Tectonophysics, 144 (1987) 37-54

Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

37

Postseismic stress and pore pressure readjustment and aftershock distributions

VICTOR C. LI *, SANDRA H. SEALE ’ and TIANQING CA0 2

’ Department of Ciwl Engineering, M. I. T., Cambridge, MA 02139 (U.S.A.)

’ Department of Earth, Atmosphere and PlanetaT Science, M.I. T., Camhrrdge, MA 02139 (U.S.A.)

(Received January 14. 1986; revised version accepted October 10. 1986)

Abstract

Li. V.C., Seale, S.H. and Cao, T.. 1987. Postseismic stress and pore pressure readjustment and aftershock distributions.

In: R.L. Wesson (Editor), Mechanics of Earthquake Faulting. Tectonophysics, 144: 37-54.

The time and spatial readjustment of stress and pore pressure due to a strike-slip rupture are investigated by means

of a model which treats the earth’s crust as a linear poro-elastic material infiltrated by fluids. The analysis is based on a

fundamental solution obtained by Rice and Cleary who treated the solid and fluid constituents as separately

compressible. The main shock is modeled as a sudden uniform stress drop and the resulting slip dislocation is

maintained in time. This suddenly introduced dislocation distribution sets up stress and pore pressure fields. Diffusion

of fluid takes place from regions undergoing compression to regions undergoing dilation. Hence the stress and pore

pressure fields evolve over time, with time scales which depend on the flow parameters of the rock medium.

It is assumed that aftershocks occur when the Coulomb stress (defined as ec = r + ~(0” + p), where 6, and 7 are

the increase in normal stress and shear stress across a potential slip surface, p is a friction coefficient, and p is the pore

pressure increase) exceeds the shear strength of the slip surface. The coseismic elastic solution confirms the suggestion

of spatial distribution of aftershocks by Stein and Lisowski. The transient solution predicts a small rotation of the

aftershock zone from the compression into the dilation region at the two ends of the main rupture. whereas the

aftershock -zones for off-fault aftershocks appear to expand and shift into the dilating regions. A new plot is developed

which provides a means to study the association between high Coulomb stress increase with off-fault aftershock

locations.

The aftershock distributions of three earthquakes (Pasinler, 1983: Borrego Mountain, 1968: and Haicheng. 1975)

are compared to the zones predicted by the model. The earthquake aftershocks are skewed towards the dilatant regions

and in some cases form distinct off-fault clusters in the areas of high Coulomb stress. The ability of the model to

provide a primary aftershock mechanism and the implications of this mechanism for material properties are discussed

in the contexts of these sample earthquakes. The model predicts the generally observed spatial expansion of the

aftershock zones and suggests an explanation for the presence or absence of off-fault aftershock clusters. However, the

inaccuracy in location of aftershocks and the small magnitude change in predicted Coulomb stress makes it difficult to

confirm or disprove the theory that aftershock distributions are controlled by fluid flow.

Introduction

It has been observed that for some earthquakes

with dominant strike-slip focal mechanism, the

spatial distribution of aftershocks has distinct pat-

terns when the aftershocks can be located accu-

rately. Some commonly observed characteristics of

these patterns include the linear extension of the

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.

38

Homestead Valley earthquake discussed by Stein

and Lisowski (1983) and Das and Scholz (1981)and

the Borrego Mountain and Managua earthquakes

discussed by Das and Scholz (1981). Although the

temporal development of such patterns has not

been well identified, a large proportion of the

aftershocks do occur with some time delay.

Attempts at explaining the spatial distributions

of aftershocks have been made by several re-

searchers. Gzovsky et al. (1974) and Das and

Scholz (1981) analyzed the stress field induced by

a plane strain Mode II (shear) crack and found

that at distances about one rupture length 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 Chin-

nery (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 in-

cluded the effect of friction proportional to in-

duced 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 pro-

portion 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 compo-

nent. The time source is due to the diffusion

process of water induced by a pressure disequi-

librium set up by the main rupture. The diffusion

process allows water to flow from the high com-

pression region to the dilatant region, causing a

time-dependent stress change. The result is a grad-

ual 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

range from days to months. We suggest that at

least for some earthquake ruptures this mecha-

nism may be responsible for the time delay of

aftershocks and full development of their spatial

patterns as mentioned earlier.

