Continuous Time Random Walk Approach Research Articles

Time-dependent dispersion coefficients for the evolution of displacement fronts in heterogeneous porous media

We present an approach for quantifying displacement fronts in heterogeneous porous media based on the concept of time-dependent apparent dispersion coefficients. The concept of constant asymptotic macrodispersion generally overestimates the area swept by a displacement front and leads to unrealistic upstream dispersion. We show that the large-scale front spreading can be captured by a one-dimensional advection–dispersion equation that is parameterized by a suitably chosen temporally evolving dispersion coefficient. For purely advective front spreading, we derive an analytical expression based on a predictive continuous time random walk approach, which applies to highly heterogeneous porous media. This analysis elucidates the variability of solute travel times as the key longitudinal spreading mechanism. It shows that the evolution of dispersion can be captured as the sum of exponentials that decay on two dominant time scales. In a particle-based picture, these scales mark the short time at which transported particles start exploring the flow variability and the large time at which the slowest particles start decorrelating their transport velocity. Based on these insights, we propose a heuristic formula that accounts for the impact of local-scale dispersion as an additional decorrelation mechanism. The heuristic expression for the longitudinal dispersion coefficient captures solute spreading for a broad range of Péclet numbers and heterogeneity variances. The proposed approach is tested against direct numerical simulations. It provides a robust and fast method for quantifying the evolution of displacement fronts in heterogeneous porous media with possible applications, for example, in groundwater contamination modelling, underground gas storage, and geothermal energy production.

Stochastic resetting and linear reaction processes: A continuous time random walk approach

We investigate a diffusion process by considering simultaneously stochastic resetting and linear reaction kinetics in the continuous time random walk approach. We first consider the formalism for a single species and then extend it to multiple species. We perform the analysis by considering a general probability density function for the random walk, allowing us to obtain various behaviors for the waiting time and jumping probability distributions. The behavior of these distributions has implications for the diffusion-like equations which emerge from this approach and can be connected to different fractional operators with singular or nonsingular kernels. We also show that diffusion-like equations can exhibit a large class of behaviors related to different processes, particularly anomalous diffusion.

Effect of Pore‐Wall Roughness and Péclet Number on Conservative Solute Transport in Saturated Porous Media

AbstractWhile modeling solute transport has been an active subject of research in the past few decades, the influence of pore‐wall roughness on contaminant migration has not yet been addressed. We therefore conduct particle tracking simulations in three porous domains that have different pore‐wall roughness characteristics. Specifically, we consider five surface fractal dimensions ds = 1.0, 1.1, 1.2, 1.4, and 1.6, and four different Péclet numbers Pe = 10, 102, 103, and 105. Overall, arrival time distributions are simulated for 60 scenarios (3 domains 5 surface fractal dimensions 4 Péclet numbers) some of which show heavy‐tailed patterns indicating non‐Fickian transport. To interpret the simulations and quantify the transport behavior, we analyze the resulting arrival time distributions by the continuous time random walk (CTRW) approach. Results show that, on average, as the surface fractal dimension increases from 1.0 to 1.6, the CTRW model parameters , an exponent showing the degree of anomalous transport, v, the average solute velocity, and t2, the cut‐off time to Fickian transport, remain nearly constant. However, the dispersion coefficient, D, increases and the characteristic transition time, t1, decreases. We found t1 and D are more sensitive to pore‐wall roughness compared to the other CTRW parameters. We also found that as the Péclet number increases from 10 to 105, on average, v and D increase, t1 and decrease, and t2 remains nearly constant. The simulations demonstrate that the exponent and the dispersion coefficient are correlated to the average solute velocity.

Solute Trapping and the Mechanisms of Non‐Fickian Transport in Partially Saturated Porous Media

AbstractWe study the upscaling of pore‐scale solute transport in partially saturated porous media at different saturation degrees. The interaction between structural heterogeneity, phases distribution and small‐scale flow dynamics induces complex flow patterns and broad probability distributions of flow, which control key aspects of transport, such as residence and arrival times, dispersion, and spatial solute distributions, as well as chemical reactions. A continuous‐time random walk (CTRW) framework that integrates the processes of advection, diffusion, and trapping in immobile zones is used to upscale and evaluate the transport of diluted solutes. Results of this model were compared to direct numerical simulations solving the advection‐diffusion equation in experimental saturation patterns. The comparison between simulations results, with different Péclet numbers (Pe), and the physics‐based upscaled CTRW approach allows for a quantitative analysis of the governing factors of transport in partially saturated porous media. This analysis shows that the fluid phase saturation decreases the advective tortuosity, the media's characteristic length, the fraction of the immobile region, and the mean trapping time. At the same time, for a given saturation degree, the normalized mean trapping time is proportional to the Pe. This suggests that the characteristic trapping length is proportional to the media's characteristic (correlation) length. Moreover, the trapping frequency decreases with increasing Pe.

