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At all scales, porous materials stir interstitial fluids as they are advected, leading to complex distributions of matter and energy. Of particular interest is whether porous media naturally induce chaotic advection at the Darcy scale, as these stirring kinematics profoundly impact basic processes such as solute transport and mixing, colloid transport and deposition and chemical, geochemical and biological reactivity. While many studies report complex transport phenomena characteristic of chaotic advection in heterogeneous Darcy flow, it has also been shown that chaotic dynamics are prohibited in a large class of Darcy flows. In this study we rigorously establish that chaotic advection is inherent to steady 3D Darcy flow in all realistic models of heterogeneous porous media. Anisotropic and heterogenous 3D hydraulic conductivity fields generate non-trivial braiding of stream-lines, leading to both chaotic advection and (purely advective) transverse dispersion. We establish that steady 3D Darcy flow has the same topology as unsteady 2D flow, and so use braid theory to establish a quantitative link between transverse dispersivity and Lyapunov exponent in heterogeneous Darcy flow. We show that chaotic advection and transverse dispersion occur in both anisotropic weakly heterogeneous and in heterogeneous weakly anisotropic conductivity fields, and that the quantitative link between these phenomena persists across a broad range of conductivity anisotropy and heterogeneity. The ubiquity of macroscopic chaotic advection has profound implications for the myriad of processes hosted in heterogeneous porous media and calls for a re-evaluation of transport and reaction methods in these systems., Comment: 35 pages, 8 figures

Periodic forcing of flow in compressible porous media is an important driver for solute dispersion and mixing in geological and engineered porous media subject for example to tides, pumping and recharge cycles, or fluid injection and withdrawal cycles with a wide range of environmental and industrial applications. The combination of periodic forcing, spatial medium heterogeneity and medium compressibility leads to intricate spatio-temporal flow, dispersion and mixing patterns. We analyze these patterns using detailed numerical simulations based on a stochastic representation of the spatial medium heterogeneity. Solute dispersion is characterized by the interface length and width, mixing in terms of the dilution index and the distribution of concentration point values. Poincar\'e maps show how the interplay of heterogeneity and compressibility leads to the creation of stable regions that inhibit the advancement and dispersion of the mixing interface, and chaotic regions that at the same time enhance solute mixing. This means that spatial heterogeneity in combination with temporal forcing can lead to the containment of solute and at the same time promote mixing.

Conceptualizing a porous media as a network of conductors sets a compromise between the oversimplifying conceptualization of the media as a bundle of capillary tubes and the computationally expensive and unobtainable detailed description of the media's geometry needed for direct numerical simulations. These models are abundantly being used to evaluate single and multiphase flow characteristics. The different flow characteristics are valuable in evaluating phenomena that may or may not be relevant for different applications. Here, we evaluate how different information about the pore space affects the ability of the network model to evaluate different flow characteristics.

Pore networks play a key role in understanding and quantifying flow and transport processes in complex porous media. Realistic pore-spaces may be characterized by singular regions, i.e., isolated subnetworks that do not connect inlet and outlet, resulting from unconnected porosity or multiphase configurations. The robust identification of these features is critical for the characterization of network topology and the solution of the set of linear equations of flow and transport. We propose a robust method based on singular value decomposition to solve for network flow and locate isolated subnetworks simultaneously. The method's performance is demonstrated for networks of different complexity.

Media classification and the construction of pore network models from binary images of porous media hinges on accurately characterizing the pore space. We present an efficient method for (i) locating critical points, that is, pore body and throat centers, and (ii) partitioning of the pore space using information on the curvature of the distance map (DM) of the binary image. Specifically, we use the local maxima and minima of the determinant map of the Hessian matrix of the DM to locate the center of pore bodies and throats. The locating step provides structural information on the pore system, such as pore body and throat size distributions and the mean coordination number. The partitioning step is based on the eigenvalues of the Hessian, rather than the DM, to characterize the pore space using either watershed or medial axis transforms. This strategy eliminates the common problem of saddle-induced over-partitioning shared by all traditional marker-based watershed methods and represents an efficient method to determine the skeleton of the pore space without the need for morphological reconstruction.