The suggestion that water flow may be respon-

sible for aftershock generation was originally pro-

posed by Nur and Booker (1972). They pointed

out that the diffusion process produces an inverse

square root time decay in pressure change similar

to the aftershock event decay law of Omori (1894).

Here we hasten to add that other physical mecha-

nisms such as stress corrosion or viscoelasticity

may also operate simultaneously, 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 genera-

tion 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 con-

stituents. The primary differences between the re-

sults 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 repre-

senting 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

due to Rice and Cleary (1976) who treated the

fluid and solid constituents as separately com-

pressible. While we have not evaluated the degree

of improvements with the slip geometry and the

material behavior adopted in the present model,

they may nevertheless be expected to be closer to

reality (in comparison to uniform slip and incom-

pressible constituents).

There are, however, many limitations in the

model. For example, the two-dimensional plane

strain deformation can only be an approximation

to reality, although this approximation may be

reasonable when the fault width is large. The

assumed smooth and constant (in time) slip distri-

bution on the main rupture is also inconsistent

with observations in surface breaks and aftershocks

on the fault. However, for locations off the fault,

the slip distribution assumed may be sensible.

In the following, we shall briefly describe the

theoretical model and discuss some numerical re-

sults based on this model. The aftershock patterns

of the 1983 Pasinler earthquake, the 1968 Borrego

Mt. earthquake and the Haicheng earthquake will

be discussed in the context of the model results,

with the latter two 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 parame-

ters precludes a strong verification of the theoreti-

cal predictions. We have also not studied thor-

oughly the sensitivity of theoretical results to the

various material parameters used in the model.

Model description and results

The model 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 (Fig. 1). The

slip magnitude 6(x) is determined from elastic

crack theory in which a uniform stress drop Au is

imposed along the length of the rupture:

S(x) = 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-

r-

39

1

Fig. 1. Geometry of fault slip; slip distrlbutlon 6(y) given by

eqn. (1) induces Coulomb stress q and pore pressure p.

ulus and V, is the undrained Poisson ratio. By

linear superposition, the time and spatial distri-

bution of the induced stresses u,, with (i, j = x,

y) and pore pressure p due to the main rupture

are:

atqx’) a,,(~, y. t)= -j-;rG,,(x-x'. y. r)rdx’

and:

p(x, 4‘. t) = -/;,G&-x'. x, r)%&$dx’

where we have already assumed that the main

rupture occurs at time zero. The Green’s function

G,,(x -x’, _Y, r) and G,(x - x’, J’, t) represents

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-

elastic medium. Such fundamental solutions have

been previously obtained by Rice and Cleary

(1976) and they have been included in Appendix

A. The factor (as( x’)/ax’) dx’ may be interpret-

ed as the accumulated slip within an element dx’

along the length of the rupture. Hence the individ-

ual stress components and pressure may be com-

puted using eqns. (1) and (2), and the numerical

implementation is described in Appendix A.

Following Stein and Lisowski (1983) we as-

sume that the occurrence of aftershocks is associ-

ated with the increase of Coulomb stress, defined

as:

u,=u,, +/J(u,.+p) (3)

L

Fig. 2. a. Contours of Coulomb stress at t = 0: shown only for J > 0. Stress contours for 1’ i 0 can be obtained by 180° rotation

about origin. b. Contours of Coulomb atress 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

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

generation implicitly assumes that the aftershock

focal mechanisms are similar to that of the main

rupture.

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 nor-

malized with respect to the half-rupture length

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

Coulomb stress is indicated by the solid contours

and decrease of Coulomb stress is indicated by the

dashed contours. Several characteristics in this

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.

This may explain the observation of aftershocks

being distributed along a zone longer than the

main rupture. The off-fault peak identified (as

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

41

(b)

Fig. 2 (continued).

close to the dilatant end of the main rupture

(point A). Between these two peaks, the Coulomb

stress is continuously enhanced with time. (The

locations of peaks that occur near 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 di-

latant 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 A, B, C, D in Fig. 2) is

plotted as a function of time (Fig. 4). Time has

been normalized by a characteristic relaxation

time, defined by t, = 12/4c where c is the diffusiv-

ity 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 picks up. Figure 5 shows the corre-

sponding change 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 -

Fig. 3. Pore pressure at time r = 0.

