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

Multiscale Porosity Microfluidics to Study Bacterial Transport in Heterogeneous Chemical Landscapes

Microfluidic models are proving to be powerful systems to study fundamental processes in porous media, due to their ability to replicate topologically complex environments while allowing detailed, quantitative observations at the pore scale. Yet, while porous media such as living tissues, geological substrates, or industrial systems typically display a porosity that spans multiple scales, most microfluidic models to date are limited to a single porosity or a small range of pore sizes. Here, a novel microfluidic system with multiscale porosity is presented. By embedding polyacrylamide (PAAm) hydrogel structures through in-situ photopolymerization in a landscape of microfabricated polydimethylsiloxane (PDMS) pillars with varying spacing, micromodels with porosity spanning several orders of magnitude, from nanometers to millimeters are created. Experiments conducted at different porosity patterns demonstrate the potential of this approach to characterize fundamental and ubiquitous biological and geochemical transport processes in porous media. Accounting for multiscale porosity allows studies of the resulting heterogeneous fluid flow and concentration fields of transported chemicals, as well as the biological behaviors associated with this heterogeneity, such as bacterial chemotaxis. This approach brings laboratory studies of transport in porous media a step closer to their natural counterparts in the environment, industry, and medicine.

Statistical-physical method for simulating the transport of microplastic-antibiotic compound pollutants in typical bay area

Microplastics and antibiotics are emerging pollutants in the environment and have received widespread attention globally. In coastal areas, microplastic and antibiotic pollution is ubiquitous and often overlapping. Microplastic-antibiotic compound pollutants that are formed through adsorption have thus become a major concern. However, modeling knowledge of microplastic transport in coastal areas is still limited, and research on the impact of compound pollutants caused by Polythene (PE)-antibiotics in such settings is in early stages. In this study, using a lattice Boltzmann method (LBM) and temporal Markov method (TMM) under a statistical-physical framework, we simulated pollutant transport and PE-antibiotic compound pollutants in coastal areas. First, a series of models are proposed, including an LBM wave-current coupling model, an LBM antibiotic transport model, an LBM particle-tracking model, a TMM microplastic transport model and the final LBM-TMM hybrid compound pollutant model. Then, the suitability and applicability of the models was validated using experimental data and numerical simulations. Finally, the models were applied to a study area, Laizhou Bay (China). The simulation results demonstrate that adsorption will reduce the concentration of antibiotics in the water environment. Within 44 days, the adsorbed antibiotic carried by PE particles migrate further, and the width of the pollution zone escalates from 234.2 m to 689.0 m.

Intermittent random walks under stochastic resetting

We analyze a one-dimensional intermittent random walk on an unbounded domain in the presence of stochastic resetting. In this process, the walker alternates between local intensive search, diffusion, and rapid ballistic relocations in which it does not react to the target. We demonstrate that Poissonian resetting leads to the existence of a non-equilibrium steady state. We calculate the distribution of the first arrival time to a target along with its mean and show the existence of an optimal reset rate. In particular, we prove that the initial condition of the walker, i.e., either starting diffusely or relocating, can significantly affect the long-time properties of the search process. Moreover, we demonstrate the presence of distinct parameter regimes for the global optimization of the mean first arrival time when ballistic and diffusive movements are in direct competition.

Analysis of the Type V Intermittency Using the Perron-Frobenius Operator

A methodology to study the reinjection process in type V intermittency is introduced. The reinjection probability density function (RPD), and the probability density of the laminar lengths (RPDL) for type V intermittency are calculated. A family of maps with discontinuous and continuous RPD functions is analyzed. Several tests were performed, in which the proposed technique was compared with the classical theory of intermittency, the M function methodology, and numerical data. The analysis exposed that the new technique can accurately capture the numerical data. Therefore, the scheme presented herein is a useful tool to theoretically evaluate the statistical variables for type V intermittency.

Time-integral iteration method for two-dimensional anomalous transport

A methodology is developed to describe time-dependent phenomena associated with nonlocal transport in complex, two-dimensional geometries. It is an extension of the ''iterative method" introduced previously to solve steady-state transport problems [Maggs and Morales, Phys. Rev. E 99, 013307 (2019)10.1103/PhysRevE.99.013307], and it is based on the ''jumping particle" concepts associated with the continuous-time random walk (CTRW) model. The method presented explicitly evaluates the time integral contained in the CTRW master equation. A modified version of the Mittag-Leffler function is used for the waiting-time probability distributions to incorporate memory effects. Calculations of the propagation of ''anomalous transport waves" in various systems, with and without memory, illustrate the technique.

