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Mechanisms of rockmass failure and prevention strategies in rockburst conditions

Xiaoping Yi and Peter K. Kaiser

Geomechanics Research Center, Laurentian University, Sudbury, Ontario, Canada P3E 2C6

ABSTRACT: The interaction of plane seismic waves with underground openings is studied by applying two analytical approaches, the dynamics and continuum mechanics approaches. It is found that low frequency waves from a seismic source hundreds of meters away can cause both rock ejections and dynamic stress concentrations at both walls of an opening. These dynamic stress concentrations, superimposed on existing high static stress concentrations, may lead to different types of rock failure including secondary rockbursts (mainly strain bursts). The high frequency, high magnitude seismic waves from nearby rockburst sources may break rock and eject rock blocks from one wall of an excavation whereas the opposite wall is not affected. Formulae for calculating the ejection velocities are developed. Based on this rock ejection model, strategies for mine design, rockmass de-stressing and support design are discussed.

1. INTRODUCTION

Rock failures in mining are referred to as falls of ground induced by gravity or seismicity and rockbursts. Rockfalls are non-violent falls of loose rocks under the influence of gravity, whereas rockbursts are of instantaneous and violent nature (Jaeger and Cook 1984). Two basic mechanisms are involved in rockbursts: (i) instantaneous slip on an existing geological feature (fault-slip) and (ii) instantaneous fracturing of highly-stressed rock. For practical purposes in hard rock mining, the following three types of rockbursts were defined: fault slip, pillar and strain energy rockbursts (Hedley 1987). Fault slip rockbursts involve primarily the mechanism of sliding, strain energy rockbursts the mechanism of fracturing, and pillarbursts often a combination of the two mechanisms.

The Ontario Ministry of Labor (1990) defines a rockburst as the instantaneous failure of rock causing an expulsion of material at the surface of an opening or a seismic disturbance to a surface or underground mine. This definition reflects the concern for the effects of rockbursts rather than their causes. Therefore, as far as support design is concerned, the expulsion of material at the surface of an opening must be prevented or contained, and a rockburst damage model based on the ejection of rock blocks is a logical and realistic basis for support design.

This paper presents results from analytical studies of rock failure mechanisms due to rockbursts. Two approaches, the dynamics and the continuum mechanics approaches, are reviewed and investigated to provide a better understanding of rock ejections into openings during rockbursts. The former approach deals with the dynamic response of discrete rock blocks and the latter deals with dynamic stress concentrations and wave reflections at the openings.

2. ROCK EJECTION MODELS

2.1 The dynamics approach

When a seismic wave propagates in an elastic continuum, a point or particle of this continuum oscillates around its stationary or equilibrium position, and this particle eventually returns to its stationary position some time after the wave has passed by. A particle is prevented from flying off and moving to a different position due to the atomic forces between adjacent particles. The continuum mechanics approach must be employed to deal with

Figure 1. An underground drift in the path of a propagating plane seismic wave.

this problem (section 2.2). On the other hand, a particle, on the free boundary of an opening, may fly off into the opening if the retaining forces are overcome due to excessive acceleration. Similarly, rock blocks on the free boundary of an opening which are separated from the surrounding rockmass by joints or fractures can be projected into the opening by seismic waves. The dynamics approach to be described in this section can be employed to study this type of problem.

Figure 1 shows a tunnel of diameter D in the path of a plane seismic wave with a wave length l. This seismic wave is assumed to be sinusoidal with positive (compressive) and

negative (tensile) pulses. If the tunnel diameter is much smaller than the wave length or Dl-1 << 1, the wave is not disturbed by the tunnel and wave reflection does not occur. Furthermore, a rock block can be considered as a particle if its size is much less than the wave length. If such a rock block is located on the wall of the tunnel facing the source and is separated from the surrounding rockmass by joints or fractures, it can be carried into the tunnel by a positive pulse of a seismic P wave and then left in the tunnel as the wave passes by. On the other hand, if the rock block is on the opposite wall, it may initially remain stationary during the compressive pulse of the P wave and is carried into the tunnel by the negative pulse. Roberts and Brummer (1988) used this approach to analyze rockburst damage mechanisms in South African mines. If the incoming seismic wave is an S wave, rock blocks at the roof and floor of the tunnel may be ejected into the tunnel.

