Abstract
We reconstructed pore structures of three porous solids that differ from each other in morphology and topology of pore space. To achieve this, we used a stochastic method based on simulated annealing and X-ray computed microtomography. Simulated annealing was constrained by the following microstructural descriptors sampled along the principal and diagonal directions: the two-point probability function for the void phase and the lineal-path functions for both void and solid phases. The stochastic method also assumed the isotropic pore structures in accordance with a recent paper (Čapek et al. in Transp Porous Media 88(1): 87–106 (2011)). With the exception of the solid with the widest pores, we made tomographic volume images in high and low resolution, which enabled us to study the effect of resolution on microstructural descriptors and effective transport properties. A comparison of the two-point probability function and the lineal-path function sampled in the principal directions revealed that the pore structures derived from the tomographic volume images were slightly anisotropic, in opposition to the assumption of the stochastic method. Besides the anisotropy, other microstructural descriptors including the pore-size function and the total fraction of percolating cells indicated that the morphological and topological characteristics of the pore structures depended on the reconstruction method and its parameters. Particularly, the pore structures reproduced using the stochastic method contained wider pores than those obtained using X-ray tomography. Deviations between the pore structures derived from low- and high-resolution tomographic volume images were also observed and imputed to partial volume artefacts. Then, viscous flow of incompressible liquid, ordinary diffusion, Knudsen flow and self-diffusion of water in the reconstructed pore spaces were simulated. As counterparts, experimental data were measured by means of permeation and Wicke–Kallenbach cells and pulsed field gradient NMR. Deviations between the simulated quantities on the one hand and experimental data on the other hand were generally acceptable, which corroborated the pore-space models. As expected, the predictions based on the tomographic models of pore space were more successful than those derived from the stochastic models. The stationary effective transport properties, i.e. the effective permeability, the effective pore size and the geometric factor, were sensitive to a bias in long-range pore connectivity. Furthermore, the time-dependent effective diffusivity was found to be especially sensitive to relatively small morphological deviations between the real and reconstructed pore structures. It is concluded that the combined predictions of the effective permeability, the effective pore size, the geometric factor and time-dependent effective self-diffusivity of water are needed for the reliable evaluation of pore-space reconstruction.
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Abbreviations
- a :
-
Voxel or pixel size
- \({\mathscr {C}}_2^{(s)}\) :
-
Two-point cluster function for the solid phase
- \({\mathscr {C}}_2^{(v)}\) :
-
Two-point cluster function for the void phase
- D :
-
Diffusivity (scalar)
- \({{\textsf {\textit{D}}}}\) :
-
Diffusivity tensor
- \(D_{ii}\) :
-
Diagonal components of the diffusivity tensor
- \({{\textsf {\textit{D}}}}^k\) :
-
Knudsen diffusivity tensor
- \(D_{ii}^k\) :
-
Diagonal components of the Knudsen diffusivity tensor
- \(D^m\) :
-
