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

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.