We examine the nonequilibrium nature of two-phase fluid displacements in a quasi-two-dimensional medium (a model open fracture), in the presence of localized constrictions ("defects"), from a theoretical and numerical standpoint. Our analysis predicts the capillary energy dissipated in abrupt interfacial displacements (jumps) across defects, and relates it to the corresponding hysteresis cycle, e.g. in pressure-saturation. We distinguish between "weak" (reversible interface displacement, exhibiting no hysteresis and dissipation) and "strong" (irreversible) defects. We expose the emergence of dissipation and irreversibility caused by spatial interactions, mediated by interfacial tension, among otherwise weak defects. We exemplify this cooperative behavior for a pair of weak defects and establish a critical separation distance, analytically and numerically, verified by a proof-of-concept experiment.

We study the mixing dynamics of a solute that is transported by advection and dispersion in a heterogeneous Darcy scale porous medium. We quantify mixing and dynamic uncertainty in terms of the mean squared solute concentration and the concentration variance. The latter measures the degree of mixing of the solute and at the same time the uncertainty around the mean concentration. Its evolution is controlled by the creation of concentration fluctuations due to solute spreading and its destruction by local dispersion. For moderate heterogeneity, this interplay can be quantified by using apparent and effective dispersion coefficients. For increasing heterogeneity we find deviations from the predicted behavior. In order to shed light on these behaviors and separate solute mixing and spreading, we decompose the solute plume into partial plumes, transport Green functions, and analyze their dynamics relative to those of the whole plume. This reveals that the variability in the dispersive scales of the Green functions in the plume and their interactions due to the strong focusing of preferential flow paths play an important role in highly heterogeneous porous media.

The continuous time random walk (CTRW) approach has been widely applied to model large-scale non-Fickian transport in the flow through disordered media. Often, the underlying microscopic transport mechanisms and disorder characteristics are not known, and their effect on large-scale solute dispersion is encoded by a heavy-tailed transition time distribution. Here we study how the microscale physics manifests in the CTRW framework, and how it affects solute dispersion. To this end, we consider transport in disordered media with random sorption and random flow properties. Both disorder mechanisms can give rise to anomalous particle transport. We present the CTRW models corresponding to each of these physical scenarios to discuss the different manifestations of microscale heterogeneity on large-scale dispersion depending on the particle injection modes. The combined impact of random sorption and advection is studied with a novel CTRW model that explicitly represents both microscale disorder mechanisms. While random advection and sorption may show similar large-scale transport behaviors, they can be clearly distinguished in their response to uniform injection conditions, and, in general, to initial particle distributions that are not flux-weighted. These findings highlight the importance of the microscale physics for the interpretation and prediction of anomalous dispersion phenomena in disordered media., Comment: 29 pages, 6 figures

In this letter, we show that pore-scale diffusiophoresis of colloidal particles along local salt gradients manifests in the macroscopic dispersion of particles in a porous medium. Despite is transient character, this microscopic phenomenon controls large-scale particle transport by altering their partitioning between transmitting and dead-end pores. It determines the distribution of residence and arrival times in the medium. Depending on the diffusiophoretic mobility, particles can be mobilized from or trapped in dead-end pores, which provides a means for the controlled manipulation of particles in porous media., Comment: Article: 4 figures, 6 pages. Supplementary Information: 3 figures, 6 pages

