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2024 | OriginalPaper | Buchkapitel

5. Menger Sponge Models

verfasst von : Gabor Korvin

Erschienen in: Statistical Rock Physics

Verlag: Springer Nature Switzerland

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Chapter Highlights

Menger Sponge (in 3D) and Sierpiński Carpet (in 2D) are the most useful fractal models of the rocks’ pore space. Sections 5.1 and 5.2 treat these models, their combinatorial and topological properties. Random and multifractal generalizations are discussed in Sect. 5.3, where the permeability of these models is also studied. A mathematical Appendix 5.4 introduces a useful measure, lacunarity, a tool to distinguish between textures of the same fractal dimension but different geometry. Petrophysical applications are summarized in Table 5.1.

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Vorheriges Kapitel Random Network Models

Nächstes Kapitel Coordination Number of Grains

Allain C, Cloitre M (1991) Characterizing the lacunarity of random and deterministic fractal sets. Phys Rev A 44(6):3552–2558CrossRef

Anderson R (1958) One-dimensional continuous curves and a homogeneity theorem. Ann Math 68:1–16CrossRef

Arizabalo RD, Oleschko K, Korvin G, Lozada M, Castrejón R, Ronquillo G (2006) Lacunarity of geophysical well logs in the Cantarell oil field, Gulf of Mexico. Geofísica Int 45(2):99–113

Atzeni C, Pia G, Sanna U, Spanu N (2008) A fractal model of the porous microstructure of earth-based materials. Constr Build Mater 22(8):1607–1613CrossRef

Bear J, Verruijt A (1987) Modeling groundwater flow and pollution. Springer, New YorkCrossRef

Beck C, Schlögl F (1993) Thermodynamics of chaotic systems: an introduction. Cambridge University Press, New YorkCrossRef

Bendat JS, Piersol AG (1966) Measurement and analysis of random data. Wiley, New York

Bird NRA, Dexter AR (1997) Simulation of soil water retention using random fractal networks. Eur J Soil Sci 48:633–641CrossRef

Bird NRA, Bartoli F, Dexter AR (1996) Water retention models for fractal soil structures. Eur J Soil Sci 47:1–6CrossRef

Bird N, Perrier E, Rieu M (2000) The water retention function for a model of soil structure with pore and solid fractal distributions. Eur J Soil Sci 55:55–63CrossRef

Boufadel MC, Lu S, Molz FJ, Lavallee D (2000) Multifractal scaling of the intrinsic permeability. Water Resour Res 36:3211–3222CrossRef

Carman PC (1937) Fluid flow through granular beds. Trans Inst Chem Eng 15:150–166

Carman PC (1939) Permeability of saturated sands, soils and clays. J Agric Sci 29:262–273CrossRef

Carman PC (1956) Flow of gases through porous media. Butterworths Scientific Publications, London

Chen Y, Zhang C, Shi M, Peterson GP (2009) Role of surface roughness characterized by fractal geometry on laminar flow in microchannels. Phys Rev E 80(2):026301CrossRef

Cihan A, Perfect E, Tyner JS (2007) Water retention models for scale-variant and scale-invariant drainage of mass prefractal porous media. Vadose Zone J. 6:786–792CrossRef

Cihan A, Sukop MC, Tyner JS, Perfect E, Huang H (2009) Analytical predictions and lattice Boltzmann simulations of intrinsic permeability for mass fractal porous media. Vadose Zone Journal 8(1):187–196CrossRef

Cousins TA (2016) Effect of rough fractal pore-solid interface on single-phase permeability in random fractal porous media. Master of Science in Engineering Thesis. The University of Texas at Austin

Cowan Kenneth Lee, 1987. The Fractal Dimension as a Petrophysical Parameter. MSc in Engineering Thesis, The University of Texas at Austin.

