Skip to main content
SpringerLink
Log in

Fractal Models of Porous Media

  • Chapter
Book cover Percolation Theory for Flow in Porous Media

Part of the book series: Lecture Notes in Physics ((LNP,volume 880))

Abstract

As pointed out in the foreword to the first edition of this book, it is the distribution of the fluids within a medium rather than the medium itself that directly controls the properties of flow, conduction, dispersion, and fluid retention. Of course the distribution of the fluids within the medium is influenced importantly by the structural properties of the medium itself. Insofar as they can be assumed to reflect the characteristics of the pore space, one can develop medium models which generate observed pressure-saturation curves. Most of these models have fractal characteristics. The present chapter reviews a number of fractal models that generate similar to identical pressure-saturation curves though with different interpretations.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 69.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 89.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Anderson, A.N., McBratney, A.B., Crawford, J.W.: Application of fractals to soil studies. Adv. Agron. 63, 1–76 (1998)

    Article  Google Scholar 

  2. Arya, L.A., Paris, J.F.: A physic-empirical model to predict the soil moisture characteristics from particle size distribution and bulk density data. Soil Sci. Soc. Am. J. 45, 1023–1029 (1981)

    Article  Google Scholar 

  3. Avnir, D., Biham, O., Lidar, D., Malcai, O.: Is the geometry of nature fractal? Science 279, 38–40 (1998)

    Article  Google Scholar 

  4. Bird, N.R.A.: Comment on “An improved fractal equation for the soil water retention curve” by E. Perfect et al. Water Resour. Res. 34, 929 (1998)

    Article  Google Scholar 

  5. Bird, N.R.A., Perrier, E., Rieu, M.: The water retention function for a model of soil structure with pore and solid fractal distributions. Eur. J. Soil Sci. 51, 55–63 (2000)

    Article  Google Scholar 

  6. Bittelli, M., Flury, M.: Errors in water retention curves determined with pressure plates. Soil Sci. Soc. Am. J. 73, 1453–1460 (2009)

    Article  Google Scholar 

  7. Brooks, R.H., Corey, A.T.: Hydraulic properties of porous media. Colorado State University, Hydrology Paper No. 3, 27 p. (1964)

    Google Scholar 

  8. Campbell, G.S.: A simple method for determining unsaturated hydraulic conductivity from moisture retention data. Soil Sci. 117, 311–314 (1974)

    Article  Google Scholar 

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

    Google Scholar 

  10. Cheng, J.-T., Pyrak-Nolte, L.J., Nolte, D.D., Giordano, N.J.: Linking pressure and saturation through interfacial areas in porous media. Geophys. Res. Lett. 31, L08502 (2004)

    Google Scholar 

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

    Article  Google Scholar 

  12. Crawford, J.W., Matsui, N., Young, I.M.: The relation between the moisture-release curve and the structure of soil. Eur. J. Soil Sci. 46, 369–375 (1995)

    Article  Google Scholar 

  13. de Gennes, P.G.: Partial filling of a fractal structure by a wetting fluid. In: Adler, D., Fritzsche, H., Ovshinsky, S.R. (eds.) Physics of Disordered Materials, pp. 227–241. Plenum, New York (1985)

    Chapter  Google Scholar 

  14. de Gennes, P.G.: Wetting: statics and dynamics. Rev. Mod. Phys. 57(3), 827–863 (1985)

    Article  Google Scholar 

  15. de Gennes, P.G.: Dynamic capillary pressure in porous media. Europhys. Lett. 5, 689–691 (1988)

    Article  Google Scholar 

  16. Deinert, M.R., Parlange, J.-Y., Cady, K.B.: Simplified thermodynamic model for equilibrium capillary pressure in a fractal porous medium. Phys. Rev. E 72, 041203 (2005)

    Google Scholar 

  17. Deinert, M.R., Dathe, A., Parlange, J.-Y., Cady, K.B.: Capillary pressure in a porous medium with distinct pore surface and pore volume fractal dimensions. Phys. Rev. E 77, 021203 (2008), pp. 3

    Article  Google Scholar 

  18. Dullien, F.A.L.: New network permeability model of porous media. AIChE J. 21, 299–307 (1975)

    Article  Google Scholar 

  19. Filgueira, R.R., Pachepsky, Ya.A., Fournier, L.L., Sarli, G.O., Aragon, A.: Comparison of fractal dimensions estimated from aggregate mass-size distribution and water retention scaling. Soil Sci. 164, 217–223 (1999)

    Article  Google Scholar 

  20. Freeman, E.J.: Fractal geometries applied to particle size distributions and related moisture retention measurements at hanford. Washington, M.A. Thesis, University of Idaho, Moscow (1995)