Binary diffusivity or self-diffusivity in the bulk fluid
- \(d_s\) :
-
Diameter of cylindrical pellet
- E :
-
“Energy” of the digitised system (6)
- \(e_0\) :
-
Contribution to E
- \(e_1\) :
-
Contribution to E
- \(e_2\) :
-
Contribution to E
- g :
-
Gradient amplitude
- \(g_m\) :
-
Maximum gradient amplitude
- \(g_r\) :
-
Reading gradient amplitude
- \(h_s\) :
-
Height of cylindrical pellet
- \(\mathscr {I}^{(v)}\) :
-
Phase (indicator) function for the void phase
- \(\mathscr {I}^{\mathscr {K}}_i\) :
-
Indicator function of percolation along direction i
- \({\mathscr {L}}^{(s)}\) :
-
Lineal-path function for the solid phase
- \({\mathscr {L}}^{(v)}\) :
-
Lineal-path function for the void phase
- \(l_i\) :
-
Number of voxels in \({\mathbb {V}}\) measured along direction i
- \({\mathbb {M}}\) :
-
Cubic measurement cell related to the local percolation theory
- M :
-
Molar mass of a gas
- \(m_t\) :
-
Total number of \({\mathbb {M}}\) randomly thrown into \({\mathbb {V}}\)
- \(m_o\) :
-
Number of FPC octants occupied by a fluid
- \(n_d\) :
-
Number of spatial dimensions
- P :
-
Pressure
- \({\mathscr {P}}\) :
-
Pore-size probability density function
- \({\mathscr {Q}}_i\) :
-
Total fraction of measurement cells percolating along direction i
- R :
-
Gas constant (here 8.31441 J mol\(^{-1}\) K\(^{-1}\))
- \(\mathbf {r}\) :
-
Position vector associated with the random walker
- \(r_i\) :
-
Component of \(\mathbf {r}\)
- \({\mathscr {S}}_1^{(v)}\) :
-
One-point probability function for the void phase
- \({\mathscr {S}}_2^{(s)}\) :
-
Two-point probability function for the solid phase
- \({\mathscr {S}}_2^{(v)}\) :
-
Two-point probability function for the void phase
- \({\mathscr {S}}_n^{(v)}\) :
-
n-point probability function for the void phase
- s :
-
Pore wall area per unit volume of \({\mathbb {V}}\)
- \(s_e\) :
-
BET-specific surface area (per unit volume of \({\mathbb {V}}\))
- T :
-
Thermodynamic temperature
- t :
-
Time
- \(t_1\) :
-
Longitudinal relaxation time
- \(\mathbf {u}\) :
-
Vector of relative displacement \(\mathbf {u} = \mathbf {x}_2 - \mathbf {x}_1\)
- u :
-
Length of a line segment, \(u = |\mathbf {u}|\)
- \(\mathbf {V}\) :
-
Macroscopic velocity vector
- \({\mathbb {V}}\) :
-
Convex region of space filled by the medium
- \(V_{ Hg }\) :
-
Specific pore volume invaded by mercury
- \(\mathbf {v}\) :
-
Vector of walker’s velocity (thermal velocity of a gas molecule)
- \(\mathbf {x}\) :
-
Position vector associated with \(\mathscr {I}^{(v)}\) and \({\mathbb {V}}\)
- \(\beta _{ii}\) :
-
Main diagonal component of the effective permeability tensor
- \(\bar{\beta }\) :
-
Average value of \(\beta _{ii}\)
- \(\beta _e\) :
-
Effective permeability determined experimentally
- \(\varDelta \) :
-
Amount of time separating the gradient pulses
- \(\delta \) :
-
Local radius of pore space
- \(\delta _n\) :
-
Duration of rectangular gradient pulses
- \(\varepsilon \) :
-
Half size of the first-passage cube
- \(\zeta \) :
-
Parameter of the stochastic reconstruction algorithm
- \(\eta \) :
-
Parameter of the merit function (6)
- \(\kappa _{ii}\) :
-
Main diagonal component of the effective pore-size tensor
- \(\bar{\kappa }\) :
-
Average value of \(\kappa _{ii}\)
- \(\kappa _e\) :
-
Effective pore size determined experimentally
- \(\varLambda \) :
-
Side length of \({\mathbb {M}}\)
- \({\upmu }\) :
-
Fluid viscosity
- \({\varvec{\xi }}\) :
-
Displacement vector associated with the random walker
- \(\xi _i\) :
-
Component of \(\varvec{{\xi }}\)
- \(\rho \) :
-
