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2024 | OriginalPaper | Buchkapitel

3. The Internal Topology of Rocks

verfasst von : Gabor Korvin

Erschienen in: Statistical Rock Physics

Verlag: Springer Nature Switzerland

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Chapter Highlights

We discuss the modern mathematical tools for studying the topology of pore space. In Sect. 3.1. Minkowski functionals are introduced through the paradigm of Mark Kac (Kac, Am Math Monthly 73:1–24, 1966) ‘Can one hear the shape of a drum?’. Euler characteristics, a powerful tool to study connectivity is treated in 3.2. The elegant and sophisticated research tool, Persistent Homology, is built up in a series of easy-to-follow steps in 3.3. The abstract constructions are illustrated with practical examples, and there are copious references in the text and in Tables (3.1, 3.4, 3.11) for their use in Petrophysics.

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Aboufoul M, Garcia A (2017) Factors affecting hydraulic conductivity of asphalt mixture. Mater Struct 50:1–16CrossRef

Aktas ME, Akbas E, Fatmaoui AE (2019) Persistence homology of networks: methods and applications. Appl Netw Sci 4(1):1–28CrossRef

Alexandroff P, Hilbert D (1932) Einfachste Grundbegriffe der Topologie. Julius Springer, BerlinCrossRef

Alqahtani N, Armstrong RT, Mostaghimi P (2018) Deep learning convolutional neural networks to predict porous media properties. SPE Asia Pacific Oil and Gas Conference and Exhibition.

Amorim E, Moreira RA, Santos FAN (2019) The Euler characteristic and topological phase transitions in complex systems. J Phys Complex 3:025003

Andreeva MV, Kalyuzhnyuk AV, Krutko VV, Russkikh NE, Taimanov IA (2021) Representative elementary volume via averaged scalar Minkowski functionals. arXiv: 2008.03727v2

Andrew M (2019) Comparing organic-hosted and intergranular pore networks: topography and topology in grains, gaps and bubbles. In: Dowey P, Osborne M, Volk H (eds) Application of analytical techniques to petroleum systems, vol 484. Geological Society, London, Special Publications, pp 241–253

Archie G (1942) The electrical resistivity log as an aid in determining some reservoir characteristics. Trans AIME 146(1):54–62CrossRef

Armstrong RT, McClure JE, Robins V, Liu Z, Arns CH, Schlüter S, Berg S (2018) Porous media characterization using Minkowski functionals: theories, applications and future directions. Transp Porous Media 2018:1–31

Arns CH (2002) The influence of morphology on physical properties of reservoir rocks. Doctor of Philosophy Thesis in Petroleum Engineering. School of Petroleum Engineering. The University of New South Wales

Arns CH, Knackstedt MA, Pinczewski WV, Mecke KR (2001) Euler-Poincaré characteristics of classes of disordered media. Phys Rev E 63(031112):1–13

Arns CH, Knackstedt MA, Mecke KR (2002) Characterising the morphology of disordered materials. In: Mecke KR, Stoyan D (eds) Lecture notes in physics, vol. 600. Springer-Verlag, Berlin-Heidelberg, pp 37–74

Arns CH, Knackstedt MA, Pinczewski WV (2002a) V_p: V_s relationships for model sandstones. Geophys Res Lett 29(8):44-1–44-4

Bachmat Y, Bear J (1986) Macroscopic modelling of transport phenomena in porous media. 1: The continuum approach. Transp Porous Media 1:213–240CrossRef

Bakke S, Øren P (1997) 3-d pore-scale modeling of sandstones and flow simulations in the pore networks. SPE J 2:136–149CrossRef

Bazaikin Y, Gurevich B, Iglauer S, Khachkova T, Kolyukhin D, Lebedev M, Lisitsa V, Reshetova G (2017) Effect of ct image size and resolution on the accuracy of rock property estimates. J Geophys Res Solid Earth 122:3635–3647

Bear J (1972) Dynamics of fluids in porous media. Parts 1–2. American Elsevier Publishing Company, New York

Bengio Y (2009) Learning deep architectures in AI. Found Trends Mach Learn 2(1):1–127CrossRef

Beran M (1965) Use of the variational approach to determine bounds for the effective permittivity in random media. Nuovo Cim 38:771–782CrossRef

Berchtold MA (2007) Modelling of random porous media using Minkowski-Functionals. Doctoral Thesis. ETH, Zürich

Berg CF (2014) Permeability description by characteristic length, tortuosity, constriction and porosity. Transp Porous Media 103:381–400CrossRef

Berryman JG, Blair SC (1986) Use of digital image analysis to estimate fluid permeability of porous materials. Application of the two-point correlation function. J Appl Phys 60(6):1930–1938

