Fractal character and mechanism of rock bursts

Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. Vol. 30, No. 4, pp. 343-350, 1993

0148-9062/93 $6.00 + 0.00 Pergamon Press Ltd

Printed in Great Britain

Fractal Character and Mechanism of Rock Bursts H. XIEt W. G. PARISEAU:I:

Rock bursts are a potential hazard to mine structures and underground personnel. Many studies have been conducted to understand the causes of rock bursts and to predict their occurrence, however, few successes have been achieved. In this paper, the associated microseismicity of rock bursts is studied by using fractal geometry and damage mechanics. Based on the number-radius relation of fractals, the distributions of previously reported micoseismic event locations were examined and found to have a fractal clustering structure. The degree of clustering of microseismic events increases with the approach of a main rock burst that corresponds to a decreasing fractal dimension. The lowest fractal dimension is generally produced near the occurrence of a rock burst. Thus, the fractal dimension has potential use as a rock burst predictor. In seismology, this fractal nature of rock bursts is consistent with the conclusion of a lower fractal dimension (or b-value)being associated with the occurrence of a main earthquake. The fractal and physical mechanisms of rock bursts are analyzed in theory using damage mechanics and the fractal concept. A strong failure (a rock burst, or an earthquake) is seen as equivalent to a fractal cluster of crackings within the rock mass. The energy release E of a fractal cluster of crackings within the rock mass increases exponentially with a decrease of the fractal dimension D in the form: D =Ci x exp[-C2E]. Thus, the fractal nature of rock bursts and earthquakes is well-explained in theory, and a better understanding of rock bursts is obtained.

tO date only a few successful predictions have been achieved [2]. The causes may be due to two aspects: the Rock bursts are experienced in underground mining at first may be that the exact physical process of rock bursts various localities in the world, causing death and injury is very complicated and too difficult to measure, and the to underground miners and damage to mine structures second may be that the initial data recorded by these (drifts, stopes, etc.). To date, many studies have been approaches are not completely utilized. Even so, the conducted to understand the causes of rock bursts and microseismic technique (or acoustic emission technique) outbursts and to predict their occurrence. The approaches is still commonly used in monitoring rock bursts. Microseismic data analysis, as an initial step, includes made to detect rock bursts include: the microgravity method, rheological method, rebound method, drilling- both the location of rock noise sources and the rate yield method, microseismic method, and so on [1]. at which the noises occur. Other analyses may include Although all of these methods have been used, none energy release rates, energy per event, and, at a more is completely reliable, and few are useful in the rapidly advanced level, analyses of the microseismic waveform advancing mining environment. The U.S. Bureau of to determine the failure mechanism and stress conditions Mines recognized microseismic technology as a potential within the rock. It has been determined that the microtool for rock burst prediction as early as 1939, however, seismic noise rate (number of microseismic events per unit of time) increases as failure approaches, and the neartDepartment of Mathematics, China University of Mining & Technol- failure noise rate may be 10-100 times the background ogy, Xuzhou, Jiangsu 221008, Peoples' Republic of China; Inter- level or stable noise rate. This increase will then be national Center of Material Physics, Academia, Sinica, Shenyang, followed by a dramatic decrease of microseismicity in a Peoples' Republic of China. :I:Department of Mining Engineering, University of Utah, Salt Lake broad region surrounding the location of the impending burst. This phenomenon is called the seismicity anomaly. City, UT 84112, U.S.A. 343 INTRODUCTION

344

XIE and PARISEAU: FRACTAL ROCK BURSTS

Therefore, the microseismic noise rate is commonly used as the precursor factor of rock bursts. Observational results [1, 3, 4, 13] have also indicated that the occurrence of a rock burst certainly corresponds to seismicity anomaly, however, reversing this case does not hold true. On the other hand, some rock bursts occur in active zones of microseismicity, while others do not. Thus, microseismic noise rate alone is not reliable in the prediction of rock bursts. In this regard, the distribution of the microseismic event locations is often overlooked. In fact, this distribution is a realistic record of the damage evolution process experienced by a rock burst. It may play an important role in the prediction of rock bursts. More recently, Coughlin and Kranz [19] have studied this distribution by using the correlation exponent method and have found that it has a fractal structure. In this paper, we use the number-radius relation to examine the distributions of microseismic event locations. We find these distributions really are fractals. A very interesting fractal character of the distributions is obtained. By using damage mechanics and fractal geometry, we analysze the fractal and physical mechanisms of rock bursts to obtain a better understanding of rock bursts.