Bulk density of a porous solid
- \(\varrho \) :
-
Surface relaxivity
- \(\tau _e\) :
-
Amount of time needed for walker’s motion in the heterogeneous FPC
- \(\tau _o\) :
-
Amount of time needed for walker’s motion in the homogeneous FPC
- \(\tau _n\) :
-
Echo time
- \(\phi _v\) :
-
Void volume fraction (porosity)
- \(\phi _{ve}\) :
-
Porosity estimated from bulk and skeletal densities
- \(\psi _{ii}\) :
-
Main diagonal component of the geometric factor tensor
- \(\bar{\psi }\) :
-
Average value of \(\psi _{ii}\)
- \(\psi _e\) :
-
Geometric factor determined experimentally
- \(\omega \) :
-
Parameter of the modified Stokes–Einstein equation (16)
References
Adler, P.M.: Porous media: geometry and transports. Butterworth/Heinemann, Boston (1992)
Arns, C.H., Knackstedt, M.A., Mecke, K.R.: Boolean reconstructions of complex materials: integral geometric approach. Phys. Rev. E 80(5), 051,303 (2009)
Bakke, S., Øren, P.E.: 3-D pore-scale modelling of sandstones and flow simulations in the pore networks. SPE J. 2(2), 136–149 (1997)
Bentz, D.P., Martys, N.S.: a Stokes permeability solver for three-dimensional porous media. Tech. Rep. No. 7416, NIST, Gaithersburg, USA (2007). http://ftp.nist.gov/pub/bfrl/bentz/permsolver
Bergman, D.J., Dunn, K.J.: Self-diffusion in a periodic porous medium with interface absorption. Phys. Rev. E 51(4), 3401–3416 (1995)
Bergman, D.J., Dunn, K.J., Schwartz, L.M., Mitra, P.P.: Self-diffusion in a periodic porous medium: a comparison of different approaches. Phys. Rev. E 51(4), 3393–3400 (1995)
Bieniek, A., Moga, A.: An efficient watershed algorithm based on connected components. Pattern Recognit. 33(6), 907–916 (2000)
Biswal, B., Manwart, C., Hilfer, R.: Three-dimensional local porosity analysis of porous media. Phys. A 255(3–4), 221–241 (1998)
Biswal, B., Manwart, C., Hilfer, R., Bakke, S., Øren, P.E.: Quantitative analysis of experimental and synthetic microstructures for sedimentary rock. Phys. A 273(3–4), 452–475 (1999)
Biswal, B., Held, R.J., Khanna, V., Wang, J., Hilfer, R.: Towards precise prediction of transport properties from synthetic computer tomography of reconstructed porous media. Phys. Rev. E 80(4), 041,301 (2009)
Čapek, P., Hejtmánek, V., Brabec, L., Zikánová, A., Kočiřík, M.: Stochastic reconstruction of particulate media using simulated annealing: improving pore connectivity. Transp. Porous Media 76(2), 179–198 (2009)
Čapek, P., Hejtmánek, V., Kolafa, J., Brabec, L.: Transport properties of stochastically reconstructed porous media with improved pore connectivity. Transp. Porous Media 88(1), 87–106 (2011)
Čapek, P., Veselý, M., Bernauer, B., Sysel, P., Hejtmánek, V., Kočiřík, M., Brabec, L., Prokopová, O.: Stochastic reconstruction of mixed-matrix membranes and evaluation of effective permeability. Comput. Mater. Sci. 89, 142–156 (2014a)
Čapek, P., Veselý, M., Hejtmánek, V.: On the measurement of transport parameters of porous solids in permeation and Wicke–Kallenbach cells. Chem. Eng. Sci. 118, 192–207 (2014b)
Chan, Kim I., Torquato, S.: Determination of the effective conductivity of heterogeneous media by Brownian motion simulation. J. Appl. Phys. 68(8), 3892–3903 (1990)
Chan, Kim I., Cule, D., Torquato, S.: Comment on “Walker diffusion method for calculation of transport properties of composite materials”. Phys. Rev. E 61(4), 4659–4660 (2000)
Coker, D.A., Torquato, S.: Simulation of diffusion and trapping in digitized heterogeneous media. J. Appl. Phys. 77(3), 955–964 (1995)
Dullien, F.A.L.: Porous media: fluid transport and pore structure, 2nd edn. Academic Press, San Diego (1992)