Danielson RE, Sutherland PL (1986) Porosity. In: Klute A (ed) Methods of soil analysis: Part 1 – Physical and mineralogical methods. American Society of Agronomy, Madison, WI, pp 443–461

Dattelbaum DM, Ionita A, Patterson BM, Branch BA, Kuettner L (2020) Shockwave dissipation by interface-dominated porous structures. AIP Adv 10(7):075016CrossRef

Delesse A (1847) Procédé mécanique pour déterminer la composition des roches. (Extrait). C.r. hebd. séanc. Acad. Sci., Paris 25:544

Doughty C, Karasaki K (2002) Flow and transport in hierarchically fractured rock. J Hydrol 263:1–22CrossRef

Du ZY, Yang ML, Liu QG, Yu M (2019) Numerical investigation on thermal conduction in fractal-like porous media. Arch Environ Sci Environ Toxicol: AESET-109. https://doi.org/10.29011/AESET-109 100009

Evertsz CJG, Mandelbrot BB (1992) Multifractal measures. In: Peitgen H-O, Jurgens H, Saupe D (eds) Chaos and fractals: new frontiers of science. Springer-Verlag, New York, pp 921–969

Fallico C, Tarquis AM, De Bartolo S, Veltri M (2010) Scaling analysis of water retention curves for unsaturated sandy loam soils by using fractal geometry. Eur J Soil Sci 61:425–436CrossRef

Freund JE (1971) Mathematical statistics. Prentice-Hall Inc., Englewood Cliffs, NJ

Friesen I, Mikula RJ (1987) Fractal dimensions of coal particles. J Colloid Interface Sci 120(1):263–271CrossRef

Garrison Jr JR, Pearn WC, von Rosenberg DW (1992) The fractal Menger sponge and Sierpinski carpet as models for reservoir rock/pore systems: I. Theory and image analysis of Sierpinski carpets. In Situ 16(4):351–406

Garrison JR Jr, Pearn WC, Rosenberg DU (1993) The fractal Menger sponge and Sierpinski carpet as models for reservoir rock pore systems: II. Image analysis of natural fractal reservoir rocks. In Situ 17:1–53

Garza-López RA, Naya L, Kozak JJ (2000) Tortuosity factor for permeant flow through a fractal solid. J Chem Phys 112(22):9956–9960CrossRef

Ghanbarian-Alavijeh B, Hunt AG (2012) Comments on “More general capillary pressure and relative permeability models from fractal geometry” by Kewen Li. J Contam Hydrol 140–141:21–23CrossRef

Ghanbarian-Alavijeh B, Hunt A (2012b) Unsaturated hydraulic conductivity in porous media: percolation theory. Geoderma 187–188:77–84CrossRef

Ghanbarian-Alavijeh B, Millán H, Huang G (2011) A review of fractal, prefractal and pore-solid-fractal models for parameterizing the soil water retention curve. Can J Soil Sci 91(1):1–14CrossRef

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

Ghanbarian B, Hunt AG, Sahimi M, Ewing RP, Skinner TE (2013b) Percolation theory generates a physically based description of tortuosity in saturated and unsaturated porous media. Soil Sci Soc Am J 77:1920–1929CrossRef

Gibson JR, Lin H, Bruns MA (2006) A comparison of fractal analytical methods on 2- and 3-dimensional computed tomographic scans of soil aggregates. Geoderma 134:335–348CrossRef

Giménez D, Rawls WJ, Lauren JG (1999) Scaling properties of saturated hydraulic conductivity in soil. Geoderma 88:205–220CrossRef

Gouyet J-F (1996) Physics and fractal structures. Springer, New York

Hentschel HGE, Procaccia I (1983) The infinite number of generalized dimensions of fractals and strange attractors. Physica 8:435–444

Hu Y, Wang Q, Zhao J, Xie S, Jiang H (2020) A novel porous media permeability model based on fractal theory and ideal particle pore-space geometry assumption. Energies 13(3):510CrossRef

Jin Y, Zhu YB, Li X, Zheng JL, Dong JB (2015) Scaling invariant effects on the permeability of fractal porous media. Transp Porous Media 109(2):433–453CrossRef

Katz AJ, Thompson AH (1985) Fractal sandstone pores: implications for conductivity and pore formation. Phys Rev Lett 54(12):1325–1328CrossRef

Kim JW, Sukop MC, Perfect E, Pachepsky YA, Choi H (2011) Geometric and hydrodynamic characteristics of three-dimensional saturated prefractal porous media determined with Lattice Boltzmann modeling. Transp Porous Media 90(3):831–846CrossRef