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FRACTAL MEASUREMENT

METHODS Rock, like many other materials, when stressed, produces an impulse generated by microcracking associated with the release of strain energy stored internally within its structure and manifested in the form of elastic waves. These elastic waves travel from the point of origin (hypocenter) within the rock to a boundary where, using acoustic emission, the location of the hypocenter (also called the microseismic event location) can be measured. From a physical point of view, the microseismic noise events occurring during the monitoring process of rock bursts characterize the microfracturing process (or damage evolution process) before rock bursting. Studies [6-11, 20] have indicated that the damage evolution process from the small-scale (a microcracking, cm) to the large-scale (an earthquake, km) is a fractal. Therefore, the rock burst which is damage and fracture with an intermediate-scale (m) may also be a fractal. The microseismic event locations construct a spatial distribution of a point set in which a point corresponds to a cracking surface or volume element in physical space. Thus, the fractal dimension of the damage evolution process experienced by a rock burst can be directly measured from the distribution of the point set. Consider a sphere with radius r at center point x over the distribution. The total number of microseismic events inside this sphere over the distribution can then be counted and is denoted by M ( r ) , as shown in Fig. 1. Thus, we obtain a set of data M(r~) associated with different radii r~ (i = 1,2 . . . . ). From fractal geometry [10, 11], there is a relation between M(r~) and r~ in the form: M ( r ) oc r ~ for the line distribution of the point set,

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for a fractal distribution with fractal dimension D. Equation (1) is also called the number-radius relation, and the fractal dimension D is called the clustering dimension which is equal to the slope of the log M ( r ) -log(r) plot (Fig. 3). In this fractal measurement, the center point x of the spheres with different radii ri is chosen as the mass center of the distribution. The fractal dimension of the distribution is also obtained from a correlation integral concept [11, 12]. The correlation integrals C ( r ) for the hypocenter distributions (p~ ,P2 . . . . . PN) as shown in Fig. 1 can be obtained in the form: C ( r ) = [No. of pair (Pi,Pj) with I P i - P j [ < r]/N2, ( i , j = 1,2, 3 . . . . . N)

(2)

where N is the total number of microseismic events over the distribution. If the distribution has a fractal structure, we can express C ( r ) as: C(r) ~ r °

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where D is a kind of fractal dimension called the correlation exponent giving the lower limit of the Hausdorff dimension [10]. This correlation exponent method [equations (2) and (3)] can be combined with the determination of epicenters in seismicity by which the coordinates of the pair (pi, pj) can be calculated. In this way, the fractal measurement can be performed by the computer using software monitoring microseismic or macroseismic events. Thus, like information on daily rock noise rate, the fractal dimension can be daily shown by the micro- or macroseismic monitoring system.

XIE and PARISEAU:

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FRACTAL CHARACTER OF ROCK BURSTS

Leighton [5] reported monitoring microseismic data of a rock-burst-prone pillar in a metallic ore vein in the galena mine, up to and through failure. The stope w a s being excavated by overhand cut-and-fill mining. A major rock burst occurred on 24 May 1979. In addition to information on the daily rock noise rate, plots of rock noise locations were compiled by computer. The plots of the rock noise locations for 5 days prior to the 24 May burst are reproduced in Fig. 2. We apply the fractal measurement method discussed in the previous section to measure the number-radius relation of the distribution of the rock noise locations. The plots of log M(r)

vs log (r) are shown in Fig. 3. From Fig. 3, it is seen that there exists a good linear correlation between log M(r) and log (r). This indicates that the distribution of the rock noise locations is a fractal and has statistical selfsimilarity. The reliabilities of linear regression are all larger than 0.96. The fractal dimensions for the 5 days prior to the 24 May burst are directly obtained from the slopes of plots in Fig. 3. Figure 4 shows the relation between the fractal dimension and the date for these 5 days. From Fig. 4, a very interesting result was found; prior to the major rock burst, the lowest fractal dimension (D = 0.4104) occurred on 24 May. This result is consistent with the conclusion in seismology that the lowest fractal dimension (or b-value) occurs near the

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may 20 may 21 may 22 may 23 may 24 DATE Fig. 4. The correlation between fractal dimensions and date for the 5 days prior to the 24 May rock burst.

