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.

[1]  Y. Hiraoka,et al.  Pore configuration landscape of granular crystallization , 2017, Nature Communications.

[2]  Bernd Hamann,et al.  Topologically Clean Distance Fields , 2007, IEEE Transactions on Visualization and Computer Graphics.

[3]  H. Vogel,et al.  On the reconstruction of structural and functional properties in random heterogeneous media , 2011 .

[4]  Dorthe Wildenschild,et al.  Image processing of multiphase images obtained via X‐ray microtomography: A review , 2014 .

[5]  Marcel G. Schaap,et al.  Comparison of pressure‐saturation characteristics derived from computed tomography and lattice Boltzmann simulations , 2007 .

[6]  K. Ball.  CONVEX BODIES: THE BRUNN–MINKOWSKI THEORY , 1994 .

[7]  J. Bear.  Dynamics of Fluids in Porous Media , 1975 .

[8]  J. W. Martin,et al.  Pore geometry and transport properties of Fontainebleau sandstone , 1993 .

[9]  Jan Prins,et al.  A novel heterogeneous algorithm to simulate multiphase flow in porous media on multicore CPU-GPU systems , 2014, Comput. Phys. Commun..

[10]  Jean Serra,et al.  Image Analysis and Mathematical Morphology , 1983 .

[11]  J. Parlange.  Porous Media: Fluid Transport and Pore Structure , 1981 .

[12]  Krzysztof M. Gorski,et al.  Minkowski functionals used in the morphological analysis of cosmic microwave background anisotropy maps , 1998 .

[13]  Dorthe Wildenschild,et al.  Efficiently engineering pore-scale processes: The role of force dominance and topology during nonwetting phase trapping in porous media , 2015 .

[14]  Martin J. Blunt,et al.  Multiphase Flow in Permeable Media: A Pore-Scale Perspective , 2017 .

[15]  M. Blunt,et al.  Pore space reconstruction using multiple-point statistics , 2005 .

[16]  Djebbar Tiab,et al.  Petrophysics: Theory and Practice of Measuring Reservoir Rock and Fluid Transport Properties , 1996 .

[17]  Clemens Bechinger,et al.  Direct relations between morphology and transport in Boolean models. , 2015, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  Diego Maza,et al.  Topological analysis of tapped granular media using persistent homology. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[19]  COMPUTATION OF MINKOWSKI MEASURES ON 2 D AND 3 D BINARY IMAGES D , 2007 .

[20]  Martin J. Blunt,et al.  Capillary trapping in sandstones and carbonates: Dependence on pore structure , 2012 .

[21]  R. Schneider.  Convex Bodies: The Brunn–Minkowski Theory: Minkowski addition , 1993 .

[22]  D. Wildenschild,et al.  An improved method for estimating capillary pressure from 3D microtomography images and its application to the study of disconnected nonwetting phase , 2018 .

[23]  Valerio Pascucci,et al.  Computing Morse-Smale Complexes with Accurate Geometry , 2012, IEEE Transactions on Visualization and Computer Graphics.

[24]  Nagel,et al.  An integral‐geometric approach for the Euler–Poincaré characteristic of spatial images , 2000, Journal of microscopy.

[25]  Victor Sofonea,et al.  Morphology of spinodal decomposition , 1997 .

[26]  Karine Le Gorju Jago,et al.  X-RAY COMPUTED MICROTOMOGRAPHY OF RUBBER , 2012 .

[27]  Klaus Mecke,et al.  Minimal surface scaffold designs for tissue engineering. , 2011, Biomaterials.

[28]  Masao Kimura,et al.  Non-empirical identification of trigger sites in heterogeneous processes using persistent homology , 2018, Scientific Reports.

[29]  B. Ninham,et al.  Curvature energy of surfactant interfaces confined to the plaquettes of a cubic lattice , 1990 .

[30]  John Doe,et al.  Vadose Zone Journal 2014 Summary of Editorial Reports , 2015 .

[31]  Claudio Perez Tamargo.  Can one hear the shape of a drum , 2008 .

[32]  Alexander G. Schwing,et al.  From connected pathway flow to ganglion dynamics , 2015 .

[33]  Alasdair N. Houston,et al.  Emergent Behavior of Soil Fungal Dynamics: Influence of Soil Architecture and Water Distribution , 2012 .