    Google Scholar 

  21. Ghanbarian, B., Hunt, A.G., Ewing, R.P., Sahimi, M.: Tortuosity in porous media: a critical review. Soil Sci. Soc. Am. J. 77, 1461–1477 (2013)

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  24. Ghanbarian-Alavijeh, B., Skinner, T.E., Hunt, A.G.: Saturation-dependence of dispersion in porous media. Phys. Rev. E 86, 066316 (2012)

    Article  Google Scholar 

  25. Gvirtzman, H., Roberts, P.V.: Pore scale spatial analysis of two immiscible fluids in porous media. Water Resour. Res. 27, 1165–1176 (1991)

    Article  Google Scholar 

  26. Hoffman, R.L.: A study of the advancing interface. I. Interface shape in liquid-gas systems. J. Colloid Interface Sci. 50, 228–241 (1975)

    Article  Google Scholar 

  27. Hunt, A.G.: Comments on “Fractal fragmentation, soil porosity, and soil water properties: I. Theory”. Soil Sci. Soc. Am. J. 71, 1418–1419 (2007)

    Article  Google Scholar 

  28. Hunt, A.G., Gee, G.W.: Water-retention of fractal soil models using continuum percolation theory: tests of Hanford Site soils. Vadose Zone J. 1, 252–260 (2002)

    Google Scholar 

  29. Hunt, A.G., Gee, G.W.: Wet-end deviations from scaling of the water retention characteristics of fractal porous media. Vadose Zone J. 2, 759–765 (2003)

    Google Scholar 

  30. Hunt, A.G., Skinner, T.E.: Hydraulic conductivity limited equilibration: effect on water retention characteristics. Vadose Zone J. 4, 145–150 (2005)

    Article  Google Scholar 

  31. Hunt, A.G., Ghanbarian, B., Saville, K.C.: Unsaturated hydraulic conductivity modeling for porous media with two fractal regimes. Geoderma 207–208, 268–278 (2013)

    Article  Google Scholar 

  32. Jullien, R., Botet, R.: Aggregation and Fractal Aggregates. World Scientific, Singapore (1987)

    Google Scholar 

  33. Katz, A.J., Thompson, A.H.: Fractal sandstone pores: implications for conductivity and pore formation. Phys. Rev. Lett. 54, 1325–1328 (1985)

    Article  Google Scholar 

  34. Kosugi, K.: Three-parameter lognormal distribution model for soil water retention. Water Resour. Res. 30, 891–901 (1994)

    Article  Google Scholar 

  35. Larson, R.G., Morrow, N.R.: Effects of sample size on capillary pressures in porous media. Powder Technol. 30, 123–138 (1981)

    Article  Google Scholar 

  36. Madadi, M., Sahimi, M.: Lattice Boltzmann simulation of fluid flow in fracture networks with rough, self-affine surfaces. Phys. Rev. E 67, 026309 (2003)

    Google Scholar 

  37. Mandelbrot, B.B.: The Fractal Geometry of Nature. Freeman, San Francisco (1982)

    Google Scholar 

  38. Millán, H., González-Posada, M.: Modelling soil water retention scaling: comparison of a classical fractal model with a piecewise approach. Geoderma 125, 25–38 (2005)

    Article  Google Scholar 

  39. Mitzenmacher, M.: A brief history of generative models for power law and lognormal distributions. Internet Math. 1(2), 226–251 (2004)

    Article  Google Scholar 

  40. Mourzenko, V.V., Thovert, J.-F., Adler, P.M.: Percolation and conductivity of self-affine fractures. Phys. Rev. E 59, 4265–4284 (1999)

    Google Scholar 

  41. Mourzenko, V.V., Thovert, J.-F., Adler, P.M.: Permeability of self-affine fractures. Transp. Porous Media 45, 89–103 (2001)

    Article  Google Scholar 

  42. Mualem, Y.: A new model for predicting the hydraulic conductivity of unsaturated porous media. Water Resour. Res. 12, 513–522 (1976)

    Article  Google Scholar 

  43. Nigmatullin, R.R., Dissado, L.A., Soutougin, N.N.: A fractal pore model for Archie’s law in sedimentary rocks. J. Phys. D, Appl. Phys. 25, 32–37 (1992)

    Article  Google Scholar 

  44. Ojeda, M., Perfect, E., Alcaniz, J.M., Ortiz, O.: Fractal analysis of soil water hysteresis as influenced by sewage sludge application. Geoderma 134, 386–401 (2006)