Blasquez I, Poiraudeau J-F (2003) Efficient processing of Minkowski functionals on a 3d binary image using binary decision diagrams. J WSCG 11(No1). ISSN 1213–6972. WSCG’2003, Feb 3–7, 2003, Plzen, Czech Republic

Bobrowski O, Skraba P (2020) Homological percolation and the euler characteristic. Phys Rev E 101:032304

Boelens MP, Tchelepi HA (2021) QuantImPy: Minkowski functionals and functions with Python. SoftwareX 16:100823CrossRef

Borges J, Pires L, Cássaro F, Roque W, Heck R, Rosa J, Wolf F (2018) X-ray microtomography analysis of representative elementary volume (REV) of soil morphological and geometrical properties. Soil and Tillage Research https://doi.org/10.1016/j.still.2018.05.004

Bubenik P (2009) Persistent homology of functions. (Ppt presentation, Cleveland State University, August 4, 2009). https://academic.csuohio.edu/bubenik_p/talks/functions.pdf

Carlsson G (2009) Topology and data. Am Math Soc Bull New Series 46(2):255–308

Carlsson G, Zomorodian A, Collins A, Guibas LJ (2005) Persistence barcodes for shapes. Int J Shape Model 11(02):149–187CrossRef

Carman PC (1936) Fluid flow through granular beds. Trans Inst Chem Eng 15:150–166

Chazal F, Michel B (2021) An Introduction to Topological Data Analysis: Fundamental and Practical Aspects for Data Scientists Front. Artif. Intell., Sec. Machine Learning and Artificial Intelligence vol. 4

Chen L, Rong Y (2010) Digital topological method for computing genus and the Betti numbers. Topol Appl 157:1931–1936

Cohen-Steiner D, Edelsbrunner H, Harer J (2007) Stability of persistence diagrams. Discret Comput Geom 37(1):103–120CrossRef

Cohen-Steiner D, Edelsbrunner H, Harer J, Mileyko Y (2010) Lipschitz functions have Lp-stable persistence. Found Comput Math 10:127–139CrossRef

Costa A (2006) Permeability-porosity relationship: a re-examination of the Kozeny-Carman equation based on a fractal pore-space geometry assumption. Geophys Res Lett 33:L02318CrossRef

Coxeter HSM (1973) Regular polytopes, 3rd edn. Dover Publication Inc., New York

Dahrabou A, Viseur S, Gonzalez-Lorenzo A, Rohmer J, Bac A, Real P, Mari J-L, Audigane P (2016) Topological comparisons of fluvial reservoir rock volumes using betti numbers: application to CO₂ storage uncertainty analysis. In: Bac A, Mari JL (eds) Computational topology in image context. CTIC 2016. Lecture notes in computer science, vol 9667. Springer, Cham, pp 101–112

Daigle H (2016) Application of critical path analysis for permeability prediction in natural porous media. Adv Water Resour 96:43–54CrossRef

Delesse MA (1847) Procédé mécanique pour déterminer la composition chimique des roches. CR Acad Sci (Paris) 25:545–565

Delgado-Friedrichs O, Robins V, Sheppard A (2014) Morse theory and persistent homology for topological analysis of 3D images of complex materials. In: Proceedings of 2014 IEEE international conference on image processing (ICIP), pp 4872–4876

Delgado-Friedrichs O, Robins V, Sheppard A (2015) Skeletonization and partitioning of digital images using discrete Morse theory. IEEE Trans Pattern Anal Mach Intell 37(3):654–666CrossRef

Dohnalik M, Jarzyna J (2015) Determination of reservoir properties through the use of computed X-ray microtomography—eolian sandstone examples. Geol Geophys Environ 41(3):223–248

Edelsbrunner H, Harer J (2008) Persistent homology—a survey. Contemp Math 453:257–282CrossRef

Edelsbrunner H, Harer JL (2010) Computational topology: an introduction. AMS Press, Providence

Edelsbrunner H, Letscher D, Zomorodian A (2002) Topological persistence and simplification. Discret Comput Geom 28(4):511–533CrossRef

Epstein N (1989) On tortuosity and the tortuosity factor in flow and diffusion through porous media. Chem Eng Sci 44:777–779CrossRef

Euler L (1758) Elementa doctrinae solidorum. Novi Commentarii Acad. Scientiarum Petropolitanae 109–140

Fabbri R, Costa LDF, Torelli JC, Bruno OM (2008) 2D Euclidean distance transform algorithms: a comparative survey. ACM Comput Surv 40(1):2-1–2-44

Feng M, Porter MA (2021) Persistent homology of geospatial data: a case study with voting. SIAM Rev 63(1):67–99

Flegg HG (1974) From geometry to topology. The English Universities Press Ltd., London

