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Effective permeabilities for flow through anisotropic microscopic geometries

Loïc Balazi, Fabian Holzberger, Stephan B. Lunowa, Malte A. Peter, Daniel Peterseim, Barbara Wohlmuth

TL;DR

The paper develops a multiscale framework to determine anisotropic effective permeabilities in dense fibre-like microstructures, motivated by blood flow through endovascular coils in aneurysms. By combining homogenisation, Representative Elementary Volumes, and cell-problem formulations, it derives a tensorial permeability that can be rotated to align with fibre orientation. Numerical validation confirms analytical expressions and shows that anisotropy significantly alters local flow direction and magnitude, with the rotated, anisotropic model (K_phi) matching resolved Stokes results more closely than isotropic models. The approach is demonstrated on a realistic coil geometry and is applicable to other fibrous porous media, enabling more realistic macro-scale hemodynamics and related engineering applications.

Abstract

This work develops a computational and theoretical framework for determining effective permeabilities in anisotropic microscopic geometries containing dense, fibre-like obstacles, motivated by the need to model flow in coiled aneurysm domains accurately. Building on homogenisation theory and fully resolved simulations in Representative Elementary Volumes (REVs), we validate the permeability model introduced in [C. Boutin, Study of permeability by periodic and self-consistent homogenisation. Eur. J. Mech. A Solids, 19(4):603-632, 2000] and propose a systematic methodology for capturing the directional variations induced by fibre orientation. The resulting permeability tensors are incorporated into macroscopic flow simulations based on the Darcy equation, enabling direct comparison of anisotropic and isotropic permeability models across several benchmark configurations. Our findings show that anisotropy has a significant impact on local flow direction and magnitude, generating directional permeability contrasts which cannot be reproduced by classical isotropic approximations. By integrating coil-induced microstructural effects into continuum-scale hemodynamic models, the proposed approach enables more realistic assessment of post-treatment aneurysm flow behaviour. Beyond this clinical application, the framework is broadly applicable to other biomedical and engineering systems involving fibrous or filamentous porous microstructures.

Effective permeabilities for flow through anisotropic microscopic geometries

TL;DR

The paper develops a multiscale framework to determine anisotropic effective permeabilities in dense fibre-like microstructures, motivated by blood flow through endovascular coils in aneurysms. By combining homogenisation, Representative Elementary Volumes, and cell-problem formulations, it derives a tensorial permeability that can be rotated to align with fibre orientation. Numerical validation confirms analytical expressions and shows that anisotropy significantly alters local flow direction and magnitude, with the rotated, anisotropic model (K_phi) matching resolved Stokes results more closely than isotropic models. The approach is demonstrated on a realistic coil geometry and is applicable to other fibrous porous media, enabling more realistic macro-scale hemodynamics and related engineering applications.

Abstract

This work develops a computational and theoretical framework for determining effective permeabilities in anisotropic microscopic geometries containing dense, fibre-like obstacles, motivated by the need to model flow in coiled aneurysm domains accurately. Building on homogenisation theory and fully resolved simulations in Representative Elementary Volumes (REVs), we validate the permeability model introduced in [C. Boutin, Study of permeability by periodic and self-consistent homogenisation. Eur. J. Mech. A Solids, 19(4):603-632, 2000] and propose a systematic methodology for capturing the directional variations induced by fibre orientation. The resulting permeability tensors are incorporated into macroscopic flow simulations based on the Darcy equation, enabling direct comparison of anisotropic and isotropic permeability models across several benchmark configurations. Our findings show that anisotropy has a significant impact on local flow direction and magnitude, generating directional permeability contrasts which cannot be reproduced by classical isotropic approximations. By integrating coil-induced microstructural effects into continuum-scale hemodynamic models, the proposed approach enables more realistic assessment of post-treatment aneurysm flow behaviour. Beyond this clinical application, the framework is broadly applicable to other biomedical and engineering systems involving fibrous or filamentous porous microstructures.

Paper Structure

This paper contains 12 sections, 3 theorems, 35 equations, 13 figures, 5 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Derivation and numerical validation of the permeability for anisotropic microscopic geometries
  3. Flow in porous media
  4. Preliminary results
  5. Numerical validation of the permeabilities
  6. Application to a coil in an aneurysm
  7. Numerical validation of the upscaled macroscopic model
  8. Setting 1: Cube cut by cylinders
  9. Setting 2: Cylinder cut by spirals
  10. Setting 3: Realistic coil in an artificial aneurysm
  11. Conclusion
  12. Algorithms for homogenization by spatial averaging

Key Result

Lemma 1

For any real number $\kappa>0$, consider a scaled unit cell $Y^\kappa =(0, \kappa)^N$. Introducing the scaled variable $z:=\kappa y$, $y \in Y$, the scaled permeability matrix satisfies where $\mathbf{K}$ is given by eq:MatrixPermeability and $\mathbf{K}^\kappa$ is defined by the same relation but replacing $Y$ by $Y^\kappa$ in eq:cellpb and eq:MatrixPermeability.

Figures (13)

  • Figure 1: Comparisons between computed permeabilities and analytical permeabilities \ref{['eq:Klong']}, \ref{['eq:Kperp_mod']} and \ref{['eq:Kiso']} for arrangements of cylinders of radius $R=0.1$.
  • Figure 2: Comparisons of permeabilities for one cylinder of different radii.
  • Figure 3: The coil under study.
  • Figure 4: The 12 chosen REVs and their respective porosities $\phi$.
  • Figure 5: Comparison of permeabilities for REVs from the real coil geometry. The analytical permeabilities are computed with $R = R_{\rm coil} = 0.178$ (Framing Coil).
  • ...and 8 more figures

Theorems & Definitions (9)

  • Lemma 1
  • proof
  • Remark 2
  • Lemma 3
  • proof
  • Remark 4
  • Lemma 5
  • proof
  • Remark 6