pressure may be computed using eqns. (1) and (2), and the numerical implementation is described in Appendix A. Following Stein and Lisowski (1983) we assume that the occurrence of aftershocks is associated with the increase of Coulomb stress, defined as:
imposed S(x)
=
along the length of the rupture: 2(L -;)A0
dm
(1)
where Au is the main rupture stress drop, I is the half-length of rupture. G is the elastic shear mod-
along the length of the rupture.
u,=u,,
+/J(u,.+p)
Hence the individ-
(3)
L
Fig. 2. a. Contours about
origin.
of Coulomb
b. Contours
stress at t = 0: shown
of Coulomb
atress
only for J > 0. Stress contours
for 1’ i 0 can be obtained
by 180°
rotation
I --* m
where ~1 is a coefficient of friction (chosen as 0.75 for the following discussion) and the pore pressure p has been included to reflect its influence on
I).Contours are not shown for I: < O.l/ for lack of numerical accuracy. Due to symmetry, we have chosen to show only the space y > 0. Increase of
reducing frictional resistance to sliding. Of course u, ,,, (I,. and p are all time dependent functions given by eqn. (2) although it might be expected that the time change of pore pressure p would have the strongest effect on a,. The association of increase a, as defined in eqn. (3) with aftershock
Coulomb stress is indicated by the solid contours and decrease of Coulomb stress is indicated by the dashed contours. Several characteristics in this
generation implicitly assumes that the aftershock focal mechanisms are similar to that of the main rupture.
This may explain the observation of aftershocks being distributed along a zone longer than the main rupture. The off-fault peak identified (as
In Fig. 2a, we show a contour plot of coseismic Coulomb stress change (normalized by the stress drop) at t = 0, immediately after the main rupture, due to a right-lateral rupture lying along the x-axis between - 1 and 1 (all length dimensions are normalized with respect to the half-rupture length
symmetrical) by Das and Scholz (1981) has been rotated into the dilatant zone and is located at about J = 1.21 from the main rupture, at an angle of 25” to the y-axis (point D). The Coulomb stress increase there attains a maximum of 15% of the stress drop. Interestingly, another peak exists
figure are worth mentioning. Along the fault, a very large Coulomb stress increase exists beyond the rupture zone at both ends as may be expected.
41
(b)
Fig. 2 (continued).
close
to the dilatant
(point stress
A). Between these two peaks, the Coulomb is continuously enhanced with time. (The
locations
of peaks
end
that
of the
occur
near
main
rupture
the plane
of
rupture depend on the particular slip distribution and thus may vary between earthquakes.) It might be expected that aftershock development will be particularly active in this sector. The long term enhanced Coulomb stress is shown in Fig. 2b. After sufficient time the off-fault peak merges with the high stress zone at the end of the fault. To understand
this behavior,
we show
in Fig. 3 the coseismic spatial distribution of pore pressure. The negative pore pressure in the dilatant zone will induce flow from the compression zone and sets up a diffusion process. Of course the induced pore pressure vanishes in the long term equilibrium state. The pore pressure change at
four locations
(labelled
plotted as a function been normalized by
A, B, C, D in Fig. 2) is of time (Fig. 4). Time has a characteristic relaxation
time, defined by t, = 12/4c where c is the diffusivity of the rock mass. The pore pressure rises with time from an initial negative value to zero in this fault quadrant. The locations close to the fault end (e.g. location A) have shorter delay time before pressure sponding
picks up. Figure 5 shows the correchange in Coulomb stress with time is
greater the closer the location is to the dilatant fault end (A, B, C, D in decreasing order). This suggests an apparent expansion of aftershock zones (Utsu, 1969) particularly if an off-fault cluster of small quakes is absent or not recognized as aftershocks of the main rupture. The Coulomb stress along the line A-D is shown in Fig. 6 at various fractions of the characteristic time. Again
-----.\
*,
-.04 .\ ‘\\ \ -.06
A/---
--“I
-\
-
-1
+1
Fig. 3. Pore pressure
0.00
,-0.4Y5 (1 $0.10
I
I
II c
t&j-0. 15 $0.20
-
F-0.2;
-
g-c.“”
.-
“-0.35
-
-0.40
at time r = 0.
