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Statistical scaling of geometric characteristics in stochastically generated pore microstructures

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Abstract

We analyze the statistical scaling of structural attributes of virtual porous microstructures that are stochastically generated by thresholding Gaussian random fields. Characterization of the extent at which randomly generated pore spaces can be considered as representative of a particular rock sample depends on the metrics employed to compare the virtual sample against its physical counterpart. Typically, comparisons against features and/patterns of geometric observables, e.g., porosity and specific surface area, flow-related macroscopic parameters, e.g., permeability, or autocorrelation functions are used to assess the representativeness of a virtual sample, and thereby the quality of the generation method. Here, we rely on manifestations of statistical scaling of geometric observables which were recently observed in real millimeter scale rock samples [13] as additional relevant metrics by which to characterize a virtual sample. We explore the statistical scaling of two geometric observables, namely porosity (ϕ) and specific surface area (SSA), of porous microstructures generated using the method of Smolarkiewicz and Winter [42] and Hyman and Winter [22]. Our results suggest that the method can produce virtual pore space samples displaying the symptoms of statistical scaling observed in real rock samples. Order q sample structure functions (statistical moments of absolute increments) of ϕ and SSA scale as a power of the separation distance (lag) over a range of lags, and extended self-similarity (linear relationship between log structure functions of successive orders) appears to be an intrinsic property of the generated media. The width of the range of lags where power-law scaling is observed and the Hurst coefficient associated with the variables we consider can be controlled by the generation parameters of the method.

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References

  1. Adler, P.M., Jacquin, C.G., Quiblier, J.A.: Flow in simulated porous media. Int J. Multiphas Flow 16(4), 691–712 (1990)

    Article  Google Scholar 

  2. Alexander, K.S.: Percolation and minimal spanning forests in infinite graphs. Ann. Probab, 87–104 (1995)

  3. Alexander, K.S., Molchanov, S.A.: Percolation of level sets for two-dimensional random fields with lattice symmetry. J. Stat. Phys. 77(3), 627–643 (1994)

    Article  Google Scholar 

  4. Arns, C.H., Knackstedt, M.A., Mecke, K.R.: Reconstructing complex materials via effective grain shapes. Phys. Rev. Lett. 91(21), 215–506 (2003)

    Article  Google Scholar 

  5. Balhoff, M.T., Thomas, S.G., Wheeler, M.F.: Mortar coupling and upscaling of pore-scale models. Computat. Geosci 12(1), 15–27 (2008)

    Article  Google Scholar 

  6. Benzi, R., Ciliberto, S., Baudet, C., Chavarria, G.R., Tripiccione, R.: Extended self-similarity in the dissipation range of fully developed turbulence. EPL Europhys. Lett. 24(4), 275 (1993)

    Article  Google Scholar 

  7. Benzi, R., Ciliberto, S., Tripiccione, R., Baudet, C., Massaioli, F., Succi, S.: Extended self-similarity in turbulent flows. Phys. Rev. E 48(1), R29 (1993)

    Article  Google Scholar 

  8. Biswal, B., Oren, P.E., Held, R.J., Bakke, S., Hilfer, R.: Modeling of multiscale porous media. Image Anal Stereol 28, 23–34 (2009)

    Article  Google Scholar 

  9. 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)

    Article  Google 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)

    Article  Google Scholar 

  11. Coker, D.A., Torquato, S.: Extraction of morphological quantities from a digitized medium. J. Appl. Phys. 77(12), 6087–6099 (1995)

    Article  Google Scholar 

  12. Duda, A., Koza, Z., Matyka, M.: Hydraulic tortuosity in arbitrary porous media flow. Phys. Rev. E 84(3), 036–319 (2011)

    Article  Google Scholar 

  13. Guadagnini, A., Blunt, M., Riva, M., Bijeljic, B.: Statistical scaling of geometric characteristics in millimeter scale natural porous media. Trans. Porous Med. 101(3), 465–475 (2014)

    Article  Google Scholar 

  14. Guadagnini, A., Neuman, S.P., Riva, M.: Numerical investigation of apparent multifractality of samples from processes subordinated to truncated fBm. Hydrol. Processes 26(19), 2894–2908 (2012)

