Arch. Min. Sci., Vol. 53 (2008), No 1, p

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1 Arch. Min. Sci., Vol. 53 (28), No 1, p EWA TAKUSKA-WĘGRZYN* APPLICATION OF STATISTICAL METHODS FOR EVALUATION OF ROCK-BURST RISKS IN COPPER ORE MINE CONDITIONS ZASTOSOWANIE METOD STATYSTYCZNYCH DO OCENY STANU ZAGROŻENIA TĄPANIAMI W WARUNKACH KOPALŃ RUD MIEDZI The paper presents a proposal for a method of rock-burst risk estimation based upon an analysis of the non-homogeneity degree of rock cracking processes. In periods preceding moments of tremors there are tendencies towards enlargement of crack sizes causing increases in the non-homogeneity degree. It is known that rock cracking is represented in the form of a seismoacoustic emission, which can be observed (registered). The non-homogeneity degree of the cracking process can be estimated on the basis of the statistical analysis of the registered seismoacoustic emission stream. It is known from experience that the registered emission constitutes nonstationary and non-homogeneous stream of events. Information about energies (extents) of processes and time intervals between effects is necessary for its complete description. The paper presents appropriate models of measures describing the non-homogeneity degree of the cracking process. These measures treated as indicative functions enable a qualitative connection between the non-homogeneity degree discussed and the rock-burst risk condition. The findings have been illustrated with examples of behaviour of these measures estimated on the basis of the registered seismoacoustic emission stream in the G-22/4 ZG Rudna Branch. Keywords: tremors, rock cracking, seismoacoustic emission, seismoacoustic effects, stream of events, non-homogeneity degree, extent of processes, time intervals between effects, rock-burst risk W pracy przedstawiono propozycję sposobu oceny zagrożenia tąpaniami, opartego na analizie stopnia niejednorodności procesów pękania skał. W okresach poprzedzających momenty wystąpienia wstrząsów występują tendencje w kierunku powiększania się rozmiarów pęknięć, powodując wzrosty stopnia niejednorodności. Wiadomo, że pękanie skał jest odwzorowywane w formie emisji sejsmoakustycznej, którą możemy obserwować (rejestrować). Stopień niejednorodności procesu pękania może być oceniany na podstawie analizy statystycznej rejestrowanego strumienia emisji sejsmoakustycznej. Z praktyki wiadomo, że rejestrowana emisja stanowi niestacjonarny i niejednorodny strumień zdarzeń. Do pełnego jej opisu * AGH UNIVERSITY OF SCIENCE AND TECHNOLOGY, FACULTY OF GEOLOGY, GEOPHYSICS AND ENVIRONMENTAL PROTECTION, DEPARTMENT OF GEOPHYSICS, AL. MICKIEWICZA 3, 3-59 KRAKÓW, POLAND

