3. Monitoring Rock Mass Stability

In 1708, Gottfried Wilhelm von Leibniz said “Time and space are not things, but order of things”, (Wiener, 1951). Or, as Albert Einstein put it “Time and space are modes by which we think, and not condition in which we live” (Forsee, 1967). Therefore, time is nature’s way to keep everything from happening all at once, (Atiyah, 1988). And John Wheeler (1911–2008) quipped that “Space is what prevents everything happening to me”. In essence, anything is possible if it happens too fast to be detected.

Time, as incorporated in the basic laws of physics, e.g. by Newton, Maxwell, or Einstein, does not distinguish between past and future, all processes are time reversible, meaning that they can proceed backwards as well as forwards through time. In his excellent book, Eddington (1928) introduced the concept of the arrow of time. He writes that “cause and effect are closely bound up with arrow of time, i.e. the cause must precede the effect. Thus in primary physics, which knows nothing of time’s arrow, there is no discrimination of cause and effect, but events are connected by a symmetrical causal relation which is the same viewed from either end”. He then states that the future is associated with more randomness and the past with less randomness.¹ The subject of arrow of time and irreversibility was thoroughly explored by the Nobel prize winner Ilya Prigogine in Prigogine (1980) and Prigogine (1997), where he states that irreversibility can no longer be identified with a mere appearance that would disappear if we had perfect knowledge. He used the concept of entropy to distinguish between reversible and irreversible processes. Only irreversible processes contribute to entropy production, e.g. chemical reactions, heat conduction, diffusion. The second law of thermodynamics then states that irreversible processes lead to a kind of “one-sidedness” of time. The positive time direction is associated with the increase of entropy. The law of entropy increase is simply a law of increasing disorganisation. In this process, the initial conditions are forgotten. As Prigogine (1997) speculates: “The big bang was an event associated with an instability within the medium that produced our universe. It marked the start of our universe but not the start of time. Although our universe has an age, the medium that produced our universe has none. Time has no beginning, and probably no end”.

On a macroscopic scale, all processes in nature are dissipative. Natural systems consist of a large number of elements which, at any given time, are not in the same state. Therefore, in order to accommodate the differences, a macroscopic system spontaneously generates local flow of energy and momentum in the form of localised small-scale events, in addition to those imposed by the external conditions. Close to equilibrium, the distribution of fluctuations is more or less random, the correlation time and the correlation length are short, and nonlinearities are mostly hidden. In other words, near equilibrium fluctuations are harmless. Away from equilibrium, the system is more susceptible to the action of intermittent intrinsic instabilities that, due to their nonlinear nature, are agents of spatial and temporal correlations. Here the distribution of fluctuations is broader than Gaussian, with slower, power law like, decays or with additional peaks, facilitating rare but larger events that dominate the behaviour of the system. The finite values of spatial and temporal correlations measure the distance from equilibrium, and as this distance grows, the influence of nonlinearities increases. When the spatial range of fluctuations increases, elements of the different parts of the system interact, and the system can generate and maintain a reproducible relationship among its distant parts. In this process of self-organisation, the system creates spatial and temporal patterns that are not directly imposed by external forces. When the range of correlations becomes comparable with the system size, the resulting coherence, or order, may influence its behaviour qualitatively, and the system may become critical and undergo bifurcation, i.e. transition to another state (Nicolis & Prigogine, 1977). The divergence of correlation length indicates that the details of the system are irrelevant to its critical behaviour.

In essence, the traditional classical science evolved around stability and certitudes, and probability was associated with ignorance, while in reality we experience fluctuations, instabilities, multiple choices, and limited predictability. Once instability is included, the meaning of “law of nature” changes, and they no longer express certitudes but rather possibilities and need to be formulated on the statistical level (Prigogine, 1996) or, as Wallerstein (1999) said, “Probability is the only scientific truth there is”.

Figure 3.1 left illustrates the stochastic analogue of bifurcation described by Nicolis and Prigogine (1989).

