Microscale Chaotic Mixing as a Driver for Chemical Reactions in Porous Media.

Mixing-induced reactions play a key role in a large range of biogeochemical and contaminant transport processes in the subsurface. Fluid flow through porous media was recently shown to exhibit chaotic mixing dynamics at the pore scale, enhancing microscale concentration gradients and controlling mixing rates. While this phenomenon is likely ubiquitous in environmental systems, it is not known how it affects chemical reactions. Here, we use refractive index matching and laser-induced fluorescence imaging of a bimolecular redox reaction to investigate the consequence of pore scale chaotic mixing on the reaction rates. The overestimation of measured reaction rates by the classical macrodispersion model highlights the persistence of incomplete mixing on the pore scale. We show that the reaction product formation is controlled by microscale chaotic mixing, which induces an exponential increase of the mixing interface and of the reaction rates. We derive a reactive transport model that captures experimental results and predicts that chaotic mixing has a first order control on reaction rates across a large range of time scales and Péclet and Damköhler numbers. These findings provide a new framework for understanding, assessing, and predicting mixing-induced reactions and their role on the fate and mobility of environmental compounds in natural porous media.

Fluid-Solid Reaction in Porous Media as a Chaotic Restart Process.

Chemical and biological reactions at fluid-solid interfaces are central to a broad range of porous material applications and research. Pore-scale solute transport limitations can reduce reaction rates, with marked consequences for a wide spectrum of natural and engineered processes. Recent advances show that chaotic mixing occurs spontaneously in porous media, but its impact on surface reactions is unknown. We show that pore-scale chaotic mixing significantly increases reaction efficiency compared to nonchaotic flows. We find that reaction rates are well described in terms of diffusive first-passage times of reactants to the solid interface subjected to a stochastic restart process resulting from Lagrangian chaos. Under chaotic mixing, the shear layer at no-slip interfaces sets the restart rate and leads to a characteristic scaling of reaction efficiency with Péclet number, in excellent agreement with numerical simulations. Reaction rates are insensitive to the flow topology as long as flow is chaotic, suggesting the relevance of this process to a broad range of porous materials.

Stretching and folding sustain microscale chemical gradients in porous media.

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.

Pore-Scale Mechanisms for Spectral Induced Polarization of Calcite Precipitation Inferred from Geo-Electrical Millifluidics.

Spectral induced polarization (SIP) has the potential for monitoring reactive processes in the subsurface. While strong SIP responses have been measured in response to calcite precipitation, their origin and mechanism remain debated. Here we present a novel geo-electrical millifluidic setup designed to observe microscale reactive transport processes while performing SIP measurements. We induced calcite precipitation by injecting two reactive solutions into a porous medium, which led to highly localized precipitates at the mixing interface. Strikingly, the amplitude of the SIP response increased by 340% during the last 7% increase in precipitate volume. Furthermore, while the peak frequency in SIP response varied spatially over 1 order of magnitude, the crystal size range was similar along the front, contradicting assumptions in the classical grain polarization model. We argue that the SIP response of calcite precipitation in such mixing fronts is governed by Maxwell-Wagner polarization due to the establishment of a precipitate wall. Numerical simulations of the electric field suggested that spatial variation in peak frequency was related to the macroscopic shape of the front. These findings provide new insights into the SIP response of calcite precipitation and highlight the potential of geoelectrical millifluidics for understanding and modeling electrical signatures of reactive transport processes.

Kinetic Study on Clogging of a Geothermal Pumping Well Triggered by Mixing-Induced Biogeochemical Reactions.

