3. Examples of the simulation of seismicity patterns by different numerical models
3.3. Clustering in space and time
3.3.4. Analysis of a set of clusters
Consider a catalogue of the local seismicity in an area of interest for a given period of time. By applying the cluster identification procedure the catalogue is decomposed into a set of
connected clusters. The mass and the shape characteristics of each cluster are evaluated according to the above formulated rules. The next step is to study the distribution of the clusters according to one or more of the cluster characteristics. The spatial distribution of the centroids of the clusters can be related to the locations of stress concentrations and seismogenic
geological structures. Additional information about the seismogenic structures can be extracted from the orientation of the principle axes of the cluster ellipsoids. The temporal evolution of spatial seismic clusters can be related to the local mining activity and can specify the input to numerical models which is one of the form of integration of modelling with monitoring.
The simplest means of evaluating the spatial distribution of seismicity is to plot the data on mine plans with the size of the symbol being proportional to some parameter obtained from the seismic event e.g. magnitude or moment. This does not allow for evaluation of the patterns.
The application of numerical or analytical models based on the theory of elasticity led to the development of parameters based on stress or strain measures that could be used to indicate the spatial variability of seismicity.
McCreary et al. (1993) suggest that reasonable estimates of seismic potential could be obtained at the Ansil Mine in Canada by consideration of modelled principal stresses. A three-
dimensional boundary element code was used to model the mine layout. Known seismic events were plotted along with the principal stress contours. The seismicity correlated to regions in which the principal stress was greater than 55 MPa and less than 80 MPa. Regions with a principal stress greater than 80 MPa did not exhibit seismicity and were considered to have failed completely. McCreary et al. (1993) commented that parameters such as the principal stress change due to mining, the stress difference and the extension strain should also be considered. Bek et al. (1996) noted that the majority of events in a deep copper mine lay on pre- existing fault planes. They plotted the stresses at each event location, obtained from an elastic model, in the Mohr Coulomb space, to obtain an estimate of the elastic stresses at which the events were occurring. They stressed the need for inelastic models to be able to address the issue of path dependence of rock failure on the evolution of seismicity.
The excess shear stress ESS can be defined as the shear stress acting on an existing or potential plane in excess of the dynamic rupture strength (Ryder, 1988). Reasonable estimates of seismic event magnitude were obtained by Ozbay et al. (1993) and Webber (1990) using the MINSIM-D program and considering the modelled ride on a plane of potential failure below the stope. The integral of the ride over the expected area of slip was used to calculate an equivalent seismic moment. Seismic location accuracy must also be considered when comparing the predictions of numerical modelling with actual seismic event locations (Webber, 1990). The extension to three dimensions can be made by considering the volume excess shear stress VESS (Spottiswoode, 1990). Comparison with seismic events in mines indicates that the maximum VESS corresponds to regions of high seismicity and that the maximum event
magnitudes can be related to the maximum VESS. The importance of explicitly including the co- seismic strain changes to model inelastic response is stressed by Spottiswoode (1990).
The energy release rate ERR is defined as the amount of strain energy released per unit area of mining. The ERR can be related to the volume of closure. Minney and Naismith (1993)
attempted to match the rate of change of stored energy with the seismicity from mining a remnant pillar. They noted, however, that the elastic models were only useful in hindsight for understanding the sequence of seismic events. A method to introduce inelastic response by limiting the maximum stress carried by pillars and the rock mass ahead of the face produced better correlations between the ERR and the locations of events (Spottiswoode, 1997) McGarr (1976) proposed that a link between seismicity and the area that had been mined out could be obtained through the relationship
V G Mo =γ ∆
where Mois the cumulative seismic moment, G, is the shear modulus and ∆Vis the volume of closure in the stope. The dimensionless parameter γ is determined from back analysis of mine seismic data (Milev and Spottiswoode, 1997) for a particular region.
In models where the fault zones are treated explicitly, the frictional properties affect the space and time distributions of slip (Cochard and Madariaga, 1996, Tse and Rice, 1986). Plots of slip, or stress drop against position on the fault for various time contours can provide insight into the space-time correlation of events for different parameter ranges.
A series of tests to investigate clustering in the particle flow code (PFC3D) were performed by (Hazzard, 1999). The numerical models were compared against physical laboratory
experiments. The model used in a test with uniaxial loading of a cylinder is shown in Figure 3.3.1a. The plot of events occurring immediately after failure is shown in Figure 3.3.1b and indicates how the events are clustered along a single failure plane.
Figure 3.3.1 Cylindrical PFC3D sample (Hazzard, 1999) showing a) cutaway view of sample and b) localisation and clustering of events along final plane immediately after failure.
Hazzard (1999) also performed a series of tests on a square sample to investigate the effect of the parameters of a proposed clustering algorithm. The sample and final set of event clusters are shown in Figure3.3.2. As shown in Figure 3.3.3, the size distribution depends on the stress path applied to the model. Test I considered hydrostatic compression followed by compression of one side of the cube to failure. Test II was an extension test, where the stress on one side was relieved until the sample failed. Figure 3.3.4 and 3.3.5 show that the cluster size also depends on the choice of the space and time windows in the clustering algorithm.
Figure 3.3.2 a) Square PFC3D sample for loading in compression or extension (test II) clusters recorded prior to failure in Test I. b) Clusters of events for tests I (Hazzard, 1999).
0.001 0.01 0.1 1
-6.2 -6 -5.8 -5.6 -5.4 -5.2 -5
Magnitude, Me
Probability of event with magnitude > Me
Test II b ~ 3.4 Test I
b ~ 4.1
Figure 3.3.3. Frequency-magnitude plots for two tests with different stress paths. Test I is a compression tests and test II is an extension test. (Hazzard, 1999).
0.001 0.01 0.1 1
-2.9 -2.7 -2.5 -2.3 -2.1 -1.9 -1.7
Magnitude, Me Prabability of Event with Magnitude > Me
Individual Cracks
2.5 3.75 5 7.5
Figure 3.3.4. Comparison of frequency-magnitude plots for events with different space windows. The value of Re in particle diameters is shown on each curve. (Hazzard, 1999).
0.001 0.01 0.1 1
-2.9 -2.7 -2.5 -2.3 -2.1 -1.9 -1.7
Magnitude, Me Probability of Event with Magnitude > Me
2.4 4.8
9.6 14.4 Individual 19.2
Cracks
Figure 3.3.5. Comparison of frequency-magnitude plots with different time windows (Te).
The value of Te in µµµµs is shown on each curve. The radius of the space window (Re) is 5 particles diameters in each case (Hazzard, 1999).