Understanding flow and transport of bacteria in porous media is crucial to technologies such as bioremediation, biomineralization or enhanced oil recovery. While physicochemical bacteria filtration is well-documented, recent studies showed that bacterial motility plays a key role in the transport process. Flow and transport experiments performed in microfluidic chips containing randomly placed obstacles confirmed that the distributions of non-motile particles stays compact, whereas for the motile strains, the distributions are characterized by both significant retention as well as fast downstream motion. For motile bacteria, the detailed microscopic study of individual bacteria trajectories reveals two salient features: (i) the emergence of an active retention process triggered by motility, (ii) enhancement of dispersion due to the exchange between fast flow channels and low flow regions in the vicinity of the solid grains. We propose a physical model based on a continuous time random walk approach. This approach accounts for bacteria dispersion via variable pore-scale flow velocities through a Markov model for equidistant particle speeds. Motility of bacteria is modeled by a two-rate trapping process that accounts for the motion towards and active trapping at the obstacles. This approach captures the forward tails observed for the distribution of bacteria displacements, and quantifies an enhanced hydrodynamic dispersion effect that originates in the interaction between flow at the pore-scale and bacterial motility. The model reproduces the experimental observations, and predicts bacteria dispersion and transport at the macroscale., Comment: 30 pages, 10 figures

Natural porous systems, such as soil, membranes, and biological tissues comprise disordered structures characterized by dead-end pores connected to a network of percolating channels. The release and dispersion of particles, solutes, and microorganisms from such features is key for a broad range of environmental and medical applications including soil remediation, drug delivery and filtration. Yet, the role of microscopic structure and flow for the dispersion of particles and solutes in such disordered systems has been only poorly understood, in part due to the stagnant and opaque nature of these microscopic systems. Here, we use a microfluidic model system that features a pore structure characterized by dead-ends to determine how particles are transported, retained and dispersed. We observe strong tailing of arrival time distributions at the outlet of the medium characterized by power-law decay with an exponent of 2/3. Using numerical simulations and an analytical model, we link this behavior to particles initially located within dead-end pores, and explain the tailing exponent with a hopping and rolling mechanism along the streamlines inside vortices within dead-end pores. These dynamics are quantified by a stochastic model that predicts the full evolution of arrival times. Our results demonstrate how microscopic flow structures can impact macroscopic particle transport.

We study the interplay of pore-scale mixing and network-scale advection through heterogeneous porous media, and its role for the evolution and asymptotic behavior of hydrodynamic dispersion. In a Lagrangian framework, we identify three fundamental mechanisms of pore-scale mixing that determine large scale particle motion, namely, the smoothing of intra-pore velocity contrasts, the increase of the tortuosity of particle paths, and the setting of a maximum time for particle transitions. Based on these mechanisms, we derive a theory that predicts anomalous and normal hydrodynamic dispersion based on the characteristic pore length, Eulerian velocity distribution and P\'eclet number.

Advective trapping occurs when solute enters low velocity zones in heterogeneous porous media. Classical local modeling approaches combine the impact of slow advection and diffusion into a hydrodynamic dispersion coefficient and many temporally non-local approaches lump these mechanisms into a single memory function. This joint treatment makes parameterization difficult and thus prediction of large scale transport a challenge. Here we investigate the mechanisms of advective trapping and their impact on transport in media composed of a high conductivity background and isolated low permeability inclusions. Breakthrough curves show that effective transport changes from a streamtube-like behavior to genuine random trapping as the degree of disorder of the inclusion arrangement increases. We upscale this behavior using a Lagrangian view point, in which idealized solute particles transition over a fixed distance at random advection times combined with Poissonian advective trapping events. We discuss the mathematical formulation of the upscaled model in the continuous time random walk and mobile-immobile mass transfer frameworks, and derive a model for large scale solute non-Fickian dispersion. These findings give new insight into transport in highly heterogeneous media.