    Article  Google Scholar 

  45. Oron, A.P., Berkowitz, B.: Flow in rock fractures: the local cubic law assumption reexamined. Water Resour. Res. 34, 2811–2825 (1998)

    Article  Google Scholar 

  46. Perfect, E.: Estimating mass fractal dimensions from water retention curves. Geoderma 88, 221–231 (1999)

    Article  Google Scholar 

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

    Article  Google Scholar 

  48. Perfect, E., Rasiah, V., Kay, B.D.: Fractal dimensions of soil aggregate-size distributions calculated by number and mass. Soil Sci. Soc. Am. J. 56, 1407–1409 (1992)

    Article  Google Scholar 

  49. Perfect, E., Kenst, A.B., Díaz-Zorita, M., Grove, J.H.: Fractal analysis of soil water desorption data collected on disturbed samples with water activity meter. Soil Sci. Soc. Am. J. 68, 1177–1184 (2004)

    Article  Google Scholar 

  50. Perrier, E., Bird, N., Rieu, M.: Generalizing a fractal model of soil structure: the pore-solid fractal approach. Geoderma 88, 137–164 (1999)

    Article  Google Scholar 

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

    Article  Google Scholar 

  52. Poon, C.Y., Sayles, R.S., Jones, T.A.: Surface measurement and fractal characterization of naturally fractured rocks. J. Phys. D, Appl. Phys. 25, 1269–1275 (1992)

    Article  Google Scholar 

  53. Rieu, M., Sposito, G.: Fractal fragmentation, soil porosity, and soil water properties: I. Theory, II. Applications. Soil Sci. Soc. Am. J. 55, 1231–1244 (1991)

    Article  Google Scholar 

  54. Rieu, M., Perrier, E.: Fractal models of fragmented and aggregated soils. In: Baveye, P., et al. (eds.) Fractals in Soil Science, pp. 169–201. CRC Press, Boca Raton (1997)

    Google Scholar 

  55. Sahimi, M.: Flow and Transport in Porous Media and Fractured Rock: From Classical Methods to Modern Approaches. Wiley-VCH, New York (2011). 709 pp.

    Book  Google Scholar 

  56. Shirazi, M.A., Boersma, L., Hart, J.W.: A unifying quantitative analysis of soil texture: improvement of precision and extension of scale. Soil Sci. Soc. Am. J. 52, 181–190 (1988)

    Article  Google Scholar 

  57. Toledo, P.G., Novy, R.A., Davis, H.T., Scriven, L.E.: Hydraulic conductivity of porous media at low water content. Soil Sci. Soc. Am. J. 54, 673–679 (1990)

    Article  Google Scholar 

  58. Turcotte, D.L.: Fractals and fragmentation. J. Geophys. Res. 91, 1921–1926 (1986)

    Article  Google Scholar 

  59. Tyler, S.W., Wheatcraft, S.W.: Application of fractal mathematics to soil water retention estimation. Soil Sci. Soc. Am. J. 53, 987–996 (1989)

    Article  Google Scholar 

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

    Article  Google Scholar 

  61. Tyler, S.W., Wheatcraft, S.W.: Fractal scaling of soil particle size-distributions: analysis and limitations. Soil Sci. Soc. Am. J. 56, 362–369 (1992)

    Article  Google Scholar 

  62. van Genuchten, M.Th.: A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J. 44, 892–898 (1980)

    Article  Google Scholar 

  63. Weitz, D.A., Stokes, J.P., Ball, R.C., Kushnick, A.P.: Dynamic capillary pressure in porous media: origin of the viscous-fingering length scale. Phys. Rev. Lett. 59, 2967–2970 (1987)

    Article  Google Scholar 

  64. Zhang, X., Knackstedt, M.A., Sahimi, M.: Fluid flow across mass fractals and self-affine surfaces. Physica A 233, 835–847 (1996)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Earth and Environmental Sciences, Wright State University, Dayton, OH, USA

    Allen Hunt

  2. Department of Agronomy, Iowa State University, Ames, IA, USA

    Robert Ewing

  3. Department of Earth and Environmental Sciences, Wright State University, Dayton, OH, USA

    Behzad Ghanbarian

Authors
  1. Allen Hunt
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Robert Ewing
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Behzad Ghanbarian
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Hunt, A., Ewing, R., Ghanbarian, B. (2014). Fractal Models of Porous Media. In: Percolation Theory for Flow in Porous Media. Lecture Notes in Physics, vol 880. Springer, Cham. https://doi.org/10.1007/978-3-319-03771-4_4

Download citation

Publish with us

Policies and ethics

Search

Navigation