For engineering applications, we may choose lD-1 > 10 as a practical limit to apply this dynamics approach. For this limit, the pulse width is 5 times the opening size, and it will be shown later that the ejection velocity of a rock block is 90 % of the peak particle velocity (section 2.3). In frequency terms, we find f < c(10D)-1. With a P wave propagation velocity of 6000 m s-1 and a tunnel diameter of D < 6 m, the frequency must be below 100 Hz. This range of frequency is typical of particle velocity waves from rockburst sources hundreds of meters away from measurement points (McGarr et al. 1981, Brink and Mountfort 1984 and Leighton 1984). This suggests that for tunnels, rock block ejections may be commonly observed if the peak particle velocity is sufficiently large. For mined stope with a size D < 60 m, rock ejection may occur if f < 10 Hz. Therefore, if the orebody between two adjacent levels is completely mined out, rocks on stope walls may be ejected by seismic waves of very low frequencies.

As shown in Figure 1, a rock block of mass M, isolated by frictionless joints may attain an initial ejection velocity vi equal to the peak particle velocity vp. The initial kinetic energy is then 0.5Mvp2. Part of this kinetic energy may be dissipated due to friction between the rock block and the surrounding rockmass and in the breaking-up of the rock block, and the rest must be absorbed by the artificial support. It must be realized that the friction between the rock block and the surrounding rockmass increases with increasing rock block thickness and the kinetic energy absorbed by the artificial support is usually well below the initial kinetic energy of 0.5Mvp2.

The general conclusion from this section is that low frequency seismic waves may cause ejections of both small and relatively large rock blocks on opposed walls of a drift.

2.2 Continuum mechanics approach

2.2.1 Dynamic stress concentration and wave reflection

A propagating planar seismic sine wave encountering a cylindrical underground tunnel (Fig. 1) may cause dynamic stress concentrations or wave reflections at the boundary of the tunnel. Two extreme situations arise depending on the relative size of the tunnel D compared to the wave length l of the seismic wave.

One extreme occurs if the tunnel is small compared to the wave length, i.e., Dl-1 approaches zero. In this situation, the loading of the tunnel due to the wave is similar to static loading and may be called quasi-static body force loading. The mathematical proof of this statement can be found in Pao and Mow (1973). The resulting dynamic stress concentrations around the tunnel due to seismic waves can be approximately derived employing the theory of static continuum mechanics.

The other extreme situation occurs if the size of the tunnel is very large compared to the wave length, that is, the ratio Dl-1 is very large. In this situation, the wall of the tunnel facing the seismic source acts as a free surface to the propagating wave and the theory of wave reflection applies. The wall on the opposite side of the tunnel and the rockmass behind it is shielded from any disturbance. Theoretical treatment of this scenario can be found in Kolsky (1963) and Brady and Brown (1985). The fact that we often observe damage only on one wall of an opening suggests that this damage mechanism may be involved. For engineering applications, we may assume that this extreme situation applies if the opening size is about 6 times the pulse width. The corresponding frequency requirement is therefore f > 3cD-1. With cp = 6000 m s-1 and D < 6 m, the lower frequency limit is of the order of 3 kHz.

For tunnels with diameters D comparable to the wave length l, the concept of wave diffraction is applicable, but the theoretical treatment is mathematically rather involved (Pao and Mow, 1973). Nevertheless, certain qualitative results can be induced and quantitative results estimated from results for the above two extremes. The concepts of dynamic stress concentration around an opening derived from quasi-static loading and stress wave reflection can be useful in mining and civil engineering, since detailed quantitative analyses are often not feasible due to a lack of detailed data on wave amplitude, wave length and rockmass properties.

2.2.2 Dynamic stress concentration

For the concept of dynamic stress concentration to apply, the relationship between opening size D and wave frequency f proposed in section 2.1 must be satisfied, i.e., f < c(10D)-1. For a plane wave containing both P and S wave components, propagating in the direction of the x axis (Fig. 1), the dynamic stresses induced in the rockmass are as follows (Brady and Brown 1985):

sx = r cpvx

sxy = r csvy

sxz = r csvz (1)

