Download Physical modeling of the formation and evolution of

TECTONOPHYSICS
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ELSEVIER
Tectonophysics 277 (1997) 57-81
Physical modeling of the formation and evolution of seismically active
fault zones
A.V. Ponomarev a,*, A.D. Zavyalov a, V.B. Smirnov
b, D.A.
Lockner c
a Institute of Seismology; U1PE RAS, Bol. Gruzinskaya 10, Moscow 123810, Russia
h Physical faculty of Moscow State Universi~, Moscow, Russia
" US Geological Survey, 345 Middlefield Rd., Menlo Park, CA 91425, USA
Received 4 March 1996; accepted 30 September 1996
Abstract
Acoustic emission (AE) in rocks is studied as a model of natural seismicity. A special technique for rock loading
has been used to help study the processes that control the development of AE during brittle deformation. This technique
allows us to extend to hours fault growth which would normally occur very rapidly. In this way, the period of most
intense interaction of acoustic events can be studied in detail. Characteristics of the acoustic regime (AR) include the
Gutenberg-Richter b-value, spatial distribution of hypocenters with characteristic fractal (correlation) dimension d, Hurst
exponent H, and crack concentration parameter Pc.
The fractal structure of AR changes with the onset of the drop in differential stress during sample deformation. The
change results from the active interaction of microcracks. This transition of the spatial distribution of AE hypocenters is
accompanied by a corresponding change in the temporal correlation of events and in the distribution of event amplitudes as
signified by a decrease of b-value. The characteristic structure that develops in the low-energy background AE is similar to
the sequence of the strongest microfracture events. When the AR fractal structure develops, the variations of d and b are
synchronous and d = 3b. This relation which occurs once the fractal structure is formed only holds for average values of d
and b. Time variations of d and b are anticorrelated. The degree of temporal correlation of AR has time variations that are
similar to d and b variations.
The observed variations in laboratory AE experiments are compared with natural seismicity parameters. The close
correspondence between laboratory-scale observations and naturally occurring seismicity suggests a possible new approach
for understanding the evolution of complex seismicity patterns in nature.
Keywords: failure; acoustic emission; fractal structure; earthquake
1. Introduction
Seismicity possesses a certain organization. The
structure of seismicity can be seen in the nonrandomness of the s p a c e - t i m e - e n e r g y distribution of
* Corresponding author. Fax:
geodin @adonis.iasnet.ru
+7
095
255-6040;
e-mail:
earthquakes; it obeys a certain statistic having fractal properties (e.g., Rikunov et al., 1987; Sadovsky,
1989; Anonymous, 1989a; Sadovsky and Pisarenko,
1991). These properties reflect a self-similarity of
seismicity which can in turn be regarded as a manifestation of general patterns of the seismic process at
different scales.
At present, the nature of seismicity structure is not
0040-1951/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved.
Pll S 0 0 4 0 - 1 9 5 1 ( 9 7 ) 0 0 0 7 8 - 4
58
A.V Ponomarev et al./Tectonophysics 277 (1997) 57-81
fully understood and has been explained using a variety of approaches based on various a-priori assumptions (e.g., Kagan and Knopoff, 1981; Gabrielov et
al., 1986; Kuksenko, 1986; Chelidze, 1987; Narkunskaya and Shnirman, 1989; Pisarenko et al., 1989;
Ito and Matsuzaki, 1990; Main et al., 1990; Ulomov,
1990; Lyubushin, 1991; Damaskinskaya et al., 1993;
Gol'dshtein and Osipenko, 1993). This variety reflects both the complexities of the problem and the
lack of a sufficient data base to properly identify
the statistical regularities governing the formation
of the observed seismicity structure. Such a data
base is difficult to obtain due to the low occurrence
rate of large earthquakes. Under natural conditions
it is practically impossible to monitor (much less to
control) the principal characteristics of the system
under study, namely, the states of stress and strain in
the lithosphere and their time variation. Furthermore,
it is not possible to repeat experiments under controlled conditions. Thus, the main requirements for a
physical experiment are not present.
One solution to this problem is to model seismicity in the laboratory by means of acoustic emission (AE) in artificial and natural materials. We
consider the acoustic activity, similar to seismicity
(Riznichenko, 1958), as the aggregate of acoustic
events in time and space. In this case the applicability of laboratory results to natural conditions
should be examined. The statistical self-similarity of
seismicity, as well as the positive results achieved
in laboratory modeling of earthquake sources and
their preparation, lends credence to laboratory modeling of earthquakes (Sobolev and Koltsov, 1988;
Sobolev, 1993; Lockner, 1993). Besides, this clarification of the correspondence between acoustic
events and their patterns in laboratory conditions
on the one hand, and the natural phenomena on
the other, can help reveal the nature of the seismic
process.
At present, a number of studies indicate a similarity of the statistical properties of the AE and
seismicity in the energy domain. It is found that
the distribution of AE pulses over energy obeys
a power law, i.e., corresponds to the GutenbergRichter frequency-magnitude relation (Mogi, 1962;
Vinogradov, 1964; Scholz, 1968; Weeks et al., 1978;
King, 1983; Hirata, 1987; Cai et al., 1988; Zavyalov
and Sobolev, 1988; Meredith et al., 1990; Zhaoyong
et al., 1990; Sammonds et al., 1992). Some studies (Vinogradov, 1964; Scholz, 1968; Weeks et al.,
1978; Cai et al., 1988; Zavyalov and Sobolev, 1988;
Meredith et al., 1990) demonstrated a decreasing
slope of the AE recurrence curve as the load on the
specimen approached a critical value. This effect is
known for seismicity as well (Mogi, 1985; Zavyalov
and Sobolev, 1988; Sidorin, 1992).
Other studies (Hirata et al., 1987; Lockner and
Byerlee, 1991; Lei et al., 1992) examined the spatial
structure of AE. They found that AE has a spatial
fractal structure typical of seismicity as well (Anonymous, 1989a; Sadovsky and Pisarenko, 1991). A
geometric parameter, the fractal dimension, is also
noted to vary during specimen failure. Time variations of the fractal dimension of seismicity, related to
changes in the state of stress, are described in several
papers published at the same time or soon after the
papers on the spatial structure of AE (Hirata, 1989;
Zhenwen et al., 1990; Smimov, 1993; Smirnov et al.,
1994).
Focal mechanism solutions of acoustic events are
reported, for example, in Lei et al. (1992) and reviewed in Lockner (1993). These mechanisms are
similar in type to those of earthquake foci.
Studies of time sequences of AE pulses reveal
effects typical of earthquake sequences. Lockner and
Byerlee (1977), Hirata (1987) and Lockner (1993)
found AE aftershock sequences obeying the Omori
law. It is thus demonstrated that acoustic aftershock
activity shows the same statistical regularities as natural aftershocks. Temporal clustering of AE pulses
which reflects the interaction of acoustic events is
observed in specimens of different sizes (Vinogradov and Mirzoev, 1968; Sobolev et al., 1982;
Nishizawa and Noro, 1990; Damaskinskaya et al.,
1993). Alexeev et al. (1993) examined sequences
of electromagnetic emission pulses (EME) in rock
samples to show that the electromagnetic emission
has fractal properties in time. The estimates of the
Hurst exponent obtained indicate that fracture formation is not a sequence of independent events;
the correlation of EME events increases with the
development of rock failure. Alexeev and Egorov
(1993) proposed a statistical model that relates the
increase of the Hurst exponent with the growth of
the concentration of microcracks. Time correlation
and temporal fractal properties are also found for the
A. V Ponomarev et aL /Tectonophysics 277 (1997) 57-81
seismic process (Kagan and Knopoff, 1981; Rikunov
et al., 1987; Smalley et al., 1987; Zhenwen et al.,
1990).
Thus, the results of AE in rock samples indicate the presence of certain structural properties of
ensembles of acoustic events that are qualitatively
and quantitatively similar to the properties of seismicity. The present paper deals with patterns in the
formation and evolution of the structure of acoustic
activity.
