Transport in Porous Media

, Volume 128, Issue 1, pp 179–200 | Cite as

Stochastic 3D Modeling of Three-Phase Microstructures for Predicting Transport Properties: A Case Study

  • M. NeumannEmail author
  • B. Abdallah
  • L. Holzer
  • F. Willot
  • V. Schmidt


We compare two conceptually different stochastic microstructure models, i.e., a graph-based model and a pluri-Gaussian model, that have been introduced to model the transport properties of three-phase microstructures occurring, e.g., in solid oxide fuel cell electrodes. Besides comparing both models, we present new results regarding the relationship between model parameters and certain microstructure characteristics. In particular, an analytical expression is obtained for the expected length of triple phase boundary per unit volume in the pluri-Gaussian model. As a case study, we consider 3D image data which show a representative cutout of a solid oxide fuel cell anode obtained by FIB-SEM tomography. The two models are fitted to image data and compared in terms of morphological characteristics (like mean geodesic tortuosity and constrictivity) as well as in terms of effective transport properties. The Stokes flow in the pore phase and effective conductivities in the solid phases are computed numerically for realizations of the two models as well as for the 3D image data using Fourier methods. The local and effective physical responses of the model realizations are compared to those obtained from 3D image data. Finally, we assess the accuracy of the two methods to predict permeability as well as electronic and ionic conductivities of the anode.


Stochastic microstructure modeling Effective conductivity Permeability Solid oxide fuel cells 3D image data 


\(\beta _1, \beta _2, \beta _3\)

Constrictivities of the three phases

\(\varepsilon _1, \varepsilon _2, \varepsilon _3\)

Volume fractions of the three phases

\(\widehat{\varepsilon }\)

Estimator for the volume fraction of a stationary random closed sets

\(\widehat{\varepsilon }^{\star }\)

Estimator for the volume fraction in the graph-based microstructure model

\(\gamma _1, \gamma _2, \gamma _3\)

Parameters of the distance measure used for the graph-based microstructure model

\(\varGamma \)

Pore–solid interface

\(\kappa \,{(}\mathrm {m}^2{)}\)


\(\widehat{\kappa }~{(}\mathrm {m}^2{)}\)

Geometrical predictor of permeability

\(\lambda _1, \lambda _2, \lambda _3~{(}\mathrm {m}^{-3}{)}\)

Intensities of the Poisson point processes

\(\mu _f~{(}{{\mathrm {kg}}}\cdot \mathrm {m}^{-1} \cdot \mathrm {s}^{-1}{)}\)

Viscosity of an incompressible Newtonian fluid

\(\nu _3\)

3-dimensional Lebesgue measure

\(\varPhi \)

probability distribution function of the standard normal distribution

\(\phi ~{(}{\mathrm {kg}}\cdot \mathrm {m}^2 \cdot \mathrm {s}^{-3} \cdot \mathrm {A}^{-1}{)}\)

Electrical potential (or ionic concentration)

\(\rho _Y, \rho _Z\)

Covariance functions of the Gaussian random fields Y and Z

\(\sigma ~{(}{\mathrm {kg}}^{-1}\cdot \mathrm {m}^{-2} \cdot \mathrm {s}^{3} \cdot \mathrm {A}^{2}{)}\)

Effective conductivity

\(\sigma _{\mathrm {sol}}~{(}{\mathrm {kg}}^{-1}\cdot \mathrm {m}^{-2} \cdot \mathrm {s}^{3} \cdot \mathrm {A}^{2}{)}\)

Intrinsic conductivity

\(\tau _1, \tau _2, \tau _3\)

Mean geodesic tortuosities of the three phases

\(\varTheta \)

Parameter space

\(\theta _{ij}~{(}\mathrm {m}^{-1}{)}\)

Parameters for modeling two-point coverage probability functions, \(i,j \in \lbrace 1,2 \rbrace \)

\(\vartheta _0~{(}\mathrm {m}^{-2}{)}, \vartheta _1~{(}\mathrm {m}^{-1}{)}\)

Intensities of point processes related to the triple phase boundary

\(\varXi _1, \varXi _2, \varXi _3\)

Random closed sets denoting the three different phases

\(b_1, b_2, b_3\)

Parameters of the beta-skeletons

\(C_1, C_2, C_3\)

Two-point coverage probability functions of the three phases


Euclidean distance between a point \(x \in \mathbb {R}^3\) and a set \(A \subset \mathbb {R}^3\)

\(d_\gamma (x,A)\)

Distance measure with parameter \(\gamma \) between a point \(x \in \mathbb {R}^3\) and a set \(A \subset \mathbb {R}^3\)

\(\mathbf {E}~{(}{\mathrm {kg}}\cdot \mathrm {m} \cdot \mathrm {s}^{-3} \cdot \mathrm {A}^{-1}{)}\)

Electrical vector field (or opposite gradient of ionic concentration)

\(\mathbf {G}~{(}{\mathrm {kg}}\cdot \mathrm {m}^{-2} \cdot \mathrm {s}^{-2}{)}\)

Macroscopic pressure gradient

\(\mathcal {G}_1, \mathcal {G}_2, \mathcal {G}_3\)

Beta-skeletons of the three phases


Function used to estimate the volume fraction in the graph-based model

\(\mathcal {H}_{k}\)

k-dimensional Hausdorff measure for \(k \in \lbrace 1,2,3 \rbrace \)

\(\mathbf {J}~{(}A{)}\)

Electrical current (or particle current)

\(L_{\mathrm {TPB}}~{(}{\mathrm {m}}^{-2}{)}\)

Expected length of the triple phase boundary per unit volume


M-factor, i.e., the ratio of effective and intrinsic conductivity


Geometrical predictor of the M-factor


Origin in the 3-dimensional Euclidean space

\(p~{(}{\mathrm {kg}}\cdot \mathrm {m}^{-1} \cdot \mathrm {s}^{-2}{)}\)

Pressure field

\(\mathbb {R}^3\)

3-dimensional Euclidean space


Coefficient of determination

\(r_{\mathrm {max}}~{(}\mathrm {m}{)}\)

Median of the volume equivalent particle radius distribution

\(r_{\mathrm {min}}~{(}\mathrm {m}{)}\)

Median radius of the characteristic bottleneck in a microstructure

\({\mathcal {S}}\)

Conductive phase

\(S_1, S_2, S_3\)

Specific surface area of the three phases

\(s_{\mathrm {GBM}}~{(}\mathrm {m}{)}\)

Smoothing parameter of the graph-based microstructure model

\(s_{\mathrm {PGM}}~{(}\mathrm {m}{)}\)

Smoothing parameter of the pluri-Gaussian microstructure model

\(u_Y, u_Z\)

Thresholds defining the excursion sets of the Gaussian random fields Y and Z

\(\mathbf {v}~{(}\mathrm {m} \cdot \mathrm {s}^{-1}{)}\)

Velocity of an incompressible Newtonian fluid

\(X_1, X_2, X_3\)

Homogeneous Poisson point processes


Gaussian random fields

\(\varDelta \)

Laplacian operator

\(\nabla \)

Gradient operator

\(\partial A\)

Boundary of a set \(A \subset \mathbb {R}^3\)


Supplementary material


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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Institute of StochasticsUlm UniversityUlmGermany
  2. 2.MINES ParisTech, Centre for Mathematical MorphologyPSL Research UniversityFontainebleauFrance
  3. 3.Institute of Computational PhysicsZHAW WinterthurWinterthurSwitzerland

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