Special care should be taken to specify the initial state of the modeled rock mass. For a continuum: the equations that determine the evolution over time of the state variables (eg the local stresses).
Introduction
- Formulation of the problem
- Model - data interaction
- Visual inspection
- Calibration
- Model optimisation
- Integration of seismic monitoring with numerical modelling
- Forward and inverse modelling
- The concept of integration
- Restrictions imposed on models by the integration paradigm
- Structure of an integrated seismic monitoring-modelling system
Forward problem: Given are the properties of the material, the initial state and the loading conditions. The concrete approach to solving this problem will depend on the structure of the model.
Review of numerical models potentially useful for integration
Generic modelling methods
- Discrete Models
- Continuum models
- Boundary Element Models and discrete fault planes
- Static vs. dynamic modelling
- Stochastic vs. deterministic models
Most of the readily available codes (as mentioned earlier) are static, and consequently the rock mass is implicitly assumed to have no inertia (zero density/infinite wave speed). The speed-dependent friction laws cause complexity in the response of the fault set in a homogeneous medium.
Numerical models currently being developed for use in SA mining Industry
- MINSIM/MAP3D
- Boundary element basics
- Simulation of seismic activity using the displacement discontinuity boundary
- MAP3Di/Point KERNEL
- MAP3Di
- Point Kernel
- DIGS
- The Integrated Damage Rheology Model (IDRM)
- Physical basis of IDRM
- Material Stability and Seismic Events in IDRM
- Typical Problems treated by the IDRM
- Integration of real seismic events in IDRM
The external load corresponds to the boundary conditions required for the uniqueness of the solution. Therefore, it is necessary to include the degree of damage in the free energy of the system.
Examples of the simulation of seismicity patterns by different numerical models
Size distribution
- Case study 2 - modelled seismic data
- Case study 1 versus case study 2 – Relative changes
Implement filtering methods for the modeled data that calculate a cumulative frequency-magnitude relationship due to the failure of the aperities over time. For the parameter assumptions used in the analyzes performed, a significant difference was observed between the modeled and observed b-values of the cumulative frequency-magnitude relationship. An analysis of the cumulative frequency-magnitude relationship derived from the observed seismic data selected along the Break fault produces the plots shown in Figures 3.1.31 and 3.1.32.
An analysis of the cumulative frequency-magnitude relationships derived from the modeled seismic data from case study 1 yields the graphs shown in Figures 3.1.33 and 3.1.34. An analysis of the cumulative frequency-magnitude relationship derived from the modeled seismic data from Case Study 2 yields the graphs shown in Figures 3.1.36 and 3.1.37. The resulting b-values of the two case studies were consistent with the b-value of the observed data.
Comparison between frequency magnitude ratios. all the modeled and observed seismic event data analyzed are from the Break fault at TauTona mine). All the modeled seismic data derived from the above assumptions showed some power law behavior. It can therefore be noted that different methods of moment calculation change the resulting b-value of the cumulative frequency-magnitude ratio.
Time distribution
An infinite cluster (or exciting cluster) is a connected cluster which includes occupied sites from opposite ends of the lattice. On the other hand, the model will certainly percolate for p=1 or p<1, but very close to 1. Instead, the percolation model exhibits a phase transition where the corresponding critical point is the percolation threshold, i.e. the value of the occupancy probability for which the model can generate an intriguing cluster for the first time.
The properties of subcritical and supercritical (spanning) clusters in percolation models can be relevant for modeling faults and specifically for choosing a distribution density of the asperities. The results showed that there was very little difference over most of the size range, although larger sizes. The model in Figure 3.2.3a uses a rate-independent model, and slip events occur once, after which no further relaxation is allowed.
The model in Figure 3.2.3 uses a viscoplastic rheological model and shows a number of slips extending across the fault surface after each initial slip.
Clustering in space and time
- Cluster identification: connected clusters
- Cluster analysis of model-generated data
- Quantitative analysis of spatial seismic clusters
- Analysis of a set of clusters
- Modelling with IDRM
An important part of the cluster analysis of seismic data (real or model generated) is the quantification of each individually connected cluster. The weights can be chosen based on the size of the events or set to one for all events in the cluster. Then the length of the major axis of the cluster ellipsoid is equal to the length of the vector (Pk - PC) and the unit vector along the major axis is collinear with (Pk - PC).
Select the point Pm which is farthest from the principle axis line. The spatial distribution of cluster centroids can be related to the locations of stress and seismogenic concentrations. Related group number 63 (out of 111) from two different perspectives: a) On the plane of the principle and the secondary axis; b) in the plane of the principle and the tertiary axis.
Another factor that affects the results of cluster analysis of data generated by the model is the geometry of the spatial discretization.
Migration of Seismicity
The integration of seismic migration and aftershock activity with numerical modeling requires an assessment of the models' ability to represent observed behavior. An initial simulation of these effects has been carried out by considering the time-dependent formation of the fracture zone around a mine face. In this case, a small area of approx. 10m × 10m covered by a random mesh of grid elements and the edge of the stope is simulated as a rectangular slot adjacent to this area.
