Transport in Porous Media

, Volume 110, Issue 1, pp 81–111 | Cite as

Prediction and Evaluation of Time-Dependent Effective Self-diffusivity of Water and Other Effective Transport Properties Associated with Reconstructed Porous Solids

  • Martin Veselý
  • Tom Bultreys
  • Mikuláš Peksa
  • Jan Lang
  • Veerle Cnudde
  • Luc Van Hoorebeke
  • Milan Kočiřík
  • Vladimír Hejtmánek
  • Olga Šolcová
  • Karel Soukup
  • Kirill Gerke
  • Frank Stallmach
  • Pavel Čapek


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.


X-ray computed microtomography Stochastic reconstruction Random-walk simulation Pulsed field gradient NMR Viscous flow Isobaric counter-current diffusion Knudsen flow 

List of symbols


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


Diffusivity (scalar)

\({{\textsf {\textit{D}}}}\)

Diffusivity tensor


Diagonal components of the diffusivity tensor

\({{\textsf {\textit{D}}}}^k\)

Knudsen diffusivity tensor


Diagonal components of the Knudsen diffusivity tensor


Binary diffusivity or self-diffusivity in the bulk fluid


Diameter of cylindrical pellet


“Energy” of the digitised system (6)


Contribution to E


Contribution to E


Contribution to E


Gradient amplitude


Maximum gradient amplitude


Reading gradient amplitude


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


Number of voxels in \({\mathbb {V}}\) measured along direction i

\({\mathbb {M}}\)

Cubic measurement cell related to the local percolation theory


Molar mass of a gas


Total number of \({\mathbb {M}}\) randomly thrown into \({\mathbb {V}}\)


Number of FPC octants occupied by a fluid


Number of spatial dimensions



\({\mathscr {P}}\)

Pore-size probability density function

\({\mathscr {Q}}_i\)

Total fraction of measurement cells percolating along direction i


Gas constant (here 8.31441 J mol\(^{-1}\) K\(^{-1}\))

\(\mathbf {r}\)

Position vector associated with the random walker


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


Pore wall area per unit volume of \({\mathbb {V}}\)


BET-specific surface area (per unit volume of \({\mathbb {V}}\))


Thermodynamic temperature




Longitudinal relaxation time

\(\mathbf {u}\)

Vector of relative displacement \(\mathbf {u} = \mathbf {x}_2 - \mathbf {x}_1\)


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)



Financial support (#P204/11/1206) from the Czech Science Foundation, Czech Republic is gratefully acknowledged.

Supplementary material

11242_2015_557_MOESM1_ESM.pdf (441 kb)
Supplementary material 1 (pdf 442 KB)


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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • Martin Veselý
    • 1
  • Tom Bultreys
    • 2
  • Mikuláš Peksa
    • 3
    • 4
  • Jan Lang
    • 3
  • Veerle Cnudde
    • 2
  • Luc Van Hoorebeke
    • 5
  • Milan Kočiřík
    • 6
  • Vladimír Hejtmánek
    • 7
  • Olga Šolcová
    • 7
  • Karel Soukup
    • 7
  • Kirill Gerke
    • 8
  • Frank Stallmach
    • 4
  • Pavel Čapek
    • 1
  1. 1.Faculty of Chemical TechnologyUniversity of Chemistry and Technology, PraguePrague 6Czech Republic
  2. 2.Department of Geology and Soil Science – UGCTGhent UniversityGentBelgium
  3. 3.Faculty of Mathematics and PhysicsCharles University in PraguePrague 2Czech Republic
  4. 4.Faculty of Physics and Earth SciencesUniversity of LeipzigLeipzigGermany
  5. 5.Department of Physics and Astronomy – UGCTGhent UniversityGhentBelgium
  6. 6.J. Heyrovský Institute of Physical Chemistry of AS CRPrague 8Czech Republic
  7. 7.Institute of Chemical Process Fundamentals of AS CRPrague 6Czech Republic
  8. 8.CSIRO Land and WaterGlen OsmondAustralia

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