Abstract
The increasing access to 3d digital images of porous media provides an ideal avenue for the determination of their transport properties, by solving the governing equations in their actual microscale geometry and evaluating the tensor coefficient that relates the mean flux and driving gradient. However, the first and puzzling question along the way is the choice of the conditions to be imposed for this resolution at the boundaries of the sample. This methodological issue is explored here with the purpose of quantifying the influence of the boundary conditions (BC) in relation with the parameters of the system (porosity, characteristic length scale of the microstructure, ratio of the phase conductivities), assessing the level of confidence associated with the predictions, devising criteria to anticipate the risk of serious artefacts, and if possible proposing ways to limit them. Although the terminology of thermal transfer is used, the developments apply to the upscaling of any transport property governed by a diffusion equation, including thermal or electrical conduction, mass diffusion or Darcy flow. Quantitative indicators are introduced for a rigorous individual or comparative assessment of conductivity tensors, and they are used in the analysis of the results of extensive calculations based on four tomographic images of various kinds of porous materials, with a broad range of conductivity contrasts, and various kinds of BC’s. Ultimately, practical criteria are proposed for the a priori and a posteriori detection of at-risk situations, and a self-diagnosing protocol is proposed to screen out the influence of the BC’s, when this is possible.
Similar content being viewed by others
References
Adler, P.M.: Porous Media: Geometry and Transports. Butterworth/Heinemann, Stoneham, MA (1992)
Alam, K., Anghelescu, M.S., Bradu, A.: Computational model of porous media using true 3-D images. In: Öchsner, A., Murch, G.E. (eds.) Heat Transfer in Multi-Phase Materials. Adv. Struct. Mater. vol. 2, Springer, Berlin (2010). https://doi.org/10.1007/8611_2010_7
Andrä, H., Combaret, N., Dvorkin, J., Glatt, E., Han, J., Kabel, M., Keehm, Y., Krzikalla, F., Lee, M., Madonna, C., Marsh, M., Mukerji, T., Saenger, E.H., Sain, R., Saxena, N., Ricker, S., Wiegmann, A., Zhan, X.: Digital rock physics benchmarks part II: computing effective properties. Comput. Geosci. 50, 33–43 (2013)
Auriault, J.-L.: Heterogeneous periodic and random media. Are the equivalent macroscopic descriptions similar? Int. J. Eng. Sci. 49, 806–808 (2011)
Bailly, D.: Vers une modélisation des écoulements dans les massifs très fissurés de type karst : étude morphologique, hydraulique et changement d'échelle. Ph.D. Thesis, Université de Toulouse (2009)
Bailly, D., Ababou, R., Quintard, M.: Geometric characterization, hydraulic behavior and upscaling of 3D fissured geologic media. Math. Comput. Simul. 79, 3385–3396 (2009)
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)
Bruggeman, D.A.G.: Berechnung verschiedener physikalischer konstanten von heterogenen substanzen. Annalen der Physik 24, 636–679 (1935)
De Lucia, M., de Fouquet C.l., Lagneau, V., Bruno, R.: Equivalent block transmissivity in an irregular 2D polygonal grid for one-phase flow: a sensitivity analysis. C. R. Geosci. 341, 327–338 (2009)
Durlofsky, L.J.: Numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media. Water Resour. Res. 27, 699–708 (1991)
Durlofsky, L.J.: Upscaling and gridding of fine scale geological models for flow simulation. In: 8th International Forum on Reservoir Simulation, Iles Borromees, Stresa, Italy (2005)
Ferry, L.: Caractérisation de résidus de combustion de câbles électriques, Technical report PTF18-30/LAF, IMT Mines Alès (2018)
Gerke, K.M., Karsanina, M.V., Katsman, R.: Calculation of tensorial flow properties on pore level: exploring the influence of boundary conditions on the permeability of three-dimensional stochastic reconstructions. Phys. Rev. E 100, 053312 (2019)
Gomez-Hermindez, J.J., Journel, A.G.: Stochastic characterization of grid-block permeabilities: from point values to block tensors. In: Guerillot D., Guillon, O. (eds.) 2nd European Conference on the Mathematics of Oil Recovery. Edition Technip, Paris, pp. 83–90 (1990)
Guan, K.M., Nazarova, M., Guo, B., Tchelepi, H., Kovscek, A.R., Creux, P.: Effects of image resolution on sandstone porosity and permeability as obtained from x-ray microscopy. Transp. Porous Med. 127, 233–245 (2019). https://doi.org/10.1007/s11242-018-1189-9
Guibert, R., Horgue, P., Debenest, G., Quintard, M.: A comparison of various methods for the numerical evaluation of porous media permeability tensors from pore-scale geometry. Math. Geosci. 48, 329–347 (2016)