B
-
-CT.45 ,
i) Li? - ‘1
-G
/ _;
A -‘ 1 loglo
Fig. 4. Pore pressure
vs. time at locations
I -3
1’
/
-2
I - !
(4ct/&2) A. B. C. and D shown
in Fig. 2a, b
D
1
43
,
I
I
I
lo’;;10 (4Ct/&!l Fig. 5. Coulomb
stress vs. time
Q,. 0. 0. 0.
0. 0. 0. 0. 0.
KORIIALIZED DISTAFICE FROM FAULT EbD ALOllG LINE A - E Fig. 6. Coulomb
stress vs. distance
along
A-E.
(0.0) to very long time scale (co). The dotted pre-stress
states. See text for more details.
The four curves
and dashed
show fluid pressure-deformation
lines indicate
effects on spatial
distribution
coupling
effects
of aftershocks
from coseismic for two different
44
the off-fault
peak at D is clearly
(and
is still maintained
seen
that
most
between
shown
at f = O.lt,).
of the Coulomb
A and D occurs within
time. For example, km, and diffusivity
for a rupture range
at t = 0
It may
stress
be
increase
one characteristic length
In the following patterns
quakes in the context
sections,
of several
we discuss
strike-slip
of these theoretical
(see, e.g. Li, 1984/1985), the characteristic time is between i to 50 yrs. If the larger diffusivity ap-
consider-
ations.
zones may
Pasinler earthquake (October 1983) The Pasinler center
earthquake
40.3” N and
event which occurred
(M, = 7.1 (NEIS);
42.2” E (USGS)) in northeast
Turkey.
It has a
well-constrained
diffusivity
of strike N42OE dip 80” which is consistent
hundred
extend
these time lengths
by a
fold.
Figure
6 offers another,
interpretation
perhaps
of aftershock
more general,
patterns.
Suppose
the
fault plane
solution
epi-
is a recent
be fully developed within half a month (O.lt,) to half a year after the main rupture. The smaller would
the
earth-
of 21= 50
of 10’ to lo5 cm2 s-’
plies, it may be seen that the aftershock
different. aftershock
with attitude with
the trend of observed surface faulting and aftershock patterns (Toksoz, 1984). The aftershock distribution shown
of 10 days after
in Fig. 7. The data
the main
rupture
were collected
is
by a
pre-rupture stress state and the material properties of the rock mass in the vicinity of the main
network
rupture are such that a 20% of Au increase in Coulomb stress is required for aftershock occur-
instruments deployed is much larger than
rence (indicated by the horizontal dashed line), then no off-fault aftershock clusters will be observed, and an expansion of the aftershock zone will be recorded. Aftershocks will be confined to about 0.11 immediately after the rupture and gradually expanded to about 0.61 (measured along the line A-E). If, however, the required minimum in Coulomb stress is 10% of Au, then immediately after the main rupture, near the fault end, the
where faulting was observed. Because the lengths of the surface faulting and the observed surface displacements (10 to 80 cm) are too small to represent the complete faulting of an earthquake of magnitude M, = 7.1, and because the most prominent fault break falls to one side of the
aftershocks
are confined
to about
0.21 and simul-
taneously an off-fault cluster occurs between 0.71 and 1.71. With time (say t = O.lt,) the near fault
of eight portable
analog
and three digital
locally. The aftershock zone the 10 to 12 km long zone
maximum intensity contour, Toksoz (1984) suggested that the primary slip may have occurred at depth. We have placed the fault line visually with respect to the aftershock gated isoseismals. Fault
locations and the elonmotion is left-lateral as
aftershock zone expands outwards. At one characteristic time, the expanded zone merges with the
indicated by the arrows. Admittedly the fault length shown in Fig. 7 is at best dubious and for this reason, we have not carried out detailed anal-
off-fault
ysis
clusters.