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

Korvin G (1993) The kurtosis of reflection coefficients in a fractal sequence of sedimentary layers. Fractals 1:263–268

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

Korvin G (2021) Allometric power laws. In: Daya Sagar B, Cheng Q, McKinley J, Agterberg F (eds) Encyclopedia of mathematical geosciences. Encyclopedia of Earth Sciences Series. Springer, Cham

Kozeny J (1927) Über Kapillare Leitung des Wassers in Boden Stizungsber. Akad. Wiss. Wien 136:271–306

Krohn CE (1988a) Sandstone fractal and Euclidean pore volume distributions. J Geophys Res: Solid Earth 93(B4):3286–3296CrossRef

Krohn CE (1988b) Fractal measurements of sandstones, shales, and carbonates. J Geophys Res: Solid Earth 93(B4):3297–3305CrossRef

Li J, Yu B (2011) Tortuosity of flow paths through a Sierpinski carpet. Chin Phys Lett 28:34701CrossRef

Li ZL, Tauraso R (2006) Problem 11208. American Mathematical Monthly 113

Li LC, Lin M, Ji L, Jiang W, Ca G (2018) Rapid evaluation of the permeability of organic-rich shale using the 3D intermingled-fractal model. SPE J 23(6):2175–2187CrossRef

Liu HH, Molz FJ (1997) Multifractal analyses of hydraulic conductivity distributions. Water Resour Res 33:2483–2488CrossRef

Liu K, Ostadhassan M, Zou J, Gentzis T, Rezaee R, Bubach B, Carvajal-Ortiz H (2018) Multifractal analysis of gas adsorption isotherms for pore structure characterization of the Bakken Shale. Fuel 219:296–311CrossRef

Mackeprang C, Myers K (2007) Problem 11208 from Volume 133 No. 3 p. 267 Menger Sponge Map-Coloring. American Mathematical Monthly

Majumdar A, Tien CL (1990) Fractal characterization and simulation of rough surfaces. Wear 136(2):313–327CrossRef

Mandelbrot BB (1974) Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier. J Fluid Mech 62:331–358CrossRef

Mandelbrot BB (1977) Fractals: form, chance, and dimensions. W. H. Freeman and Co., San Francisco, California

Mandelbrot BB (1982) The fractal geometry of nature. W.H. Freeman, New York

Mandelbrot BB (1989) Multifractal measures, especially for the geophysicist. PAGEOPH 131:5–42CrossRef

Meakin P (1998) Fractals, scaling and growth far from equilibrium. Cambridge University Press, Cambridge, UK

Menger K (1932) Kurventheorie. Teubner, Leipzig

Miyazima S, Eugene Stanley H (1987) Intersection of two fractal objects: Useful method of estimating the fractal dimension. Phys Rev B Condens Matter 35(16):8898–8900

Mortensen NA, Okkels F, Bruus H (2005) Reexamination of Hagen-Poiseuille flow: shape dependence of the hydraulic resistance in microchannels. Phys Rev E 71:057301CrossRef

Mostafa ME (2008) Finite cube elements method for calculating gravity anomaly and structural index of solid and fractal bodies with defined boundaries. Geophys J Int 172:887–902CrossRef

Mukhopadhyay S, Sahimi M (2000) Calculation of effective permeabilities of field-scale porous media. Chem Eng Sci 55:4495–4513CrossRef

OEIS®, 2022. A135918 - Genus of stage-n Menger sponge. The On-Line Encyclopedia of Integer Sequences

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

Patil GP (1962) Maximum likelihood estimation for generalized power series distributions and its application to the truncated binomial distribution. Biometrika 49:227–237CrossRef

Paz-Ferreiro J, da Luz L, Lado M, Vázquez EV (2013) Specific surface area and multifractal parameters of associated nitrogen adsorption and desorption isotherms in soils from Santa Catarina Brazil. Vadose Zone J 12:185–192CrossRef

Perfect E (2005) Modeling the primary drainage curve of prefractal porous media. Vadose Zone J 4:959–966CrossRef

Perfect E, Gentry W, Sukop MC, Lawson JE (2006) Multifractal Sierpinski carpets: Theory and application to upscaling effective saturated hydraulic conductivity. Geoderma 134:240–252CrossRef

