Physical modeling of the formation and evolution of seismically active fault zones

ELSEVIER Tectonophysics 277 (1997) 57-81

TECTONOPHYSICS II

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 nonran- domness of the space- t ime-energy distribution of

* Corresponding author. Fax: +7 095 255-6040; e-mail: geodin @ adonis.iasnet.ru

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 mani- festation 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; Narkun- skaya 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 appli- cability 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 clar- ification 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 Gutenberg- Richter 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 (Anony- mous, 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 re- viewed 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 (Vino- gradov 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

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https://www.researchgate.net/publication/228077888_The_Role_of_Acoustic_Emission_in_the_study_of_Rock_Fracture?el=1_x_8&enrichId=rgreq-88968008-f8e0-4fcf-a8bc-1321c836bd5c&enrichSource=Y292ZXJQYWdlOzIyMzA2MzIwMjtBUzoxODg1MTc5MDA2Mjc5NjlAMTQyMTk1NzU3ODQzMA==

A. V Ponomarev et aL /Tectonophysics 277 (1997) 57-81 59

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 specimen- machine 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 remov- ing 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

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 chan- nels 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 er- ror 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. Ponomarev et 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 t ime-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 mm 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-uni- formly, 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 mm 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, small- amplitude 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 61

°

A E 3 6 "

oo o o

i i .

¢ M

600 - -

500 - - ~ " " - ~ , E A E 400 -- ~ |

300 - - ~ 100

200 -- 50 100 - -

0 J-'=' f 0 i

1.o ~ ' I ' I ! ' I ' I ' li

0.8 ' ' "Q 0.6

i i

2.8 ,

2.0 i 1.0 ~~_ , ~ o . 8 ' '

0.6 I

0.8

0,6 I 1,o_

e~ e~ 0.0 -6

- 1 . 0 ~ I

20000 24000 28000 32000 36000 40000 44000 48000 t ime , 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

: f

,-?, "1o

b

AE39 o

o', MPa ~ EAE -

400 ~ 100

300

200 0 1.2

I I ] ; I I I i J

1.5 I I I I i I I ~ d

1.0 0.8 0.6 0.4

3.0

2.5

2.0

0.8 I 0.6

0.4 0.8 ~ 0.6

0.4 1.0 ~-

0.0 ~

-1.0 5000

i t

T

I I t. I -"- I ! I I 8000 9000 10000 11000 12000 13000

time, s

I I

I I

t I 6000 7000

Fig. 1 (continued).

to their reporting level in the catalogues. Table 1 con- tains the adopted lowest reported amplitude classes, sample sizes and the parameters of the recurrence curve for AE pulses.

3. Methods of data analysis To characterize the structure of acoustic activity

a number of statistical parameters from catalogues

A.V. Ponomarev et al./Tectonophysics 277 (1997) 57-81 63

or, M P a

C

AE42 O

i

EAE

..Q

"O

: f

t"o

-6

600 500 ~ ~ 400 300 200 100 ~

1.2 --t 1.0 -- 0.8 0.6 t ~ 3 . 0 ~ ~

2.5

2.0 t ~ I 0 . 8 _

0.6 ~

o.4 I ~

0 . 8 _ 0.6 0.4 1 I 1 . 0 ~

0.0

-1 .o ~ I 4000 6000

I I r

I I

= r

~ 80 I I J

40

i ~ J , i , j l 4 _ _ 1 I

p

p

I

, j

8000 10000 12000 14000 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.

64 A. V Ponomarev et a l . / Tectonophysics 277 (1997) 57-81

Z

10000

1000

100

10

,"~'~ :----.: AE38 b--o:8o

\ t I

1 2 3 4 5 K o

1 0 0 0 0 = • ,-" ",.,.

b 091

1000

z I O0 ,

i,

1o ,

I , , , , i , , , , i , , , , i , , , , t , , , , i

0 1 2 3 4 5

K 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.

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 - AK/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

Table 1 Representative class Kmi n of catalogues of acoustic events and the parameters of the recurrence law: N =- A x 10 -h(K 2)

Specimen Krnin Sample size b A (c I q)

AE36 2.0 30649 0.80 ± 0.01 2.01 + 0.04 AE39 1.9 12584 0.95 4- 0.01 2.7 4- 0.1 AE42 1.4 27575 0.91 4- 0.01 1.45 4- 0.02

described in (Sadovsky and Pisarenko, 1991). The recurrence law (1) is represented in the form

log Nj ---- - -bAKj + log A (2)

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 (3)

A K i n 10M1 + TAro(b) = 0

where the following notations used: r+n

M o = E m j j=r

r+n

M1 = E j m j

j =r (4) r+n

~o(b) = y ] 10 -jb~xK

j ~ r r+n

~o'(b) = - A K I n 10 ~--'~. j lO -kbAK

j=r

m j being the histogram of earthquake distribution over energy intervals (the number of earthquakes in the j th interval).