    Article  Google Scholar 

  15. Guadagnini, A., Riva, M., Neuman, S.P.: Extended power-law scaling of heavy-tailed random air-permeability fields in fractured and sedimentary rocks. Hydrol. Earth Syst Sci 16(9), 3249–3260 (2012)

    Article  Google Scholar 

  16. Hilfer, R.: Local porosity theory and stochastic reconstruction for porous media. In: Statistical Physics and Spatial Statistics, pp. 203–241. Springer (2000)

  17. Hilfer, R.: Review on scale dependent characterization of the microstructure of porous media. Trans. Porous Media 46(2–3), 373–390 (2002)

    Article  Google Scholar 

  18. Hilfer, R., Zauner, T.: High-precision synthetic computed tomography of reconstructed porous media. Phys. Rev. E 84(6), 062–301 (2011)

    Article  Google Scholar 

  19. Hyman, J.D., Smolarkiewicz, P.K., Winter, C.: Heterogeneities of flow in stochastically generated porous media. Phys.Rev. E 86, 056–701 (2012). doi:10.1103/PhysRevE.86.056701

    Article  Google Scholar 

  20. Hyman, J.D., Smolarkiewicz, P.K., Winter, C.L.: Pedotransfer functions for permeability: a computational study at pore scales. Water Resour. Res. 49 (2013). doi:10.1002/wrcr.20170

  21. Hyman, J.D., Winter, C.L.: Hyperbolic regions in flows through three-dimensional pore structures. Phys. Rev. E 88, 063–014 (2013)

    Article  Google Scholar 

  22. Hyman, J.D., Winter, C.L.: Stochastic generation of explicit pore structures by thresholding Gaussian random fields. J Comput. Phys. 277(0), 16–31 (2014)

    Article  Google Scholar 

  23. Iassonov, P., Gebrenegus, T., Tuller, M.: Segmentation of X-ray computed tomography images of porous materials: a crucial step for characterization and quantitative analysis of pore structures. Water Resour. Res 45, 9 (2009)

    Google Scholar 

  24. Latief, F.E., Biswal, B., Fauzi, U., Hilfer, R.: Continuum reconstruction of the pore scale microstructure for fontainebleau sandstone. Physica A 389(8), 1607–1618 (2010)

    Article  Google Scholar 

  25. Lemaitre, R., Adler, P.M.: Fractal porous media iv: three-dimensional Stokes flow through random media and regular fractals. Transport in Porous Med. 5(4), 325–340 (1990)

    Article  Google Scholar 

  26. Mandelbrot, B.B., Van Ness, J.W.: Fractional Brownian motions, fractional noises and applications. SIAM Rev 10(4), 422–437 (1968)

    Article  Google Scholar 

  27. Manwart, C., Torquato, S., Hilfer, R.: Stochastic reconstruction of sandstones. Phys. Rev. E 62, 893–899 (2000). doi:10.1103/PhysRevE.62.893

    Article  Google Scholar 

  28. Matyka, M., Khalili, A., Koza, Z.: Tortuosity-porosity relation in porous media flow. Phys. Rev. E 78 2(026), 306 (2008)

    Google Scholar 

  29. Mecke, K.R.: Integral geometry in statistical physics. Int. J. Mod. Phys. B 12(09), 861–899 (1998)

    Article  Google Scholar 

  30. Neuman, S.P., Guadagnini, A., Riva, M., Siena, M.: Recent advances in statistical and scaling analysis of earth and environmental variables. In: Advances in Hydrogeology, pp. 1–25 Springer (2013)

  31. Okabe, H., Blunt, M.: Prediction of permeability for porous media reconstructed using multiple-point statistics. Phys. Rev E 70(6), 066–135 (2004)

    Article  Google Scholar 

  32. Oostrom, M., Mehmani, Y., Romero-Gomez, P., Tang, Y., Liu, H., Yoon, H., Kang, Q., Joekar-Niasar, V., Balhoff, M., Dewers, T., Tartakovsky, G., Leist, E., Hess, N., Perkins, W., Rakowski, C., Richmond, M., Serkowski, J., Werth, C., Valocchi, A., Wietsma, T., Zhang, C.: Pore-scale and continuum simulations of solute transport micromodel benchmark experiments. Computat. Geosci., 1–23 (2014)