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The system at time \(t_{1}\) is described by a sharp unimodal peak around its mean value \(\left \langle X\right \rangle \). In other words, the system is firmly in the current state, and the internal fluctuations and/or the external influences are weak. At time \(t_{2}\) due to increased strength of fluctuations and/or changes in boundary conditions, the range of spatial correlation increased, and the system has a flat distribution, exploring regions of phase space that are far from the most likely value, preparing for bifurcation, i.e. a qualitative change of its behaviour due to a small smooth change in the parameter values. At this stage, one would expect seismic activity to develop spatial correlations. For example, the seismic response to blasting would have far wider spatial range than at the time \(t_{1}\). This is a state of marginal stability. Such correlations can only arise in systems in which the regime of Poissonian fluctuation, or disorder, has been overcome, i.e. there is an increased coherence and associated lower dimensionality of the system. In a Poissonian regime, the inter-event times are distributed with the same probability density function, i.e. the process is memoryless, and the vanishing of the covariance in the joint probability distribution in two sub-volumes implies the complete absence of spatial correlations, i.e. no interactions between events. Such a state can therefore be considered as the prototype of disorder. At time \(t_{3}\), the system transitioned into a multi-hump regime beyond bifurcation and is more likely to be at \(\left \langle X_{2}\right \rangle \) than \(\left \langle X_{1}\right \rangle \).

In general, the approach to the critical point and the nature of the instability depend on the complexity of the system that depends, in turn, on the degree of order or disorder in the system. A highly entropic state is random and not so complex, and a low entropic state is not complex because of its regularities (Crutchfield & Young, 1989; Kaneko & Tsuda, 2001), see Fig. 3.1 right. Therefore one would expect the state of marginal stability, state at time \(t_{2}\), to represent high complexity where the system itself hesitates what to do next. For complexity to feature, the system needs to be open to interaction with the environment where there is a net flow of energy. Closed systems would proceed towards an ordered or disordered equilibrium state, which is not complex.

Mining is an open system where transitions between ordered and disordered states are driven by stress and strain gradients imposed by rock extraction and by resulting seismic and aseismic deformation. Moreover, mining is not a spontaneous process. Rock extraction takes place at a particular place, at a particular time, and at a particular rate which are all highly variable compared to the tectonic regime. This interferes with the process of self-organisation. The average rate of deformation induced by mining is at least two orders of magnitude greater than the average slip rate of tectonic plates. The bulk of seismic activity in mines starts with rock extraction, increases with the extraction ratio of the ore body, and stops rather quickly with cessation of mining. Larger events tend to occur after the extraction ratio, or the depth of mining, or both reach a certain level. The complexity of the rock mass response to mining is driven mainly by two competing spatio-temporal processes of excitation and relaxation. In terms of the stress-strain behaviour of a given volume of rock, it would be the mode switching and the balance between strain hardening and softening that drives complexity. Since softer, heterogeneous systems spend more time after the peak stress and dissipate more, they generate more entropy at the costs of complexity. Thus, it may be that the relative drop in complexity, due to increase in entropy, gives rise to lower seismic hazard and more visible alerts. Increase in disorder leads to slower transitions and to diffused instabilities, while highly homogeneous systems may crack in one go with little warning or precursory behaviour.

An important agent in the development of spatial and temporal correlations in highly stressed rock is seismic activity itself. By breaking numerous asperities, seismic events smooth the system, allowing transfer of stresses over larger distances and thus paving the way for even larger events. Disorder, on the other hand, plays a stabilising role and, to a degree, can be engineered into the system by scattered layout and/or by a scattered sequence of mining or blasting. Disordered directions of local stresses and a slower loading rate, i.e. slower rate of mining, may also play a role, since it promotes healing and thus stress roughening.

Assuming that large seismic events are those small ones that were not stopped soon enough, the one way to manage seismic hazard is to engineer a mine layout that, together with the natural structures, is “rough” enough to limit the extent of ruptures. It has been postulated that mining scenarios that introduce spatial heterogeneity, or roughness, may de-correlate the system and be less likely to develop larger dynamic instabilities (e.g. van Aswegen & Mendecki, 1999; Handley et al., 2000; Mendecki, 2001, Figure 8; Mendecki, 2005).

However, there is a degree of confusion regarding the role disorder plays in rock mass stability. Since order is prerequisite of human survival, the impulse to produce orderly arrangements is inbred by evolution. In common speech, order describes organisation, structural regularity, and is associated with stability. We can achieve more if we act in organised manner, and therefore, there is a tendency to design mining a layout with simple geometrical shapes with straight lines. When in the late 80s I argued for a disordered or scattered mining layout in South African gold mines, a senior mine executive replied “a long-wall is a long-wall is a long-wall”, which in an essence promoted straight lines. It took a few years and countless rockbursts to change that perception.