Flannery, B.P., Deckman, H.W., Roberge, W.G., D’Amico, K.L.: Three-dimensional x-ray microtomography. Science 237(4821), 1439–1444 (1987)
Fredrich, J.T., Menendez, B., Wong, T.F.: Imaging the pore structure of geomaterials. Science 268(5208), 276–279 (1995)
Gerke, K.M., Karsanina, M.V., Vasilyev, R.V., Mallants, D.: Improving pattern reconstruction using directional correlation functions. EPL 106(6), 66,002 (2014)
Gonzalez, R.C., Woods, R.E.: Digital image processing, 3rd edn. Pearson Prentice Hall, Upper Saddle River (2008)
Haynes Jr, H.W.: The experimental evaluation of catalysts effective diffusivity. Catal. Rev. Sci. Eng. 30(4), 563–627 (1988)
Hidajat, I., Singh, M., Cooper, J., Mohanty, K.K.: Permeability of porous media from simulated NMR response. Transp. Porous Media 48(2), 225–247 (2002)
Hilfer, R.: Local porosity theory and stochastic reconstruction for porous media. In: Mecke, K., Stoyan, D. (eds.) Statistical physics and spatial statistics, lecture notes in physics, vol. 254, pp. 203–241. Springer, Berlin (2000)
Hilfer, R., Manwart, C.: Permeability and conductivity for reconstruction models of porous media. Phys. Rev. E 64(2), 021,304 (2001)
Hürlimann, M.D., Helmer, K.G., Latour, L.L., Sotak, C.H.: Restricted diffusion in sedimentary rocks. Determination of surface-area-to-volume ratio and surface relaxivity. J. Magn. Reson. A 111(2), 169–178 (1994)
Ioannidis, M.A., Kwiecien, M.J., Chatzis, I.: Electrical conductivity and percolation aspects of statistically homogeneous porous media. Transp. Porous Media 29, 61–83 (1997)
Jiao, Y., Stillinger, F.H., Torquato, S.: Modeling heterogeneous materials via two-point correlation functions: basic principles. Phys. Rev. E 76(3), 031,110 (2007)
Jiao, Y., Stillinger, F.H., Torquato, S.: Modeling heterogeneous materials via two-point correlation functions. II. Algorithmic details and applications. Phys. Rev. E 77(3), 031,135 (2008)
Jiao, Y., Stillinger, F.H., Torquato, S.: A superior descriptor of random textures and its predictive capacity. Proc. Natl. Acad. Sci. USA 106(42), 17,634–17,639 (2009)
Jin, G., Torres-Verdín, C., Toumelin, E.: Comparison of NMR simulations of porous media derived from analytical and voxelized representations. J. Magn. Reson. 200(2), 313–320 (2009)
Kainourgiakis, M.E., Kikkinides, E.S., Steriotis, T.A., Stubos, A.K., Tzevelekos, K.P., Kanellopoulos, N.K.: Structural and transport properties of alumina porous membranes from process-based and statistical reconstruction techniques. J. Colloid Interface Sci. 231, 158–167 (2000)
Kikkinides, E.S., Politis, M.G.: Linking pore diffusivity with macropore structure of zeolite adsorbents. Part I: three dimensional structural representation combining scanning electron microscopy with stochastic reconstruction methods. Adsorption 20(1), 5–20 (2014a)
Kikkinides, E.S., Politis, M.G.: Linking pore diffusivity with macropore structure of zeolite adsorbents. Part II: simulation of pore diffusion and mercury intrusion in stochastically reconstructed zeolite adsorbents. Adsorption 20(1), 21–35 (2014b)
Latour, L.L., Mitra, P.P., Kleinberg, R.L., Sotak, C.H.: Time-dependent diffusion coefficient of fluids in porous media as a probe of surface-to-volume ratio. J. Magn. Reson. A 101(3), 342–346 (1993)
Latour, L.L., Kleinberg, R.L., Mitra, P.P., Sotak, C.H.: Pore-size distributions and tortuosity in heterogeneous porous media. J. Magn. Reson. A 112(1), 83–91 (1995)