Linking mixing and flow topology in porous media: An experimental proof

Transport processes in porous media are controlled by the characteristics of the flow field which are determined by the porous material properties and the boundary conditions of the system. This work provides experimental evidence of the relation between mixing and flow field topology in porous media at the continuum scale. The setup consists of a homogeneously packed quasi-two-dimensional flow-through chamber in which transient flow conditions, dynamically controlled by two external reservoirs, impact the transport of a dissolved tracer. The experiments were performed at two different flow velocities, corresponding to Péclet numbers of 191 and 565, respectively. The model-based interpretation of the experimental results shows that high values of the effective Okubo-Weiss parameter, driven by the changes of the boundary conditions, lead to high rates of increase of the Shannon entropy of the tracer distribution and, thus, to enhanced mixing. The comparison between a hydrodynamic dispersion model and an equivalent pore diffusion model demonstrates that despite the spatial and temporal variability in the hydrodynamic dispersion coefficients, the Shannon entropy remains almost unchanged because it is controlled by the Okubo-Weiss parameter. Overall, our work demonstrates that under highly transient boundary conditions, mixing dynamics in homogeneous porous media can also display complex patterns and is controlled by the flow topology.

Intermittency Reinjection in the Logistic Map

Just below a Period-3 window, the logistic map exhibits intermittency. Then, the third iterate of this map has been widely used to explain the chaotic intermittency concept. Much attention has been paid to describing the behavior around the vanished fixed points, the tangent bifurcation, and the formation of the characteristic channel between the map and the diagonal for type-I intermittency. However, the reinjection mechanism has not been deeply analyzed. In this paper, we studied the reinjection processes for the three fixed points around which intermittency is generated. We calculated the reinjection probability density function, the probability density of the laminar lengths, and the characteristic relation. We found that the reinjection mechanisms have broader behavior than the usually used uniform reinjection. Furthermore, the reinjection processes depend on the fixed point.

Contributions of biofilm-induced flow heterogeneities to solute retention and anomalous transport features in porous media

Microbial biofilms are ubiquitous within porous media and the dynamics of their growth influence surface and subsurface flow patterns which impacts the physical properties of porous media and large-scale transport of solutes. A two-dimensional pore-scale numerical model was used to evaluate the impact of biofilm-induced flow heterogeneities on conservative transport. Our study integrates experimental biofilm images of Paenibacillus 300A strain in a microfluidic device packed with cylindrical grains in a hexagonal distribution, with mathematical modeling. Biofilm is represented as a synthetic porous structure with locally varying physical properties that honors the impact of biofilm on the porous medium. We find that biofilm plays a major role in shaping the observed conservative transport dynamics by enhancing anomalous characteristics. More specifically, when biofilm is present, the pore structure in our geometry becomes more spatially correlated. We observe intermittent behavior in the Lagrangian velocities that switches between fast transport periods and long trapping events. Our results suggest that intermittency enhances solute spreading in breakthrough curves which exhibit extreme anomalous slope at intermediate times and very marked late solute arrival due to solute retention. The efficiency of solute retention by the biofilm is controlled by a transport regime which can extend the tailing in the concentration breakthrough curves. These results indicate that solute retention by the biofilm exerts a strong control on conservative solute transport at pore-scale, a role that to date has not received enough attention.

Reactive Transport: A Review of Basic Concepts with Emphasis on Biochemical Processes

Reactive transport (RT) couples bio-geo-chemical reactions and transport. RT is important to understand numerous scientific questions and solve some engineering problems. RT is highly multidisciplinary, which hinders the development of a body of knowledge shared by RT modelers and developers. The goal of this paper is to review the basic conceptual issues shared by all RT problems, so as to facilitate advancement along the current frontier: biochemical reactions. To this end, we review the basic equations to indicate that chemical systems are controlled by the set of equilibrium reactions, which are easy to model, but whose rate is controlled by mixing. Since mixing is not properly represented by the standard advection-dispersion equation (ADE), we conclude that this equation is poor for RT. This leads us to review alternative transport formulations, and the methods to solve RT problems using both the ADE and alternative equations. Since equilibrium is easy, difficulties arise for kinetic reactions, which is especially true for biochemistry, where numerous challenges are open (how to represent microbial communities, impact of genomics, effect of biofilms on flow and transport, etc.). We conclude with the basic eleven conceptual issues that we consider fundamental for any conceptually sound RT effort.

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-β, 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.

The Multi-Advective Water Mixing Approach for Transport through Heterogeneous Media

Finding a numerical method to model solute transport in porous media with high heterogeneity is crucial, especially when chemical reactions are involved. The phase space formulation termed the multi-advective water mixing approach (MAWMA) was proposed to address this issue. The water parcel method (WP) may be obtained by discretizing MAWMA in space, time, and velocity. WP needs two transition matrices of velocity to reproduce advection (Markovian in space) and mixing (Markovian in time), separately. The matrices express the transition probability of water instead of individual solute concentration. This entails a change in concept, since the entire transport phenomenon is defined by the water phase. Concentration is reduced to a chemical attribute. The water transition matrix is obtained and is demonstrated to be constant in time. Moreover, the WP method is compared with the classic random walk method (RW) in a high heterogeneous domain. Results show that the WP adequately reproduces advection and dispersion, but overestimates mixing because mixing is a sub-velocity phase process. The WP method must, therefore, be extended to take into account incomplete mixing within velocity classes.

