Refinement of a Poroelastic Model for Zero Porosity: Finite Element Implementation and Investigation of Fluid Mechanics in the Perivascular Space

TL;DR

This work addresses a fundamental limitation of conventional poroelasticity, where vanishing porosity makes the fluid-motion equations ill-posed. It develops a mixture-theory–based reformulation that remains well-posed as $\phi_f\to 0$, verifies the approach with the method of manufactured solutions, and implements a finite-element method within an ALE framework. As a testbed, the authors apply the model to peristaltic flow in the brain's perivascular space, showing that literature parameters can push the system toward extreme, nonphysical zero-porosity states unless tissue deformation is accounted for. The results demonstrate that a deformable solid framework is essential to capture fluid-structure interactions in CNS transport and highlight the inadequacy of purely Darcy-based models for these multiphysics problems.

Abstract

In conventional formulations of poroelasticity, when the porosity approaches zero or vanishes in some parts of the poroelastic domain, if only temporarily, the governing equations degenerate to those for the solid phase thereby inhibiting a suitable determination of the fluid velocity field. To address this challenge, we reformulated a poroelastic model based on mixture theory to accommodate scenarios with zero porosity. We verified our model using the method of manufactured solutions and demonstrated its ability to handle extreme conditions in a sample test problem. As an application of our framework, we investigated peristaltic flow in the perivascular space of a penetrating arteriole in brain. Our analysis revealed that some literature-suggested parameters can drive the model to predict extreme non-physiological conditions. We further demonstrated that these extreme conditions can be somewhat mitigated by accounting for the deformation of the surrounding brain tissue.

Figures (11)

• Figure 1: Current configuration $B_t$ of a biphasic mixture, along with the reference configurations of solid and fluid components, $B_\mathrm{s}$ and $B_\mathrm{f}$, respectively. Positions within $B_\mathrm{s}$ and $B_\mathrm{f}$ will be labeled as $\boldsymbol{\mathbf{X}}_\mathrm{s}$ and $\boldsymbol{\mathbf{X}}_\mathrm{f}$, respectively, while $\boldsymbol{\mathbf{x}}$ signifies the position within $B_t$. The boundaries of $B_\mathrm{s}$, $B_\mathrm{f}$, and $B_t$ are $\partial B_\mathrm{s}$, $\partial B_\mathrm{f}$, and $\partial B_t$, respectively. These boundaries are oriented by their respective outward unit normal fields $\boldsymbol{\mathbf{n}}_\mathrm{s}$, $\boldsymbol{\mathbf{n}}_\mathrm{f}$, and $\boldsymbol{\mathbf{n}}$.
• Figure 2: Plot of $\kappa_\mathrm{s}/\overline{\kappa}_\mathrm{s}$ as a function of $\phi_{\mathrm{f}}/\phi_{\mathrm{f}_\mathrm{c}}$ to illustrate the behavior of the permeability $\kappa_\mathrm{s}$ of the solid component as defined in Eq. \ref{['equ:definition_ks']}. The permeability remains constant when $\phi_{\mathrm{f}} > \phi_{\mathrm{f}_\mathrm{c}}$.
• Figure 3: Model verification via the method of manufactured solutions, with $\phi_{f} = \np{0.5}$ in the first row and $\phi_{f} = \np{0}$ in the second row. \ref{['fig:mms phif=0.5']} and \ref{['fig:mms phif=0']} show radial and axial components of the velocity field for the solid (left) and fluid phase (right) for $\phi_{f} = \np{0.5}$ and $\phi_{f} = \np{0}$, respectively. As expected, both components of the fluid velocity converge to those of the solid phase under the limit case condition. \ref{['fig:conv_rates phif=0.5']} and \ref{['fig:conv_rates phif=0']} represent convergence rates for relative $\triangle$-norm (all fields' $L^2$-norm, pressure's and solid displacement's $H^1$-semi-norms, and fluid velocity's $H^{div}$-semi-norm) of the error $e^\square_\triangle=\|\square-\blacksquare\|_\triangle/\|\square\|_\triangle$ between the numerical solution $\blacksquare$ and exact solution $\square$ at time $t = \np[s]{5e-2}$ for two cases of $\phi_{f} = \np{0.5}$ and $\phi_{f} = \np{0}$, respectively, obtained using P1 interpolation for all fields and a cubic bubble augmentation for the solid displacement (i.e., a MINI element Arnold1984AStableFiniteEl).
• Figure 4: Squeezing test for a full cylinder modeled as a 2D axisymmetric domain. \ref{['fig:ExampleProblemSetup']} Problem setup. \ref{['fig:ExampleProblemInitial']} Initial and \ref{['fig:ExampleProblemFinal']} deformed configuration. \ref{['fig:ExampleProblemPorosityTime']} Evolution of porosity over time at a point located at the center of the squeezing zone along the symmetry axis, i.e. at the coordinate $\bigl(\np{0},\left(Z_0+Z_1\right)/2\bigr)$, and the average porosity over the radial line passing through the same point, as shown in Fig. \ref{['fig:ExampleProblemSetup']} in red color. In the initial state, the fluid is homogeneously distributed. Under the applied traction, the fluid redistributes, resulting in fluid-free regions and regions where fluid and solid coexist.
• Figure 5: \ref{['fig:ExampleProblem Filtration Velocity Norm']} Magnitude of the filtration velocity $\|\boldsymbol{\mathbf{v}}_\mathrm{flt}\|$ at $t=\np[s]{2}$. It can be observed that the filtration velocity approaches zero wherever the porosity vanishes. \ref{['fig:ExampleProblem Relative Velocity Norm']} Magnitude of the relative velocity $\|\boldsymbol{\mathbf{v}}_\mathrm{f} - \boldsymbol{\mathbf{v}}_\mathrm{s}\|$ at $t = \np[s]{2}$. In regions where the porosity vanishes, the fluid and solid velocities tend to converge. It is worth noting that in regions far from the squeezing zone, both the solid and fluid velocities are zero. \ref{['fig:ExampleProblem Traditional Formulation']} The porosity $\phi_\mathrm{f}$ obtained using the traditional poroelastic formulation. The simulation stops computing in the middle ($t=\np[s]{5e-2}$) because the traditional formulation is numerically unstable as the porosity becomes very small ($\np{5e-4}$).

Theorems & Definitions (10)

• Remark 1: Concerning $\nabla$ and $\nabla\!\cdot\!$
• Remark 2: The need for a modified model for vanishing porosity
• Remark 3: Concerning traction boundary conditions
• Remark 4: The filtration velocity and its dependence on porosity
• Remark 5: On the boundedness of physical quantities
• Remark 6: On the regularity of $\boldsymbol{\mathbf{u}}_{\mathrm{s}}$
• Remark 7: On the regularity of $p$
• Remark 8: Consistency with the strong form
• Remark 9: On the unusual presentation of the weak forms
• Remark 10: On finite element spaces