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

Dentz, Marco; Lester, Daniel; Borgne, Tanguy Le; Barros, Felipe P. J. de

doi:10.1103/physreve.94.061102

Cited by 26 publications

(52 citation statements)

References 56 publications

(94 reference statements)

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“…In the wait first model [19,37,38], where the particle is localized in space, and then makes an instantaneous displacement, big jumps can be generated only by the second process since the particle is at rest at the moment of observation T , and the tail of P (R, T ) is given by B 1 only. Another possible extension where the big jump principle can be applied is the model where the motion of the particle is not ballistic [39] . An important example of accelerated motion will be discussed in detail in Section III.…”

Section: Lévy Walks: the Jump Rate And The Big Jumpmentioning

confidence: 99%

Single-big-jump principle in physical modeling

2019

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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. PACS numbers:

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Section: Lévy Walks: the Jump Rate And The Big Jumpmentioning

confidence: 99%

Single-big-jump principle in physical modeling

2019

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“…In order to investigate the impact of strong heterogeneity of velocity point values, we consider a velocity distribution that is characterized by power-law behavior at low velocities [77,72]…”

Section: Heterogeneitymentioning

confidence: 99%

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

Comolli

Dentz

2017

Eur. Phys. J. B

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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. We derive analytical expressions for the asymptotic scaling of the moments of the spatial particle distribution and first arrival time distribution (FATD), and perform numerical particle tracking simulations of the coupled CTRW to capture the full average transport behavior. 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. arXiv:1707.05560v2 [physics.flu-dyn]

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“…As shall be shown, we observe three main types of transport structures (open, closed, and chaotic) in periodically forced aquifers over the flow parameter space Q ¼ T×G×C. In this section we briefly review these visualization tools, classify and describe these transport structures, and show how they change with T, G, and C. In the absence of sources and sinks, steady Darcy flow is typified by open streamlines and an absence of stagnation points (Bear, 1972), leading to slow mixing and limited transport dynamics (Dentz et al, 2016;Sposito, 2001). Conversely, unsteady Darcy flows can break these topological constraints due to transient switching of streamlines (Lester et al, 2009Metcalfe, Lester, Ord, Kulkarni, Rudman, et al, 2010;Neupauer et al, 2014;Trefry et al, 2012), leading to a much richer set of possible transport structures.…”

Section: Transport Structures Of Periodically Forced Aquifersmentioning

confidence: 99%

When Do Complex Transport Dynamics Arise in Natural Groundwater Systems?

Lester

Trefry³

et al. 2020

Water Resources Research

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In a recent paper (Trefry et al., 2019, https://doi.org/10.1029/2018wr023864), we showed that the interplay of aquifer heterogeneity and poroelasticity can produce complex transport in tidally forced aquifers, with significant implications for solute transport, mixing, and reaction. However, what was unknown was how broadly these transport dynamics can arise in natural groundwater systems and how these dynamics depend upon the aquifer properties and tidal and regional flow characteristics. In this study we answer these questions through parametric studies of these governing properties. We uncover the mechanisms that govern complex transport dynamics and the bifurcations between transport structures that depend upon changes in the governing parameters, and we determine the propensity for complex dynamics to occur in natural aquifer systems. These results clearly demonstrate that complex transport structures and dynamics may arise in natural tidally forced aquifers around the world, producing solute transport and mixing behavior that is very different to that of the conventional Darcy flow picture.

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

Cited by 26 publications

References 56 publications

Single-big-jump principle in physical modeling

Single-big-jump principle in physical modeling

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

When Do Complex Transport Dynamics Arise in Natural Groundwater Systems?

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