We implement the Multi-Rate Mass Transfer (MRMT) model for mobile-immobile transport in porous media within the open-source finite volume library \textsc{OpenFOAM}\reg \citep{Foundation2014}. Unlike other codes available in the literature [Geiger, S., Dentz, M., Neuweiler, I., SPE Reservoir Characterisation and Simulation Conference and Exhibition (2011); Silva, O., Carrera, J., Dentz, M., Kumar, S., Alcolea, A., Willmann, M., Hydrology and Earth System Sciences 13, (2009)], we propose an implementation that can be applied to complex three-dimensional geometries and highly heterogeneous fields, where the parameters of the MRMT can arbitrarily vary in space. Furthermore, being built over the widely diffused OpenFOAM\reg library, it can be easily extended and included in other models, and run in parallel. We briefly describe the structure of the < multiContinuumModels > library that includes the formulation of the MRMT based on the works of [Haggerty, R., Gorelick, S.M., Water Resources Research 31, (1995)] and [F. Municchi and M. Icardi Phys. Rev. Research 2, 013041, (2020)]. The implementation is verified against benchmark solutions and tested on two- and three-dimensional random permeability fields. The role of various physical and numerical parameters, including the transfer rates, the heterogeneities, and the number of terms in the MRMT expansions is investigated. Finally, we illustrate the significant role played by heterogeneity in the mass transfer when permeability and porosity are represented using Gaussian random fields., Comment: 28 pages, 13 figures, 3 tables

Mathematical models based on probability density functions (PDF) have been extensively used in hydrology and subsurface flow problems, to describe the uncertainty in porous media properties (e.g., permeability modelled as random field). Recently, closer to the spirit of PDF models for turbulent flows, some approaches have used this statistical viewpoint also in pore-scale transport processes (fully resolved porous media models). When a concentration field is transported, by advection and diffusion, in a heterogeneous media, in fact, spatial PDFs can be defined to characterise local fluctuations and improve or better understand the closures performed by classical upscaling methods. In the study of hydrodynamical dispersion, for example, PDE-based PDF approach can replace expensive and noisy Lagrangian simulations (e.g. trajectories of drift-diffusion stochastic processes). In this work we derive a joint position-velocity Fokker-Planck equation to model the motion of particles undergoing advection and diffusion in in deterministic or stochastic heterogeneous velocity fields. After appropriate closure assumptions, this description can help deriving rigorously stochastic models for the statistics of Lagrangian velocities. This is very important to be able to characterise the dispersion properties and can, for example, inform velocity evolution processes in Continuous Time Random Walk (CTRW) dispersion models. The closure problem that arises when averaging the Fokker Planck equation shows also interesting similarities with the mixing problem and can be used to propose alternative closures for anomalous dispersion.

We present a unified approach to those observables of stochastic processes under reset that take the form of averages of functionals depending on the most recent renewal period. We derive solutions for the observables, and determine the conditions for existence and equality of their stationary values with and without reset. For intermittent reset times, we derive exact asymptotic expressions for observables that vary asymptotically as a power of time. We illustrate the general approach with general and particular results for the power spectral density, and moments of subdiffusive processes. We focus on coupling of the process and reset via a diffusion-decay process with microscopic dependence between transport and decay. In contrast to the uncoupled case, we find that restarting the particle upon decay does not produce a probability current equal to the decay rate, but instead drastically alters the time dependence of the decay rate and the resulting current., Comment: 15 pages, 2 figures, corrected copying error in (S42), homogenize terminology, v3 add references, improve title

We develop a time domain random walk approach for conservative solute transport in heterogeneous media where medium properties vary over a distribution of length scales. The spatial transition lengths are equal to the heterogeneity length scales, and thus determined by medium geometry. We derive analytical expressions for the associated transition times and probabilities in one spatial dimension. This approach determines the coarse-grained solute concentration at the interfaces between regions; we derive a generalized master equation for the evolution of the coarse-grained concentration and reconstruct the fine-scale concentration using the propagator of the subscale transport mechanism. The performance of this approach is demonstrated for diffusion under random retardation in power-law media characterized by heavy-tailed lengthscale and retardation distributions. The coarse representation preserves the correct late-time scaling of concentration variance, and the reconstructed fine-scale concentration is essentially identical to that obtained by direct numerical simulation by random walk particle tracking.