Eq. 3 isreduced to the equation:

j=r

where X = 10 -bAK. Eq. 5 has the solution in the interval (0, 1) if the following conditions are fulfilled:

{ mr < M0 (6)

M1 < (r + n/2)Mo


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

(7)

Var/~ - ~o (/~) TO~

where

r+, (8) ~0"(/~) = (AKin 10) 2 ~ 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 (9) do = -lira/__, 0 log 1

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 sep- arated 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

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 = - ~(t) (12)

r 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_<~ - minX( t , r)l_<t<_r (14)

The range then increases with increasing interval duration r. Standard deviation S is estimated by:

1 S = {~(t) - G} 2 (15) t=l

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 approxi- mates 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 over- lapped 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) = I r i - 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) -- (18) lauF

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 = -- li (19) n

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 67

Table 2 Parameters of time windows

Specimen Sample size Window, Shift, no. of events no. of events

AE36 30649 1000 500 AE39 12584 500 500 AE42 27575 1000 1000

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 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 anoma- lies 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 3 Parameters of acoustic regime: average values and 90% confidence intervals

Specimen b d Hr Ht d - 3b

'Flat" part of loading curve AE36 . . . . . AE39 1.03 4- 0.02 2.86 4- 0.07 0.58 4- 0.02 0.56 + 0.02 -0 .24 4- 0.09 AE42 1.08 :[- 0.03 2.80 4- 0.02 0.58 4- 0.02 0.58 4- 0.03 -0 .50 4- 0.07

Falling part of loading curve AE36 0.74 4- 0.02 2.3l 4- 0.03 0.66 4- 0.02 0.74 4- 0.01 0.07 4- 0.09 AE39 0.7 4- 0.1 2.4 ± 0.4 0.62 :t: 0.1 0.72 4- 0.1 0.4 4- 0.2 AE42 0.68 4- 0.02 2.23 4- 0.05 0.62 4- 0.04 0.67 4- 0.03 0.20 4- 0.08


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

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

:ff

1.0

0.8 I 0.6 0.4

1.8

1.6

1.4 F-- L

1.0 ~- 0.8 ~

0.6

0.4 24000

AE36

28000 32000 36000 40000 44000 t i m e , s

AE42 1.2 ~- 1.0 .Q 0.8 ~ 0.6 0.4 2.0

1.8 ~ "o 1.6

1.4 ~ -

\ \

i _ i i I

I i i F 1.0 ?- L o608

0.4 I i t ] 1 i t 8000 10000 12000 14000

t i m e , s 16000

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 69

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 Ham- burger, 1989; Smirnov and Lyusina, 1992) (Fig. 4). The clustering function is defined as:

q(3t) g -- (23)

qo(St)

g 5 ~ I

AE36

I

"~ 3 ~ - 45000 S ]

' x \ j , \ ', \ / '~ \

I ~24o-3~oo~--i ~'\ \ ? , ~ , ~ ' ~ V ~

1

< Clustering >

0 - - r - - . . . . . . F ' ' ~- . . . . J ~ . . . . . . . I - ' . . . . . . . I , , , 7

0 1 10 100 1000 t ime distance ~t, s

Fig. 4. Clustering function of acoustic events m experiment AE36 for two periods.

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

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.

5.1. Interaction of acoustic events

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

70

Table 4 Correlation matrixes

A.V. Ponomarev et al./Tectonophysics 277 (1997) 57-81

b d Hr Ht d-3b

AE36 b 1 d - 0 . 6 ± 0.1 Hr - 0 . 2 4- 0.1 Ht - 0 . 4 4- 0.1 d-3b - 0 . 95 ± 0.01

AE39 b 1 d 0.8 4-0.1 Hr -0 .1 4- 0.2 H, - 0 . 6 ± 0 . 1 d-3b - 0 . 8 ± 0.1

AE42 b 1 d 0.88 ± 0.04 Hr - 0 . 2 4- 0.2 Ht - 0 . 6 ± 0.1 d-3b - 0 . 93 ± 0.03

- 0 . 6 4- 0.1 - 0 . 2 ± 0.1 - 0 . 4 ± 0.1 -0 .95 i 0.01 1 0.3 4- 0.1 0.4 4- 0.1 0.78 ± 0.05

0.34-0.1 1 0.64-0.1 0.34-0.1 0 . 4 ± 0 . 1 0 . 6 ± 0 . 1 1 0 . 4 ± 0 . 1

0.78 ± 0.05 0.3 4- 0.1 0.4 4- 0.1 1

0.8 ± 0.1 -0 .1 ± 0.2 - 0 . 6 ± 0.1 - 0 . 8 4- 0.1 1 0.1 ± 0.2 - 0 . 2 ± 0.2 - 0 . 3 ± 0.2