XIE and PARISEAU:

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XIE and PARISEAU:

FRACTAL ROCK BURSTS

time of a strong earthquake [11]. For example, D = 0.45 for the M = 7.2 earthquake of Songpan, and D = 0.43 for the M = 6.3 earthquake of Dengkou in China in 1976 (see Chap. 16 in Ref. [11]). This result also indicates that the occurrence of a minimum fractal dimension of microseismic event locations may be a precursor to a major rock burst. Kneisley [3] summarized a U.S. Bureau of Mines microseismic study in a deep, western U.S. coal mine that has historically experienced face bumps and floor bursts in both the room-and-pillar and longwall sections. The mine, located in western Colorado, operated at a depth of nearly 914 m (3000 It), with other active mine workings being located approx. 122 m (400 ft) above the study panel. The 3.048-m-thick coalbed had been mined using the advancing longwall mining method. Prior to 1983, stress relief by volley firing was practiced to destress the longwall face corners and mining sections at depths exceeding 610m (2000ft). Soon after mining began on the longwall panel, face bumps occurred. From Feb. to Sept. 1983, several coal bumps and floor bursts occurred as a result of either mining or destressing (Fig. 8a). General microseismic activity was summarized using both the microseismic event rate and location of these seismic sources. Microseismic source locations for each month of this study (from Feb. to Sept. 1983) are reproduced in Figs 5 and 6. The microseismic event rate and several related coal bumps and floor bursts are shown in Fig. 8a. From these data, the microseismicity can be broadly divided into three intervals [3]: (1) Feb.April, a period marked by headgate area face bumps and the first major floor burst on 20 April; (2) May-late Aug., when intensive face and floor destressing began, but no bumps or bursts occurred; and (3) late Aug.Sept., a period of gradually increasing microseismic activity, and floor bursts occurred in September. Note that activity increases of equal or greater magnitude were also recorded in the second interval. This indicates that the microseismic event rate is really not reliable in the prediction of rock bursts. By using the fractal measurement method, we measure the distributions of microseismic event locations as shown in Figs 5 and 6. The measurement results are illustrated in Fig. 7. Excellent 2.4|

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FRACTAL AND PHYSICAL MECHANISMS OF ROCK BURSTS

The physical process of a rock burst can be described by damage mechanics, developed in recent years [14]. First, local cracking of rock may uniformly occur where a local damage tensile stress state is produced because of the natural defects of rock [15] even though under normally high compressive stress induced by mining activities. The cracking then increases as stress further increases, and crackings begin to cluster within certain zones in the rock mass. The crack density (damage degree or number of cracks per unit volume) in the cluster zone is significantly greater than the background crack density that developed within the rock mass prior to clustering formation (called background zone). The cluster zone is also called a severe damage zone where the material has only very low stiffness. A significant increase in microseismic event rate can be recorded by acoustic emission. If further mining activities induce a tensile stress state because of specific geologic features (such as faults, folds, dikes and joints), the cluster zone becomes the origin of a burst, and a strong failure will suddenly occur. If further mining activities induce a high compressive stress state, the cluster zone needs to experience a damage "healing" process (crack closing) in which a partial amount of strain energy is absorbed by damage healing. This corresponds to the presence of a distinct quiet period of microseismic events, or the