Dispersion of motile bacteria in a porous medium

Understanding flow and transport of bacteria in porous media is crucial to technologies such as bioremediation, biomineralization and 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 bacteria 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 modelled 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 combined effect of pore-scale flow variability and bacterial motility. The model reproduces the experimental observations, and predicts bacteria dispersion and transport at the macroscale.

Effect of ultrasound amplitude and frequency on nanoparticle diffusion in an agarose hydrogel

Exposure of nanoparticles in a porous medium, such as a hydrogel, to low-intensity ultrasound has been observed to dramatically enhance particle penetration rate. Enhancement of nanoparticle penetration is a key issue affecting applications such as biofilm mitigation and targeted drug delivery in human tissue. The current study used fluorescent imaging to obtain detailed experimental measurements of the effect of ultrasound amplitude and frequency on diffusion of nanoparticles of different diameters in an agarose hydrogel, which is often used as a simulant for biofilms and biological tissues. We demonstrate that the acoustic enhancement occurs via the phenomenon of oscillatory diffusion, in which a combination of an oscillatory flow together with random hindering of the particles by interaction with hydrogel proteins induces a stochastic random walk of the particles. The measured variation of acoustic diffusion coefficients with amplitude and frequency were used to validate a previous statistical theory of oscillatory diffusion based on the continuous time random walk approach.

Pore-scale Transport in Rocks of Different Complexity Modeled by Random Walk Methods

We investigate pore-scale transport of a passive solute in three types of reservoir rocks of distinctly different heterogeneity characteristics, using both random walk particle tracking (RWPT), a Lagrangian method free of numerical dispersion, and time domain random walk (TDRW), a regular-lattice-based approximation of the advection–diffusion equation. The transport behavior is probed in terms of solute breakthrough curves for a large range of values of the Péclet number, and for both flux-weighted and uniform injection conditions. We compare the two numerical modeling approaches and thereby highlight the impact of the distribution of pore-scale flow velocities on the large-scale transport behavior. We discuss the properties of two numerical approaches and analyze the influence of the numerical resolution of the velocity field on the simulated transport behaviors. A main difference consists in the presence of numerical dispersion in the TDRW method in contrast to the RWPT method. We find that this feature does not affect the simulated large-scale transport compared to the RWPT method, because numerical dispersion is negligible compared to the impact of the broad spectra of velocity fluctuations, which leads to heavy-tailed breakthrough curves and broad peak behaviors determined by hydrodynamic dispersion. The two random walk methods can be equivalently used to simulate hydrodynamic transport at pore scale. The direct simulations provide breakthrough curves that cover up to ten orders of magnitude in time. The tailing behavior is directly related to the distribution of pore-scale flow velocities using a continuous time random walk approach. The comparison of breakthrough curves and velocity distributions for three different rock types indicates that the solute breakthrough behavior can be used to infer the pore-scale velocity distribution and the medium structure.

Transport Upscaling under Flow Heterogeneity and Matrix-Diffusion in Three-Dimensional Discrete Fracture Networks

We investigate the combined effects of network scale flow variability and retention due to matrix-diffusion on the scaling behavior of transport through fractured media. Two of the principal mechanisms controlling the transport of solutes through fractured low-permeability media are broad distributions of flow velocities and retention times in the solid matrix. We study the relative impact of these two processes under different initial conditions using a set of three-dimensional discrete fracture network simulations. We use these simulations to develop and calibrate an upscaled continuous time random walk (CTRW) approach for advective transport based on an Ornstein-Uhlenbeck model for the particle velocities that accounts for the fracture-matrix coupling using a compound Poisson process. This CTRW model can be conditioned on the initial solute distribution and allows to observe late-time scaling behavior at distances beyond what is feasible to observe using high-fidelity direct numerical simulations. We determine that the initial distribution of particles leads to marked differences in the persistent long-term scale behavior in the solute travel time distributions, even those undergoing retention due to matrix diffusion through implementation and analysis of the model.