Garboczi EJ (1990) Permeability, diffusivity, and microstructural parameters: a critical review. Cem Concr Res 20:591–601CrossRef

Ghanbarian B, Hunt AG, Ewing RP, Sahimi M (2013) Tortuosity in porous media: a critical review. Soil Sci Soc Am J 77:1461–1477CrossRef

Ghanbarian B, Hunt AG, Ewing RP, Skinner TE (2014) Universal scaling of the formation factor in porous media derived by combining percolation and effective medium theories. Geophys Res Lett 41(11):3884–3890CrossRef

Ghrist R (2008) Barcodes: the persistent topology of data. Bull Am Math Soc 45:61–75CrossRef

Ghrist RW (2014) Elementary applied topology, vol 1, 10th edn. Createspace, Seattle

Gilmanov R, Kalyuzhnyuk A, Taimanov I, Yakovlev A (2018) Topological characteristics of digital models of geological core. In: 2nd international cross-domain conference for machine learning and knowledge extraction (CD-MAKE), Hamburg, Germany, pp 273–281

Gong L, Nie L, Xu Y (2020) Geometrical and topological analysis of pore space in sandstones based on X-ray computed tomography. Energies 13:3774

Hadwiger H (1957) Vorlesungen über Inhalt, Oberfläche und Isoperimetrie. Springer, BerlinCrossRef

Haruzi P, Katsman R, Halisch M, Waldmann N, Spiro B (2021) Supplement of Benchmark study using a multi-scale, multi-methodological approach for the petrophysical characterization of reservoir sandstones. Suppl Solid Earth 12:665–689

Hashin Z, Shtrikman S (1962) A variational approach to the theory of the elastic behaviour of polycrystals. J Mech Phys Solids 10(4):343–352CrossRef

Hatcher A (2002) Algebraic topology. Cambridge University Press, Cambridge, UK

Hausmann J-C (1995) On the Vietoris–Rips complexes and a cohomology theory for metric spaces. In: Prospects in topology: proceedings of a conference in honour of william browder, annals of mathematics studies, vol 138. Princeton University Press, Princeton, NJ, pp 175–188

Felix H (1914) Grundzüge der Mengenlehre. Veit & Co, Leipzig

Herring AL, Harper EJ, Andersson L, Sheppard A, Bay BK, Wildenschild D (2013) Effect of fluid topology on residual nonwetting phase trapping: implications for geologic CO₂ sequestration. Adv Water Resour 62:47–58CrossRef

Herring AL, Andersson L, Schlüter S, Sheppard A, Wildenschild D (2015) Efficiently engineering pore-scale processes: the role of force dominance and topology during nonwetting phase trapping in porous media. Adv Water Resour 79:91–102CrossRef

Herring AL, Robins V, Sheppard AP (2019) Topological persistence for relating microstructure and capillary fluid trapping in sandstones. Water Resour Res 55(1):555–573CrossRef

Huber N (2018) Connections between topology and macroscopic mechanical properties of three-dimensional open-pore materials. Front Mater 5:69

Hyde ST, Barnes IS, Ninham BW (1990) Curvature energy of surfactant interfaces confined to the plaquettes of a cubic lattice. Langmuir 6:1055–1062CrossRef

Ioannidis MA, Chatzis I (2000) On the geometry and topology of 3D stochastic porous media. J Colloid Interface Sci 229:323–334CrossRef

Ioannidis MA, Lang E (1998) Microgeometry and topology of statistically homogeneous porous media. Trans Ecol Environ 17:223–230

Ivonin D, Kalnin T, Grachev E, Shein E (2020) Quantitative analysis of pore space structure in dry and wet soil by integral geometry methods. Geosciences 10(9):365

Jiang F, Tsuji T, Shirai T (2018) Pore geometry characterization by persistent homology theory. Water Resour Res 54(6):4150–4163CrossRef

Jiang H, Arns CH (2020) A fast FFT method for 3D pore-scale rock-typing of heterogeneous rock samples via Minkowski functionals and hydraulic attributes. Web Conf 146: 04002 (2020) SCA 2019

Kac M (1966) Can one hear the shape of a drum? Am Math Monthly 73:1–24CrossRef

Kaczynski T, Mischaikow K, Mrozek M (2006) Computational homology, vol 157. Springer Science Business Media

Katz A, Thompson A (1986) Quantitative prediction of permeability in porous rock. Phys Rev B34(11):8179CrossRef

Kellerer HG (1984) Minkowski functionals of Poisson processes. Z Wahr Verw Gebiete 67:3–84CrossRef

Khachkova TS, Bazaikin YV, Lisitsa VV (2020) Use of the computational topology to analyze the pore space changes during chemical dissolution. Numer Methods Program 21:41–55

