A Unified Theory for Chaotic Mixing in Porous Media: from Pore Networks to Granular Systems

TL;DR

This work develops a unified theory of chaotic mixing in porous media that spans both continuous and discrete solid phases. By linking pore-scale topology to fluid stretching and folding through advection maps and hyperbolic manifolds, it shows that chaotic mixing arises from a shared mechanism despite differing geometries, including both continuous pore-branch/merger dynamics and discrete grain contacts. The authors derive Lyapunov-based predictions for mixing in both classes, including a general formulation λ∞ = f ln(r) that captures how folding frequency and per-event stretch govern mixing rates, and they quantify discontinuous mixing via the mix-norm Φ(c). The framework enables predictive design of porous architectures with tunable mixing and transport properties, with broad implications for reactive transport, filtration, and biological/geochemical processes in complex porous media.

Abstract

Recent studies have revealed the central role of chaotic stretching and folding at the pore scale in controlling mixing within porous media, whether the solid phase is discrete (as in granular and packed media) or continuous (as in vascular networks and open porous structures). Despite its widespread occurrence, a unified theory of chaotic mixing across these diverse systems remains to be developed. Furthermore, previous studies have focused on fluid stretching mechanisms but the folding mechanisms are largely unknown. We address these shortcomings by presenting a unified theory of mixing in porous media. We thus show that fluid stretching and folding (SF) arise through the same fundamental kinematics driven by the topological complexity of the medium. We find that mixing in continuous porous media manifests as discontinuous mixing through a combination of SF and cutting and shuffling (CS) actions, but the rate of mixing is governed by SF only. Conversely, discrete porous media involves SF motions only. We unify these diverse systems and mechanisms by showing that continuous media represents an analog of discrete media with finite-sized grain contacts. This unified theory provides insights into the generation of pore-scale chaotic mixing and points to design of novel porous architectures with tuneable mixing and transport properties.

Figures (17)

• Figure 1: Various porous networks as examples of continuous porous media: (a) tofu microstructure Huang:2018aa, (b) gyroidal tissue scaffold Melchels:2009aa, (c) ceramic foam (https://filterceramic.com/alu-ceramic-foam-filter), (d) vascular network of the heart Huang:2009aa. (e) Mixing of dyes in a 3D micromixer Therriault:2003aa.
• Figure 2: Various granular matter as examples of discrete porous media: (a) granular sandstone El-Bied:2002aa, (b) corn kernels, (c) packed corks, (d) glass beads. (e) Mixing of a continuously injected dye plume through a random glass bead pack Heyman:2020aa. Fluid is index-matched to the beads and only a few beads are shown in grey at 40% of their true diameter.
• Figure 3: Characteristics of chaotic mixing in discrete porous media. (a) Numerically reconstructed trajectories of tracer particles, taken from PIV experiments within a glass bead pack (adapted from Souzy:2020aa). (b) Numerically computed skin friction field $\mathbf{u}(\mathbf{x})$ over the surface of a sphere for steady 3-D Stokes flow within a bead pack (other spheres not shown) with node $\mathbf{x}_p^n$ (green) and saddle $\mathbf{x}_p^s$ (black) points, and 1D stable $\mathcal{W}_{1\text{D}}^s$ and unstable $\mathcal{W}_{1\text{D}}^u$ manifolds (black lines). Inset: the same sphere with streamlines shown close to the surface, indicating separation of streamlines in the vicinity of the 2D unstable manifold $\mathcal{W}_{2\text{D}}^U$. Image courtesy Regis Turuban, Scuola Internazionale Superiore di Studi Avanzati, Italy. (c) Sequences of experimental dye trace images for steady flow in a random bead pack at different distances $x$ downstream from the injection point in terms of the bead diameter $d$. These images show that bead contacts systematically trigger stretching and folding of fluid elements leading to the formation of sharp cusps in the dye filament. Numbers label fixed spheres and arrows depict directions of fluid stretching (adapted from Heyman:2020aa).
• Figure 4: Characteristics of chaotic mixing in continuous porous media. (a) An archetypal element of an open (continuous) porous network involving a connected pore branch and merger. (b) Numerical simulation of fluid mixing of a diffusive scalar in (a), illustrating the formation of striated material distributions due to fluid stretching and folding which arises at (c) the saddle-type stagnation point ($\mathbf{x}_p^s$) in the skin friction field. (d) Experimental images of dyed fluids at the inlet (top) and outlet (bottom) of the macroscopic analogue of the pore branch and merger shown in (a). (e) Detail of dyed fluid distributions near the macroscopic pore merger and (f) cross-section of the dye distribution exiting the pore merger, which agrees well with the scalar distribution shown in (b) (adapted from Lester:2019aa).
• Figure 5: Schematic of the structure of the skin friction field $\mathbf{u}$ surrounding type I-IV critical points (black dots) on a portion (bounded by the dotted lines) of the fluid boundary $\partial\Omega$ and the associated stable $\mathcal{W}^s$ and unstable $\mathcal{W}^u$ manifolds. The interior 2D manifolds for type III, IV critical points are shown as light blue surfaces. Arrows indicate the eigenvectors of the skin friction gradient tensor, and the double arrows on the streamlines reflect the sum $\eta_1+\eta_2+2\eta_3=0$. Adapted from Lester:2016aa.