We study the mixing in the presence of convective flow in a porous medium. Convection is characterized by the formation of vortices and stagnation points, where the fluid interface is stretched and compressed enhancing mixing. We analyze the behavior of the mixing dynamics in different scenarios using an interface deformation model. We show that the scalar dissipation rate, which is related to the dissolution fluxes, is controlled by interfacial processes, specifically the equilibrium between interface compression and diffusion, which depends on the flow field configuration. We consider different scenarios of increasing complexity. First, we analyze a double-gyre synthetic velocity field. Second, a Rayleigh-B\'enard instability (the Horton-Rogers-Lapwood problem), in which stagnation points are located at a fixed interface. This system experiences a transition from a diffusion controlled mixing to a chaotic convection as the Rayleigh number increases. Finally, a Rayleigh-Taylor instability with a moving interface, in which mixing undergoes three different regimes: diffusive, convection dominated, and convection shutdown. The interface compression model correctly predicts the behavior of the systems. It shows how the dependency of the compression rate on diffusion explains the change in the scaling behavior of the scalar dissipation rate. The model indicates that the interaction between stagnation points and the correlation structure of the velocity field is also responsible for the transition between regimes. We also show the difference in behavior between the dissolution fluxes and the mixing state of the systems. We observe that while the dissolution flux decreases with the Rayleigh number, the system becomes more homogeneous. That is, mixing is enhanced by reducing diffusion. This observation is explained by the effect of the instability patterns.

Kinetic Monte Carlo methods such as the Gillespie algorithm model chemical reactions as random walks in particle number space. The inter-reaction times are exponentially distributed under the assumption that the system is well mixed. We introduce an arbitrary inter-reaction time distribution, which may account for the impact of incomplete mixing on chemical reactions, and in general stochastic reaction delay, which may represent the impact of extrinsic noise. This process defines an inhomogeneous continuous time random walk in particle number space, from which we derive a generalized chemical master equation. This leads naturally to a generalization of the Gillespie algorithm. Based on this formalism, we determine the modified chemical rate laws for different inter-reaction time distributions. This framework traces Michaelis--Menten-type kinetics back to finite-mean delay times, and predicts time-nonlocal macroscopic reaction kinetics as a consequence of broadly distributed delays. Non-Markovian kinetics exhibit weak ergodicity breaking and show key features of reactions under local non-equilibrium., Comment: 12 pages, 2 figures, changed spelling of name of first author

This paper studies the mechanisms of dispersion in the laminar flow through the pore space of a $3$-dimensional porous medium. We focus on pre-asymptotic transport prior to the asymptotic hydrodynamic dispersion regime, in which solute motion may be described by the average flow velocity and a hydrodynamic dispersion coefficient. High performance numerical flow and transport simulations of solute breakthrough at the outlet of a sand-like porous medium evidence marked deviations from the hydrodynamic dispersion paradigm and identifies two distinct regimes. The first regime is characterized by a broad distribution of advective residence times in single pores. The second regime is characterized by diffusive mass transfer into low-velocity region in the wake of solute grains. These mechanisms are quantified systematically in the framework of a time-domain random walk for the motion of marked elements (particles) of the transported material quantity. The model is parameterized with the characteristics of the porous medium under consideration and captures both pre-asymptotic regimes. Macroscale transport is described by an integro-differential equation for solute concentration, whose memory kernels are given in terms of the distribution of mean pore velocities and trapping times. This approach quantifies the physical non-equilibrium caused by a broad distribution of mass transfer time scales, both advective and diffusive, on the representative elementary volume (REV). Thus, while the REV indicates the scale at which medium properties like porosity can be uniquely defined, this does not imply that transport can be characterized by hydrodynamic dispersion.