Khanamiri HH, Torsæter O (2018) Fluid topology in pore scale two-phase flow imaged by synchrotron X-ray microtomography. Water Resour Res 54:1905–1917CrossRef

Kim J (2012) Phase-field models for multi-component fluid flows. Commun Comput Phys 12:613–661CrossRef

Kirkpatrick S (1973) Percolation and conduction. Rev Modern Phys 45(4):574–588

Koestel J, Larsbo M, Jarvis N (2020) Scale and REV analyses for porosity and pore connectivity measures in undisturbed soil. Geoderma 366:114206CrossRef

Kondic L, Goullet A, O’Hern CS, Kramar M, Mischaikow K, Behringer RP (2012) Topology of force networks in compressed granular media. Europhys Lett 97(5):54001CrossRef

Kong TY, Rosenfeld A (1989) Digital topology: introduction and survey. Comput Vision, Graph Image Process 48(3):357–393CrossRef

Konkle SF, Moran PJ, Hamann B, Joy KI (2003) Fast methods for computing isosurface topology with Betti numbers. In: Post F, Nielsen GM, Bonneau GP (eds) Data visualization: the state of the art. proc. Dagstuhl seminar on scientific visualization. Kluwer Academic Publishers, Dordrecht, The Netherlands, pp 363–376

Korvin G (1982) Axiomatic characterization of the general mixture rule. Geoexploration 19(4):267–276CrossRef

Korvin G (1992) Fractal models in the earth sciences. Elsevier, Amsterdam

Korvin G (2016) Permeability from microscopy: review of a dream. Arab J Sci Eng 41(6):2045–2065CrossRef

Korvin G, Oleschko K, Abdulraheem A (2014) A simple geometric model of sedimentary rock to connect transfer and acoustic properties. Arab J Geosci 7:1127–1138

Korvin G, Sterligov B, Oleschko K, Cherkasov S (2013) Entropy of shortest distance (ESD) as pore detector and pore-shape classifier. Entropy 15(6):2384–2397CrossRef

Kozeny J (1927) Über kapillare Leitung des Wassers im Boden. Sitzungsber Akad Wiss (Wien) 136:271–306

Kramár M, Goullet A, Kondic L, Mischaikow K (2013) Persistence of force networks in compressed granular media. Phys Rev E 87:042207CrossRef

Kramár M, Goullet A, Kondic L, Mischaikow K (2014a) Evolution of force networks in dense particulate media. Phys Rev E 90:052203CrossRef

Kramár M, Goullet A, Kondic L, Mischaikow K (2014b) Quantifying force networks in particulate systems. Phys D 283:37–55CrossRef

Legland D, Kiêu K, Devaux M-F (2011) Computation of Minkowski measures on 2D and 3D binary images. Image Anal Stereol 26:83CrossRef

Lehmann P, Berchtold M, Ahrenholz B, Tölke J, Kaestner A, Krafczyk M, Flühler H, Künsch H (2008) Impact of geometrical properties on permeability and fluid phase distribution in porous media. Adv Water Resour 31(9):1188–1204CrossRef

Li M, Tang YB, Bernabé Y, Zhao JZ, Li XF, Bai XY, Zhang LH (2015) Pore connectivity, electrical conductivity, and partial water saturation: network simulations. J Geoph Res Solid Earth 120(6):4056–4068

Likos CN, Mecke KR, Wagner H (1995) Statistical morphology of random interfaces in microemulsions. J Chem Phys 102:9350–9361CrossRef

Lisitsa V, Bazaikin Y, Khachkova T (2020) Computational topology-based characterization of pore space changes due to chemical dissolution of rocks. Appl Math Model 88:21–37CrossRef

Liu Z, Herring AVR, Armstrong R (2017) Prediction of permeability from Euler characteristic of 3D images. The International Symposium of the Society of Core Analysts

Liu Z, Herring A, Arns C, Berg S, Armstrong RT (2017b) Pore-scale characterization of two-phase flow using integral geometry. Transp Porous Media 118(1):99–117CrossRef

Lozano-Durán A, Borrell G (2016) Algorithm 964: an efficient algorithm to compute the genus of discrete surfaces and applications to turbulent flows. ACM Trans Math Softw 42(4):Article 34

Macdonald IG (1995) Symmetric functions and hall polynomials, 2nd edn. Clarendon Press, OxfordCrossRef

Marafini E, La Rocca M, Fiori A, Battiato I, Prestininzi P (2020) Suitability of 2D modelling to evaluate flow properties in 3D porous media. Transp Porous Media 134:315–329

Martys NS, Torquato S, Bentz DP (1994) Universal scaling of fluid permeability for sphere packings. Phys Rev E 50:403CrossRef