where sx, sy and sz are normal stresses, sxy and sxz shear stresses, vx, vy and vz particle velocities, cp and cs longitudinal and shear wave velocities, and r and n the density and Poison's ratio of the rockmass. A typical hard rock has a propagation velocity ratio of cpcs-1 = 1.73 (for a Poison's ratio of n = 0.25) with cp = 6 km s-1 and r = 2600 kg m-3. For a P wave with a peak particle velocity of vx = 1 m s-1, the peak dynamic stresses are sx = 15.6, sy = sz = 5.2 MPa, and for an S wave with a peak particle velocity of vy = 1 m s-1, the peak dynamic shear stress is sxy = 8.9 MPa. If a positive or compressive P wave pulse is followed by a negative or tensile pulse, for example, the peak tensile stresses for vx = 2 m s-1 are sx = -31.2, sy = sz = -10.4 MPa. The typical tensile strength of intact hard rocks is about 20 MPa (Jaeger and Cook 1979). Taking into account that the strength increases with increasing strain rate, such rock should sustain a peak particle velocity of a few meters per second. This agrees with the suggestion by Brune (1970) that the upper limit of peak particle velocity at the sources of earthquakes is of the order of 1 m s-1. Since a fractured rockmass can not sustain any significant tensile stress, failure is inevitable if the rockmass is in the path of a tensile seismic wave.

It is important to realize that the dynamic stresses due to seismic waves are superimposed on the pre-existing static stresses. Furthermore, there are dynamic stress concentrations around an opening when a seismic wave encounters an underground opening. These dynamic stress concentrations can be obtained by employing the continuum mechanics approach. Since these stresses are functions of time, underground openings are subject to cyclic loadings.

The general conclusion from this sub-section is that low frequency seismic waves can cause dynamic stress concentrations around underground tunnels. If these dynamic stress concentrations are of sufficient magnitudes and are superimposed on high existing static stress concentrations, rockmass failures may occur or secondary rockbursts may be triggered.

2.3 Rock block ejection velocities

Ejection velocities for the following two situations will be presented in this section: (i) the opening size is much smaller than the wave length and there is no wave reflections (the wave does not "see" the opening), and (ii) the opening size is much larger than the wave length and the theory for wave reflection applies. If a plane compressive wave with a local peak particle velocity vpp propagates normal to a free surface of the opening, the reflected wave is tensile and the peak particle velocity of the free surface is 2vpp (Brady and Brown 1985). A rock block at the free surface may be ejected if the resulting ejection velocity is sufficiently large. The initial ejection velocity of a rock block can be determined by calculating the momentum trapped in the block immediately before the ejection (Wasley 1973 and Kolsky 1963).

Yi and Kaiser (1991) derived the ejection velocities for the two special cases shown in Figures 2a and b. For this purpose, three assumptions were made: (i) a plane sine wave can be approximated by a simpler triangular shape, (ii) the frictional and tensile resistances of the five surfaces of a rock block can be ignored, and (iii) the rock block is in contact with the rockmass such that the incoming seismic wave is transmitted completely across the interface. With the first assumption, the momentum of the wave is underestimated, but linear equations are derived. This underestimation is compensated by the effects of the second assumption which mostly likely introduces the largest error, particularly for thick rock blocks where block boundary friction and energy loss in rock fracturing may dominate. The third assumption is satisfied if the wave peak particle displacement is much larger than the joint aperture. Therefore, the derived ejection velocities represent upper limits which may be used for practical support design.

If a P compressive wave, whose wave length is much smaller than the opening size, approaches the tunnel roof from above (Figure 2a), wave reflection occurs and the reflected wave is tensile. A rock block is ejected when the net stress on the interface, as a result of the incident compressive and reflected tensile waves, becomes tensile. The ratio of rock block initial ejection velocity vi to the local peak particle velocity vpp was derived by Yi (1993):

h lp-1 < 0.5 (2)

where h is the thickness of the rock block and lp the P wave length. The ejection velocity becomes negative for h lp-1 > 0.5. It can be seen from the above equation that vi = 2 vpp if the rock block is reduced to the size of a particle, vi = vpp if h = 0.25lp, and vi = 0 if h = 0.5lp. The concept of rock block ejection by compressive waves was developed for the Hopkinson Pressure Bar Experiment (Kolsky 1963). This phenomenon was also

Figure 2. Ejection of rock blocks in the roof of an underground opening by (a) a compressive P wave from above and (b) a vertically oscillating S wave from a horizontal direction.

demonstrated by Watson and Sanderson (1979) who applied concentrated explosive blasting charges on concrete slabs and observed tension-induced slabbing and spalling.