We study the causes of anomalous dispersion in Darcy-scale porous media characterized by spatially heterogeneous hydraulic properties. Spatial variability in hydraulic conductivity leads to spatial variability in the flow properties through Darcy's law and thus impacts on solute and particle transport. We consider purely advective transport in heterogeneity scenarios characterized by broad distributions of heterogeneity length scales and point values. Particle transport is characterized in terms of the stochastic properties of equidistantly sampled Lagrangian velocities, which are determined by the flow and conductivity statistics. The persistence length scales of flow and transport velocities are imprinted in the spatial disorder and reflect the distribution of heterogeneity length scales. Particle transitions over the velocity length scales are kinematically coupled with the transition time through velocity. We show that the average particle motion follows a coupled continuous time random walk (CTRW), which is fully parameterized by the distribution of flow velocities and the medium geometry in terms of the heterogeneity length scales. The coupled CTRW provides a systematic framework for the investigation of the origins of anomalous dispersion in terms of heterogeneity correlation and the distribution of heterogeneity point values. Broad distributions of heterogeneity point values and lengths scales may lead to very similar dispersion behaviors in terms of the spatial variance. Their mechanisms, however are very different, which manifests in the distributions of particle positions and arrival times, which plays a central role for the prediction of the fate of dissolved substances in heterogeneous natural and engineered porous materials.

We investigate tracer transport on random discrete fracture networks that are characterized by the statistics of the fracture geometry and hydraulic conductivity. While it is well known that tracer transport through fractured media can be anomalous and particle injection modes can have major impact on dispersion, the incorporation of injection modes into effective transport modelling has remained an open issue. The fundamental reason behind this challenge is that---even if the Eulerian fluid velocity is steady---the Lagrangian velocity distribution experienced by tracer particles evolves with time from its initial distribution, which is dictated by the injection mode, to a stationary velocity distribution. We quantify this evolution by a Markov model for particle velocities that are equidistantly sampled along trajectories. This stochastic approach allows for the systematic incorporation of the initial velocity distribution and quantifies the interplay between velocity distribution and spatial and temporal correlation. The proposed spatial Markov model is characterized by the initial velocity distribution, which is determined by the particle injection mode, the stationary Lagrangian velocity distribution, which is derived from the Eulerian velocity distribution, and the spatial velocity correlation length, which is related to the characteristic fracture length. This effective model leads to a time-domain random walk for the evolution of particle positions and velocities, whose joint distribution follows a Boltzmann equation. Finally, we demonstrate that the proposed model can successfully predict anomalous transport through discrete fracture networks with different levels of heterogeneity and arbitrary tracer injection modes.

We study modeling of two-phase flow in highly heterogeneous fractured and porous media. The flow behaviour is strongly influenced by mass transfer between a highly permeable (mobile) fracture domain and less permeable (immobile) matrix blocks. We quantify the effective two-phase flow behaviour using a multirate rate mass transfer (MRMT) approach. We discuss the range of applicability of the MRMT approach in terms of the pertinent viscous and capillary diffusion time scales. We scrutinize the linearization of capillary diffusion in the immobile regions, which allows for the formulation of MRMT in the form of a non-local single equation model. The global memory function, which encodes mass transfer between the mobile and the immobile regions, is at the center of this method. We propose two methods to estimate the global memory function for a fracture network with given fracture and matrix geometry. Both employ a scaling approach based on the known local memory function for a given immobile region. With the first method, the local memory function is calculated numerically, while the second one employs a parametric memory function in form of truncated power-law. The developed concepts are applied and tested for fracture networks of different complexity. We find that both physically based parameter estimation methods for the global memory function provide predictive MRMT approaches for the description of multiphase flow in highly heterogeneous porous media.

This paper derives and analyzes continuous time random walk (CTRW) models in radial flow geometries for the quantification of non-local solute transport induced by heterogeneous flow distributions and by mobile-immobile mass transfer processes. To this end we derive a general CTRW framework in radial coordinates starting from the random walk equations for radial particle positions and times. The particle density, or solute concentration is governed by a non-local radial advection-dispersion equation (ADE). Unlike in CTRWs for uniform flow scenarios, particle transition times here depend on the radial particle position, which renders the CTRW non-stationary. As a consequence, the memory kernel characterizing the non-local ADE, is radially dependent. Based on this general formulation, we derive radial CTRW implementations that (i) emulate non-local radial transport due to heterogeneous advection, (ii) model multirate mass transfer (MRMT) between mobile and immobile continua, and (iii) quantify both heterogeneous advection in a mobile region and mass transfer between mobile and immobile regions. The expected solute breakthrough behavior is studied using numerical random walk particle tracking simulations. This behavior is analyzed by explicit analytical expressions for the asymptotic solute breakthrough curves. We observe clear power-law tails of the solute breakthrough for broad (power-law) distributions of particle transit times (heterogeneous advection) and particle trapping times (MRMT model). The combined model displays two distinct time regimes. An intermediate regime, in which the solute breakthrough is dominated by the particle transit times in the mobile zones, and a late time regime that is governed by the distribution of particle trapping times in immobile zones.