The sustainability of ground-source geothermal systems can be severely impacted by microbially mediated clogging processes. Biofouling of water wells by hydrous ferric oxide is a widespread problem. Although the mechanisms and critical environmental factors associated with clogging development are widely recognized, effects of mixing processes within the wells and time scales for clogging processes are not well characterized. Here we report insights from a joint hydrological, geochemical, and metagenomics characterization of a geothermal doublet in which hydrous ferric oxide and hydrous manganese oxide deposits had formed as a consequence of mixing shallow groundwater containing dissolved oxygen and nitrate with deeper, anoxic groundwater containing dissolved iron (FeII) and manganese (MnII). Metagenomics identify distinct bacteria consortia in the pumping well oxic and anoxic zones, including autotrophic iron-oxidizing bacteria. Batch mixing experiments and geochemical kinetics modeling of the associated reactions indicate that FeII and MnII oxidation are slow compared to the residence time of water in the pumping well; however, adsorption of FeII and MnII by accumulated hydrous ferric oxide and hydrous manganese oxide in the well bore and pump riser provides "infinite" time for surface-catalyzed oxidation and a convenient source of energy for iron-oxidizing bacteria, which colonize the surfaces and also catalyze oxidation. Thus, rapid clogging is caused by mixing-induced redox reactions and is exacerbated by microbial activity on accumulated hydrous oxide surfaces.

Mixing and reaction kinetics in porous media: an experimental pore scale quantification.

We propose a new experimental set up to characterize mixing and reactive transport in porous media with a high spatial resolution at the pore scale. The analogous porous medium consists of a Hele-Shaw cell containing a single layer of cylindrical solid grains built by soft lithography. On the one hand, the measurement of the local, intrapore, conservative concentration field is done using a fluorescent tracer. On the other hand, considering a fast bimolecular reaction A + B → C occurring as A displaces B, we quantify the rate of product formation from the spatially resolved measurement of the pore scale reaction rate, using a chemiluminescent reaction. The setup provides a dynamical measurement of the local concentration field over 3 orders of magnitude and allows investigating a wide range of Péclet and Damköhler numbers by varying the flow rate within the cell and the local reaction rate. We use it to study the kinetics of the reaction front between A and B. While the advection-dispersion (Fickian) theory, applied at the continuum scale, predicts a scaling of the cumulative mass of product C as MC ∝ √t, the experiments exhibit two distinct regimes in which the produced mass MC evolves faster than the Fickian behavior. In both regimes the front rate of product formation is controlled by the geometry of the mixing interface between the reactants. Initially, the invading solute is organized in stretched lamellae and the reaction is limited by mass transfer across the lamella boundaries. At longer times the front evolves into a second regime where lamellae coalesce and form a mixing zone whose temporal evolution controls the rate of product formation. In this second regime, the produced mass of C is directly proportional to the volume of the mixing zone defined from conservative species. This interesting property is indeed verified from a comparison of the reactive and conservative data. Hence, for both regimes, the direct measurement of the spatial distribution of the pore scale reaction rate and conservative component concentration is shown to be crucial to understanding the departure from the Fickian scaling as well as quantifying the basic mechanisms that govern the mixing and reaction dynamics at the pore scale.

Ferriphaselus amnicola strain GF-20, a new iron- and thiosulfate-oxidizing bacterium isolated from a hard rock aquifer.

Ferriphaselus amnicola GF-20 is the first Fe-oxidizing bacterium isolated from the continental subsurface. It was isolated from groundwater circulating at 20 m depth in the fractured-rock catchment observatory of Guidel-Ploemeur (France). Strain GF-20 is a neutrophilic, iron- and thiosulfate-oxidizer and grows autotrophically. The strain shows a preference for low oxygen concentrations, which suggests an adaptation to the limiting oxygen conditions of the subsurface. It produces extracellular stalks and dreads when grown with Fe(II) but does not secrete any structure when grown with thiosulfate. Phylogenetic analyses and genome comparisons revealed that strain GF-20 is affiliated with the species F. amnicola and is strikingly similar to F. amnicola strain OYT1, which was isolated from a groundwater seep in Japan. Based on the phenotypic and phylogenetic characteristics, we propose that GF-20 represents a new strain within the species F. amnicola.