2. Experimental method and raw data
2.1. Experimental method
It is well known that brittle rocks fail under
homogeneous compression by the development of
macroscopic fractures. It is often difficult to study
the kinetics of this process due to the rapid, unstable
fracture growth at velocities approaching the elastic
wave speed. Use of rigid loading frames can significantly increase the stability of specimens during
the 'overcritical' deformation stage by decreasing
the amount of stored elastic energy in the specimenmachine system (Wawersik and Brace, 1971; Peng
and Johnson, 1972; Stavrogin and Protosenya, 1985).
However, even in such cases many crystalline rocks,
and granite in particular, store sufficient elastic energy for a rapid, uncontrolled failure.
An alternate approach consists of rapidly removing elastic energy from the loading system as the
fracture grows, by means of a fast-acting press with
servo-control and feedback of acoustic activity (AE
rate). This technique has been tested in some experiments (Terada et al., 1984) and has been used in
recent studies of the process of failure preparation in
granite and sandstone samples (Lockner et al., 1991,
1992a). The most recent experiments managed to extend the final phase of fracture formation to a period
of many minutes, succeeding in stable development
of macrofractures whose growth can be considered
quasistatic.
The present paper uses data obtained in three
independent experiments, AE36, AE39 and AE42.
The apparatus and method of the experiment are described in detail in Lockner and Byerlee (1980) and
Lockner et al. (1992a). Cylindrical specimens 76.2
mm in diameter and 190.5 mm in length made of
59
Westerly granite were deformed by axial loading under the conditions of constant confining pressure of
50 MPa. Six identical resonance (approximately 0.8
MHz) piezoelectric sensors were mounted directly
on the surface of each sample to receive acoustic signals. One of the sensors situated near the mid-plane
of the sample was used to control the feedback that
changed the rate of loading depending on the current
acoustic activity. The threshold value of AE rate used
in the feedback circuit was maintained at a level of
a few tens of events per second during the entire
experiment. The feedback system was designed so
that increasing AE rate automatically resulted in an
immediate decrease or reversal of axial deformation
rate. The maximum unloading rate of the loading
system, about 6 MPa/s, was adequate to provide a
stable and controlled growth of macrofracture in all
experiments.
The acquisition system for acoustic events
recorded digitally the relative first arrival times and
amplitudes of first signal peaks tbr each of the six
piezosensors. Timing resolution of the digital channels in determining first arrivals was 0.05 /zs. AE
sources were located by an algorithm that used the
times of first compressional arrivals at the sensors
and minimized the travel time residuals. The velocity of elastic waves was determined independently
during the experiments at different loading levels by
means of four additional piezosensors, the ray paths
of ultrasonic monitoring making angles of 30, 50 and
90 degrees with the vertical axis of the specimen.
This permitted calculation of velocity variations for
different directions as the loading progressed to provide a more accurate location of the AE sources.
Acoustic events that had travel time residuals above
5 #s were excluded from the total AE catalogue as
having too high location errors. The net location error was typically 1-3 mm (see appendix in Lockner
et al. (1992a) for a more complete discussion).
The catalogues thus obtained included about
46,000 reliably located events for experiments AE36,
AE42 and about 20,300 events for AE39. The relative energy of an acoustic event was estimated from
the amplitude of the first signal peak. The amplitudes
at different sensors were adjusted for geometrical
spreading to a common distance 10 mm from the
hypocenter and then averaged. Thus, the energy estimate was compensated for the effects of anisotropy,
60
A.V. Ponomarevet al./Tectonophysics 277 (1997) 57-81
network geometry, and other features of structure
and instrumentation.
2.2. Raw data
Fig. 1 illustrates the loading history. Each experiment comprises three stages of loading: (1) the
rise of differential stress up to a maximum; (2) a
stage of nearly constant stress; and finally, (3) a
drop in stress due to incipient failure. Below we
shall call the two last stages the 'fiat' and falling
parts of the loading curve. A previous analysis of
time-space features of the AE distribution showed
that failure nucleated in the central part of each
specimen near its surface at or after reaching the
maximum load (Lockner et al., 1992a). Before the
appearance of this cluster of events, the microfractures are distributed over the specimen volume rather
chaotically. After sufficient damage accumulates in
the samples, failure localizes in the plane of future
rupture. This localization is accompanied by a relatively abrupt decrease in the axial load and, without
feedback control, a considerable increase in acoustic
activity. The rupture forms through the specimen at
an angle of 200-25 ° to the loading axis, acoustic
emission gradually decaying as the unloading rate
decreases. Location of AE indicates that the rupture propagates at a velocity of a few microns per
second as a thin, 1-5 m m wide, crescent-shaped
damage zone. The typical width of this zone along
its propagation direction amounts to 10-30 mm,
the length depending on rupture duration. Thus, for
specimens AE39 and AE42 this dimension equaled
70-80 mm. After the axial load dropped to 80%
of the peak value, the experiment was stopped, the
load was removed and the macrorupture stopped
growing.
If the experiment is allowed to run to completion,
the rupture will grow until it has bisected the sample.
In experiment AE36 the rupture reached the lower
base of the specimen in the interval 31,500-32,500
s at a load of about 0.55 of the maximum. At this
point the rupture plane intersected one of the steel
loading pieces and the axial stress again increased
(Fig. la). The load continued to increase for about
1.5 h (up to 37,500 s) and resulted in the formation
of new feather macrofractures in the lower part of
the specimen. This was accompanied by intensive
acoustic emission at the final stage of the experiment
(43,000-46,000 s). Along with the AE activation
mentioned, relatively small (30-40 MPa) short stress
drops were repeatedly observed (Fig. la) which were
also accompanied by outbreaks of acoustic activity
(28,400 s, 31,600-32,500 s, 44,000 s). These phenomena show that the rupture grows rather non-uniformly, being accelerated and slowed due to failure
of minor inhomogeneities.
Thus, while experiment AE36 demonstrates a total and complex evolution of a macrorupture, experiments AE39 and AE42 mainly show the initial
stage of its formation. The latter experiments typically have stages of 'constant' differential stress
almost 5 times as long as that in experiment AE36.
This is due to the use of more sensitive piezosensors
and a decreased threshold of AE rate in the feedback circuit. Accordingly, while about 1400 events
were recorded by this stage in experiment AE36, the
corresponding segment of the loading path resulted
in 13,000 recorded events in experiment AE39 and
20,000 events in experiment AE42.
2.3. Preliminary processing and selection of
catalogues of acoustic events
Raw catalogues of acoustic events were reduced
to a standard format: time, x,y,z-coordinates and energy class of each event. The quantity K = 2log U,
where U is AE pulse amplitude when reduced to the
distance 10 m m from the hypocenter was taken as
the energy class I of the acoustic event. The quantity U 2 for AE pulses is, to a first approximation,
proportional to the energy of an event.
Due to attenuation of acoustic waves, smallamplitude AE pulses are lost by the monitoring system. A threshold amplitude above which AE event
sampling was considered complete was determined
by analysis of the recurrence curve (Fig. 2) for each
experiment. Acoustic events were selected according
I W e u s e the concept of energy class rather than the more
traditional seismic magnitude 'scale' because energy class is a
more direct representation of event energy E: K = log E; in
our case K -,~ log E + const. Energy class is used in Russia
to represent earthquake magnitudes. When necessary, we have
used the following relation between energy class and Richter
magnitude: K = 1.5M + const.
A. V. Ponomarev et aL / Tectonophysics 277 (1997) 57-81
°
oo
61
o
AE36
i
"
i
.
¢M
600 - 500 - - ~ " " - ~
400 -300 - -
,
~
EAE
|
~ 100
200 -100 - -
50
0
i
1.o
J-'='
~ '
0.8
"Q 0.6
I
'
I!
'
I
'
I
'
li
'
'
i
i
2.8
,
2.0
1.0 ~~_
,
o.8
'
f
0
i
~
'
0.6
I
0.8
1,o_
0,6
e~
I
e~ 0.0
-6
-1.0
20000
~
I
24000 28000 32000 36000 40000 44000 48000
time, s
Fig. l. Variation of differential stress a (loading history), release of acoustic energy EAE and parameters of acoustic regime structure
in experiments: (a) AE36, (b) AE39, (c) AE42. The top of the figure presents schemes of macrofractures based on the location of AE.