But rock subjected to excessive stress also ruptures along straight lines. Yes, in aseismic mines, an orderly design may facilitate better financial outcome, but in deep hard rock mines it is the order or the smoothness of the mine layout and geological structures that promote more frequent and more destructive rockbursts. Put simply, the entropy or disorder measures the degree to which energy is mixed up inside a system, and therefore, it scales positively with the quantity of energy no longer available to do physical work.

This introduction explains why this chapter is about the degree of rock mass stability rather than prediction or forecasting. Keilis-Borok (1994) described the earthquake generating part of the solid Earth as a hierarchical nonlinear dissipative system. The same can be said about a rock mass subjected to mining. It consists of a hierarchy of geological structures under load, random heterogeneities, patches of rock resisting deformation where stresses are increasing, patches of fractured rock where stresses are decreasing, and passive volumes of rock not influenced by loading. Such a system shows partial self-similarity, fractality, and self-organisation, and its parts may remain in a sub-critical state even after larger events.

The surface which bounds all stress states that correspond to elastic deformation in six-dimensional stress space is called the yield or damage surface. The evolution of the yield surface with continued deformation specifies the manner in which the material hardens or softens during the deformation. The direction of the inelastic strain increment beyond the yield surface is determined by the inelastic potential. The inelastic potential surface is always normal to the inelastic strain rate vector \(\dot {\epsilon }_{in}\). The associated flow or normality rule applies to the material where yield and potential surfaces coincide, that is when the inelastic strain rate vector is always normal to the yield surface. Under the associated flow with a smooth yield surface, one would expect a strain softening behaviour before the instability. This could be described by the classical instability criterion formulated by Hadamard in 1903 (Truesdell & Noll, 2004), and rediscovered by Pearson (1956) and Hill (1958), where \(\dot {\epsilon }_{e}\) and \(\dot {\epsilon }_{in}\) are elastic and inelastic strain rates, \(\dot {\sigma }\) is the rate of change of stress, and dV  the volume of interest. The second inequality in Eq. (3.1) states that for unstable deformation of dV  the inner product of the next increment of stress with the next increment of strain should be negative, For the elastic strain increments \(d\sigma d\epsilon _{e}>0\), therefore the deformation will be unstable only when the inelastic term balances the elastic term, so that there is a net strain softening. Near the end of the hardening regime, however, almost all further strain increments are inelastic. As stress and strain are tensors, there are many different components that may be influential in causing the instability.

At some stage, the system may cease to deform in a homogeneous mode and undergoes bifurcation. In general, bifurcation refers to a qualitative change of the object under study due to a change of parameters on which the object depends. In this case, it is the non-uniqueness of the deformation path. A bifurcation mode may or may not be amplified by continued deformation past the bifurcation point. After the displacement corresponding to the bifurcation point, there may be more than one incremental displacement field that satisfies the equilibrium and boundary conditions. If \(\dot {\sigma }_{1}\) and \(\dot {\sigma }_{2}\) are the stress rate fields corresponding to two such solutions, with \(\dot {\epsilon }_{1}\) and \(\dot {\epsilon }_{2}\) the respective strain rates, then the necessary condition for bifurcation to occur is that there exists an incremental displacement field such that \(\intop _{V}\Delta \dot {\sigma }\Delta \dot {\epsilon } = 0\), where \(\Delta \dot {\sigma } = \dot {\sigma }_{1}\)- \(\dot {\sigma }_{2}\) and \(\Delta \dot {\epsilon } = \dot {\epsilon }_{1} - \dot {\epsilon }_{2}\).