Stretching, coalescence, and mixing in porous media.

We study scalar mixing in heterogeneous conductivity fields, whose structural disorder varies from weak to strong. A range of stretching regimes is observed, depending on the level of structural heterogeneity, measured by the log-conductivity field variance. We propose a unified framework to quantify the overall concentration distribution predicting its shape and rate of deformation as it progresses toward uniformity in the medium. The scalar mixture is represented by a set of stretched lamellae whose rate of diffusive smoothing is locally enhanced by kinematic stretching. Overlap between the lamellae is enforced by confinement of the scalar line support within the dispersion area. Based on these elementary processes, we derive analytical expressions for the concentration distribution, resulting from the interplay between stretching, diffusion, and random overlaps, holding for all field heterogeneities, residence times, and Péclet numbers.

Flow intermittency, dispersion, and correlated continuous time random walks in porous media.

We study the intermittency of fluid velocities in porous media and its relation to anomalous dispersion. Lagrangian velocities measured at equidistant points along streamlines are shown to form a spatial Markov process. As a consequence of this remarkable property, the dispersion of fluid particles can be described by a continuous time random walk with correlated temporal increments. This new dynamical picture of intermittency provides a direct link between the microscale flow, its intermittent properties, and non-Fickian dispersion.

Electrical Signatures of Diffusion-Limited Mixing: Insights from a Milli-fluidic Tracer Experiment.

We investigate how diffusion-limited mixing of a layered solute concentration distribution within a porous medium impacts bulk electrical conductivity. To do so, we perform a milli-fluidic tracer test by injecting a fluorescent and electrically conductive tracer in a quasi two-dimensional (2D) water-saturated porous medium. High resolution optical- and geoelectrical monitoring of the tracer is achieved by using a fluorimetry technique and equipping the flow cell with a resistivity meter, respectively. We find that optical and geoelectrical outputs can be related by a temporal re-scaling that accounts for the different diffusion rates of the optical and electrical tracers. Mixing-driven perturbations of the electrical equipotential field lines cause apparent electrical conductivity time-series, measured perpendicularly to the layering, to peak at times that are in agreement with the diffusion transport time-scale associated with the layer width. Numerical simulations highlight high sensitivity of such electrical data to the layers' degree of mixing and their distance to the injection electrodes. Furthermore, the electrical data correlate well with time-series of two commonly used solute mixing descriptors: the concentration variance and the scalar dissipation rate.

Spatial Markov model of anomalous transport through random lattice networks.

Flow through lattice networks with quenched disorder exhibits a strong correlation in the velocity field, even if the link transmissivities are uncorrelated. This feature, which is a consequence of the divergence-free constraint, induces anomalous transport of passive particles carried by the flow. We propose a Lagrangian statistical model that takes the form of a continuous time random walk with correlated velocities derived from a genuinely multidimensional Markov process in space. The model captures the anomalous (non-Fickian) longitudinal and transverse spreading, and the tail of the mean first-passage time observed in the Monte Carlo simulations of particle transport. We show that reproducing these fundamental aspects of transport in disordered systems requires honoring the correlation in the Lagrangian velocity.

Anomalous kinetics in diffusion limited reactions linked to non-Gaussian concentration probability distribution function.

We investigate anomalous reaction kinetics related to segregation in the one-dimensional reaction-diffusion system A + B → C. It is well known that spatial fluctuations in the species concentrations cause a breakdown of the mean-field behavior at low concentration values. The scaling of the average concentration with time changes from the mean-field t(-1) to the anomalous t(-1/4) behavior. Using a stochastic modeling approach, the reaction-diffusion system can be fully characterized by the multi-point probability distribution function (PDF) of the species concentrations. Its evolution is governed by a Fokker-Planck equation with moving boundaries, which are determined by the positivity of the species concentrations. The concentration PDF is in general non-Gaussian. As long as the concentration fluctuations are small compared to the mean, the PDF can be approximated by a Gaussian distribution. This behavior breaks down in the fluctuation dominated regime, for which anomalous reaction kinetics are observed. We show that the transition from mean field to anomalous reaction kinetics is intimately linked to the evolution of the concentration PDF from a Gaussian to non-Gaussian shape. This establishes a direct relationship between anomalous reaction kinetics, incomplete mixing and the non-Gaussian nature of the concentration PDF.