From the above discussion, it should be clear that the stress state prior to the main rupture, the material properties (particularly fracture properties). and the main shock stress drop dictate the presence or absence of off-fault clusters, although in both cases, aftershock zone expansion (with time) may be observed. Furthermore, the area1 size and location of the off-fault clusters depend on the rupture length, while the time scale of expansion and merging with the off-fault cluster depend on the diffusivity of the rock mass. Because these dependent parameters can be different from earthquake to earthquake, the exact temporal and spatial pattern of aftershocks may also look quite
for
this
case.
However.
two
loops
of
aftershocks can be seen to skew towards the dilatant quadrants (compare with Fig. 2). There is no distinctive clustering at an off-fault location, and this may be indicative of the case of required (for aftershock occurrence) Coulomb stress alteration exceeding the induced off-fault peak Coulomb reference
stress alteration, as described to the dashed line in Fig. 6.
earlier
in
Borrego Mountain earthquake (April 1968) The main shock of the 1968 Borrego Mountain earthquake is shown in Fig. 8, together with the spatial distribution of 533 aftershocks for 91 days
45 40.6
40.5
40.4
~
40.3
z ;: 4
40.2
/
Narman
/
..
\
: .,z:\.
40.1
40
39.9 LONGITUDE
Fig. 7. Aftershock
distributions
Note the concentration
of the Pasinler
of aftershocks
earthquake
in the dilatant
(after
ToksGz. 1984). Symbol
star indicates
location
of the main shock.
quadrants
15’
30’
116”OO’
45’ I-
\
j’-
‘..
Q
25 KM
0 -1.
s,
I
“..,,
X”
i’LONGITUDE Fig. 8. Aftershock the dilatant
distributions
quadrants.
of the Borrego
Mountain
earthquake
(after
Hamilton,
1972). Note the off-fault
clusters
shifted
into
Bon-ego
Aftershocks
METERS Fig. 9. Early aftershocks the basis for choosing
of the Borrego
Mountain
the half fault length,
earthquake
in rotated
I = 20 km. Arrow
at origin
after the main rupture (from Hamilton, 1972). The length of rupture (dashed line in Fig. 8) is esti-
chosen
rupture
length
tion
is
of main shock.
zone ruptures..
Without
a better
we have used the 5 day aftershocks
with an arrow. The
of 40 km is longer
5 days after the main shock. This distribution
location
observed surface break of 31 km (Fig. 8). The early-day aftershocks defining the rupture zone is a generally accepted notion, especially for subduc-
mated to be 40 km from the extent of the aftershock distribution within 5 days of the main event (Fig. 9). The location of the main shock at the origin of Fig. 9 is indicated
coordinates, indicates
rupture
than the
plane
for the Borrego
to define
Mountain
quake. Figure 8 shows the tendency
Borrcgo
alternative, the
earth-
of the off-fault
Aftershocks
iI 0
c
00
0
-i.s
Fig. 10. Borrego at 0.81 i length
Mountain
1y 11.51associated
I = 20 km. Diffusivity
aftershocks
-‘l
plotted
with positive c = 0.1 m* s-’
-6.5
in a distance-Coulomb
Coulomb used.
stress change
_l
4
00
015
1
stress space. Note the clear clustering of approximately
10% of stress drop
2
1.5
of off-fault
of main
shock.
aftershocks Half
fault
47
aftershocks the right
to fall into lateral
fault
also quite distinctive, approximately Coulomb
the dilatant motion.
quadrant
for
The clustering
and the clusters
in the location
Coulomb
is
aftershocks,
are located
of peak
induced
stress. This may be seen by comparing
Fig. 8 to Fig. 2a. To further
investigate
the correlation
between Random
u 000
as a function
as shown
obtained
using the following
in Fig.