The crack height (stop width) was set to one meter and the fractures were allowed to activate before the slot. To control for migration effects, the distance of the center of each sequentially activated grid element from the center of the stop page was calculated and averaged within each time step. This will enable the program to consider forward modeling the potential for the formation and interaction of new seismic events, based on a set of input seismic events and a known and planned mine layout.
A dot kernel analysis of the stop showing the distance of events to the stop over time with size indicated by the size of the circles.
Seismic Monitoring: Requirements and Limitations
Requirements
The shape of the source cannot be defined routinely and is often assumed to be circular or at best rectangular above the fault plane. The deviatoric component's eigenvalues and corresponding eigenvalues describe, respectively, the magnitude and orientation of the principal moment axis (neglecting gravity) acting at the source. Characteristic length, l, associated with the narrowest dimension of the source, which is inversely proportional to the higher dominant frequency of the velocity spectrum at the source, fo, also known as the second corner frequency from the displacement spectrum, .
It follows from the linearity of the basic equations that Fc and u must be proportional to each other, thus. Thus, the relative contribution of fracture work to seismic energy increases with a decrease in fracture size. The faster the stress fluctuations, the greater their contribution to the radiated energy due to the presence of the time derivative of stress.
In the far field, the P and S wave contributions to the total radiated energy are proportional to the integral of the square of the P and S velocity spectrum.
Uncertainties
The estimation of the uncertainty depends on the method used to solve the system of station equations. Where h is the estimated hypocenter, C is the covariance matrix associated with the matrix of the partial derivatives evaluated at the estimated hypocenter. The value of k? depends on the distribution of the residuals, which are usually assumed to be normally distributed.
This type of simulation was required to generate large numbers of the replicated time series and inversions. The alternative approach is to take the form of the probability distribution and then estimate the matrix for the mean, variance, and covariance. The mean value and the variance of the normal distribution are given by W and c2 respectively.
In what follows, we will assume that the underlying probability density functions (pdf) of the estimated seismic parameters are Gaussian, with the variances directly related to the standard deviations.
Limitations
General issues and future developments
- Validation of model representations of seismicity
- Definition of a seismic event in the context of a particular model
- Quasi-static Boundary Element Tessellation/space filling codes
- IDRM
- Computational geometry issues in numerical modelling practice
- The continuum limit and finite-size scaling
- Error analysis of real and modelled data
- Types of errors
- Errors due to the discretization (grid-size and grid-quality effects)
- Specifying the initial and boundary conditions for a particular model
- Constitutive relations
- Theoretical derivation of constitutive relations
- Empirical constitutive relations
All numerical models relevant to the mining industry fall into one of the above categories. The smaller the grid (ie, the finer the mesh), the smaller the minimum size of seismic events generated by the model. In the boundary integral scheme, the shape of the elements is directly related to the errors in the numerical integration (which again uses interpolation techniques).
Choosing a numerical approximation of the solution to a mathematical problem formulated in the model. A very important source of systematic errors in observational data is related to the design of the measuring equipment. The most difficult of these is to quantify the errors due to the choice of basic equations embedded in the numerical model.
The errors due to the shape of the individual grid cells are treated in the same way.
Conclusions
Requirements imposed by the integration on the numerical models
Numerical models that are suitable for seismic integration
- Existing models
- Future model development
Future developments should explore the possibility of formulating a fully dynamic version of the DDM to allow for dynamic fault slip and associated wave propagation effects. This would have to be integrated with deformations dominated by slow creeps and would allow a much richer inclusion of waveform characteristics in the integration process. Hybrid methods, such as the integrated damage rheology model (IDRM) described in this report, promise detailed dynamic simulation of complex seismic resource processes and their integration with mine planning.
At the quasi-dynamic level, damage rheology modeling is conceptually suitable for integration with real seismic data by converting seismic events into corresponding additional loads on rocks. Non-linear continuum models, such as the finite element method (FEM) or other forms of finite difference models, may also prove to be suitable tools for developing a hybrid seismic damage model. In these cases, numerical strategies must be used to address absorbing boundary problems for mining applications.
Recommendations for integration in practice
A conceptual view of the integration of seismic monitoring with numerical modeling for the needs of the mining industry is the first step towards developing the new generation of mine design and planning tools.
Practical aspects of hybridizing the boundary integral method with damage rheology modeling for seismic data simulation. van Aswegen, G., Durrheim, R. and Ortlepp, W.D. Proceedings of the 5th Symposium on Rock Blasts and Seismicity in Mining, September 2001, South Africa. An analysis of the asperity model under viscoplastic displacement loading and its integration with quantitative seismology. Modeling of damages and instabilities in the rock mass by means of a non-linear rheological model. and Ortlepp, W.D. Proceedings of the 5th Symposium on Rock Blasts and Seismicity in Mining, September 2001, South Africa.
On the relationship between seismic moment and stress drop in the presence of stress and force heterogeneity, Journal of Geophysical Research, Vol. Mining Safety Advisory Committee to the South African Department of Minerals and Energy (SIMRAC), Johannesburg, South Africa. Quantitative analysis of seismic activity associated with the extraction of a tailings pile in a moderately deep level gold mine.
Modeling notch formation mechanisms in the URL mine-by-test tunnel using bonded assemblies of circular particles.
