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# Stretching and folding sustain microscale chemical gradients in porous media

Edited by Andrea Rinaldo, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland, and approved April 21, 2020 (received for review February 20, 2020)

## Significance

Porous media flows are central to environmental, industrial, and biological systems, where they transport molecules, particles, and microorganisms and trigger important biogeochemical reactions. There is increasing evidence that many of these processes are highly sensitive to chemical gradients below the pore-scale. However, it is unknown how porous architectures control microscale concentration heterogeneities. Here, we provide high-resolution experimental images that fully resolve three-dimensional pore-scale mixing dynamics. We show that grain contacts control the folding of fluid elements in the pore space, which, in addition to fluid stretching at stagnation points, leads to the exponential enhancement of microscale concentration gradients. We use these insights to derive a kinematic model linking mixing rates to pore geometry, opening perspectives for reactive transport modeling.

## Abstract

Fluid flow in porous media drives the transport, mixing, and reaction of molecules, particles, and microorganisms across a wide spectrum of natural and industrial processes. Current macroscopic models that average pore-scale fluctuations into an effective dispersion coefficient have shown significant limitations in the prediction of many important chemical and biological processes. Yet, it is unclear how three-dimensional flow in porous structures govern the microscale chemical gradients controlling these processes. Here, we obtain high-resolution experimental images of microscale mixing patterns in three-dimensional porous media and uncover an unexpected and general mixing mechanism that strongly enhances concentration gradients at pore-scale. Our experiments reveal that systematic stretching and folding of fluid elements are produced in the pore space by grain contacts, through a mechanism that leads to efficient microscale chaotic mixing. These insights form the basis for a general kinematic model linking chaotic-mixing rates in the fluid phase to the generic structural properties of granular matter. The model successfully predicts the resulting enhancement of pore-scale chemical gradients, which appear to be orders of magnitude larger than predicted by dispersive approaches. These findings offer perspectives for predicting and controlling the vast diversity of reactive transport processes in natural and synthetic porous materials, beyond the current dispersion paradigm.

Fluid mixing in porous media plays a key role in a range of natural and industrial systems (1⇓–3). In these confined environments, mixing enables or limits reactions controlling the degradation of contaminants in the subsurface; the cycles of biogeochemical elements such as nitrogen, iron, and carbon; and the sequestration of

Recent theories (19, 20) have suggested that laminar flow through three-dimensional porous media may possess the basic ingredients for chaotic advection (e.g., the exponential deformation of fluid elements), which would represent a possible mechanism for the enhancement of microscale chemical gradients and the persistence of incomplete mixing at the pore-scale. These chaotic dynamics may have particularly important consequences for microbial processes, a broad range of which are hosted in porous environments (21). Biological processes in turbulent flows have been shown to be deeply altered by chaotic advection, which promotes coexistence of competitive microbial species (22) and affects the chemotactic responses of microorganisms (23). However, whether such chaotic dynamics can spontaneously develop in laminar flows through porous media remains on open question.

A key experimental barrier to the direct imaging of solute advection in three-dimensional porous materials is their predominantly opaque nature. While X-ray microtomography technologies have progressed significantly (24), they still cannot resolve the fine structures produced below pore-scale. In contrast, use of visible-spectrum refractive-index matching between the solid and the fluid phases represents a viable alternative to observe solute mixing, as obtained with hydrogel beads in water (25). However, as molecular diffusion eventually masks the deformation of dyed fluid elements, a direct measurement of fluid deformation in random porous media is an outstanding challenge. Here, we overcome these limitations by performing high-resolution laser imaging of the evolution of a low-diffusivity fluorescent dye plume through a column of optically transparent borosilicate spheres via high-precision refractive-index matching (Fig. 1). This technique allows reconstruction of the three-dimensional dye plume at unprecedented resolution, thus providing direct experimental observation of pore-scale fluid deformation and mixing in porous media. These data reveal the hitherto unknown role of grain contacts in controlling folding and stretching of fluid elements, a mechanism that generates strong chaotic advection and significantly enhances chemical gradients at the microscale. Since grain contacts are inherent to all granular porous materials, we deduce that chaotic mixing is ubiquitous in flow through all such materials, potentially impacting a large range of fluid-borne phenomena in natural and engineered systems.