+1

0.00 I I

,-0.4Y5 - (1 $0.10 -

II

t&j-0. 15 - c

$0.20 -

F-0.2; - B

g-c.“” .-

“-0.35 -

-0.40 -

-CT.45 A /

1’ i) Li? , / I I

- ‘1 -G _ ; -‘ 1 -3 - 2 - ! D 1

loglo (4ct/&2)

Fig. 4. Pore pressure vs. time at locations A. B. C. and D shown in Fig. 2a, b

43

Q,.

0.

0.

0.

0.

0.

0.

0.

0.

, I I

I

lo’;;10 (4Ct/&!l

Fig. 5. Coulomb stress vs. time

KORIIALIZED DISTAFICE FROM FAULT EbD ALOllG LINE A - E

Fig. 6. Coulomb stress vs. distance along A-E. The four curves show fluid pressure-deformation coupling effects from coseismic

(0.0) to very long time scale (co). The dotted and dashed lines indicate effects on spatial distribution of aftershocks for two different

pre-stress states. See text for more details.

44

the off-fault peak at D is clearly shown at t = 0

(and is still maintained at f = O.lt,). It may be

seen that most of the Coulomb stress increase

between A and D occurs within one characteristic

time. For example, for a rupture length of 21= 50

km, and diffusivity range of 10’ to lo5 cm2 s-’

(see, e.g. Li, 1984/1985), the characteristic time is

between i to 50 yrs. If the larger diffusivity ap-

plies, it may be seen that the aftershock zones may

be fully developed within half a month (O.lt,) to

half a year after the main rupture. The smaller

diffusivity would extend these time lengths by a

hundred fold.

Figure 6 offers another, perhaps more general,

interpretation of aftershock patterns. Suppose the

pre-rupture stress state and the material properties

of the rock mass in the vicinity of the main

rupture are such that a 20% of Au increase in

Coulomb stress is required for aftershock occur-

rence (indicated by the horizontal dashed line),

then no off-fault aftershock clusters will be ob-

served, and an expansion of the aftershock zone

will be recorded. Aftershocks will be confined to

about 0.11 immediately after the rupture and grad-

ually 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

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

aftershock zone expands outwards. At one char-

acteristic time, the expanded zone merges with the

off-fault clusters.

From the above discussion, it should be clear

that the stress state prior to the main rupture, the

material properties (particularly fracture proper-

ties). 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 expan-

sion and merging with the off-fault cluster depend

on the diffusivity of the rock mass. Because these

dependent parameters can be different from earth-

quake to earthquake, the exact temporal and spa-

tial pattern of aftershocks may also look quite

different. In the following sections, we discuss the

aftershock patterns of several strike-slip earth-

quakes in the context of these theoretical consider-

ations.

Pasinler earthquake (October 1983)

The Pasinler earthquake (M, = 7.1 (NEIS); epi-

center 40.3” N and 42.2” E (USGS)) is a recent

event which occurred in northeast Turkey. It has a

well-constrained fault plane solution with attitude

of strike N42OE dip 80” which is consistent with

the trend of observed surface faulting and

aftershock patterns (Toksoz, 1984). The aftershock

distribution of 10 days after the main rupture is

shown in Fig. 7. The data were collected by a

network of eight portable analog and three digital

instruments deployed locally. The aftershock zone

is much larger than the 10 to 12 km long zone

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

maximum intensity contour, Toksoz (1984) sug-

gested that the primary slip may have occurred at

depth. We have placed the fault line visually with

respect to the aftershock locations and the elon-

gated isoseismals. Fault motion is left-lateral as

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-

ysis for this case. However. two loops of

aftershocks can be seen to skew towards the di-

latant 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 alter-

ation exceeding the induced off-fault peak

Coulomb stress alteration, as described earlier in

reference to the dashed line in Fig. 6.