Acoustic energy was calculated in attribute units in the intervals of 100 s in experiment AE36 and 10 s in experiments AE39 and AE42.
The following parameters of acoustic regime are used: b-value for AE-events energy class K ~. logE + const distribution (Eq. 5); fractal
(correlation) dimension d of AE-events distribution in the specimen volume (Eq. 10); Hurst exponents for the series of distances (Hr)
and time intervals (Ht) between the consequent AE-events (Eq. 17). Vertical dashed lines separate 'flat' from 'decreasing stress' regions
in (b) and (c) and indicate moments of significant failures in (a); anomalous variations of acoustic regime parameters are accentuated by
thick curves in (a).
62
A. V. Ponomarev et al. / Tectonophysics 277 (1997) 57-81
b
AE39
o
o', M P a
~
EAE
-
400 ~
100
300
200
0
1.2
1.0
0.8
0.6
0.4
I
I
]
;
I
I
I
i
J
I
I
I
I
i
I
~I
d
3.0
2.5
2.0
1.5
0.8 I
i
t
0.6
0.4
:f
"1o
I
I
I
I
t
I
0.8 ~
0.6
0.4
1.0 ~-
,-?,
T
0.0 ~
-1.0
I
I t.
I -"- I !
I
I
5000 6000 7000 8000 9000 10000 11000 12000 13000
time, s
Fig. 1 (continued).
to their reporting level in the catalogues. Table 1 contains the adopted lowest reported amplitude classes,
sample sizes and the parameters of the recurrence
curve for AE pulses.
3. Methodsofdataanalysis
To characterize the structure of acoustic activity
a number of statistical parameters from catalogues
63
A.V. Ponomarev et al./Tectonophysics 277 (1997) 57-81
C
O
AE42
..Q
i
or, M P a
EAE
600
500~ ~
400
300
200
100 ~
~ 80
I
I
J
40
I
I
r
1.0
-1.2--t
0.8
0.6
t
3.0~
~
~
I
I
=
"O
r
2.5
2.0
t
~
I
0.8_
0.6
~
o.4
:f
I
0.8_
0.6
~
i
~
J
,
i
,
jl
4__1
I
p
0.4
1.0~
t"o
-6
1
I
p
0.0
I
-1.o
4000
~
I
6000
,
8000
10000 12000 14000
j
16000
time, s
Fig. 1 (continued).
of acoustic events are used, namely the slope of
the recurrence curve b, fractal (correlation) dimension of the hypocenter set d, the Hurst exponent
H, and a parameter representing fracture concentration Pc.
A. V Ponomarev et a l . / Tectonophysics 277 (1997) 57-81
64
Table 1
Representative class Kmin of catalogues of acoustic events and
the parameters of the recurrence law: N =- A x 10-h(K 2)
,"~'~ :----.:AE38b--o:8o
10000
1000
Z
100
\t
I
10
Specimen
Krnin
Sample size
b
A (c Iq)
AE36
AE39
AE42
2.0
1.9
1.4
30649
12584
27575
0.80 ± 0.01
0.95 4- 0.01
0.91 4- 0.01
2.01 + 0.04
2.7 4- 0.1
1.45 4- 0.02
described in (Sadovsky and Pisarenko, 1991).
The recurrence law (1) is represented in the form
log Nj ---- - - b A K j + log A
1
2
3
4
5
K
o
•,-"
=
10000
",.,.
b 091
1000
where j = r,r -t- 1. . . . . r = n, Nj is the mean
number of earthquakes with classes in the interval
[K0 + A K ( j - 1/2), Ko + A K ( j + 1/2)]) normalized to the time interval T examined, AK is the
interval width for the histogram, (n + 1) is the number of histogram intervals. The estimate of maximum
likelihood for b and A are found from the equations:
M0 - TAro(b) = 0
z
,
I O0
(2)
(3)
A K i n 10M1 + TAro(b) = 0
i,
where the following notations used:
1o
r+n
,
Mo=E
mj
j=r
r+n
I
,
0
,
,
,
i
1
,
,
,
,
i
,
,
,
2
,
i
3
,
,
,
,
t
,
,
,
,
i
5
4
M1 = E
K
jmj
j =r
r+n
b
Fig. 2. Frequency-energy class relations for acoustic events in
experiments: (a) AE36 and AE39, (b) AE42. Energy class K
is defined as K = 21ogU ~ logE 4- const, where U is AE
pulse amplitude when reduced to a distance of 10 mm from the
hypocenter and E is the energy of the AE-event.
~o(b) = y ]
(4)
10 -jb~xK
j~r
r+n
~o'(b) = - A K I n
10 ~--'~. j l O -kbAK
j=r
3.1. Parameters of the recurrence law for acoustic
events
The law of recurrence for acoustic events has the
form
log N = - b ( K - Ko) + log A
(1)
where N is the number of events per unit time with
energy class in the range (K - A K / 2 , K + AK/2).
The b and A parameters are estimated by the method
of maximum likelihood. The estimates of maximum
likelihood are obtained according to the algorithm
being the histogram of earthquake distribution
over energy intervals (the number of earthquakes in
the jth interval).
Eq. 3 isreduced to the equation:
mj
j=r
where X = 10 -bAK. Eq. 5 has the solution in the interval (0, 1) if the following conditions are fulfilled:
{
mr < M0
M1 < (r + n/2)Mo
(6)
A.V. Ponomarev et al./Tectonophysics 277 (1997) 57-81
Given the constraints of Eq. 6, Eq. 5 can be solved
numerically, giving b-values. Then the estimate of
maximum likelihood for A can be expressed as
= Mo/T~p(D). The variances of b and A can be
calculated from:
V a r A - AqJ"(/~)
TO
Var/~ -
(7)
~o(/~)
TO~
where
r+,
~0"(/~) = ( A K i n 10) 2 ~
(8)
j210-JAKt;
j~r
3.2. Fractal dimension
To measure the fractal dimension of a set of
events, two estimates are most often used: box counting and correlation dimensions (Feder, 1987). The
box counting dimension is directly estimated using
the concept of fractal dimension:
log No
do = -lira/__, 0 log 1
(9)
where No is the number of cells of size l which
include at least one element from the set.
The correlation dimension is understood to be:
log C(1)
d2 = limt~ 0 - (10)
log 1
Here C(1) is the correlation integral:
C ( 1 ) = P( ri - r j <1)
(11)
m ( m - 1)
where P ( . . . ) is the number of pairs of events separated by a distance less than l, and m is the total
number of events. It is known that d2 < do, the
equality being true for so-called homogeneous fractals (Feder, 1987).
Note that Eqs. 9 and 10 require finding limits for
l --+ 0, operations that cannot be done when do and
d2 are estimated from observed data. For this reason
do and d2 are estimated by introducing the concept
of a scaling region, i.e., the range of I where log P or
log C are linear functions of log I. The scaling region
may be restricted from below by the accuracy of
65
the data, from above by the size of the spatial region
examined, or by actual changes in the structure of the
set examined (changes of slope in the log P versus
log 1 and log C versus log 1 functions).
The estimates of cell dimension need rather large
sample sizes to ensure statistical representativeness
(for example, 10,000 to 20,000 events as discussed in
Sadovsky and Pisarenko (1991). This circumstance
does not permit one to detect time variations of the
cell dimension from the data available. Estimation of
the correlation integral requires much smaller samples (Shuster, 1988; Anonymous, 1989b) so that the
correlation dimension is used to solve the present
problem. The penalty for using the correlation dimension is that it is not strictly speaking a fractal
dimension, but can be interpreted as approximating
the fractal cell dimension. The correlation dimension
is estimated by plotting histogram 1l and, on its
basis, the dependence of log C on log l, finding the
scaling region (or regions) and a robust regression
estimate of do (the index 2 will be omitted below).
The algorithm is discussed in detail in Sidorin and
S mirnov (1995).
3.3. Hurst exponent
Natural processes whose observation results in
time series of measurements can be studied by the
statistical method of normalized range or the Hurst
method (Feder, 1987).