### Three-Dimensional Imaging of Mixing Patterns in Porous Media.

We observed three-dimensional fluid deformation and solute mixing in laminar flows through monodispersed random bead packs of diameters *A*) and advected downstream by the porous flow at the mean longitudinal advection velocity u. Cross-stream concentration patterns of the dye plume are imaged in the pore-space via a translational scan using a laser sheet and a camera. The dye cross-section rapidly evolves into a highly elongated (Fig. 2*B*) and striated filamentous structure (Fig. 2*C*) due to transverse stretching and folding of fluid elements in pores (Movies S1–S3). The combination of a highly viscous fluid mixture and a high-molecular-weight dye results in laminar flows of low diffusivity, characterized by Reynolds numbers on the order of Re = *SI Appendix*, Table S1). The deformation of the dye plume (Fig. 1*A*) thus closely shadows that of the advected fluid, facilitating direct visualization of pore-scale fluid deformation. We use spline fitting on the images to reconstruct the backbone of the cross-sectional dye footprint, called a filament, and estimate its total length

The mean total filament elongation *SI Appendix*, Table S1), exhibits clear exponential growth with normalized longitudinal distance *topological entropy* of the flow (26). Via the central limit theorem, μ is related to the mean λ and variance *Materials and Methods*)*C*). These fluid deformations are the hallmarks of chaotic mixing, thus permitting exponential elongation of material elements in a finite-sized domain.

### The Role of Grain Contacts in Folding.

Folding of dye filaments is consistently initiated downstream of contact points between two beads (Fig. 1*B* and *SI Appendix*, Fig. S1). The cusp-shaped geometry near grain contacts means that when crossing a contact point (Fig. 1, *B.0*), fluid elements are first compressed in the direction joining the two bead centers and stretched in the perpendicular direction. Downstream of the contact point, the direction of compression and stretching are exchanged and a cusp forms locally in the dye filament (Fig. 1, *B.1*). This cusp is stretched in the following pore space, leading to a folded filament made of two straight segments (Fig. 1, *B.2*). This stretching and folding process is repeated sequentially as the folded filament encounters other contact points (Fig. 1, *B.3*), leading to thin solute dye foliations that are the hallmarks of chaotic advection (26) (Fig. 2*C*).

Recent studies (19, 20) identified the role of separation and reattachment points on open grain boundaries (saddle points) in generating exponential stretching of fluid elements. Here, we uncovered the distinct role of contact points between grains in generating systematic folding of fluid elements. Simulations of laminar flow in periodic bead packings (*SI Appendix*, section A) show that attracting and repelling stream surfaces (unstable and stable manifolds) produced by these saddles indeed control stretching of material lines in the pore space. We found that these manifolds intersect orthogonally at grain contacts (*SI Appendix*, Fig. S2 and Movie S5), where both the local flow velocity and the stretching rate vanish and manifold stabilities are exchanged, so that repelling stream surfaces become attracting and vice versa. Hence, over a contact point, the local flow structure imparts finite curvature to fluid elements, which results in the sharp folds observed experimentally (Fig. 1). The repetition of this basic stretching and folding sequence over successive contact points offers a simple geometric framework to relate stretching rates to granular structure.

### Linking Stretching Statistics to the Porous Structure.

Sharp folding of dye filament in between contact points produces a number *Inset*). Hence, the average advection distance between two successive contact points is statistically constant and equal to **1** as

As shown by Eq. **2**, the strength of chaotic advection is entirely governed by the spatial frequency *Materials and Methods* that in isotropic packings, **3** is also applicable to nonisotropic packings with a prefactor that quantifies the distribution of orientations of the contact lines joining bead centers with respect to the mean flow direction (*Materials and Methods*). Insertion of the experimental values **3** yields *Materials and Methods*). Eq. **3** provides a quantitative link between microscopic fluid stretching rates and porous media structural properties.

## Discussion

### Stretching and Folding Sustain Microscale Chemical Gradients.

Repeated sequences of stretching and folding leads to exponential compression of fluid elements that can sustain concentration gradients at the pore-scale (Fig. 2). These concentration gradients are locally controlled by the balance between diffusive spreading rate (

### Microscale Mixing Model.

The experiments in this study have used high Péclet numbers to uncover the rate and kinematics of fluid deformation in porous media. These results may be extended to the prediction of macroscopic mixing rates and concentration statistics at arbitrary Péclet numbers via lamellar mixing models that couple stretching and diffusion (31⇓–33). In *SI Appendix*, section C, we derive such a mixing model and compare its predictions in terms of dye-concentration statistics to the experimental data. The lamellar model successfully captures the measured exponential decay of the mean maximum solute concentration of dye filaments with longitudinal distance *SI Appendix*, section B). From the normalized Batchelor scale (Eq. **5**), pore-scale concentration fluctuations predicted by the lamellar model will persist for all Péclet numbers larger than 5. In this range, macroscopic dispersion models fails to resolve pore-scale concentration gradients, leading to incorrect predictions of a broad range of reactive transport dynamics (3, 16⇓–18). Coupling lamellar mixing models with reactive processes is therefore a promising avenue to capture the effect of pore-scale incomplete mixing on biogeochemical dynamics.