Matheron G (1975) Random sets and integral geometry. Wiley, New York

MATHWORKS (2022) 3-D volumetric image processing—MATLAB & Simulink. Available online: https://www.mathworks.com/help/images/3d-volumetric-image-processing.html?s_tid=CRUX_lftnav

McClure JE, Armstrong RT, Berrill MA, Schlüter S, Berg S, Gray WG, Miller CT (2018) A geometric state function for two-fluid flow in porous media. Phys Rev Fluids 3:084306CrossRef

Mecke KR (1998) Integral geometry and statistical physics. Int J Mod Phys B 12:861–899CrossRef

Mecke KR (2000) Additivity, convexity, and beyond: applications of Minkowski functionals in statistical physics. In: Mecke KR, Stoyan D (eds) Statistical physics—the art of analyzing and modeling spatial structures. Lecture notes in physics, vol 554. Springer, Berlin, pp 111–184

Mecke KR (2001) Exact moments of curvature measures in the Boolean Model. J Stat Phys 102(5/6):1343–1381CrossRef

Mecke K, Arns C (2005) Fluids in porous media: a morphometric approach. J Phys Condens Matter 17(9):503–534

Mecke KR, Seyfried A (2002) Strong dependence of percolation thresholds on poly-dispersity. Europhys Lett 58:28–34CrossRef

Mecke KR, Stoyan D (eds) (2000) Statistical physics—the art of analyzing and modeling spatial structures. Lecture notes in physics, vol 554. Springer, Berlin

Mecke K, Wagner H (1991) Euler characteristic and related measures for random geometric sets. J Stat Phys 64(3–4):843–850CrossRef

Mecke KR, Buchert T, Wagner H (1994) Robust morphological measures for large-scale structure in the Universe. Astron Astrophys 288:697–704

Michielsen K, De Raedt H (2001) Integral-geometry morphological image analysis. Phys Rep General Maths 347(2001):461–538

Miles RE (1976) Estimating aggregate and overall characteristics from thick sections by transmission microscopy. J Microscopy 107(3):227–233CrossRef

Milton GW (1982) Bounds on the elastic and transport properties of two-component composites. J Mech Phys Solids 30(3):177–191CrossRef

Moon C, Mitchell SA, Heath JE, Andrew M (2019) Statistical inference over persistent homology predicts fluid flow in porous media. Water Resour Res 55:9592–9603CrossRef

Munch E (2017) A user’s guide to topological data analysis. J Learn Anal 4(2):47–61

Munkres JR (1993) Elements of algebraic topology. Addison Wesley, Menlo Park, CAL

Nair P, Mühlbauer S, Roy S, Pöschel T (2021) Can Minkowski tensors of a simply connected porous microstructure characterize its permeability? Phys Fluids 33: 042010

Nakahara M (2003) Geometry, topology and physics. Institute of Physics Publishing, Bristol-Philadelphia

Nishiyama N, Yokoyama T (2017) Permeability of porous media: role of the critical pore size. J Geophys Res Solid Earth 122:6955–6971CrossRef

Noiriel C, Gouze P, Bernard D (2004) Investigation of porosity and permeability effects from microstructure changes during limestone dissolution. Geophys Res Lett 31:L24603CrossRef

Ohser J, Mücklich F (2000) Statistical analysis of microstructure in materials science. John Wiley and Sons Ltd., New York

Ohser J, Nagel W, Schladitz K (2002) The Euler number of discretized sets—on the choice of adjacency in homogeneous lattices. In: Mecke KR, Stoyan D (eds) Morphology of condensed matter. Springer, Heidelberg

Joachim O, Nagel W, Schladitz K (2003) The Euler number of discretised sets—surprising results in three dimensions. Image Anal Stereol 22:11–19

Okuma G, Kadowaki D, Hondo T, Tanaka S, Wakai F (2017) Interface topology for distinguishing stages of sintering. Sci Rep 7:11106

Okuma G, Tanaka S, Wakai F (2022) Domain coarsening in viscous sintering as a result of topological pore evolution. J Eur Ceramic Soc 42:729–733

Oleschko K (1998–1999) Delesse principle and statistical fractal sets: Part 1. Dimensional equivalents. Soil Tillage Res 49:255–266; Part 2. Unified fractal model for soil porosity. Ibid 52:247–257

Ott H, Kharrat A, Borji M, Clemens T, Arnold P (2019) Screening of EOR potential on the pore scale by statistical and topological means. SCA 2019-011

Ott H, Kharrat A, Borji M, Arnold P (2020) Fluid-phase topology of complex displacements in porous media. Phys Rev Res 2:023240CrossRef

Otter N, Porter MA, Tillmann U, Grindrod P, Harrington HA (2017) A roadmap for the computation of persistent homology. EPJ Data Sci 6:1–38CrossRef

Øren PE, Bakke S, Arntzen OJ (1998) Extending predictive capabilities to network models. SPE J 3:324–336CrossRef