2 24 konieczna jest informacja o energiach (rozmiarach) zjawisk oraz odstępach czasu między zjawiskami. W pracy przedstawiono odpowiednie modele miar opisujących stopień niejednorodności procesu pękania. Miary te traktowane jako funkcje wskaźnikowe umożliwiają jakościowe powiązanie omawianego stopnia niejednorodności ze stanem zagrożenia tąpaniami. Uzyskane wyniki zostały zilustrowane przykładami zachowania się tych miar estymowanych na podstawie rejestrowanego strumienia emisji sejsmoakustycznej w Oddziale G-22/4 ZG Rudna. Słowa kluczowe: wstrząsy górnicze, pękanie skał, emisja sejsmoakustyczna, zjawiska sejsmoakustyczne, strumień zdarzeń, stopień niejednorodności, rozmiary zjawisk, odstępy czasu między zjawiskami, zagrożenie tąpaniami. 1. Introduction The main objective of this paper is to present possibilities of application of statistical methods for development of methods of rock-burst risk monitoring, in particular of tremor risk monitoring. The method of analysis has been adjusted to the geological and mining conditions in copper ore mines. It is known that the sufficient size of samples analysed is the basic principle of correct application of statistical methods. This principle is met in the case of seismoacoustic emission because it is characterized by high activity of effects. The seismoacoustic emission is generated by cracking processes occurring in the rockmass, caused by mining. Knowing the correct structure of these processes has a direct impact on the quality of their description. It is known from geomechanical theories (Jaeger & Cook, 1969) that individual rockmass cracks are characterized by a high level of indeterminism and should be treated as random events. Therefore it is assumed that cracking processes have stochastic structure and should be modeled in the form of random streams of events. Thus concluding about their course should be performed on the basis of probabilistic methods. These processes cannot be directly observed, however, as it is well known, they can be analyzed on the basis of properly registered seismic emission. The frequency band of registering and distances between sensors should ensure such size of registered effects as is necessary for correct performance of statistical analyses. On the basis of research carried out in copper ore mine conditions it has been found that this band should be within the range of 3 and 5 Hz whereas distances between the sensors should not exceed the distance between 1 and 2 m. With research it may turn out that the band limits may be changed. However excessive increase in its upper limit leads to increase in the share of various noises in registered records. Nevertheless the lower band limit determines the radius (in terms of typology) of the area observed. A seismic emission registered in such a way will be referred to as a seismoacoustic emission in further considerations. It is described in probabilistic terms. At the beginning an assumption is made that individual events of the cracking process or cracks are equated with specific seismoacoustic effects. It is also assumed that statistical parameters, in particular probability distributions of specific characteristics, describe both the cracking processes and the seismoacoustic emission they cause. More precisely, these distributions are described by the same models. Seismoacoustic effects

3 are identified on the basis of signals registered with a system of multiple sensors, at least of two sensors. It should be emphasized that registrations performed with single sensors may contain too many noises. These assumptions have been confirmed in numerous surveys published (Cianciara, 1999; Cianciara et al., 26) and constitute the basis for concluding on the course of cracking processes. Research on cracking processes should be carried out on the basis of crack sizes. They are however difficult to define because they are not directly observable. Therefore their sizes can only be estimated on the basis of analysis of the registered seismoacoustic emission. The study assumes a concept that crack sizes should be estimated on the basis of energy of seismoacoustic effects (Cianciara et al., 26) or on the basis of time intervals between subsequent effects (Cianciara & Cianciara, 26). It is assumed that in terms of seismoacoustics, crack sizes are proportional to the square of norms of signals representing the effects. However in the case of estimation of sizes on the basis of time intervals between effects, linear statistical dependence connecting them with the energy of effects is used (Cianciara & Cianciara, 26). These dependence can be presented as follows: u k F, where: u k the time intervals between effects, E F,k the physical energy of effects, E F, the reference energy, ε k random values, δ and σ coeficients. 25 EF, k log k (1.1) E However their expected values meet the following linear dependency: where: M means the expected value. E F, k M [ U] M log (1.2) EF, During research on seismoacoustic emission generated in conditions of copper ore mines it was found that these two characteristics allow to conclude on the direction of the cracking process development. For clarity one should add that the energy discussed herein is not treated as a physical parameter but it represents a mathematical notion. However it does not change the fact that it is fully suitable for statistical analyses and yields good results. Cracking processes are treated as non-homogenous stream of random events of specific course dynamics. Research on their dynamics is carried out on the basis of analysis of experimental data registered in specific time intervals T, the so called information