If the bifurcation grows, it may be symmetrical or asymmetrical. A symmetrical bifurcation (e.g. barrelling of the specimen) is stable, so \(d\sigma d\epsilon >0\), i.e. the deformation is stable when it is easier to start deformation elsewhere rather than to pursue it where it has begun. Asymmetrical bifurcation, characteristic of materials that show frictional and dilatational responses to stress, results in strain localisation in single, conjugate, or quasi-periodic multiple shear bands (Rudnicki & Rice, 1975; Hobbs et al., 1990; Muhlhaus et al., 1992). The possibility of localisation arises when one or more stress components within a volume are able to decrease with increasing strain (Cundall, 1990). Here the deformation may be unstable. Alternatively, strain localisation can be caused by relative strain softening, where material within a localised strain zone hardens at a lower rate than the adjacent body of rock (Hobbs et al., 1990; Jessell & Lister, 1991). The localisation is driven by the release of strain energy from an unloading region outside the localisation zone. Strain localisation may occur under both strain softening and strain hardening material response and is promoted by non-associated flow and/or by the presence of vertices on the yield surface (Hobbs et al., 1990; Olson, 1992).

Fredrich et al. (1989) conducted the first laboratory study to quantify all the constitutive parameters for pressure-sensitive dilatant materials in both the brittle and semi-brittle regime. They measured an internal friction coefficient, a dilatancy factor, being the ratio between the increments of inelastic volume strain and the inelastic shear strain, and a hardening modulus for Carrera marble deformed at room temperature and at different confining pressures. At a confining pressure of 5 MPa, they showed a brittle failure mode, and at 40–190 MPa, semi-brittle failure was observed. In the brittle region, shear localisation was not evident until the sample was deformed well into the post-failure region. In the semi-brittle regime, with increasing confining pressure, the hardening modulus increased, whereas the friction coefficient, dilatancy factor, and Poisson ratio decreased. Thus, the bifurcation model predicts that shear localisation is inhibited in the semi-brittle regime as long as the rock continues to strain harden.

Wawersik et al. (1990) investigated localisation of deformation theoretically and in the laboratory. They found that deviation from normality, i.e. associated flow, promotes localisation before peak stress for plane-strain compression. Triaxial deformation was characterised by a broad peak followed by gradual strain softening, with breakdown instability occurring only well beyond the peak stress.

In the case where the nucleation zone is localised and is developing far from the free surface, the shear band(s) are constrained by both elastic and, to a limited extent, inelastic deformation in the less deformed rock surrounding the shear zone. In such a case, hardening of the system undergoing deformation is to be expected even if the material softens within the nucleation zone. The critical states of stability of a three-dimensional body can generally occur only if the compression stress is of the same order of magnitude as the shear modulus of the material (Bazant & Cedolin, 1991).

Whether or when the process of strain softening and/or strain localisation becomes unstable depends on the energy release and the dissipation inside the softening zone and on the exchange of mechanical energy between the softening zone and the ambient rock. Stress decrease associated with softening within the nucleation zone induces some elastic un-straining and redistribution of stresses in the ambient rock. Energy released by elastic rebound of the surrounding rock is fed into the softening region and accelerates its deformation. The amount of energy fed into the nucleation zone correlates positively with its size and with the strain rate or, rather, rate of un-straining in the ambient rock. As soon as an input of elastic energy exceeds the energy dissipation due to the inelastic processes in the growing nucleation zone, the system becomes unstable.

One should distinguish between the stability of deformation defined by the stress-strain response of the material and the stability of the system being deformed. For a given loading, the necessary condition for instability of a system, e.g. a tunnel, stope, mined out fault, or dyke, is that the volume of net softening needs to reach a critical size, which introduces a size scale into the instability criterion. The necessary condition for the dynamic instability is that this critical volume needs to be overloaded suddenly, which introduces a critical time scale.

There is no universal rule how to determine the critical length scale; however, one can try for a proxy to gauge the proximity of the system to the critical point. Some of the proxies are based on the assumption that the stability of the rock mass subjected to mining can be related to its stiffness, i.e. its ability to resist deformation with increasing stress. While the overall stiffness of the rock mass is being maintained, the seismic response to mining, measured by the cumulative inelastic co-seismic deformation, e.g., seismic potency P, is expected to be proportional to the effective volume mined, \(\sum P\sim V_{meff}\). As mining progresses, the overall stiffness of the rock mass is being degraded and the rate of potency release may increase. With further degradation in stiffness, the response may become nonlinear with accelerating potency release, frequently associated with an increase in activity rate, signifying potential for larger instability. The dynamics of such instability depends on the ratio of the stiffness of the potentially unstable volume of rock to the stiffness of the surrounding rock mass. The higher this ratio the more energy will be released per unit of inelastic deformation at the source.