Distribution of liquid flow in a pore network during evaporation

The variation of the distribution of the liquid flow in porous media during evaporation is still a puzzle. We resolve it with the pore network modeling approach. The distribution of the evaporation-induced liquid flow in a pore network composed of about 2.5 million pores is determined. The probability density function of the magnitude of the normalized liquid flow rate is obtained. For the low normalized liquid flow rate, the probability density function is power-lawlike. The power-law exponent depends on both the liquid saturation and the location of the moving meniscus in the main liquid cluster. The evaporation-induced liquid flow in the pores in the pore network can be correlated. Whether the liquid flow distributions in various zones in the pore network are similar or not relies significantly on the location of the moving meniscus in the main liquid cluster. The functions depicting the relation between the power-law exponent and the local liquid saturation for the zones adjacent to and away from the open side of the pore network are different. These findings from the pore scale studies provide insights into developing the accurate continuum model for evaporation in porous media.

Calculation of the Statistical Properties in Intermittency Using the Natural Invariant Density

We use the natural invariant density of the map and the Perron–Frobenius operator to analytically evaluate the statistical properties for chaotic intermittency. This study can be understood as an improvement of the previous ones because it does not introduce assumptions about the reinjection probability density function in the laminar interval or the map density at pre-reinjection points. To validate the new theoretical equations, we study a symmetric map and a non-symmetric one. The cusp map has symmetry about x=0, but the Manneville map has no symmetry. We carry out several comparisons between the theoretical equations here presented, the M function methodology, the classical theory of intermittency, and numerical data. The new theoretical equations show more accuracy than those calculated with other techniques.

Pore-Scale Mixing and the Evolution of Hydrodynamic Dispersion in Porous Media

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 intrapore 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 in terms of the characteristic pore length, Eulerian velocity distribution, and Péclet number.

Correlated continuous-time random walk in a velocity field: Anomalous bifractional crossover

The diffusion of space-time correlated continuous-time random walk moving in the velocity field, which includes the fluid flowing freely and the fluid flowing through porous media, is investigated in this paper. Results reveal that it presents anomalous diffusion merely caused by space-time correlation in the freely flowing fluid, and the bias from the velocity field only supplies a standard advection, which is verified by the corresponding generalized diffusion equation which includes a standard advection term. However, the diffusion in the fluid flowing through porous media, i.e., the mean squared displacement, can display a bifractional form of which one originates from space-time correlation and the other one originates from dispersive bias caused by sticking of the porous media. The fractional advection term emerging in the corresponding generalized diffusion equation confirms the results. Moreover, the coexistence of correlation and dispersive bias result in crossover phenomenon in-between the diffusive process at an intermediate timescale, but just as the definition of diffusion, the one owning the largest diffusion exponent always prevails at large timescales. However, since the two fractional diffusions originate from a different mechanism, even if it owns the smaller diffusion exponent, that one can dominate the diffusion if it fluctuates much stronger than the other one, which no longer obeys the previous conclusion.

A review of spatial Markov models for predicting pre-asymptotic and anomalous transport in porous and fractured media

Heterogeneity across a broad range of scales in geologic porous media often manifests in observations of non-Fickian or anomalous transport. While traditional anomalous transport models can successfully make predictions in certain geological systems, increasing evidence suggests that assumptions relating to independent and identically distributed increments constrain where and when they can be reliably applied. A relatively novel model, the Spatial Markov model (SMM), relaxes the assumption of independence. The SMM belongs to the family of correlated continuous time random walks and has shown promise across a wide range of transport problems relevant to natural porous media. It has been successfully used to model conservative as well as more recently reactive transport in highly complex flows ranging from pore scales to much larger scales of interest in geology and subsurface hydrology. In this review paper we summarize its original development and provide a comprehensive review of its advances and applications as well as lay out a vision for its future development.