4 26 windows, which are moved in time by a set step. During the research conducted it has been found that the cracking process dynamics observed in copper ore mines is characterized by significantly lower speed compared to the one in hard coal mines. It can be confirmed by the analysis of activity of effects, which in the case of copper ore mines is even up to ten times lower that the one registered in hard coal mines. To be able to obtain adequate effectiveness of estimates one should use relatively large windows, the size of which, as appears from practice, should be even a few 24-hour periods. Therefore, it is possible to examine the course of rockmass cracking process only on the basis of seismoacoustic emission as it is characterized by a sufficient activity. 2. The non-homogeneity degree of the cracking process Tendencies in cracking process development can be examined by treating them as a random streams of events. One of the factors is the characteristics of non-homogeneity of the streams in question. This notion is taken from probabilistics and it states that a stream of events is homogeneous if it has independent and stationary increments (Kowalenko et al., 1989). If the increments concerned are non-stationary, the stream of events is non-homogeneous. The theoretical analysis of cracking processes on the basis of geomechanical theories shows that their courses are characterized by a significant degree of non-homogeneity (Cornell, 1968; Jaeger & Cook, 1969). The non-homogeneoity in question concerns changes in the state of stress, which cause the presence of tendencies in development of crack sizes. Examination of the degree of non-homegeneity is conducted by analysis of seismoacoustic emission, by determination of probability distributions and the expected value of energy of effects and time intervals between effects. It is assumed that the statistical distribution of emission characteristics is governed by the Weilbull s law (Cianciara, 2), which can be presented in the following form: for F ( ) (2.1) exp[ ( ) ] for where: β and γ are parameters, and β >, γ > while ς means values of individual characteristics, which means: in the case of energy log, whereas in the case of time E E u intervals between effects. u The distribution of probability density is described by the formulas (Cianciara et al., 25): for f ( ) (2.2) ( ) exp[ ( ) ] for

5 And the expected value of the random variable ζ is expressed as follows (Cianciara at al., 25): 27 M [ ] ( ) (2.3) where: Γ is the Euler gamma function. In the case of time intervals between effects (ζ = U) the window size T divided by the expected value M[U] represents the expected number of effects Nˆ T of the stream of events concerned, that is: 1 ˆ T ˆ ˆ N T ˆ (2.4) ( ˆ ) In consequence a model is defined, which describes the measure of non-homogeneity of the cracking process (Cianciara et al., 24). In conditions of copper ore mines good results are obtained by using measures in the form of a quotient of emission stream expected values to the Poisson process expected value treated as the standard. In this case the numeric value of non-homegeneity degree ϑ(t) is estimated on the basis of emission characteristics ς according to the following model: NT ( T) (2.5) T M [] where: M the functional determining the value, T the information window, N T the number of effects registered in the window T. The numeric values of this measure are determined on the basis of emission registered in the information window T. Moving this window by a step we obtain the course of the degree of non-homogeneity in the form of time-dependent function. This function reflects the tendencies in rockmass cracking process development. An example graph of this function is presented in fig. 1. Calculations have been made on the basis of emission registered in the G-22/4 ZG Rudna Branch. It appears from the analysis of the experimental material registered in conditions of copper ore mines that the emission is well described by a model of non-homogeneous and locally stationary stream of events. In the case of locally homogeneous stream its parameter λ is variable in time λ(t) and it reflects the geomechanical condition of the rockmass. Therefore a correlation function describing grouping of effects (Takuska-Węgrzyn, 1999) has been used for examination of emission nature in terms of its connection with geomechanical parameters. The following dependency (Takuska-Węgrzyn, 1999) was used to develop the correlation function K(T) estimation algorithm: K( T) M ( ; T ) M ( T) los (2.6)

6 E E Degree of in homogeneity E6 5.1E4 8.8E5 1.3E5 5.8E Time [yr-m-d] Fig. 1. The course of function describing the non-homogeneity degree of the cracking process ϑ(t; T) determined in the form of the expected value of sizes of seismoacoustic effects. Estimations have been performed on the basis of emission registered in the G-22/4 ZG Rudna Branch Rys. 1. Przebieg funkcji opisującej stopień niejednorodności procesu pękania ϑ(t,t) wyznaczonej w formie wartości oczekiwanej rozmiarów zjawisk sejsmoakustycznych. Estymacje prowadzono na podstawie emisji rejestrowanej w Oddziale G-22/4 ZG Rudna In practice this parameter can be estimated in accordance to this algorithm, selecting signals in a specific time window T (of N T size) out of the data sequence and next moving it with the inflow of measurement information along the data sequence by a step. This way the continuous change of value of this parameter in the form of time-dependent function is obtained. The problem of determination of the K(T) parameter basically comes down to estimation of disturbances in distributions of time intervals between effects in the T window in relation to a purely random process (Poisson s). An example course of the correlation function is presented in fig. 2. Calculations have been performed on the basis of emission registered in the G-22/4 ZG Rudna Branch.