Using a Lagrangian framework, we discuss different conceptualizations of the blockscale dispersion tensor. We distinguish effective and ensemble blockscale dispersion, which measure the impact of subscale velocity fluctuations on solute dispersion. Ensemble dispersion quantifies subscale velocity fluctuations between realizations, which overestimates the actual velocity variability. Effective dispersion on the other hand quantifies the actual blockscale velocity variability and thus reflects the impact of subscale velocity fluctuations on mixing and spreading. Based on these concepts, we quantify the impact of subscale velocity fluctuations on solute particle spreading and determine the governing equations for the coarse-grained concentration distributions. We develop analytical and semi-analytical expressions for the average and variance of the blockscale dispersion tensor in 3D flow fields as a function of the structural parameters characterizing the subsurface. Our results illustrate the relevance of the blockscale, the initial scale of the solute body and local-scale dispersion in controlling the uncertainty of the plume's dispersive behavior. The analysis performed in this work has implications in numerical modeling (i.e. grid design) and allows to quantify the uncertainty of the blockscale dispersion tensor.

We develop a continuous time random walk (CTRW) approach for the evolution of Lagrangian velocities in steady heterogeneous flows based on a stochastic relaxation process for the streamwise particle velocities. This approach describes persistence of velocities over a characteristic spatial scale, unlike classical random walk methods, which model persistence over a characteristic time scale. We first establish the relation between Eulerian and Lagrangian velocities for both equidistant and isochrone sampling along streamlines, under transient and stationary conditions. Based on this, we develop a space continuous CTRW approach for the spatial and temporal dynamics of Lagrangian velocities. While classical CTRW formulations have non-stationary Lagrangian velocity statistics, the proposed approach quantifies the evolution of the Lagrangian velocity statistics under both stationary and non-stationary conditions. We provide explicit expressions for the Lagrangian velocity statistics, and determine the behaviors of the mean particle velocity, velocity covariance and particle dispersion. We find strong Lagrangian correlation and anomalous dispersion for velocity distributions which are tailed toward low velocities as well as marked differences depending on the initial conditions. The developed CTRW approach predicts the Lagrangian particle dynamics from an arbitrary initial condition based on the Eulerian velocity distribution and a characteristic correlation scale., Comment: 13 pages, 4 figures

Mixing fronts, where fluids of different chemical compositions mix with each other, are typically subjected to velocity gradients, ranging from the pore scale to the catchment scale due to permeability variations and flow line geometries. A common trait of these processes is that the mixing interface is strained by shear. Depending on the P\'eclet number $Pe$, which represents the ratio of the characteristic diffusion time to the characteristic advection time, and the Damk\"ohler number $Da$, which represents the ratio of the characteristic diffusion time to the characteristic reaction time, the local reaction rates can be strongly impacted by the dynamics of the mixing interface. This impact has been characterized mostly either in kinetics-limited or in mixing-limited conditions, that is, for either very low or very high $Da$. Here the coupling of shear flow and chemical reactivity is investigated for arbitrary Damk\"ohler numbers, for a bimolecular reaction and an initial interface with separated reactants. Approximate analytical expressions for the global production rate and reactive mixing scale are derived based on a reactive lamella approach that allows for a general coupling between stretching enhanced mixing and chemical reactions. While for $PeDa$, we uncover a "strong stretching" scenario where new scaling laws emerge from the interplay between reaction kinetics, diffusion, and stretching. The analytical results are validated against numerical simulations. These findings shed light on the effect of flow heterogeneity on the enhancement of chemical reaction and the creation of spatially localized hotspots of reactivity for a broad range of systems ranging from kinetic limited to mixing limited situations.