Predicting Vertical Concentration Profiles in the Marine Atmospheric Boundary Layer With a Markov Chain Random Walk Model

In an effort to better represent aerosol transport in mesoscale and global-scale models, large eddy simulations (LES) from the National Center for Atmospheric Research (NCAR) Turbulence with Particles (NTLP) code are used to develop a Markov chain random walk model that predicts aerosol particle profiles in a cloud-free marine atmospheric boundary layer (MABL). The evolution of vertical concentration profiles are simulated for a range of aerosol particle sizes and in a neutral and an unstable boundary layer. For the neutral boundary layer we find, based on the LES statistics and a specific model time step, that there exist significant correlation for particle positions, meaning that particles near the bottom of the boundary are more likely to remain near the bottom of the boundary layer than being abruptly transported to the top, and vice versa. For the unstable boundary layer, a similar time interval exhibits a weaker tendency for an aerosol particle to remain close to its current location compared to the neutral case due to the strong nonlocal convective motions. In the limit of a large time interval, particles have been mixed throughout the MABL and virtually no temporal correlation exists. We leverage this information to parameterize a Markov chain random walk model that accurately predicts the evolution of vertical concentration profiles. The new methodology has significant potential to be applied at the subgrid level for coarser-scale weather and climate models, the utility of which is shown by comparison to airborne field data and global aerosol models.

Plume deformation, mixing, and reaction kinetics: An analysis of interacting helical flows in three-dimensional porous media

Heterogeneity and macroscopic anisotropy of porous media play an important role for dilution and reaction enhancement of conservative and reactive plumes. In this study, we perform numerical simulations to investigate steady-state flow and transport in three-dimensional heterogeneous porous media. We consider two macroscopic anisotropic inclusions resulting in helical flows with twisting streamlines in a three-dimensional flow-through domain. The inclusions are obtained by alternating two layers of angled slices of coarse and fine porous media with different hydraulic conductivity. We investigate flow and transport scenarios considering different geometry and relative position of the two anisotropic inclusions yielding helical flow fields with different extent of interaction. We use metrics of stretching and folding to characterize the flow field and entropy-based metrics for the analysis of the conservative and reactive transport problems. The outcomes show that the two helices result in different patterns of twisting streamlines, which cause distinct deformation of the plumes. However, mixing and reaction enhancement could not be directly related to the extent of the flow field deformation: Configurations with strong deformation can result in only moderate mixing enhancement, whereas configurations with limited deformation of the flow field can lead to significant mixing of the solute plume. Finally, we explore the impact of different degradation rates on reactive transport and the role of reaction kinetics on the entropy balance for a reactant undergoing transport and mixing-controlled degradation in the twisting flow fields. The results show that strong mixing enhancement due to helical flow increases the importance of the reaction kinetics that becomes the rate-limiting process for solute reactive transport.

Trait-specific dispersal of bacteria in heterogeneous porous environments: from pore to porous medium scale

The dispersal of organisms controls the structure and dynamics of populations and communities, and can regulate ecosystem functioning. Predicting dispersal patterns across scales is important to understand microbial life in heterogeneous porous environments such as soils and sediments. We developed a multi-scale approach, combining experiments with microfluidic devices and time-lapse microscopy to track individual bacterial trajectories and measure the overall breakthrough curves and bacterial deposition profiles: we, then, linked the two scales with a novel stochastic model. We show that motile cells of Pseudomonas putida disperse more efficiently than non-motile mutants through a designed heterogeneous porous system. Motile cells can evade flow-imposed trajectories, enabling them to explore larger pore areas than non-motile cells. While transported cells exhibited a rotation in response to hydrodynamic shear, motile cells were less susceptible to the torque, maintaining their body oriented towards the flow direction and thus changing the population velocity distribution with a significant impact on the overall transport properties. We also found, in a separate set of experiments, that if the suspension flows through a porous system already colonized by a biofilm, P. putida cells are channelled into preferential flow paths and the cell attachment rate is increased. These two effects were more pronounced for non-motile than for motile cells. Our findings suggest that motility coupled with heterogeneous flows can be beneficial to motile bacteria in confined environments as it enables them to actively explore the space for resources or evade regions with unfavourable conditions. Our study also underlines the benefit of a multi-scale approach to the study of bacterial dispersal in porous systems.

Anomalous diffusion in an electrolyte saturated paper matrix

Diffusion of colored dye on water saturated paper substrates has been traditionally exploited with great skill by renowned water color artists. The same physics finds more recent practical applications in paper-based diagnostic devices deploying chemicals that react with a bodily fluid yielding colorimetric signals for disease detection. During spontaneous imbibition through the tortuous pathways of a porous electrolyte saturated paper matrix, a dye molecule undergoes diffusion in a complex network of pores. The advancing front forms a strongly correlated interface that propagates diffusively but with an enhanced effective diffusivity. We measure this effective diffusivity and show that it is several orders of magnitude greater than the free solution diffusivity and has a significant dependence on the solution pH and salt concentration in the background electrolyte. We attribute this to electrically mediated interfacial interactions between the ionic species in the liquid dye and spontaneous surface charges developed at porous interfaces, and introduce a simple theory to explain this phenomenon.