7 E6 Correlation coefficient E4 1.3E Time [yr-m-d] Fig. 2. The graph of the course of the correlation function K(t;T) describing the non-homogeneity degree of the cracking process estimated on the basis of the seismoacoustic emission stream registered in the G-22/4 ZG Rudna Branch Rys. 2. Wykres przebiegu funkcji korelacyjnej K(t,T) opisującej stopień niejednorodności procesu pękania estymowanej na podstawie strumienia emisji sejsmoakustycznej rejestrowanej w Oddziale G-22/4 ZG Rudna 3. Conclusion According to the applicable geomechanical theories, rock-burst risk and in consequence occurence of tremors is caused by adequate increases in the state of stress. Increases in stress in turn cause rockmass cracking, which proceeds in compliance with specific laws. For instance crack sizes comply with the exponential Gutenberg-Richter s law (Gutenberg & Richter, 1954). Also time intervals between cracks are subject to specific changes. It causes disturbances in the distribution of seismic effects in the registered stream of emissions. These disturbances can be followed systematically by analysing of the non-homogeneity degree of the registered stream of seismoacoustic emission. The non-homogeneity degree concerned is a probabilistic notion and cannot be estimated by means of statistical analysis. This study confirms practical possibility of concluding on

8 3 the rockmass condition in circumstances of copper ore mines on the basis of an analysis of the non-homogeneity degree of the registered seismoacoustic emission. It is illustrated by fig. 1 and 2, which present the courses of the non-homogeneity degree estimated by two methods, on the basic of analysis of the expected value of emissions and grouping of seismoacoustic effects. The figures show that the times of tremor occurrences are preceded by increases in the course of the function. REFERENCES Cianciara B., Emisja sejsmiczna jako nośnik informacji o rozwoju procesu pękania górotworu. Geoinformatica Polonica no. 1. Cianciara A., 2. System monitorowania zagrożenia wystąpieniem silnych wstrząsów w oparciu o analizę emisji sejsmoakustycznej. Doctoral dissertation. Biblioteka Główna AGH. Kraków. Cohen A.C., Maximum likelihood estimation in the Weibull distribution based on complete and on censored samples. Technometrics 7, Cox D.R., Some statistical methods related with series of event. J. Roy. Soc. B, 17, 2. Goszcz A., Elementy mechaniki skał oraz tąpania w polskich kopalniach węgla i miedzi. PAN IGSMiE Kraków. Gutenberg B., Richter C.F., Seismicity of the earth and associated phenomenon. 2nd edition, Princeton University Press, Princeton. Kowalenko I.N., Kuzniecow N.J., Szurienkow W.M., Procesy stochastyczne. Poradnik PWN. Warszawa. Lasocki S., Weibull distribution as a model for sequense of seismic events induced by mining. Acta Geophys. Pol. 41, no. 2. Lehmann E.L., Teoria estymacji punktowej. PWN Warszawa. Marcak H., Geofizyczne modele rozwoju procesu niszczenia górotworu poprzedzające tąpnięcia i wstrząsy w kopalniach. Publs. Inst. Geophys. Pol. Ac. Sc. M-6 (176) s Marcak H., Wpływ struktury górotworu naruszonego robotami górniczymi na tworzenie sekwencji wstrząsów górniczych. Biblioteka Szkoły Eksploatacji Podziemnej, Seria Wykłady nr 8, CPPGSMiE PAN Kraków. Nur A., A Note of the Constitutive Law for Dilatancy. Pageoph. Vol Takuska-Węgrzyn E., A method ofevalution of the threat of bursts based on an analysis of the concentration of seismoacustic phenomen (in Polish). Quarterly, AGH-UST. Geology, vol. 24, Bull. 2. Received: 26 June 27