[34]  Pierre M. Adler,et al.  The formation factor of reconstructed porous media , 1992 .

[35]  I. Battiato,et al.  The Impact of Pore‐Scale Flow Regimes on Upscaling of Immiscible Two‐Phase Flow in Porous Media , 2018, Water Resources Research.

[36]  L. Santaló.  Integral geometry and geometric probability , 1976 .

[37]  Leila De Floriani,et al.  A discrete morse-based approach to multivariate data analysis , 2016, SIGGRAPH Asia Symposium on Visualization.

[38]  Markus Hilpert,et al.  Pore-morphology-based simulation of drainage in totally wetting porous media , 2001 .

[39]  S. Hassanizadeh,et al.  Micromodel study of two‐phase flow under transient conditions: Quantifying effects of specific interfacial area , 2014 .

[40]  Alkiviades C. Payatakes,et al.  True-to-mechanism model of steady-state two-phase flow in porous media, using decomposition into prototype flows , 2001 .

[41]  P. Mazur,et al.  Non-equilibrium thermodynamics, , 1963 .

[42]  F. Stillinger,et al.  A superior descriptor of random textures and its predictive capacity , 2009, Proceedings of the National Academy of Sciences.

[43]  Cass T. Miller,et al.  Stability of a fluid-fluid interface in a biconical pore segment. , 2003, Journal of colloid and interface science.

[44]  K R Mecke,et al.  Euler-Poincaré characteristics of classes of disordered media. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[45]  Mohammad Saadatfar,et al.  Percolating length scales from topological persistence analysis of micro‐CT images of porous materials , 2016 .

[46]  Harry Vereecken,et al.  VZJ Launches New Paper Type: “Priority Communications” , 2013 .

[47]  Thompson,et al.  Quantitative prediction of permeability in porous rock. , 1986, Physical review. B, Condensed matter.

[48]  Dorthe Wildenschild,et al.  Effect of fluid topology on residual nonwetting phase trapping: Implications for geologic CO 2 sequestration , 2013 .

[49]  Melissa L. Knothe Tate,et al.  Whither flows the fluid in bone?" An osteocyte's perspective. , 2003 .

[50]  V. Grimm.  Ten years of individual-based modelling in ecology: what have we learned and what could we learn in the future? , 1999 .

[51]  Peter John Wood,et al.  Ieee Transactions on Pattern Analysis and Machine Intelligence Theory and Algorithms for Constructing Discrete Morse Complexes from Grayscale Digital Images , 2022 .

[52]  J. McClure,et al.  Beyond Darcy's law: The role of phase topology and ganglion dynamics for two-fluid flow. , 2016, Physical review. E.

[53]  F. J. Caniego,et al.  Morphological Functions to Quantify Three‐Dimensional Tomograms of Macropore Structure in a Vineyard Soil with Two Different Management Regimes , 2013 .

[54]  H. Hadwiger.  Vorlesungen über Inhalt, Oberfläche und Isoperimetrie , 1957 .

[55]  Valerio Pascucci,et al.  Understanding the Structure of the Turbulent Mixing Layer in Hydrodynamic Instabilities , 2006, IEEE Transactions on Visualization and Computer Graphics.

[56]  Hans-Jörg Vogel,et al.  Soil‐structure development including seasonal dynamics in a long‐term fertilization experiment , 2011 .

[57]  Dietmar W Hutmacher,et al.  Assessment of bone ingrowth into porous biomaterials using MICRO-CT. , 2007, Biomaterials.

[58]  Klaus Mecke,et al.  Utilizing Minkowski functionals for image analysis: a marching square algorithm , 2008 .

[59]  Frieder Enzmann,et al.  Real-time 3D imaging of Haines jumps in porous media flow , 2013, Proceedings of the National Academy of Sciences.

[60]  A Sheppard,et al.  Morphological clues to wet granular pile stability. , 2008, Nature materials.

[61]  Hans-Jörg Vogel,et al.  In Situ Visualization and Quantification of Three‐Dimensional Root System Architecture and Growth Using X‐Ray Computed Tomography , 2014 .

[62]  Black hole entropy and the dimensional continuation of the Gauss-Bonnet theorem. , 1993, Physical review letters.

[63]  Yukio Matsumoto.  An Introduction to Morse Theory , 2001 .