The method of normalized range was initially developed by Hurst to solve problems connected with
the study of the Nile discharge and the accumulation of water resources. Later it was used to analyze
time series of various natures: meteorology data, the
number of Sun spots, the growth of tree rings, etc.
Let some time series ~(t) be given, t being understood as a discrete time which takes on integer
values. The mean value of ~(t) for the time period r
is equal to
,±
G = r
~(t)
(12)
t=l
Then, X(t) is the accumulated deviation of current ~(t) values around the mean G:
t
X(t, r) = Z
{~(u) - G}
(13)
66
A.V. Ponomarev et a l . / Tectonophysics 277 (1997) 57-81
Let us refer to the difference between the maximum and minimum values of the accumulated X
deviations by the name 'range'
D ( r ) = maxX(t, r)l_<t_<~ - m i n X ( t , r)l_<t<_r
(14)
The range then increases with increasing interval
duration r. Standard deviation S is estimated by:
S =
1
{~(t) - G} 2
t=l
(15)
It has been proved that for many time series the
observed normalized range D / S is well described by
the empirical relation:
D/S = (r/2) H
(16)
where H is the Hurst exponent. The value of H is
determined from the slope of the curve that approximates the dependence of log D / S on log r:
log D / S = H log r + const
(17)
For a 'purely' random process producing a sequence of independent trials, H = 1/2. For H > 1/2
the time series has a persistent behavior. In that case,
the present tendency of a process is supported, i.e.,
if, during some time, an increase in a parameter is
observed, then one can expect its increase during the
subsequent period of the same duration as well. For
H < 1/2 the process has an anti-persistent behavior: after a period of increase for a variable one can
expect a decrease.
Deviations of the Hurst exponent from the value
H = 1/2 can be regarded as an indicator of temporal
correlation of states of the physical system which
generates the process ~(t). For H > 1/2 the correlation of states is such that the tendencies of change in
the system are supported and for H < 1/2 they are
inhibited.
The present paper uses the following algorithm
for calculating H. Empirically a window that included T points of an observed time series was
chosen. The window was moved along the time
series at intervals of At. The window T was overlapped with no intersections of smaller windows
TI < "(2 < - . . ~ T. For each window rk the mean
D / S was calculated and then a log ( D / S ) k versus
log rk curve was plotted and fitted by Eq. 17. Its
slope determined the value of H for the time interval
coinciding with T.
This study includes two calculations of the Hurst
exponent for time series based on the following
parameters of acoustic activity:
(a) The distance between hypocenters of two successive acoustic events: ~(t) = Iri - r i _ l [ , H = Hr.
(b) The time interval between two successive
acoustic events ~(t) = t i - - ti-I, H = Hr.
3.4. Fracture concentration parameter
The fracture concentration parameter Pc at time t
is calculated from:
N,I/3
Pc(t) --
lauF
(18)
where N, = n~ V is the volume density of seismic
events that took place in the loaded sample until t, V
is the volume of the sample, and Ia~r is the averaged
fracture length (Sobolev and Zavyalov, 1980). The
parameter Pc has a clear physical meaning, being
the ratio of the average distance between fractures to
their mean length.
The mean length of the fractures accumulated in
the object is:
lavr = -n
li
(19)
i=1
Here 1 i is the length of the unit fracture. Since
the lengths of fractures were not measured during
the experiments, we assumed Ei cx l~, where Ei is
the acoustic energy released during the formation
of a fracture of length li (Sadovsky et al., 1987).
Assuming that Ei ~x U 2, where Ui is the amplitude
of the acoustic pulse, we adopted:
li = oU 2/3
(20)
where 0 is a constant.
Substituting Eqs. 19 and 20 into Eq. 18, we
obtain:
n2/3
Pc = v
-
(21)
vY 3
i=1
where
v = _1 VI/3
(22)
Calculations of Pc/v were made according to
Eq. 21 so that Pc was determined to within the
unknown constant factor v.
A.V. Ponomarev et al,/Tectonophysics 277 (1997) 57-81
Table 3 shows that the values d = 2.8, Hr = 0.6
and/4, = 0.6 correspond to the 'flat' part of the loading curve. The value d ~ 3 indicates that the distribution of acoustic events throughout the specimen is
nearly random and uncorrelated. The values Hr ~ 0.5
and H, ~ 0.5 indicate a weak temporal correlation of
acoustic activity: acoustic emission is close to a sequence of independent trials. Thus, during this period
there is little structure to the acoustic activity. The
slope of the recurrence curve b ~ 1 is considerably
higher than the typical value b ~ 0.5.
The values of b and d decrease and those of
/4,- and Ht increase during the falling part of the
loading curve. That means that at this stage a certain structure of acoustic activity arises: the acoustic
events cluster in space and the acoustic process acquires a temporal correlation of its states. The values
Hr > 1/2 and /4, > 1/2 indicate a persistent behavior of the acoustic process. This means that the
tendencies of variation for Iri -- r i _ l l and Iti -- ti II
are self-supported (Feder, 1987).
Experiment AE36 permits detailed examination
of the acoustic activity for the falling segment of the
loading curve. Fig. la clearly shows typical anomalies of the b-value before the larger failures (shown
by dashed lines in Fig. la): the b-value increases
some time before them and decreases immediately
before them. This kind of variation in b is recorded
for seismic activity as well and is used as a precursory phenomenon (Zavyalov and Sobolev, 1988;
Sobolev et al., 1991). Fig. la shows that other characteristics of acoustic activity also undergo variations
synchronously with the variations of b. In these cases
the changes of d, Ht and Hr take place in anti-phase
with b.
Table 2
Parameters of time windows
Specimen
Sample size
Window,
no. of events
Shift,
no. of events
AE36
AE39
AE42
30649
12584
27575
1000
500
1000
500
500
1000
67
4. Results
The chosen statistical parameters of acoustic activity were calculated in time windows that contained
a given number of events and 'sliding' along the time
axis at a prescribed interval (Table 2). The results are
shown in Fig. 1. The abscissas of the points in these
figures are the ends of the respective time windows
(they are equal to the moment of the last event in
each window).
In Fig. lb and c a change in the level of the chosen
parameters of acoustic activity is distinctively seen
(decreasing b and d and increasing Ht and Hr). This
change is associated with the passage of the loading
curves from the 'flat' part to the falling part (Fig. lb,
c). The respective values of the parameters are given
in Table 3.
In experiment AE42 (Fig. lc) the decrease in d
begins prior to the decrease in the b-value during
the transition from the flat to the falling part of the
loading curve. The decrease in d clearly precedes the
activation and the drop in load related to it.
Fig. 1 and Table 3 also present the deviations
of empirical data from the relation d = 3b. The
meaning of this parameter will be examined when
discussing the results.
Table 3
Parameters of acoustic regime: average values and 90% confidence intervals
Specimen
b
'Flat" part of loading curve
AE36
.
AE39
1.03 4- 0.02
AE42
1.08 :[- 0.03
Falling part of loading curve
AE36
0.74 4- 0.02
AE39
0.7 4- 0.1
AE42
0.68 4- 0.02
.
d
Hr
.
2.86 4- 0.07
2.80 4- 0.02
.
0.58 4- 0.02
0.58 4- 0.02
2.3l 4- 0.03
2.4 ± 0.4
2.23 4- 0.05
0.66 4- 0.02
0.62 :t: 0.1
0.62 4- 0.04
Ht
d - 3b
0.56 + 0.02
0.58 4- 0.03
- 0 . 2 4 4- 0.09
- 0 . 5 0 4- 0.07
0.74 4- 0.01
0.72 4- 0.1
0.67 4- 0.03
0.07 4- 0.09
0.4 4- 0.2
0.20 4- 0.08
.
68
A.V. Ponomarev et al./ Tectonophysics 277 (1997) 57-81
failure region. At first it was three-dimensional and
then converted in a narrow layer - - it became nearly
two-dimensional (fault-like). To exclude this obvious
effect the same statistical parameters were calculated
only for a 6-ram-thick layer, defined by the failure
plane (Fig. 3). The fractal dimension was calculated
After failure the b-value increases (similarly to
the case of seismicity), while d, Ht and Hr decrease,
and their values return to their initial levels.
