Transport in Porous Media

, Volume 101, Issue 3, pp 465–475 | Cite as

Statistical Scaling of Geometric Characteristics in Millimeter Scale Natural Porous Media

Article

Abstract

We analyze statistical scaling of structural attributes of two millimeter scale rock samples, Estaillades limestone and Bentheimer sandstone. The two samples have different connected porosities and pore structures. The pore-space geometry of each sample is reconstructed via X-ray micro-tomography at micrometer resolution. Directional distributions of porosity and specific surface area (SSA), which are key Minkowski functionals (geometric observables) employed to describe the pore-space structure, are calculated from the images, and scaling of associated order-\(q\) sample structure functions of absolute incremental values is analyzed. Increments of porosity and SSA tend to be statistically dependent and persistent (tendency for large and small values to alternate mildly) in space. Structure functions scale as powers \(\xi (q)\) of directional separation distance or lag, \(s\), over an intermediate range of \(s\), displaying breakdown in power law scaling at large and small lags. Powers \(\xi \!\!\left( q \right) \) of porosity and SSA inferred from moment and extended self-similarity (ESS) analyses of limestone and sandstone data tend to be quasi-linear and nonlinear (concave) in \(q\), respectively. We observe an anisotropic behavior for \(\xi (q)\), which appears to be mild for the porosity of the sandstone sample while it is marked for both porosity and SSA of the limestone rock sample. The documented nonlinear scaling behavior is amenable to analysis by viewing the variables as samples from sub-Gaussian random fields subordinated to truncated fractional Brownian motion or fractional Gaussian noise.

Keywords

Microstructure Scaling Extended self-similarity  Structure functions  Pore scale characterization 

Notes

Acknowledgments

Funding from MIUR (Italian Ministry of Education, Universities and Research-PRIN2010-11; project: “Innovative methods for water resources under hydro-climatic uncertainty scenarios”) is acknowledged. The authors are grateful to M. Siena for providing the algorithm and code for the extraction of porosity and SSA from the digitized binary images. The authors thank S.P. Neuman for stimulating discussions.