Perrier EMA, Bird NRA (2002) Modelling soil fragmentation: the pore solid fractal approach. Soil Tillage Res. 64:91–99CrossRef

Perrier E, Rieu M, Sposito G, Marsily G (1996) Models of the water retention curve for soils with a fractal pore size distribution. Water Resour Res 32:3025–3031CrossRef

Perrier E, Bird N, Rieu M (1999) Generalizing the fractal model of soil structure: the PSF approach. Geoderma 88:137–164CrossRef

Pfeifer P, Avnir D, Farin D (1984) Scaling behaviour of surface irregularity in the molecular domain: from adsorption studies to fractal catalysis. J Stat Phys 36:699–716CrossRef

Pia G (2016) High porous Yttria-stabilized Zirconia with aligned pore channels: morphology directionality influence on heat transfer. Ceram Int 42(10):11674–11681CrossRef

Pia G, Casnedi L (2017) Heat transfer in high porous alumina: experimental data interpretation by different modelling approaches. Ceram Int 43(12):9184–9190CrossRef

Pia G, Sanna U (2013a) Intermingled fractal units model and electrical equivalence fractal approach for prediction of thermal conductivity of porous materials. Appl Therm Eng 61:186–192CrossRef

Pia G, Sanna U (2013b) A geometrical fractal model for the porosity and thermal conductivity of insulating concrete. Constr Build Mater 44:551–556CrossRef

Pia G, Sanna U (2014a) An Intermingled Fractal Units Model and method to predict permeability in porous rock. Int J Eng Sci 75:31–39CrossRef

Pia G, Sanna U (2014b) An Intermingled Fractal Units Model to evaluate pore size distribution influence on thermal conductivity values in porous materials. Appl Therm Eng 65(1/2):330–336CrossRef

Pia G, Sassoni E, Franzoni E, Sanna U (2014) Predicting capillary absorption of porous stones by a procedure based on an intermingled fractal units model. Int J Eng Sci 82:196–204

Pia G, Casnedi L, Ionta M et al (2015) On the elastic deformation properties of porous ceramic materials obtained by pore-forming agent method. Ceram Int 41(9):11097–11105CrossRef

Pia G, Casnedi L, Sanna U (2016a) Porosity and pore size distribution influence on thermal conductivity of Yttria-stabilized Zirconia: experimental findings and model predictions. Ceram Int 42(5):5802–5809CrossRef

Pia G, Siligardi C, Casnedi L, Sanna U (2016b) Pore size distribution and porosity influence on sorptivity of ceramic tiles: from experimental data to fractal modelling. Ceram Int 42(8):9583–9590CrossRef

Plotnick RE, Gardner RH, Hargrove WW, Prestegaard K, Perlmutter M (1996) Lacunarity analysis: a general technique for the analysis of spatial patterns. Phys Rev E 53(5):5461–5468CrossRef

Power WL, Tullis TE, Weeks JD (1988) Roughness and wear during brittle faulting. J Geophys Res: Solid Earth 93(B12):15268–15278CrossRef

Ren W, Zhou H, Zhong J, Xue D, Wang C, Liu Z (2022) A multi-scale fractal approach for coal permeability estimation via MIP and NMR methods. Energies 15(8):2807CrossRef

Rieu M, Sposito G (1991a) Fractal fragmentation, soil porosity and soil water properties: I Theory. Soil Sci Soc Am J 55:1231–1238CrossRef

Rieu M, Sposito G (1991b) Fractal fragmentation, soil porosity and soil water properties: II Applications. Soil Sci Soc Am J 55:1239–1244CrossRef

Rieu M, Sposito G (1991c) Relation pression capillaire-teneur en eau dans les milieux poreux fragmentés et identification du caractère fractal de la structure des sols. Metals Technology 1:462–467

Roy A, Perfect E, Dunne WM, Odling N, Kim J-W (2010) Lacunarity analysis of fracture networks: evidence for scale-dependent clustering. J Struct Geol 32(10):1444–1449CrossRef

Saucier A (1992a) Effective permeability of multifractal porous media. Physica A 183:381–397CrossRef