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phenomenon of a seismicity anomaly. This is the "quiet period" of damage in the cluster zone. After the quiet period of damage, a new local damage tensile stress state can again be formed in the cluster zone under continuous high compressive stress. The cracks at this time very easily coalescence, and a strong failure (a burst) may suddenly occur. However, it does not assure that a rock burst actually occurs in the cluster zone. The background zone may be early to suddenly undergo failure (a burst) because enough strain energy may be stored during the quiet period of damage in the cluster zone. This may be the reason why some rock bursts occur in active zones of seismicity while others do not, as reported by Haramy and McDonnell [1]. Although the physical process experienced by a rock burst is very complicated, mathematically, the microseismic events of this process are only simple geometry--fractal clustering geometry. The microseismic events are almost uniformly distributed in the high-stress area in the period far from rock bursting, and have a higher fractal dimension. The more the clustering of microseismic events occurs near the rock burst, the lower the fractal dimension that will be produced. This is a fractal geometrical mechanism of rock bursts.

Since each of the microseismic events in the 2-D case corresponds to a cracking surface element area (or a crack island area) At (note that a crack volume will be considered if in 3-D case), the strain energy E~ stored in the rock masses at least needs to be dissipated for the occurrence of this cracking of rock can be expressed as: E, = ~,A,

(4)

where 7s is the surface free,energy of rock. Since the measured results given in the previous section show that the distribution of cracking events or microseismic events is a fractal, from the basic definition of fractal geometry [18], the number of cracking events (or crack islands) with cracking surface area greater than A~ should thus satisfy the fractal distribution as follows: n = no A , -

D/2.

(5)

Note that n0--* 1, as seismic events (or cracking events) approach a cluster. Since a rock burst is actually equivalent to a fractal cluster of crackings within the rock, the total energy release of rock bursting thus becomes: E, = ~,,A~n = ~,A~ - D/2

(6)

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(7)

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XIE and PARISEAU: FRACTAL ROCK BURSTS

349

Table 1. The relation between strain energy release and fractal Table2. The b-values, fractal dimensions and microseismicactivity dimension at the $3-860L miningpanel of the Sunagawacoal mine in Japan D 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 Seismic Seismic Fractal Et/7, 1.93 1.80 1.68 1.57 1.46 1.37 1.28 1.19 1.11 !.04 No. energya releaserate" dimensions of events" (kJ) (J/m2) b-Values= (D -- 2b) 51 54.8 50.2 0.96 1.92 73 209.7 60.6 0.67 ! .34 3 123 10.7 6.1 1.50 3.00 y = 40.2067 * 10^(-1.2013x)

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where the scale of At is (m) ~ because our analyses are performed in the intermediate-scale of mines, and the fractal dimension D is the range 0-2.0. Similarly, for the 3-D case, equation (7) can be written as:

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(8)

where V~ is the scale of cracking volume elements, and the fractal dimension ranges from 0 to 3.0. Based on equation (7), the required energy release ratio Et/?s for different fractal dimensions can be calculated where the scale of Aj is chosen as (2 m2). The calculated values are listed in Table 1. The tendency of Et/?, to vary in relation to the fractal dimension is shown in Fig. 9. From Fig. 9, it is seen that the lower the fractal dimension, the more strain energy needs to be released. This corresponds to the occurrence of a main rock burst. For the 3-D case, i.e. D e [0, 3], the same tendency of energy release Et to vary in relation to the fractal dimension can be obtained from equation (8). The theoretical results [equations (7) and (8), Fig. 9] are consistent with the fractal description of earthquakes defined by the Gutenberg-Richter relation (also called the magnitude-frequency relation) given by: log N = a - b M ,

(9)

where a and b are constants for a region and N is the frequency of magnitude M. Aki [16] and Turcotte [17] have shown that equation (9) has a fractal behavior for earthquakes, and the b-value in equation (9) is strongly related to the fractal dimension of earthquakes in the form: D

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Fig. I0. The relation between fractal dimension and seismic energy release obtained by microseismicmeasurements.

