Porous Media Characterization Using Minkowski Functionals: Theories, Applications and Future Directions

  • Ryan T. ArmstrongEmail author
  • James E. McClure
  • Vanessa Robins
  • Zhishang Liu
  • Christoph H. Arns
  • Steffen Schlüter
  • Steffen Berg


An elementary question in porous media research is in regard to the relationship between structure and function. In most fields, the porosity and permeability of porous media are properties of key interest. There is, however, no universal relationship between porosity and permeability since not only does the fraction of void space matter for permeability but also the connectivity of the void fraction. With the evolution of modern day X-ray microcomputed tomography (micro-CT) and advanced computing, it is now possible to visualize porous media at an unprecedented level of detail. Approaches in analyzing micro-CT data of porous structures vary in the literature from phenomenological characterization to network analysis to geometrical and/or topological measurements. This leads to a question about how to consistently characterize porous media in a way that facilitates theoretical developments. In this effort, the Minkowski functionals (MF) emerge from the field of statistical physics where it is evident that many physical processes depend on the geometry and topology of bodies or multiple bodies in 3D space. Herein we review the theoretical basis of the MF, mathematical theorems and methods necessary for porous media characterization, common measurement errors when using micro-CT data and recent findings relating the MF to macroscale porous media properties. This paper is written to provide the basics necessary for porous media characterization and theoretical developments. With the wealth of information generated from 3D imaging of porous media, it is necessary to develop an understanding of the limitations and opportunities in this exciting area of research.


Minkowski functionals X-ray microcomputed tomography Pore morphology Euler characteristic Persistence homology 



We thank the Tyree X-Ray Laboratory in the School of Minerals and Energy Resources Engineering, UNSW for assistance with image collection and data processing. Professors Stephen Foster (fiber-reinforced concrete) and Melissa Knothe-Tate and Dr. Tzong-Tyng Hung (mouse leg) are acknowledged for graciously sharing their microtomography data. We thank Ji-Youn Arns and Zhenghuai Guo for their persistence in contrast optimization for the mouse leg and visualization of the fiber-reinforced concrete samples. Funding was provided from the Australian Research Council Discovery Grant DP160104995. VR is supported by ARC Future Fellowship FT140100604. An award of computer time was provided by the Department of Energy INCITE program. This research also used resources of the Oak Ridge Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC05-00OR22725.


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Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.School of Minerals and Energy Resoureces EngineeringUniversity of New South WalesSydneyAustralia
  2. 2.Advanced Research ComputingVirginia TechBlacksburgUSA
  3. 3.Research School of Physics and EngineeringAustralian National UniversityCanberraAustralia
  4. 4.Helmholtz-Centre for Environmental ResearchHalleGermany
  5. 5.Shell Global Solutions International B.V.RijswijkThe Netherlands
  6. 6.Department of Earth Science and Engineering, Department of Chemical EngineeringImperial College LondonLondonUK

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