Correlations within polyprotein forced unfolding dwell-times introduce sequential dependency

Polyproteins, comprised from proteins arrayed in tandem, respond to mechanical loads through partial unfolding and extension. This response to tension that enables their physiological function is related to the ability to dynamically regulate their elasticity. The unique arrangement of their individual mechanical components (proteins and polymeric linkers), and the interactions between them eventually determines their performance. The sequential unfolding-times within a polyprotein are inherently assumed to be independent and identically distributed (iid), thus expected to follow an exponential distribution. Nevertheless, a large body of literature using single molecule force spectroscopy (SMFS) provides evidence that forced unfolding-times of N proteins within a polyprotein do not follow the exponential distribution. Here we use SMFS with Atomic Force Microscopy to measure the unfolding kinetics of Poly-(I91)8 at 180 pN. The unfolding time-intervals were statistically analysed using three common approaches, all exhibiting an N-effect: hierarchical behavior with non-identical unfolding time distributions. Using continuous time random walk approach indicates that the unfolding times display subdiffusive features. Put together with free-energy reconstruction of the whole unfolding polyprotein, we provide physical explanation for this nontrivial behavior, according to which the elongating polypeptide chain with each unfolding event intervenes with the sequential unfolding probabilities and correlates them.

Mechanisms, Upscaling, and Prediction of Anomalous Dispersion in Heterogeneous Porous Media

AbstractWe study the upscaling and prediction of large‐scale solute dispersion in heterogeneous porous media with focus on preasymptotic or anomalous features such as tailing in breakthrough curves and spatial concentration profiles as well as nonlinear evolution of the spatial variance of the concentration distribution. Spatial heterogeneity in the hydraulic medium properties is represented in a stochastic modeling approach. Direct numerical Monte Carlo simulations of flow and advective particle motion combined with a Markov model for streamwise particle velocities give insight in the mechanisms of preasymptotic and asymptotic solute transport in terms of the statistical signatures of the medium and flow heterogeneity. Based on the representation of equidistantly sampled particle velocities as a Markov process, we derive an upscaled continuous time random walk approach that can be conditioned on the flow velocities and thus hydraulic conductivity in the injection region. In this modeling framework, we identify the Eulerian velocity distribution, advective tortuosity, and the correlation length of particle velocities as the key quantities for large‐scale transport prediction. Thus, the upscaled model predicts the spatial concentration profiles, their first and second centered moments, and the breakthrough curves obtained from direct numerical Monte Carlo simulations in spatially heterogeneous conductivity fields. The presented approach allows to relate the medium and flow properties to large‐scale preasymptotic and asymptotic solute dispersion.

Effect of colloids on non-Fickian transport of strontium in sediments elucidated by continuous-time random walk analysis

Understanding the influence of colloids on radionuclide migration is of significance to evaluate environmental risks for radioactive waste disposals. In order to formulate an appropriate modelling framework that can quantify and interpret the anomalous transport of Strontium (Sr) in the absence and presence of colloids, the continuous time random walk (CTRW) approach is implemented in this work using available experimental information. The results show that the transport of Sr and its recovery are enhanced in the presence of colloids. The causes can be largely attributed to the trap-release processes, e.g. electrostatic interactions of Sr, colloids and natural sediments, and differences in pore structures, which gave rise to the varying interstitial velocities of dissolved and, if any, colloid-associated Sr. Good agreement between the CTRW simulations and the column-scale observations is demonstrated. Regardless of the presence of colloids, the CTRW modelling captures the characteristics of non-Fickian anomalous transport (0 < β < 2) of Sr. In particular, a range of 0 < β < 1, corresponding to the cases with greater recoveries, reveal strongly non-Fickian transport with distinctive earlier arrivals and tailing effects, likely due to the physicochemical heterogeneities, i.e. the repulsive interactions and/or the macro-pores originating from local heterogeneities. The results imply that colloids can increase the Sr transport as a barrier of Sr sorption onto sediments herein, apart from often being carriers of sored radionuclides in aqueous phase. From a modelling perspective, the findings show that the established CTRW model is valid for quantifying the non-Fickian and promoted transport of Sr with colloids.