Sato et al. [4] have found that the magnitude-frequency relation of microseismic activity associated with hydraulic mining also follows the GutenbergRichter relation [equation (9)]. Their measured results of the b-values and microseismic activity at the $3-860L mining panel of the Sunagawa coal mine in Japan are listed in Table 2. Here the fractal dimensions are calculated directly from equation (1 1) based on the b-values given in Table 2. The varying tendency of seismic energy release with the fractal dimension is shown in Fig. 10. From Fig. 10, it is seen that the measured results obtained by Sato et al. have the same tendency as those of the theoretical analyses given by equations (7) and (8). From Figs 9 and 10, we find that the energy release will exponentially increase with the decrease of the fractal dimension. After generalizing the regression equations shown in Figs 9 and 10, we can conclude a general formulation of the energy release to vary in relation to the fractal dimension as follows: D = C, x e x p [ - C 2 E ] ,

(12)

where C, and C2 are constants varying with the region and the measurement scale, and the fractal dimension ranges from 0.0 to 3.0. Equation (12) explains, in theory, a fractal and physical mechanism of why the lower fractal dimension corresponds to the occurrence of a main rock burst in mines, and a main earthquake in seismology. A lower fractal

350

XIE and PARISEAU: FRACTAL ROCK BURSTS

d i m e n s i o n value indicates the f o r m a t i o n o f larger c r a c k ing surfaces o r v o l u m e s within the r o c k mass, i n d i c a t i n g p o s s i b l e r o c k instability. M o r e strain energy needs to be released, c o r r e s p o n d i n g to the o c c u r r e n c e o f a m a i n rock burst. T h e theoretical a n a l y s e s in this section also shows t h a t it is p o s s i b l e to p r e d i c t the o c c u r r e n c e o f m a i n r o c k b u r s t s by a decrease in the fractal d i m e n s i o n for the d i s t r i b u t i o n o f m i c r o s e i s m i c event locations. CONCLUSIONS A r o c k b u r s t is a localized m a c r o f r a c t u r e o f brittle r o c k in an u n d e r g r o u n d o p e n i n g with an i n t e r m e d i a t e scale between a m i c r o c r a c k i n g (cm) a n d an e a r t h q u a k e (km). The physical process e x p e r i e n c e d by a rock b u r s t is a d a m a g e e v o l u t i o n process from m i c r o f r a c t u r i n g to a s u d d e n m a c r o f r a c t u r e . By using acoustic emission, the d i s t r i b u t i o n o f m i c r o f r a c t u r i n g before r o c k b u r s t i n g is t r a n s f o r m e d into the spatial d i s t r i b u t i o n o f a p o i n t set in which each o f the points, i.e. each o f the m i c r o s e i s m i c events c o r r e s p o n d s to a f r a c t u r i n g surface or v o l u m e e l e m e n t in the physical space. U s i n g fractal g e o m e t r y , we find t h a t the d i s t r i b u t i o n o f this p o i n t set is a clustering fractal a n d has g o o d statistical self-similarity. T h e fractal d i m e n s i o n decreases with the e v o l u t i o n o f r o c k m i c r o fracturing. Specifically, the lowest fractal d i m e n s i o n is g e n e r a l l y p r o d u c e d n e a r the o c c u r r e n c e o f a m a i n rock burst. In o r d e r to e x p l a i n the fractal c h a r a c t e r o f rock bursts, the fractal a n d physical m e c h a n i s m s o f rock bursts have been a n a l y z e d in theory. A r o c k b u r s t is seen to be e q u i v a l e n t to a fractal cluster o f m i c r o f r a c t u r i n g s within the r o c k mass. The a n a l y t i c a l results are in g o o d agreem e n t with the m e a s u r e d ones a n d s h o w that the fractal d i m e n s i o n ( D ) o f the cluster o f c r a c k i n g s within the r o c k m a s s is e x p o n e n t i a l l y related to the strain energy release ( E ) in the form: D = C l × exp(-C2E).

Thus, the r e a s o n that the lower fractal d i m e n s i o n corres p o n d s to the o c c u r r e n c e o f a m a i n rock b u r s t in mines, a n d a m a i n e a r t h q u a k e in s e i s m o l o g y is, in theory, explained. R e s e a r c h results given in this p a p e r indicate t h a t it m a y be p o s s i b l e to p r e d i c t the o c c u r r e n c e o f m a i n r o c k b u r s t s by the decrease in the fractal d i m e n s i o n o f the d i s t r i b u t i o n o f m i c r o s e i s m i c event locations.

Accepted for publication 23 February 1993.

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