References

  1. Adler, P.M.: Porous Media: Geometry and Transports. Butterworth-Heinemann, Boston (1992)Google Scholar
  2. Benzi, R., Ciliberto, S., Baudet, C., Chavarria, G.R., Tripiccione, R.: Extended self-similarity in the dissipation range of fully developed turbulence. Europhys. Lett. 24, 275–279 (1993a). doi: 10.1209/0295-5075/24/4/007 CrossRefGoogle Scholar
  3. Benzi, R., Ciliberto, S., Tripiccione, R., Baudet, C., Massaioli, F., Succi, S.: Extended self-similarity in turbulent flows. Phys. Rev. E 48, R29–R32 (1993b). doi: 10.1103/PhysRevE.48.R29 CrossRefGoogle Scholar
  4. Bijeljic, B., Raeini, A., Mostaghimi, P., Blunt, M.J.: Predictions of non-Fickian solute transport in different classes of porous media using direct simulation on pore-scale images. Phys. Rev. E. 87, 013011 (2013a). doi: 10.1103/PhysRevE.87.013011 CrossRefGoogle Scholar
  5. Bijeljic, B., Mostaghimi, P., Blunt, M.J.: Insights into non-Fickian solute transport in carbonates. Water Resour. Res. 49(5), 2714–2728 (2013b). doi: 10.1002/wrcr.20238 CrossRefGoogle Scholar
  6. Blunt, M.J., Bijeljic, B., Dong, H., Gharbi, O., Iglauer, S., Mostaghimi, P., Paluszny, A., Pentland, C.: Pore-scale imaging and modelling. Adv. Water Resour. 51, 197–216 (2013)CrossRefGoogle Scholar
  7. Buades, A., Coll, B., Morel, J.-M.: A non-local algorithm for image denoising. CVPR ’05 Proceedings of the 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR’05) 2, pp. 60–65, 2005Google Scholar
  8. Buades, A., Coll, B., Morel, J.-M.: Nonlocal image and movie denoising. Int. J. Comput. Vision 76, 123–139 (2008)CrossRefGoogle Scholar
  9. Castle, J.W., Molz, F.J., Lu, S., Dinwiddie, C.L.: Sedimentology and fractal-based analysis of permeability data, John Henry member, Straight Cliffs formation (upper Cretaceous), Utah, U.S.A. J. Sediment. Res. 74(2), 270–284 (2004)CrossRefGoogle Scholar
  10. Chakraborty, S., Frisch, U., Ray, S.S.: Extended self-similarity works for the Burgers equation and why. J. Fluid Mech. 649, 275–285 (2010). doi: 10.1017/S0022112010000595 CrossRefGoogle Scholar
  11. Coker, D.A., Torquato, S.: Extraction of morphological quantities from a digitized medium. J. Appl. Phys. 77(121), 6087–6099 (1995)CrossRefGoogle Scholar
  12. Dashtian, H., Jafari, G.R., Sahimi, M., Msihi, M.: Scaling, multifractality, and long-range correlations in well log data of large-scale porous media. Physica A 390, 2096–2111 (2011). doi: 10.1016/j.physa.2011.01.010 CrossRefGoogle Scholar
  13. Deshpande, A., Flemings, P.B., Huang, J.: Quantifying lateral heterogeneities in fluvio-deltaic sediments using three-dimensional reflection seismic data: Offshore Gulf of Mexico. J. Geophys. Res. 102(B7), 15385–15401 (1997)CrossRefGoogle Scholar
  14. Di Federico, V., Neuman, S.P.: Scaling of random fields by means of truncated power variograms and associated spectra. Water Resour. Res. 33, 1075–1085 (1997). doi: 10.1029/97WR00299 CrossRefGoogle Scholar
  15. Di Federico, V., Neuman, S.P., Tartakovsky, D.M.: Anisotropy, lacunarity, upscaled conductivity and its covariance in multiscale fields with truncated power variograms. Water Resour. Res. 35(10), 2891–2908 (1999)CrossRefGoogle Scholar
  16. Feder, J.: Fractals. Plenum Press, New York (1988)CrossRefGoogle Scholar
  17. Guadagnini, A., Neuman, S.P., Schaap, M.G., Riva, M.: Anisotropic Statistical Scaling of Soil and Sediment Texture in a Stratified Deep Vadose Zone near Maricopa, Arizona, Geoderma  214–215, 217–227 (2014). doi: 10.1016/j.geoderma.2013.09.008
  18. Guadagnini, A., Neuman, S.P.: Extended self-affinity of signals exhibiting apparent multifractality. Geophys. Res. Lett. 38, L13403 (2011). doi: 10.1029/2011GL047727 CrossRefGoogle Scholar
  19. Guadagnini, A., Neuman, S.P., Riva, M.: Numerical investigation of apparent multifractality of samples from processes subordinated to truncated fBm. Hydrol. Process. 26, 2894–2908 (2012). doi: 10.1002/hyp.8358 CrossRefGoogle Scholar
  20. Hilfer, R.: Review on scale dependent characterization of the microstructure of porous media. Transport Porous Med. 46, 373–390 (2002)CrossRefGoogle Scholar
  21. Kravchenko, A.N., Martín, M.A., Smucker, A.J.M., Rivers, M.L.: Limitations in determining multifractal spectra from pore-solid soil aggregate images. Vadose Zone J. 8, 220–226 (2009)CrossRefGoogle Scholar
  22. Latief, F.D.E., Biswal, B., Fauzi, U., Hilfer, R.: Continuum reconstruction of the pore scale microstructure for Fontainebleau sandstone. Physica A 389, 1607–1618 (2010)CrossRefGoogle Scholar
  23. Leonardis, E., Chapman, S.C., Foullon, C.: Turbulent characteristics in the intensity fluctuations of a solar quiescent prominence observed by the Hinode Solar Optical Telescope. Astrophys. J. 745, 185 (2012). doi: 10.1088/0004-637X/745/2/185 CrossRefGoogle Scholar
  24. Meerschaert, M.M., Kozubowski, T.J., Molz, F.J., Lu, S.: Fractional laplace model for hydraulic conductivity. Geophys. Res. Lett. 31, L08501 (2004). doi: 10.1029/2003GL019320 CrossRefGoogle Scholar