10. These
i the earthquake
catalogue
triplet
y,, t,). After normalizing
from stress
plots
procedure:
aftershock (x,,
distance
of Coulomb
of
are
For each provides
Aftershocks
08 O _R
v-
”
B”
0
00
O”B
0
0
‘0
ts, O cs0
Too -0
.nn
0
0
O
0
!
0
00
00 0 0 01 0
80 0
0
o”oo
.
!
-2x104
-1 B 0
0
0
0 003
3” g
0 o0o rnc
0
$OO”
0
0
10’
0
0 %J 0 0
0 !
+..
-10’
0
0
-
“_ 80
au 0
.m.
oooo 0
0
n
!
.
q
4
@ .
2x104
Meters
(a)
Random
distributions
of aftershocks
with same number
Also I = 20 km and c = 1.0 m2 s-r. in the positively
case shown in Fig. IOa, b.
stressed
region
Aftershocks
of events (533) and cover same spatial
This random
catalogue
is seen for this random
produces
case, in contrast
area as for the Borrego
the distance-Coulomb to the distinct
a
with the half
-Oo
0
in (b). No clustering
the occurrence
we replot the aftershock
increase,
8
-
earthquake.
and
trace
0
Fig. 11. (a) Random
increase
the fault
0
Mountain
stress
stress plot shown
clustering
for the actual
48
rupture (x,/l,
length y,/l,
and
t/t,)
relaxation
time
t,, the triplet
may then be used as arguments
in eqn. (3) to calculate
the normalized
Coulomb
stress (u,/Aa),. Each circle in Fig. 10 represents a pair ( 1y 1,/I, (~,/Au)~). In other words, each aftershock in the catalogue is plotted at its absolute normalized
distance
from the rupture
stress. The tight clustering 0.1 - 0.15Au suggests stress
increase
and Anderson, In
order
the Coulomb stress computed for the event. The time distribution of the aftershocks is lost in this
pattern
representation;
aftershocks
however,
off-fault
clusters
are en-
Figure Mountain
that the required
for aftershock
at uc = Coulomb
occurrence
in this
region is 1.5 to 2.3 bar, based on an estimated stress drop of 15 bar for the main shock (Kanamori
versus
hanced.
of aftershocks
of
1975). to bring
out
the
subtleties
in Fig. 10, we used an artificial
spatially
uniform
random
of the catalogue
distribution
with the same number
of
of events in an
area of 50 km x 50 km which covers all of the real 10 shows
such pairs
aftershock
from
catalogue.
the Borrego
The
distinctive
features to look for in these plots are the association of off-fault aftershocks at about half a fault length distance on either side of the main rupture ( 1y I/l = 1) with
positive
increase
Aftershock
in Coulomb
aftershocks,
shown
in
Fig.
lla.
The
result
of
applying the same procedure of reducing these data to the u,/Au -y/I plane is shown in Fig. 1 I b. The right-ward opening half-funnel shape is an artifact of the increasing spatial occupation the positively stressed zone with distance y/l,
Aftershock
Migration
of as
Migration
x
0
Ldk??-i
0
0
Distance
Aftershock
Aftershock
Migration
i 0
Day 11-29
0
Fig. 12. Four “time-windows” the main rupture
of the Borrego
km
Day 30-91 0
0
of logarithmically Mountain
Rupture,
Migration
s 8 0
0
From Initial
equal time periods
earthquake.
showing
number
of earthquakes
as a function
of distance
from
49
seen in the stress contour the real catalogue fault
aftershocks
tive Coulomb
associating
much slip
These
to the details and
to
(Fig.