Pore-scale velocities in three-dimensional porous materials with trapped immiscible fluid

We study and document the influence of wetting and nonwetting trapped immiscible fluid on the probability distribution of pore-scale velocities of the flowing fluid phase. We focus on drainage and imbibition processes within a three-dimensional microcomputed tomographic image of a real rock sample. The probability distribution of velocity magnitude displays a heavier tail for trapped nonwetting than wetting fluid. This behavior is a signature of marked changes in the distribution and strength of preferential flow paths promoted by the wettability property of the trapped fluid. When the latter is wetting the host solid matrix, high-velocity areas initially present during single-phase flow conditions are mainly characterized by increased or decreased velocity magnitudes, and the velocity field remains correlated with its counterpart associated with the single-phase case. Otherwise, when the trapped fluid is nonwetting, features that are observed to prevail are appearance and disappearance of high-velocity areas and a velocity field that is less correlated to the one obtained under single-phase conditions.

Multifractal Intermittency in Granular Flow through Bottlenecks

We experimentally analyze the intermittent nature of granular silo flow when the discharge is controlled by an extracting belt at the bottom. We discover the existence of four different scenarios. For low extraction rates, the system is characterized by an on-off intermittency. When the extraction rate is increased the structure functions of the grains velocity increments, calculated for different lag times, reveal the emergence of multifractal intermittency. Finally, for very high extraction rates that approach the purely gravitational discharge, we observe that the dynamics become dependent on the outlet size. For large orifices the behavior is monofractal, whereas for small ones, the fluctuations of the velocity increments deviate from Gaussianity even for very large time lags.

Estimating the accuracy of the random walk simulation of mass transport processes

The mass transport processes always accompany the flow phenomena and have attracted many researchers. A lot of numerical methods have been developed to study them. These numerical methods can be classified into the Eulerian and the Lagrangian approaches. The Lagrangian approach has advantages in high stability and simplicity over the Eulerian approach, but suffers from heavy computational cost. In this paper, we are mainly concerned with the trade-offs between the accuracy and computational cost when applying the random walk method, which is a Lagrangian approach for examining the mass transport scenario. We introduce a linear model to assess the accuracy of the random walk method in several computational configurations. Studies on computational parameters, i.e. the size of time step and number of particles, are conducted with the focus on estimation of the longitudinal dispersion coefficient D_L in steady flows. The results show that the proposed linear model can satisfactorily explain the computational accuracy, both in sample and out-of-sample. Furthermore, we find a constant dimensionless parameter, which quantifies a generic relationship between the accuracy and the number of particles regardless of the flow and diffusion conditions. This dimensionless parameter is of theoretic value and offers guidelines for choosing the correct computational parameters to achieve the required numerical accuracy.

Single-big-jump principle in physical modeling

The big-jump principle is a well-established mathematical result for sums of independent and identically distributed random variables extracted from a fat-tailed distribution. It states that the tail of the distribution of the sum is the same as the distribution of the largest summand. In practice, it means that when in a stochastic process the relevant quantity is a sum of variables, the mechanism leading to rare events is peculiar: Instead of being caused by a set of many small deviations all in the same direction, one jump, the biggest of the lot, provides the main contribution to the rare large fluctuation. We reformulate and elevate the big-jump principle beyond its current status to allow it to deal with correlations, finite cutoffs, continuous paths, memory, and quenched disorder. Doing so we are able to predict rare events using the extended big-jump principle in Lévy walks, in a model of laser cooling, in a scattering process on a heterogeneous structure, and in a class of Lévy walks with memory. We argue that the generalized big-jump principle can serve as an excellent guideline for reliable estimates of risk and probabilities of rare events in many complex processes featuring heavy-tailed distributions, ranging from contamination spreading to active transport in the cell.

A spatial Markov model for upscaling transport of adsorbing-desorbing solutes

The Spatial Markov Model (SMM) is an upscaled model with a strong track record in predicting upscaled behavior of conservative solute transport across hydrologic systems. Here we propose an SMM that can account for reactive linear adsorption and desorption processes and test it on a simple benchmark problem: flow and transport through an idealized periodic wavy channel. The methodology is built using trajectories that are obtained from a single high resolution random walk simulation of conservative transport across one periodic element. Our approach encodes information about where a particle starts at the inlet, where it leaves at the outlet, how long it takes to cross the domain and one additional piece of information, the number of times a particle strikes the boundary, with the objective of predicting large scale transport with arbitrary linear adsorption and desorption rates. Our benchmark problem demonstrates that predictions made with our proposed SMM agree favorably with results from direct numerical simulations, which resolve the full transport problem.