[64]  Klaus Mecke,et al.  Statistical Physics and Spatial Statistics , 2000 .

[65]  Afra Zomorodian,et al.  The Theory of Multidimensional Persistence , 2007, SCG '07.

[66]  P. A. Slotte,et al.  Description of Free Energy for Immiscible Two‐Fluid Flow in Porous Media by Integral Geometry and Thermodynamics , 2018, Water Resources Research.

[67]  Konstantin Mischaikow,et al.  Analysis of Kolmogorov flow and Rayleigh–Bénard convection using persistent homology , 2015, 1505.06168.

[68]  Dorthe Wildenschild,et al.  Defining a novel pore-body to pore-throat “Morphological Aspect Ratio” that scales with residual non-wetting phase capillary trapping in porous media , 2018, Advances in Water Resources.

[69]  Jason I. Gerhard,et al.  Measurement and prediction of the relationship between capillary pressure, saturation, and interfacial area in a NAPL‐water‐glass bead system , 2010 .

[70]  Peyman Mostaghimi,et al.  Time-Lapsed Visualization and Characterization of Shale Diffusion Properties Using 4D X-ray Microcomputed Tomography , 2018 .

[71]  Eric J. Anderson,et al.  Pairing computational and scaled physical models to determine permeability as a measure of cellular communication in micro- and nano-scale pericellular spaces , 2008 .

[72]  M. C. Leverett,et al.  Capillary Behavior in Porous Solids , 1941 .

[73]  Stig Bakke,et al.  Reconstruction of Berea sandstone and pore-scale modelling of wettability effects , 2003 .

[74]  Frieder Enzmann,et al.  Connected pathway relative permeability from pore-scale imaging of imbibition , 2016 .

[75]  D. Wildenschild,et al.  Pore‐scale displacement mechanisms as a source of hysteresis for two‐phase flow in porous media , 2016 .

[76]  Topology in Dynamics, Differential Equations, and Data , 2016 .

[77]  H. Weyl.  On the Volume of Tubes , 1939 .

[78]  M. Ferri,et al.  Betti numbers in multidimensional persistent homology are stable functions , 2013 .

[79]  Frank Mücklich,et al.  Statistical Analysis of Microstructures in Materials Science , 2000 .

[80]  H. Vogel,et al.  Water Flow and Solute Transport in Unsaturated Sand—A Comprehensive Experimental Approach , 2015 .

[81]  Herbert Edelsbrunner,et al.  Computational Topology - an Introduction , 2009 .

[82]  Cass T. Miller,et al.  Influence of phase connectivity on the relationship among capillary pressure, fluid saturation, and interfacial area in two-fluid-phase porous medium systems. , 2016, Physical review. E.

[83]  Martin J. Blunt,et al.  Residual CO2 imaged with X‐ray micro‐tomography , 2011 .

[84]  M. Blunt,et al.  Pore space reconstruction of vuggy carbonates using microtomography and multiple‐point statistics , 2007 .

[85]  M. Blunt,et al.  Pore-scale imaging and modelling , 2013 .

[86]  H. Vogel,et al.  Denitrification in Soil Aggregate Analogues-Effect of Aggregate Size and Oxygen Diffusion , 2018, Front. Environ. Sci..

[87]  M. Blunt,et al.  Measurements of the capillary trapping of super‐critical carbon dioxide in Berea sandstone , 2011 .

[88]  Martin J Blunt,et al.  Pore-network extraction from micro-computerized-tomography images. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[89]  R. Armstrong,et al.  Critical capillary number: Desaturation studied with fast X‐ray computed microtomography , 2014 .

[90]  S. Whitaker.  The method of volume averaging , 1998 .

[91]  Peyman Mostaghimi,et al.  Signature of non-Fickian solute transport in complex heterogeneous porous media. , 2011, Physical review letters.

[92]  Dietmar W Hutmacher,et al.  Analysis of 3D bone ingrowth into polymer scaffolds via micro-computed tomography imaging. , 2004, Biomaterials.

[93]  Muhammad Sahimi,et al.  Flow and Transport in Porous Media and Fractured Rock: From Classical Methods to Modern Approaches , 1995 .

[94]  Klaus Mecke,et al.  Integral Geometry in Statistical Physics , 1998 .