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

40.1

40

39.9

/ Narman / . . : .,z:\. \

LONGITUDE

Fig. 7. Aftershock distributions of the Pasinler earthquake (after ToksGz. 1984). Symbol star indicates location of the main shock.

Note the concentration of aftershocks in the dilatant quadrants

j’-

i’-

30’ 15’ 116”OO’ 45’

I-

0 -1. I

‘..

25 KM

\

Q s, “..,,

X”

LONGITUDE

Fig. 8. Aftershock distributions of the Borrego Mountain earthquake (after Hamilton, 1972). Note the off-fault clusters shifted into

the dilatant quadrants.

Bon-ego Aftershocks

METERS

Fig. 9. Early aftershocks of the Borrego Mountain earthquake in rotated coordinates, 5 days after the main shock. This distribution is

the basis for choosing the half fault length, I = 20 km. Arrow at origin indicates location of main shock.

after the main rupture (from Hamilton, 1972). The observed surface break of 31 km (Fig. 8). The

length of rupture (dashed line in Fig. 8) is esti- early-day aftershocks defining the rupture zone is

mated to be 40 km from the extent of the a generally accepted notion, especially for subduc-

aftershock distribution within 5 days of the main tion zone ruptures.. Without a better alternative,

event (Fig. 9). The location of the main shock at we have used the 5 day aftershocks to define the

the origin of Fig. 9 is indicated with an arrow. The rupture plane for the Borrego Mountain earth-

chosen rupture length of 40 km is longer than the quake. Figure 8 shows the tendency of the off-fault

Borrcgo Aftershocks

iI 0

c

00 _l

4 0 00

-i.s -‘l -6.5 015 1 1.5 2

Fig. 10. Borrego Mountain aftershocks plotted in a distance-Coulomb stress space. Note the clear clustering of off-fault aftershocks

at 0.81 i 1 y 11.51 associated with positive Coulomb stress change of approximately 10% of stress drop of main shock. Half fault

length I = 20 km. Diffusivity c = 0.1 m* s-’ used.

47

aftershocks to fall into the dilatant quadrant for Coulomb stress increase and the occurrence of the right lateral fault motion. The clustering is aftershocks, we replot the aftershock distance from also quite distinctive, and the clusters are located the fault trace as a function of Coulomb stress

approximately in the location of peak induced increase, as shown in Fig. 10. These plots are Coulomb stress. This may be seen by comparing obtained using the following procedure: For each Fig. 8 to Fig. 2a. aftershock i the earthquake catalogue provides a

To further investigate the correlation between triplet (x,, y,, t,). After normalizing with the half

Random Aftershocks

u 000 8 -Oo

0 08

O 0

_R

- v-

0

B” 0

” 0

00

0 O” B

0

0 0 0 -1 ‘0

Too -0 0

g

O 0 0 00 O 0 0 0 0 0 0 80 o”oo 0 B o0 o 0 au 0 0 0 n

003 0

3” 0 0 $OO” oooo 0 %J “_ 80 0 4

@ ts, cs !

0 0 0 0 .nn 0 !

0 1 . rnc 0

- .m.

q

. .

-2x104 -10’ +.. !

10’ !

0 2x104

(a) Meters

Random Aftershocks

Fig. 11. (a) Random distributions of aftershocks with same number of events (533) and cover same spatial area as for the Borrego

Mountain earthquake. Also I = 20 km and c = 1.0 m2 s-r. This random catalogue produces the distance-Coulomb stress plot shown

in (b). No clustering in the positively stressed region is seen for this random case, in contrast to the distinct clustering for the actual

case shown in Fig. IOa, b.

48

rupture length and relaxation time t,, the triplet

(x,/l, y,/l, t/t,) 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 ( 1 y 1 ,/I, (~,/Au)~). In other words, each

aftershock in the catalogue is plotted at its ab-

solute normalized distance from the rupture versus

the Coulomb stress computed for the event. The

time distribution of the aftershocks is lost in this

representation; however, off-fault clusters are en-

hanced.