Ubiquity of anomalous transport in porous media: Numerical evidence, continuous time random walk modelling, and hydrodynamic interpretation

Anomalous transport in porous media is commonly believed to be induced by the highly complex pore space geometry. However, this phenomenon is also observed in porous media with rather simple pore structure. In order to answer how ubiquitous can anomalous transport be in porous media, we in this work systematically investigate the solute transport process in a simple porous medium model with minimal structural randomness. The porosities we consider range widely from 0.30 up to 0.85, and we find by lattice Boltzmann simulations that the solute transport process can be anomalous in all cases at high Péclet numbers. We use the continuous time random walk theory to quantitatively explain the observed scaling relations of the process. A plausible hydrodynamic origin of anomalous transport in simple porous media is proposed as a complement to its widely accepted geometric origin in complex porous media. Our results, together with previous findings, provide evidence that anomalous transport is indeed ubiquitous in porous media. Consequently, attentions should be paid when modelling solute transport by the classical advection-diffusion equation, which could lead to systematic error.

Blood flow and transport in the human placenta

The placenta is a multi-functional organ that exchanges blood gases and nutrients between a mother and her developing fetus. In humans, fetal blood flows through intricate networks of vessels confined within villous trees, the branches of which are bathed in pools of maternal blood. Fluid mechanics and transport processes play a central role in understanding how these elaborate structures contribute to the function of the placenta, and how their disorganization may lead to disease. Recent advances in imaging and computation have spurred significant advances in simulations of fetal and maternal flows within the placenta, across a range of lengthscales. Models describe jets of maternal blood emerging from spiral arteries into a disordered and deformable porous medium, and solute uptake by fetal blood flowing through elaborate three-dimensional capillary networks. We survey recent developments and emerging challenges in modeling flow and transport in this complex organ.

Upscaling Mixing in Highly Heterogeneous Porous Media via a Spatial Markov Model

In this work, we develop a novel Lagrangian model able to predict solute mixing in heterogeneous porous media. The Spatial Markov model has previously been used to predict effective mean conservative transport in flows through heterogeneous porous media. In predicting effective measures of mixing on larger scales, knowledge of only the mean transport is insufficient. Mixing is a small scale process driven by diffusion and the deformation of a plume by a non-uniform flow. In order to capture these small scale processes that are associated with mixing, the upscaled Spatial Markov model must be extended in such a way that it can adequately represent fluctuations in concentration. To address this problem, we develop downscaling procedures within the upscaled model to predict measures of mixing and dilution of a solute moving through an idealized heterogeneous porous medium. The upscaled model results are compared to measurements from a fully resolved simulation and found to be in good agreement.

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.

Biased continuous-time random walks for ordinary and equilibrium cases: facilitation of diffusion, ergodicity breaking and ageing

We examine renewal processes with power-law waiting time distributions (WTDs) and non-zero drift via computing analytically and by computer simulations their ensemble and time averaged spreading characteristics. All possible values of the scaling exponent α are considered for the WTD ψ(t) ∼ 1/t1+α. We treat continuous-time random walks (CTRWs) with 0 < α < 1 for which the mean waiting time diverges, and investigate the behaviour of the process for both ordinary and equilibrium CTRWs for 1 < α < 2 and α > 2. We demonstrate that in the presence of a drift CTRWs with α < 1 are ageing and non-ergodic in the sense of the non-equivalence of their ensemble and time averaged displacement characteristics in the limit of lag times much shorter than the trajectory length. In the sense of the equivalence of ensemble and time averages, CTRW processes with 1 < α < 2 are ergodic for the equilibrium and non-ergodic for the ordinary situation. Lastly, CTRW renewal processes with α > 2-both for the equilibrium and ordinary situation-are always ergodic. For the situations 1 < α < 2 and α > 2 the variance of the diffusion process, however, depends on the initial ensemble. For biased CTRWs with α > 1 we also investigate the behaviour of the ergodicity breaking parameter. In addition, we demonstrate that for biased CTRWs the Einstein relation is valid on the level of the ensemble and time averaged displacements, in the entire range of the WTD exponent α.

Pore-scale statistics of flow and transport through porous media

Flow in porous media is known to be largely affected by pore morphology. In this work, we investigate the effects of pore geometry on the transport and spatial correlations of flow through porous media in two distinct pore structures arising from three-dimensional assemblies of overlapping and nonoverlapping spheres. Using high-resolution direct numerical simulations (DNS), we perform Eulerian and Lagrangian analysis of the flow and transport characteristics in porous media. We show that the Eulerian velocity distributions change from nearly exponential to Gaussian distributions as porosity increases. A stretched exponential distribution can be used to represent this behavior for a wide range of porosities. Evolution of Lagrangian velocities is studied for the uniform injection rule. Evaluation of tortuosity and trajectory length distributions of each porous medium shows that the model of overlapping spheres results in higher tortuosity and more skewed trajectory length distributions compared to the model of nonoverlapping spheres. Wider velocity distribution and higher tortuosity for overlapping spheres model give rise to non-Fickian transport while transport in nonoverlapping spheres model is found to be Fickian. Particularly, for overlapping spheres model our analysis of first-passage time distribution shows that the transport is very similar to those observed for sandstone. Finally, using three-dimensional (3D) velocity field obtained by DNS at the pore-scale, we quantitatively show that despite the randomness of pore-space, the spatially fluctuating velocity field and the 3D pore-space distribution are strongly correlated for a range of porous media from relatively homogeneous monodisperse sphere packs to Castlegate sandstone.