In the previous section, the wave reflection approach was assumed to be applicable for l< D/3. Since h < lp/2 (Eq. 2), the wave reflection approach is applicable for h < D/6 (less than 1 m for a 6 m tunnel) or for 2h <lp< D/3 (3cpD-1 < f < 0.5cph-1, for D = 6 m, 1500 < f < 3000 Hz). Since seismic waves of frequencies in this range attenuate fast with propagation distance (section 4), only high frequency waves from close-by seismic sources can cause ejections of small rock blocks with dimensions less than one sixth of opening size. For D = 60 m and h = 1 m, we obtain 300 < f < 3000 Hz. Frequencies higher than 300 Hz have been obtained for particle velocity waves measured at locations hundreds of meters away from seismic sources (McGarr et al. 1981, Brink and Mountfort 1984 and Leighton 1984). Therefore, if the orebody between two adjacent levels is completely mined out, ejections of fairly large rock blocks (up to 10 m for a 60 m high open stope) from stope walls may be triggered. In reality, this would occur only if the large rock block was already separated from the surrounding rockmass before the arrival of the seismic wave.

If a vertically oscillating S wave propagates horizontally (Figure 2b), no wave reflection occurs and a rock block at the roof may be ejected by the vertical oscillations. The initial ejection velocity can be expressed as (Yi 1993):

L ls-1 < 1 (3)

where vps and ls are the peak particle velocity and wave length of the S wave respectively and L is the width of the rock block. Equations (2) and (3) are plotted in Figure 3.

Figure 3. Ratio of ejection velocity to ppv versus ratio of rock block thickness h or length L to wave length relationships.

If a P compressive wave, whose wave length is much larger than the opening size, approaches the tunnel roof from above (Figure 2a), wave reflection does not occur and a formula similar to equation (3) can be obtained:

h lp-1 < 1 (4)

From equations (3) and (4), it follows that the ejection velocity vi of a rock block is a function of the block size. The maximum value of vi is equal to the peak particle velocity and is obtained when the block size to wave length ratio approaches zero. The ejection velocity is higher than 0.9 vps or 0.9 vpp if the opening size D < 0.1l or f < 0.1 cpD-1 (f < 100 Hz for a 6 m tunnel). For engineering purposes, the rock block ejection velocity may be set equal to the peak particle velocity for frequencies below this limit.

3. ROCK FAILURE MECHANISMS AND STRATEGIES FOR FAILURE PREVENTION IN ROCKBURST CONDITIONS

According to the analyses in the previous section, the rockmass behind an opening may be completely or partially shielded from the seismic waves. While seismic waves with frequency below about 200 Hz are not likely to be affected by tunnels several meters in diameter, the rockmass behind a mined-out stope may be shielded from seismic waves with much higher frequencies. Measurements of seismic waves at locations hundreds of meters away from the source indicate that their dominant frequencies are typically less than 500 Hz (Gibowicz 1984, Fernandez and Heever 1984, McGarr et al. 1981, Brink and Mountfort 1984, and Leighton 1984). The higher frequency waves may have been attenuated or dissipated in the rockmass, or shielded by underground openings located between the sources and the measurement points. High frequency waves, while being blocked, may cause rock ejections at these openings. Seismic waves with frequencies below about 500 Hz can cause ejections of rocks at both small and large openings and induce dynamic stress concentrations around relatively small openings.

For rockburst research purposes, it is unfortunate that strong motion measurements providing peak particle velocities close to the sources have not been measured extensively. For example, the peak particle velocities measured by McGar et al (1981) were around 10 mm s-1. We know neither the amplitudes nor frequencies of these waves at locations very close to seismic sources. However, it may be postulated that frequencies higher than several kilohertz can only be associated with sufficiently large amplitudes at locations very close to a rockburst source. Nevertheless, the following rock failure mechanisms in rockbursts may be suggested:

Seismic waves from seismic sources several hundred meters away can cause both dynamic stress concentrations and ejections of large and small rock blocks at underground openings. These dynamic stress concentrations, superimposed on the existing high static stress concentrations around openings, can cause different types of rock failures including secondary rockbursts near the openings (sub-sources). The higher frequency waves from these nearby rockbursts (say within tens of meters) can cause ejections of relatively small rock blocks.