Papadopoulos L, Porter MA, Daniels KE, Bassett DS (2018) Network analysis of particles and grains. J. Complex Netw 6:485–565CrossRef

Patania A, Vaccarino F, Petri G (2017) Topological analysis of data. EPJ Data Sci 6(1):1–6CrossRef

Pérez-Rosales C (1982) On the relationship between formation resistivity factor and porosity. SPE J 22(04):531–536

Pleijel A (1954) A study of certain Green’s functions with applications in the theory of vibrating membranes. Ark Mat 2(1954):553–539CrossRef

Porter M, Wildenschild D, Grant G, Gerhard J (2010) Measurement and prediction of the relationship between capillary pressure, saturation, and interfacial area in a NAPL-water-glass bead system. Water Resour Res 46:W08512CrossRef

Pothuaud L, Rietbargen BV, Mosekilde L, Beuf O, Levitz P, Benhamou CL, Majumdar S (2002) Combination of topological parameters and bone volume fraction better predicts the mechanical properties of trabecular bone. J Biomech 35:1091–1099CrossRef

Prokhorov D, Lisitsa V, Bazaikin Y (2021) Digital image reduction for analysis of topological changes in pore space during chemical dissolution. International Conference on Computational Science ICCS 2021, pp 382–393

Protter MH (1987) Can one hear the shape of a drum? Revisited. SIAM Rev 29(2):185–197

Renshaw CE (1995) On the relationship between mechanical and hydraulic apertures in rough-walled fractures. J Geophys Res 100:629–636

Robins V (1999) Towards computing homology from finite approximations. Topol Proc 24:503–532

Robins V, Wood PJ, Sheppard AP (2011) Theory and algorithms for constructing discrete Morse complexes from grayscale digital images. IEEE Trans Pattern Anal Mach Intell 33(8):1646–1658CrossRef

Robins V, Saadatfar M, Delgado-Friedrichs O, Sheppard AP (2016) Percolating length scales from topological persistence analysis of micro-CT images of porous materials. Water Resour Res 52(1):315–329CrossRef

Robinson J, Slater L, Johnson T, Shapiro A, Tiedeman C, Ntarlagiannis D, Johnson C, Day-Lewis F, Lacombe P, Imbrigiotta T, Lane J (2016) Imaging pathways in fractured rock using three-dimensional electrical resistivity tomography. Groundwater 54(2):186–201CrossRef

Roy S, Tarafdar S (1997) Archie’s law from a fractal model for porous rocks. Phys Rev 55:8038–8041CrossRef

Saadatfar M, Takeuchi H, Robins V, Francois N, Hiraoka Y (2017a) Pore configuration landscape of granular crystallization. Nat Commun 8:15082CrossRef

Saadatfar M, Takeuchi H, Hanifpour M, Robins V, Francois N, Hiraokam Y (2017b) Granular compaction and the topology of pore deformation. EPJ web of conferences, powders & grains, p 16009

Sahimi M (1985) Flow and transport in porous media and fractured rock: from classical methods to modern approaches. VCH, New York

Santaló LA (1976) Integral geometry and geometric probability. Addison-Wesley, Reading, MA

Schindelin J, Rueden CT, Hiner MC, Eliceiri KW (2015) The Image J ecosystem: an open platform for biomedical image analysis. Mol Reprod Dev 82:518–529CrossRef

Schlüter S, Berg S, Rücker M, Armstrong R, Vogel HJ, Hilfer R, Wildenschild D (2016) Pore-scale displacement mechanisms as a source of hysteresis for two-phase flow in porous media. Water Resour Res 52(3):2194–2205CrossRef

Schmalzing J, Kerscher M, Buchert T (1995) Minkowski functionals in cosmology. astro-ph/9508154(05 OCT 1995): 1–10

Schmidhuber J (2015) Deep learning in neural networks: ano. Neural Netw 61:85–117CrossRef

Scholz C (2014) Fluss und Transport in mikrofluidischen porösen Medien . Von der Fakultät Mathematik und Physik der Universität Stuttgart zur Erlangung der Würde eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigte Abhandlung

Scholz C, Wirner F, Götz J, Rüde U, Schröder-Turk GE, Mecke K, Bechinger C (2012) Permeability of porous materials determined from the Euler characteristic. Phys Rev Lett 109:264504CrossRef

Scholz C, Wirner F, Klatt M, Hirneise D, Schröder-Turk G, Mecke K, Bechinger C (2015) Direct relations between morphology and transport in Boolean models. Phys Rev E 92:043023CrossRef