Understanding anomalous transport and reaction kinetics due to microscopic physical and chemical disorder is a long-standing goal in many fields including geophysics, biology, and engineering. We consider reaction-diffusion characterized by fluctuations in both transitions times and decay rates. We introduce and analyze a model framework that explicitly connects microscopic fluctuations with the mescoscopic description. For broad distributions of transport and reaction time scales we compute the particle density and derive the equations governing its evolution, finding power-law decay of the survival probability, and spatially heterogeneous decay that leads to subdiffusion and an asymptotically stationary surviving-particle density. These anomalies are clearly attributable to non-Markovian effects that couple transport and chemical properties in both reaction and diffusion terms., Comment: Explain model and applications in more detail. 19 pages, 6 figures

Fluid deformation and strain history are central to wide range of fluid mechanical phenomena ranging from fluid mixing and particle transport to stress development in complex fluids and the formation of Lagrangian coherent structures (LCSs). To understand and model these processes it is necessary to quantify Lagrangian deformation in terms of Eulerian flow properties, currently an open problem. To elucidate this link we develop a Protean (streamline) coordinate transform for steady three-dimensional (3D) flows which renders both the velocity gradient and deformation gradient upper triangular. This frame not only simplifies computation of fluid deformation metrics such as finite-time Lyapunov exponents (FTLEs) and elucidates the deformation structure of the flow, but moreover explicitly recovers kinematic and topological constraints upon deformation such as those related to helicity density and the Poincar\'{e}-Bendixson theorem. We apply this transform to several classes of steady 3D flow, including helical and non-helical, compressible and incompressible flows, and find random flows exhibit remarkably simple (Gaussian) deformation structure, and so are fully characterised in terms of a small number of parameters. As such this technique provides the basis for the development of stochastic models of fluid deformation in random flows which adhere to the kinematic constraints inherent to various flow classes., Comment: 18 pages

Under steady flow conditions, the topological complexity inherent to all random 3D porous media imparts complicated flow and transport dynamics. It has been established that this complexity generates persistent chaotic advection via a three-dimensional (3D) fluid mechanical analogue of the baker's map which rapidly accelerates scalar mixing in the presence of molecular diffusion. Hence pore-scale fluid mixing is governed by the interplay between chaotic advection, molecular diffusion and the broad (power-law) distribution of fluid particle travel times which arise from the non-slip condition at pore walls. To understand and quantify mixing in 3D porous media, we consider these processes in a model 3D open porous network and develop a novel stretching continuous time random walk (CTRW) which provides analytic estimates of pore-scale mixing which compare well with direct numerical simulations. We find that chaotic advection inherent to 3D porous media imparts scalar mixing which scales exponentially with longitudinal advection, whereas the topological constraints associated with 2D porous media limits mixing to scale algebraically. These results decipher the role of wide transit time distributions and complex topologies on porous media mixing dynamics, and provide the building blocks for macroscopic models of dilution and mixing which resolve these mechanisms., Comment: 36 pages

We study the relation between flow structure and fluid deformation in steady two-dimensional random flows. Beyond the linear (shear flow) and exponential (chaotic flow) elongation paradigms, we find a broad spectrum of stretching behaviors, ranging from sub- to superlinear, which are dominated by intermittent shear events. We analyze these behaviors from first principles, which uncovers stretching as a result of the non-linear coupling between Lagrangian shear deformation and velocity fluctuations along streamlines. We derive explicit expressions for Lagrangian deformation and demonstrate that stretching obeys a coupled continous time random walk, which for broad distributions of flow velocities describes a L\'evy walk for elongation. 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, which can have strong anomalous diffusive character., Comment: 12 pages, 5 figures