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

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


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.


Natural rock Heterogeneous pore geometry Persistent homology Rock mineralization Random walk 



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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© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.International Institute for Carbon-Neutral Energy Research, Kyushu UniversityFukuokaJapan
  2. 2.Department of Earth Resources EngineeringKyushu UniversityFukuokaJapan
  3. 3.Yamaguchi UniversityYamaguchiJapan
  4. 4.Institute of Fluid Science, Tohoku UniversitySendaiJapan
  5. 5.Institute of Mathematics for Industry, Kyushu UniversityFukuokaJapan

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