Levitz, P.: Off-lattice reconstruction of porous media: critical evaluation, geometrical confinement and molecular transport. Adv. Colloid Interface Sci. 76–77, 71–106 (1998)
Lymberopoulos, D.P., Payatakes, A.C.: Derivation of topological, geometrical, and correlational properties of porous media from pore-chart analysis of serial section data. J. Colloid Interface Sci. 150(1), 61–80 (1992)
Mair, R.W., Wong, G.P., Hoffmann, D., Hürlimann, M.D., Patz, S., Schwartz, L.M., Walsworth, R.L.: Probing porous media with gas diffusion NMR. Phys. Rev. Lett. 83(16), 3324–3327 (1999)
Masschaele, B., Cnudde, V., Dierick, M., Jacobs, P., Van Hoorebeke, L., Vlassenbroeck, J.: UGCT: new X-ray radiography and tomography facility. Nucl. Instrum. Methods Phys. Res. Sect. A. Accelerators, Spectrometers, Detectors Assoc. Equip. 580(1), 266–269 (2007)
Masschaele, B., Dierick, M., Van Loo, D., Boone, M.N., Brabant, L., Pauwels, E., Cnudde, V., Van Hoorebeke, L.: HECTOR: a 240 kv micro-CT setup optimized for research. J. Phys. Conf. Ser. 463(012), 012 (2013)
Mills, R.: Self-diffusion in normal and heavy water in the range \(1-45^{\circ }\). J. Phys. Chem. 77(5), 685–688 (1973)
Mitra, P.P., Sen, P.N., Schwartz, L.M.: Short-time behavior of the diffusion coefficient as a geometrical probe of porous media. Phys. Rev. B 47(14), 8565–8574 (1993)
Novák, V., Štěpánek, F., Kočí, P., Marek, M., Kubíček, M.: Evaluation of local pore sizes and transport properties in porous catalysts. Chem. Eng. Sci. 65(7), 2352–2360 (2010)
Okabe, H., Blunt, M.J.: Prediction of permeability for porous media reconstructed using multiple-point statistics. Phys. Rev. E 70(6), 066,135 (2004)
Olayinka, S., Ioannidis, M.A.: Time-dependent diffusion and surface-enhanced relaxation in stochastic replicas of porous rock. Transp. Porous Media 54(3), 273–295 (2004)
Øren, P.E., Bakke, S.: Process based reconstruction of sandstones and prediction of transport properties. Transp. Porous Media 46(2–3), 311–343 (2002)
Otsu, N.: A threshold selection method from gray-level histograms. IEEE Trans. Syst. Man Cybern. SMC 9(1), 62–66 (1979)
Park, I.S., Do, D.D., Rodrigues, A.E.: Measurement of the effective diffusivity in porous media by the diffusion cell method. Catal. Rev. Sci. Eng. 38(2), 189–247 (1996)
Price, W.S.: Pulsed-field gradient nuclear magnetic resonance as a tool for studying translational diffusion: Part 1. Basic theory. Concepts Magn. Reson. 9(5), 299–336 (1997)
Reid, R.C., Prausnitz, J.M., Poling, B.E.: The properties of gases and liquids, 4th edn. McGraw-Hill, Boston (1987)
Rozman, M.G., Utz, M.: Efficient reconstruction of multiphase morphologies from correlation functions. Phys. Rev. E 63(6), 066,701 (2001)
Sahimi, M.: Heterogeneous materials I. Linear transport and optical properties, interdisciplinary applied mathematics, vol. 22. Springer, New York (2003)
Sahimi, M.: Flow and transport in porous media and fractured rock: from classical methods to modern approaches, 2nd edn. Wiley-VCH, Weinheim (2011)
Sahimi, M., Stauffer, D.: Efficient simulation of flow and transport in porous media. Chem. Eng. Sci. 46(9), 2225–2233 (1991)
Shearing, P.R., Golbert, J., Chater, R.J., Brandon, N.P.: 3D reconstruction of SOFC anodes using a focused ion beam lift-out technique. Chem. Eng. Sci. 64(17), 3928–3933 (2009)
Sonka, M., Hlavac, V., Boyle, R.: Image processing, analysis, and machine vision, 2nd edn. Brooks/Cole, Pacific Grove (1999)
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(14), 2001–2004 (1994)