0.1 -4-0.2 1 0 .44-0 .2 0 . 2 ± 0 . 2 - 0 . 2 4 - 0 . 2 0 .44-0 .2 1 0.7 d: 0.1 - 0 . 3 4- 0.2 0.2 ± 0.1 0.7 4- 0.l 1

0.88 + 0.04 - 0 . 2 + 0.2 - 0 . 6 + 0.1 -0 .93 i 0.03 1 - 0 . 2 ± 0.2 - 0 . 4 -4- 0.2 - 0 . 6 4- 0.1

-0 .2-4-0 .2 1 0 . 0 ± 0 . 2 0.1 ± 0 . 2 - 0 . 4 ± 0.2 0.0 ± 0.2 1 0.7 ± 0 i - 0 . 6 ± 0 . 1 0.1 ± 0 . 2 0.74-0.1 1

(1994) have analyzed experiment AE36 (summariz- ing 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). Ac- cording 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 ~ - . . . . . h . . . . . I . . . . . . ~ . . . . ! . . . . . T - . . . . . . . . . . . . . . . . . . . . . . . . .

__ I J J !

i

i i i i i

4 . . . . . I . . . . . ~ . . . . . I . . . . . T

r I I r

3 - . . . . i . . . . . . . . . . . . . . 1 -

2 . . . . ~ . . . . i . . . . . ~ . . . . . . . . r -

I t

i i i i

0 . 0 1 . . . . . i . . . . . , . . . . . . ~ . . . . . . . . . • _

9 . . . . I . . . . I . . . . . . % . . . . . I . . . . . ' - - -

. . . . . I . . . . . I - - - - - i " . . . . . . J . . . . . r "

. . . . I . . . . . . . . . . 1 . . . . p . . . . . I - - -

4 - ~ - ~ - ~ - , , I , , , , r , , , ~ i i J i i i , ~ i 1

2 0 0 0 0 2 5 0 0 0 3 0 0 0 0 3 5 0 0 0 4 0 0 0 0 4 5 0 0 0

t,s Fig. 5. Crack concentration parameter Pc in experiment AE36.

the cumulative AE events to the cumulative microcrack density from direct crack counting statistics. He found that the AE events significantly underes- timated 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.


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 persis- tency 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 Ku- pers (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 neigh- borhood. 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 uni- form 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


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~ (26)

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 man- ner 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

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)

Taking logarithms, we have:

logN = - b K + d l o g L + d t l o g T + B ]

K = log E / (32)

b = (otd + f i d t ) / a

B = b log E - d log )~ - d t log 0

Eq. 32 represents the generalized frequency- magnitude relation, taking into account the fractal property of seismicity as described in postulates 1- 3. 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 ab - ad - bdt = 0. If we substitute the values ot = 1 and a = 3, based on known statistical and theoretical relations we obtain:

d = 3 b - f ld (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,

74 A. V, Ponomarev et a l . / Tectonophysics 277 (1997) 57-81

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 (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 frequency- magnitude 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

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

3.5

3.0

2.5

2.0

--(3

1.5

1.0

0.5

0.0

C

/ /

/

B

: ."z, "

° / v v

3

IQOQQ

VVVVV 4

.

_ _ 8

0.0 0.2 0.4 0.6 0.8 1.0 1.2

b Fig. 6. Scat ter d i a g r a m for d- and b-values: 1 = AE36 , 2 = AE39 , 3 = AE42 , 4 = Javakhet p la teau (Smirnov, 1993), 5 = Caucasus (Smirnov, 1993), 6 = var ious areas o f the wor ld (Kei l i s -Borok et al., 1989), 7 = Nurek water reservoi r (Smirnov et al., 1994), 8 =

d = 3 b .

A. V. Ponomarev et al. / Tectonophy sic s 277 (1997) 57-81 75

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 time- variations 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-

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. Weak- ening 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


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0.4

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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


Table 5 Correlation matrix for the Javakhet plateau

b d Hr Ht d-3b

b 1 -0.4 -4- 0.1 -0.3 -4- 0.2 -0.6 -4- 0.1 -0.86 ± 0.05 d 0.4±0.1 1 0.3±0.2 0.04-0.2 0.8+0.1 Hr -0.3 4-0.2 0.3 ±0.2 1 0.7 ±0.1 0.04-0.2 Ht -0.65:0.1 0.0±0.2 0.7±0.1 1 0.4±0.2 d-3b -0.86 ± 0.05 0.8 ± 0.1 0.0 ± 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. 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.

(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

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 at- tention and support throughout the study. We are very grateful to reviewers of the paper P.R. Sam- monds, K. Mair, P. Reasenberg and an anonymous reviewer whose comments greatly improved the clar- ity of the text. The work was made with the financial support of the Russian Foundation for Basic Re- search and International Science Foundation.

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Physical modeling of the formation and evolution of seismically active fault zones

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