  33. Porter, M.L., Wildenschild, D.: Image analysis algorithms for estimating porous media multiphase flow variables from computed microtomography data: a validation study. Computat. Geosci. 14(1), 15–30 (2010)

    Article  Google Scholar 

  34. Quiblier, J.A.: A new three-dimensional modeling technique for studying porous media. J.Colloid Interf. Sci 981, 84–102 (1984)

    Article  Google Scholar 

  35. Riva, M., Neuman, S., Guadagnini, A.: Sub-Gaussian model of processes with heavy-tailed distributions applied to air permeabilities of fractured tuff. Stochastic Environmental Research and Risk Assessment 27(1), 195–207 (2013)

    Article  Google Scholar 

  36. Riva, M., Neuman, S.P., Guadagnini, A., Siena, M.: Anisotropic scaling of berea sandstone log air permeability statistics. Vadose Zone Journal 12, 3 (2013)

    Article  Google Scholar 

  37. Romero, P., Gladkikh, M., Azpiroz, G. Computat. Geosci 13(2), 171–180 (2009). doi:10.1007/s10596-008-9098-6

    Article  Google Scholar 

  38. Siena, M., Guadagnini, A., Riva, M., Bijeljic, B., Nunes, J.P.P., Blunt, M.J.: Statistical scaling of pore-scale Lagrangian velocities in natural porous media. Phys. Rev. E 90 2(023), 013 (2014)

    Google Scholar 

  39. 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(1), 29–42 (2012)

    Article  Google Scholar 

  40. Siena, M., Hyman, J.D., Riva, M., Guadagnini, A., Winter, C.L., Smolarkiewicz, P.K., Gouze, P., Sadhukhan, S., Inzoli, F., Guédon, G., Colombo, E.: Direct numerical simulation of fully-saturated flow in natural porous media at the pore scale: a comparison of three computational systems. Comput. Geosci., 1–15 (2015). doi:10.1007/s10596-015-9486-7

  41. Siena, M., Riva, M., Hyman, J.D., Winter, C.L., Guadagnini, A.: Relationship between pore size and velocity probability distributions in stochastically generated porous media. Phys. Rev. E 89(003), 000 (2014)

    Google Scholar 

  42. Smolarkiewicz, P.K., Winter, C.L.: Pores resolving simulation of Darcy flows. J. Comput. Phys 229(9), 3121–3133 (2010)

    Article  Google Scholar 

  43. Tartakovsky, A.M., Meakin, P., Scheibe, T.D., Eichler West, R.M.: Simulations of reactive transport and precipitation with smoothed particle hydrodynamics. J.Comput. Phys. 222(2), 654–672 (2007)

    Article  Google Scholar 

  44. 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)

    Article  Google Scholar 

  45. Yao, J., Frykman, P., Kalaydjian, F., Thovert, J.F., Adler, P.M.: High-order moments of the phase function for real and reconstructed model porous media A comparison. J. Colloid Interf. Sci 156(2), 478–490 (1993)

    Article  Google Scholar 

  46. Yeong, C., Torquato, S.: Reconstructing random media. Phys. Rev. E 57(1), 495 (1998)

    Article  Google Scholar 

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Authors and Affiliations

  1. Computational Earth Sciences (EES-16) Earth and Enviromental Sciences Division and The Center for Nonlinear Studies, Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA

    Jeffrey D. Hyman

  2. Dipartmento di Ingegneria Civile e Ambientale, Politecnico di Milano, Piazza L. Da Vinci 32, 20133, Milano, Italy

    Alberto Guadagnini

  3. Department of Hydrology and Water Resources, University of Arizona, Tucson, AZ, 85721-0011, USA

    Alberto Guadagnini

  4. Department of Hydrology and Water Resources, Program in Applied Mathematics, University of Arizona, Tucson, AZ, 85721-0011, USA

    C. Larrabee Winter

Authors
  1. Jeffrey D. Hyman
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  2. Alberto Guadagnini
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  3. C. Larrabee Winter
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Correspondence to Jeffrey D. Hyman.

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Hyman, J.D., Guadagnini, A. & Winter, C.L. Statistical scaling of geometric characteristics in stochastically generated pore microstructures. Comput Geosci 19, 845–854 (2015). https://doi.org/10.1007/s10596-015-9493-8

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