We calculated the fractal dimension for the whole
specimen and decreasing of d-value reflects in a
varying degree simply the conversion of shape of the
AE36
1.0
0.8 I
0.6
0.4
1.8
1.6
1.4 F-L
1.0 ~-
:ff
0.8 ~
0.6
0.4
24000
28000
32000
36000
time,
40000
44000
s
AE42
1.2 ~1.0
.Q 0.8~
0.6
0.4
2.0
"o
1.8
1.6
\
i
_
i
i
I
i
]
1
\
I
~
1.4 ~ -
i
F
o608
1.0 ?L
0.4
8000
I
i
10000
t
12000
time,
i
14000
t
16000
s
Fig. 3. Parameters of the structure of the acoustic regime within the 6-ram-thick layer corresponding to the rupture plane in experiments:
(a) AE36, (b) AE42,
A. V Ponomarev et al. / Tectonophysics 277 (1997) 57-81
for a two-dimensional set of 'epicenters' of acoustic
events - - the projection of hypocenters on the failure
plane. According to the figure, the character of the
parameter variations of the acoustic regime structure
determined for the entire specimen is also observed
within the narrow fault layer.
The decrease of the d-value from approximately 2
to the value 1.7 indicates that during rupture formation not only the shape of the failure area is converted,
but also spatial clustering of events within the forming fault zone occurs. This suggests that during this
nucleation stage the acoustic regime acquires fractal
properties. The upper limit for scaling of the correlation integral was approximately 20-25 ram. This
value determines the upper limit for the typical size
of the spatial region of fractal structure formation.
In the time domain the acoustic regime is characterized by the presence of event clusters of 5-10
rain duration. The analysis was made on the basis
of the so-called clustering function (Eneva and Hamburger, 1989; Smirnov and Lyusina, 1992) (Fig. 4).
The clustering function is defined as:
q(3t)
qo(St)
g --
(23)
AE36
g
I
'x \
\ /
j,\
This statistical analysis of acoustic activity indicates the formation of a fractal structure of acoustic
activity during the deformation of rock specimens.
The structure arises at the stage of falling differential
stress, when the specimen releases its stored elastic
energy.
45000 S ]
',
'~
\
I~24o-3~oo~--i
5.1. Interaction of acoustic events
~'\
\?,
~, ~ ' ~ V ~
1
Clustering
<
0 --r-0
......
F
1
'
' ~- ....
J ~
10
where q(3t) is the number of pairs with the interval
3t between the events of catalogue and qo(6t) is
one for the Poisson process. Fig. 4 illustrates that
g > 1 for periods less then 5-10 min, indicating that
the density of events for time intervals of 5-10 min
duration is higher than predicted for random events.
We wish to note an essential difference in the
variations of b, d, Hr and Ht between experiments
AE39, AE42 and AE36. In experiments AE39 and
AE42 (during the transition from the 'flat' part of
the loading curve to the falling part) the changes
of b occur in phase with d and in anti-phase with
Hr. In experiment AE36 (at the falling part of the
loading curve) the situation is different: b and d
change in anti-phase, and Hr and Ht change in phase
with d and in anti-phase with b. Table 4 presents the
respective correlation matrixes.
One also notes an unexpected similarity between
the curves of d - 3b and of Hr. This similarity is
unexpected, because the calculations of b and d use
only the coordinates and energy classes of acoustic
events and not their occurrence times within the
window, while the calculations of Ht use only the
times of events and not their spatial location or
energy.
5. Discussion
5~ I
" ~3 ~ -
69
.......
>
I-
100
'
.......
I
,,,7
1000
time distance ~t, s
Fig. 4. Clustering function of acoustic events m experiment
AE36 for two periods.
The formation of a structure of acoustic activity,
i.e., the departure of the acoustic process from the
sequence of independent trials, indicates a rising
interaction between microcracks. This is directly
shown by the increase in the Hurst exponent that
accompanies the formation of the spatial structure of
acoustic activity.
A visual inspection of the distribution of acoustic
events shows that the period of increasing structure
coincides with the nucleation of the macrorupture.
Lockner et al. (1992a) and Reches and Lockner
A.V. Ponomarev et al./Tectonophysics 277 (1997) 57-81
70
Table 4
Correlation matrixes
b
d
Hr
Ht
d-3b
AE36
b
d
Hr
Ht
d-3b
1
- 0 . 6 ± 0.1
- 0 . 2 4- 0.1
- 0 . 4 4- 0.1
- 0 . 9 5 ± 0.01
- 0 . 6 4- 0.1
1
0.34-0.1
0.4±0.1
0.78 ± 0.05
- 0 . 2 ± 0.1
0.3 4- 0.1
1
0.6±0.1
0.3 4- 0.1
- 0 . 4 ± 0.1
0.4 4- 0.1
0.64-0.1
1
0.4 4- 0.1
- 0 . 9 5 i 0.01
0.78 ± 0.05
0.34-0.1
0.4±0.1
1
AE39
b
d
Hr
H,
d-3b
1
0.8 4-0.1
- 0 . 1 4- 0.2
-0.6±0.1
- 0 . 8 ± 0.1
0.8 ± 0.1
1
0.1 -4-0.2
-0.24-0.2
- 0 . 3 4- 0.2
- 0 . 1 ± 0.2
0.1 ± 0.2
1
0.44-0.2
0.2 ± 0.1
- 0 . 6 ± 0.1
- 0 . 2 ± 0.2
0.44-0.2
1
0.7 4- 0.l
- 0 . 8 4- 0.1
- 0 . 3 ± 0.2
0.2±0.2
0.7 d: 0.1
1
AE42
b
d
Hr
Ht
d-3b
1
0.88 ± 0.04
- 0 . 2 4- 0.2
- 0 . 6 ± 0.1
- 0 . 9 3 ± 0.03
0.88 + 0.04
1
-0.2-4-0.2
- 0 . 4 ± 0.2
-0.6±0.1
- 0 . 2 + 0.2
- 0 . 2 ± 0.2
1
0.0 ± 0.2
0.1 ± 0 . 2
- 0 . 6 + 0.1
- 0 . 4 -4- 0.2
0.0±0.2
1
0.74-0.1
- 0 . 9 3 i 0.03
- 0 . 6 4- 0.1
0.1 ± 0 . 2
0.7 ± 0 i
1
(1994) have analyzed experiment AE36 (summarizing some other empirical and theoretical results) to
suggest a scenario of the formation of shear rupture.
They suggested that the nucleation of a rupture is
controlled by interaction of microfractures: rupture
occurs when the concentration of microfractures becomes high enough for active microcrack interaction.
Note that this principle is consistent with the concept of avalanche-unstable rupture formation (AUF)
(Myachkin et al., 1975). In the early stages of loading, newly formed microfractures are sufficiently far
from one another so that their respective stress fields
do not overlap and the microdefects do not interact.
As more microfractures accumulate, the average distance between them decreases, and the stress fields
associated with individual microcracks begin to overlap, resulting in microfracture interaction. When the
density of the dominant microfractures reaches a
critical level, the system of microfractures becomes
unstable and a larger defect develops. This situation
characterizes the transition of the failure process into
a localized stage (Kuksenko, 1986; Kuksenko et al.,
1996).
The transition to active interaction in a system of
defects with accompanying loss of stability is statistically characterized by the concentration criterion of
failure. According to this criterion, failure arises at
a certain ratio of the average distance between the
defects to their average size, i.e., at a certain value of
the parameter of failure concentration Pc. Zavyalov
and Sobolev (1988) estimated Pc for different failure
scales.
Fig. 5 presents P~/v as a function of time calculated from Eq. 21 for specimen AE36. It is seen from
this figure that a jump of Pc occurs at the time of the
formation of acoustic structure (about 25,000 s). We
take the value Pc/n = 1.8 × 10 -2 at the jump as the
critical value for Westerly granite deformed under
these conditions. The critical value of the parameter
of failure concentration typical of seismic activity
is Pc* (Zavyalov and Sobolev, 1988). Then Eq. 21
yields v and, knowing the specimen volume V, the
volume of 0 is:
= 0.17 mm/mV z/3
(24)
The minimum amplitude of AE pulses completely
reported in the experiment is 10 mV (Table 1). According to Eq. 20, the size of the respective sources
of acoustic events is 0.8 rnm. This estimate is the
upper limit for the size of cracks emitting the lowest
reported acoustic signals.