Biofilms in 3D porous media: Delineating the influence of the pore network geometry, flow and mass transfer on biofilm development

This study investigates the functional correspondence between porescale hydrodynamics, mass transfer, pore structure and biofilm morphology during progressive biofilm colonization of a porous medium. Hydrodynamics and the structure of both the porous medium and the biofilm are experimentally measured with 3D particle tracking velocimetry and micro X-ray Computed Tomography, respectively. The analysis focuses on data obtained in a clean porous medium after 36 h of biofilm growth. Registration of the particle tracking and X-ray data sets allows to delineate the interplay between porous medium geometry, hydrodynamic and mass transfer processes on the morphology of the developing biofilm. A local analysis revealed wide distributions of wall shear stresses and concentration boundary layer thicknesses. The spatial distribution of the biofilm patches uncovered that the wall shear stresses controlled the biofilm development. Neither external nor internal mass transfer limitations were noticeable in the considered system, consistent with the excess supply of nutrient and electron acceptors. The wall shear stress remained constant in the vicinity of the biofilm but increased substantially elsewhere.

Suspended particle transport through constriction channel with Brownian motion

It is well known that translocation events of a polymer or rod through pores or narrower parts of micro- and nanochannels have a stochastic nature due to the Brownian motion. However, it is not clear whether the objects of interest need to have a larger size than the entrance to exhibit the deviation from the dynamics of the surrounding fluid. We show by numerical analysis that the particle injection into the narrower part of the channel is affected by thermal fluctuation, where the particles have spherical symmetry and are smaller than the height of the constriction. The Péclet number (Pe) is the order parameter that governs the phenomena, which clarifies the spatio-temporal significance of Brownian motion compared to hydrodynamics. Furthermore, we find that there exists an optimal condition of Pe to attain the highest flow rate of particles relative to the dispersant fluid flow. Our finding is important in science and technology from nanopore DNA sequencers and lab-on-a-chip devices to filtration by porous materials and chromatography.

Controls on eDNA movement in streams: Transport, Retention, and Resuspension

Advances in detection of genetic material from species in aquatic ecosystems, including environmental DNA (eDNA), have improved species monitoring and management. eDNA from target species can readily move in streams and rivers and the goal is to measure it, and with that infer where and how abundant species are, adding great value to delimiting species invasions, monitoring and protecting rare species, and estimating biodiversity. To date, we lack an integrated framework that identifies environmental factors that control eDNA movement in realistic, complex, and heterogeneous flowing waters. To this end, using an empirical approach and a simple conceptual model, we propose a framework of how eDNA is transported, retained, and resuspended in stream systems. Such an understanding of eDNA dispersal in streams will be essential for designing optimized sampling protocols and subsequently estimating biomass or organismal abundance. We also discuss guiding principles for more effective use of eDNA methods, highlighting the necessity of understanding these parameters for use in future predictive modeling of eDNA transport.

Transition from non-Fickian to Fickian longitudinal transport through 3-D rough fractures: Scale-(in)sensitivity and roughness dependence

Understanding transport in rough fractures from non-Fickian to Fickian regimes and the prediction of non-Fickian transport is critical for the development of new transport theories and many practical applications. Through computational experiments that fall within the macrodispersion regime, we first simulated and analyzed solute transport through synthetic rough fractures with stationary geometrical properties (i.e., fracture roughness σ_b/〈b〉 and correlation length λ, where b refers to aperture with its standard deviation σ_b and arithmetic mean 〈b〉) across increasing fracture longitudinal transport domain length L, with L/λ ranging from 2.5 to 50. The results were used to determine how solute transport behavior evolves with increasing scale in the longitudinal direction. Moreover, a set of correlated fractures with aperture fields following normal and log-normal distributions was created to further identify and quantify the dependence of non-Fickian transport on roughness. We found that although persistent intermittent velocity structures were present, the breakthrough curves (BTCs) and residence time distributions showed diminishing early arrival and tailing, features of non-Fickian transport, with increasing longitudinal L/λ, ultimately converging to a Fickian transport regime given σ_b/〈b〉 remained constant. Inverse analysis of the experimental BTCs with the advection-dispersion equation (ADE) model showed that the dispersion coefficient (D) was non-trivially scale-dependent. Simulation results for rough fractures with varying σ_b/〈b〉 and L/λ indicated that the ratio of fluid velocity to transport velocity fitted to the ADE model depends on σ_b/〈b〉 and L/λ. The continuous time random walk (CTRW) performed much better across all transport scales, and resulted in scale-independent fitted parameters, i.e., β in the memory function. The fitted β is proportional to σ_b/〈b〉but is insensitive to L/λ. Therefore, bulk longitudinal solute transport across the pre-asymptotic and asymptotic regimes can be estimated based on the CTRW model parameterized by measurable fracture physical properties.