theory does
are
are presumably
breaking
of “barriers”
Hence
the aftershocks
investigate
the
with
the
as described
by
lying close to
time-dependence
of after-
shock occurrence, we checked to see if the fluid flow rate may be correlated directly with the time occurrence of aftershocks. One method of doing this is by varying the relaxation time t, (= r2/4c) through controlling the diffusivity c. If there is direct correlation, then the clustering effect shown in Fig. 10 will be optimized. Unfortunately the characteristic be sensitive
pattern enough
in Fig. 10 does not appear to to show major distinction
through two orders of magnitude change in c. This is apparently due to the small change in Coulomb stress magnitude condition,
the day after the main shock and persisted
for some time afterwards. enough number
If the catalogue
is long
(i.e., beyond 91 days) we expect that the of aftershocks everywhere will eventually
subside at later time periods. These observations are consistent with the time-dependent enhancement of Coulomb
stress by fluid diffusion.
Haicheng
earthquake
(February 1975)
of the seismic
associated
the horizontal axis (y = 0) in Figs. 10 and 11 should be disregarded for the present discussion. To
peared
on or very close to the
aftershocks
more sensitive distributions.
Aki (1984).
with posi-
catalogue
that our present
the aftershocks
rupture.
postseismic
distinctly
for the artificial
It should be noted main
itself with the off-
stress change (Fig. 10) in contrast
the distribution lib). not predict
plot in Fig. 2. However,
distinguishes
The Haicheng lateral
earthquake
strike-slip
best known
motion)
recent
(M, = 7.3 with left-
in China
great
ruptures
is one of the because
of the
successful short term prediction by Chinese scientists. The spatial pattern of 1239 (M > 3.0) aftershock events for 1421 days after the main rupture is shown in Fig. 13. The fault trace is obtained by a least square fit of aftershocks visually judged to be close to or on the fault. While surface
breakage
due to the Haicheng
earthquake
has been reported to run as much as 70 km long (Gu et al., 1976) the early aftershocks within 5 hrs after the main rupture occupy a length of only 60 km (Fig. 14). We have adopted the shorter estimate as the actual rupture length for this analysis. The fault length estimated from the aftershock zone
5 days after
from the undrained
to the drained
same
(60 km) although
and may also be related
to the inaccu-
occurred especially close to the main shock. This fault trace and rupture length is plotted as the
racies in aftershock locations. An alternative method to show the time-depen-
the main
rupture
many
more
remains events
the have
dotted line on Fig. 13. Note that the epicentre (white star in Fig. 13 and arrow in Fig. 14) of the main shock is not located at the middle of the
dent component of aftershock patterns is by means We divided the 91 day of “ time-windows”. aftershock catalogue from Hamilton (1972) into four logarithmically equal periods (day 1-3, day 4-10, day 11-29, and day 30-91). The number of aftershocks as a function of distance from the main rupture is plotted in Fig. 12. It is seen that there is a general decay in the number of
stress increase scribed earlier
aftershocks from the main rupture outwards. Two other features should be noted. A great number of
aftershocks, and the result is shown in Fig. 15. A cluster of aftershocks is seen at approximately one
aftershocks
half-rupture-length
occur with some time delay,
i.e., not
coseismically. In the third and fourth period (day 11-29 and day 30-91) the number of events both near the main rupture and particularly off-fault (24-32 km from main-rupture) actually increases. Indeed Hamilton (1972) and Allen and Nordquist (1972) have shown that the off-fault clusters ap-
assumed rupture (and there is no reason to do so). It is shown in Fig. 13 that the off-fault aftershocks are again skewed towards the dilatant quadrant. The procedure for correlating Coulomb and aftershock occurrence dewas used to study the Haicheng
away
accompanied
by
a
Coulomb stress increase of about 10 to 15% of the stress drop. This amounts to CI== 5 to 8 bar necessary for triggering the off-fault aftershocks, for an estimated stress drop of 53 bar (Cipar, 1979). To show the time dependence of aftershock occurrence, the 333 days after the main rupture
50
Hoicheng
1 I.
.