Hashin, Z., Shtrikman, S.: A variational approach to the theory of the effective magnetic permeability of multiphase materials. J. Appl. Phys. 33, 3125–3131 (1962)
Haussener, S., Coray, P., Lipinski, W., Wyss, P., Steinfeld, A.: Tomography-based heat and mass transfer characterization of reticulate porous ceramics for high-temperature processing. J. Heat Transf. 132, 023305 (2010)
Henriette, A., Jacquin, C.G., Adler, P.M.: The effective permeability of heterogeneous porous media. Phys. Chem. Hydrodyn. 11, 63–80 (1989)
Khan, F., Enzmann, F., Kersten, M., Wiegmann, A., Steiner, K.: 3D simulation of the permeability tensor in a soil aggregate on basis of nanotomographic imaging and LBE solver. Soils Sediments 12, 86–96 (2012)
Landauer, R.: Electrical conductivity in inhomogeneous media. AIP Conf. Proc. 40, 2–43 (1978)
Lang, P.S., Paluszny, A., Zimmerman, R.W.: Permeability tensor of three-dimensional fractured porous rock and a comparison to trace map predictions. J. Geophys. Res. Solid Earth 119, 6288–6307 (2014)
Long, J.C.S., Remer, J.S., Wilson, C.R., Witherspoon, P.A.: Porous media equivalents for networks of discontinuous fractures. Water Resours. Res. 18, 645–658 (1982)
Malinouskaya, I., Mourzenko, V.V., Thovert, J.-F., Adler, P.M.: Wave propagation through saturated porous media. Phys. Rev. E 77, 066302 (2008)
Mostaghimi, P., Blunt, M.J., Bijeljic, B.: Computations of absolute permeability on micro-CT images. Math. Geosci. 45, 103–125 (2013)
Petrasch, J., Schrader, B., Wyss, P., Steinfeld, A.: Tomography-based determination of the effective thermal conductivity of fluid-saturated reticulate porous ceramics. J. Heat Transf. 130, 032602 (2008)
Petrasch, J., Meier, F., Friess, H., Steinfeld, A.: Tomography based determination of permeability, Dupuit-Forchheimer coefficient, and interfacial heat transfer coefficient in reticulate porous ceramics. Int. J. Heat Fluid Flow 29, 315–326 (2008)
Piller, M., Schena, G., Nolich, M., Favretto, S., Radaelli, F., Rossi, E.: Analysis of hydraulic permeability in porous media: from high resolution x-ray tomography to direct numerical simulation. Transp. Porous Med. 80, 57–78 (2009)
Pouya, A., Fouché, O.: Permeability of 3D discontinuity networks: new tensors from boundary-conditioned homogenisation. Adv. Water Resour. 32, 303–314 (2009)
Quintard, M., Whitaker, S.: One- and two-equation models for transient diffusion processes in two-phase systems. In: Hartnett J.P., Irvine, T.F. (eds.), Advances in Heat Transfer, Vol. 23, pp. 369–464 (1993). https://doi.org/10.1016/S0065-2717(08)70009-1
Ralston, A., Rabinowitz, P.A.: First Course in Numerical Analysis. Dover Publications, Mineola, N.Y. (2001)
Renard, P., de Marsily, G.: Calculating equivalent permeability: a review. Adv. Water Resour. 20, 253–278 (1997)
Shi, J., Boyer, G., Thovert, J.-F.: Simulation of the pyrolysis of charring polymers: influence of the porous media properties. In: Proceedings of European Symposium of Fire Safety Science, ESFSS 2018, Nancy, France (2018)
Spanne, P., Thovert, J.-F., Jacquin, C.J., Lindquist, W.B., Jones, K.W., Adler, P.M.: Synchrotron computed microtomography of porous media. Topology and transports. Phys. Rev. Lett. 73, 2001–2004 (1994)
Thovert, J.-F., Adler, P.M.: Grain reconstruction of porous media: application to a Bentheim sandstone. Phys. Rev. E 83, 056116 (2011)
Thovert, J.-F., Yousefian, F., Spanne, P., Jacquin, C.G., Adler, P.M.: Grain reconstruction of porous media: application to a low-porosity Fontainebleau sandstone. Phys. Rev. E 63, 61307–61323 (2001)
Wen, X.H., Durlofsk, L.J., Edwards, M.G.: Use of border regions for improved permeability upscaling. Math. Geol. 35, 521–547 (2003)
Wiener, O.: Die Theorie des Mischkörpers für das Feld des stationären Strömung. Erste Abhandlung: Die Mittelwertsätze für Kraft, Polarisation und Energie, Abhandl. d.K.S. Gesellsch. d. Wisseensch. Math.-Phys., 32, 509–604 (1912)
Wu, X., Hou, T., Efendiev, Y.: Analysis of upscaling absolute permeability. Discrete Contin. Dyn. Syst. Ser. B 2, 185–204 (2002)
Acknowledgements
We gratefully thank Laurent Ferry (C2MA) for preparing the samples of thermally degraded polymers and Pascal Laheurte (LEM3) for providing their tomographic images. This work pertains to the French Government Programme Investissements d’Avenir (LABEX INTERACTIFS, reference ANR-11-LABX-0017-01).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Shi, J., Boyer, G., Mourzenko, V.V. et al. On the Influence of Boundary Conditions when Determining Transport Coefficients from Finite Samples of Porous Media: Assessment for Tomographic Images of Real Materials. Transp Porous Med 132, 561–590 (2020). https://doi.org/10.1007/s11242-020-01404-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11242-020-01404-1