Various corrections can be made that might provide a more accurate estimate of the microcracks that
produce the detected AE. Lockner (1993) compared
A.V. Ponomarev et aL ITectonophysics 277 (1997) 57-81
71
Pc/v
0 . 1 0
~ -
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Fig. 5. Crack concentrationparameter Pc in experimentAE36.
the cumulative AE events to the cumulative microcrack density from direct crack counting statistics.
He found that the AE events significantly underestimated the total number of microcracks that were
added to the sample in the pre-nucleation period.
The estimate was approximately 600 new microcracks for every recorded AE event. Therefore the
real value of Pc/v should be less than the estimate
done according to Eq. 21.
Furthermore, because of the stress shielding effects of the steel endplugs, AE is concentrated in the
central region of the sample. So the effective volume
V, as used in our calculation, should be about 1/2 of
the total sample volume.
Finally, it has been argued (Lockner et al., 1992b;
Lockner, 1993; Reches and Lockner, 1994) that Pc*
for laboratory experiments is apparently less than
P~* = 10 obtained for seismicity.
If we assume that the average undetected crack
size in no larger than the average length of detected
cracks, we have from Eq. 21 ~ < 0.017mm/mV 2/3.
Accordingly the size of microcracks emitting the
lowest reported signals of 10 mV is about 0.08 mm.
This value is in remarkably good agreement with
direct estimates of the size distribution of microfractures in Westerly granite samples during rupture
nucleation (Hadley, 1975; Lockner et al., 1992a;
Lockner, 1993). Thus, the formation of a non-random structure in acoustic activity coincides in time
with development of the critical concentration of defects, i.e., the structure arises when a strong defect
interaction takes place.
The interaction of defects that occurs through their
stress fields is statistically expressed in the fact that
a given acoustic event changes the probability of the
next event in some volume. The linear dimension of
this volume is approximately equal to the interaction distance of defects R. It is clear that the interaction distance depends on the defect size. Suppose that
defects interact effectively when their concentration
reaches the critical value. Then it follows from the
definition of the fracture concentration parameter that
R = 0.5P~l
(25)
where Pc* is the critical value of the fracture concentration parameter and l is the defect size.
72
A.V. Ponomarevet al./Tectonophysics 277 (1997) 57-81
The influence of an acoustic event on the next
event can be both positive (increased probability)
and negative (decreased probability). The mechanism of positive influence involves a redistribution
of stresses in the vicinity of the event. Both mechanisms are seen to act approximately in the same
volume controlled by the source dimension (Reches
and Lockner, 1994). The formation and evolution
of a fractal system of defects are presumably the
result of defect action. Note that in our case another
mechanism of negative influence results from the
method of loading: the occurrence of an event decreases the average loading rate due to the feedback
with the machine. Whereas the presence of the first
two mechanisms in the tectonosphere of the Earth
is evident enough, the last mechanism may not have
a natural counterpart and was developed mainly for
experimental convenience.
It should be noted that the above considerations
remain, at this point, hypotheses. The available data
are still not sufficient for a clear understanding of the
contribution of each of these mechanisms to the formation of acoustic structure. However, the hypotheses are consistent with the observed experimental
results.
5.2. Formation o f fractal structure
What is the nature of the observed decrease in d,
i.e., how is the fractal defect structure formed during
failure? At present there is no definitive answer to
this question. One may however suppose that such a
structure arises as a result of the positive, reinforcing
interaction of acoustic events. In our experiments
this cooperative interaction is indicated by persistency of acoustic activity during the transition to the
'structured' phase (which is indicated by an increase
in the Hurst exponent).
The positive influence of acoustic events on each
other leads to an instability in the system of defects,
and instability is required in a number of models for
formation of fractal objects (Feder, 1987; Pietronero
and Tozatti, 1988). In particular, Pietronero and Kupers (1988) describe an analysis of a model of aggregation of objects (particles and later particle clusters)
whose probability of clustering is determined not
only by their proximity to each other, but also by
the 'population' of other objects: the presence of
existing objects increases the probability of clustering (the formation of a new object) in their neighborhood. Essentially, Pietronero and Kupers (1988)
introduce a concentration criterion of aggregation.
This model was used to describe the formation of a
fractal structure in the evolution of an initially homogeneous set of particles which is accompanied by
a decreasing fractal dimension. The dependence of
the dimension on time is rather similar to the results
obtained in experiments AE39 and AE42.
Another example of the formation of a fractal
structure in a system with positive influence of events
is given by the concept of self-organized criticality
(SOC) (Ito and Matsuzaki, 1990). This concept is
used to model seismicity in a medium with constant
energy supply. Points of the medium are set in correspondence with values of some parameter z which
increases in a random way. When some threshold
is exceeded at a point, the value of z decreases at
that point and a microfailure occurs, the drop in z
being distributed among neighbouring points with
some probability (positive influence). Then, a rule is
employed to check the values of z at these points.
If z reaches the critical value anywhere, the entire
procedure is repeated for each respective point. The
totality of 'failed' points are regarded as an earthquake source. Within the framework of such a model
Ito and Matsuzaki (1990) find the formation of a
fractal field of failure (for a homogeneous and uniform initial distribution of z) which is accordingly
followed by a decrease in the fractal dimension.
Thus, the known mechanisms of the formation
of fractal structure suggest that this process may be
caused by a positive influence of acoustic events.
5.3. Long-term relation between b and d.
The mechanism of negative influence of earthquakes on each other permits one, under certain
assumptions, to explain the in-phase variation of b
and d. The negative influence, or inhibition of earthquakes, is the result of a reduction in stress in some
vicinity of the earthquake source.
Smirnov (1995) put forward the following postulates:
(1) The earthquake with the source size l0 results
in a stress drop in some vicinity R of the source. As a
result the earthquakes with the source size l ~ l0 are
73
A.V. Ponomarev et al./Tectonophysics 277 (1997) 57-81
inhibited during some time interval r in this vicinity.
According to the principle of self-similarity of seismicity we shall assume a power law dependence of
R and r on/0:
R = ~,l~
interval T. Substituting Eqs. 26 and 27 for R and r
we find that
N =
x
x
E(ctd+fld')/aE -(°td+fld')/a
(31)
(26)
Taking logarithms, we have:
r = 01~o
(27)
where )~, a, 0 and fl are parameters.r is the time
necessary for the surroundings to restore the ability
for a region to fail. In the theory of active media,
this time is known as a time of refractoriness, or
simply refractoriness. We shall apply this term to the
lithosphere. The physical nature of refractoriness lies
in the fact that the elastic stress increases to some
critical value - - the limit of strength - - While the
value of the stress is below that critical value: failure
will not occur.
(2) The seismicity has a fractal structure. If we
cover a region of size L with cells of the size A, then
the number of cells containing earthquakes, is given
by:
n = (L/A) d
(28)
where d = const. So the seismogenic zone is represented by a fractal with dimension d. It is possible to explain the fractal structure of seismicity
by the structure of the fault system (or some other
inhomogeneities) of the lithosphere, having fractal
properties (Turcotte, 1992).
Earthquakes are also distributed in a fractal manner over a time interval T: M non-empty intervals of
length 8 occur in the interval of length T, so that
M =
(29)
The parameter d, is the fractal time dimension.
(3) The size of the earthquake source determines
the energy of the earthquake:
g = el;'
(30)
According to the first postulate, the region of
influence R resulting from an earthquake cannot
overlap during time interval r. If we cover the region
of size L without overlapping earthquakes, the number of such earthquakes is given by Eq. 28, setting
A = R. Thus ( L / R ) a earthquakes can occur during
time interval r and ( L / R ) a × ( T / t ) at during time
logN = -bK
+dlogL +dtlogT
K = log E
+ B]
/
(32)
b = (otd + f i d t ) / a
B = b log E - d log )~ - d t log 0
Eq. 32 represents the generalized frequencymagnitude relation, taking into account the fractal
property of seismicity as described in postulates 13. Notice that Eq. 32 coincides with the generalized
frequency-magnitude relation obtained empirically
by Keilis-Borok et al. (1989).
