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Modeling and Fitting of Three-Dimensional Mineral Microstructures by Multinary Random Fields

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Abstract

Modeling a mineral microstructure accurately in three dimensions can render realistic mineralogical patterns which can be used for three-dimensional processing simulations and calculation of three-dimensional mineral quantities. The present study introduces a flexible approach to model the microstructure of mineral material composed of a large number of facies. The common plurigaussian method, a valuable approach in geostatistics, can account for correlations within each facies and in principle be extended to correlations between the facies. Assuming stationarity and isotropy, founded on a new description of this model, formulas for first- and second-order characteristics, such as volume fraction, correlation function and cross-correlation function can be given by a multivariate normal distribution. In this particular situation, based on first- and second-order statistics, a fitting procedure can be developed which requires only numerical inversion of several one-dimensional monotone functions. The paper describes the whole workflow. The covariance structure is quickly obtained from two-dimensional particle pixel images using Fourier transform. Followed by model fitting and sampling, where the resulting three-dimensional microstructure is then efficiently represented by tessellations. The applicability is demonstrated for the three-dimensional case by modeling the microstructure from a Mineral Liberation Analyzer image data set of an andesitic basalt breccia.

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Acknowledgements

The presented study was part of the projects AMREP (Funding-nr: 033R119B) and REE NAM XE (Funding-nr: 033R120B) both funded by the BMBF (Federal Ministry of Education and Research, Germany) and the latter also by MOST (Ministry of Science and Technology) of Vietnam in terms of a collaboration with Hanoi University of Mining and Geology (HUMG) and Hung Hai Group (HHG). Both projects are part of the research program CLIENT ’International Partnerships for Sustainable Technologies and Services for Climate Protection and the Environment’.

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Authors and Affiliations

  1. Helmholtz Institute Freiberg for Resource Technology, Helmholtz-Zentrum Dresden - Rossendorf, Chemnitzer Str. 40, 09599, Freiberg, Germany

    Jakob Teichmann, Thomas Heinig & Karl Gerald van den Boogaart

  2. Institute for Geophysics and Geoinformatics, TU Bergakademie Freiberg, Gustav-Zeuner Str. 12, 09599, Freiberg, Germany

    Peter Menzel

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  1. Jakob Teichmann
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  2. Peter Menzel
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  4. Karl Gerald van den Boogaart
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Correspondence to Jakob Teichmann or Karl Gerald van den Boogaart.

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Teichmann, J., Menzel, P., Heinig, T. et al. Modeling and Fitting of Three-Dimensional Mineral Microstructures by Multinary Random Fields. Math Geosci 53, 877–904 (2021). https://doi.org/10.1007/s11004-020-09871-4

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  • DOI: https://doi.org/10.1007/s11004-020-09871-4

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