### Porous Materials as Mixers.

From Eq. **3**, the mixing efficiency of steady laminar flows through random bead packs (defined as the ratio of the average stretching rate to the average strain rate) is found to be 3% (*SI Appendix*, section D). This value is comparable to the performance of industrial mixers (26) and an order of magnitude larger than that of microfluidic chaotic mixers (34), thus opening opportunities for exploiting the mixing properties of porous materials. Chaotic advection is known to both increase dispersion transverse to the mean flow direction and retard longitudinal dispersion (35). It also alters the transport of finite-sized particles such as colloids and microorganisms (36) and may thus control their clustering in the pore space and deposition on grain boundaries. In relating stretching rates to the porous microstructure, Eq. **3** offers a possible pathway to the design of engineered porous materials with optimum mixing characteristics. This concept may find important applications in the design of heat exchangers; packed bed filters and reactors, where transverse dispersion and mixing act to enhance process efficiency; and for continuous flow chemistry (15), such as pressure-driven chromatography, where product selectivity and yield strongly depend upon the minimization of longitudinal dispersion. These applications would first require a validation Eq. **3** over a large range of packing geometries.

## Conclusions

Using high-resolution experimental imaging of microscale mixing in three-dimensional granular media, we have demonstrated the existence of efficient stretching and folding of fluid elements at the pore-scale. We use these insights to develop a stochastic model for the prediction of the Lyapunov exponent from the geometric properties of the grain pack and validate this model against experimental observations. The formalization of these observations into a chaotic-mixing model, coupling stretching and diffusion, demonstrates that incomplete mixing persists at pore-scale for Péclet numbers above 5. This model captures the the evolution of microscale chemical gradients, opening perspectives for understanding, predicting, and controlling a large spectrum of physical, chemical, and biological processes, in natural and engineered porous systems.

The discovery of systematic and efficient chaotic mixing in single phase laminar flows through random bead packs—the archetype of porous media—calls for deeper investigation of this phenomenon in a broad range of systems, including polydisperse packings, consolidated soils and rocks, and more complex flows, such as multiphase, inertial, or non-Newtonian flows. While these cases may act to increase the rate of mixing due to the introduction of new flow phenomena, the fundamental kinematics described in this study must persist as they arise from the underlying topology of the grain pack. The investigation of these system-specific modulations of chaotic mixing in porous matter form promising research perspectives.

## Materials and Methods

### Experimental Protocol.

The porous column consists of a vertically oriented rectangular column of cross-section 48 × 48 mm (Fig. 5), containing monodisperse borosilicate glass beads (Sigmund Lindner GmbH) of diameter *SI Appendix*, Table S1. The coordinates of the bead centers are determined via a three-dimensional Hough transform on the image stack obtained by the translational laser scan, where the background fluid fluorescence allows distinguishing the grains. From these coordinates, several structural properties of the porous media are obtained: ϕ, the solid volume fraction (the ratio of volume occupied by the beads over the total column volume);

### Reconstruction of the Dye-Filament Backbone.

In each cross-stream section of the solute dye plume, a one-dimensional backbone of the dye filament is reconstructed via the adjustment of spline curves, and its total length L is computed (Figs. 2 and 6*A*). In experimental run II (Fig. 2 and Movie S1), the tracking is possible until downstream distance *C*), where

### Distribution of Elongations.

The sequential stretching and folding process leading to exponential growth of the total filament backbone length L implies that the length l of a fluid element follows a multiplicative random process (28, 32), such that l grows as *stretching rate* of mean λ and variance

### Number of High-Curvature Regions.

As shown in Fig. 1*B*, when the filament is advected through contact points between beads, localized regions of very high curvature develop in the filament backbone. We define cusps as isolated regions of the filament backbone where the curvature κ of the spline curve exceeds the threshold *C*) that the total number

### Prediction of the Mean Stretching Rates from the Porous Media Properties.

Based on the consistently observed sequence of stretching and folding in the pore space and its control by grain contacts (Fig. 1), we derive a general expression for the magnitude of the Lyapunov exponent in random granular media as a function of the coordination number **2** as *Materials and Methods*). Thus, we estimate **6** and **7** provide the value of the mean fold distance **2**, the Lyapunov exponent reads**8** yields **8** yields

### Data Availability.

All data needed to evaluate the conclusions in this paper are available in the main text or in *SI Appendix*.

## Acknowledgments

This research was funded by the Europe Research Council grant ReactiveFronts (Grant 648377), the Agence Nationale de la Recherche (ANR) project Subsurface Mixing and Reactions (Grant ANR-14-CE04-0003), and the Contrat de Plan État-Région project BUFFON. We thank J. Jimenez-Martinez and B. Geraud for their contributions in the early experimental design and construction.

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: joris.heyman{at}univ-rennes1.fr.

Author contributions: T.L.B. designed research; J.H. performed the experiments; J.H. and R.T. contributed to experimental and numerical tools; J.H., D.R.L., Y.M., and T.L.B. analyzed data; and J.H., D.R.L., Y.M., and T.L.B. wrote the paper.

The authors declare no competing interests.

This article is a PNAS Direct Submission.

This article contains supporting information online at https://www.pnas.org/lookup/suppl/doi:10.1073/pnas.2002858117/-/DCSupplemental.

Published under the PNAS license.

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