Nanoparticles transport in heterogeneous porous media using continuous time random walk approach

Abstract Reservoir heterogeneities and reactive processes strongly affect the fluid flow in porous media. The mechanism of transport and reaction of the fluid varies from pore-to core-scales, leading to the discrepancies between theoretical predictions and experimental observations, which are caused by the anomalous diffusion. Also, the reactive processes in porous media may alter reservoir properties in different spatial and temporal scale, varying subsequent transport and reaction behaviors. In this article, the coupled transport and reaction process in heterogeneous porous media is studied. The Continuous Time Random Walk (CTRW) framework incorporating particle tracking (PT) method is proposed to simulate the transport and reaction processes at pore-scale and show macroscopic behavior at core-scale. Utilizing the CTRW-PT approach, the motion of solute particles is described as a combination of random independent spatial and temporal increments in each walk step. The particle transition distance and time are chosen from a joint space-time probability density function by a stochastic process. The modeling of reaction is followed by the modeling of transport and updates the change of porous media with its effect on following transport and reaction. Simulations of non-reactive tracer and reactive nanofluid flow under different heterogeneity, injection flow rate and rate of reaction were performed. By adjusting the ensemble parameter β and t 2 the effects of the above factors on flow behavior are accounted. The concentration profiles with time are displayed to show the effect of heterogeneity, flow rate and rate of reaction on anomalous behavior. Higher level of heterogeneity, higher flow rate and rate of reaction are captured by smaller β and larger t 2 quantitatively, representing more anomalous flow behavior. The breakthrough curves from CTRW-PT simulation well match the experimental as well as Advection-Diffusion-Reaction Equation (ADRE) modeling results in most cases. The CTRW results also better match the experimental results than the ADRE as the anomalous behavior becomes significant. It is also found that CTRW-PT simulation as a statistical approach is more computationally efficient than solving the ADRE.

Linking Structural and Transport Properties in Three‐Dimensional Fracture Networks

AbstractWe investigate large‐scale particle motion and solute breakthrough in sparse three‐dimensional discrete fracture networks characterized by power law distributed fracture lengths. The three networks we consider have the same fracture intensity values but exhibit different percolation densities, geometric properties, and topological structures. We considered two different average transport models to predict solute breakthrough, a streamtube model and a Bernoulli continuous time random walk model, both of which provide insights into the flow fields within the networks. The streamtube model provides acceptable predictions at short distances in two of the networks but fails in all cases to predict breakthrough times at the outlet plane, which indicates that particle motion in such fracture networks cannot be characterized by a constant velocity between the inlet and control plane at which the breakthrough curve is detected. Rather, the structure of the network requires that frequent velocity transitions be made as particles move through the system. Despite the relatively broad distribution of fracture radii and relatively small number of independent velocity transitions, the continuous time random walk approach conditioned on the initial velocity distribution provides reasonable predictions for the breakthrough curves at different distances from the inlet. The application of these averaged transport models provides a richer understanding of the link from the fracture network structure to flow and transport properties.

On the ability of the Generalized Continuum Transport Model to properly capture dispersion

Abstract The ability of the Generalized Continuum Transport model to describe dispersion is studied through the comparison of the breakthrough curves with an analytical solution of the linear advection-dispersion Equation. First, a velocity distribution due to Taylor dispersion in a capillary tube is related to the dispersion coeficient of the advection-dispersion equation. The same distribution is applied to the Generalized Continuum Transport model, where the dispersive flux term is not included as the term proportional to the concentration gradient. In the second stage the velocity distribution is obtained from the transition probability introduced through the Continuous Time Random Walk approach. The approaches support the idea that the Generalized Continuum Transport model captures velocity variations naturally through the parameter space. The results confirm that a proper selection of the parameter space, including its size, leads to more physical transport description as well as accurate quantification.

Continuous-time random-walk approach to supercooled liquids: Self-part of the van Hove function and related quantities.

From equilibrium molecular dynamics (MD) simulations of a bead-spring model for short-chain glass-forming polymer melts we calculate several quantities characterizing the single-monomer dynamics near the (extrapolated) critical temperature [Formula: see text] of mode-coupling theory: the mean-square displacement g0(t), the non-Gaussian parameter [Formula: see text] and the self-part of the van Hove function [Formula: see text] which measures the distribution of monomer displacements r in time t. We also determine these quantities from a continuous-time random walk (CTRW) approach. The CTRW is defined in terms of various probability distributions which we know from previous analysis. Utilizing these distributions the CTRW can be solved numerically and compared to the MD data with no adjustable parameter. The MD results reveal the heterogeneous and non-Gaussian single-particle dynamics of the supercooled melt near [Formula: see text]. In the time window of the early [Formula: see text] relaxation [Formula: see text] is large and [Formula: see text] is broad, reflecting the coexistence of monomer displacements that are much smaller ("slow particles") and much larger ("fast particles") than the average at time t, i.e. than [Formula: see text]. For large r the tail of [Formula: see text] is compatible with an exponential decay, as found for many glassy systems. The CTRW can reproduce the spatiotemporal dependence of [Formula: see text] at a qualitative to semiquantitative level. However, it is not quantitatively accurate in the studied temperature regime, although the agreement with the MD data improves upon cooling. In the early [Formula: see text] regime we also analyze the MD results for [Formula: see text] via the space-time factorization theorem predicted by ideal mode-coupling theory. While we find the factorization to be well satisfied for small r, both above and below [Formula: see text] , deviations occur for larger r comprising the tail of [Formula: see text]. The CTRW analysis suggests that single-particle "hops" are a contributing factor for these deviations.