Saucier A (1992b) Scaling of the effective permeability in multifractal porous media. Physica, A 191:289–294CrossRef

Saucier A (1996) Scaling properties of disordered multifractals. Physica A 226:34–63CrossRef

Schönhöfer P (2014) Fractal Geometries: Scaling of Intrinsic Volumes. Masterarbeit aus der Physik, Institut für Theoretische Physik I, Friedrich-Alexander-Universität Erlangen-Nürnberg

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

Sergeyev and D., , 2009 Sergeyev DY (2009) Valuating the exact infinitesimal values of area of Sierpinski’s carpet and volume of Menger’s sponge. Chaos Soliton. Fract 2:3042–3046

Sukop MC, Perfect E, Bird NRA (2001) Water retention of prefractal porous media generated with the homogeneous and heterogeneous algorithms. Water Resour Res 37:2631–2636CrossRef

Tee GJ (2015) Tunnel numbers for fractal polyhedra, 1–3. Appendix A in the online version of H. Molina-Abral, P. Real, A. Nakamura and R. Klette, Connectivity calculus of fractal polyhedrons, Pattern Recognition 48 No. 4 (April 2015):1146–1156

Tennekoon L, Boufadel MC, Lavalle D, Weaver J (2003) Multifractal anisotropic scaling of hydraulic conductivity. Water Resour Res 39(7):1193CrossRef

Toledo PG, Novy RA, Davis HT, Scriven LE (1990) Hydraulic conductivity of porous media at low water content. Soil Sci Soc Am J 54:673–679CrossRef

Turcotte DL (1993) Fractals and chaos in geology and geophysics. Cambridge University Press, Cambridge UK

Tyler SW, Wheatcraft SW (1990) Fractal processes in soil water retention. Water Resour Res 26:1047–1054CrossRef

Valdés-Parada FJ, Porter ML, Wood BD (2011) The role of tortuosity in upscaling. Transp Porous Med 88:1–30CrossRef

Veneziano D, Essiam AK (2003) Flow through porous media with multifractal hydraulic conductivity. Water Resour Res 39(6):1166CrossRef

Veneziano D, Essiam AK (2004) Nonlinear spectral analysis of flow through multifractal porous media. Chaos, Solitons FrActals 19:293–307CrossRef

Vita MC, De Bartolo S, Fallico C, Veltri M (2012) Usage of infinitesimals in the Menger’s Sponge model of porosity. Appl Math Comput 218(16):8187–8195

Wang BY, Jin Y, Chen Q, Zheng JL, Zhu YB, Zhang XB (2014) Derivation of permeability–pore relationship for fractal porous reservoirs using series-parallel flow resistance model and lattice Boltzmann method. Fractals 22:1440005CrossRef

Wheatcraft SW, Tyler SW (1988) An explanation of scale-dependent dispersivity in heterogeneous aquifers using concepts of fractal geometry. Water Resour Res 24:566–578CrossRef

Wheatcraft SW, Sharp GA, Tyler SW (1991) Fluid flow and solute transport in fractal heterogeneous porous media. In: Bear J, Corapcioglu MY (eds) Transport processes in porous media., Kluwer Academic, Dordrecht, The Netherlands, pp 695–722

Xia YX, Cai JC, Wei W, Hu XY, Wang X, Ge XM (2018) A new method for calculating fractal dimensions of porous media based on pore size distribution. Fractals 26:1850006CrossRef

Li and Jia-lin, 2009 Xing-Shang L, Xu J-l (2009) A model of void distribution in collapsed zone based on fractal theory. Procedia Earth Planet Sci 1:203–210

Yu BM, Li JH (2001) Some fractal characters of porous media. Fractals 9:365–372CrossRef

Zhao et al., 2017 Zhao Y, Zhu G, Dong Y, Danesh NN, Chen Z, Zhang T (2017) Comparison of low-field NMR and microfocus X-ray computed tomography in fractal characterization of pores in artificial cores. Fuel 210:217–226

Zhou Y, Wu S, Li Z, Zhu R, Xie S, Zhai X, Lei L (2021) Investigation of microscopic pore structure and permeability prediction in sand-conglomerate reservoirs. J Earth Sci 32(4):818–827

Titel: Menger Sponge Models
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_5