The writers realize that the above postulates may
be over-restrictive. In particular, the postulate of
stress drop in the form of an unconditional inhibition
of subsequent earthquakes seems to be too strong
a requirement. Clearly at some point a probabilistic
formulation would be desirable. We believe, however, that the above form of postulates is justified
for the present state of the study. This formulation
clearly expresses the mechanism of negative influence of earthquakes. We consider it as a starting
point for further development of the model.
The b- and d-values, included in Eq. 32, are the
exponents of seismicity self-similarity in energy and
space, respectively. As it follows from Eq. 32, they
are connected by a relation a b - a d - bdt = 0. If
we substitute the values ot = 1 and a = 3, based on
known statistical and theoretical relations we obtain:
d = 3b-
fld
(33)
Typical values of b and d are at present known
with different degrees of reliability. The b-value has
been estimated by many authors using extensive
seismological data to a high degree of confidence.
It may be said that b ~ 0.4 - 0.6 for background
seismicity (or b m ~ 0.6 -- 0.9, were bm = 1.5b is the
familiar b-value for Gutenberg-Richter magnitudes).
According to a number of estimates, we have d
1.5 on average (Anonymous, 1989a; Keilis-Borok et
al., 1989; Sadovsky and Pisarenko, 1991; Smirnov,
A. V, Ponomarev et a l . / Tectonophysics 277 (1997) 57-81
74
1993). Substituting these values into Eq. 33 we find,
that b ~ - 0 . 3 - 0.3. If we accept that b ~ 0,
Eq. 33 simplifies to the well-known relation d = 3b
or d = 2bm (Aki, 1981). Furthermore, according to
Eq. 27, in this case the refractoriness time of the
region producing the earthquakes is independent of
the size of the region:
r = 0 = const
of b and d fall along the straight line d = 3b,
forming three clusters of data points. The evolution
of acoustic activity over time proceeds from cluster
A ('flat' part of the loading curve) to cluster B
(falling part of the loading curve). This transition
reflects the formation of spatial structure of acoustic
activity with a corresponding decrease in dimension
d and a nearly synchronous decrease in b.
Thus, within the framework of the postulates,
the spatial structure of defects largely controls the
energy structure. In this sense, the relation d = 3b
corresponds to some 'stable' dynamic state of the
failure process. In this case, the stress field and the
system of defects have reached a kind of steady-state
condition.
It is a well-known idea that the size distribution
of failures is determined by the inhomogeneities in
the medium. In this case the d-value characterizes
the distribution of individual events by size, while
the dimension presented in Fig. 6 characterizes the
size distribution of clusters of events. The fulfillment
of the condition d = 3b in this case for the 'cluster'
dimension means that, in terms of seismicity, the size
(34)
If the lithosphere refractoriness is the time required for stress to increase to the local strength
limit, the fact that r is independent of l0 means that
the magnitude of the earthquake does not depend
on tectonic activity (the rate of tectonic deformation). Then earthquake magnitude is controlled by
other factors, for example, geometrical structure of
the lithosphere (Fukao and Furumoto, 1985; Main
et al., 1990). In this case, the size distribution of
lithosphere elements would control the frequencymagnitude relation for earthquakes.
Fig. 6 presents the scatter diagram for the values
of b and d as found from acoustic and various
seismic data. It is seen that on the whole the values
3.5
IQOQQ
3.0
VVVVV
4
.
B
2.5
: ."z,
_ _ 8
"
2.0
v
--(3
C
v
/°
1.5
/
1.0
0.5
/
3
/
0.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
b
Fig. 6. Scatter d i a g r a m f o r d- a n d b-values: 1 = A E 3 6 , 2 = A E 3 9 , 3 = A E 4 2 , 4 = J a v a k h e t plateau (Smirnov, 1993), 5 = C a u c a s u s
(Smirnov, 1993), 6 = various areas o f the w o r l d (Keilis-Borok et al., 1989), 7 = N u r e k w a t e r reservoir ( S m i r n o v et al., 1994), 8 =
d=3b.
A. V. Ponomarev et al. / Tectonophy sic s 277 (1997) 57-81
distribution of individual events and of their groups
obey the same statistics. The physical explanation
of this can be found in Kagan and Knopoff (1981).
These authors model the source of an earthquake
as a set of microruptures that obey the statistic
of an ensemble of earthquakes. Thus, the earthquake
source becomes similar to a fault zone. Sadovsky and
Pisarenko (1991) use a similar approach to estimate
the maximum possible earthquake source size and
magnitude of an earthquake by the size of 'seismic
spots' or groups of earthquakes.
Cluster C in Fig. 6 corresponds to the seismicity
in the tectonosphere of the Earth. It has even lower
values of d and b. It is hard to tell whether this is a
consequence of an even more 'developed' structure
of inhomogeneities in the tectonosphere or whether
it reflects conditions of failure different from those
in the experiment. The results presented are mainly
obtained for shallow seismicity corresponding to
failure in a thin layer, practically a two-dimensional
object, while experiments with specimens involve
failure of three-dimensional objects. The nature of
the differences between clusters B and C requires
further study.
5.4. Temporal variations of the fractal structure of
acoustic activity
The results obtained in experiment AE36 indicate
that the structure of acoustic activity that formed
during failure continues to evolve over time. The relation between the variations of structural parameters
is different from that at the time of fault formation.
In particular, temporal variations of b and d occur
in anti-phase (Table 4, Fig. la). As a result, the
condition d = 3b fails for the temporal variations
of d and b. A similar empirical result is known at
the seismicity scale as well (Hirata, 1989; Henderson
et al., 1992, 1994; Smirnov, 1993; Smirnov et al.,
1994).
Fig. 6 demonstrates that the anticorrelated timevariations of b and d are concentrated around the average values lying on the line d = 3b. This suggests
that the medium structure controls the energy structure over the long term and the condition d = 3b is
correct only for long-term average values. The failure of the condition d = 3b and the change of sign
in the relation between b and d mean that the tem-
75
poral variations of acoustic structure are controlled
by a mechanism different from that examined in the
previous section. It is possible that this mechanism
also acts at the stage of structure formation, but its
effect is overshadowed by the formation of the main
fractal structure. The hypothesis that d = 3b for
the long-term seismicity structure can be tested by
studying d-b diagrams containing data of the different seismoactive regions which have different fault
structures.
Note that experiment AE36 was the only one in
which the rupture reached the steel loading piston at
the base of the specimen, preventing sliding along
the rupture surface, and leading to a second increase
in differential stress (Fig. la). As a result, a new
stress field formed in the specimen with feather
fractures around the main rupture. This circumstance
compelled us to regard this stage as a new stage
of failure under modified conditions. However, the
variations of the parameters of acoustic structure
persist in this case: d and Ht vary in anti-phase
with b and all parameters undergo bay-like changes
before a major failure (about 44,000 s), indicating
the persistence of the main regularities of acoustic
activity during a later increase in load.
Recalling that the falling part of the loading curve
was studied in only one experiment, one can hardly
assert that the identified features of acoustic activity
are universal. At the same time, in our opinion,
experiment AE36 reflects the general features of a
model based on the ideas of block structure. One
may suppose that under real conditions the growth of
a rupture runs against block boundaries, after which
a new stress state arises which controls the activation
of new fractures. In this regard, it seems worthwhile
to study the dynamics of the failure structure during
the falling part of the loading curve (Sadovsky and
Pisarenko, 1991, pp. 27-28).
Main (1992) showed the importance of a correlation between d and b for understanding the character
of the failure process. He considered a modified
Griffith criterion for a two-dimensional fractal set
of aligned elliptical cracks with a long-range interaction potential (i.e. with positive feedback in the
failure process). In this system the derivate gd/gb
influences to a mechanical hardening or weakening effect. Main (1992) discussed two models for
spatial-temporal evolution of the seismicity due to:
76
A.V. Ponomarev et al. / Tectonophysics 277 (1997) 57-81
(A) progressive alignment of epicenters along an incipient fault plane; and (B) clustering of epicenters
around a potential nucleation point on an existing
fault trace. In the first case the correlation between
d and b (the positive sign of 3d/3b) leads to the
mechanical weakening associated with short-range
clustering of cracks in thin layer and a gradual loss
of one of the degrees of freedom of the system on a
large scale. In case (B) the anticorrelation of d and b
leads to weakening associated with long-range clustering on the fractal system of jogs and asperities.
