Abstract
Micro-seismic events within a mine have the potential to disturb planned daily operations as unpredictable events can lead to safety concerns. Mines are typically large-scale underground endeavours which cannot easily be modelled on the macro scale. One increasingly popular technique for simulating activity in mines is the material point method (MPM) which is a numerical algorithm designed for analysing large deformations. One attraction of the MPM over many other numerical strategies is that it does not require repeated remeshing of the underlying computational grid but this means that the MPM is not particularly suited to analysis of small-scale events. In an effort to address this dichotomy, in this work we propose an adaptation to the regular grid used in MPM together with a novel time-stepping strategy. This assists greatly in the solution of large modeling problems typical of those arising in mining contexts. We present a new approach of extracting mining induced micro-seismic events from a numerical model by converting simulated plastic strain to seismic potency with the elastoplastic Coulomb criterion used to describe tensile and shear fracturing. Our results are benchmarked against a standard geotechnical problem in order to validate our formulation. The potential practical usefulness of the model is demonstrated by applying it to a real case study consisting of 4 months of recorded micro-seismic events taken from a mine in Tasmania.
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Availability of Data and Material
Mine seismic data is the property of the mine and will not be shared.
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Abbreviations
- L :
-
Three-dimensional grid cell length
- \(L_i\) :
-
Level i grid cell size
- t :
-
Time
- \(\Delta {t}\) :
-
Time step
- \(S_l\) :
-
Shape functions used to interpolate values between MPM particles and grid nodes
- \(G_l\) :
-
Derivatives of shape functions \(S_i\)
- \(C_L\) :
-
A list of sorted grid cells ranging from \(L_n\) to \(L_0\)
- \(m_l\) :
-
Mass of node l
- \(\mathbf{v} _l\) :
-
Velocity of node l
- \(\mathbf{v} _p\) :
-
Velocity of particle p
- \(\mathbf{f} _l\) :
-
Force acting on node l
- \(\sigma _p\) :
-
Stress of particle p
- \(\sigma _\mathrm{{r}}\) :
-
Radial stress
- \(\sigma _\theta\) :
-
Tangential stress
- \(\sigma _\infty\) :
-
Uniform applied stress in Kirces test
- d :
-
Radius of opening in Kirches test
- \(V_p\) :
-
Particle volume l
- \(\mathbf{b}\) :
-
External body forces acting on a particle
- \(m_p\) :
-
Mass of particle p
- \(\mathbf{a} _p\) :
-
Acceleration of node l
- \(\mathbf{f} _\mathrm{{d}}\) :
-
Damping force to remove energy from the MPM system
- \(\mathbf{x} _p\) :
-
Position of particle p
- \(\dot{\varepsilon }_{p}\) :
-
Strain increment of particle p
- \(\Delta \sigma {_p}^\mathrm{{p}}\) :
-
Stress drop of particle p due to constitutive relation
- P :
-
Seismic potency
- \(\bar{u}\) :
-
Average slip calculated over seismic source area
- \(A_\mathrm{{s}}\) :
-
Seismic source area
- \(P_\mathrm{{s}}\) :
-
Seismic potency of a single dislocation seismic source
- \(\Delta \varepsilon\) :
-
Strain change
- \(\Delta \sigma\) :
-
Stress change
- \(\mu\) :
-
Rigidity of rock mass
- \(V_\mathrm{{s}}\) :
-
Seismic source volume
- \(\Delta \varepsilon ^\mathrm{{p}}\) :
-
Plastic strain change
- V :
-
Volume
- \(\sigma _1\) :
-
Major principal stress
- \(\sigma _2\) :
-
Intermediate principal stress
- \(\sigma _3\) :
-
Minor principal stress
- \(P^\mathrm{{T}}\) :
-
Total potency of a group of failed particles
- \(P_{ij}\) :
-
Major principal stress
- \(\Delta \varepsilon ^\mathrm{{p}}_{ij}\) :
-
Tensor of plastic strain change
- \(X_{ij}\) :
-
Arbitrary tensor in closeness test
- \(Y_{ij}\) :
-
Arbitrary tensor in closeness test
- D :
-
Mine datum (depth from surface)
- \(\alpha\) :
-
A constant scaling factor ranging between 0 and 1 to account for a proportion of aseismic potency
- b :
-
Power-law exponent in Gutenberg–Richter law
- a :
-
A constant in the Gutenberg–Richter law
- N :
-
Number of events in Gutenberg–Richter law
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Acknowledgements
We are indebted to Kevin Stacey from Renison Bell mine for allowing us to use and show the mine data. We are also grateful to the referees whose numerous suggestions led to substantial improvements to this paper. We are indebted to the editors for their guidance and advice.
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All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by GB. The first draft of the manuscript was written by GB and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.
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Basson, G., Bassom, A.P. & Salmon, B. Simulating Mining-Induced Seismicity Using the Material Point Method. Rock Mech Rock Eng 54, 4483–4503 (2021). https://doi.org/10.1007/s00603-021-02522-y
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DOI: https://doi.org/10.1007/s00603-021-02522-y