.
I
.
.
’
Aftershocks
1
-.
I
”
.
’
, 1 --
I
.
.
.
.
I
1121.5
SOKM
0
0
.
.
.
.
t
.
.
I.
.
123
122.5
122
!,
I..
L
123.5
Longitude
Fig. 13. Aftershock
distributions
of the Haicheng
earthquake.
Note
the off-fault
clusters
of aftershocks
shifted
into the dilatant
quadrants Hoicheng
Aftershocks
I
I
I
0
1
0
k
0
0
0
0
0
Q1
I
k---f--
I
METERS Fig. 14. Early aftershocks the basis for choosing
of the Haicheng
the half fault length
earthquake
in rotated
coordinates,
I = 30 km for this earthquake.
5 hrs after main shock located
at origin (arrow).
This is
51
Halcheng cy
Aftershocks
,‘.“!‘.‘.I‘..’
.~‘~l’~.‘l~‘.~~~~‘~
3 --* -
0 It
Fig. 15. Haicheng associated
aftershocks
with positive
are divided
a.5
0
-<
plotted
Coulomb
in a distance-Coulomb
stress change
into four logarithmically
stress space. Note the clear clustering
of approximately
equal periods
1.5
1
of aftershocks
10 to 15% of main shock stress drop.
present.
These general
at 0.81<
1y 1 < 1.41
I = 30 km, c = 10 m2 s-‘.
findings
are consistent
with
(day 1-4, day 5-18, day 19-78 and day 79-333) and the number of aftershocks are shown in Fig. 16 as a function of distance from the main rup-
observations of aftershock distributions. Three earthquakes having distinctive aftershock patterns are shown. They are the Pasinler, the
ture. As for the Borrego Mountain earthquake, the off-fault aftershocks appear to show up distinctly in the third period (day 19-78) after some time
Borrego Mountain and the Haicheng earthquakes. The latter two are analyzed in a bit more detail. Tight clustering of off-fault aftershocks where
delay.
Coulomb stress increases are seen in a new plot which merges theoretical calculations with
Concluding discussions
aftershock
data. The Borrego
shown to require A model
is presented
of postseismic
stress and
pore pressure readjustment in time and space. The model is based on a theory by Rice and Cleary (1976) of coupled deformation-diffusion process in a linear poro-elastic medium subjected to sudden loading. In the present study, the sudden loading is associated with a seismic rupture modelled as a uniform stress drop on the rupture surface, resulting in a continuous distribution of dislocations. These imposed dislocations set up a stress and pore pressure pute the time-dependent
field from which we comCoulomb stress. The con-
an increase
Mountain in Coulomb
region
is
stress of
2 bars to trigger the off-fault aftershocks, while the Haicheng earthquake apparently requires 5-8 bars. While the time-dependence of fluid flow provides a plausible explanation for the delay of aftershocks, the results presented have not been able to correlate them irrefutably. Even so, the “time window” plots (Fig. 12 and Fig. 16) show that many aftershocks occur with some time delay and that clusters may form subsequent to and off the main fault. As an additional example, Hutton et al. (1980) noted that aftershocks off the main fault
emerged
as clusters
6 hrs after
the main
is made by increase with
shock of the 1979 Homestead Valley earthquake. A purely elastic rheology cannot explain such
As general results, we found that the aftershock zone expands with time in the dilating quadrant. Depending on initial stress field (before rupture), rupture magnitude and length, and material parameters, off-fault clusters may or may not be
time-transient phenomenon. The time and spatial distribution of aftershocks generated by postseismic stress and pore pressure readjustments due to fluid flow may be consistent with such observed aftershock patterns. Slight modifications of the calculated stress
nection to aftershock distributions associating high Coulomb stress aftershock
occurrence.