Numerical Investigation of Thermal Counterflow of He II Past Cylinders

We investigate numerically, for the first time, the thermal counterflow of superfluid helium past a cylinder by solving with a finite volume method the complete so-called two-fluid model. In agreement with existing experimental results, we obtain symmetrical eddies both up- and downstream of the obstacle. The generation of these eddies is a complex transient phenomenon that involves the friction of the normal fluid component with the solid walls and the mutual friction between the superfluid and normal components. Implications for flow in a more realistic porous medium are also investigated.

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.

Magnetic resonance imaging measurements evidence weak dispersion in homogeneous porous media

We measure the dispersion coefficient through homogeneous bead or sand packings at different flow rates from direct magnetic resonance imaging (MRI) visualizations of the transport characteristics of a pulse of paramagnetic nanoparticles. Through two-dimensional imaging we observe homogeneous dispersion inside the sample, but we show that entrance effects may induce significant radial heterogeneities, which would affect the interpretation of the breakthrough curve. Another MRI approach then provides quantitative measurements of the evolution in time of the longitudinal particle distribution in the sample. These data can be analyzed to deduce the coefficient of dispersion independently of entrance effects. The values obtained for this "effective" dispersion coefficient are almost ten times lower than the commonly accepted values.

Testing the limits of the spatial Markov model for upscaling transport: The role of nonmonotonic effective velocity autocorrelations

The spatial Markov model is a Lagrangian random walk model, widely and successfully used for upscaling transport in heterogeneous flows across a broad range of problems. It is particularly useful at early or pre-asymptotic times when many other conventional upscaling approaches may not be valid. However, as with all upscaled models, it must have its limits. In particular, the question of what the smallest scale at which it can be legitimately applied, without violating implicit assumptions, remains. Here we address this issue by considering one of the most classical transport upscaling problems: Taylor dispersion in a bounded shear flow. We demonstrate that the smallest scale for the spatial Markov model depends on the transverse width of the domain, the variability of the flow field as quantified by a coefficient of variation, and the competition of longitudinal and transverse diffusion coefficients. We show that this scale is a factor of the Peclet number smaller than the classical Taylor dispersion scale, meaning that for advection-dominated systems where Peclet numbers are large, this model can be applied at much smaller scales than classical Taylor-Aris dispersion theories.

Stochastic transport in the presence of spatial disorder: Fluctuation-induced corrections to homogenization

Motivated by uncertainty quantification in natural transport systems, we investigate an individual-based transport process involving particles undergoing a random walk along a line of point sinks whose strengths are themselves independent random variables. We assume particles are removed from the system via first-order kinetics. We analyze the system using a hierarchy of approaches when the sinks are sparsely distributed, including a stochastic homogenization approximation that yields explicit predictions for the extrinsic disorder in the stationary state due to sink strength fluctuations. The extrinsic noise induces long-range spatial correlations in the particle concentration, unlike fluctuations due to the intrinsic noise alone. Additionally, the mean concentration profile, averaged over both intrinsic and extrinsic noise, is elevated compared with the corresponding profile from a uniform sink distribution, showing that the classical homogenization approximation can be a biased estimator of the true mean.

Solvable continuous-time random walk model of the motion of tracer particles through porous media

We consider the continuous-time random walk (CTRW) model of tracer motion in porous medium flows based on the experimentally determined distributions of pore velocity and pore size reported by Holzner et al. [M. Holzner et al., Phys. Rev. E 92, 013015 (2015)PLEEE81539-375510.1103/PhysRevE.92.013015]. The particle's passing through one channel is modeled as one step of the walk. The step (channel) length is random and the walker's velocity at consecutive steps of the walk is conserved with finite probability, mimicking that at the turning point there could be no abrupt change of velocity. We provide the Laplace transform of the characteristic function of the walker's position and reductions for different cases of independence of the CTRW's step duration τ, length l, and velocity v. We solve our model with independent l and v. The model incorporates different forms of the tail of the probability density of small velocities that vary with the model parameter α. Depending on that parameter, all types of anomalous diffusion can hold, from super- to subdiffusion. In a finite interval of α, ballistic behavior with logarithmic corrections holds, which was observed in a previously introduced CTRW model with independent l and τ. Universality of tracer diffusion in the porous medium is considered.