Figure 10 shows such pairs from the Borrego

Mountain aftershock catalogue. The distinctive

features to look for in these plots are the associa-

tion of off-fault aftershocks at about half a fault

length distance on either side of the main rupture

( 1 y I/l = 1) with positive increase in Coulomb

x

Aftershock Migration

0 0

i

Aftershock Migration

0 Day 11-29

0 0 0

stress. The tight clustering of aftershocks at uc =

0.1 - 0.15Au suggests that the required Coulomb

stress increase for aftershock 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

and Anderson, 1975).

In order to bring out the subtleties of the

pattern in Fig. 10, we used an artificial catalogue

of spatially uniform random distribution of

aftershocks with the same number of events in an

area of 50 km x 50 km which covers all of the real

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 of

the positively stressed zone with distance y/l, as

Aftershock Migration

Ldk??-i 0

Distance From Initial Rupture, km

Aftershock Migration

s 8 Day 30-91

0 0

Fig. 12. Four “time-windows” of logarithmically equal time periods showing number of earthquakes as a function of distance from

the main rupture of the Borrego Mountain earthquake.

49

seen in the stress contour plot in Fig. 2. However,

the real catalogue distinguishes itself with the off-

fault aftershocks associating distinctly with posi-

tive Coulomb stress change (Fig. 10) in contrast to

the distribution for the artificial catalogue (Fig.

lib).

It should be noted that our present theory does

not predict the aftershocks on or very close to the

main rupture. These aftershocks are presumably

much more sensitive to the details of the seismic

slip distributions. and are associated with the

postseismic breaking of “barriers” as described by

Aki (1984). Hence the aftershocks lying close to

the horizontal axis (y = 0) in Figs. 10 and 11

should be disregarded for the present discussion.

To investigate the 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 pattern in Fig. 10 does not appear to

be sensitive enough to show major distinction

through two orders of magnitude change in c. This

is apparently due to the small change in Coulomb

stress magnitude from the undrained to the drained

condition, and may also be related to the inaccu-

racies in aftershock locations.

An alternative method to show the time-depen-

dent component of aftershock patterns is by means

of “ time-windows”. We divided the 91 day

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

aftershocks from the main rupture outwards. Two

other features should be noted. A great number of

aftershocks 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-

peared the day after the main shock and persisted

for some time afterwards. If the catalogue is long

enough (i.e., beyond 91 days) we expect that the

number of aftershocks everywhere will eventually

subside at later time periods. These observations

are consistent with the time-dependent enhance-

ment of Coulomb stress by fluid diffusion.

Haicheng earthquake (February 1975)

The Haicheng earthquake (M, = 7.3 with left-

lateral strike-slip motion) in China is one of the

best known recent great ruptures because of the

successful short term prediction by Chinese scien-

tists. 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 visu-

ally 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 esti-

mate as the actual rupture length for this analysis.

The fault length estimated from the aftershock

zone 5 days after the main rupture remains the

same (60 km) although many more events have

occurred especially close to the main shock. This

fault trace and rupture length is plotted as the

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

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

stress increase and aftershock occurrence de-

scribed earlier was used to study the Haicheng

aftershocks, and the result is shown in Fig. 15. A

cluster of aftershocks is seen at approximately one

half-rupture-length away accompanied by a

Coulomb stress increase of about 10 to 15% of the

stress drop. This amounts to CI= = 5 to 8 bar neces-

sary 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

, 1 --

1

Hoicheng Aftershocks

1 I. . . I . . ’ 1 -. ” I . ’

0 0 SOKM

I . . . . I . . . . t . . . I. I.. !, L

121.5 122 122.5 123 123.5

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 1

0 0

0 0 0 0 0

k Q1 I I k---f--

METERS Fig. 14. Early aftershocks of the Haicheng earthquake in rotated coordinates, 5 hrs after main shock located at origin (arrow). This is

the basis for choosing the half fault length I = 30 km for this earthquake.

51

cy

3 -- * -

0 It -<

Halcheng Aftershocks

,‘.“!‘.‘.I‘..’ .~‘~l’~.‘l~‘.~~~~‘~

0 a.5 1 1.5

Fig. 15. Haicheng aftershocks plotted in a distance-Coulomb stress space. Note the clear clustering of aftershocks at 0.81< 1 y 1 < 1.41

associated with positive Coulomb stress change of approximately 10 to 15% of main shock stress drop. I = 30 km, c = 10 m2 s-‘.

are divided into four logarithmically equal periods

(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-

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

delay.