  25. Molz, F., Boman, G.: Further evidence of fractal structure in hydraulic conductivity distributions. Geophys. Res. Lett. 22(18), 2545–2548 (1995)CrossRefGoogle Scholar
  26. Molz, F., Liu, H., Szulga, J.: Fractional Brownian motion and fractional Gaussian noise in subsurface hydrology: a review, presentation of fundamental properties, and extensions. Water Resour. Res. 33(10), 2273–2286 (1997)CrossRefGoogle Scholar
  27. Neuman, S.P., Di Federico, V.: Multifaceted nature of hydrogeologic scaling and its interpretation. Rev. Geophys. 41(3), 1014 (2003). doi: 10.1029/2003RG000130 CrossRefGoogle Scholar
  28. Neuman, S.P.: Apparent/spurious multifractality of data sampled from fractional Brownian/Lévy motions. Hydrol. Process. 24, 2056–2067 (2010a). doi: 10.1002/hyp.7611 Google Scholar
  29. Neuman, S.P.: Apparent/spurious multifractality of absolute increments sampled from truncated fractional Gaussian/Lévy noise. Geophys. Res. Lett. 37, L09403 (2010b). doi: 10.1029/2010GL043314 CrossRefGoogle Scholar
  30. Neuman, S.P.: Apparent multifractality and scale-dependent distribution of data sampled from self-affine processes. Hydrol. Process. 25, 1837–1840 (2011). doi: 10.1002/hyp.7967 CrossRefGoogle Scholar
  31. Neuman, S.P., Guadagnini, A., Riva, M., Siena, M.: Recent advances in statistical and scaling analysis of earth and environmental variables. In: Mishra, P.K., Kuhlman, K.L. (eds.) Recent Advances in Hydrogeology, pp. 11–15. Springer Science+Business Media, New York (2013)Google Scholar
  32. Nikora, V.I., Goring, D.G.: Extended self-similarity in geophysical and geological applications. Math. Geol. 33(3), 251–271 (2001). doi: 10.1023/A:1007630021716 CrossRefGoogle Scholar
  33. Okabe, H., Blunt, M.J.: Prediction of permeability for porous media reconstructed using multiple-point statistics. Phys. Rev. E. 70, 066135 (2004). doi: 10.1103/PhysRevE.70.066135 CrossRefGoogle Scholar
  34. Okabe, H., Blunt, M.J.: Pore space reconstruction using multiple-point statistics. J. Petrol. Sci. Eng. 46, 121–137 (2005). doi: 10.1016/j.petrol.2004.08.002 CrossRefGoogle Scholar
  35. Okabe, H., Blunt, M.J.: Pore space reconstruction of vuggy carbonates using microtomography and multiple-point statistics. Water Resour. Res. 43, W12S02 (2007). doi: 10.1029/2006WR005680 CrossRefGoogle Scholar
  36. Pape, H., Clauser, C., Iffland, J.: Permeability prediction based on fractal pore-space geometry. Geophysics 64(5), 1447–1460 (1999)CrossRefGoogle Scholar
  37. Paz Ferreiro, J., Wilson, M., Vidal Vázquez, E.: Multifractal description of nitrogen adsorption isotherms. Vadose Zone J. 8, 209–219 (2009)CrossRefGoogle Scholar
  38. Paz Ferreiro, J., Miranda, J.G.V., Vidal Vázquez, E.: Multifractal analysis of soil porosity based on mercury injection and nitrogen adsorption. Vadose Zone J. 9(2), 325–335 (2010)CrossRefGoogle Scholar
  39. Perfect, E., Kay, B.D.: Applications of fractals in soil and tillage research: a review. Soil Tillage Res. 36, 1–20 (1995)CrossRefGoogle Scholar
  40. Riva, M., Neuman, S.P., Guadagnini, A., Siena, M.: Anisotropic scaling of Berea sandstone log air permeability statistics. Vadose Zone J. 12(3), (2013b). doi: 10.2136/vzj2012.0153
  41. Riva, M., Neuman, S.P., Guadagnini, A.: Sub-Gaussian model of processes with heavy tailed distributions applied to permeabilities of fractured tuff. Stoch. Environ. Res. Risk Assess. 27, 195–207 (2013a). doi: 10.1007/s00477-012-0576-y CrossRefGoogle Scholar
  42. Sahimi, M., Yortsos, Y.C.: Applications of fractal geometry to porous media: a review. SPE 20476, Society of Petroleum Engineers (1991)Google Scholar
  43. Samorodnitsky, G., Taqqu, M.S.: Stable Non-Gaussian Random Processes. Chapman & Hall, New York (1994)Google Scholar
  44. Siena, M., Guadagnini, A., Riva, M., Neuman, S.P.: Extended power-law scaling of air permeabilities measured on a block of tuff. Hydrol. Earth Syst. Sci. 16, 29–42 (2012). doi: 10.5194/hess-16-29-2012 CrossRefGoogle Scholar
  45. Stumpf, M.P.H., Porter, M.A.: Critical truths about power laws. Science 335, 665–666 (2012)CrossRefGoogle Scholar
  46. Tennekoon, L., Boufadel, M.C., Lavallee, D., Weaver, J.: Multifractal anisotropic scaling of the hydraulic conductivity. Water Resour. Res. 39(7), 1193 (2003). doi: 10.1029/2002WR001645 CrossRefGoogle Scholar
  47. Wildenschild, D., Sheppard, A.P.: X-ray imaging and analysis techniques for quantifying pore-scale structure and processes in subsurface porous medium systems. Adv. Water Resour. 51, 217–246 (2013)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • A. Guadagnini
    • 1
    • 2
  • M. J. Blunt
    • 1
    • 3
  • M. Riva
    • 1
    • 2
  • B. Bijeljic
    • 3
  1. 1.Dipartimento di Ingegneria Civile e AmbientalePolitecnico di MilanoMilanoItaly
  2. 2.Department of Hydrology and Water ResourcesUniversity of ArizonaTucsonUSA
  3. 3.Department of Earth Science and EngineeringImperial CollegeLondonUK

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