Schröder-Turk GE, Mickel W, Kapfer SC, Klatt MA, Schaller FM, Hoffmann MJF, Kleppmann N, Armstrong P, Inayat A, Hug D, Reichelsdorfer M, Peukert W, Schwieger W, Mecke K (2011) Minkowski tensor shape analysis of cellular, granular and porous structures. Adv Mater 23:2535–2553CrossRef

Schröder-Turk G, Mickel W, Kapfer S, Schaller F, Breidenbach B, Hug D, Mecke K (2013) Minkowski tensors of anisotropic spatial structure. New J Phys 15(8):083028

Schwartz LM, Martys N, Bentz DP, Garboczi EJ, Torquato S (1993) Cross-property relations and permeability estimation in model porous media. Phys Rev E 48:4584–4591CrossRef

Sen PN, Straley C, Kenyon WE (1990) Surface-to-volume ratio, charge density, nuclear magnetic relaxation, and permeability in clay-bearing sandstones. Geophysics 55:61–69CrossRef

Serra J (1982) Image analysis and mathematical morphology. Academic Press, London

Slotte PA, Berg CF, Khanamiri HH (2020) Predicting resistivity and permeability of porous media using Minkowski functionals. Transp Porous Media 131:705–722CrossRef

Smalley JJ (1967) A simple model of a diagenetic system. Sedimentology 8:27–33CrossRef

Soares JA, Rayane P, de Lopes A (2017) A model for permeability of carbonate rocks based on pore connectivity and pore size. In: 15th international congress of the brazilian geophysical society held in Rio de Janeiro, Brazil, 31 July to 3 Aug 2017

Sossa-Azuela JH, Cuevas-Jiménez EB, Zaldivar-Navarro D (2011) Alternative way to compute the Euler number of a binary image. J Appl Res Technol 9(3):335–341CrossRef

Stauffer D, Aharony A (1992) Introduction to percolation theory, 2nd edn. Taylor and Francis, London

Stoyan D, Kendall WS, Mecke J (1989) Stochastic geometry and its applications. Akademie Verlag, Berlin

Sudakov O, Burnaev E, Koroteev D (2019) Driving digital rock towards machine learning: Predicting permeability with gradient boosting and deep neural networks. Comput Geosci 127:91–98CrossRef

Sun WC, Andrade JE, Rudnicki JW (2011) Multiscale method for characterization of porous microstructures and their impact on macroscopic effective permeability. Int J Numer Methods Eng 88:1260–1279CrossRef

Suzuki A, Miyazawa M, Okamoto A, Shimizu H, Obayashi I, Hiraoka Y, Tsuji T, Kang PK, Ito T (2020) Inferring fracture forming processes by characterizing fracture network patterns with persistent homology. Comput Geosci 143:104550CrossRef

Suzuki A, Miyazawa M, Minto JM, Tsuji T, Obayashi I, Hiraoka Y, Ito T (2021) Flow estimation solely from image data through persistent homology analysis. Sci Rep 11(1):1–13CrossRef

Takahashi T, Clark AH, Majmudar T, Kondic L (2018) Granular response to impact: topology of the force networks. Phys Rev E 97:012906CrossRef

Thakur MM, Kim F, Penumadu D, Herring A (2021) Pore space and fluid phase characterization in round and angular partially saturated sands using radiation-based tomography and persistent homology. Transp Porous Media 137:131–155CrossRef

Thovert J-F, Wary F, Adler PM (1990) Thermal conductivity of random media and regular fractals. J Appl Phys 68:3872–3883CrossRef

Thovert J-F, Yousefian F, Spanne P, Jacquin CG, Adler PM (2001) Grain reconstruction of porous media: Application to a low-porosity Fontainebleau sandstone. Phys Rev E 63:061307CrossRef

Tiab D, Donaldson EC (2004) Petrophysics: theory and practice of measuring reservoir rock and fluid transport properties. Elsevier, Oxford, UK

Torquato S (1984) Bulk properties of two-phase disordered media. I. Cluster expansion for the effective dielectric constant of dispersions of penetrable spheres. J Chem Phys 81(11):5079–5088

Torquato S (2002a) Statistical description of microcstructures. Ann Rev Mater Res 32:77–111CrossRef

Torquato S (2002b) Random heterogeneous materials: microstructure and macroscopic properties. Interdisciplinary applied mathematics, vol 16. Springer, New York

Torquato S, Stell G (1983) Microstructure of two-phase random media. III. the n-point matrix probability functions for fully penetrable spheres. J Chem Phys 79:1505–1510CrossRef

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. Mathematics for Industry, vol 28. Springer, Singapore, pp 95–109

Ushizima DM, Morozov D, Weber GH, Bianchi AGC, Sethian JA, Wes Bethel E (2012) Augmented topological descriptors of pore networks for material science. IEEE Trans Visual Comput Graphics 18(12):2041–2050CrossRef