This concept cannot be applied directly to our
experiments because it is based on an assumption
that the distance between cracks is much more than
the crack size. This condition appears to be violated
at least during the transition from diffuse to localized failure. However, these theoretical results can
be considered as a qualitative tendency. According to
this model, the first stage of the failure process (the
formation of the fault) represented by experiments
AE39 and AE42, corresponds to case (A) of Main's
model. Positive correlation of d and b time-variations results in mechanical weakening at this stage.
The second stage (evolution of faults following formation) in experiment AE36 corresponds to model
(B); and the anticorrelation of d and b during this
stage provides mechanical weakening again. Weakening during failure of the rock sample is consistent
with common tenets of damage mechanics and these
experiments are consistent with Main's idea.
The departure from the relation d - 3b may be
related to a redistribution of strength and stress according to failure size (Smirnov, 1995). This is seen
as follows. From Eq. 33 one has:
d - 3b = -/3d,
(35)
and 13 = 0 for d = 3b. This relation implies that the
'life time' of a region does not depend on the size of
that region: r -----691~o= const.
If we accept that r = const, is due to the
stress build up until the strength limit is reached (or
longevity, in terms of kinetic strength), then the failure of d = 3b implies a redistribution of strength and
stress over the size scale. If 13 > 0, then r is smaller at
smaller scales, because stresses concentrate at smaller
scales. This situation may possibly occur during the
growth and concentration of small fractures in some
spatial region, which thereby diminishes its effective
strength. If/3 < 0, then stresses concentrate at higher
scales. It is possible that this situation arises after large
events, when, as a result of failure in a large volume,
individual sections become overloaded.
When b ~ 0, it follows from Eq. 35 that the
variations of d - 3b should be related to those of the
time structure of acoustic activity. Such a connection
is in fact observed. The Hurst parameter Ht varies
in phase with d - 3b. We did not investigate the
theoretical relation between Ht and dr, but both
characterize the degree of time correlation between
states of the acoustic process.
Departures from the relation d = 3b have the
highest values in time intervals following the largest
acoustic events, i.e. they manifest themselves in a
'transient' acoustic mode caused by large dynamic
events which affect the stress pattern in some region. This circumstance agrees with the hypothesis
that the departure from the relation d = 3b reflects
a transitory distortion of a balance between stress
structures and the system of defects.
Fig. 7 presents the results of a calculation of statistical characteristics we chose for the seismicity
of the Javakhet plateau in the Caucasus. A regional
catalogue of earthquakes is used where the events
with energy classes of K > 7.0, from 1962 to 1983,
and of K > 6.7, from 1984 to 1990, are completely
reported. The selected data comprise 8893 earthquakes. The parameters were calculated in a window
of 500 events 'sliding' over time at intervals of 250
events.
Comparison of Fig. 7 and Fig. la indicates a similarity of time variations of structure parameters for
seismic and acoustic activity. In particular, changes
of the parameters occur some time before the 'main'
events of the region (the Spitak and Racha earthquakes) as well as increased values of d, Hr, Ht and
d - 3b and a decreased value of b which occur during
the 'main' event.
Table 5 shows that the correlation of structure parameters for seismicity is similar to the situation in
experiments AE36 and AE42 (Table 4). This circumstance is quite consistent with the fact that the variations of seismicity reflect the variations of the fractal
structure already formed rather than its formation.
It is worth mentioning that the period of formation of the earthquake source can involve complex
and rapid fluctuation of the parameters considered
A.V. Ponomarev et al./Tectonophysics 277 (1997) 57-81
77
K
3
4
17
21
r~ ,li t i,~,lll,l~L,,,iI 1 J,l,I ,,~ i ' ,l~i,dl,! ,, , ,,i,i
10
0.8 m
..Q
0.6 L
0.4-0.2
1.9
1.6
"0
1.3
1.0 ~
J 0.7
1.0
m
0.8
0.6
0.4
m
F
I
I
t
J
I
65
70
75
_~
r
i
I
1
i
80
85
90
1.0
:£
0.8
0.6
0.4
1.0
0.0
--
-1.0
--
-6
60
95
time, year
Fig. 7. Parameters of the seismicity structure for the Javakhet plateau. The top of the figure presents the largest earthquakes of the area: 1
= Dmanisi, 2 = Paravan, 3 = Spitak, 4 = Racha.
here. In this regard, Arefyev and Tatevosyan (1991)
and Dorbath et al. (1992) investigated the structure
and seismic regime of the Spitak earthquake region
(Dec. 7, 1988, M = 7), analyzing aftershock activity
in detail. They found transient clustering of shocks
which they used to construct a multisegment model
of rupture propagation. They also showed that the
strong events can temporarily disturb self-similarity
78
A.V. Ponomarev et al./Tectonophysics 277 (1997) 57-81
Table 5
Correlation matrix for the Javakhetplateau
b
d
Hr
Ht
d-3b
b
d
Hr
Ht
d-3b
1
0.4±0.1
-0.3 4-0.2
-0.65:0.1
-0.86 ± 0.05
-0.4 -4-0.1
1
0.3 ±0.2
0.0±0.2
0.8 ± 0.1
-0.3 -4-0.2
0.3±0.2
1
0.7±0.1
0.0 ± 0.2
-0.6 -4-0.1
0.04-0.2
0.7 ±0.1
1
0.4 ± 0.2
-0.86 ± 0.05
0.8+0.1
0.04-0.2
0.4±0.2
1
of the failure process in the vicinity of the earthquake focus. Similar properties have been found in
the parameters of AR.
Thus, experiments carried out can be regarded as
physical modeling of the origin (AE39 and AE42)
and evolution (AE36) of a seismogenic zone. The
correspondence between this laboratory model and
natural systems needs further research, but the above
preliminary results seem promising.
(6) The degree of temporal correlation of
the acoustic regime undergoes variations in time.
Changes in temporal correlation of events are associated with changes of d and b.
(7) The character of variations of the structure
of the acoustic activity is similar to that of seismicity, indicating that this is a promising approach for
physical modeling of seismicity.
Acknowledgements
6. Summary
(1) Loading of rock specimens using negative
feedback makes it possible to reveal regularities
in the formation and evolution of the structure of
acoustic activity.
(2) A fractal structure of acoustic activity forms
during the failure of rock specimens. Formation of
the structure is caused by active interaction of micro-defects and thus reflects the state of the medium.
Structuring of acoustic activity is observed at the
stage of falling differential stress, when the specimen returns the elastic energy stored in it.
(3) Spatial structuring of acoustic activity is accompanied by an increase in the temporal correlation
of the acoustic regime and a decrease of the b-value.
(4) The structure of acoustic activity undergoes
temporal variations that agree with the sequence
associated with the large-scale failure. This circumstance, in particular, shows that the parameters of
structure may be useful as failure predictors.
(5) After the development of the acoustic regime
structure, the changes of fractal dimension d and of
the frequency-magnitude relation parameter b occur
in phase, and their values satisfy the relation d = 3b.
For a structure already formed this relation seems to
be fulfilled only for average d and b values, their
temporal variations taking place in anti-phase.
The experimental data were obtained in the
Laboratory of Rock Mechanics, US Geological
Survey, with participation of S.A. Stanchits and
V.S. Kuksenko within Project 02.09-12, Direction
IX, Russian-American agreement on Environmental
Protection Research. The writers are grateful to J.D.
Byerlee for fruitful discussion and critical remarks.
The writers thank G.A. Sobolev for his constant attention and support throughout the study. We are
very grateful to reviewers of the paper P.R. Sammonds, K. Mair, P. Reasenberg and an anonymous
reviewer whose comments greatly improved the clarity of the text. The work was made with the financial
support of the Russian Foundation for Basic Research and International Science Foundation.
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