Concluding discussions

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 sud-

den loading. In the present study, the sudden

loading is associated with a seismic rupture mod-

elled 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 field from which we com-

pute the time-dependent Coulomb stress. The con-

nection to aftershock distributions is made by

associating high Coulomb stress increase with

aftershock occurrence.

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

present. These general findings are consistent with

observations of aftershock distributions.

Three earthquakes having distinctive aftershock

patterns are shown. They are the Pasinler, the

Borrego Mountain and the Haicheng earthquakes.

The latter two are analyzed in a bit more detail.

Tight clustering of off-fault aftershocks where

Coulomb stress increases are seen in a new plot

which merges theoretical calculations with

aftershock data. The Borrego Mountain region is

shown to require an increase in Coulomb 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

shock of the 1979 Homestead Valley earthquake.

A purely elastic rheology cannot explain such

time-transient phenomenon. The time and spatial

distribution of aftershocks generated by post-

seismic stress and pore pressure readjustments due

to fluid flow may be consistent with such observed

aftershock patterns.

Slight modifications of the calculated stress

Aftershock Migration Aftershock Migration

Fig. 16. Four “time-windows” of logarithmically equal time periods showing number of aftershocks as a function of distance from

the main rupture for the Haicheng ‘earthquake.

fields are possible by assuming an impermeable

fault, as was recently suggested by Rudnicki

(1985). Also, variations in the coefficient of fric-

tion would change the computed stress contours.

Furthermore, direct comparisons 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).

Acknowledgements

The authors would like to thank D. Veneziano

and J. Rudnicki for helpful discussions. Support

from the National Science Foundation, the United

States Geological Survey and the Exxon Educa-

tion Foundation is gratefully acknowledged.

APPENDIX A

Numerical implementation of eqn. (2)

Based on the slip distribution (1) the accu-

mulated slip within an element dx’ along the

length of the rupture is:

&3(x’) dx, = 2(1- v")Au x’ -___ 3X' G JS dx’

(‘41) In normalized form, eqn. (2) may be written as:

‘J_ 20-v”) (I 1 -

A0 G / -1 G,,(F-T', j, i)

53

2(1-v,) 1 (AZ)

P -= AU G / -1

G,(X-X’, j, 7)

X (KS dx’

where X = x/l, j = y/l, i = ct/12, and from Rice

and Cleary (1976):

277r( 1 - V”)(l - V)

i

i

l-v 4ct ’ sin 0 2 exp( --r*/4ct) - - - _

v, - v r2

x [l - exp( -r2/4ct)])

x ( cos e l-v

L --F[l-exp(-r2/4cr)l) )

”

sin 8 y [l - exp( -r*2/4ct)] - ei i ”

{sin 8n-‘[l - ex p( --*2/4ct)l I

(A3)

where r = \lm and 0 = tan-‘(y/x) and n =

3(V” - v)/[2B(l+ V”)(l - v)].

To obtain the stress components in a rectangu-

lar coordinate system, the following transforma-

tion rules may be used:

G 1.X

i :I

cos% sin28 - 2 sin e cos e

G,, = sin*B sin28 2 sin e cos e

G,, sin e cos e -sin e cos e c0s2e - sin20 1

GV

x Go,,

: I

(A4)

G rB

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 - x”) singular term in the

integrand. Hence:

~,,(X”, Y, t>

P(% Y, i> A0

= 2(1 i vu) 3 F Gp(X - Xk, j, i)Xk k=l

(A5) where X, = cos(an/N), N = 1,. . , N - 1, Xk =

cos x(2k - 1)/2N, k = l,..., N, and N is the

number of collocation points chosen to the degree

of accuracy required. Equation (A5) has been used

to compute the Figs. 2 to 6. The numerical values

used for the material constants introduced in eqn.

(A3) are v = 0.2, vu = 0.33 and n = 0.3. These val-

ues also give B = 0.61.

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