Network-driven anomalous transport is a fundamental component of brain microvascular dysfunction.

Blood microcirculation supplies neurons with oxygen and nutrients, and contributes to clearing their neurotoxic waste, through a dense capillary network connected to larger tree-like vessels. This complex microvascular architecture results in highly heterogeneous blood flow and travel time distributions, whose origin and consequences on brain pathophysiology are poorly understood. Here, we analyze highly-resolved intracortical blood flow and transport simulations to establish the physical laws governing the macroscopic transport properties in the brain micro-circulation. We show that network-driven anomalous transport leads to the emergence of critical regions, whether hypoxic or with high concentrations of amyloid-ß, a waste product centrally involved in Alzheimer's Disease. We develop a Continuous-Time Random Walk theory capturing these dynamics and predicting that such critical regions appear much earlier than anticipated by current empirical models under mild hypoperfusion. These findings provide a framework for understanding and modelling the impact of microvascular dysfunction in brain diseases, including Alzheimer's Disease.

Concentration statistics for transport in random media.

We study the ensemble statistics of the particle density in a random medium whose mean transport dynamics describes a continuous time random walk. Starting from a Langevin equation for the particle motion in a single disorder realization, we derive evolution equations for the n-point moments of concentration by coarse graining and ensemble averaging the microscale transport problem. The governing equations describe multidimensional continuous time random walks whose waiting time distribution is given in terms of the disorder distribution. We find that the concentration is not self-averaging even for normal mean behavior. The relative concentration variance for anomalous is larger than for normal mean behavior. These results may have some impact on risk and extreme value analysis in stochastic dynamic systems.

Spatial Markov processes for modeling Lagrangian particle dynamics in heterogeneous porous media.

We investigate the representation of Lagrangian velocities in heterogeneous porous media as Markov processes. We use numerical simulations to show that classical descriptions of particle velocities using Markov processes in time fail because low velocities are much more strongly correlated in time than high velocities. We demonstrate that Lagrangian velocities describe a Markov process at fixed distances along the particle trajectories (i.e., a spatial Markov process). This remarkable property has significant implications for modeling effective transport in heterogeneous velocity fields: (i) the spatial Lagrangian velocity transition densities are sufficient to fully characterize these complex velocity field organizations, (ii) classical effective transport descriptions that rely on Markov processes in time for the particle velocities are not suited for describing transport in heterogeneous porous media, and (iii) an alternative effective transport description derives from the Markovian nature of the spatial velocity transitions. It expresses particle movements as a random walk in space time characterized by a correlated random temporal increment and thus generalizes the continuous time random walk model to transport in correlated velocity fields.

Effective transport in random shear flows.

We obtain an effective transport description for the superdiffusive motion of random walkers in stratified flow by projection of the process on the direction of stratification. The effective dimensionally reduced motion is shown to describe a correlated random walk characterized by the Lagrangian velocity correlation. We analyze the projected motion through exact analytical solutions for the distribution density for an arbitrary correlated Gaussian noise and derive an evolution equation for the one-point and conditional two-point displacement densities. The latter gives an explicit effective equation for superdiffusive transport in stratified random flow and demonstrates that the displacement density has a Gaussian scaling form for all times.

Fluid Deformation in Random Steady Three Dimensional Flow.