Aftershock
Fig. 16. Four “time-windows” the main rupture
Migration
Aftershock
of logarithmically
for the Haicheng
equal time periods
showing
number
Migration
of aftershocks
as a function
of distance
from
‘earthquake.
fields are possible by assuming an impermeable fault, as was recently suggested by Rudnicki (1985). Also, variations in the coefficient of friction would change the computed Furthermore, direct comparisons
stress contours. with field obser-
vations are hampered by complexities in the slip distributions of the main rupture, heterogeneity in the pre-existing stress field, and existence of neighboring branch faults, as in the case of the Borrego Mountain earthquake (Fig. 8).
States Geological Survey and the Exxon Education Foundation is gratefully acknowledged. APPENDIX
A
Numerical implementation
of eqn. (2)
Based on the slip distribution mulated slip within an element length
of the rupture
&3(x’) -___
dx,
is:
2(1-
=
3X'
(1) the accudx’ along the
v")Au G
x’
dx’
JS
Acknowledgements
The authors would like to thank D. Veneziano and J. Rudnicki for helpful discussions. Support from the National Science Foundation, the United
(‘41) In normalized (I
‘J_
A0
-
form, eqn. (2) may be written
20-v”) G
/
1 G,,(F-T', -1
j,
i)
as:
53
Y, i>
P(%
A0 P -= AU
2(1-v,)
1
G
/ -1
X
(KS
where X = x/l,
(AZ)
G,(X-X’,
2(1 i vu) 3 F Gp(X - Xk,
=
j, 7)
i)Xk
j,
k=l
(A5)
dx’
j = y/l,
i = ct/12,and from Rice
and Cleary (1976):
where
N = 1,. . , N - 1,
X, = cos(an/N),
cos x(2k - 1)/2N, number
k = l,...,
of collocation
and
chosen
Equation
N
is the
to the degree
of accuracy
required.
to compute
the Figs. 2 to 6. The numerical
used for the material
277r( 1 - V”)(l - V)
N,
points
Xk =
(A5) has been used
constants
values
introduced
in eqn.
(A3) are v = 0.2, vu = 0.33 and n = 0.3. These values also give B = 0.61.
i
sin 0 2 exp( --r*/4ct) i
x [l - exp(
’
l-v 4ct - - _ v, - v r2
References Aki, K.. 1984. Asperities,
-r2/4ct)])
quakes.
l-v x ( cos e --F[l-exp(-r2/4cr)l) L ”
)
J. Geophys.
Allen, C.R. and Nordquist, and late aftershocks in “The
sin 8 y
[l - exp( -r*2/4ct)]
i
{sin 8n-‘[l
- ei ”
Geol.
Borrego
Surv.
I (A3)
and 0 = tan-‘(y/x)
and n =
V”)(l - v)]. 3(V” - v)/[2B(l+ To obtain the stress components in a rectangular coordinate system, the following transformation rules may be used: G 1.X
G,,
=
i G,, :I
cos%
sin28
- 2 sin e cos e
sin*B
sin28
2 sin e cos e
sin e cos e
-sin e cos e
Chinnery, Cipar,
caused
~,,(X”,
Y, t>
by
Office,
The
faulting
of
Res.. 79: 2037-2044
stress
changes
that
accompany
Bull. Seismol. Sot. Am., 53: 921-932. processes
of the
observations C.H.,
shear
Haicheng,
of p and
1981. Off-fault
stress?
F. and Gupta. integral
M.B..
Kudreshova,
Bull.
China
s waves.
aftershock
Seismol.
Sot.
Bull.
clusters Am..
H-D.
Chen.
February Hamilton,
G.D., 1972. On the numerical equations,
71:
Osokina, results).
D.N..
Y-T,
Gao.
and
Study of Seismic Regime,
X-L
and
Liaoming
R.B., 1972. Aftershocks
Mountain
A.A.
and earthquake
(in Russian).