Anomalous dispersion in correlated porous media: a coupled continuous time random walk approach

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.

Two‐dimensional finite element method solution of a class of integro‐differential equations: Application to non‐Fickian transport in disordered media

SummaryA finite element method is developed to solve a class of integro‐differential equations and demonstrated for the important specific problem of non‐Fickian contaminant transport in disordered porous media. This transient transport equation, derived from a continuous time random walk approach, includes a memory function. An integral element is the incorporation of the well‐known sum‐of‐exponential approximation of the kernel function, which allows a simple recurrence relation rather than storage of the entire history. A two‐dimensional linear element is implemented, including a streamline upwind Petrov–Galerkin weighting scheme. The developed solver is compared with an analytical solution in the Laplace domain, transformed numerically to the time domain, followed by a concise convergence assessment. The analysis shows the power and potential of the method developed here. Copyright © 2017 John Wiley &amp; Sons, Ltd.

Modified cumulative distribution function in application to waiting time analysis in the continuous time random walk scenario

The continuous time random walk model plays an important role in modelling of the so-called anomalous diffusion behaviour. One of the specific properties of such model is the appearance of constant time periods in the trajectory. In the continuous time random walk approach they are realizations of the sequence called waiting times. In this work we focus on the analysis of waiting time distribution by introducing novel methods of parameter estimation and statistical investigation of such a distribution. These methods are based on the modified cumulative distribution function. In this paper we consider three special cases of waiting time distributions, namely α-stable, tempered stable and gamma. However, the proposed methodology can be applied to broad set of distributions—in general it may serve as a method of fitting any distribution function if the observations are rounded. The new statistical techniques are applied to the simulated data as well as to the real data of concentration in indoor air.

Continuous time random walks for the evolution of Lagrangian velocities

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.

Non-Fickian Transport Under Heterogeneous Advection and Mobile-Immobile Mass Transfer

We study the combined impact of heterogeneous advection and mobile–immobile mass transfer on non-Fickian transport using the continuous-time random walk (CTRW) approach. The CTRW models solute transport in heterogeneous media as a random walk in space and time. Our study is based on a d-dimensional CTRW model that accounts for both heterogeneous advection and mass transfer between mobile and immobile zones, to which we also refer as solute trapping. The flow heterogeneity is mapped into the distribution of advective transition times over a characteristic heterogeneity scale. Mass transfer into immobile zones is quantified by a trapping rate and the distribution of particle return times. The total particle transition time over a characteristic heterogeneity scale then is given by the advective time and the sum of trapping times over the number of trapping events. We establish explicit integro-partial differential equations for the evolution of the concentration and discuss the relation to the multirate mass transfer approach, specifically the relation between the trapping time distribution and the memory function. We then analyze the signatures of anomalous transport due to advective heterogeneity and trapping in terms of spatial moments and first passage times or breakthrough curves. The behaviors for different disorder scenarios are analyzed analytically and through random walk particle tracking simulations. Assuming that advective mass transfer is faster than diffusive, we identify three regimes of distinct transport behaviors, which are separated by the characteristic trapping rate and trapping times. (1) At early times, we identify a pre-asymptotic time regime that is fully determined by advective heterogeneity and which is characterized by superlinear growth of longitudinal dispersion. (2) For longitudinal dispersion, we identify an intermediate regime of strong superlinear diffusion. This regime is determined by the combined effect of advective heterogeneity and trapping. (3) At larger time, the asymptotic trapping-driven regime shows the signatures of diffusion in immobile zones, which leads to both sub- and superlinear dispersion. These results shed some new light on the mechanism of non-Fickian transport and their manifestation in spatial and temporal solute distributions.