[95]  S. Bakke,et al.  Process Based Reconstruction of Sandstones and Prediction of Transport Properties , 2002 .

[96]  Peyman Mostaghimi,et al.  Porosity and permeability characterization of coal: A micro-computed tomography study , 2016 .

[97]  Daniel A. Klain.  A short proof of Hadwiger's characterization theorem , 1995 .

[98]  Mark L. Rivers,et al.  Using X-ray computed tomography in hydrology: systems, resolutions, and limitations , 2002 .

[99]  Ryan T. Armstrong,et al.  Linking pore-scale interfacial curvature to column-scale capillary pressure , 2012 .

[100]  Vanessa Robins,et al.  Morse theory and persistent homology for topological analysis of 3D images of complex materials , 2014, 2014 IEEE International Conference on Image Processing (ICIP).

[101]  Olaf Delgado-Friedrichs,et al.  Skeletonization and Partitioning of Digital Images Using Discrete Morse Theory , 2015, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[102]  Hans-Jörg Vogel,et al.  Quantification of soil structure based on Minkowski functions , 2010, Comput. Geosci..

[103]  Afra Zomorodian,et al.  Computing Persistent Homology , 2004, SCG '04.

[104]  Klaus Mecke,et al.  Additivity, Convexity, and Beyond: Applications of Minkowski Functionals in Statistical Physics , 2000 .

[105]  Eric J Anderson,et al.  Bone as an inspiration for a novel class of mechanoactive materials. , 2009, Biomaterials.

[106]  Gunther H. Weber,et al.  Augmented Topological Descriptors of Pore Networks for Material Science , 2012, IEEE Transactions on Visualization and Computer Graphics.

[107]  R. Hilfer,et al.  Macroscopic capillarity and hysteresis for flow in porous media. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[108]  D. Wildenschild,et al.  X-ray imaging and analysis techniques for quantifying pore-scale structure and processes in subsurface porous medium systems , 2013 .

[109]  Christoph H. Arns,et al.  Super resolution reconstruction of μ-CT image of rock sample using neighbour embedding algorithm , 2018 .

[110]  U. Rüde,et al.  Permeability of porous materials determined from the Euler characteristic. , 2012, Physical review letters.

[111]  William G. Gray,et al.  Geometric state function for two-fluid flow in porous media , 2018, Physical Review Fluids.

[112]  Karlis Gross,et al.  Structure and properties of clinical coralline implants measured via 3D imaging and analysis. , 2006, Biomaterials.

[113]  David Cohen-Steiner,et al.  Stability of Persistence Diagrams , 2005, Discret. Comput. Geom..

[114]  Joachim Ohser,et al.  MESH FREE ESTIMATION OF THE STRUCTURE MODEL INDEX , 2011 .

[115]  Kristel Michielsen,et al.  Integral-geometry morphological image analysis , 2001 .

[116]  M. Kramár,et al.  Quantifying force networks in particulate systems , 2013, 1311.0424.

[117]  Herbert Edelsbrunner,et al.  Hierarchical Morse—Smale Complexes for Piecewise Linear 2-Manifolds , 2003, Discret. Comput. Geom..

[118]  Hans-Jörg Vogel,et al.  A numerical experiment on pore size, pore connectivity, water retention, permeability, and solute transport using network models , 2000 .

[119]  Christoph H. Arns,et al.  Pore-Scale Characterization of Two-Phase Flow Using Integral Geometry , 2017, Transport in Porous Media.

[120]  Klaus Mecke,et al.  Euler characteristic and related measures for random geometric sets , 1991 .

[121]  Herbert Edelsbrunner,et al.  Topological persistence and simplification , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[122]  William B. Haines,et al.  Studies in the physical properties of soil. V. The hysteresis effect in capillary properties, and the modes of moisture distribution associated therewith , 1930, The Journal of Agricultural Science.

[123]  Gunnar E. Carlsson,et al.  Topology and data , 2009 .

[124]  J Ohser,et al.  On the analysis of spatial binary images , 2001, Journal of microscopy.

[125]  H. Edelsbrunner,et al.  Persistent Homology — a Survey , 2022 .

[126]  A. Gray,et al.  Modern Differential Geometry of Curves and Surfaces with Mathematica, Third Edition (Studies in Advanced Mathematics) , 2006 .