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Mathematical Modeling of Rock Pore Geometry and Mineralization: Applications of Persistent Homology and Random Walk

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Agriculture as a Metaphor for Creativity in All Human Endeavors (FMfI 2016)

Part of the book series: Mathematics for Industry ((MFI,volume 28))

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Abstract

Mathematical methods used to model heterogeneous pore geometry of natural rocks and their temporal evolution (mineralization processes) are explored. Recent development of X-ray microcomputed tomography enables high-resolution (micrometers) pore geometry of rock to be obtained. Nevertheless, exploring the complex spatial distribution of pore bodies, and relating this information to hydraulic and elastic properties, remains a challenge. In this study, persistent homology is first applied to describe heterogeneous rock pores, which captures the appearance and disappearance of topological features. The persistence diagram derived from this analysis shows the characteristic features of rock pore. Next, random walk is used to model rock mineralization processes. The results show that rock pore evolution is successfully modeled using random walk by defining the probability of mineral precipitation and dispersion degree in each grid cell of a modeled rock body. The mineralization parameter can be flexibly changed and a short computation time used when using random walk; this approach may thus be practical when simulating rock evolution processes such as long-term chemical reactions in a reservoir.

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Acknowledgements

This research was improved by discussions in the Study Group Workshop in 2016 and is supported by a joint project between the International Institute for Carbon-Neutral Research (I2CNER) and Institute of Mathematics for Industry (IMI), Kyushu University. This work was partially supported by JSPS through a Grant-in-Aid for Science Research on Innovative Area (no.JP15H01143; JP17H05318). T.S. is partially supported by JSPS Grant-in-Aid (26610025, 26287019) and JST CREST Mathematics (15656429).

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Authors and Affiliations

  1. International Institute for Carbon-Neutral Energy Research, Kyushu University, Fukuoka, Japan

    Takeshi Tsuji & Fei Jiang

  2. Department of Earth Resources Engineering, Kyushu University, Fukuoka, Japan

    Takeshi Tsuji

  3. Yamaguchi University, Yamaguchi, Japan

    Fei Jiang

  4. Institute of Fluid Science, Tohoku University, Sendai, Japan

    Anna Suzuki

  5. Institute of Mathematics for Industry, Kyushu University, Fukuoka, Japan

    Tomoyuki Shirai

Authors
  1. Takeshi Tsuji
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  2. Fei Jiang
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  3. Anna Suzuki
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  4. Tomoyuki Shirai
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Corresponding author

Correspondence to Takeshi Tsuji .

Editor information

Editors and Affiliations

  1. Mathematics, Information and Statistics, CSIRO, Acton, Aust Capital Terr, Australia

    Robert S. Anderssen

  2. La Trobe University, Melbourne, Victoria, Australia

    Philip Broadbridge

  3. Kyushu University, Fukuoka, Japan

    Yasuhide Fukumoto

  4. Kyushu University, Fukuoka, Japan

    Kenji Kajiwara

  5. Queensland University of Technology, Brisbane, Queensland, Australia

    Matthew Simpson

  6. Queensland University of Technology, Brisbane, Queensland, Australia

    Ian Turner

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Tsuji, T., Jiang, F., Suzuki, A., Shirai, T. (2018). Mathematical Modeling of Rock Pore Geometry and Mineralization: Applications of Persistent Homology and Random Walk. In: Anderssen, R., Broadbridge, P., Fukumoto, Y., Kajiwara, K., Simpson, M., Turner, I. (eds) Agriculture as a Metaphor for Creativity in All Human Endeavors. FMfI 2016. Mathematics for Industry, vol 28. Springer, Singapore. https://doi.org/10.1007/978-981-10-7811-8_11

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  • DOI: https://doi.org/10.1007/978-981-10-7811-8_11

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  • Publisher Name: Springer, Singapore

  • Print ISBN: 978-981-10-7810-1

  • Online ISBN: 978-981-10-7811-8

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