The deformation of elementary fluid volumes by velocity gradients is a key process for scalar mixing, chemical reactions and biological processes in flows. Whilst fluid deformation in unsteady, turbulent flow has gained much attention over the past half century, deformation in steady random flows with complex structure - such as flow through heterogeneous porous media - has received significantly less attention. In contrast to turbulent flow, the steady nature of these flows constrains fluid deformation to be anisotropic with respect to the fluid velocity, with significant implications for e.g. longitudinal and transverse mixing and dispersion. In this study we derive an ab initio coupled continuous time random walk (CTRW) model of fluid deformation in random steady three-dimensional flow that is based upon a streamline coordinate transform which renders the velocity gradient and fluid deformation tensors upper-triangular. We apply this coupled CTRW model to several model flows and find these exhibit a remarkably simple deformation structure in the streamline coordinate frame, facilitating solution of the stochastic deformation tensor components. These results show that the evolution of longitudinal and transverse fluid deformation for chaotic flows is governed by both the Lyapunov exponent and power-law exponent of the velocity PDF at small velocities, whereas algebraic deformation in non-chaotic flows arises from the intermittency of shear events following similar dynamics as that for steady two-dimensional flow.

Cross-borehole flowmeter tests for transient heads in heterogeneous aquifers.

Cross-borehole flowmeter tests have been proposed as an efficient method to investigate preferential flowpaths in heterogeneous aquifers, which is a major task in the characterization of fractured aquifers. Cross-borehole flowmeter tests are based on the idea that changing the pumping conditions in a given aquifer will modify the hydraulic head distribution in large-scale flowpaths, producing measurable changes in the vertical flow profiles in observation boreholes. However, inversion of flow measurements to derive flowpath geometry and connectivity and to characterize their hydraulic properties is still a subject of research. In this study, we propose a framework for cross-borehole flowmeter test interpretation that is based on a two-scale conceptual model: discrete fractures at the borehole scale and zones of interconnected fractures at the aquifer scale. We propose that the two problems may be solved independently. The first inverse problem consists of estimating the hydraulic head variations that drive the transient borehole flow observed in the cross-borehole flowmeter experiments. The second inverse problem is related to estimating the geometry and hydraulic properties of large-scale flowpaths in the region between pumping and observation wells that are compatible with the head variations deduced from the first problem. To solve the borehole-scale problem, we treat the transient flow data as a series of quasi-steady flow conditions and solve for the hydraulic head changes in individual fractures required to produce these data. The consistency of the method is verified using field experiments performed in a fractured-rock aquifer.

Coupled continuous-time random walks for fluid stretching in two-dimensional heterogeneous media.

We study the relation between flow structure and fluid deformation in steady flows through two-dimensional heterogeneous media, which are characterized by a broad spectrum of stretching behaviors, ranging from sub- to superlinear. We analyze these behaviors from first principles, which uncovers intermittent shear events to be at the origin of subexponential stretching. We derive explicit expressions for Lagrangian deformation and demonstrate that stretching obeys a coupled continuous-time random walk, which for broad distributions of flow velocities becomes a Lévy walk. The derived model provides a direct link between the flow and deformation statistics, and a natural way to quantify the impact of intermittent shear events on the stretching behavior.

Anomalous transport on regular fracture networks: Impact of conductivity heterogeneity and mixing at fracture intersections.

We investigate transport on regular fracture networks that are characterized by heterogeneity in hydraulic conductivity. We discuss the impact of conductivity heterogeneity and mixing within fracture intersections on particle spreading. We show the emergence of non-Fickian transport due to the interplay between the network conductivity heterogeneity and the degree of mixing at nodes. Specifically, lack of mixing at fracture intersections leads to subdiffusive scaling of transverse spreading but has negligible impact on longitudinal spreading. An increase in network conductivity heterogeneity enhances both longitudinal and transverse spreading and leads to non-Fickian transport in longitudinal direction. Based on the observed Lagrangian velocity statistics, we develop an effective stochastic model that incorporates the interplay between Lagrangian velocity correlation and velocity distribution. The model is parameterized with a few physical parameters and is able to capture the full particle transition dynamics.