A comprehensive program for controlling rockburst hazards in a mine must aim at eliminating the occurrences of large-magnitude seismic events and possible sources of secondary sub-events. Should rockbursts occur, damages must be contained employing appropriate support systems. Mine design methodologies aiming at minimizing stress concentrations in the rockmass could reduce the occurrences of primary rockbursts, rock destressing around an opening could reduce the occurrences of secondary rockbursts, and an effective support must help contain ejected rock blocks. Since part of the kinetic energy of an ejected rock block is dissipated through friction, the prime goal of rock support is to maintain the integrity of an opening in order to minimize the energy transferred to artificial support.

4. DISCUSSION

The amplitude A of a seismic wave with wave length l at a distance x from the source can be expressed as (Kavetsky et al. 1990, Jaeger and Cook 1979, and Dobrin 1976):

(5)

where Ao is the amplitude at distance xo from the source, and the attenuation coefficient a = pQ-1l-1. Q is the coefficient of internal friction which characterizes the inelastic absorption by the rockmass. This attenuation coefficient a increases with decreasing wave length. The theoretical value for the exponent a is 0, 0.5 and 1 for a plane (planar source), cylindrical (column source) and spherical (point source) waves respectively. For a fault-slip type rockburst, therefore, a increases from 0 to 1 as the propagation distance x from the geometric center of the source area increases from zero to infinity. From equation (5), the following remarks can be made: (i) higher frequency seismic waves attenuate faster with the propagation distance or lower frequency waves propagate further away from rockburst sources, and (ii) the attenuation of a seismic wave increases non-linearly as it propagates further away from the source (since a increases from 0 to 1).

The effect of frequency on damages of surface structures from blast vibrations was recognized by US Bureau of Mines for the OSM regulation (RI 8507) where, e.g., the peak particle velocity (ppv) limits were 5 and 50 mm s-1 for 1 and 100 Hz frequencies respectively (Atlas Powder 1987). The rock ejection model described in this paper can be used to explained why the ppv limit was increased for increasing frequency. In this model, the rock ejection velocity is the controlling parameter. As seen from equations (2) to (4), for the same ppv, the ejection velocity decreases with increasing frequency (i.e., decreasing wave length). Therefore, the ppv limit for a higher frequency must be increased to maintain the same ejection velocity. In rockburst applications, the frequency effects of parameters such as the ppv have often been ignored for simplicity reasons or for specific applications (narrow range of dominant frequency). This can be seen from the ppv versus propagation distance relationships developed by McGarr et al. (1981) and Hedley (1990), where the exponential term involving the attenuation coefficient a in equation (5) was ignored. In an underground mine, there exist numerous tunnels and mined-out stopes. Apart from being attenuated in the rockmass, high frequency waves can also be shielded by underground openings. Therefore, only low frequency waves can reach openings beyond stopes and large tunnels. Since large rock blocks can be ejected by these low frequency waves (section 3), large scale ground falls induced by far away seismic sources are often observed in the field, and support design methods must be developed to deal with these extremely dangerous conditions.

5. CONCLUDING REMARKS

Low frequency seismic waves can cause ejections of both small and relatively large rock blocks on opposing walls of underground openings located both close to and far away from seismic sources. The initial ejection velocities can be obtained from equations (3) and (4) if the local peak particle velocity is known. These seismic waves may also cause dynamic stress concentrations around these openings and induce secondary rockbursts (of mainly strain energy type). The dynamic stresses may be calculated from equation (1). High frequency waves from close-by seismic sources may cause ejections of relatively small rock blocks on only one wall of an excavation and the ejection velocity is given by equation (2). The opposite wall should not be affected. Mine design and rockmass de-stressing aiming at reducing or eliminating extraordinary static stress concentrations may reduce the occurrences of large seismic events and secondary rockbursts. Effective artificial support systems must then be applied to contain ejectable rock blocks, and support design must aim at maintaining the integrity of all rock block interfaces in order to facilitate energy dissipation and minimize the energy imparted to artificial support.

ACKNOWLEDGEMENTS

Financial support from MRD (Mining Research Directorate) and NSERC (Natural Science and Engineering Research Council) is gratefully acknowledged. Comments received from Drs R. K. Brummer and D. McCreath were extremely helpful for preparing the manuscript.

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