Another guiding idea in the interpretation of seismic activity is the concept of self-organisation into critical state, i.e. a state at which the correlation length becomes comparable with the system size. Intermittently, the correlations length may reach or even exceed the system size creating conditions conducive for larger instabilities. Here one would expect an increase in the mean distance between consecutive events \(\left \langle d\right \rangle \). It is assumed that the growth of long-range correlations within the rock mass allows for progressively larger events to be generated. Zoller et al. (2001) tested the critical state concept for earthquakes in California in terms of the spatial correlation range and found a scaling relation \(\log R \sim 0.7~\mbox{m}\) between the main shock magnitude m and the critical region R. However, in mines, seismic activity follows rock extraction, and a scattered mining operation may obscure spatial correlation and influence the distribution of distances between events.

Since the development of the nucleation zone is associated with overall strain softening, sources of seismic events located within the nucleation zone would be, on average, of a slower or softer nature. This effect would be magnified in cases where the nucleation zone interfaces through the fractured zone with the opening. In the laboratory experiments, crack branching and distributed damage associated with the fracture are observed only if the strength variation in the source region is greater than the stress concentration (Labuz et al., 1985; Labuz et al., 1985; Cox & Paterson, 1990). Outside, or at the interface of the nucleation zone, one would expect seismic events of a harder or faster nature characterised by higher apparent stress. The overall size of the nucleation zone increases with the size of the potential instability (Scholz, 1990) and with the degree of inhomogeneity and decreases with the increase in the strain rate (Kato et al., 1992).

One can consider an interpretation of instability in terms of the relative stiffness of the components of the system—an instability will occur when the stiffness of the nucleation volume is equal to or exceeds the unloading stiffness of the surrounding rock, see Fig. 3.2 left. Stuart (1981) stated that, for earthquakes, the proximity to instability can be measured by the ratio of the stiffness of the fault zone to that of the elastic surroundings—stability decreases as this ratio approaches unity.

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The stress and the strain jumps at the point of instability, \(\Delta \sigma \) and \(\Delta \epsilon \), depend on the properties of the rock within and, to a certain extent, outside the nucleation volume. The stronger and more homogeneous the rock mass the higher the stress drop and the higher the ratio of the stress drop over strain drop associated with instability. In particular, instability is predicted much nearer the peak load for localised nucleation volumes and for states of deviatoric simple shear than for states of axisymmetric compression (Rudnicki, 1977).

An increase in the rate of co-seismic deformation before instability can be reflected by an increase in the rate of seismic events, an increase in the number of mid-size events or, for the same rate and sizes of events, by the softer nature of individual events, occurring within the nucleation volume. A small but statistically significant decrease in seismic activity (quiescence) has been observed at the beginning of the strain softening stage, followed by an increase just before the instability (Brady, 1977b; Brady, 1977a; Main et al., 1992).

Figure 3.2 right illustrates the acceleration of deformation preceding instability. It is assumed that the surrounding rock mass is stiffer than the nucleation zone and undergoes strain hardening at the time when the nucleation zone softens. As a result, for the same increments of loading stress, \(\sigma _{2}-\sigma _{1}\), and \(\sigma _{3}-\sigma _{2}\), the nucleation volume experiences considerably larger increments of displacement, \(u_{1}\), \(u_{2}\), \(u_{3}\), than the surrounding rock (see Rudnicki, 1988). The longer the surrounding rock maintains its stiffness the later and less dynamic is the instability.

Seismic events can routinely be quantified by the following four independent parameters derived from recorded waveforms: the time of the event, t, location, \(X=x,y,z\), seismic potency, P, and the radiated seismic energy, E. From seismic potency and energy, one can derive apparent stress, \(\sigma _{A} = E/P\), energy index, EI, i.e. the ratio of the observed radiated seismic energy of that event E, to the average energy \(\bar {E}(P) = 10^{d\log P+c}\) radiated by events of the observed potency P, for a given area of interest ((van Aswegen & Butler, 1993)), and apparent volume, \(V_{A} = \mu P^{2} / E\), (Mendecki, 1993).

From the four independent quantities derived from waveforms, we can derive a number of seismicity parameters related to co-seismic deformation and associated changes in the strain rate, stress, and rheology of the process.