Vejdemo-Johansson M, Skraba P (2016) Topology, big data and optimization. In: Emrouznejad A (ed) Big data optimization: recent developments and challenges. Springer, Cham, Switzerland, pp 147–176CrossRef

Vietoris L (1927) Über den höheren Zusammenhang kompakter Räume und eine Klasse von zusammenhangstreuen Abbildungen. Math Ann 97:454–472CrossRef

Vogel HJ (1997a) Digital unbiased estimation of the Euler-Poincaré characteristic in different dimensions. Acta Stereol 16(2):97–104

Vogel HJ (1997b) Morphological determination of pore connectivity as a function of pore size using serial sections. Eur J Soil Sci 48:365–377CrossRef

Vogel HJ (2000) A numerical experiment on pore size, pore connectivity, water retention, permeability, and solute transport using network models. Eur J Soil Sci 51:99–105CrossRef

Vogel H-J (2002) Morphology of condensed matter, physics and geometry of spatially complex systems. In: Mecke K, Stoyan D (eds) Topological characterization of porous media. Springer-Verlag, Berlin-Heidelberg, pp 75–92

Vogel H-J, Roth K (2001) Quantitative morphology and network representation of soil pore structure. Adv Water Resour 24:233–242CrossRef

Vogel H-J, Cousin I, Roth K (2002) Quantification of pore structure and gas diffusion as a function of scale. Eur J Soil Sci 53(3):465–473CrossRef

Vogel HJ, Weller U, Schlüter S (2010) Quantification of soil structure based on Minkowski functions. Comput Geosci 36(10):1236–1245CrossRef

Vogel LE, Makowski D, Garnier P, Vieublé-Gonod L, Coquet Y, Raynaud X, Nunan N, Chenu C, Falconer R, Pot V (2015) Modeling the effect of soil meso- and macropores topology on the biodegradation of a soluble carbon substrate. Adv Water Resour 83:123–136CrossRef

Volkhonskiy D, Muravleva E, Sudakov O, Orlov D, Burnaev E, Koroteev D (2021) Reconstruction of 3D porous media from 2D slices. arXiv: 1901.10233v4 [cs.CV] 6 Aug 2021

Waldir LR, de Souza ACA, Barbieri DX (2009) The Euler-Poincaré characteristic applied to identify low bone density from vertebral tomographic images. Rev Bras Reumatol 49(2):140–152

Wang Z, Bovik AC, Sheikh HR, Simoncelli EP (2004) Image quality assessment: from error visibility to structural similarity. IEEE Trans Image Processing 13:600–612CrossRef

Wanner T, Fuller ER, Saylor DM (2010) Homology metrics for microstructure response fields in polycrystals. Acta Materialia 58(1):102–111

Weinberger S (2011) What is … persistent homology? Not Am Math Soc 58(36):36–39

Weissberg HL (1963) Effective diffusion coefficient in porous media. J Appl Phys 34:2636–2639CrossRef

Weyl H (1911) Über die asymptotische Verteilung der Eigenwerte. Gött Nachr 1911:110–117

Wirner F (2015) Flow and transport of colloidal suspensions in porous media. Von der Fakultät Mathematik und Physik der Universität Stuttgart zur Erlangung der Würde eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigte Abhandlung

Yang Y, Yang H, Tao L, Yao J, Wang W, Zhang K, Luquot L (2019) Microscopic determination of remaining oil distribution in sandstones with different permeability scales using computed tomography scan. J Energy Resour Technol 141(9):092903

Yonezawa F, Cohen MH (1983) Granular effective medium approximation. J Appl Phys 54(6):2895–2899CrossRef

Yoon H, Dewers TA (2013) Nanopore structures, statistically representative elementary volumes, and transport properties of chalk. Geophys Res Lett 40:4294–4298CrossRef

Zhang D, Zhang R, Chen S, Soll WE (2000) Pore scale study of flow in porous media: Scale dependency, REV, and statistical REV. Geophys Res Lett 27:1195–1198CrossRef

Zhao Y (2019) Application of Euler-Poincaré characteristic in the prediction of permeability of porous media. Intell Autom Soft Comput 25(4):835–845

Zomorodian AJ (2005) Topology for computing. Cambridge monographs on applied and computational mathematics. Cambridge University Press, Cambridge

Zomorodian A, Carlsson G (2005) Computing persistence homology. Discrete Comput Geom 33(2):249–274CrossRef

Titel: The Internal Topology of Rocks
verfasst von: Gabor Korvin
Verlag: Springer Nature Switzerland
Buch: Statistical Rock Physics
Print ISBN: 978-3-031-46699-1

Electronic ISBN: 978-3-031-46700-4

Copyright-Jahr: 2024
DOI: https://doi.org/10.1007/978-3-031-46700-4_3