Stallmach, F., Galvosas, P.: Spin echo NMR diffusion studies. In: Webb, G.A. (ed.) Annual reports on NMR spectroscopy, vol. 61, pp. 51–131. Elsevier Academic Press Inc., San Diego (2007)
Stallmach, F., Kärger, J.: The potentials of pulsed field gradient NMR for investigation of porous media. Adsorption 5(2), 117–133 (1999)
Tahmasebi, P., Sahimi, M.: Reconstruction of three-dimensional porous media using a single thin section. Phys. Rev. E 85(6), 066,709 (2012)
Tahmasebi, P., Sahimi, M.: Cross-correlation function for accurate reconstruction of heterogeneous media. Phys. Rev. Lett. 110(7), 078,002 (2013)
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(6), 061,307 (2001)
Torquato, S.: Random heterogeneous materials: microstructure and macroscopic properties. Springer, New York (2002)
Torquato, S., Chan, Kim I.: Effective simulation technique to compute effective properties of heterogeneous media. Appl. Phys. Lett. 55(18), 1847–1849 (1989)
Torquato, S., Chan, Kim I., Cule, D.: Effective conductivity, dielectric constant, and diffusion coefficient of digitized composite media via first-passage time equations. J. Appl. Phys. 85(3), 1560–1571 (1999)
Toumelin, E., Torres-Verdín, C., Sun, B., Dunn, K.J.: Random-walk technique for simulating NMR measurements and 2D NMR maps of porous media with relaxing and permeable boundaries. J. Magn. Reson. 188(1), 83–96 (2007)
Valfouskaya, A., Adler, P.M.: Nuclear-magnetic-resonance diffusion simulations in two phases in porous media. Phys. Rev. E 72(5), 056,317 (2005)
Valfouskaya, A., Adler, P.M., Thovert, J.F., Fleury, M.: Nuclear-magnetic-resonance diffusion simulations in porous media. J. Appl. Phys. 97(8), 083,510 (2005)
Valfouskaya, A., Adler, P.M., Thovert, J.F., Fleury, M.: Nuclear magnetic resonance diffusion with surface relaxation in porous media. J. Colloid Interface Sci. 295(1), 188–201 (2006)
Vlassenbroeck, J., Dierick, M., Masschaele, B., Cnudde, V., Van Hoorebeke, L., Jacobs, P.: Software tools for quantification of X-ray microtomography at the UGCT. Nucl. Instrum. Methods Phys. Res. Sect. A. Accelerators Spectrometers Detectors Assoc. Equip. 580(1), 442–445 (2007a)
Vlassenbroeck, J., Masschaele, B., Dierick, M., Cnudde, V., De Witte, Y., Pieters, K., Van Hoorebeke, L., Jacobs, P.: Recent developments in the field of X-ray nano- and micro-CT at the centre for X-ray tomography of the Ghent university. Microsc. Microanal. 13, 184–185 (2007b)
Weickert, J., ter Haar Romeny, B.M., Viergever, M.A.: Efficient and reliable schemes for nonlinear diffusion filtering. IEEE Trans. Image Process. 7(3), 398–410 (1998)
Yeong, C.L.Y., Torquato, S.: Reconstructing random media. Phys. Rev. E 58(1), 495–506 (1998a)
Yeong, C.L.Y., Torquato, S.: Reconstructing random media. II. Three-dimensional media from two-dimensional cuts. Phys. Rev. E 58(1), 224–233 (1998b)
Zachary, C.E., Torquato, S.: Improved reconstructions of random media using dilation and erosion processes. Phys. Rev. E 84(5), 056,102 (2011)
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Financial support (#P204/11/1206) from the Czech Science Foundation, Czech Republic is gratefully acknowledged.
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Veselý, M., Bultreys, T., Peksa, M. et al. Prediction and Evaluation of Time-Dependent Effective Self-diffusivity of Water and Other Effective Transport Properties Associated with Reconstructed Porous Solids. Transp Porous Med 110, 81–111 (2015). https://doi.org/10.1007/s11242-015-0557-y
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DOI: https://doi.org/10.1007/s11242-015-0557-y