4. 1975. Acta Geophys.
earthquake
Lomakin. ruptures
Regional
pp. 113-124
of Haicheng,
solution
Q. Appl. Math., 29: 5255534.
B.B., 1974. Stresses,
Stinitz, Kishniev. mechanism
term in the
Printing
1669-1675.
Gu.
singular
Gov.
strain following
J. Geophys.
Source from
Das, S. and Scholz,
(A4)
- x”)
earthquake
of April 9. 1968”.
Seismol. Sot. Am., 69 (6): 190331916.
Gzovsky,
The Gauss-Chebyshev integration scheme (Erdogan and Gupta, 1972) is most suitable for solving eqn. (A2) as it takes advantage of the presence of the l/$(1 integrand. Hence:
J., 1979.
of singular
: GrB I
U.S.
1963.
foci (modelling
Go,,
Pap.
faulting.
earthquake
GV x
Earthquake
787,
main shock
Mountain
Mountain
medium. M.A.,
Erdogan,
c0s2e - sin20 1
J.M., 1972. Foreshock.
Booker, J.R., 1974. Time dependent a porous
of earth-
D.C.. pp. 16-23.
strike-slip
where r = \lm
and characteristics
of the Borrego
Prof.
Washington,
- ex p( --*2/4ct)l
barriers
Res., 89: 5867-5872.
Yi. Z.,
Province. Sinica,
1976.
Focal
earthquake
of
19 (4): 270-285.
of the Borrego
Mountain
from April 12 to June 12, 1968, in the Borrego earthquake
of April
Pap. 787, U.S. Gov. Printing
9, 1968. Geol.
Office, Washington,
Surv. Prof. D.C., pp.
31-54. Hull,
S., 1983. The mechanics
of aftershock.
R83-6, Dep. of Civil Engineering, Hutton,
L.K., Johnson,
C.E., Pechmann,
J.W., Cole. D.H. and Gernan, tions for the Homestead
110-114.
No. Mass.
J.C.. Ebel, J.E., Gwen,
P.T., 1980. Epicentral
Valley earthquake
15. 1979. Calif. Geol.. May:
Res. Rep.
M.I.T.. Cambridge,
sequence,
locaMarch
54
Kanamori,
H. and Anderson,
some empirical
relations
D.L., 1975. Theoretical in seismology.
basis of
Bull. Seismol.
Sot.
of rock masses. Nur,
R. and
Estimation J.R.,
pore fluid flow? Science, Omori,
F., 1894. Investigation
Comm..
2: 103-139
Rice, J.R. and Cleary,
diffusitity
122: 545-559.
1972. Aftershocks
caused
by the
of aftershocks.
Rep. Earth
Inv.
M., 1976. Some basic
for fluid-saturated
pressible
constituents.
elastic porous
Rev. Geophys.
stress diffusion media with com-
Space
Phys.,
14 (2):
1985.
Coupled
accompanying
sequence,
postseismic
deformations.
M.N.,
1984.
Earth,
Atmosphere
bridge,
Mass.
Utsu.
control
J. Geophys.
Seismicity Rep., Earth and
5th
New York.
California:
T., 1969. Aftershocks
Fat. Sci.. Hokkaido
diffusion fault.
M., 1983. The 1979 Homestead
earthquake
studies in Turkey.
deformation-fluid
slip on an impermeable
Ewing Symposium,
Stein, R. and Lisowski,
Toksoz,
175: 885-887.
(in Japanese).
solutions 227-241.
of in-situ hydraulic
Pure Appl. Geophys.,
Booker,
J.W.,
effects Maurice
Am., 65: 1073-1095. Li, V.C., 1984/1985.
Rudnicki,
and
and
Valley and
Res.. 88: 6477-6490. earthquake
Resources
Planetary
of aftershocks
Sciences,
earthquake
prediction
Laboratory, M.I.T., statistics
Univ., Jpn.. 3 (3): 129-195.
Dep. Cam(I). J.